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ICMT 2ICMT 2

II International Conference on Mathematics Textbooks

Research and Development

TIMETABLE &

ABSTRACTSRio de Janeiro

May, 7-11, 2017

Sociedade Brasileira de Matemática Aplicada e ComputacionalSociedade Brasileira de Matemática Aplicada e ComputacionalUFPE

Organização: Apoio: Financiamento:

International Programme Committee (IPC)

X Rúbia Amaral (UNESP, Brazil) � Se retary

X Ubiratan d'Ambrosio (UNIAN, Brazil) � Honorary President

X Mar elo Borba (UNESP, Brazil)

X Rute Borba (Universidade Federal de Pernambu o, Brazil)

X Mar os Cherinda (Universidade Pedagógi a de Moçambique)

X Lianghuo Fan (University of Southampton, UK) � Co- hair

X Vi tor Giraldo (Universidade Federal do Rio de Janeiro, Brazil) � Lo al Chair

X Patri io Herbst (University of Mi higan, USA)

X Marja van den Heuvel-Panhuizen (Universiteit Utre ht, Netherlands)

X Abdellah El Idrissi (É ole Normale Supérieure de Marrake h, Moro o)

X Diana Jaramillo Qui eno (Universidad de Antioquia, Colombia)

X Cyril Julie (University of the Western Cape, South Afri a)

X Gabriele Kaiser (Universität Hamburg, Germany)

X Alexander Karp (Tea hers College, Columbia University, USA)

X Jeremy Kilpatri k (University of Georgia, USA)

X Jian Liu (Beijing Normal University, China)

X Eizo Nagasaki (National Institute for Edu ational Poli y Resear h, Japan)

X Mi hael Otte (UNIAN, Brazil)

X Johan Prytz (Uppsala Universitet, Sweden)

X Sebastian Rezat (Universität Paderborn, Germany)

X Angel Ruiz (Universidad de Costa Ri a, Costa Ri a)

X Kenneth Ruthven (University of Cambridge, UK)

X Gert S hubring (UFRJ, Brazil/Universität Bielefeld, Germany) � Chair

Lo al Orgnization Committe (LOC)

X Lourdes Werle de Almeida (UEL)

X Rúbia Amaral (UNESP) � Co- hair

X Gladson Antunes (UNIRIO)

X Fran k Bellemain (UFPE)

X Marilena Bittar (UFMS)

X Ulisses Dias (UFRJ)

X Már ia Fusaro (UFRJ)

X Vi tor Giraldo (UFRJ) � Chair

X Ver�ni a Gitirana (UFPE)

X Viní ius Mano (UFRJ)

X Bruna Moustapha (UNIRIO)

X Carmen Mathias (UFSM)

X João Frederi o Meyer (UNICAMP)

X Wellerson Quintaneiro (CEFET-RJ)

X Cydara Ripoll (UFRGS)

X Wal y Santos (UFRJ)

X Fábio Simas (UNIRIO)

X Ralph Teixeira (UFF)

Contents

Conferen e Themes 3

Timetable 5

Overall Timetable 7

Day Timetable 8

Oral Communi ations � Session 1 13

















Oral Communi ations in Symposium A 19

Oral Communi ations in Symposium B 19

Oral Communi ations in Symposium C 20

Poster Presentations 21

Abstra ts 23

Plenaries 25

Symposia 31

Workshops 69

Workshops for Tea hers 77

Oral Communi ations 95

Poster Presentations 153

Campus Maps 169

CONFERENCE THEMES

1. Textbook resear h ( on epts, issues, methods, dire tions, et .)

2. Textbook analysis ( hara teristi s, treatment of ontents and/or pedagogy, et .)

3. Analysis of histori al textbooks

4. Textbook use (by tea hers, by students, and/or by other parties)

5. Textbooks and student a hievement

6. Textbook development (domain/ ompeten e analyses, tea hing traje tories, task design, format of

presenting the � ontent� to the student, format of presenting the � ontent� to the tea hers (tea her

guides))

7. Textbook poli ies (governmental edu ational poli y about textbooks, distribution, market strategies)

8. Evolution of textbooks in the light of new digital te hnologies (in luding integration of ICT tools

and innovation, e-textbook)

9. Other dis iplines in mathemati s textbooks & mathemati s in textbooks of other dis iplines

10. All other relevant issues about mathemati s textbooks

Note: Oral ommuni ation sessions were organized taking into onsideration the main Conferen e themes

asso iated with the papers. The Conferen e theme asso iate with ea h paper is indi ated by the orre-

sponding number next to the title.

3

TIMETABLE

Overall Timetable

7

Day Timetable

Sunday, May 7

th, 2017

16:00 Registration

17:00 UNIRIO

17:00 Opening Session

18:00 UNIRIO

18:00 Plenary Talk

19:30 UNIRIO

Mathemati s Textbooks for the Million � Brazil's Mathemati s Textbooks Assessment Program

João Bos o Pitombeira (Universidade Federal de Mato Grosso do Sul, Brazil)

19:30 Re eption

21:30 UNIRIO

8

Monday, May 8

th, 2017

9:00 Plenary Talk

10:30 Auditório Roxinho

The Diagrams of Eu lid's Elements

Ken Saito (Osaka Prefe ture University, Japan)

10:30 Co�ee Break

11:00

11:00 Oral Communi ations � Session 1

13:00 C-116

Oral Communi ations � Session 2

C-119


LIG 1


LIG 2

13:00 Lun h

14:30

14:20 Poster Presentations

16:30 Salão Nobre do CCMN

16:30 Co�ee Break

17:00

17:00 Symposium A

20:00 C-116

Textbook use by Tea hers and Students � Results and Methods

Sebastian Rezat (University of Paderborn, Germany), Rudolf Sträÿer (University of Giessen,

Germany)

Symposium B

C-119

Dedu tive Reasoning, Arguing and Proof in Textbooks

Luísa Rodriguez Doering, Cydara Cavedon Ripoll (Universidade Federal do Rio Grande do Sul,

Brazil)

Tuesday, May 9

th, 2017

9:00 Plenary Talk


A Multimodal Approa h for Theorising and Analysing Mathemati s Textbooks

Kay O'Halloran (Curtin University, Perth Western Australia)

10:30 Co�ee Break

11:00


13:00 C-116


C-119


LIG 1


LIG 2

13:00 Lun h

14:30

9

Wednesday, May 10

th, 2017

9:00 Plenary Talk


An International Comparison on Sele tion of Contents and Di� ulty Level of Examples in High

S hool Mathemati s Textbooks

Jianpan Wang (East China Normal University, China)

10:30 Co�ee Break

11:00


13:00 C-116


C-119


LIG 1


LIG 2

13:00 Lun h

14:30

14:30 Workshop A

16:30 C-116

Reading Geometri ally: The Analysis of Geometri Diagrams in Textbooks

Leslie Dietiker, Meghan Riling, Aaron Brakonie ki

Workshop B

C-119

Adapting So io ultural Resear h Frameworks For Brazil: The Um Livro Aberto Proje t

Vi tor Giraldo (UFRJ, Brazil), Meril Rasmussen (UBC, Canada), Fabio Simas (UNIRIO,

Brazil); Augusto Teixeira (IMPA, Brazil)

Workshop for Tea hers A

B-106-A

Uma Possibilidade Teóri o Metodológi a para Análise de Livros Didáti os por meio da Her-

menêuti a de Profundidade e dos Paratextos Editorias

Mirian Maria Andrade (UTFPR), Bruno Alves Dassie (UFF), Leandro Josué de Souza (UNESP-

Bauru)

Workshop for Tea hers B

B-106-B

O Guia do PNLD e o Pro esso de Es olha de Livros Didáti os

Marilena Bittar, Danielly Regina Kaspary dos Anjos, Adriana Barbosa Oliveira (UFMS)

Workshop for Tea hers C

Salão Nobre

O Ensino de Frações no Ensino Fundamental I: Atividades do �Livro Aberto de Matemáti a�

Humberto Bortolossi (UFF), Vi tor Giraldo (UFRJ), Letí ia Rangel (UFRJ), Wanderley

Rezende (UFF), Cydara Ripoll (UFRGS), Wellerson Quintaneiro (CEFET-RJ), Fábio Simas

(UNIRIO)

Workshop for Tea hers D

LIG 1

Produzindo Livros Dinâmi os para o Ensino de Matemáti a

Kátia Coelho da Ro ha, Laís de Almeida Pereira (PPGEMat-UFRGS)

Workshop for Tea hers E

LIG 2

Metodologias de Análise Horizontal

Henrique N. Sá Earp (Uni amp), Diana T. Amaro (IFSP-Bragança Paulista)

10

14:30 Workshop for Tea hers F

16:30 D 220

Lições de Matemáti a num Jornal da Cidade do Porto (Portugal) em 1853: um Episódio

Pe uliar na Formação de Professores

Hélder Pinto (CIDMA-UA-Portugal)

16:30 Co�ee Break

17:00

17:00 Symposium C

20:00 Salão Nobre do CCMN

Tea her-Resour e Use Around the World

Janine Remillard (University of Pennsylvania, USA), Hendrik Van Steenbrugge (Mälardalen

University, Sweden), Lu Trou he (É ole Normale Supérieure de Lyon, Fran e)

Thursday, May 11

th, 2017

9:00 Plenary Talk


Ele troni vs. Paper Textbook Presentations of Various Aspe ts of Mathemati s

Zalman Usiskin (University of Chi ago, USA)

10:30 Co�ee Break

11:00


13:00 C-116


C-119


LIG 1


LIG 2


B-106-B

13:00 Lun h

14:30

14:30 Workshop C

16:30 C-116

Writing Prompts and Mathemati s Textbooks

Christine M. Joseph (East Carolina University, USA), Jenifer J. S hneider (South Florida,

USA)

Workshop D

C-119

Textbook Analysis of New Math and Contemporary German S hool Books from Ernest's

Interest Groups Perspe tive

Rainer Kaenders (Rheinis he Friedri h-Wilhelms University Bonn, Germany), Ysette Weiss

(Johannes Gutenberg University Mainz, Germany)

Workshop for Tea hers G

LIG 1

Análise de Livros Didáti os Brasileiros

Andréia Dal in, Luisa Rodríguez Doering, Cydara Cavedon Ripoll (UFRGS)

11

14:30 Workshop for Tea hers H

16:30 LIG 2

Os Livros Didáti os de Matemáti a: Um Olhar De olonial sobre os Conhe imentos, Imagens

e Per epções sobre os Povos Afri anos e Afro-Brasileiros

Maysa Ferreira da Silva, Wilker Solidade da Silva (UFPR)

Workshop for Tea hers I

Salão Nobre

Alternativas Complementares ao Livro Didáti o: Desenvolvimento de Materiais Paradidáti os

Claudiomir Feustler Rodrigues de Siqueira (IFRS-Canoas)

Workshop for Tea hers J

B-106-A

Guia Digital do PNLD: Metare urso para a Es olha do Livro Didáti o

Ri ardo Tibúr io, Rogério Igná io, Fran k Bellemain, Rosilângela Lu ena, Ver�ni a Gitirana

(UFPE)

Workshop for Tea hers K

B-106-B

Uma Abordagem do Con eito de Hipérbole om Base na Análise de Livros Didáti os do Ensino

Médio

Nora Olinda Cabrera Zúñiga, Mariana Lima Vilela, Nayara Katherine Duarte Pinto (UFMG)

16:30 Co�ee Break

17:00

17:00 Closing Session


12


Monday, May 8

th, 2017

11:00 � 13:00

C-116

11:00 A Semioti Interpretation of the Con ept of Derivative in a Textbook (1)

Lourdes Maria Werle de Almeida, Karina Alessandra Pessoa da Silva

11:30 Textbook for Tea hers' Learning to Tea h (1)

Yuriko Yamamoto Baldin, Apare ida Fran is o da Silva

12:00 A Methodologi al Tool for Analysing Textbooks (1)

Marilena Bittar

12:30 Open Math Book Proje t (1)

Humberto José Bortolossi, Letí ia Cavendon Rangel, Cydara Ripoll, Fabio Luiz Borges Simas,

Augusto Teixeira


Monday, May 8

th, 2017

11:00 � 13:00

C-119

11:00 How Combinatorial Situations Are Represented In Brazilian Primary And Middle S hool Text-

books (2)

Rute Borba, Marilena Bittar, Juliana Montenegro, Dara Silva

11:30 On Prevalen e of Images in High S hool Geometry Textbooks (2)

Megan Cannon, Mile Kraj evski

12:00 When is an exploration exploratory? A omparative analysis of geometry lessons (2)

Leslie Dietiker, Andrew Ri hman

12:30 Tables In Textbooks For Elementary S hool Grades 4 And 5 (2)

Betânia Evangelista, Gilda Guimarães


Monday, May 8

th, 2017

11:00 � 13:00

LIG 1

11:00 Tea hers' Per eptions of the Standards for Mathemati al Pra ti e by Textbook Type (4)

Jon Davis, Corey Drake, Je�rey Choppin, Amy Roth M Du�e

11:30 Maths Textbook Analysis in The Tea hers Training Curri ulum � The Experien e at Uni amp

(4)

Henrique Sá Earp

12:00 Multipli ative Situations in Brazilian Mathemati Textbooks Approa hes to De imal Numbers

(4)

Ver�ni a Gitirana, Paula Moreira Baltar Bellemain, Ernani Martins dos Santos

12:30 A Comparison of Di� ulty Levels of Problems In luded in Mathemati s Textbooks and Prob-

lems Used in Real Classroom (4)

Chunlian Jiang, Qin Li, Chunxia Qi

13


Monday, May 8

th, 2017

11:00 � 13:00

LIG 2

11:00 A Webdo ument Stru ture as Support for the Elaboration of Digital Dida ti Material for the

Tea hing of Mathemati s (8)

Fran k Bellemain, Ver�ni a Gitirana, Rogerio Igna io

11:30 Engineering Pro ess in the Development of a Digital Guide of the Brazilian Program of Text-

books (PNLD) (8)

Fran k Bellemain, Ver�ni a Gitirana, Rogério da Silva Igná io, Rosilângela Lu ena, Pedro

Alessio

12:00 IDDM: Intera tive Digital Dida ti Material? (8)

Apare ida Santana de Souza Chiari, Mar elo de Carvalho Borba

12:30 Learning Mathemati s with Videos (8)

Liliane Xavier Neves, Mar elo de Carvalho Borba, Hannah Dora Gar ia La erda


Tuesday, May 9

th, 2017

11:00 � 13:00

C-116

11:00 Counting and Computation in Mozambi an Textbooks for Grade 1 (1)

Jan Draisma

11:30 Comparing Mathemati s Curri ulum and Textbooks between China and England: Findings

From the Shanghai Maths Proje t (1)

Lianghuo Fan, Ming Ni, Huiping Xu, Jing Xu

12:00 The Comparison of Mathemati al Competen ies in Textbooks from Brazil and Singapore (1)

Weverton Ataide Pinheiro, Kin Hang Lei, Feng-Lin Lu, Tai-Yih Tso

12:30 The Tea hing of Logarithm: Methodology Di�eren es Between Brazilian and Ameri an Text-

books (1)

Crístiam Wallao Rosa, Ri ardo Fajardo


Tuesday, May 9

th, 2017

11:00 � 13:00

C-119

11:00 The Con ept of Sampling in Mathemati s Textbooks in the 5th and 9th Grades: An Analysis

of the Proposed A tivities (2)

Tâmara Marques da Silva Gomes, Gilda Guimarães

11:30 Opportunity to Learn Problem Solving in Dut h Primary S hool Mathemati s Textbooks (2)

Mar van Zanten, Marja van den Heuvel-Panhuizen

12:00 Mathemati s and its Usefulness: a Brief Analysis of Textbooks in Mozambique (2)

Abdul arimo Ismael

12:30 Argumentation and Mathemati al Proof: Some Problems in Brazilian Textbooks (2)

Ana Paula Jahn, Aline dos Reis Matheus

14


Tuesday, May 9

th, 2017

11:00 � 13:00

LIG 1

11:00 On a Timeline for the Complete A eptan e of the Symbols N, Z, Q, R and C in Textbooks

(3)

Ri ardo Fajardo

11:30 Ratio and Proportions 1716-2016 (3)

Kristín Bjarnadóttir

12:00 Textbooks Revealing the Development of a Con ept � The Case of the Number Line (1708-

1829) (3)

Mar ello Amadeo

12:30 Russian Post-Revolutionary Mathemati s Textbooks: A Short Lived History (3)

Alexander Karp


Tuesday, May 9

th, 2017

11:00 � 13:00

LIG 2

11:00 Proje t-Based Math-S i Interdis iplinary Curri ulum Design and its Implement: A Case of

�Explore the Super-Earth� (9)

Qi Chunxia, Ma Fu, He Shengqing, Zhang Huiying

11:30 Authenti ity of Contextualized Problems Related to A elerated Motions: A Case Study of

Mexi an Se ondary-S hool Mathemati s Textbooks (9)

Gabriel Guzmán, Josip Slisko, Adrián Corona, Lidia Aurora Hernández

12:00 Intera tive Textbooks: The Case of Fra tions (9)

Stefan Ho h, Kristina Reiss, Frank Reinhold, Bernhard Werner, Jürgen Ri hter-Gebert

12:30 The Chara teristi s of Junior High S hool Mathemati s Textbooks in Japan Right After WWII

(3)

Naomi hi Makinae


Wednesday, May 10

th, 2017

11:00 � 13:00

C-116

11:00 How Books from the 6th to the 9th Grade Propose Horizontal Treatment of Combinatori s

(2)

Ana Paula Barbosa de Lima, Rute Borba

11:30 Fra tions in Textbooks and Mathemati s Tea hers' Spe ialized Knowledge (2)

Beatriz F. Litoldo, Marieli V. R. Almeida, Pris ila Kabbaz Alves da Costa, Miguel Ribeiro

12:00 Textbook Analyis: A Case for Ugandan Textbooks (2)

Imma ulate Kizito Namukasa

12:30 One-step Multipli ation and Division Word Problems in the 3rd Grade Textbooks in Bosnia

and Herzegovina (2)

Karmelita Pjani¢

15


Wednesday, May 10

th, 2017

11:00 � 13:00

C-119

11:00 Norwegian Tea hers Use of Resour es for Planning Instru tion (4)

Camilla Normann Justnes, Olaug Ellen Lona Svingen

11:30 Tea her Fidelity De isions and the Quality of Ena ted Lessons (4)

Ok-Kyeong Kim

12:00 Adaptation of Mathemati s Textbook to Daily Situations of Students in Mozambique: Case

of 8th Class (4)

Saide Issufo Momade

12:30 The Pro ess of Choosing and Using the Tea hing Book by Tea hers of Middle S hool in

Itaituba West of Pará (4)

José Antonio de Oliveira Junior


Wednesday, May 10

th, 2017

11:00 � 13:00

LIG 1

11:00 Statisti s Edu ation in Textbooks: Brazil's National Textbook Program and the Tea hers'

Manuals (6)

Natalia Amorim, Gilda Guimarães

11:30 Redesigning Open Tasks in Mathemati s Textbooks (6)

Antonio José Lopes, Joaquin Giménez, Yuly Marsela Vanegas

12:00 Linguisti , Cultural and Pedagogi Dimensions of Geometry: Navigating Textbook Develop-

ment in a Cross-National Proje t (6)

Candia Morgan, Teresa Smart, Natalya Panikarskaya, Arman Sultanov

12:30 The Con eption and Development of Textbooks for Distan e Learning Courses: The Case

Study of a Tea her Edu ation Course (6)

Már ia Maria Fusaro Pinto


Wednesday, May 10

th, 2017

11:00 � 13:00

LIG 2

11:00 Authoring Online Open Textbooks with Mathbook XML (8)

Robert A. Beezer, David Farmer

11:30 Mathemati s Textbooks in EAD: Chaterezing Them as Di�erent Tea hers' Instrument (8)

José Wilson Pereira, Ver�ni a Gitirana

12:00 Tea hers' Perspe tive of Textbook Choi e: The Digital Guide as a Meta-Resour e (8)

Ri ardo Tibur io dos Santos

12:30 Mapping the Terrain � Making the Textbook's Content & Pedagogy Tangible (8)

Mi hal Yerushalmy, Shai Olsher

16


Thursday, May 11

th, 2017

11:00 � 13:00

C-116

11:00 Text Analysis on Fra tional Cal ulus Con ept (3)

Adrian Muñoz Oroz o, Flor Monserrat Rodríguez Vásquez, Martín Patri io Ár iga Alejandre

11:30 Textbooks on Des riptive Geometry Transmitted to Brazil: Re eption and Di�usion (3)

Thiago Ma iel de Oliveira, Viní ius Mendes Couto Pereira

12:00 A Study About Transmissions of Cal ulus Textbooks to Brazil (3)

Vini ius Mendes Couto Pereira

12:30 How to Analyze Histori al Textbooks from a Dida ti al Perspe tive? The Dida ti analysis as

a Methodologi al Propose (3)

Miguel Pi ado, Luis Ri o


Thursday, May 11

th, 2017

11:00 � 13:00

C-119

11:00 Some Remarks on Textbooks Analysis: Hermeneuti s of Depth and Editorial Paratexts (1)

Leandro Josué de Souza, Mirian Maria Andrade, Bruno Alves Dassie

11:30 The Textbook in Mathemati s: Findings from a Systemati Review (1)

Natasja Steen, Matilde Stenhøj Madsen

12:00 Textbook Analysis in University Tea her Edu ation (2)

Ysette Weiss

12:30 Not my Kind of Algorithm: Comparing use of Shared Language in Mathemati s Textbooks

and Computer S ien e Instru tional Materials (9)

Kathryn Ri h, Carla Stri kland, T. Andrew Binkowski


Thursday, May 11

th, 2017

11:00 � 13:00

LIG 1

11:00 So ial Representations of Learning Mathemati s by Tea hers of Early Years of Elementary

S hool and The Choi e of the Textbook (4)

Patri ia Bastos Fosse Peres

11:30 Interbreeding Paradigms in Resear h on Mathemati s Knowing and Learning (4)

Thorsten S heiner, Már ia Maria Fusaro Pinto

12:00 Does a Proje t-Based Learning Material Improve Students' Understanding of Rational Num-

ber? A Case Study (5)

Qi Chunxia, Wang Ruilin, Lin Mengwei, Zhang Huying

12:30 ENEM and Mathemati s Textbooks for High S hool: an Analysis of the Volume of Geometri

Solids (5)

Katy Leão, Rosilângela Lu ena, Ver�ni a Gitirana

17


Thursday, May 11

th, 2017

11:00 � 13:00

LIG 2

11:00 Integrating the Con lusions of Tea hers' Feedba k into the New Mathemati s Textbooks (6)

Gergely Wints he, Dániel Katona

11:30 Brazil's Mathemati s Textbooks: An Overview of the Government Poli ies (7)

Lu as Carato Mazzi, Rúbia Bar elos Amaral

12:00 Assessing a new El Salvador Mathemati s Textbook: Case study � How Does It Contribute to

Improve Learning? (7)

Fran is o Mejía, Norihiro Nishikata

12:30 The Embodiment of Core Competen y in Textbook Compiling (7)

Liu Xiaomei, Ma Fu, Zhang Huiying


Thursday, May 11

th, 2017

11:00 � 13:00

B-106-B

11:00 Making Digital Manipulatives for Math E-Textbooks: An Overview of the Nowadays Major

User-Friendly Systems (8)

Humberto José Bortolossi

11:30 Rea hing out to Classrooms with Learning Materials through the Textbooks of Mathemati s

(6)

A.K. Rajput, Dharam Parkash

12:00 Mathemati s Textbooks as Di�erent Instruments for Di�erent Tea hers (8)

Ver�ni a Gitirana, Rosilângela Lu ena

12:30 An Analyti al Framework for Studying the Impa t of Te hnology on the Use of Mathemati s

Resour es in Tea hing and Learning (8)

Ida Mok, Lianghuo Fan

18

Oral Communi ations in Symposium A

Monday, May 8

th, 2017

17:00 � 20:00

C-116

17:00 Opening, Introdu tion

17:10 Reading Mathemati s Textbooks: Di�erent Reading Styles

Margot Berger

17:40 Analyzing Classroom Work: Students' Use of Ele troni Textbooks

Kristina Reiss, Stefan Ho h, Frank Reinhold, Bernhard Werner, Jürgen Ri hter-Gebert

18:00 Developing Metadata Categories for Curri ular Materials: Lenses to Study Digital Textbooks-

Tea hers-Relations

Shai Olsher, Mi hal Yerushalmy

18:30 Break

18:40 Uses of Dynami Textbooks in Undergraduate Mathemati s Classrooms

Vilma Mesa, Angeliki Mali, UTMOST team

19:10 Pedagogi al Fun tions of Intera tive Texts

Elena Naftaliev

19:40 Dis ussion: Methodologi al Issues

Oral Communi ations in Symposium B

Monday, May 8

th, 2017

17:00 � 20:00

C-119


17:10 The Eu lidean Division in the First S hool Years

Luisa Rodríguez Doering, Cydara Cavedon Ripoll, Janete Ja inta Carrer Soppelsa

17:20 Proofs and Arguments on Integers Numbers in the Brazilian Math Textbooks

Miguel Melendo Be k

17:30 The Introdu tion to Algebra in Textbooks

Sandro de Azevedo Carvalho, Cydara Cavedon Ripoll

17:40 Dis ussion

18:00 Area Formula Dedu tions for Plane Figures in Textbooks

Luisa Rodríguez Doering, Fran iele Mar iane Meinerz

18:10 Anality Geometry in Third Year High S hool Textbooks

Luisa Rodríguez Doering, Platão Gonçalves Terra Neto

18:20 Dis ussion

18:40 Di�erent Types of Reasoning in Geometry

Lu as Carato Mazzi

19:00 Tea her Views about Argumentation and Mathemati al Proof in S hool

Lilian Nasser, Carlos Augusto Aguilar

19:20 Dis ussion

19

Oral Communi ations in Symposium C

Wednesday, May 10

th, 2017

17:00 � 20:00

Salão Nobre


Papers on Tea hers' Intera tions with Resour es and Related Classroom Ena tments:

17:10 Design in Use: From Author-Intended to Written to Ena ted Lesson in Sweden

Hendrik Van Steenbrugge, Nina Jansson, Fredrik Blomqvist, Andreas Ryve

17:25 Analyzing Tea hers' Colle tive Engagement with Resour es Through the Lens of Their Do -

umentational Traje tories: The Case of Fren h Tea hers Fa ing a New Curri ulum

Katiane de Moraes Ro ha, Lu Trou he, Ghislaine Gueudet

17:40 An Analysis of the Engagement of Pre-Servi e Tea hers with Curri ulum Resour es in Brazil

Cibelle Assis, Ver�ni a Gitirana

17:55 From Written to Ena ted Lessons: A U.S. Tea her's Mobilization of a Mathemati al Modeling-

Based Algebra Unit

Luke Reinke

18:10 Dis ussion

Papers on Tea her Capa ity and Learning in Relation to Resour e Use:

18:25 Disaggregating Tea hers' Pedagogi al Design Capa ity (PDC) in South Afri a

Moneoang Leshota

18:40 Knowledge of Curri ulum Embedded Mathemati s: Exploring a Criti al Domain of Tea hing

in the U.S.

Janine T. Remillard, Ok-Kyeong Kim

18:55 An Investigation of Chinese Mathemati s Tea hers' Resour es Work in Colle tives and Their

Professional Development

Chongyang Wang, Lu Trou he, Birgit Pepin

19:10 Interdis iplinary Program for Professional Development in Mathemati s Tea hing in Mexi o

Daniela Reyes-Gasperini

19:25 Resour es for Tea hing

Jose Luis Cortina, Jana Visnovska

19:40 Dis ussion

20

Poster Presentations

Monday, May 8

th, 2017

14:30 � 16:30

Salão Nobre

The Mathemati al Problem Solving in Textbooks

Maria Madalena Dullius, Geovana Luiza Kliemann

The Colle tive Produ tion of Digital Dida ti Materials as Part of Initial Formation of Mathemati s

Tea hers: A Case Study Involving Digital Textbook Produ tion

Rogério da Silva Igná io

Geometri Constru tions with Rule and Compass in Brazilian Textbooks

Ana Paula Jahn, Mar os Alves dos Santos

Mathemati s Textbook Stru ture in El Salvador with the Aim of In reasing Learning Time as Mu h as

Possible

Félix Abraham Guevara Menjívar, Wendy Stefanía Rodríguez Argueta

Histori al Approa h of the Topi s Matri es, Determinants and Linear Systems Present in the Textbooks

Daniel Martins Nunes

Analysis of Brazilian Textbooks in Two Stages: Pre-Modern Times (1950) And Modern (1960 And

1970)

Alexandre Souza de Oliveira

The Types of Problems Addressed from the Content of Fun tions in the Dida ti Book: Math Context

and Appli ations

José Ant�nio de Oliveira Junior

Books Tasks' Adaptation To Di�erential And Integral Cal ulus Classes

Nélvia Santana Ramos, William José Gonçalves, André Luis Trevisan

Tea hing Probability in Early S hool Years: The Approa h in Brazilian Textbooks

Mi haelle Renata Moraes de Santana, Rute Borba

Colligative Properties and Mathemati al Signs Presented in the Dida ti Books of Chemistry: A Semi-

oti Analysis

Karina Alessandra Pessoa da Silva, Lourdes Maria Werle de Almeida

Cognitive Demands to Compare Mathemati s Textbooks in Di�erent Countries

Xhevdet Thaqi, Joaquin Giménez Antonio José Lopes

A Virtual Spa e as an Alternative to Di�eren ial and Integral Cal ulus Book

André Luis Trevisan, Alan Fran o do Couto, Mar ele Tavares Mendes

The Pres ribed Curri ulum for Combinatori s and What Is Presented in 5th Grade Brazilian Textbooks

Glau e Vilela, Rute Borba

The Comparative Study of Statisti al Course Di� ulty Degree in Junior High S hool Mathemati s

Textbooks of China, the US and Australia

Jianbo Wang

The E�e t of Litera y in the New Hungarian Mathemati s Textbooks

Gergely Wints he, Gergely Szmerka

Use of the Balan e Model in Learning Task by Authors of Mexi an Se ondary-S hool Mathemati s

Textbooks: Its Presen e and Authenti ity

Yolanda Zamora, Josip Slisko, Adrián Corona, María Ara eli Juárez

Pro edural Knowledge Development Model In Chinese Classrooms � Examples Beyond the S hool

Textbook

Jinyu Zhang

21

ABSTRACTS

Plenaries

A MULTIMODAL APPROACH FOR THEORISING AND

ANALYSING MATHEMATICS TEXTBOOKS

Kay O'Halloran

S hool of Edu ation at Curtin University, Perth, Australia

Multimodal analysis has emerged in re ent de ades as an interdis iplinary area of resear h, arising out

of linguisti s, semioti s and drawing on other relevant �elds (in this ase, mathemati s) to study the

ontributions and intera tions of linguisti and non-linguisti resour es (e.g. spoken and written language,

image, gesture, sound, page layout and website design) in the ommuni ation of meaning (Jewitt, 2014;

Jewitt, Bezemer, & O'Halloran, 2016). From this perspe tive, mathemati al knowledge is onstru ted

using language, images and mathemati al symbolism whi h integrate in spe i� ways in mathemati al texts

(O'Halloran, 2015a, 2015b). In this talk, I provide an overview of the semioti nature of ea h resour e,

in luding their underlying organisation that enables them to ful�l di�erent fun tions in mathemati s.

Following this, I undertake an analysis of a mathemati s text in order to demonstrate how linguisti ,

visual and symboli hoi es ombine to solve mathemati s problems, and the expansions of meaning that

take pla e during this pro ess. I on lude with a dis ussion of the di� ulties of tea hing and learning

mathemati s (O'Halloran, 2000, 2015a) with spe i� referen e to mathemati s textbooks.

Referen es

Jewitt, C. (Ed.). (2014). The Routledge Handbook of Multimodal Analysis (2nd ed.). London:

Routledge.

Jewitt, C., Bezemer, J. & O'Halloran, K. L. (2016). Introdu ing Multimodality. London & New

York: Routledge.

O'Halloran, K. L. (2000). Classroom Dis ourse in Mathemati s: A Multisemioti Analysis. Linguis-

ti s and Edu ation, 10(3), 359-388.

O'Halloran, K. L. (2015a). The Language of Learning Mathemati s: A Multimodal Perspe tive.

The Journal of Mathemati al Behaviour, 40 Part A, 63-74.

O'Halloran, K. L. (2015b). Mathemati s as Multimodal Semiosis. In E. Davis & P. J. Davis (Eds.),

Mathemati s, Substan e, and Surmise (pp. 287-303). Berlin: Springer.

MATHEMATICS TEXTBOOKS FOR THE MILLION � BRAZIL'S

MATHEMATICS TEXTBOOKS ASSESSMENT PROGRAM

João Bos o Pitombeira

Universidade Federal do Mato Grosso do Sul, Brazil

We will dis uss Brazil's mathemati s textbooks assessment program, whi h sele ts the textbooks freely

distributed by the Ministry of Edu ation. This program distributed 6 million mathemati s textbooks

among high s hool students in 2015. We will analyse the methodology used in the assessment program,

its legal framework, how the programme's results are known by mathemati s tea her in s hools all over

the ountry and the way they hoose the textbooks they will use in their lassrooms. Finally, we dis uss

the programme's in�uen e on the textbooks � their quality, ontents and formatting.

27

THE DIAGRAMS OF EUCLID'S ELEMENTS

Ken Saito

Osaka Prefe ture University, Japan

Although Eu lid's Elements are no longer dire tly used in s hool mathemati s, Eu lid's style deeply per-

meates in the edu ation of mathemati s. Our use of diagrams in elementary geometry, espe ially the use

of labelled points to designate geometri al obje ts su h as triangle ABC, omes from the Elements.

The study of diagrams in Greek mathemati al texts is rather a new �eld, for diagrams alled the

attention of s holars only after 2000. The �rst result of these re ent resear hes was that the diagrams we

�nd in the translations of the Elements available today, are quite di�erent from those in the best extant

manus ripts, the sour es of the text of the Elements. The former all depend on the riti al Greek edition

of the Elements by Danish s holar Johan Ludvig Heiberg, published in 1880's. So Heiberg was by no

means riti al for the diagrams in his riti al edition.

In the manus ript diagrams, we often see isos eles triangles in a proposition valid for any triangle, and

re tangles where any parallelogram is intended. This tenden y may be alled overspe i� ation (an angle

whi h an be right is often drawn as a right angle, and two sides whi h an be equal are drawn as if they

were equal).

Another hara teristi typi al to manus ript diagrams, whi h is somewhat ontradi tory to what is

just said, is that they are indi�erent to metri al a ura y: two equal lines or equal areas do not always

appear equal in the diagrams. Sometimes a right angle may be drawn as a ute or obtuse.

Heiberg's diagrams are always a urate, and avoid over-spe i� ation. For example, in the so- alled

Pythagorean theorem, proposition I.47, the triangle is re tangular but not isos eles in Heiberg, while the

manus ripts draw it as an isos eles triangle, mostly also re tangular, but sometimes obtuse-angled.

We know that Heiberg opied the diagrams of geometri al books of the Elements from an edition

of 1820's, edited by Ernst Ferdinand August, a Prussian gymnasium tea her. August, without a ess

to manus ripts, systemati ally eliminated over-spe i� ations and metri al ina ura y in the diagrams of

previous editions, apparently from pedagogi al intention.

So today we read the geometry of the Elements a ompanied by the diagrams intended for gymnasium

students of 19th entury.

As for the arithmeti al books of the Elements, whi h are also a ompanied by diagrams, the situation

I have found is di�erent from geometry, and is no less interesting. In short, Heiberg invented himself all

the diagrams for arithmeti , but this was a disaster. They are mu h less helpful for the understanding of

the text than those found in the manus ripts.

In my le ture, I will ompare the diagrams of the Elements in the manus ripts, in earlier editions, in

Heiberg's edition, and examine their expe ted roles in di�erent periods.

ELECTRONIC VS. PAPER TEXTBOOK PRESENTATIONS OF

VARIOUS ASPECTS OF MATHEMATICS

Zalman Usiskin

University of Chi ago, USA

Textbooks now appear in paper form, in ele troni form for omputers, tablets, or phones, and in hybrids

moving among the platforms. Whether an adaptation of a pre-existing paper text or an entirely new book,

some aspe ts of mathemati s seem better suited for traditional paper book presentations while others seem

better suited for ertain kinds of ele troni formats. Based in large part on our urrent work in adapting

our own existing paper textbooks for se ondary s hools for several digital formats, this talk dis usses the

platforms with regard to the presentation of various aspe ts of mathemati s, in luding: vo abulary and

notation, dedu tion, modelling, algorithms, and representations.

28

AN INTERNATIONAL COMPARISON ON SELECTION OF CONTENTS

AND DIFFICULTY LEVEL OF EXAMPLES IN HIGH SCHOOL

MATHEMATICS TEXTBOOKS

Jianpan Wang

East China Normal University, China

Based on our resear h proje t �A Comparative Study on High S hool Mathemati s Textbooks of Certain

Countries� (a China's national key resear h proje t in edu ational s ien es), this talk fo uses on the

following two aspe ts:

1. On the sele tion of ontents. The sele tion and presentation of ontents has fundamental impor-

tan e for mathemati s textbooks. In this relation, the presentation will provide an analysis of the

features of ontent sele tion in the textbooks of di�erent ountries with fo us on four major areas of

mathemati s, i.e., algebra (in luding fun tions), geometry, probability and statisti s, and al ulus,

illustrate (with tables) the similarities and di�eren es of di�erent ountries in the sele tion of on-

tents, and delineate (with examples) di�erent ways of presenting a number of mathemati s ontents

in the textbooks of di�erent ountries.

2. On the di� ulty level of mathemati al examples. Mathemati al examples are an important om-

ponent of mathemati s textbooks and their di� ulty level largely re�e ts the expe tation of the

textbooks for the students. The presentation will provide a omparison of the di� ulty level of

mathemati s examples provided in the sele ted textbooks from di�erent ountries, whi h is arried

out in �ve dimensions in luding ba kground level, mathemati s ognitive level, reasoning level, op-

eration level and knowledge overage level. The talk will end with an analysis on the omposite

di� ulty level of the examples presented in the textbooks.

Referen es

Jiansheng Bao (2002). A omparative study on the omposite di� ulty of intended mathemati s

urri ulum in Chinese and UK junior high s hools. Global Edu ation, vol. 31, no. 9. 48-52.

Jianpan Wang & Jiansheng Bao (2014). Cross-national omparison on examples in high s hool

mathemati s textbooks. Global Edu ation, vol. 43, no. 8. 101-110. In Chinese.

Jianpan Wang & Jianyue Zhang (2014). An international omparison of ore ontent in high s hool

mathemati s textbooks. Curri ulum, Tea hing Material and Method, vol. 34 no. 10. 112-119. In

Chinese.

Jianpan Wang (ed.) (2015). An International Comparison on High S hool Mathemati s Textbooks.

Shanghai: East China Normal University Press. In Chinese.

29

Symposia

SYMPOSIUM A

TEXTBOOK USE BY TEACHERS AND STUDENTS � RESULTS AND METHODS

Sebastian Rezat

University of Paderborn, Germany

srezat�math.uni-paderborn.de

Rudolf Sträÿer

University of Giessen, Germany

rudolf.straesser�uni-giessen.de

Textbooks o�er opportunities to learn, whi h have to be put into pra ti e by tea hers and students.

Tea hers and students are seen as the two major users of textbooks. The textbook is the mediating

artefa t, whi h provides a link between the goals, the knowledge and the beliefs of tea hers and students.

Besides these tea her and student related variables textbook use is also in�uen ed by additional so ial and

ultural in�uen es su h as the in�uen e of institutions, espe ially the s hool, the in�uen e of peers, family

and tutors as well as the one from those interested in mathemati s edu ation in general (the �noosphere�

sensu Chevallard) and �nally the in�uen e of onventions, norms and the publi image of mathemati s

(Rezat & Sträÿer, 2012).

Previous resear h has identi�ed the use of textbooks by tea hers and students as the ore unit of

analysis in order to understand and measure the e�e ts of textbooks on tea hing and learning mathemati s.

The symposium in the frame of the International Conferen e on Mathemati s Textbook Resear h and

Development 2017 in Rio de Janeiro (Brazil) intends to summarize the state of the art of resear h into the

use of traditional and ele troni mathemati s textbooks and related methods in order to develop future

perspe tives of this strand of resear h.

We all for papers on studies of textbook use and related in�uen es and interrelations that fo us on

questions su h as

1. How are traditional and intera tive/digital mathemati s textbooks used by tea hers?

2. How is tea hers' preparatory work in�uen ed by mathemati s textbooks?

3. How do tea hers' goals, knowledge and beliefs guide tea hers' adaptations of mathemati s text-

books?

4. How is lassroom instru tion in�uen ed by mathemati s textbooks?

5. How is textbook use in�uen ed by so ial and ultural aspe ts (as depi ted in the bottom of the

so io-dida ti al tetrahedron)?

6. How are traditional and intera tive/digital mathemati s textbooks used by students?

7. What are the e�e ts of traditional or intera tive/digital mathemati s textbooks on students' learn-

ing?

8. What are the di�eren es in students' learning a tivities/tea hers' a tivities when working with tra-

ditional ompared to intera tive/digital textbooks?

Results from theoreti al and empiri al studies on the use of mathemati s textbooks will be omple-

mented by dis ussing methodologi al hallenges of investigating the textbook in use and its e�e ts on

tea hers and students. The use of intera tive/digital textbooks may o�er new methodologi al hallenges

and additional potentials in terms of resear h methodology. Consequently, the symposium an also provide

spa e for the dis ussion of questions like

M1): What are appropriate methods to investigate tea hers' and/or students' use of mathemati s text-

books?

M2): How to investigate intera tions and interrelations of students' and tea hers' use of mathemati s

textbooks?

33

M3): What about additional hallenges and potentials when analysing the use of intera tive/digital text-

books?

Referen es

Rezat, Sebastian, and Rudolf Sträÿer. 2012. �From the dida ti al triangle to the so io-dida ti al

tetrahedron: artifa ts as fundamental onstituents of the dida ti al situation.� ZDM � The Inter-

national Journal on Mathemati s Edu ation 44 (5): 641-651. doi:10.1007/s11858-012-0448-4

Rezat, Sebastian, & Rudolf Sträÿer. 2015. �Methodologi al issues and hallenges in resear h on

mathemati s textbooks.� Nordi Studies in Mathemati s Edu ation 20 (3-4): 247-266.

ORAL COMMUNICATIONS IN SYMPOSIUM A

READING MATHEMATICS TEXTBOOKS: DIFFERENT READING STYLES

Margot Berger

University of Witwatersrand

margot.berger�wits.a .za

I present a taxonomy for ategorising the ways in whi h �ve di�erent students read a se tion of a hapter

in a mathemati s textbook. This taxonomy is used to suggest a relationship between its ategories and

the student's ability to solve mathemati s problems based on the text.

Resear h on the way in whi h learners read mathemati s textbooks is s ant. Ex eptions to this are

based on frameworks from literary theory. Shepherd, Seldon and Seldon (2012) adapted the Constru tively

Responsive Reading Framework from literary theory to explore why many mathemati s undergraduate

students do not read their mathemati s texts in a useful way. Similarly, Weinberg and Wiesner (2011)

adapted reader-oriented theory (deriving from a bran h of literary riti ism) to illuminate the relationship

between reader and mathemati al text. Berger (2016) developed a framework, spe i� to mathemati s

dis ourse, for exploring the relationship between ena ted dis ourse (the student's way of reading the text)

and the written dis ourse (the text).

In this paper, I develop the latter framework so as to ategorise di�erent approa hes to reading

mathemati s by di�erent learners. The relationship between the written and ena ted dis ourse is theorised

in terms of Sfard's (2008) theory of ommognition. A ording to this theory, individual development is

understood as the individualization of olle tive or so ial a tivity (in this ase, the so ial a tivity is the

dis ourse of the textbook). Within this theory I indu tively develop ategories of analysis for ways in whi h

students may read maths text (in textbooks) based on several observation of di�erent students reading

mathemati s texts.

The empiri al �eld is �ve in-servi e mathemati s tea hers who are enrolled in a self-studying ourse on

Pre al ulus. These tea hers are on eptualised as mathemati s learners both within the ourse and within

this resear h; this is be ause their prior knowledge of mathemati s is often weak and not rigorous. These

learners were individually video-taped while reading and studying a sub- hapter of a parti ular Pre al ulus

textbook (Sullivan 2014). This sub- hapter addresses operations with logarithms and hange of bases.

It was hosen be ause it looks at mathemati al on epts with whi h the students are familiar, but from

a more advan ed perspe tive. For example, they know pro edures for working with logarithms but are

not familiar with proofs of theorems around logarithms. The trans ripts of all �ve learners are analysed

in terms of the ways in whi h they set about reading the text and doing the pres ribed exer ises whi h

appear at the end of the sub- hapter. In parti ular, attention is paid to the opportunities their styles of

reading a�ord them in terms of using the textbook as a resour e for learning.

In my taxonomy, styles of reading involve the depth of reading - lose reading with or without expli it

onne tions, s anning, skimming or avoiding.

34

• `Close reading with onne tions' is hara terised by a omprehensive reading of all the text as well

as the making of expli it onne tions to prior knowledge or to other parts of the text.

• `Close reading without onne tions' is hara terised by a omprehensive reading of all the text

without expli it onne tions.

• `S anning' is hara terised by the reader looking for spe i� keywords or information.

• `Skimming' is hara terised by a qui k look at the text in whi h no expli it onne tions or explana-

tions are presented.

• `Avoiding' is pe uliar to mathemati s reading: all theory and proofs are avoided. All attention is

fo used on pro edures.

In my study, Abby exhibits many traits of a lose reader with onne tions. She �rst reads through

the entire sub- hapter, i.e. text, theorems, proofs, de�nitions and worked examples (WE), highlighting

what is new or unfamiliar to her. She frequently and expli itly relates what she is reading to what she has

previously read. For example, she notes whi h property of logs the proof, worked example, et .is using.

Abby only starts doing exer ises when she has read the entire sub- hapter. When doing these exer ises,

she frequently explains how their solution relates to parti ular properties or theorems. She is able to do

all exer ises very well.

Solly is ategorised as a lose reader without onne tions. That is, like Abby, he reads through

all the text very arefully, highlighting what is new to him and frequently explaining what he has read, in

his own words. However, unlike Abby, he does not expli itly refer to properties or theorems whi h he may

have previously en ountered, as he reads. Like Abby, he only starts doing exer ises when he has read all

the text. He impli itly refers to some properties of logs when doing exer ises.

Paul is a s anner. He starts by reading the exer ises and looking ba k in the text to see whether

there is a WE, proof or property whi h relates to the exer ise. �At �rst I look at the questions and then

from there I went ba k to look at the. . . I looked at the properties basi ally" (line 23). He su essfully

ompletes most exer ises.

Tom is a skimmer. Like Paul, he starts by reading the exer ises, returning to the text on a need to

know basis. However, unlike Paul, he is not always able to a ess an appropriate WE, proof or Theorem

in the text. As he says: �. . . what maybe might make it di� ult for me to understand was that the way I

wrote it. . . it's quite di�erent to what was done in the book. And it makes it di� ult for me to understand

what the textbook has explained to me" (line 31). He makes many errors in the exer ises and is unable

to use the book to assist him although he tries.

Judy is an avoider. She does not read any proofs or WEs. For example, when telling the interviewer

why she will not read the proof of the property: loga ar = r she says: �but I think, I think I understand. I

understand be ause I know this rule" (line 162). Judy tries some WEs on her own but she does not he k

them adequately against the WEs in the textbook. She makes several errors in the exer ises but usually

does not realise that she has made these errors.

In the small sample looked at above, there seems to be a strong relationship between style of reading

and the student's ability to solve problems based on the text. Mu h more resear h is required to tease

out the exa t nature of this relationship.

Referen es

Berger, Margot. 2016. �Reading and Learning from Mathemati s Textbooks: An Analyti Frame-

work.� In Pro eedings of the 40th Conferen e of the International Group for the Psy hology of

Mathemati s Edu ation, edited by Csaba Csikos, Attila Raus h, and Judit Szitanyi, 2:83-90. Szeged,

Hungary.

Sfard, Anna. 2008. Thinking as Communi ating: Human Development, the Growth of Dis ourses,

and Mathematizing. New York: Cambridge University Press.

35

Shepherd, Mary D, Annie Selden, and John Selden. 2012. �University Students' Reading of Their

First-Year Mathemati s Textbooks.� Mathemati al Thinking and Learning 14 (3): 226-56.

Sullivan, M. 2014. Pre al ulus. Ninth Edition. Boston: Pearson Edu ation.

Weinberg, A, and E. Wiesner. 2011. "Understanding Mathemati s Textbooks through Reader-

Oriented Theory." Edu ational Studies in Mathemati s 76: 49-63.

USES OF DYNAMIC TEXTBOOKS IN UNDERGRADUATE MATHEMATICS

CLASSROOMS

Vilma Mesa

University of Mi higan

vmesa�umi h.edu

Angeliki Mali

University of Mi higan

anglmali�umi h.edu

UTMOST team

We present preliminary �ndings from a pilot study ondu ted within the ontext of the Undergraduate

Tea hing and Learning in Mathemati s with Open Software and Textbooks (UTMOST) proje t. This

proje t lays the ground work for understanding the a�ordan es and hallenges of developing and using

open sour e learning platforms in the tea hing and learning of liner algebra and abstra t algebra in

undergraduate mathemati s. Motivating the UTMOST Proje t are two foundational questions: (a) How

do students and tea hers use textbooks? and (b) How an we develop textbooks that will improve tea hing

and learning? Primarily, we seek to develop data olle tion proto ols and instruments, test pro esses of

analysis, and �ne-tune dissemination strategies. The out omes of this proje t are a set of instruments

for data olle tion for a larger study that will also investigate orrelations of resour e use and student

learning.

Context

The UTMOST proje t fo uses on the development and use of open sour e omputational resour es

in the tea hing and learning of mathemati s at undergraduate level. The proje t ontinues work in the

development of four inter onne ted suites of te hnologi al resour es: a) SageMathCloud (SMC), a web

appli ation that provides a s ienti� omputing environment for ollaboration among groups of people

(e.g., a university lass); b) Sage Cells, live Sage omputations whi h an be embedded in any web

page (Sage is a free, open-sour e omputer algebra system); ) MathBook XML (MBX), a new textbook

format that seamlessly in orporates omputation; and d) Stru tured LATEX to XML (SL2X), a method

of onverting existing textbooks to the new MBX format. Notably, SMC tra ks a student's time spent

at various tasks su h as viewing a textbook, using the intera tive features of a textbook, working on

exer ises, editing LATEX do uments, dis ussing the ourse with a fellow student, or dis ussing the ourse

with the instru tor. UTMOST improves these resour es and has a resear h omponent asso iated with

their use in real settings. The proje t investigates the ways in whi h textbooks in the open sour e platform

are used by tea hers and students in Linear Algebra and Abstra t Algebra over three semesters. There

are three overar hing aims of the resear h omponent of the proje t:

1. To identify the tea hers' and students' use of textbooks that are in the open sour e platform, and

the tea hers' and students' use of identi al PDF versions of the same textbooks

2. To ontrast the uses of these resour es (same ontent, di�erent platforms) and

3. To propose measures of student learning that would potentially identify the impa t of these resour es.

Theoreti al and Analyti al Underpinnings

The proje ts seeks to determine the extent to whi h instru tors o�oad, adapt, or improvise as they

design and ena t instru tion (Brown 2009) via an analysis of tea hers logs and individual and ross- ase

36

study analysis (Yin 2003). We hara terize the resour es used by instru tors and students depending on

availability of multi-media (e.g., animations, graphi s), intera tivity with text (e.g., designing ontent),

so ialization (e.g., via blogs, hats), ustomization of learning experien e (e.g., supporting students with

additional needs), ongoing assessment of students' progress (e.g., automati grading), ost e�e tiveness

(e.g., faster and heaper to revise), and a essibility (e.g., a�ordability, broadband a ess) as des ribed

in (Choppin et al. 2014). This hara terization is instrumental for di�erentiating how the dynami and

the non-dynami textbooks are deployed by the users and will provide the ontext for di�erentiating

use. We seek to identify instru tors' utilization s hemes (Rabardel and Waern 2003, Mesa and Gri�ths

2012) using a do umentational approa h (Gueudet and Trou he 2009). We identify students' utilization

s hemes, di�erentiating by in- lass and out-of- lass use, and generating patterns that might be asso iated

with their performan e, using both their bi-weekly logs, and their responses as ase parti ipants, and a

modi�ed version of data olle tion as des ribed by Rezat (2007, 2012, 2013).

Methods

In the pilot phase we olle t in depth data from four lassrooms, three using the dynami version of the

textbooks via observations, interviews, and self-reports of use, and ompare those data with automati ally

generated data that an be aptured via the online platform. Other forms of data in lude tea her surveys

and logs of work, student surveys and logs of work, video-re ordings of tea hers' planning sessions and

lesson ena tment, interviews with tea hers, fo us groups with students and tests for students' performan e.

By the time we present in May, we will be able to draw on our pilot data (4 tea hers at three universities

in two states). As far as we know, this will be the �rst time it will be possible to dire tly study the use

of omputational resour es and to dire tly monitor instru tor and student use of textbooks in a manner

that aptures the �ne details of how, when, and for how long users view a parti ular example, de�nition,

or proof. SMC makes these apabilities possible.

Referen es

Brown, Matthew W. 2009. �The tea her-tool relationship: Theorizing the design and use of ur-

ri ulum materials." In Mathemati s tea hers at work, edited by Janine T. Remillard, Beth Herbel-

Eisenmann and Gwendolyn Lloyd, 17-36. New York: Routledge.

Choppin, Je�rey, Cynthia Carson, Zenon Borys, Catheleen Cerosaletti, and Robert Gillis. 2014. �A

typology for analyzing digital urri ula in mathemati s edu ation." International Journal of Edu a-

tion in Mathemati s, S ien e and Te hnology 2 (1):11-25.

Gueudet, Ghislaine, and Lu Trou he. 2009. �Towards new do umentation systems for mathemati s

tea hers?" Edu ational Studies in Mathemati s 71:199-218.

Mesa, Vilma, and Brett Gri�ths. 2012. �Textbook mediation of tea hing: An example from tertiary

mathemati s instru tors." Edu ational Studies in Mathemati s 79 (1):85-107.

Rabardel, Pierre, and Yvonne Waern. 2003. �From artefa t to instrument" Intera ting with Com-

puters 15 (5):641-645.

Rezat, Sebastian. 2007. �The use of mathemati s textbooks by students." 31st Conferen e of the

Psy hology of Mathemati s Edu ation, Korea.

Rezat, Sebastian. 2012. �Intera tions of tea hers' and students' use of mathemati s textbooks." In

From text to 'lived' resour es: Mathemati s urri ulum materials and tea her development, edited

by Ghislaine Gueudet, Birgit Pepin and Lu Trou he, 239-245. Dordre th: Springer.

Rezat, Sebastian. 2013. �The textbook-in-use: Students' utilization s hemes of mathemati s text-

books related to self-regulated pra ti ing." ZDM International Journal of Mathemati s Edu ation

45:659-670. doi: 10.1007/s11858-013-0529-z.

Yin, Robert K. 2003. Case study resear h design and methods. Thousand Oaks, CA: Sage.

37

PEDAGOGICAL FUNCTIONS OF INTERACTIVE TEXTS

Elena Naftaliev

A hva A ademi College, Israel

elenanaftaliev�gmail. om

Te hnologi al development has aused hanges in learning environments in general and textbooks in par-

ti ular. The limitations and the potential of the presentational media were always part of the mathemati s

ulture. Des ribing the ommuni ation of Greek mathemati ians, Netz (1999) suggests that the limita-

tions of the media available (wetted sand, dusted surfa e or wax tablets) were essentially similar to those

of modern books: �Diagrams, as a rule, were not drawn on site. The limitations of the media available

suggest rather, the preparation of the diagram prior to the ommuni ative a t- a onsequen e of the

inability to erase." (p.16). An intera tive diagram (ID), is a relatively small unit of an intera tive text (in

e-textbook or another material).The ID`s omponents are the given example, its representations (verbal,

visual and other) and intera tive tools. A stati diagram that presents spe i� information presents a point

of view thus impli itly engaging the viewer in meaningful interpretations. An ID presents information and

expli itly requires the viewer to take a tion and hange the text within given limitations.

�Visions of the future of the textbook raise questions about the pedagogi al fun tions of this edu ational

form: What is it that textbooks provide pedagogi ally and epistemologi ally, besides a reminder of the

weight of the past? How might they hange in the future, and how ould su h hanges serve the interests

of publishers, authors, students, and edu ators?" (Friesen, 2013, p.2). Intera tive textbooks are envisioned

as allowing the learner and tea her to approa h texts in an exploratory mode, rather than simply re eiving

it in a �xed, prepa kaged form. There are profound di�eren es between the traditional page in math

textbooks that appears on paper and the new page that derives its prin iples of design and organization

from the s reen and the a�ordan es of te hnology. This issue requires s holars to develop lens for analyzing

pedagogi al design and tea hing-learning pro esses with intera tive texts. Therefore we developed and

elaborated semioti framework for pedagogi al fun tionality of ID that would allow an orderly dis ussion

of the subje t (Naftaliev & Yerushalmy, 2017). There are three fun tions of ID in the framework: the

presentational fun tion, the orientational fun tion and the organizational fun tion (Table 1).

The presentational fun tion fo uses on what and how is being illustrated by the diagram. Three types

of examples are widely used: spe i� , random, and generi examples.

Spe i� examples present the exa t data of the a tivity of whi h they are part of. They serve as a

dynami illustration that helps analyze the situation without being able to hange the information. Random

examples are spe i� examples generated within given onstraints, presenting di�erent information at

various times and for di�erent users. In a generi example, the diagram is stru tured to be representative;

it presents a situation that an be part of the given task, but it is not intended to present the spe i� data

of the a tivity but to help learners be ome a quainted with the generi views of the example through a

pro ess of inquiry. The tone in whi h the text addresses the learner is subje t to design de isions having to

do with the orientational fun tion. �Sket hiness" vs. �rigorousness" of diagrams is an important fa tor in

reader orientation. An a urate ID has ri hness of detail, but ompleteness of detail in sket h means that

the user has to work in order to see through the whole, to make onta t with and examine details. The

IDs' design made it possible to address the given graphs as a sket h, but at the same time the sket h an

be intera tively unfolded into a detailed a urate diagram, whi h auses students to hange their fo us

from data testing to hoosing the ne essary data.

38

The organizational fun tion looks at the system of relations de�ning wholes and parts and spe i� ally

at how the elements of text ombine together. IDs an be designed to fun tion in three di�erent ways:

Illustrating, Elaborating, Guiding. Illustrating IDs are simply-operated, unsophisti ated representations.

They are intended to orient the student's thinking to the stru ture and obje tives of the a tivity by usually

o�ering a single representation and relatively simple a tions. The important omponents in the design of

the Elaborating IDs are ri h tools and linked representations that enable various dire tions in the sear h

for a solution. We use the term Guiding IDs in relation to guided inquiry. This kind of ID provides the

means for students to explore new ideas. In addition to providing resour es that promote inquiry, they

also set the boundaries and provide a framework for the pro ess of working with the task.

The goals of the resear h were: What are the hara teristi s of a tivity onsisted of reading and

solving tasks whi h are presented as IDs, as a way for students to explore texts of the task? How do

the hara terizations of pro esses vary in a ordan e to the three designed organizational fun tions of

IDs: illustrating, elaborating, and narrating? We analysed the work of 13-14 year-old students in task-

based interviews fo used on three major �elds in the s hool-algebra urri ulum (Modeling, Formulating

mathemati al phenomena, Manipulating). The hosen sequen es intended to redu e the e�e t of spe i�

algebra ontent on resear h on lusions. Ea h series in ludes a preliminary task and three omparable

tasks; ea h was designed upon a semioti framework.

A ross our studies we had found that similar tasks with di�erent IDs should be onsidered as di�erent

learning settings. We found that even the minimal intera tion designed in the illustrating ID an be helpful

in onsolidating relevant knowledge that is not adequately stru tured yet. Students who worked with the

ID looked for ways to bypass the designed onstrains: they hanged the representation of the data in the

given example and expanded the given representations or built new ones. Regarding guiding IDs we found

that it an be a form of instru tion toward development of new mathemati al ideas. The guiding IDs'

design limits the student's a tion and at the same time provides an open spa e for student's ideas. The

various linking tools and representations in the elaborating IDs lead to di�erent problem-solving pro esses

and a variety of solutions. The di�eren es between methods were manifest in the variety of the signi� ant

items in the examples, in the representations students hose to work with, in the order of preferen e of

the various representations and in the hoi e to use or not to use the in luded tools.

Referen es

Friesen, Norm. 2013. �The past and likely future of an edu ational form a textbook ase.� Edu a-

tional Resear her 42, (9): 498-508.

Naftaliev, Elena, and Yerushalmy Mi hal. 2017. �Engagement with Intera tive Diagrams: The Role

Played by Resour es and Constraints.� In Digital Te hnologies in Designing Mathemati s Edu ation

Tasks, edited by Leung, A. & Ba aglini-Frank, A., 153-173. Springer International Publishing.

Netz, Reviel. 1999. The Shaping of Dedu tion in Greek Mathemati s: A Study in Cognitive History.

Cambridge University Press, UK.

DEVELOPING METADATA CATEGORIES FOR CURRICULAR MATERIALS: LENSES

TO STUDY DIGITAL TEXTBOOKS-TEACHERS-RELATIONS

Shai Olsher Mi hal Yerushalmy

University of Haifa, Israel

olshers�edu.haifa.a .il mi halyr�edu.haifa.a .il

Digital textbooks and urri ulum resour es an provide many opportunities for tea her engagement in

the form of organizing ontent and generating novel mathemati al experien es for their learners. When

attempting to study the intera tions of tea hers with digital urri ular materials, we explore ways to support

39

the tea her's potential role as a artographer (Remillard, 2016) � having to generate a representation of a

domain of urri ular materials, and also to set paths that would help learners get better a quainted with

the mathemati al terrain.

Our obje t of mapping is a digital textbook, for whi h we use the broad de�nition of Pepin et. al

(2015) for an evolving e-textbook: an evolving stru tured set of digital resour es, dedi ated to tea hing,

initially designed by di�erent types of authors, but open for re-design by tea hers, both individually and

olle tively. While the evolved ontent (tasks, tools, learning obje ts in general) is an important aspe t

of the quality of the book we view the properties of the � olle tion� su h as: stru ture, balan e, and

sequen ing to be ru ial to the oheren e and quality of the book. Taking into onsideration the fa t that

textbooks have some representation of their ontent, e.g. table of ontent, or a site map, when given the

opportunity tea hers hange textbooks to be more a essible (Olsher & Even, 2014).

In most ases methodology used is omparative analysis of ontent analysis or ase studies of long

observation on a single or few tea hers, and the analysis of textbooks is done a ording to the resear h

question. Resear hers determine what they wish to ompare, and then the materials and use ases

are oded a ordingly � from number of pages taught in a ertain topi to nuan es of ommon ore

standards that are manifested in tasks and their ena tment. Our methodologi al approa h enables an

expansion of this pra ti e by adding the tea hers' perspe tive to the resear her's ategories of oding, by

supporting tagging of urri ular materials by tea hers using a browser extension that enables to tag any

webpage a ording to prede�ned ategories, in a ontext whi h holds textbooks and urri ular materials

for tea hing as datasets. This study in ludes development and design based resear h of the browser

extension. The initial oding ategories and rational were designed, among other aspe ts, to address

te hnology related hara teristi s. These ategories serve as an initial state when attempting to listen to

and learn from tea hers while still providing them with a literature based staring point. As the tea hers

begin to use the extension, we gather their remarks about the ategories, and modify the extension

to �t with the ategories tea hers suggest as relevant for them to use in tagging and representing the

urri ular materials. The dynami analysable database enables us to tra k versions of the meta-data

ategories for tagging reated and modi�ed by the tea hers upon the basi given version. This analysis

an reveal ommonalities and di�eren es among various audien es of edu ators: among the tea hers,

among resear hers or between tea hers and resear hers in a wide range of dimensions: from relevant and

irrelevant ategories to ommuni ating information about urri ular resour es.

Referen es

Olsher, S., & Even, R. 2014. "Tea hers editing textbooks: Changes suggested by tea hers to the

math textbook they use in lass". In Pro eedings of the International Conferen e on Mathemati s

Textbook Resear h and Development (ICMT-2014) edited by K. Jones, C. Bokhove, G. Howson, &

L. Fan. Southampton, GB: University of Southampton.

Pepin, B., Gueudet, G., Yerushalmy, M., Trou he, L. & Chazan, D. 2015. "etextbooks in/for Tea h-

ing and Learning Mathemati s: A Disruptive and Potentially Transformative Edu ational Te hnol-

ogy". In Handbook of International Resear h in Mathemati s Edu ation, edited by L. English, & D

Kirshner, 636-661. New York: Taylor & Fran is.

Remillard, J. 2016. "Keeping an eye on the tea her in the digital urri ulum ra e". In Digital

Curri ula in S hool Mathemati s, edited by Bates, M. and Usiskin, Z., 195-204. Charlotte, NC:

Information age publishing.

ANALYZING CLASSROOM WORK: STUDENTS' USE OF ELECTRONIC TEXTBOOKS

Kristina Reiss, Stefan Ho h, Frank Reinhold, Bernhard Werner, Jürgen Ri hter-Gebert

Te hni al University of Muni h

kristina.reiss�tum.de, stefan.ho h�tum.de, frank.reinhold�tum.de, werner�ma.tum.de, ri hter�ma.tum.de

40

�ALICE: fra tions� (Adaptive Learning in an Intera tive Computer-supported Environment) is an ele troni

intera tive textbook on fra tions. This learning environment aims at assisting students' work with fra tions

and understanding their problem-solving pra ti es. Moreover, it supports tea hers in diagnosing students'

learning pro esses. Based on the work of 155 6th-graders who used this e-book during a four-week lass

on fra tions we provide pro ess data allowing the analysis of students' a tions in some detail. In parti ular,

the data show students' answers as well as their ways to these answers. A ordingly, it is possible to get

insights in their understanding or misunderstanding of fra tions. In this presentation, we will introdu e

typi al strategies of students' when a quiring the on ept of fra tion. Based on theoreti al onsiderations,

we will provide empiri al eviden e for these strategies and will dis uss bene�ts for mathemati s tea hers

and the mathemati s lassroom.

41

SYMPOSIUM B

DEDUCTIVE REASONING, ARGUING AND PROOF IN TEXTBOOKS

Luísa Rodriguez Doering Cydara Cavedon Ripoll

Universidade Federal do Rio Grande do Sul, Brazil

ydara�mat.ufrgs.br ldoering�mat.ufrgs.br

There is international re ognition of the importante of reasoning and proof in students' learning of math-

emati s at all levels of edu ation, and of the di� ulties met by students and tea hers in this area. Indeed,

many students fa e di� ulties with reasoning about mathemati al ideas and onstru ting or understanding

mathemati al arguments that meet the standard of proof. Tea hers also fa e di� ulties with reasoning

and proof, and existing urri ulum materials tend to o�er inadequate support for lassroom work in this

area. (THEMATIC STUDY GROUP 18 REASONING AND PROOF in MATHEMATICS EDUCATION,

INTERNATIONAL CONGRESS ON MATHEMATICS EDUCATION � ICME 13, July 2016)

In the International Congress on Mathemati s Edu ation ICME 13 there were reports that in many

ountries an emphasis on proof is reappearing in mathemati s urri ula. Hen e it is of major importan e

that the textbooks support that tenden y. Do textbooks in lude ontexts, ontents, results that meet the

standard of proof? If yes, are the arguments appropriate for students at the proposed level? Are they

orre t? Are they oherent to the previous ontents of the textbook?

We invite submissions on reports referring to the situation in the author�s ountry on erning reasoning,

arguing and proof in the textbooks, illustrated by ex erpts on�rming the situation. We also invite

submissions of reports on experien es fo using on mathemati al arguing in the lassroom, based or not

on textbooks, using or not symboli language.

The 180 minutes of this symposium will be used for short presentations followed by dis ussions aiming

to get an overview of the state of the art on erning reasoning in textbooks.

ORAL COMMUNICATIONS IN SYMPOSIUM B

PROOFS AND ARGUMENTS ON INTEGERS NUMBERS IN THE BRAZILIAN 7TH

GRADE MATH TEXTBOOKS

Miguel M. Be k

Universidade Federal do Rio Grande do Sul, Brazil

miguel.be k�ufrgs.br

The present work is the result of the survey arried out on mathemati al statements (propositions, orol-

laries, theorems and others) present in the mathemati al textbooks of the National Plan of the Dida ti

Book (PNLD 2016) orresponding to the 7th grade of Elementary Edu ation (11 year-old-students) on

Integer Numbers. This study seeks to map the mathemati al statements that are presented to the students

and how the authors prove or demonstrate the validity of those statements. The results obtained were

analyzed in terms of the mathemati al stru ture of the set Z. It turned out that there is almost total

absen e of de�nitions as already pointed out by ALMEIDA (2015), the arguments are based on pun tual

examples and non-generalized statements. With respe t to addition and subtra tion between integers, the

arguments are redu ed to geometri exempli� ations.

Referen es

ALMEIDA, T.B. Uma Revisitação aos Conjuntos Numéri os no Ensino Médio, Dissertação de

Mestrado da UFRGS, Programa de Pós-Graduação em Ensino de Matemáti a, Universidade Federal

do Rio Grande do Sul, 2013. Disponível em http://hdl.handle.net/10183/133662. A esso: 15 dez

2016.

42

BRASIL, Ministério da Edu ação. 1998. Parâmetros urri ulares na ionais: matemáti a. Brasília:

MEC/SEF Brasil.

BRASIL, Ministério da Edu ação. 2016. PNLD 2017: Matemáti a � Ensino Fundamental anos

�nais. Brasília: Ministério da Edu ação, Se retaria de Edu ação Bási a.

THE INTRODUCTION TO ALGEBRA IN TEXTBOOKS

Sandro de Azevedo Carvalho

Instituto Federal de Edu ação, Ciên ia e Te nologia Sul-Rio-Grandense

sandro arvalho�sapu aia.ifsul.edu.br

Cydara Cavedon Ripoll

Universidade Federal do Rio Grande do Sul

ydara�mat.ufrgs.br

In many Brazilian textbooks algebrai expressions are introdu ed with no onne tion to arithmeti or to

mathemati al thinking. In those books, for instan e, instead of dealing with symbols as generi numbers,

the operations with algebrai expressions are de�ned, and new nomen lature and new rules are presented,

like �similar terms" and �we an only add or subtra t similar terms", respe tively.

If we ompute with algebrai expressions without making use of the properties of the arithmeti

operations, we loose an unique opportunity of using mathemati al thinking with the students, mainly the

one related to abstra tion.

In fa t, algebra in Elementary S hool is losely related to the mathemati al way of arguing. Using

symbols as representatives of numbers, we an establish arithmeti assertions and justify them using

generi thinking, that is, arguing making use of those symbols and not of parti ular instan es.

Algebra enables us to write shortly and oherently general relations (...). It is an essential tool for

proving properties, des ribing patterns and solving problems. (...) For that, abstra tion is ne essary

to free us from ontexts and parti ular ases.

(Martinez 2013, p. 16)

An analysis on erning the introdu tion to �algebrai expressions" in Brazilian mathemati s textbooks

approved in the Plano Na ional do Livro Didáti o (PNLD � the Brazilian national textbook programme for

free distribution of textbooks to all publi s hools.) was made by Carvalho (2010). It in ludes in oheren es

that were found in those textbooks, and it is also remarked that those books do not follow the suggestions

made on the o� ial do ument Parâmetros Curri ulares Na ionais para o Ensino Fundamental (PCN � the

Brazilian standards do ument for the �rst eight years of Elementary S hool), for instan e,

Through the exploration of everyday problems, the student will re ognize di�erent uses of algebra

(generalizing arithmeti patterns, establishing relations between two magnitudes, modeling, solving

problems whi h are arithmeti ally di� ult, and will represent problems using equations and inequalities

(distinguishing parameters, variables, unknowns, getting in onta t with formulas), (...).

(Se retaria de Edu ação Fundamental, 1998, p. 50-1).

In Carvalho (2010), and also in the present text, the authors want to strengthen that the �rst onta t

of the student with algebra should o ur through the use of symbols that represent numbers, whi h Usiskin

(1999) alls generalized arithmeti s.

In this work, we analyse books from other ountries with respe t to the introdu tion of algebra,

in luding textbooks from Portugal, Chile, Fran e, and Cape Verde and also books from Fran e and Chile

written for Elementary S hool tea hers.

In those books we re ognized, by one hand, some similarities with the Brazilian textbooks with respe t

to riti ized items (dividing algebrai expressions without paying attention to the values of the variable

43

that make the denominator equal to zero; no mention to properties of the operations, restri ting the

argues to rules like �we an only add or subtra t similar terms"; no de�nition or inadequate de�nition

of �algebrai expression"). By the other hand, we ould also �nd some suggestions related to the �rst

onta t with algebra in favor that the symbols should be interpreted as numbers and the operations with

algebrai expressions should be based on the properties of the arithmeti operations.

Referen es

Carvalho, Sandro A. 2010. �Pensamento genéri o e expressões algébri as no Ensino Fundamen-

tal". (Masters diss.,Universidade Federal do Rio Grande do Sul) A essed: O tober 21, 2016.

http://www.lume.ufrgs.br/handle/10183/29352.

Se retaria de Edu ação Fundamental. 1998. �Parâmetros urri ulares na ionais: matemáti a".

Brasília: MEC/SEF Brasil. A essed: O tober 21, 2016.

http://portal.me .gov.br/seb/arquivos/pdf/matemati a.pdf.

Usiskin, Zalman. 1999. �Con eptions of S hool Algebra and Uses of Variables". In Algebrai

Thinking, Grades K-12: Readings from NCTM's S hool-Based Journals and Other Publi ations,ed.

Barbara Moses (Reston, Va.: National Coun il of Tea hers of Mathemati s), 7-13.

Martinez, Salomé and Varas, Maria L. 2013. Álgebra para Futuros Profesores de Edu a ión Bási a,

Proye to FONDEF. (Ed. SM Chile S.A.).

AREA FORMULA DEDUCTIONS FOR PLANE FIGURES IN TEXTBOOKS

Luisa Rodríguez Doering

UFRGS, Brazil

ldoering�mat.ufrgs.br

Fran iele Mar iane Meinerz

EMEF Heitor Villa Lobos, Brazil

fran ielemeinerz�hotmail. om

Mathemati al argumentation in the lassroom has been gaining prominen e in the last years, and is

present in important do uments about Mathemati s tea hing. Although, a ording to the Parâmetros

Curri ulares Na ionais (National Curri ulum Parameters), one of the general obje tives for the tea hing

of Mathemati s in elementary edu ation is that the students are presented to situations that favor the

pro ess of "des ribe, represent and present results a urately, and argue about their onje tures, making

use of oral language and establishing relationships between it and di�erent mathemati al representations"

(Brasil, 1997), studies indi ate that the involvement of students in a tivities related to the development of

argumentation are not usual in lassrooms (Nunes and Almoloud, 2013). In fa t, in addition to students

not being exposed to a tivities related to argumentation, students often also do not have the opportunity

to be involved in the tea hing of geometry, whi h boils down to the tea hing of measures, however "it is

a fa t that geometri questions tend to arouse the interest of adoles ents and young people in a natural

and spontaneous way. In addition, it is a fertile �eld of problem situations that favors the development of

the ability to argue and build demonstrations. (Brasil, 1998).

The argumentation and the dedu tive reasoning must be motivations for the valorization and the

insertion of geometry in the s hool environment (Búrigo, 2005). Thus, believing that geometry is a good

�eld for the insertion of argumentation in the lassroom, in our work we seek to investigate the ontent of

area al ulation of plane �gures in relation to argumentation and dedu tion of mathemati al formulas in

textbooks. We analyzed four textbooks from di�erent olle tions. In the olle tions analyzed, the ontent

starts in di�erent years: 6th, 7th, or 8th grade, with revisions in volumes of later years. In the four works

analyzed, all the authors use the same order of presentation of the �gures whose area is to be al ulated:

re tangle, square, parallelogram, triangle, trapeze and lozenge. This order is natural: the formula for

al ulating the area of the parallelogram an be obtained from the area of the re tangle, the areas of the

44

triangle and the trapeze an be obtained from the area of the parallelogram, and the area of the lozenge

an be obtained from the area of the triangle, or the area of the re tangle.

About the argumentation in the dedu tions of these formulas, we an observe some super� iality,

be ause, in some instan es, fundamental details are omitted, whi h ould en ourage the development of

the students' mathemati al thought pro esses. With regard to the dedu tion of the formula for the area

of the re tangle, for example, whi h we believe to be one of the most important dedu tions to be made,

sin e it is from it that we an dedu e all the other formulas, none of the analyzed authors makes use of

generi thinking: numeri al examples are given, where one an observe the number of squares (units of

area) that form the re tangle, but in only one of the textbooks the author re alls that to ount these

squares, one an use the idea of multipli ation. In the example given, the author shows a re tangle with 5

m of base and 3 m of height, and only informs that to ount the squares we an multiply 5 by 3, without

making any allusion to the meaning of re tangular arrangement asso iated with multipli ation (not even

is there a suggestion ommented in tea her's book): the re tangle divided into 5 olumns, with 3 units

of area in ea h, obtaining 5× 3 = 15 units of area. The generi thought ould now easily be employed to

extend what was done earlier, namely, dividing a re tangle into b olumns, with a squares in ea h olumn,

we would have a total of b×a squares, and then dedu t the formula for al ulating the area of a re tangle

in a generi form for natural numbers. We believe that this argumentation would be within the rea h of

students, as already shown in Meinerz (2015). We also believe that this would be a good starting point

to later extend the formulas to rational and real measures, making use of the on ept of these numbers.

In the other dedu tions of the area formulas of the triangle, parallelogram, trapeze and lozenge, some

authors go from the numeri al example to the respe tive formula of area al ulation without justi� ations,

and others ompose and de ompose generi �gures, omitting justi� ations that are within the rea h of

students and that validate the argumentation, ompleting the mathemati al thinking. Hen e, we believe

that for the argumentative development of the students we need textbooks that present more omplete

dedu tions of the formulas for the al ulation of areas of plane �gures, or that o�er a tivities that involve

the students and en ourage the dedu tive thinking and the argumentation in the lassroom. Thus, in

our work we present a more detailed analysis of these ontents, and we suggest a tivities that allow the

students to onje ture and to dedu e some formulas for the al ulation of areas of some plane �gures.

Referen es

Bian hini, E. 2014. 7

a. edição. Matemáti a. 7

o. Ano. São Paulo: Moderna.

Brasil. 1998. Ministério da Edu ação. Se retaria de Edu ação Fundamental. Parâmetros Curri u-

lares Na ionais: Matemáti a. (3

o. e 4o. i los do ensino fundamental). Brasília: MEC.

Brasil. 1997. Ministério da Edu ação. Se retaria de Edu ação Fundamental. Parâmetros Curri u-

lares Na ionais: Matemáti a (Ensino Fundamental). Brasília: MEC.

Búrigo, Elisabete., Z. 2005. Para que ensinar e aprender Geometria no Ensino Fundamental? Um

exer í io de re�exão sobre o urrí ulo. In Teorias e fazeres da es ola em mudança. Editora UFRGS,

Porto Alegre.

Dante, Luiz. R. 2012. Projeto Teláris: Matemáti a. 6

o. ano. São Paulo: Áti a.

Iezzi, Gelson. 2013. Dol e, Osvaldo. Ma hado, Antonio. Matemáti a e realidade. Ensino Funda-

mental. São Paulo: Editora Atual.

Leonardo, F. M. D. 2010. Projeto Araribá: matemáti a 8

o. ano. Obra Coletiva, Editora Moderna, 3.

Meinerz, Fran iele. M. 2015. �O estudo da área via omposição e de omposição de �guras planas:

uma possibilidade para inserção da argumentação na es ola bási a�. (Undergraduate diss., Univer-

sidade Federal do Rio Grande do Sul).

A essed: November 14, 2016. http://hdl.handle.net/10183/134199

Nunes, J., & Almouloud, S. 2013. AG.2013 O modelo de Toulmin e a análise da práti a da

argumentação em matemáti a. Edu ação Matemáti a Pesquisa. São Paulo, 15(2), 487-512.

45

THE EUCLIDEAN DIVISION IN THE FIRST SCHOOL YEARS


UFRGS, Brazil


Cydara Cavedon Ripoll

UFRGS, Brazil


Janete Ja inta Carrer Soppelsa

EMEF Madre Feli idade, Brazil

jsopelsa�gmail. om

The ideas related to the division of natural numbers appear in the �rst years of elementary s hool. At this

level, division is a omplex operation for the students be ause, when � ompared to the other elementary

operations, the division of natural numbers di�ers in the following sense. While with addition, subtra tion

and multipli ation we have two initial values and obtain a third value, whi h is the result of the operation,

the division of natural numbers gives us two resulting values: the quotient and the remainder.� (Ripoll,

Rangel, Giraldo, 2016, p. 104)

In the six textbooks olle tions that we analysed, division is introdu ed in the 2nd year of elementary

s hool, right after multipli ation, and they all onsider both the meaning of equal parts as well as the

meaning of measure (that is, how many times the divisor �ts in the dividend).

In all but one of the analysed olle tions, the �rst examples of division involve always a dividend that

is a multiple of the divisor, therefore, in all ases the remainder is equal to zero, and there is no mention

of the remainder in those �rst examples. Hen e, the student is led to think that division also has only one

out ome, the quotient. Despite the lear intention of alling the attention to the lose and undeniable

relation between multipli ation and division, we would like to o�er some issues to be onsidered when

tea hing Eu lidean division.

Is it natural to start the dis ussion on division of natural numbers only with dividends that are multiples

of the divisors?

Are the divisions with zero remainder a tually the most frequent ases of division in the students' real

life?

Shouldn't the division have an �independent� life from the multipli ation, in ontrast to what happens

with addition and subtra tion?

When division is onsidered as a partition, we an see there is no emphasis on the aspe t �equal parts�

implied by the term �division�. It should be noted that it is not always natural, for 2nd-year-students,

that �division� should be in equal parts, hen e, at least, this term should eli it a onsideration from the

student, as suggested, for instan e, in Figure 1.

Still related to the little emphasis to the onvention implied by the term division, we remark that not

only it implies �partition in equal parts� but also �remainder < divisor�, whi h is an important hypothesis to

be onsidered in the onstru tion of division algorithms. In none of the analysed textbooks we ould �nd

any referen e to the fa t that, for instan e, although both equalities 17 = 2× 5 + 7 and 17 = 3× 5 + 2are true, only the se ond one derives from the division in N (Eu lidean division). Therefore, in those

textbooks, the hypotheses for this operation, whi h are a ru ial part of mathemati al reasoning, are not

emphasized.

Many authors of textbooks for Elementary S hool, right after introdu ing the division operation and

giving some examples, all of them with zero remainder, end up omparing division as an inverse operation

to multipli ation, and some of them even in lude a se tion entitled �Multipli ation and Division: inverse

operations� and in it emphasize that �between multipli ation and division happens the same as between

addition and subtra tion.� This last assertion is wrong in the number universe N, and it suggests to the

students that in all divisions in N the remainder is equal to zero. Figure 2 shows a situation whi h is

familiar to the students, and whi h demonstrates that this assertion is false. The Eu lidean division is the

only division that makes sense in the universe N, and very often the remainder plays an essential role in

the problem we are trying to solve. Thus, this mis on eption should be avoided in elementary edu ation.

�In the majority of the elementary s hool textbooks, the meaning of the remainder in an Eu lidean

division relies only on the ideas of a leftover, and no further or more detailed exploration is devoted to

46

its use; furthermore, the majority of the exer ises proposed are related to dire t omputations or point

almost ex lusively to the ideas asso iated the quotient of a division.� (Soppelsa, 2016)

In the analysed textbooks, we ould also �nd little appeal to the representation with �gures in the

omputation of divisions with non-zero remainder. It should be stressed that this representation an help

students a lot, both in visualizing the omputation (see Figure 3, for instan e) and in establishing the

inverse pro ess, whi h not only helps to verify the orre tness of the omputation of the division, but also

o�ers an opportunity for the generi thinking, whi h leads them to re ognize the relationship between the

terms dividend = divisor × quotient+ rest of the division and to justify its validity with words.

In our work, we will present a proposal to address the Eu lidean division that takes into a ount the

issues we dis ussed above and that stimulate the mathemati al thinking of the students of the �rst years

of elementary s hool.

Referen es

Isolani, C. M. M. et al. 2005. Coleção Construindo o Conhe imento � Matemáti a, 1

a. série. São

Paulo: IBEP.

Ripoll, C.; Giraldo, V.; Rangel, L. 2016. Livro do Professor de Matemáti a na Edu ação Bási a,

Vol. 1, Coleção Matemáti a para o Ensino. Rio de Janeiro: SBM.

Soppelsa, J. J. 2016. Divisão eu lidiana: um olhar para o resto (Masters diss.,Universidade Federal

do Rio Grande do Sul) A essed: November 14, 2016. http://hdl.handle.net/10183/148203

ANALITYC GEOMETRY IN THIRD YEAR HIGH SCHOOL TEXTBOOKS


UFRGS, Brazil


Platão Gonçalves Terra Neto

FET Liberato Salzano Vieira da Cunha

platao.neto�liberato. om.br

A ording to the parameter of the National Curri ulum (Brasil, 2006, p. 128), the third year of Brazilian

high s hool ontemplates the tea hing of analyti geometry, with the goal of algebrai ally representing

geometri forms of the Cartesian plane. Su h representations involve equations and oordinates as a

means to give the student a distin t way to deal with geometry. Besides, the national urri ulum also

re ommends that:

The tea hing of analyti geometry should be presented in an arti ulate manner with algebra, enlarging

the ability of visualization. It is important to value not only the algebrai manipulation, whi h many

times has no meaning for the student, but also to emphasize the geometri meaning of the oe� ients

in the equations (of the line and the ir le), of parallel lines and of perpendi ular lines, among others.

Su h representations involve equations and oordinates as a means to give the student a distin t way

to deal with geometry.

(Brasil, 2016, p.563)

47

In this arti le, we analyze the hapter devoted to analyti geometry in four mathemati s textbooks for

the third year of high s hool that belong to the Brazilian National Textbook Program. It is interesting

to stress the operational importan e of the textbook in the Brazilian edu ation system and keep in mind

that su h an analysis should be a onstant task. Choppin (2002) expresses the need of this task when he

states that:

A textbook is not a book that we read, but an instrument that we use. The omplexity of the textbook,

and therefore, its analysis, derives from the fa t that it takes on multiple fun tions (and with the

passing of time, these tasks are always more and more) with its bene� iaries (students, tea hers,

families,. . . ) whose expe tations vary a ording to time (tea her preparing alone his lasses, tea her

le turing, et .). It is the seizing of the ons ien e of the dynami dimension of the textbook (it only

really exists through the uses we give it!) that la ks most analyses.

(Choppin, 2002, p. 22-23)

For our analysis, we hose the following topi s: the Cartesian plane, points (middle point and distan e

between points), lines (equations and distan e between points and lines), and triangles (area formulas).

In ea h of these topi s, we initially verify how the ontent is presented: through appli ations of the

mathemati s, the histori al ontext, or theoreti al results. Next we verify if the main results are simply

presented or if there are some justi� ations or dedu tions and, in this ase, whether the used argumentation

is omplete, that is, if it onstitutes a proof (in mathemati s as a s ien e). We lose ea h topi he king

whether the proposed a tivities instigate the mathemati al thinking of the student.

The analysis of these topi s grew out of a resear h for the Master's degree in Mathemati s Edu ation

in whi h we also asked ourselves if the textbook allows the integration between the results of plane and

analyti geometry, sin e Duval (2011), in his Theory of Registers of Semioti Representation explains

the importan e of the use of more than one type of register, sin e �. . . usually, the swit hing of registers

onstitutes a fundamental dida ti ognitive variable: it fa ilitates onsiderably the learning� (p. 81).

An interesting approa h we found in the dedu tion of the triangle area in the Cartesian plane is

exempli�ed in the following �gure:

Figura 1: A numeri al-geometri al approa h to the triangle area al ulation

This method of al ulating an area is based on the al ulation of the area of a larger re tangle minus

the area of three smaller triangles, and it is presented as an introdu tory example of the �Triangle� topi in

one of the analyzed books. In this ase, the author uses results of plane geometry to solve issues in analyti

geometry and, later on, builds up a triangle area formula � through determinants � algebrai ally, making

use of a plane geometry argument to develop a formula for the al ulus of area through the determinants

of analyti geometry.

Our analysis, however, shows a big gap in the textbooks with respe t to the presentation of the

dedu tion of formulas involved in analyti geometry. What we observed is that the textbooks develop the

�rst two topi s (Cartesian plane and points) in greater detail, whereas, on e they move on to the latter

topi s (lines and triangles), they seem to assume that some of the steps of the dedu tion of formulas an

be skipped. Moreover, the use of �gures of plane geometry for the presentation of formulas is gradually

48

redu ed, transforming the later ontents in an array of formulas, without the bene�t for the student � and

the tea her � of a better understanding of he obtained result.

Referen es

Brasil. Ministério da Edu ação e do Desporto. Orientações Edu a ionais Complementares aos

Parâmetros Curri ulares Na ionais: Ensino Médio. Brasília. MEC, 2006.

Disponível em: http://portal.me .gov.br/seb/arquivos/pdf/Cien ias Natureza.pdf. A esso em: 30

Nov. 2016.

Brasil. Ministério da Edu ação e do Desporto. Base Na ional Comum Curri ular: Ensino Médio.

Brasília. MEC, 2016. Disponível em: http://basena ional omum. me .gov.br/do umentos/bn -

2versao.revista.pdf. A esso em: 30 Nov. 2016.

Choppin, Alain. O historiador e o livro es olar. Tradução de Maria Helena Camara

Bastos. Disponível em: http://seer.ufrgs.br/index.php/asphe/arti le/view/30596/pdf. A esso em:

20 out. 2016

Duval, Raymond Ver e ensinar a matemáti a de outra forma: entrar no modo matemáti o de pensar:

os registros de representações semióti as. São Paulo: PROEM, 201

Paiva, Manoel. Matemáti a Paiva. São Paulo: Moderna, 2013, v. 3.

DIFFERENT TYPES OF REASONING IN GEOMETRY

Lu as Carato Mazzi

University of Campinas (UNICAMP), Brazil

l mazzi�outlook. om

In 1985, Brazil developed a National Program of Textbooks (PNLD). It has the obje tive of evaluating,

buying and free distributing the textbooks that are going to be used in publi s hools. First, a ommittee

hosen by this program evaluate the books and de ide whi h ones is quality produ ed to possibly be used

at s hools. Se ond, the program reates a guide that presents the most important ontent and approa h

of the approved textbooks, so the tea hers an hoose whi h one they want to work with. Finally, the

government sends the textbooks to all publi s hools. �S hool textbooks have re eived in reasing attention

in the international resear h ommunity of mathemati s edu ation over the last de ades. Nevertheless,

mathemati s textbook resear h as a �eld of resear h is still at an early stage of development� (Fan, 2010,

765).

Based on those assumptions, this paper aims to present a PhD resear h that has the main goal to

analyse six olle tions of Brazilian's mathemati s textbooks approved by PNLD, more spe i� ally, high

s hool textbooks, in a total of 18 books. Its fo us is to omprehend the di�erent fun tions attributed to

and/or played by proof in geometry, also to identify how di�erent types of reasoning - dedu tive, indu tive,

abdu tive and reasoning by analogy - appears in hapters of Geometry. Reid and Knipping (2010) onsiders

re ent resear h in mathemati s edu ations has made onsiderable ontributions to widening our knowledge

of the kinds of reasoning involved and related to proving, however it is still a new �eld of resear h that

needs to be explored.

Referen es

Fan, Lianghuo. 2013. �Textbook resear h as s ienti� resear h: towards a ommon ground on issues

and methods of resear h on mathemati s textbooks". ZDM Mathemati s Edu ation, 45: 765-777.

Reid, David A.; and Knipping, Christine. 2010. Proof in Mathemati s Edu ation: resear h, learning

and tea hing. Wolfville: Sense Publishers.

49

TEACHER VIEWS ABOUT ARGUMENTATION AND MATHEMATICAL PROOF IN

SCHOOL

Lilian Nasser

Projeto Fundão - IM/UFRJ, Brazil

lnasser.mat�gmail. om

Carlos Augusto Aguilar Jr.

COLUNI/UFF

arlosaugustobolivar�hotmail. om

This arti le relates a resear h about argumentation and proof in the mathemati s lassroom at Basi S hool

and the tea her's view on this hallenge: how to develop in students of this level of edu ation the ability

to argue and prove in Mathemati s. We believe that the development of dedu tive reasoning depends

more on tea hers' a tions than on dida ti al materials. Although the Brazilian Curri ular Parameters

(PCN) re ommend that �Mathemati s intervenes strongly with the formation of intelle tual apa ities,

the stru turing of the thought and the development of pupil's dedu tive reasoning" (BRASIL, 1997, p.15),

tea hers have the power of designing the tea hing sequen e of a tivities presented to students, de iding

how to approa h argumentations and justi� ations.

The new Brazilian National urri ulum, still under dis ussion, re ommends that

the student must be motivated, in his s hooling lifetime, to question, to formulate, to test and to

validate hypotheses, to sear h ounter examples, to model situations, to verify the adequa y of the

answer given to a problem, to develop languages and, as a onsequen e, to onstru t forms to think

that lead him to re�e t and to a t in a riti al way about the questions fa ed daily.

(BRASIL, BNCC, 2016, p. 131)

The investigation, ondu ted in two stages, intended to determine how tea hers appre iate and a ept

the various levels and types of argumentation presented by students of elementary and se ondary s hool

(AGUILAR JR., 2012; AGUILAR JR & NASSER, 2014).

First, a form with questions, going beyond the simple appli ation of known results and al ulations,

and require deeper logi al-dedu tive reasoning, through argumentation and justi� ation, was given to

students. The questions had been applied in three publi s hools, to 124 students of 8th and 9th grades

of basi edu ation, from 12 to 17 years of age. The on lusion was that the level of argumentation of

this group is still ingenuous and informal. The great majority of the given answers presented empiri al

hara ter, sin e the veri� ation of the truth was based on examples, what Bala he� (1988) alls naive

empiri ism.

In the se ond stage, some sele ted responses from this form have been ategorized a ording to the

models and types of proof established by Bala he� (1988) and by Harel and Sowder (1998) to assemble the

form that was applied to 59 tea hers, following a methodology undertaken by Hoyles (1997), in Britain.

In this form, the parti ipants of the inquiry evaluated the answers given by the pupils, justifying the

marks given. In the analyses arried through, we verify that tea hers have great in lination for arguments

approa hing to the formal proof. More in ipient and ingenuous proposals of argumentation had not been

so well valued, and pragmati answers (BALCHEFF, 1998), were onsidered una eptable as proofs, under

the point of view of the mathemati al rigor.

Data analysis, whi h also took into a ount 10 undergraduate students who parti ipated in ompleting

the form, indi ates that tea hers in this group, in general, are not in lined to foster the development of

a tivities in the lassroom, in order to onstru t skills and abilities to argue and prove in mathemati s.

Their preferen e, in terms of evaluating students' responses, is for arguments loser to Bala he�'s model

of on eptual proof (1988), whi h is the rigorous and formal proof, supported, a epted and pra ti ed at

the A ademy.

Textbooks ould help in this issue, sin e they a t as guides for the tea hing pra ti e. Martins and

Mandarino (2014) analyzed the Guide of Textbooks of 2011, fo using on the hapters on erning Geom-

etry. They on luded that the 11 olle tions evaluated from the Guide, in general, presented a tivities

and explanations that mixed pragmati and intelle tual justi� ations, deta hing the absen e of a tivities

exploring the inquiry pro ess, essential and fundamental in the onstru tion of the mathemati s knowledge.

50

But, even in the ases where reasoning and argumentation is emphasized in mathemati s textbooks,

we do not believe that the majority of Brazilian tea hers adopt this kind of a tivity in their pra ti e.

Referen es

AGUILAR JUNIOR, C. A. 2012. �Postura de Do entes quanto aos tipos de Argumentação e Prova

Matemáti a apresentados por alunos do Ensino Fundamental". Master's Degree Dissertation -

PEMAT, Instituto de Matemáti a, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil.

AGUILAR JUNIOR, C. A and NASSER, L. 2014. " Study on Tea her Views in Relation to Argu-

mentation and Mathemati al Proof in S hool". Bolema, Rio Claro (SP), v. 28 (50), p. 1012-1031.

BALACHEFF, N. 1988. "Aspe ts of proof in pupils' prati e of s hool mathemati s". In: PIMM, D.

(Ed.). Mathemati s, tea hers and hildren, Hodder & Stoughton: London, GB, p. 216-235.

BRASIL. 1997. Parâmetros Curri ulares Na ionais: Matemáti a. Se retaria de Ensino Fundamental

- SEF/MEC. Brasília, Brasil.

BRASIL. 2016. Base Na ional Comum Curri ular. MEC, Brasília, Brasil.

HAREL, G. and SOWDER, L. 1998. "Types of Student's Justi� ations". The Mathemati s Tea her,

v. 91, (8), p. 670-675.

HOYLES, C. 1997. �The Curri ular Shaping of Students' Approa hes to Proof". For the Learning

of Mathemati s, 17 (1), p. 7-15.

MARTINS, R. B. and MANDARINO, M. C. F. 2014. "Argumentação, prova e demonstração

em geometria: análise de oleções de livros didáti os dos anos �nais do Ensino Fundamental".

Boletim GEPEM, N

o

62 - jan/jul. 2013, p. 101-115. Rio de Janeiro (RJ). (Disponível em:

http://doi.editora ubo. om.br/10.4322/gepem.2014.026. A esso em 11. nov 2016).

51

SYMPOSIUM C

TEACHER-RESOURCE USE AROUND THE WORLD

Janine Remillard

University of Pennsylvania, USA

janiner�upenn.edu

Hendrik Van Steenbrugge

Mälardalen University, Sweden

hendrik.van.steenbrugge�mdh.se

Lu Trou he

ENS de Lyon, Fran e

lu .trou he�ens-lyon.fr

Purpose

This symposium presents resear h from di�erent ountries and s hool systems around the world,

seeking to examine di�erent aspe ts of tea hers' intera tions with and use of resour es, fa tors that

in�uen e them, and their onsequen es for instru tion. Representing studies of tea her' use of resour es

from China, Fran e, South Afri a, Sweden, and the United States, the symposium explores the relationship

between individual and olle tive tea her apa ity and the design of resour es as fa tors that shape the

ena ted urri ulum. The following questions will guide the session:

1. How might we understand the pro esses by whi h tea hers engage with urri ulum resour es to

design instru tion?

2. How does tea her apa ity in�uen e tea hers' urri ulum designs and how is this apa ity developed

and enhan ed?

3. How do resour e features ontribute to tea hers' urri ulum designs?

4. How does olle tive use in�uen e tea hers' design de isions over time?

5. How do tea hers' urri ulum designs ontribute to the ena ted urri ulum?

Guiding Frameworks

The work represented in the session is supported by a parti ipatory perspe tive on tea hers' resour e

use, whi h views it as a dynami pro ess, involving interpretation, appropriation, and design (Gueudet &

Trou he, 2009; Remillard, 2005). Resting on so io- ultural analyses of the agent-tool relationship (Vygot-

sky, 1978), this perspe tive on eptualises resour es as ultural tools that mediate tea hers' urri ulum

design work and are produ ts of this work. Tea hers' intelle tual and ultural resour es also mediate this

pro ess. Finally, tea hers' urri ulum design work o urs in a ontext, often with other tea hers, and

unfolds over time, leading to new designs, new apa ities, and new urri ulum ena tments.

Signi� an e and theme

The symposium �ts within the onferen e theme of "textbook use by tea hers," however, we onsider

instru tional and urri ular resour es that go beyond the textbook, in luding resour es designed to guide,

support, and enhan e mathemati s tea hing and learning in s hools and re-sour es generated by tea hers

as they design instru tion. In 2016, the types of resour es used by tea hers are diverse and in lude print,

digital, and online tools. The symposium will onsider tea hers' use of resour es a ross di�erent ultural

ontexts and types of resour es.

Symposium Parti ipants and Organization

The symposium is intended to highlight the work of early- areer resear hers. Four resear hers have

agreed to submit abstra ts by O t. 31, 2016, as detailed below. Up to 5 additional papers an be

a epted. The session will begin with a 10-minute introdu tion to the themes of the session, followed

by 15-minute presentations of ea h paper, and on luding with su� ient time devoted to questions and

dis ussion among the audien e around the guiding questions, moderated by the session hairs.

Referen es

Gueudet, Ghislaine, and Trou he, Lu , 2009. �Towards new do umentation systems for mathemati s

tea hers?� Edu ational Studies in Mathemati s 71(3), 199-218.

52

Brown, Matthew W. (2009). �The tea her-tool relationship: Theorizing the design and use of

urri ulum materials.� In J. T. Remillard, et al., (Eds.), Mathemati s tea hers at work: Conne ting

urri ulum materials and lassroom instru tion (pp. 17-36). New York: Routledge.

Remillard, Janine T. 2005. �Examining key on epts in resear h on tea hers' use of mathemati s

urri ula.� Review of Edu ational Resear h 75(2), 211-246.

Vygotsky, Lev S. 1978. Mind in So iety. Cambridge, MA: Harvard University Press.

ORAL COMMUNICATIONS IN SYMPOSIUM C

AN ANALYSIS OF THE ENGAGEMENT OF PRE � SERVICE TEACHERS WITH

CURRICULUM RESOURCES IN BRAZIL

Cibelle Assis

Universidade Federal da Paraíba, Brazil

ibelle�d e.ufpb.br

Ver�ni a Gitirana

Universidade Federal de Pernambu o, Brazil

veroni a.gitirana�gmail. om

The importan e of Mathemati s textbooks in tea hers' pra ti e is in ontestable. In fa t, regarding di�erent

edu ational grades or di�erent ountries, mathemati s textbooks play fundamental roles (Choppin, 2004),

being used frequently by tea hers in di�erent ways (Rezat, 2012).

As in many ountries, in Brazil, tea hers are heavily in�uen ed by textbooks. They have being

fundamental to tea hers de ision on whi h ontents must be taught as well as the instru tional approa h

to be developed in lass (Lajolo, 1996).

Brazil States S hools re eives textbooks from a Federal Government program alled PNLD � Programa

Na ional do Livro Didáti o. It is a national program responsible for the pro ess of sele tion, evaluation

and hoi e of the textbook. The mathemati s tea hers� team of ea h s hool hoose the textbook they

whi h to use from a set of mathemati s textbooks, submitted by the editors, evaluated by a national

ommission appointed by the Government, this evaluation being published in a guide (Brazil, 2016). In

general, the PNLD Guide ontains a review for ea h olle tion of textbooks approved by the government

and texts presenting the riteria of analysis, and general re�e tion on the olle tions. Ea h review, in the

ase of mathemati s, omprises analysis of: mathemati s ontent for ea h themes; ontents hoi e and

distribution; tea hing methodology; stru ture; tea hers guides; and their potentialities and la ks regarding

the pedagogi al pra ti e. In addition, the textbooks must attend the urri ulum riteria expressed in

Diretrizes Curri ulares Na ionais Gerais para a Edu ação Bási a (Brazil, 2013).

Other urri ular referen es available at s hools and whi h are used by tea hers and pre�servi e tea hers

at universities are the Parâmetros Curri ulares Na ionais � PCN for the Fundamental Level (Brasil, 1998)

and for the High S hool (Brasil, 2000). These do uments are also used to give tea her and their s hool

ommunity the support for dis ussion and development of edu ational proje ts and also referen es for

analyzing and sele ting dida ti materials and te hnologi al resour es.

Therefore, a resour e is neither isolated (Gueudet & Trou he, 2009, p. 205) nor neutral (Kieran et

al., 2012), that means that other resour es omplement the textbook's use and tea hers engage their

own believes and knowledges for intera ting with them in di�erent ways. In that sense, Remillard (2012)

on eptualized modes and forms of engagement for what the tea hers do in their transa tions with a

urri ulum resour e: how they engage in it, infuse meaning and make sense of its o�erings.

During the pre�servi e ourse or in their daily routine at s hool, the tea hers have onta t with a

set of urri ulum resour es in luding textbooks, books, PCN and PNLD Guide. Nonetheless, in order

to prepare a math lass, for example, how are the pro esses by whi h tea hers engage with urri ulum

and resour es: similar pro esses, due to the textbook, or a diversity of pro esses, due to the diversity of

urri ulum resour es?

53

In order to answer this question, we supervised three pre�servi e tea hers pro ess while designing their

resour es for spe i� mathemati s lasses. Ea h pre�servi e tea her hose a publi s hool and a subje t,

and also produ ed a worksheet to be used in lass. Interviews were undertaken with these tea hers and a

report was demanded about the initial intentions when building their worksheets.

The data analysis had some steps: (a) identi� ation of the involved urri ulum resour es; (b) try to

understand how they were integrated within the worksheet, fo using on its purpose. The results show

that ea h of the pre�servi e tea hers took into onsideration the textbook adopted by the referen e

s hool, PNLD Guide, and PCN referen es. Nevertheless, the pro ess by whi h ea h one engaged with

these urri ular resour es has its parti ularity. They revealed personal urri ulum resour es and forms

of engagement. Consequently, all together indi ates that the pro esses by whi h tea hers engage with

urri ulum resour es to design instru tion are hara terized by their diversity eviden ed by the di�erent

roles assigned to these resour es.

In Brazil, the textbooks� role is of ourse important due to theirs potentialities and also ultural issues,

but due to presen e of other resour es, that role an hange, espe ially during the training ourses when

pre servi e tea hers have the opportunity to dis uss possible uses of textbooks and ontributions of other

resour es.

In this Symposium we also propose to dis uss our methodology for olle ting and analyzing data and

to re�e t how it is possible to apply our results to in�servi e mathemati s tea hers.

Referen es

Brasil. 1998. Ministério da Edu ação e do Desporto. Se retaria de Edu ação Fundamental.

�Parâmetros urri ulares na ionais: Matemáti a". Brasília: MEC/SEF.

Brasil. 2000. Ministério da Edu ação. Se retaria de Edu ação Média e Te nológi a. �Parâmetros

urri ulares na ionais: Matemáti a. Ensino Médio". Brasília: MEC.

Brasil. 2013. �Diretrizes Curri ulares Na ionais Gerais da Edu ação Bási a". Brasília: MEC, SEB,

DICEI.

Brasil. 2016. Ministério da Edu ação. Se retaria de Edu ação Fundamental. �Guia de livros

didáti os PNLD 2017: Apresentação. Ensino fundamental anos �nais". Brasília: MEC/SEF.

Choppin, Alain. 2004. �História dos livros e das edições didáti as: sobre o estado da arte". Edu ação

e Pesquisa 30(3) 549�566.

Gueudet, Ghislaine, and Trou he, Lu . 2009. �Towards new do umentation systems for mathemati s

tea hers?" Edu ational Studies in Mathemati s 71(3),199�218.

Lajolo, Marisa. 1996. �Livro Didáti o: um (quase) manual de usuário". Em Aberto, 16(69), 3�9.

Kieran, Tanguay, and Armando Solares. 2012. �Resear her�Designed Resour es and Their Adap-

tation Within Classroom Tea hing Pra ti e: Shaping Both the Impli it and the Expli it". In From

Text to �Lived" Resour es, edited by Ghislaine Gueudet et al, 189� 213. New York: Springer.

Remillard, Janine. 2012. �Modes of Engagement: Understanding tea hers� Transa tions with Math-

emati s Curri ulum Resour es". In From Text to �Lived" Resour es, edited by Ghislaine Gueudet et

al, 105� 121. New York: Springer.

Rezat, Sebastian. 2012. �Intera tions of Tea hers� and Students� use of Mathemati s Textbooks".

In From Text to �Lived" Resour es, edited by Ghislaine Gueudet et al, 231� 245. New York: Springer.

54

RESOURCES FOR TEACHING

Jose Luis Cortina

Universidad Pedagogi a Na ional, Mexi o

j ortina�upn.mx

Jana Visnovska

The University of Queensland, Australia

j.visnovska�uq.edu.au

We analyze the role played by an instru tional sequen e (Cortina, Visnovska, and Zúñiga 2014), in ef-

fe tively supporting a professional development ollaboration with Irene, a Mexi an publi s hool tea her

who, in a relatively short period of time, modi�ed signi� antly her instru tional pra ti es, and su essfully

supported her �fth grade students in making sense of some important fra tion notions. The instru -

tional sequen e is based on a onje tured learning traje tory on fra tions as measures, and was developed

through a series of lassroom design experiments, following Cobb and olleagues' methodologi al guidelines

(Gravemeijer and Cobb 2006).

We analyze the role played by the instru tional resour e in the 6-month professional development

ollaboration between Irene and the �rst author and hara terize three layers of support it provided. First,

during six one-hour preparation sessions, the instru tional sequen e was a resour e for the �rst author

in helping Irene improve on her understanding of fra tions as measures, and in familiarizing her with the

di�erent aspe ts of instru tional pra ti es that pla e student mathemati al reasoning at the enter of

de ision-making in the lassroom. Se ond, instru tional sequen e guided Irene and the �rst author as

they ollaboratively planned the eighteen weekly instru tional sessions in response to Irene's students'

progress. Third, the resour e guided Irene in supporting students' sense making of important fra tion

notions during lassroom intera tions.

We explain how the support the instru tional sequen e provided at ea h of the three layers ontributed

to its viability as a resour e for Irene's tea hing. During the preparation sessions, the sequen e be ame a

resour e to support Irene's mathemati al learning, and to dis uss with her di�eren es in how students might

reason when engaging in spe i� instru tional a tivities. During o-planning of the instru tional sessions,

the sequen e be ame a resour e for Irene's (a) assessment of students' prior parti ipation, (b) anti ipation

of student reasoning in the oming session, and ( ) planning responses to students' ontributions and

emerging ideas. Finally, during the instru tional sessions, the sequen e be ame a resour e for making

in-the-moment de isions about guiding lassroom intera tions, making them more manageable.

In dis ussing the signi� an e of our analysis, we refer to the state of mathemati al learning in Mexi o,

parti ularly with regard to pupils living in harsh so ial and e onomi al ir umstan e. For more than �fteen

years, national and international assessments have shown a rather disturbing image. For instan e, in a

re ent assessment ondu ted by the National Institute of Edu ational Evaluation (Instituto Na ional para

la Evalua ión de la Edu a ión 2015), 60.5

Mexi o's ase is not an ex eption in Latin Ameri a. Data form PISA (Organisation for E onomi

Co-operation and Development 2013) suggests that Argentina, Brazil, Chile, Colombia, Costa Ri a, Peru,

and Uruguay fa e similar hallenges.

In Mexi o, more than a de ade of e�orts at the publi poli y level in luded hanging the urri ulum,

developing new o� ial textbooks, implementing high-stakes tests, and establishing programs of spe ial

rewards for good tea hers. Yet, these e�orts did not result in a lear indi ation of improvement of students'

learning or performan e.

In this ontext, the ase we analyze sheds light into the role that instru tional sequen es that are a

produ t of areful design and experimentation in lassrooms, an play in supporting tea hers who work with

low performing students. With adequate support, Irene engaged in ambitious and e�e tive instru tional

pra ti e, and made a noti eable di�eren e in the quality of her students' mathemati al learning. While

our ase highlights the resour e use by a single tea her, the des ribed layers of support are onsistent

with our earlier analysis of the role that similar instru tional sequen e played in supporting the learning

of a group of 12 middle s hool tea hers in the USA over 5 years (Visnovska, Cobb, and Dean 2012).

We dis uss impli ations of our analysis for on eptualization and design of edu ative urri ulum materials

(Davis and Kraj ik 2005).

55

Referen es

Cortina, J. L., J. Visnovska, and C. Zúñiga. 2014. "Unit fra tions in the ontext of proportional-

ity: Supporting students' reasoning about the inverse order relationship." Mathemati s Edu ation

Resear h Journal 26:79-99.

Davis, E. A. and J. S. Kraj ik. (2005). "Designing edu ative urri ulum materials to promote

tea her learning." Edu ational Resear her 34(3): 3-14.

Gravemeijer, K., and P. Cobb. 2006. "Design resear h from a learning design perspe tive." In

Edu ational design resear h: The design, development and evaluation of programs, pro esses and

produ ts, edited by J. van den Akker, K. Gravemeijer, S. M Kenney and N. Nieveen, 45-85. New

York: Routledge.

Instituto Na ional para la Evalua ión de la Edu a ión. 2015. Plan Na ional para la Evalua ión de

los Aprendizajes (Planea). Resultados na ionales 2015, 6

o

de primaria y 3

o

de se undaria, lenguaje

y omuni a ión, y matemáti as. Méxi o, D. F.: Autor.

Organisation for E onomi Co-operation and Development. 2013. PISA 2012 results: What students

know and an do - student performan e in mathemati s, reading and s ien e Vol. 1. Paris: Author.

Visnovska, J., P. Cobb, and C. Dean. 2012. "Mathemati s tea hers as instru tional designers:

What does it take?" In From text to 'lived' resour es: Mathemati s urri ulum materials and tea her

development, edited by G. Gueudet, B. Pepin and L. Trou he, 323-341. Dordre ht: Springer.

DISAGGREGATING TEACHERS' PEDAGOGICAL DESIGN CAPACITY (PDC) IN SOUTH

AFRICA

Moneoang Leshota

National University of Lesotho

mj.leshota�nul.ls

Leshota des ribes an analyti al approa h that disaggregates tea hers' PDC. The study situated in so io-

ultural theory and notion of mediating artefa ts, regards tea hing as mediation between the mathemati s

taught and the learners. In the pro ess, tea hers and resour es (pres ribed textbook in this ase) forge

relationships whi h in�uen e the out ome of instru tion. Resour es a�ord tea hers with ontent to be

taught as well as an instru tional approa h . How tea hers utilise the a�ordan es is the basis for the

analyti al approa h adopted in this study.

The analyti al approa h adopted utilises a lesson as a unit of analysis instead of a tea her to allow

patterns of use a ross all twenty lessons regardless of the tea her. Using data from twenty (20) lessons

of seven (7) grade 10 tea hers, the analysis identi�es the omissions and inje tions (insertions) made by

tea hers in instru tion as key to de�ning their apa ity for pedagogi design. Further ategorization of the

omissions into produ tive and riti al omissions; and inje tions into robust and distra tive, illuminate two

key issues pertaining to tea hers' mobilisation of the textbook. Firstly, they illuminate di�erent `usages' of

the textbook by tea hers as ta it use that leads to weak tea her-textbook relationships versus deliberate

use leading to strong and intimate tea her-textbook relationships.

Referen es

Brown, Matthew. �The Tea her-Tool Relationship: Theorizing the Design and Use of Curri ulum

Materials." Chap. 2 In Mathemati s Tea hers at Work: Conne ting Curri ulum Materials and

Classroom Instru tion, edited by Janine T. Remillard, Beth A. Herbel-Eisenmann and Gwendolyn

M. Lloyd. Routledge, 2009.

56

Gibson, James J. �The Theory of A�ordan es." In Per eiving, A ting, and Knowing: Toward an

E ologi al Psy hology, edited by R Shaw and J Bransford. 67-84: Erlbaum, 1977.

Leshota, Moneoang. �The Relationship between Textbook A�ordan es and Mathemati s' Tea hers'

Pedagogi al Design Capa ity (Pd )." University of the Witwatersrand, 2015.

Leshota, Moneoang, and J Adler. �The Analysis of Tea hers' Mobilisation of the Textbook A�or-

dan es." Paper presented at the International Conferen e on Mathemati s Textbook Resear h and

Development (ICTM-2014), University of Southampton, 2014.

Polanyi, M. The Ta it Dimension. An hor Books, 1967.

Remillard, Janine T. �Examining Key Con epts in Resear h on Tea hers' Use of Mathemati s Cur-

ri ula." Review of Edu ational Resear h 75, no. 2 (2005): 211-46.

Vygotsky, L. S. Mind in So iety. Cambridge, MA: Harvard University Press, 1978.

Werts h, James V. Mind as A tion. Mind as A tion. New York, NY US: Oxford University Press,

1998.

FROM WRITTEN TO ENACTED LESSONS: A U.S. TEACHER'S MOBILIZATION OF A

MATHEMATICAL MODELING-BASED ALGEBRA UNIT

Luke T Reinke

University of North Carolina at Charlotte

lreinke�un .edu

Numerous s holars propose that students an develop deep understandings of mathemati al on epts

through solving ontextualized problems (CPs) (e.g. Freudenthal, 1991), but resear h suggests that

tea hers may la k a nuan ed understanding of how CPs an support learning of new mathemati s (Lee,

2012). This understanding is a key omponent of the pedagogi al design apa ity (Brown, 2009) needed

to e�e tively implement any urri ulum that introdu es and develops mathemati al ideas through mathe-

mati al modelling or other forms of CPs. This ase study hara terizes the role of ontextualized problems

in one ena ted algebra unit to provide insight into how the tea her's apa ities and design pra ti es medi-

ated students' experien es with the modelling-based urri ulum. Spe i� ally, I ask, how does the tea her's

apa ity for understanding the use of CPs in developing students' mathemati al understanding in�uen e

her design de isions, and how do the tea her's design de isions ontribute to the ena ted urri ulum?

Theoreti al Framework

I frame my study using Stein, Remillard and Smith's (2007) des ription of the temporal phases of

urri ulum use: a tea her transforms the written urri ulum into the [tea her℄ intended urri ulum when

she designs the lesson during planning. Then, the tea her and students intera t with ea h other and the

written urri ulum during the lesson to reate the ena ted urri ulum. Using this framework, I tra e a

modeling-based algebra unit from the written urri ulum phase through to the ena ted urri ulum.

Methods

The dataset in ludes the tea her interviews, the written urri ulum, the tea hers' plans, and trans ripts

of lassroom dis ourse. I ode ea h segment of the written text a ording to whether it referen es (a)

ontextualized problems or examples, (b) non- ontextualized problems or examples, or ( ) general state-

ments about mathemati al on epts or pro edures that do not referen e spe i� problems or examples. I

used these odes to hara terize the way the developers sequen ed the di�erent types of task in the text.

Then, I ode ea h portion of the tea hers' plans to determine the extent to whi h her plans follow the

sequen es in the written urri ulum. I also oded ea h utteran e in the trans ript of the lassroom to

identify whether lassroom dis ourse around one type of task in luded referen es to other types.

57

Results

In the written urri ulum, the sequen e pro eeded from (a) ontextualized problems, (b) to non-

ontextualized problems, to ( ) tasks whi h asked students to make generalizations about mathemati al

on epts and pro edures. There were few instan es where the authors asked students to re�e t ba k on

previously worked CPs. The tea her identi�ed a barrier to learning that had arisen in previous years with

other students; students did not naturally see onne tions between the problems set in di�erent ontexts.

To mitigate this phenomenon, she supplemented the written urri ulum with tasks that asked students to

re�e t a ross previous ontextualized problem-solving experien es.

At other points in the unit, the tea her resequen ed or omitted parti ular portions of the urri ulum so

that the ontextualized problems were no longer used as introdu tions. Be ause the units were sequen ed

with the intent that students would leverage ontextualized problem solving experien es to help them

solve non- ontextualized problems, students were left to work the non- ontextualized problem without a

key resour e intended by the developers. Interviews revealed that she made these hanges be ause she

had not observed students leveraging the ontextualized experien es in the past, and she believed that

some of her students learned best when non- ontextualized experien es ame �rst.

Dis ussion

The tea her's design apa ity has learly been developed through her repeated use of the urri u-

lum. These previous ena tments developed two key omponents of her pedagogi al design apa ity: (1)

the ability to identify barriers that might prevent the intended learning opportunities to unfold during

ena tment, and (2) the ability to design supplemental tasks to mitigate these di� ulties.

Her reordering of the urri ulum, though, represents the presen e of omponent (1) and the absen e of

omponent (2). Here, she identi�ed that the developers' intent was not realized in previous ena tments,

but instead of making instru tional moves that helped students leverage and see the onne tion between

ontextualized and non- ontextualized tasks, she instead modi�ed the urri ulum in su h a way that it

no longer aligned with the desired intent. This was due both to a la k of re ognition of how to onne t

students' work between the two types of tasks (she demonstrated low design apa ity in this area), and her

belief that for some students, the sequen e was not the most e�e tive. I hypothesize that, in ases where

tea her beliefs on�i t with developers' intent, a tea her with high design apa ity and the ne essary

autonomy would sele t di�erent urri ulum materials that align better with her beliefs and orresponding

pra ti es.

Referen es

Brown, Matthew W. (2009). �The tea her-tool relationship: Theorizing the design and use of

urri ulum materials." In J. T. Remillard, et al., (Eds.), Mathemati s tea hers at work: Conne ting

urri ulum materials and lassroom instru tion (pp. 17-36). New York: Routledge

Freudenthal, Hans. 1991. Revisiting mathemati s edu ation. China Le tures Springer.

Lee, Ji-Eun. 2012. �Prospe tive elementary tea hers' per eptions of real-life onne tions re�e ted in

posing and evaluating story problems." Journal of Mathemati s Tea her Edu ation 15 (6):429-452

Stein, Mary Kay, Remillard, Janine., & Smith, Margaret. (2007). How urri ulum in�uen es student

learning. In F. Lester (Ed.), Se ond handbook of resear h on mathemati s tea hing and learning

(pp. 319-369). Charlotte, NC: Information Age Publishing.

KNOWLEDGE OF CURRICULUM EMBEDDED MATHEMATICS: EXPLORING A

CRITICAL DOMAIN OF TEACHING IN THE U.S.

Janine Remillard

University of Pennsylvania

janiner�upenn.edu

Ok-kyeong Kim

Western Mi higan University

Ok-Kyeong.kim�wmi h.edu

58

Knowledge used in tea hing is urrently re eiving tremendous attention, both in resear h and pra ti e.

Sin e Shulman's (1986, 1987) seminal work proposing a knowledge base for tea hing, resear hers have

ontinued to explore questions about ontent-spe i� knowledge for tea hing. Our work examines a

domain of the work of tea hing not widely explored in the literature-using urri ulum resour es to design

mathemati s instru tion-and proposes a framework for identifying the mathemati al demands of the work

and the knowledge tea hers draw on to navigate them. We use the term knowledge of urri ulum embedded

mathemati s (KCEM) to refer to the knowledge a tivated by tea hers when reading and interpreting

mathemati al tasks, instru tional designs, and representations in mathemati s urri ulum materials. We

do not propose KCEM as a new form of mathemati s knowledge for tea hing (Ball, Thames, & Phelps,

2008), but as a spe i� aspe t of tea hing in whi h models of ontent-spe i� tea hing knowledge,

proposed by Shulman (1986; 1897) and others (Ball et al.; Mason & Spen e, 1999; Rowland, 2005), are

a tivated.

Analyses of Content-Spe i� Tea her Knowledge

S holarly e�orts to on eptualize spe ialized tea her knowledge have been driven by two key aims:

to make mu h of the invisible work of tea hing visible and to un over spe ialized forms of knowledge

that might be developed in tea hers. These e�orts, for the most part, have taken a pra ti e-based

approa h, by analyzing the tasks tea hers engage in when planning, ena ting, or evaluating the out omes

of instru tion. Among them, there is general agreement that tea hing is a omplex a tivity requiring

knowledge, deliberation, and skilled a tion (Shulman, 1987).

Methods for Con eptualizing KCEM

Our e�orts to on eptualize KCEM involved empiri ally-grounded analysis of spe i� tasks required

of tea hers when using urri ulum materials. We examined omponents of the tea her's guides and

interviewed tea hers when planning lessons with urri ulum resour es to onsider two related questions:

1. How are mathemati al ideas embedded in mathemati s tea her's guides?

2. What do tea hers need to reason about in order to surfa e and make produ tive use of these ideas

when planning with the guides?

We also onsulted the literature to understand how this emerging onstru t might be positioned within

existing on eptions of ontent-spe i� tea her knowledge.

Dimensions of Knowledge of Curri ulum Embedded Mathemati s

Based on our analysis, we propose four overlapping dimensions of KCEM. Foundational mathemati al

ideas refers to understanding of the foundational mathemati al ideas that underlie instru tional formats.

Representations and onne tions a ross them involves making onne tions a ross di�erent representations

of the same mathemati al idea, in luding models, stories, and symboli representations. Understanding

problem omplexity involves the ability to ategorize and order tasks by in reasing di� ulty as well as

the ability to identify possible points of onfusion for learners asso iated with a given task. Re ognizing

mathemati al learning pathways in a urri ulum in ludes understanding how a parti ular goal is situated

within a set of ideas that develop over time.

Signi� an e of KCEM and future resear h

The four dimensions of KCEM o�er spe i� ations of aspe ts of tea hers' work dire tly related to the

nature and design of a major tool of the trade. They an be used to identify spe i� types of knowledge

important to tea hing and situations where they are typi ally deployed. KCEM hara terizes a primary

domain of mathemati s tea hing and o�ers a framework that allows resear hers and edu ators to prioritize

spe i� types and uses of knowledge for tea hing dis ussed in the literature. Further, the four dimensions

point to routine features of urri ulum resour es around whi h knowledge development might o ur.

When made expli it, these dimensions an be examined by tea hers to support their understanding of the

mathemati al omponents of these tools.

59

Referen es

Ball, Deborah Leowenberg, Thames, Mark, & Phelps, Geo�. 2008. Content knowledge for tea hing:

What makes it spe ial? Journal of Tea her Edu ation, 59(5), 389-407.

Mason, John, & Spen e, Mary. 1999. Beyond mere knowledge of mathemati s: The importan e of

knowing-to a t in the moment. Edu ational Studies in Mathemati s, 38, 135-161.

Remillard, Janine T. 2005. �Examining key on epts in resear h on tea hers' use of mathemati s

urri ula." Review of Edu ational Resear h 75(2), 211-246.

Rowland, Tim. 2013. The knowledge quartet: The genesis and appli ation of a framework for

analyzing mathemati s tea hing and deepening tea hers' mathemati s knowledge. Journal of Edu-

ation, 1(3), 15-43.

Shulman, L. S. (1986). Those who understand: Knowledge growth in tea hing. Edu ational

Resear her, 15(2), 4-14.

Shulman, L. S. (1987). Knowledge and tea hing: Foundations of the new reform. Harvard Edu a-

tional Review, 57(1), 1-21.

INTERDISCIPLINARY PROGRAM FOR PROFESSIONAL DEVELOPMENT IN

MATHEMATICS TEACHING IN MEXICO

Daniela Reyes-Gasperini

Mathemati s Edu ation Department of Center for Resear h and

Advan e Studies of the National Polyte hni Institute

dreyes� investav.mx

Abstra t

Sin e a few years, Mexi o again fa es a urri ular hange. Currently, it analyzes and restru tures of way

of deep mathemati al knowledge of the urri ulum. PIDPDM, with support in the So ioepistemologi al

Theory (Cantoral, 2013). We proposed various ourses that promote hanges in relation with knowledge,

through the problematization of s hool mathemati s (psm). Tea hers perform, subsequently, the pro esses

of intervention into the lassroom (Reyes-Gasperini, Cantoral, & Montiel, 2015). The psm promotes for

tea hers a distin t look at tea hing resour es, su h as developing a on rete idea about the design of

learning situations, favoring the transversality of mathemati s, a epting the diversity of students' answers

to the same question. This training aims to impa t tea hers' a tivity in their own lassrooms with their

students. PIDPDM program a ompanies the tea hing pro ess in order to support urri ulum hanges,

towards implementation in lassrooms.

Introdu tion

The Mexi an s hool system, at the high-s hool level, is divided into te hnologi al subsystems, general

and preparation for work that is grouped in several departments: Department of General High-S hool

(DGB), Department of General Te hnologi al and Industrial Edu ation (DGETI), Department of National

Te hni al Edu ation and High-S hool (COLBACH). All of them propel a tions for professional tea her

development with support of the High S hool Subse retary - SEMS. The PIDPDM is a program that takes

ourses, workshops and seminars for professional tea her development in mathemati s and is dire ted to

all the subsystems. For su h purpose, we elaborate a spe i� strategy that in ludes aspe ts of the so ial

onstru tion of mathemati al knowledge as one of the sour es for the redesign on the part of the tea hers.

For that, introdu ing the pms as an indispensable resour e for instru tional mathemati s in the lassroom.

This is a fundamental element for the foundational design whi h looks for an alternative intervention

60

of tea hers' professional development, one that emphasizes the hange of tea hers' relationships with

mathemati al knowledge, as well as their autonomy and leadership attitudes for the transformation and

edu ational improvement. This resour e would be fun tional for all tea hers, who subsequently developed

designs of intervention for the mathemati s lassroom where a tivities will be ontextualized in the ulture

of the students.

The problematization of s hool mathemati s (psm)

The psm is a me hanism for ollaboration and profound, professional questioning and understanding

of s hool mathemati s that traditionally is worked in their lassrooms. It abandons the Mexi an tradition

of tea hing themes in universal ontexts for the signi� ation of mathemati al on epts.

The Ante edent: the ase of Oaxa a, Mexi o

During the three years an a ademi team onforming to edu ational mathemati s, we worked on the

design and development of the Master's in Mathemati s Tea her Edu ation in the Junior High of the

State Institute of Publi Edu ation of Oaxa a, in the Federal Tea hers' College of Oaxa a, Mexi o. We

arried out a work of problematization of s hool mathemati s that treats the diverse themes with the

hara teristi s that were fundamental for tea her empowerment (Reyes-Gasperini, 2016): Proportionality,

representational fun tion, trigonometri al modelation, Thinking and Variational Language before linear

and nonlinear. The results obtained were set forth in eviden e that the parti ipating professors, as results

of the psm, were able to in orporate into tasks that showed their professional development and were

used as resour es- theoreti al solutions belonging to the �eld. Among them were 1) used as resour es

spe ialized bibliographies to onfront that whi h the textual books given as a fa t, as for example refutes

phrases su h as this: 'when a quantity in reases and the other too, this orresponds to a relation of dire t

proportionality' 2) They question the plans of study and proposal of lesson plans that provides the same

Se retary of Publi Edu ation; 3) It is a tively in orporated in the dis iplinary ommunity of Edu ational

Mathemati s parti ipating in national and international ongresses and onsulting spe ialized literature

(journals ISI and S opus); 4) designs founded theoreti ally ontextualized to the reality of their students

and leading them to their experimental stage.

The urrent proje ts of professional tea her development sin e the PIDPDM

During the year 2016 we performed an intervention in three major proje ts with a total of six-thousand

parti ipating tea hers. The �rst, analyzed as the Prueba PLANEA (national test to measure the su ess of

high-s hool students), in order to design strategi learning situations for the psm involved in the transversal

manner; this had the purpose of knowing diverse strategies to approa h ommon questions based on the

development of mathemati al thinking. The se ond, approa hed the urri ular materials that realized the

problematization of thoughts: sto hasti , algebrai , arithmeti (numeri sense), in analyti al geometry,

in trigonometry, variational, geometri and fun tional; whose obje tive will be the deepening of those

mathemati al topi s onsidered to be the national test for the Mathemati s tea her. When the tea hers

return to fa e their evaluation, they will have new tools whi h they an defend, to give their answers. The

third, analyzed learning situations designed under a so ioepeistemologi al foundation, in order that the

tea hers later design and support their own situations to implement them in the lassroom.

The PIDPDM has realized diverse edu ational interventions in the frame of the national proposal

of professional tea her development in mathemati s whose results are translated into an improvement

in the evaluation of tea her performan e, in the improvement of student performan e in their national

evaluations, in the in orporation of tea her in a spe i� ommunity that dis usses the problemati s of

the profession and in the fundamental theoreti al design and implementation of learning situations of

ontextualized learning that a ompany the new urri ular proposal.

Referen es

Cantoral, R. 2013. Teoría So ioepistemológi a de la Matemáti a Edu ativa: estudios sobre on-

stru ión so ial del ono imiento. España: Gedisa.

61

Reyes-Gasperini, D., Cantoral, R., & Montiel, G. (2015). Supporting maths tea hers in the so ial

onstru tion of knowledge. In In lusive edu ation in a tion: Empowering tea hers: empowering

learners (example 116). European Agen y for Spe ial Needs and In lusive Edu ation: UNESCO.

Disponible en http://www.in lusive-edu ation-in-a tion.org/example-116.html

Reyes-Gasperini, D (2016). Empoderamiento do ente desde una visión so ioepistemológi a: una

alternativa para la transforma ión y la mejora edu ativa. Do toral thesis, Cinvestav, Méxi o.

DOCUMENTATIONAL TRAJECTORIES AS A MEANS TO UNDERSTAND TEACHERS'

ENGAGEMENT WITH RESOURCES: THE CASE OF FRENCH TEACHERS FACING A

NEW CURRICULUM

Katiane de Moraes Ro ha

ENS de Lyon, Fran e

mr.katiane�gmail. om

Lu Trou he

ENS de Lyon, Fran e


Ghislaine Gueudet

Université Rennes 1, Fran e

ghislaine.gueudet�espe-bretagne.fr

The new digital resour es provide new means for designing and sharing tea hing materials. In Fran e, these

new means are extensively used. A good example is Sésamath asso iation (http://www.sesamath.net) that

designs ollaboratively online resour es, among them free mathemati s e-textbooks. These phenomena

motivated a national resear h proje t, ReVEA (Living Resour es for Tea hing and Learning, 2014-2018,

http://anr-revea.fr) that seeks to understand whi h resour es are used by tea hers and how. This proje t

a ts as in ubator of on epts (Trou he 2016). Spe ially, it leads to on eptual developments for the

do umentational approa h to dida ti s (Gueudet and Trou he 2012). In this approa h, tea hers are

onsidered as a tors to �nd and design their resour es, along what is named their do umentation work.

Gueudet and Trou he (2012) propose the term resour e in a omprehensive a eptation, as everything

nourishing tea hers' work and the term do ument as a hybrid entity, resulting of the tea her do umentation

work, asso iating a resour e developed for a given dida ti al goal and a s heme piloting the pro ess of

design and usage of this resour e. The on ept of s heme is grounded in Vergnaud's work (2009, 88),

meaning �the invariant organization of a tivity for a ertain lass of situations". The kernel of s hemes

is onstituted by operational invariants, meaning �knowledge in a tion", guiding tea hers do umentation

work, and, in the same time, spurred by it.

In their day-to-day work, tea hers fa e situations that requires re�e tion on pra ti e, leading them to

build their raft knowledge (Ruthven 2015). This type of knowledge �refers to the professional knowledge

that arises from and in their day-to-day lassroom tea hing" (Ruthven 2015, 523). This knowledge takes

into a ount stru turing features of lassroom pra ti e: working environment, a tivity format, urri ulum

s ript, time e onomy and resour e system (Ruthven 2014). We propose to ombine this approa h with the

do umentational approa h to dida ti s, introdu ing the notion of raft do umentational knowledge as the

professional knowledge that arises from and in tea hers' do umentation work. This raft do umentational

knowledge develops throughout the development of s hemes.

Tea hers' do umentation work develops in so ial ontext, in a way depending on their own history

with tea hing resour es. We propose a new on ept, do umentational traje tory, to model the tea her's

history with resour es, de�ning it (Ro ha 2016) as the interplay, over the time, between events that

a tea her meets and resour es, this interplay being so ially situated, be ause it happens in s hools or

olle tives, or be ause the events or the resour es themselves are so ial produ ts. We retain the broad

de�nition of thought olle tive existing when �two or more people are ex hanging thoughts� (Fle k 1934,

44) and generating a thought style � hara terized by ommon features in the problems of interest to a

thought olle tive, by the judgment whi h the thought olle tive onsiders evident, and by the methods

whi h it applies as a means of ognition" (Fle k 1934, 99). In this sense, to understand how tea hers

build their raft do umentational knowledge, the analysis of their do umentational traje tory through a

olle tive lens takes an important pla e. Then, we dis uss here: how tea hers do umentational traje tory

onditions, and is sensitive to, their do umentation work?

62

For this purpose, we follow two tea hers, Anna and Viviane, in di�erent middle s hools in Fran e using

Sésamath textbooks. Our methodologi al framework is inspired by re�exive investigation (Gueudet and

Trou he, 2012), that proposes long-term follow-up of tea hers leading them to re�e t about their own

work. We developed a new tool, a Re�e tive Mapping of Do umentational Traje tory (RMDT, see in

Ro ha, 2016), using the own eyes of a tea her to des ribe her history with resour es.

This work takes pla e in a riti al moment in Fran e, a new urri ulum for grades 1 to 9, being

implemented in September 2016, proposing deep hanges: the urri ulum is designed over y les of three

years instead of one year; new topi s, mainly algorithmi s and programing, are introdu ed. We follow

Anna and Viviane's preparation of the progression related to this new topi (algorithmi and programming)

in their ordinary environment. We asked them to map their do umentational traje tory. We ross the

data from their preparation of a new lesson with the mapping of their do umentational traje tory. Our

preliminary �ndings eviden e the importan e of tea hers' olle tive work, over a long period, for feeding

their resour e systems and developing their raft do umentational knowledge.

Referen es

Fle k, Ludwik. 1981. Genesis and Development of a S ienti� Fa t. Chi ago: University of Chi ago

Press (original edition, 1934).

Gueudet, Ghislaine, and Lu Trou he. "Tea hers' Work with Resour es: Do umentational Geneses

and Professional Geneses." In From Text to "Lived" Resour es, edited by Ghislaine Gueudet, Birgit

Pepin, and Lu Trou he, 23-41. Dordre ht: Springer Netherlands, 2012.

Ro ha, Katiane D.M. "Uses of Online Resour es and Do umentational Traje tories: the Case of

Sésamath." Paper presented at 13th International Congress on Mathemati al Edu ation. Germany,

Hamburg, July 24-31, 2016.

Ruthven, Kenneth. "Constituting Digital Tools and Materials as Classroom Resour es: The Example

of Dynami Geometry." In From Text to "Lived" Resour es, edited by Ghislaine Gueudet, Birgit

Pepin, and Lu Trou he, 83-103. Dordre ht: Springer Netherlands, 2012.

Ruthven, Kenneth. 2015. "Knowledge Creation through Dialogi Intera tion between the Pra ti es

of Tea hing and Resear hing." In Handbook of International Resear h in Mathemati s Edu ation,

edited by Lyn D. English and David Kirshner, 523-39. London: Routledge.

Vergnaud, Gérard. 2009. The theory of on eptual �elds. Human development, 52(2): 83-94.

Trou he, Lu . 2016. Des olle tifs produ teurs et partageurs de ressour es, et leurs a teurs - Pro�ls

et traje toires, Livrable ReVEA 4.2. 2016. A essed O tober 14, 2016. http://www. fem.asso.fr/a tus-

revea/livrables/livrable-revea-4.2.

DESIGN IN USE: FROM AUTHOR-INTENDED TO WRITTEN TO

TEACHER-INTENDED LESSON IN SWEDEN

Hendrik Van Steenbrugge Nina Jansson Fredrik Blomqvist Andreas Ryve

Mälardalen University, Sweden

hendrik.van.steenbrugge�mdh.se nina.jansson�mdh.se fredrik.blomqvist�mdh.se andreas.ryve�mdh.se

INTRODUCTION

In many Swedish mathemati s lasses, mu h of the intera tion is lo ated between students and their

textbook, and tea hers have a responsive rather than a proa tive role (Hemmi & Ryve, 2015). This is

re�e ted in the type of support in the tea her's guide (Remillard, Van Steenbrugge, & Bergqvist, 2016).

The proje t reported about in this presentation aims to promote more student � student and tea her

63

� students intera tions into daily mathemati s lasses in elementary s hool. As urri ulum programs

are widely used in Sweden (Boesen et al., 2014), the main vehi le to promote this hange is a new

urri ulum program. Chara teristi for the new urri ulum program is the in lusion of �open response and

reengagement lessons", edu ative support for tea hers to plan for and ena t these lessons, and a digital

omponent to support tea hers and the urri ulum design pro ess. We will report about the �rst two of

these hara teristi s.

THEORETICAL BACKGROUND

Building on Stein, Remillard and Smith (2007) and Brown, Pitvore , Ditto, and Kelso (2009), we

understand urri ulum use as unfolding over the following temporal phases: author�intended lesson, writ-

ten lesson, tea her�intended lesson, and ena ted lesson. Brown et al. (2009) underlined the signi� an e

to di�erentiate between the author�intended and written lesson as their study revealed that tea hers'

�delity to the written lesson did not guarantee �delity to the author�intended lesson. The relationship

author�intended lesson � written lesson � tea her�intended lesson is a major on ern in the design pro ess

of the program.

CURRENT STUDY

We report about initial stages in the design pro ess of a new Swedish urri ulum program for elementary

s hool mathemati s. The aim is twofold. First, we aim to identify a) how the anti ipated supports play

out during tea her use, b) fa tors that in�uen e how tea hers take up the supports, and ) how we

an improve tea her support. Se ond, learning from our try�outs, we aim to improve understanding of

translating our initial intentions into the written materials.

The data sour es we draw on onsist of tea her interviews to identify tea her orientations toward

tea hing and learning mathemati s and urri ulum programs, written lessons, des riptions of tea hers'

plans for lesson ena tment, observations of ena ted lessons, and post�interviews with tea hers. Three

�rst�grade and three�se ond grade tea hers parti ipate in this study.

Des ription of modules and envisioned tea her support

The two modules onsist of four lessons and fo us on one mathemati al topi (grade one: subtra tion;

grade two: multipli ation). The �rst two lessons in ea h module introdu e students to ollaboration and

having dis ussions. The third and fourth lessons relate to an open response (lesson 3) and reengagement

lesson (lesson 4). These lessons are distin t from traditional Swedish lessons due to their problem�based

and formative assessment approa h.

To help tea hers plan for and ena t the modules, we in luded support to understand the instru tional

sequen e � both between as within individual lessons. We did so be ause we anti ipate that tea hers have

to modify the written lessons and be ause Choppin (2011) underlined the importan e of tea hers' related

understanding for produ tive urri ulum adaptations. Support in relation to the instru tional sequen e

over the lessons in lude an introdu tion to the module onsisting of a des ription of the lesson goals, how

the module onne ts to previous and future ontent, and how the lessons in the module relate to ea h

other. A tea her vignette des ribes how a � titious tea her made a produ tive adaptation. These two

types of support have been des ribed earlier in a study about redesigning an existing s ien e urri ulum

program (Davis et al., 2014). Within ea h lesson, a) lesson goals are des ribed as well as the expe ted

mastery level, b) preview boxes summarize the up oming a tivity, and ) expli it des riptions are in luded

about how spe i� lesson omponents relate to one another and to previous/future lessons, and about

where to insert additional a tivities. Inspiration for these supports omes from studies of how urri ulum

authors ommuni ate with tea hers in Ameri an, Swedish and Flanders urri ulum programs (Atanga &

Reinke, 2013; Van Steenbrugge, Remillard, & Bergqvist, 2015).

Referen es

Atanga, N., & Reinke, L. (2013). An Analysis of Authors' Communi ation of Transparen y and

Rationale for Design in Five Elementary Mathemati s Curri ulum Guides. Paper presented at the

AERA annual meeting, San Fran is o.

64

Boesen, J., Helenius, O., Bergqvist, E., Bergqvist, T., Lithner, J., Palm, T., & Palmberg, B. (2014).

Developing mathemati al ompeten e: From the intended to the ena ted urri ulum. The Journal

of Mathemati al Behavior, 33, 72-87.

Brown, S. A., Pitvore , K., Ditto, C., & Kelso, C. R. (2009). Re on eiving Fidelity of implementa-

tion: An Investigation of Elementary Whole-Number Lessons. Journal for Resear h in Mathemati s

Edu ation, 40(4), 363-395.

Choppin, J. M. (2011). Learned adaptations: Tea hers' understanding and use of urri ulum re-

sour es. Journal of Mathemati s Tea her Edu ation, 14(5), 331-353.

Davis, E. A., Palin sar, A. S., Arias, A. M., Bisma k, A. S., Marulis, L. M., & Iwashyna, S. K.

(2014). Designing Edu ative Curri ulum Materials: A Theoreti ally and Empiri ally Driven Pro ess.

Harvard Edu ational Review, 84(1), 24-52.

Hemmi, K., & Ryve, A. (2015). E�e tive mathemati s tea hing in Finnish and Swedish tea her

edu ation dis ourses. Journal of Mathemati s Tea her Edu ation, 18(6), 501-521.

Remillard, J. T., Van Steenbrugge, H., & Bergqvist, T. (2016). A ross- ultural analysis of the

voi e of six tea her's guides from three ultural ontexts. Paper presented at the AERA annual

meeting, Washington, DC.

Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How urri ulum in�uen es student learning.

In F. K. Lester (Ed.), Se ond handbook of resear h on mathemati s tea hing and learning (pp.

319-370). Greenwi h, CT: Information Age Publishing.

Van Steenbrugge, H., Remillard, J. T., & Bergqvist, T. (2015). How mathemati s urri ulum

programs an be designed to to a omodate a �exible restru turing by the tea her. Paper presented

at the AERA annual meeting, Chi ago.

AN INVESTIGATION OF CHINESE MATHEMATICS TEACHERS' RESOURCES WORK

IN COLLECTIVES AND THEIR PROFESSIONAL DEVELOPMENT

Chongyang Wang

East China Normal University, China

Chongyang.wang�ens-lyon.fr

Lu Trou he

ENS de Lyon, Fran e


Birgit Pepin

TU/e, Eindhoven, Netherlands

b.e.u.pepin�tue.nl

The fast development of te hnology enri hes the types of resour es that tea hers an a ess and use,

and the platforms for storing and pro essing su h resour es. This is likely to hange tea hers' working

modes, and at the same time provides hallenges for dealing with these resour es. A ording to Adler

(2000), a resour e ould be anything with the potential to �re-sour e" the tea her's a tivity. We keep

the de�nition of resour e in our resear h as something en ompassing materials and elements intervening

�upstream" of tea hing, su h as emails with tea her olleagues, websites onsulted, students' work et .

(Gueudet, Pepin, & Trou he, 2012). Adler (2000) also stressed that more resour es do not ne essarily

lead to better pra ti e; a �re-sour ed" tea her should be a tea her a ting with material and so io- ultural

resour es, and not simply a tea her surrounded by material resour es. The on eptualization of tea hers'

work with/on resour es has been further developed by Gueudet et al. (2012). They name the intera tions

between tea hers and resour es do umentation work, and in this do umentation approa h to dida ti s,

the tea her's professional growth is �generated" through a do umentational genesis. The result, developed

along this genesis, is named a do ument omposed of resour es ombined to rea h a tea hing obje tive

and of a s heme (Vergnaud, 1998) onne ting this obje tive, the organization of a tion, and professional

knowledge. The resour es a tea her works with onstitute her resour e system.

We situate tea her's do umentation work in olle tives and pay parti ular attention to the so io- ultural

fa tors su h as support and onstrains, as human work always en ompasses a ultural, histori al and so ial

65

reality (Engeström, 1987). Through do umentation work, tea hers are linked not only to resour es, but

also to other tea hers. In terms of theoreti al frames, we ombine the do umentational approa h to

dida ti s to investigate the genesis/development of resour es; and a tivity theory (Engeström, 1987) to

study in olle tives the in�uential so ial- ultural and histori al fa tors (e.g. the institutional requirements;

in�uen es from their olleagues in olle tives).

We propose the notion of do umentation expertise (DE) to represent the expertise in tea her do -

umentation work, the s hemes for better resour e work whi h onsist by skills (organization of a tion)

and related knowledge. Brown (2009, p. 29) de�ned tea her's ability to per eive and mobilize urri -

ular resour es in order to raft instru tional episodes as pedagogi al design apa ity (PDC), and Pepin,

Gueudet, and Trou he (2017) developed this further to de�ne mathemati s-dida ti al design apa ity. Fur-

ther study of Remillard (2016) detailed the pro edures of per eiving the urri ular resour es (the same way

for non- urri ular resour es) as identifying the mathemati al purpose or point underlying mathemati al-

instru tional obje ts: PDC is the ability to per eive the underlying stru tures of the designed resour es

with respe t to their instru tional intensions. Compared with PDC, DE involves a wider range of re-

sour es, ompassing a omplete pro ess of do umentation genesis: resour es design, implementation, and

re�e tion.

We are interest to �nd: How do mathemati s tea hers show and develop their DE in their olle tive

do umentation work?

Within a olle tive working ulture des ribed by Yang (2010), a Tea hing Resear h Group (TRG)

represents the s hool-based tea her organization and provides institutional supports for tea hers' olle tive

work. It has been a knowledged as a ru ial fa tor for explaining the top performan e of Chinese students

in international evaluations (Wang, 2013). This is also eviden ed by one of our previous studies with three

Chinese mathemati s tea hers (Pepin, Xu, Trou he, Wang, 2016): the olle tive plays an important role

in tea her professional growth, regardless the di�eren es in working habits and resour e preferen es.

The PhD resear h (Wang, 2016) reported in this paper explores the DE of mathemati s tea hers

in olle tives in two di�erent ontexts: China; and Fran e. This is being explored in two dimensions:

the individual resour es work of sele ted mathemati s tea hers; and the in�uen es from the olle tives

on those tea hers' do umentation work in front of new hallenges: The Fren h ase is situated in the

national urri ulum reform, and we pay attention on the sele ted tea hers' olle tive resour e work for a

new tea hing topi , algorithmi .

From February to April 2017, another data olle tion will take pla e in a Chinese middle s hool.

We hypothesise that the TRG is likely to o�er the main resour es for the mathemati s tea hers and to

ontribute to their resour e systems. Moreover, the tea hers' individual s hemes are likely to have an

important olle tive omponent, and DE is likely to be developed through/in olle tive work. In terms of

methodology, we will validate the data olle tion instruments (e.g. observations; re�e tive interviews) we

developed in Fran e for this �new" ontext.

Referen es

Adler, J. (2000). Con eptualising resour es as a theme for tea her edu ation, Journal of Mathe-

mati s Tea her Edu ation, 3, 205-224.

Brown, M.W. (2009). The tea her-tool relationship: Theorizing the design and use of urri ulum

materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd, (Eds.), Mathemati s tea hers

at work: Conne ting urri ulum materials and lassroom instru tion (pp. 17-36). New York:

Routledge.

Engeström, Y. (1987). Learning by expanding. An a tivity-theoreti al approa h to developmental

resear h. Helsinki: Orienta-Konsultit.

Gueudet, G., Pepin, B., & Trou he, L. (Eds.) (2012). From text to 'lived' resour es: Mathemati s

urri ulum materials and tea her development. New York: Springer.

66

Pepin, B., Gueudet, G., & Trou he, L. (2017- under review). Mathemati s tea hers as designers:

digital resour es as windows for investigating tea hers' (mathemati al) dida ti al design apa ity.

ZDM- Mathemati s Edu ation Resear h.

Pepin, B., Xu, B., Trou he, L., & Wang, C. (2016). Developing a deeper understanding of math-

emati s tea hing expertise: Chinese mathemati s tea hers' resour e systems as windows into their

work and expertise. Edu ational Studies in Mathemati s, http://rd u.be/koXk

Remillard, J. (2016). Understanding tea her-resour e intera tions: per eiving urri ulum resour es.

ICME-13, Hamburg, July.

Wang, J. (2013). Mathemati s edu ation in China: tradition and reality. Najing: Jiangsu Edu ation

Publishing House.

Wang, C. (2016). Analyzing tea hers' expertise, resour es and olle tive work throughout Chinese

and Fren h windows. ICME-13, Hamburg, July.

Yang, X. (2010). Con eption and hara teristi s of expert mathemati s tea hers in China. New

York: Springer.

67

Workshops

WORKSHOP A

READING GEOMETRICALLY: THE ANALYSIS OF GEOMETRIC DIAGRAMS IN

TEXTBOOKS

Leslie Dietiker Meghan Riling Aaron Brakonie ki

Boston University, USA

dietiker�bu.edu mriling�bu.edu brak�bu.edu

In Dietiker and Brakonie ki (2014), we drew from Pimm (2006) to onsider the question What does a

diagram ask of its reader? We proposed dimensions of reading geometri diagrams gleaned from analyzing

the geometri tasks in multiple elementary and se ondary textbooks, in luding traditional and reform

urri ula from multiple ountries. These dimensions represent distin t aspe ts of geometri diagrams that

students are expe ted to pay attention to and interpret as they negotiate the meanings of mathemati al

tasks. Our primary on ern is that the ways in whi h students are expe ted to interpret geometri

diagrams appears to hange; the geometri diagrams in tasks in elementary textbooks expe t students to

make di�erent assumptions about the information in the diagram than those in high s hool textbooks.

In this workshop, we will share what we have learned from our re ent study of the expe tations for how

students read geometri diagrams. We will introdu e a framework des ribing the expe tations for students

while reading geometri diagrams and will explain several examples from U.S. textbooks to demonstrate.

Then, parti ipants will work in groups to use the framework to analyze several geometri diagrams from

a mixture of textbooks for di�erently-aged students. We will end this session with a dis ussion on the

impli ations these hanging expe tations have for the tea hing and learning of geometry. With eviden e

that students are expe ted to develop sophisti ated ways to negotiate meaning from diagrams, we argue

that edu ators need to raft opportunities for students to develop strategies for reading geometri diagrams.

In the study that will be dis ussed, we ompared the geometri diagrams found in a Grade 1 U.S.

textbook with the diagrams of a U.S. high s hool geometry textbook. To ode the reading expe tations

of the geometri diagrams, we developed six odes to des ribe the geometri reading of diagrams, identi-

fying: (1) how the reader is expe ted to interpret the diagram as something (e.g., a real life obje t or a

representation of a set of obje ts), (2) whether dedu tive reasoning from the diagram is required to solve

the task, (3) if the reader needs to mentally redraw the diagram to answer the task, (4) whether the reader

needs to interpret onventional markings to omplete the task, (5) if the reader is required to read the

diagram at all to answer the task, and (6) whether the diagram had the potential for being dimensionally

onfusing. Using these odes, we report how the expe tations of reading geometri ally hange as students

progress through s hool.

Referen es

Dietiker, Leslie, and Aaron Brakonie ki. 2014. �Reading Geometri ally: The Negotiation of the

Expe ted Meaning of Diagrams in Geometry Textbooks." In Pro eedings of the International Con-

feren e on Mathemati s Textbook Resear h and Development, edited by Keith Jones, Christian

Bokhove, Geo�rey Howson, and Lianghuo Fan, 191-96. University of Southampton, UK.

Pimm, David. 2006. �Drawing on the Image in Mathemati s and Art." In Mathemati s and the

Aestheti : New Approa hes to an An ient A�nity, edited by Nathalie Sin lair, David Pimm, and

William Higginson, 160-90. New York, NY: Springer.

71

WORKSHOP B

ADAPTING SOCIOCULTURAL RESEARCH FRAMEWORKS FOR BRAZIL: THE UM

LIVRO ABERTO PROJECT

Vi tor Giraldo

UFRJ, Brazil

vi tor.giraldo�ufrj.br

Meril Rasmussen

UBC, Canada

merilx�gmail. om

Fabio Simas

UNIRIO, Brazil

fabio.simas�uniriote .br

Augusto Teixeira

IMPA, Brazil

augusto�impa.br

The Um Livro Aberto proje t (https://www.umlivroaberto. om/wp/) in Brazil is working towards reating

a ollaborative, open-sour e digital and print textbook for middle s hool through an iterative pro ess where

pra ti ing tea hers are able to respond and ontribute to an evolving set of improving urri ular materials

(Simas & Teixeira, 2016). To support this pro ess, we have mobilized a team of resear hers aimed at

reating a meaningful pro ess of developmental resear h that an inform the work of our writers while

simultaneously supporting the e�orts of pra ti ing tea hers in the �eld as they implement these new

materials. Drawing broadly on urrent international models (Even & Olsher, 2012; Gravemeijer & van

Eerde, 2009; Ruthven, 2012; Pepin, Gueudet & Trou he, 2013), the Um Livro Aberto proje t is, at its

ore, an e�ort to improve the quality of mathemati s edu ation in Brazil by leveraging te hnology to

fa ilitate ollaboration and understanding between pra ti ing tea hers and university-based mathemati s

edu ators and mathemati ians. In this workshop, we seek to engage with, and learn from, our international

olleagues, o�ering up our experien es thus far as a ase study through whi h to dis uss methodologi al,

strategi , and pra ti al issues that arise out of our e�orts to adapt emerging so io ultural frameworks to the

developing-world ontext of Brazil. We look broadly to the emerging frameworks and methodologies as well

as the pra ti al, real-world examples su h as the Sésamath Proje t in Fran e (http://www.sesamath.net/)

and the Integrated Mathemati s Wiki-book Proje t in Israel. This English-language, resear h-fo used

workshop will be omplemented by a parallel workshop in Portuguese aimed at Brazilian tea hers who are

involved as partners in the Um Livro Aberto proje t as well as a several related resear h paper presentations.

Referen es

Even, Ruhama, and Shai Olsher. 2012. �Tea hers as parti ipants in textbook development: The

Integrated Mathemati s Wiki-book Proje t.� In Pro eedings of the Chais Conferen e on Instru tional

Te hnologies Resear h 2012: Learning in the Te hnologi al Era, edited by Y. Eshet-Alkalai, A. Caspi,

S. Eden, N. Geri, Y. Yair, and Y. Kalman, 30-34. Raanana: The Open University of Israel.

Freudenthal, Hans. 1973. Mathemati s as an edu ational task. Netherlands: Springer.

Gravemeijer, Koeno, and Dolly van Eerde. 2009. �Design resear h as a means for building a knowl-

edge base for tea hers and tea hing in mathemati s edu ation.� The Elementary S hool Journal

109(5): 510-524.

Ruthven, Kenneth. 2012. �Constituting Digital Tools and Materials as Classroom Resour es: The

Example of Dynami Geometry.� In From text to 'lived' resour es: Mathemati s urri ulum materials

and tea her development. edited by Ghislaine Gueudet, Birgit Pepin, and Lu Trou he, 83-103. New

York: Springer. DOI: 10.1007/978-94-007-1966-8

Pepin, Birgit, Ghislaine Gueudet, and Lu Trou he. 2013. �Investigating Textbooks as Cru ial

Interfa es between Culture, Poli y and Tea her Curri ular Pra ti e: Two Contrasted Case Studies

in Fran e and Norway.� ZDM Mathemati s Edu ation. 45:685-698 DOI 10.1007/s11858-013-0526-2

Simas, Fabio, and Augusto Teixeira. 2016. Livro Aberto de Matemati a. Rio de Janeiro: IMPA.

Sour ed from:

https://www.umlivroaberto. om/wp/ and https://www.umlivroaberto. om/livro/lib/exe/fet h.php?

media=projeto_livro_didati o_versao_grande_publi o.pdf

72

WORKSHOP C

WRITING PROMPTS AND MATHEMATICS TEXTBOOKS

Christine M. Joseph

East Carolina University

joseph 15�e u.edu

Jenifer J. S hneider

University of South Florida

s hneidj�usf.edu

Various organizations, su h as the National Coun il of Tea hers of Mathemati s (NCTM), National Re-

sear h Coun il (NRC), and members of the Coun il of Chief State S hool O� ers (CCSSO) and the

National Governor's Asso iation, Center for Best Pra ti es (NGA Center), have produ ed standards do -

uments that highlight the use of writing in the mathemati s lassroom. The NRC formulated the Strands

of Mathemati al Pro� ien y (NRC, 2001). Furthermore, members of the Coun il of Chief State S hool

O� ers (CCSSO) and the National Governors Asso iation, Center for Best Pra ti es (NGA Center) de-

veloped ommon standards for all states where ommuni ation is embedded throughout the ontent

re ommendation (CCSS, 2010).

The NCTM Prin iples and Standards for S hool Mathemati s (2000) states that the ontent strands

should be taught through mathemati al pro esses. Similarly, the Common Core State Standards for Math-

emati s (CCSSM) also states that ontent strands should be taught through a series of eight mathemati al

pra ti es. Whether the pro esses are utilized in isolation or as a onne ted omponent, the pro ess of

writing an be demonstrated throughout these strands. For example, in order to problem solve one an

write an explanation or des ription of the problem solving pro ess by reasoning and proving one's mathe-

mati al thinking. Students an also write to des ribe the pro ess of onne ting the mathemati s ontent

in addition to providing an explanation of a parti ular mathemati al representation.

The textbook publishing industry, as well as urri ulum proje ts funded by the National S ien e Foun-

dation (NSF), moved qui kly to develop urri ulum materials (i.e., textbooks) to align to standards

re ommendations from these various organizations. Publishers realize that in addition to the standards

do uments, the most ommon in�uen e on ontent appears to be the textbook/ urri ulum program (Weis,

Pasley, Smith, Banilower, & He k, 2003). Thus, the mathemati s textbook is typi ally resear hed as the

dominant tool in lassroom instru tion (Tarr et al., 2008).

Resear hers note the a�e tive and ognitive bene�ts of writing in mathemati s (Senk & Thompson,

2003). However, the language in the types of prompts has not been investigated. The language of prompts

and usage of prompts dire tly in�uen e lassroom opportunities for students to develop mathemati al

thinking. In order to onstru t a response in mathemati s, the student must be able to omprehend

the prompt while produ ing pre ise language to respond to the prompt. O'Connell et al. (2005) note

that words are the building blo ks for ontent understanding, emphasizing that in order to ommuni ate,

it is important for students to understand the words that express that ontent. The omprehension of

mathemati s en ompasses not only vo abulary terms, but also the understanding of symbols (Thompson

et al., 2008). These types of vo abulary have the potential to make the omprehension of mathemati s

a omplex pro ess.

The PSSM (NCTM, 2000) pla es an emphasis on vo abulary under the pro ess strand of ommu-

ni ation by re ommending that students use mathemati al vo abulary to express mathemati al ideas in

a pre ise manner. However, there are only two studies fo used on the instru tional impli ations of lan-

guage and vo abulary in mathemati s textbooks for middle grades learners (Haggarty & Pepin, 2002;

Herbel-Eisenmann, 2007). The Haggarty and Pepin (2002) study examined and ompared the layout of

the mathemati al textbooks used in Fran e, Germany and England. Additionally, the study investigated

the opportunities students had to perform mathemati al pro esses through the use of the vo abulary and

language in the dire tions. The Herbel-Eisenmann (2007) study examined one middle s hool National

S ien e Foundation (NSF) funded textbook for the �voi e" of that parti ular textbook. More importantly

the resear her examined the linguisti hoi es (i.e., use of imperatives, pronouns, modal verbs and ex-

pressions) developed by the textbook authors in order to understand the role of the reader and how the

73

relationship between the reader and the author is onstru ted.

In light of these �ndings, the presenters investigated elementary mathemati s textbooks from widely

used textbook publishing ompanies. The areas explored were developed from resear h in writing and

textbook use. The development of a tool in the form of a matrix assisted with the oding of prompts in

these parti ular areas. The �ndings lead to professional development workshops and further resear h and

resear h presentations in the area of tea her support and prompt modi� ation.

The presenters in this workshop will present and review a modi�ed version of the matrix for parti ipants

to utilize in investigating how writing is treated in mathemati s textbooks. The presenters will also

demonstrate how to utilize a developed tool for planning for writing in mathemati s The �nal portion of

the workshop will onsist of the presenters and parti ipants developing (from textbook samples) tasks that

are di�erentiated in a ademi vo abulary as well as modifying the task for di�erent purposes of writing.

Ea h one of these workshop omponents has had national re ognition in onferen e presentations and

manus ripts for publi ation.

Referen es

Coun il of Chief State S hool O� ers. (a essed 2010) Common ore standards for mathemati s.

Retrieved from http://www. orestandards.rg/assets/CCSSI_Math%20Standards.pdf.

Haggarty, Linda., & Pepin, Birgit. 2002. An investigation of mathemati s textbooks and their

use in English, Fren h and German lassrooms: Who gets an opportunity to learn what? British

Edu ational Resear h Journal, 28(4), 567-590.

Herbel-Eisenmann, Beth. 2007. From intended urri ulum to written urri ulum: Examining the

�voi e" of a mathemati s textbook. Journal for Resear h in Mathemati s Edu ation, 38(4), 344-369.

National Coun il of Tea hers of Mathemati s, ed. 2000. Prin iples and standards for s hool math-

emati s. Vol. 1. National Coun il of Tea hers of Mathemati s.

O'Connell, Susan R., Celia Baemon, Jennifer M. Beyea, Susan S. Denvir, Leila A. Dowdall, Nikki G.

Friedland, and Joelle D. Ward. 2005. �Aiming for Understanding: Lessons Learned about Writing

in Mathemati s." Tea hing Children Mathemati s 12 (4),192.

Senk, Sharon L., and Denisse R. Thompson. 2003. �S hool mathemati s urri ula: Re ommenda-

tions and issues." Standards-based s hool mathemati s urri ula: What are they: 3-27.

Tarr, James E., Robert E. Reys, Barbara J. Reys, Os ar Chavez, Je�rey Shih, and Steven J. Osterlind.

2008. �The impa t of middle-grades mathemati s urri ula and the lassroom learning environment

on student a hievement." Journal for Resear h in Mathemati s Edu ation, 247-280.

Weiss, I. R., J. D. Pasley, P. S. Smith, E. R. Banilower, and D. J. He k. 2003. �Looking inside the

lassroom: a study of K-12 mathemati s and s ien e edu ation in the US Chapel Hill." NC: Horizon

Resear h.

WORKSHOP D

TEXTBOOK ANALYSIS OF NEW MATH AND CONTEMPORARY GERMAN SCHOOL

BOOKS FROM ERNEST'S INTEREST GROUPS PERSPECTIVE

Rainer Kaenders

Rheinis he Friedri h-Wilhelms University Bonn

R.Kaenders�uni-bonn.de

Ysette Weiss

Johannes Gutenberg University Mainz

weisspid�uni-mainz.de

Textbooks in German middle and se ondary s hools have hanged onsiderably within the past forty years.

Math tea hing went from New Math to the ontemporary tea hing pra ti e. What has hanged pre isely

74

in textbooks during this hange? Is there less mathemati s now than there used to be during the German

New Math? After the dis reditation of the New Math Movement � may it be rightly or not, we annot

be satis�ed by just looking at how far the language in textbooks is similar to the language in modern

mathemati s.

The answer to the question, whether there is more or less mathemati s today than forty years ago,

we have to larify what one means with `mathemati s'? As Paul Ernest (1991, 2000) pointed out in

detail at the example of the British so iety, the question about the nature of mathemati s and about the

orresponding ideals of mathemati s tea hing is ne essarily related to one's a�liation to one or several

interest groups with respe t to mathemati s tea hing. In parti ular, the normative riteria for the tea hing

of mathemati s are not a matter that mathemati ians an larify among themselves.

Ernest (lo . it.) distinguishes between �ve su h interest groups. Although the German politi al

situation is di�erent from the British one, the basi dire tions of politi al orientation are omparable.

1. Industrial Trainers

�Radi al 'New Right', onservative politi ians and petty bourgeois: A quiring basi mathemati al

skills and numera y and so ial training in obedien e (authoritarian, basi skills entred)�,

2. Te hnologi al Pragmatists

�Merito rati industry- entred industrialists, managers, et ., New Labour: Learning basi skills and

learning to solve pra ti al problems with mathemati s and information te hnology (industry and

work entred)�,

3. Old Humanist Mathemati ians

�Conservative mathemati ians, preserving rigour of proof and purity of mathemati s: Understanding

and apability in advan ed mathemati s, with some appre iation of mathemati s (pure mathemati s

entred)�,

4. Progressive Edu ators

�Professionals, liberal edu ators, welfare state supporters: Gaining on�den e, reativity and self

expression through maths ( hild- entred progressivist)�,

5. Publi Edu ators

�Demo rati so ialists and radi al reformers on erned with so ial justi e and inequality: Empower-

ment of learners as riti al and mathemati ally literate itizens in so iety (empowerment and so ial

justi e on erns)�.

In his analysis, Ernest (1991, p. 131�) des ribes in detail many fa ets of ea h respe tive interest

group: Politi al Ideology, View of Mathemati s, Moral Values, Theory of So iety, Theory of the Child,

Theory of Ability, Mathemati al Aims, Theory of Learning, Theory of Tea hing Mathemati s, Theory of

Resour es, Theory of Assessment in Maths, Theory of So ial Diversity.

In the workshop, we onsider tasks in several textbook series for German pre-university edu ation in

the light of Ernest's interest groups perspe tive. Here in Figure 1 we give an example.

Figure 1: Task from Mathematik Neue Wege SII � Allgemeine Ausgabe 2011, Analysis II, p. 71.

75

�Polynomial garage doorway: The underground parking is only a essible via a steep doorway. The

passage should be as smooth as possible and the maximal slope should not be too big, su h that it

is traversable by an ordinary passenger ar. Make a planning by using an appropriate fun tion graph.

Compare your solution with those of others.� (Translation by the authors).

To get a detailed view onto the respe tive task, we �rst have to put it in a bigger ontext as far as

their ontent and the expe ted student a tivity is on erned. We ould think of several a tivities:

• planning a garage doorway for a spe ial garage,

• modelling the design of garage doorways in general,

• pra ti e to deal with polynomial fun tions,

• give a meaning to the dis ussion of polynomial fun tions,

• preparing for the next exam.

• ea h ase, we ask for possible motives and operationalisations of the respe tive task and onsider

the tools that the students are supposed to use while working on the task. In this way, we get an

impression of the intended mathemati al awareness ( f. Kaenders & Kvasz, 2011 and Kaenders,

Kvasz, L. & Weiss-Pidstryga h, 2011). In the spe trum of aspe ts that appear by this analysis, we

try to re ognize the fa ets of the interest groups as Ernest had des ribed them.

Referen es

Ernest, Paul. 1991. The Philosophy of Mathemati s Edu ation. London: Falmer Press.

Ernest, Paul. 2000. Why tea h mathemati s? In Why Learn Maths?, edited by Steve Bramall, and

John White, (Eds.), 1-14. London: Bedford Way Papers.

Kaenders, Rainer, Kvasz, Ladislav, and Weiss-Pidstryga h, Ysette. 2011. Mathemati al awareness

by linguisti analysis of variable substitution. CERME 7, Rzeszów, Poland.

Kaenders, Rainer, and Kvasz, Ladislav. 2011. Mathematis hes Bewusstsein. In Mathematik verste-

hen � philosophis he und didaktis he Perspektiven, edited by Katja Lengnink, Gregor Ni kel and,

Rudolf Wille, 71-85. Wiesbaden: Vieweg+Teubner Verlag.

76

Workshops for Tea hers

WORKSHOP FOR TEACHERS A

UMA POSSIBILIDADE TEÓRICO METODOLÓGICA PARA ANÁLISE DE LIVROS

DIDÁTICOS POR MEIO DA HERMENÊUTICA DE PROFUNDIDADE E DOS

PARATEXTOS EDITORIAS

Mirian Maria Andrade

Universidade Te nológi a Federal do Paraná

andrade.mirian�gmail. om

Bruno Alves Dassie

Universidade Federal Fluminense

badassie�gmail. om

Leandro Josué de Souza

Universidade Estadual Paulista �Júlio de Mesquita Filho� - Bauru

leandrojosue�gmail. om

A Hermenêuti a de Profundidade (HP) surge, basi amente, omo uma teoria da interpretação de textos,

tomando textos em sentido amplo, isto é, um onjunto de símbolos que são riações humanas arregadas

de intenções ujo signi� ado é atribuído por aquele que o lê, num pro esso hermenêuti o, interpretativo.

É por essa lente que Ri oeur (e vários �lósofos seus ontemporâneos) onsidera um texto, omo uma

produção humana inten ional (ou que possa ser assumida omo tal), materializada em símbolos (Andrade

e Oliveira 2011) e, portanto, aberta à interpretação. Thompson propõe o Referen ial Metodológi o da

Hermenêuti a de Profundidade para analisar a ideologia de formas simbóli as nos meios de omuni ação

de massa (Thompson 1995). Na Edu ação Matemáti a a HP vem sendo utilizada omo uma orien-

tação metodológi a para analisar textos didáti os e não didáti os, onsiderando estes textos omo formas

simbóli as. Formas simbóli as são ações, falas, imagens e textos produzidos e re onhe idos omo signi-

� ativos para os sujeitos envolvidos nos ontextos de produção, emissão e re epção. Podemos sintetizar

que, formas simbóli as, são riações humanas inten ionais.

A HP se apresenta, omo método, em três fases (dimensões, movimentos ou momentos) interligadas e

on omitantes, que podem ser sinteti amente nomeadas omo: Análise Só io-Históri a, Análise Formal ou

Dis ursiva e Interpretação/Reinterpretação. Na análise só io-históri a a intenção é analisar o ontexto em

que a forma simbóli a foi pensada, produzida e, posteriormente, ir ulou. Na análise formal ou dis ursiva

o fo o entral do exer í io analíti o é o "objeto de estudo" em si, é o momento de olhar para as estruturas

do objeto e de questionar sobre a intenção deste. Por �m, há a fase de interpretação/reinterpretação que,

além de osturar, ontinuamente, todos os momentos da análise (pois nessa fase ressalta-se a atribuição

de signi� ados), trata de registrar uma apreensão geral de todo o pro esso interpretativo. É o momento

de apresentar um possível signi� ado, na ompreensão do hermeneuta, da forma simbóli a (Thompson

1995).

Em nossos movimentos de pesquisa, no entanto, per ebemos que pode haver uma signi� ativa on-

tribuição, em termos de análises, quando os Paratextos Editoriais, de a ordo om Genette (Genette 2009),

são hamados a efetivar exer í ios analíti os junto da Hermenêuti a de Profundidade. Isso se dá, sobre-

tudo, quando a forma simbóli a é um livro (didáti o ou não). Dois elementos são entrais no que diz

respeito a essa opção: (a) o Referen ial Metodológi o da Hermenêuti a de Profundidade é pensado para

analisar formas simbóli as quaisquer, enquanto a proposta dos paratextos de Genette dirige-se mais pro-

priamente à análise de livros - uma forma simbóli a parti ular; (b) uma leitura mais atenta das disposições

de Genette nos aponta que pode haver dissonân ias entre os on eitos que fundamentam essas disposições

em relação às que sustentam a proposta de Thompson, mas tratar dessas dissonân ias ompete a outros

possíveis estudos e não interfere no modo omo os mobilizamos em nossas pesquisas.

E o que são, então, Paratextos Editoriais de a ordo om Genette? Um paratexto é, segundo Genette,

aquilo por meio de que um texto se torna livro. Segundo esse autor, é por meio do paratexto que o

texto deixa de ser um texto bruto e passa a ser um livro. Podemos listar diversos elementos en ontrados

num livro (ou externos a ele, mas que se rela ionam a ele) que podem ser lassi� ados omo paratextos,

segundo a visão de Genette: o nome do autor, os títulos e os subtítulos, a data da obra, os releases, as

dedi atórias, as epígrafes, a instân ia prefa ial, as notas de rodapé, listas de obras do mesmo autor, notas

79

do autor ou do editor, menções de preço, onversas e entrevistas sobre o livro, formato, orrespondên ias

ao autor, as ilustrações, as apas, os anexos et . (Genette 2009).

Permitindo que sejam estabele idas aproximações om outros métodos de investigação, a HP amplia

as possibilidades do leitor ao mesmo tempo em que se ampliam suas fronteiras. A aproximação entre a HP

e os paratextos editoriais tem se mostrado bastante fun ional nas nossas investigações. Por isso, é nosso

interesse, tratar dessas questões teóri o metodológi as, para análise de livros, do modo omo as temos

mobilizado e das possibilidades que elas podem apresentar para a análise de livros didáti os. Para tanto,

é nossa intenção abordar o uso do Referen ial Metodológi o da Hermenêuti a de Profundidade (HP) e

dos Paratextos Editoriais nas pesquisas em Edu ação Matemáti a, realizar uma dis ussão teóri a sobre

estes referen iais metodológi os, bus ando apresentar os resultados de algumas pesquisas em Edu ação

Matemáti a que se fundamentaram nestas teorias; te er uma dis ussão sobre a aproximação da on epção

de Paratextos Editorias na análise de textos a partir do Referen ial Metodológi o da HP, visando apresentar

as possibilidades e as limitações desta aproximação. Compõem, também, esse nosso interesse, realizar uma

atividade práti a, onde possamos, por meio do desenvolvimento de um ensaio de análise hermenêuti a,

apresentar omo efetivamente este exer í io analíti o pode a onte er segundo nossas fundamentações

teóri as e nossas experiên ias. Pretendemos que este ensaio possa ser realizado om tre hos de livros

didáti os e do manual do professor. Nossa es olha por este tipo de material deve-se a dois fatores:

primeiro, a limitação do tempo de duração, que não permite exer í ios de análise muito longos. Segundo,

o livro didáti o é um material de utilização do professor de Matemáti a da es ola bási a, de fá il a esso,

e o manual do professor, pode ser entendido, a partir de Genette, omo um paratexto editorial do livro

didáti o.

Referên ias

Andrade, Mirian M., e Oliveira, Fábio D. 2011. �A análise de textos didáti os em História da

Edu ação Matemáti a�. I Congresso Ibero-Ameri ano de História da Edu ação Matemáti a, Covilhã,

Portugal. Maio 26 - 29. A essado em 19 de outubro, 2016.

http://www.apm.pt/�les/177852_C54_4dd7a40f 6b6a.pdf.

Genette, Gérard. 2009. Paratextos Editoriais. Cotia, SP: Ateliê Editorial.

Thompson, John B. 1995. Ideologia e Cultura Moderna: Teoria so ial ríti a na era dos meios de

omuni ação de massa. Petrópolis, RJ: Editora Vozes.

WORKSHOP FOR TEACHERS B

O GUIA DO PNLD E O PROCESSO DE ESCOLHA DE LIVROS DIDÁTICOS

Marilena Bittar Danielly Regina Kaspary dos Anjos Adriana Barbosa Oliveira

Universidade Federal de Mato Grosso do Sul

marilenabittar�gmail. om kaspary.d�gmail. om drideoliveira7�gmail. om

Periodi amente é pre iso es olher, em uma lista forne ida pelo Programa Na ional do Livro Didáti o

(PNLD), o livro a ser utilizado por três anos nas es olas públi as do Brasil. Professores e gestores têm,

quase sempre, o mesmo problema diante de si: de idir a obra a ser usada, normalmente em um espaço de

tempo bastante urto. Tendo em vista o fato de que o livro didáti o é um elemento entral de apoio aos

professores no preparo de suas aulas e, muitas vezes, esse é o úni o material de estudos a sua disposição,

é ne essário que haja atenção e uidado nesse pro esso de es olha para que essa obra auxilie, de fato,

o professor em sua tarefa otidiana. Assim, onsideramos de fundamental importân ia que o professor

tenha um olhar ríti o sobre esse re urso didáti o e leve em onsideração suas on epções de ensino e de

aprendizagem no momento da es olha da oleção.

80

A ompra e a distribuição de livros didáti os no Brasil são assistidas pelo Programa Na ional do

Livro Didáti o (PNLD), om �nan iamento próprio do Fundo Na ional de Desenvolvimento da Edu ação

(FNDE). Esse programa é responsável também pela avaliação das obras submetidas em editais que, se

aprovadas ompõem a lista de oleções que podem ser es olhidas pelos professores e es olas públi as.

Após o pro esso de avaliação, é apresentado à omunidade, om espe ial atenção aos professores, o Guia

do Livro Didáti o. Esse do umento ontém a des rição e análise de ada obra aprovada, om o objetivo

de subsidiar a es olha da oleção pelos professores.

No entanto, o que temos notado em ursos de formação e em onversas om pro�ssionais que vivem o

ambiente es olar, é que o momento de es olha da oleção de livros didáti os é pou o valorizado e por vezes

sofre prejuízo dos problemas que se apresentam no dia-a-dia da es ola. Por isso, observamos professores

que nun a tiveram a oportunidade de folhear e analisar o Guia e também não tiveram a oportunidade de

se olo arem em um momento de estudo para essa seleção. A es olha da oleção, por esses motivos, é

onduzida por fatores que pou o estão ligados àquilo que o professor a redita.

Consideramos que a ação de es olher a oleção que mais se aproxima da realidade em que se vive e

das on epções de ensino e aprendizagem que se tem, pode oportunizar um momento ri o de dis ussão

e formação. E é na elaboração de um momento omo esse que se origina a o� ina aqui apresentada.

Para tanto, es olhemos realizar um trabalho de análise das obras aprovadas no PNLD 2016, destinado

aos anos ini iais do ensino fundamental. Essa análise onsiste em entender alguns ritérios avaliativos

utilizados pelo Programa e entender também o próprio Guia, para que ele possa se tornar efetivamente um

instrumento de apoio ao professor. Bus aremos, om isso, promover uma experiên ia de leitura dinâmi a

do Guia, bus ando levar em onsideração o urto tempo que os professores e gestão têm e destinam à

es olha de uma oleção. Essa experiên ia servirá omo um ensaio do que pode ser feito nas es olas e,

om isso, temos omo pretensão inspirar formas de tornar esse momento mais frutífero.

Nossa experiên ia om o trabalho de análise de livros didáti os sob o olhar de teorias da Didáti a da

Matemáti a, espe ialmente a Teoria Antropológi a do Didáti o, desenvolvida por Chevallard (1998), que

permite entender e des rever as es olhas matemáti as e didáti as apresentadas em livros didáti os nos

indi a que ompreender as es olhas dos autores dos livros didáti os permite também entender e questionar

a nossa práti a omo professores.

Apresentamos a seguir algumas ações que ompõem a ideia de nossa o� ina:

• Breve apresentação do PNLD e dos elementos que onstituem uma resenha;

• Entrega, em material impresso para ada parti ipante, de re ortes das resenhas das oleções aprovadas

pelo PNLD 2016;

• Dis ussões em grupos sobre o tópi o �Visão Geral" ontido nas resenhas;

• Eleição, pelos grupos, de uma oleção que mais se aproxima daquilo que o grupo a redita para o

ensino e aprendizagem da Matemáti a;

• Debate no grupo geral;

• Análise do tópi o �Em sala de aula", ontido no Guia;

• Nova eleição pelos grupos da oleção a ser es olhida;

• Debate quanto às possíveis mudanças das es olhas das oleções e os ritérios que onduziram para

essa de isão.

A formação de professores (tanto a ini ial omo a ontinuada) há muito sofre in�uên ias das on-

epções behavioristas que, em geral, onsideram o professor omo um sujeito a ser modi� ado (em suas

on epções) por meio de ursos no modelo magistral. Esta tendên ia, entretanto, vem sendo ontestada

desde os anos 1980 e diversos modelos alternativos têm sido propostos e prati ados, sempre olo ando o

professor no entro do pro esso e om papel ativo em sua formação. Essa é a perspe tiva que assumimos

e, nesse sentido, a o� ina que propomos visa, representar um �espaço de formação mútua" em que � ada

81

professor é hamado a desempenhar, simultaneamente, o papel de formador e formando" (NÓVOA, 1995,

p.26).

Referên ias

Brasil, Ministério da Edu ação. PNLD 2016: guia digital/Ministério da Edu ação - Se retaria de

Edu ação Bási a � SEB. Brasília, DF: 2016. MEC, 2007.152 p.

Chevallard, Yves. 1998. �Analyse des pratiques enseignantes et dida tique des mathematiques:

L'appro he anthropologique." Re her hes em Dida tique des Mathématiques 19(2): 221-266. A es-

sado no site http://yves. hevallard.free.fr/spip/spip/arti le.php3?id_arti le=27 no dia 15 de Maio

de 2013, om paginação 1- 29.

Nóvoa, Antonio. 1995. A formação dos professores e pro�ssão do ente. In: Nóvoa, A. (org.) Os

professores e sua formação. Lisboa: Dom Quixote, p. 13-33.

WORKSHOP FOR TEACHERS C

O ENSINO DE FRAÇÕES NO ENSINO FUNDAMENTAL I: ATIVIDADES DO �LIVRO

ABERTO DE MATEMÁTICA�

Humberto Bortolossi

UFF

hjbortol�vm.u�.br

Vi tor Giraldo

UFRJ

vi tor.giraldo�ufrj.br

Letí ia Rangel

UFRJ

leti iarangel�ufrj.br

Wanderley Rezende

UFF

wmrezende�id.u�.br

Cydara Ripoll

UFRGS


Wellerson Quintaneiro

CEFET-RJ

profmatwellerson�gmail. om

Fábio Simas

UNIRIO


O projeto �Um Livro Aberto de Matemáti a" (https://www.umlivroaberto. om/wp/) visa à produção

de livros didáti os de matemáti a om li ença aberta a partir de um trabalho olaborativo envolvendo

professores universitários e da es ola bási a. O projeto é an orado e a ompanhado por pesquisa ientí� a

e por ações de formação e de desenvolvimento pro�ssional do professor de matemáti a. Re onhe endo o

papel e a relevân ia do livro didáti o na práti a do professor (Even & Olsher, 2012, 2014; Pepin, Gueudet &

Trou he, 2013), do entes em atuação na sala de aula da es ola bási a são onvidados a apli ar e a avaliar

todo material produzido, em um pro esso de interação que os quali� a omo olaboradores do material

didáti o. Como etapa piloto do projeto, a �m de identi� ar e de avaliar os vários problemas logísti os,

metodológi os, de design e de onteúdo que ertamente seriam enfrentados, foi produzido um livro sobre

frações, tema re onhe idamente desa�ador no ontexto es olar (Streeftland, 1991), dirigido ao Ensino

Fundamental I. Nesta o� ina, pretendemos reproduzir parte da dinâmi a de interação om os professores

que têm apli ado e avaliado o livro de frações a partir da realização de atividades propostas no material.

Este workshop, em língua portuguesa, é dirigido para professores da es ola bási a e será omplementado

por um workshop paralelo, em língua inglesa, fo ado em pesquisa e onduzido por professores par eiros

no projeto. (Professores que atuam no primeiro segmento do Ensino Fundamental I são espe ialmente

bem-vindos!).

Referên ias

Even, Ruhama, and Shai Olsher. 2012. �Tea hers as parti ipants in textbook development: The

Integrated Mathemati s Wiki-book Proje t." In Pro eedings of the Chais Conferen e on Instru tional

Te hnologies Resear h 2012: Learning in the Te hnologi al Era, edited by Y. Eshet-Alkalai, A. Caspi,

S. Eden, N. Geri, Y. Yair, and Y. Kalman, 30-34. Raanana: The Open University of Israel.

82

Even, Ruhama, and Shai Olsher. 2014. �Tea hers Editing Textbook: Changes Suggested by Tea h-

ers to the Math Textbook They Use in Class." In Pro eedings of the International Cnferen e on

Mathemati s Textbook Resear h and Development 2014 (ICMT-204). University of Southampton,

UK.

Pepin, Birgit, Ghislaine Gueudet, and Lu Trou he. 2013. �Investigating Textbooks as Cru ial

Interfa es between Culture, Poli y and Tea her Curri ular Pra ti e: Two Contrasted Case Studies

in Fran e and Norway." ZDMMathemati s Edu ation. 45:685-698 DOI 10.1007/s11858-013-0526-2

Stree�and, Leen. Fra tions in Realisti Mathemati s Edu ation: A Paradigm of Developmental

Resear h. Dordre ht: Kluwer A ademi Publishers, 1991.

WORKSHOP FOR TEACHERS D

OFICINA: PRODUZINDO LIVROS DINÂMICOS PARA O ENSINO DE MATEMÁTICA

Kátia Coelho da Ro ha Laís de Almeida Pereira

PPGEMat - Universidade Federal do Rio Grande do Sul

prof.laispereira�gmail. om katia oelhoro ha�gmail. om

O livro didáti o é um importante re urso de sala de aula desde os primórdios da edu ação, om a as ensão

das te nologias ele também foi in orporado ao meio digital. No entanto, vemos que seu formato e-book

muitas vezes está totalmente vin ulado à sua forma impressa, des onsiderando as possibilidades de agregar

ao seu onteúdo atividades mais dinâmi as e que podem favore er a onstrução de on eitos.

O a esso ao omputador na es ola é ada vez mais frequente e seu uso pode torná-lo um interessante

re urso para que os alunos visualizem o que o orre om as �guras quando as movemos, e em um nível

mais avançado per ebam omo as propriedades matemáti as se rela ionam na imagem. Conforme Valente

(2005, p. 11):

Hoje, a utilização de omputadores na Edu ação é muito mais diversi� ada, interessante e desa�adora,

do que simplesmente a de transmitir informação ao aprendiz. O omputador pode ser também

utilizado para enrique er ambientes de aprendizagem e auxiliar o aprendiz no pro esso de onstrução

do seu onhe imento.

Assim, disponibilizar um livro dinâmi o onde o aluno onstrua seu próprio onhe imento onsiderando

seus onhe imentos prévios pode tornar esse re urso mais atrativo e prin ipalmente favorável ao pro esso

de aprendizagem. O uso de ferramentas interativas, segundo Gravina e Basso (2012, p. 14), �(...)

in orporam sistemas dinâmi os de representação na forma de objetos on reto-abstratos. São on retos

porque existem na tela do omputador e podem ser manipulados e são abstratos porque respondem às

nossas elaborações e onstruções mentais".

A proposta dessa o� ina é vin ular mídias digitais para o ensino de matemáti a à riação de um

ambiente virtual de aprendizagem olaborativa estruturado omo um livro. O ambiente es olhido para

hospedar o livro é o PBWorks, um ambiente wiki que permite o ompartilhamento de imagens, vídeos e

re ursos do Geogebra, riando assim, um ambiente que hamamos de �Livro Dinâmi o". A opção pelo uso

do termo livro está rela ionada à forma omo o material foi organizado e o termo dinâmi o por possuir

vídeos e re ursos manipuláveis, inviáveis em um livro impresso onven ional. Os re ursos manipuláveis

foram riados no software Geogebra, um software de Geometria Dinâmi a om o qual é possível onstruir

apliquetas dinâmi as que permitem a manipulação e a interação om on eitos matemáti os.

Defendemos uma proposta de livro em que o aluno seja levado a observar, explorar propriedades para

que possa onstruir o onhe imento de forma ativa, por isso propomos a riação do livro om atividades

dinâmi as a ompanhadas de questionamentos e possibilidades de registros das impressões dos alunos

83

através de Formulários do Google in orporado à Wiki. Desta forma, esta o� ina tem omo objetivo apre-

sentar este re urso à omunidade do ente e permitir aos parti ipantes riar essa ferramenta in orporando

a esta re ursos omo apliquetas do Geogebra, Vídeos e Formulários do Google.

A o� ina terá duração de 2 horas. No primeiro momento será feita uma dis ussão sobre o uso do livro

didáti o e de te nologias em sala de aula om apoio no referen ial teóri o utilizado. Após será apresentado

um Livro Dinâmi o elaborado pelas autoras e utilizados om alunos sobre o Teorema de Pitágoras. Este

livro foi onstruído em um ambiente wiki ao qual foi in orporado re ursos omo vídeo, Formulários Google,

apliquetas GeoGebra e imagens. A �gura 1 apresenta a página ini ial do Livro Dinâmi o sobre o Teorema

de Pitágoras e uma das atividades nele in orporadas, dando a ideia de um Livro Dinâmi o.

Fonte: produzido pelas autoras.

Figura 1: Livro Dinâmi o

No segundo momento será a vez dos parti ipantes da o� ina onstruírem o seu espaço wiki no PBWorks

a partir das orientações das tutoras. Cada parti ipante poderá es olher um assunto e bus ar vídeos,

imagens entre outros que queiram in luir em seu ambiente virtual, tudo será feito om a ompanhamento,

instruções e disponibilização de tutoriais.

Referên ias

Gravina, Maria Ali e, and BASSO, Mar us Vini ius de Azevedo. 2012. �Mídias Digitais na Edu ação

Matemáti a." In: gravina, Maria Ali e et al (Org.). Matemáti a, Mídias Digitais e Didáti a: tripé

para formação do professor de Matemáti a. Porto Alegre: Evangraf, Cap. 1. p. 11-36.

Valente, José Armando. 2005. Informáti a na Edu ação no Brasil:Análise e Contextualização

Históri a. In: J.A. Valente (org.). O Computador na So iedade do Conhe imento. Brasília: Estação

Palavra - USP. p. 11-28.

Disponível em: http://www. dominiopubli o.gov.br/download/texto/me003150.pdf. A esso em: 1

maio 2015.

84

WORKSHOP FOR TEACHERS E

METODOLOGIAS DE ANÁLISE HORIZONTAL

Henrique N. Sá Earp

Universidade Estadual de Campinas

henrique.saearp�ime.uni amp.br

Diana T. Amaro

Instituto Federal de Edu ação, Ciên ia e Te nologia de São Paulo - Bragança Paulista

dianatamaro�ifsp.edu.br

Des rição e Objetivos

Esta o� ina dedi a-se à formulação de metodologias de análise omparativa de livros didáti os de

Matemáti a. A proposta baseia-se na experiên ia da dis iplina de Análise de Livros e Material Didáti o

de Matemáti a [1℄[2℄, estruturada e ministrada pelo primeiro proponente desde 2014 para o urso de

Li en iatura em Matemáti a (Noturno) do Instituto de Matemáti a, Estatísti a e Computação Cientí� a

(IMECC) da Universidade Estadual de Campinas (Uni amp).

O termo análise horizontal foi unhado durante a práti a da dis iplina, para designar a omparação

entre dois ou mais livros quanto a sua abordagem de um mesmo onteúdo. Trata-se da elaboração e

apli ação de uma grade de avaliação que permita a es olha do melhor, simulando a ne essidade pro�ssional

on reta de arbitrar entre opções determinadas pelas ir unstân ias, a despeito dos méritos propedêuti os

de ada livro em si.

Espera-se ontribuir para um engajamento estruturado na es olha dos livros didáti os, ofere endo um

instrumento de exame, argumentação e protagonismo. A o� ina é voltada a até dezesseis (16) professores

de Matemáti a ou li en iandos em anos �nais, não havendo pré-requisitos teóri os.

Atividades

Parte I (30min) Revisaremos de forma su inta as etapas e ritérios do PNLD. Em seguida, apresentaremos

para dis ussão os prin ípios da análise horizontal, utilizando omo exemplos os materiais produzidos pelas

turmas da dis iplina na Uni amp entre 2014 e 2016 [1℄[2℄.

Parte II (60min) Os parti ipantes serão divididos em grupos de 4 integrantes para omparar um par de

livros aprovados no PNLD em anos re entes, disponibilizados pelos organizadores, a er a de um mesmo

onteúdo. Os grupos deverão formular uma metodologia prévia e uma divisão de tarefas.

Parte III (30min) Os grupos apresentarão oralmente sua on lusão.

Justi� ativa e fundamentação teóri a

O Programa Na ional do Livro Didáti o [3℄ postula que o livro dialoga om o professor e om o aluno

e trans ende o desenvolvimento de ompetên ias deste, para o upar lugar entral no trabalho daquele.

Desta am-se o balizamento da ementa e a preparação das aulas, tanto na explanação dos onteúdos

quanto na formulação de atividades, exer í ios e avaliações. No nosso ontexto so ial, não-raro o livro é

também meio de formação omplementar do próprio do ente.

Os editais do PNLD almejam prover livros-texto e a ervos relevantes a todas as es olas públi as. As

resenhas dos títulos aprovados pelo MEC integram o Guia do Livro Didáti o (GLD), a partir do qual

ada es ola elege os livros segundo suas próprias premissas pedagógi as. O PNLD re omenda que a

es olha envolva os professores, e pesquisas apontam que, de fato, os do entes elen am as três opções

om base no GLD e no material de divulgação, quando esta hega até as es olas públi as. Contudo,

Zambon et al (2013) e Tolentino Neto (2003) apontam que a demora das editoras no envio do material

de divulgação, o des onhe imento do GLD e dos pro edimentos do PNLD, a sobre arga de trabalho dos

do entes e ausên ia de um momento dedi ado à es olha dos livros didáti os omprometem o pro esso.

Nestas ondições, não há uniformidade de ritérios e a de isão dá-se uni amente em função da in�uên ia

das editoras e da subjetividade dos envolvidos.

85

Análise Horizontal

A omparação entre dois livros a er a de um mesmo onteúdo visa a es olher o melhor. A primeira

distinção dá-se entre os itens de onteúdo presentes em ambos, ditos onteúdos pares, e aqueles que

o orrem em apenas um dos textos, os onteúdos díspares. A metodologia de análise deve estruturarse

em torno destes eixos, prevendo duas grades de valoração. Sobre os onteúdos pares deve-se pontuar

a estruturação editorial do apítulo, ênfases e estratégias de abordagem, rigor matemáti o e exer í ios,

estes quanto à quantidade, adequação pedagógi a e distribuição no texto. Para os onteúdos díspares,

pondera-se apenas a sua relevân ia, ou seja, é veri� ado se ada item é essen ial ou ao menos interessante

para o aprofundamento do onteúdo. É importante desta ar que a análise horizontal não visa ao exame

ríti o do texto em si, apenas à es olha pragmáti a � sob ritérios previamente estabele idos � entre as

opções disponíveis. [2℄

Sobre os proponentes

Diana T. Amaro é li en iada em Matemáti a (2014) e aluna do Mestrado Pro�ssional em Matemáti a

(PROFMAT) na Uni amp. Atua, desde 2014, omo do ente de matemáti a nos Cursos Té ni os Integra-

dos ao Ensino Médio no IFSP � Bragança Paulista.

Henrique N. Sá Earp é ba harel (2001) e mestre (2002) em Matemáti a pela UFRJ, mestre em Físi a

Teóri a pela Universidade de Cambridge (2004) e doutor em Matemáti a pelo Imperial College London.

É Professor Adjunto II do IMECC-Uni amp.

Referên ias

1. Página da dis iplina: http://www.ime.uni amp.br/ ma225/Apresenta ao.htm.

2. Manual da dis iplina: http://www.ime.uni amp.br/ ma225/manual_da_dis iplina.pdf

3. http://www.fnde.gov.br/programas/livro-didati o/livro-didati o-histori o. [04/11/2016℄.

4. PNLD 2017: guia de livros didáti os � ensino fundamental, anos �nais, MEC, 2015.

http://www.fnde.gov.br/programas/livro-didati o/guias-do-pnld/es olha-pnld-2017. [04/11/2016℄.

5. TOLENTINO NETO, Luiz C. B. de. O pro esso de es olha do livro didáti o de Ciên ias por

professores de 1

a. a 4

a. séries, Dissertação de Mestrado, Fa uldade de Edu ação, USP, 2003.

6. ZAMBON, L. B.; TERRAZZAN, E. A. Políti as de material didáti o no Brasil: organização dos

pro essos de es olha de livros didáti os em es olas públi as de edu ação bási a. Revista Brasileira

de Estudos Pedagógi os, v. 94, n. 237, p. 585-602, 2013.

http://rbep.inep.gov.br/index.php/rbep/arti le/view/379 [04/11/2016℄.

WORKSHOP FOR TEACHERS F

LIÇÕES DE MATEMÁTICA NUM JORNAL DA CIDADE DO PORTO (PORTUGAL) EM

1853: UM EPISÓDIO PECULIAR NA FORMAÇ�O DE PROFESSORES

Hélder Pinto

CIDMA � UA, Portugal

hbmpinto1981�gmail. om

A Asso iação Industrial Portuense foi fundada em 1849 na idade do Porto (Portugal). A edu ação

industrial foi uma das suas primeiras prioridades tendo esta asso iação riado a Es ola Industrial do Porto

em 1852. Em 1853, António Luís Soares, professor desta es ola e da A ademia Polité ni a do Porto,

86

publi ou várias lições de aritméti a no jornal da asso iação � Da exposição dos elementos da arithmeti a

na aula de instru ção primaria da Asso iação. Estas lições eram dirigidas a futuros professores primários

que deveriam reproduzi-las aos seus alunos por diversas lo alidades do norte de Portugal. O autor in lui

várias onsiderações sobre a importân ia de propagar a instrução bási a de matemáti a, quer para os

operários fabris quer para os trabalhadores do omér io, duas atividades importantes para a idade do

Porto nessa épo a. Observe-se a tabela a seguir onde se apresenta a estrutura destas lições (tabela 1).

Note-se que António Luís Soares (Porto, 1805-1875) foi professor da A ademia Polité ni a do Porto

desde 1836 (Primeira adeira: Aritméti a, Geometria Elementar, Trigonometria e Álgebra elementar) e

professor da Es ola Industrial do Porto desde 1852 (Aritméti a, Álgebra e Geometria). Existem pou os

dados disponíveis sobre este professor e, ex eto o texto que aqui se apresenta, existe apenas uma referên ia

(Carvalho, pp. 9-10) para um outro texto (Exposição dos elementos de Aritméti a para uso dos estudantes

do Colégio de Santa Bárbara na idade de Pelotas, Pelotas, Brasil, 1849), um livro de aritméti a publi ado

no Brasil (não nos foi possível en ontrar nenhuma ópia desse texto). S ipião de Carvalho a�rma ainda

que António Luís Soares esteve no Brasil entre 1847 e 1851, mas não indi a quais os motivos dessa sua

estadia.

Neste workshop vamos fazer uma leitura a ompanhada destas lições de aritméti a realçando as suas

prin ipais ara terísti as. Uma das ara terísti as mais pe uliares deste texto é o fa to de dar um grande

relevo às unidades de medida (peso, omprimento, áreas, volumes, et .) bem omo às suas apli ações

ao omér io e à indústria. Neste ponto faz-se a apologia do novo sistema métri o e das suas múltiplas

vantagens em relação às antigas unidades portuguesas. Por outro lado, é também interessante o modo

omo este texto foi es rito: em formato normal apare ia o que deveria ser reproduzido aos alunos e em

formato itáli o as indi ações pedagógi as destinadas apenas aos professores que iriam reproduzir estas

aulas.

Note-se que é bastante pe uliar o fa to de estas aulas se terem publi ado num jornal e não, por

exemplo, num livro a ser adquirido apenas pelos interessados, o que pare e ser indi ativo da intenção de

propagar estes onhe imentos bási os de matemáti a por diversos públi os-alvo.

87

Neste workshop iremos então analisar om mais pormenor estas aulas bem omo as suas poten ial-

idades, quer históri as quer didáti as, realçando ainda algumas onsequên ias do fa to de se tratar de

aulas promovidas por uma asso iação industrial (des reveremos também om um pou o mais de pormenor

esta asso iação).

Agrade imento: Este trabalho foi �nan iado pelo CIDMA � Centro de Investigação e Desenvolvimento

em Matemáti a e Apli ações e pela FCT - Fundação para a Ciên ia e Te nologia, no âmbito do projeto

UID/MAT/04106/2013.

Referên ias

Alves, Jorge Fernandes. 1996. �O emergir das asso iações industriais no Porto (meados do sé ulo

XIX)�. In Análise So ial, vol. 31, n.

o. 136/137, pp. 527-544. Lisboa: Instituto de Ciên ias So iais.

Alves, Luís Alberto Marques. 2006. �ISEP: Identidade de uma Es ola om Raízes Oito entistas�.

In Sísifo, Revista de Ciên ias da Edu ação, n.

o. 1, pp. 57-70. Lisboa: Universidade de Lisboa.

Carvalho, S ipião. 1937. �A Matemáti a na A ademia Polité ni a do P�rto�. In O Ensino na

A ademia Polité ni a, pp. 1-31. Porto: Universidade do Porto.

Fernandes, Rogério. 1993. �Génese e Consolidação do Sistema Edu ativo Na ional (1820-1910)�.

In O Sistema de Ensino em Portugal, Sé ulos XIX-XX ( oord. Maria Cândida Proença), pp. 23-46.

Lisboa: Edições Colibri. Jornal da Asso iação Industrial Portuense. 1853.

Pinto, Hélder. 2013. A Matemáti a na A ademia Polité ni a do Porto. Tese de Doutoramento.

Lisboa: Universidade de Lisboa.

Serra, António Dias da Costa. 1989. História do Instituto Industrial do Porto onvertido no Instituto

Superior de Engenharia do Porto, em 1974. Porto.

Torgal, Luís Reis. 1993. �A Instrução Públi a�. In História de Portugal, o Liberalismo (1807-1890),

Vol. V (dir. José Mattoso), pp. 609-651. Lisboa: Cír ulo de Leitores.

WORKSHOP FOR TEACHERS G

ANÁLISE DE LIVROS DIDÁTICOS BRASILEIROS

Andréia Dal in Luisa Rodríguez Doering Cydara Cavedon Ripoll

Universidade Federal do Rio Grande do Sul

andreia.dal in�ufrgs.br ldoering�mat.ufrgs.br ydara�mat.ufrgs.br

A análise de livros didáti os por parte de professores da Edu ação Bási a vem se onstituindo omo uma

práti a relativamente re ente no Brasil. Embora as políti as que regularizam a es olha e oferta de livros

didáti os para estudantes das es olas públi as tenham origem ainda em 1929, é somente a partir de 1996

que é ini iado o pro esso de avaliação pedagógi a dos livros ins ritos para o Programa Na ional do Livro

Didáti o (PNLD), om a publi ação do primeiro �Guia de Livros Didáti os" de 1

a. a 4

a. séries, que foi

ampliado para as demais séries nos anos subsequentes.

Segundo o Ministério da Edu ação e Cultura do Brasil, os livros foram avaliados seguindo ritérios

previamente dis utidos, e esse pro edimento foi sendo aperfeiçoado e é apli ado até hoje. Os livros que

apresentam erros on eituais, desatualizações, pre on eito ou dis riminação de qualquer tipo ou, ainda,

que ontêm frases ou en aminhamentos que induzem a erros, são ex luídos do Guia do Livro Didáti o.

Além dessa avaliação do governo, é também importante que os professores das es olas públi as pratiquem

a análise de livros didáti os, pois estes, além de serem onsultados no pro esso de es olha de quais livros

aprovados serão distribuídos gratuitamente nas suas es olas, tal análise deve levar em onta também

outros itens não ontemplados no Guia. Nesta perspe tiva é que propomos esta o� ina, destinada aos

88

professores brasileiros, objetivando apresentar e exer itar elementos de análise que possam ontribuir para

uma melhor quali� ação do professor e aprimorar o pro esso de análise e de es olha dos livros didáti os

de matemáti a a serem utilizados nas salas de aula brasileiras.

Esta O� ina será ministrada em português, om duração de 120 minutos, os quais serão distribuídos

em três momentos.

No primeiro momento (20 minutos), pretende-se apresentar e onven ionar uma lista de itens para

serem onsiderados pelo grupo na análise que será prati ada durante a O� ina; tais ítens deverão in luir:

análise do onteúdo, análise da linguagem matemáti a, análise das imagens, bem omo suas interrelações.

Nesse momento, serão in luídos exemplos para bem ilustrar ada ítem.

No segundo momento (40 minutos), pretende-se propor ao grupo 2 ex ertos om material para ser

analisado, preferivelmente em pequenos grupos (de 3 professores), um versando sobre divisão eu lidiana

nos anos in iais e outro versando sobre expressões algébri as no segundo segmento do Ensino Fundamental.

O ter eiro momento (60 minutos), será destinado à apresentação dos trabalhos realizados por ada

grupo, a ompanhada de uma dis ussão oletiva sobre as análises realizadas. Como fe hamento pretende-se

dis utir a lista de ítens onven ionada no primeiro momento, oportunizando seu aprimoramento.

Referên ias

Dal in, Andréia. 2002. �Um olhar sobre o paradidáti o de Matemáti a". Dissertação de Mestrado.

UNICAMP: Fa uldade de Edu ação. Campinas, São Paulo.

Ministério da Edu ação. 1998.�"Plano Na ional do Livro Didáti o". Brasília: MEC. Brasil. A esso:

11 de Novembro de 2016. http://portal.me .gov.br/seb/arquivos/pdf/livro03.pdf.

Ripoll, Cydara C., �Mal ditas frases en ontradas em livros didáti os de Matemáti a para a Es ola

Bási a". A esso: 25 de Outubro de 2016. www.mat.ufrgs.br/∼ ydara/mal_ditas.pdf.

Trevisan, Andréia. 2014. �O que as imagens dos livros didáti os de matemáti a nos dizem sobre

multi ulturalismo?" Edu ação Matemáti a em revista. v. 16, n.2.

Em: http://revistas.pu sp.br/index.php/emp/arti le/view/16673. A esso: 11 de Novembro de

2016.

WORKSHOP FOR TEACHERS H

OS LIVROS DIDÁTICOS DE MATEMÁTICA: UM OLHAR DECOLONIAL SOBRE OS

CONHECIMENTOS, IMAGENS E PERCEPÇÕES SOBRE OS POVOS AFRICANOS E

AFRO-BRASILEIROS

Maysa Ferreira da Silva Wilker Solidade da Silva

Universidade Federal do Paraná

mayfsil�hotmail. om mayfsil�hotmail. om

O Livro Didáti o (LD), desde sua implantação e expansão omo políti a de governo em 1938, se tornou

gradativamente a ferramenta mais utilizada pelos professores brasileiros para dire ionar o desenvolvimento

do onteúdo em sala de aula e além dela. Consequentemente, o LD se torna, om a ampliação do ensino

institu ionalizado públi o no Brasil, um me anismo possível de engendramento de estratégias ideológi as

erigidas pelo Estado, arregando as pretensões e interesses de um determinado momento históri o.

Do umentos o� iais que onstituem a dis iplina es olar de Matemáti a histori amente trazem em

seu bojo o dis urso da neutralidade políti a e so ial, re�etindo nos materiais e LD uma perpetuação

de um modelo hegem�ni o de onhe imento. Quando questionamos essa estrutura, prin ipalmente num

viés étni o-ra ial, é possível ompreender omo tal on epção tem se onstituído em um provo ador de

diferenças so iais, naturalizadas e fortale idas. O não expor a existên ia de onhe imento para além da

89

interpretação dada pela herança euro êntri a fortale e o que Silva (2012, p.113) onsidera omo estratégia

de dis urso �que atua ativamente para onstruir sentidos" e perpetuar posições de poder na o ultação de

desigualdades.

Por ser o LD uma mídia, estando na formação dos novos idadãos o efeito prin ipal de sua dimensão,

as relações entre ideologias e a onstrução so ial se entrelaçam na estrutura que ompreende a edição e

utilização do LD no espaço es olar. A metodologia empregada parte dos estudos de John B. Thompson

(2011) sobre o uso da Hermenêuti a de Profundidade (HP), uma das teorias de referên ia desse autor para

ompreensão e interpretação de ideologias ontidas nas produções midiáti as. Os modos de operação e

estratégias da ideologia são base para a re�exão da interação entre sentido e poder na vida so ial, omo

também fundamentam a orientação da HP omo método de pesquisa. Para Thompson (2011) estas

podem ser delineadas em três fases distintas, e orientam os aminhos para se interpretar o objeto: �Só io

Históri o", �Formal-Des ritivo" e �Interpretação/Reinterpretação".

A partir da análise por esse viés propomos debater as relações da onstrução do LD om a estrutura

edu a ional a que este se remete, bem omo a estrutura formal que dire iona o que é ou não fundamental

para um LD de Matemáti a que não privilegie uma úni a fonte de onhe imento. Nesta perspe tiva,

adotando uma postura de olonial para se ompreender a � olonialidade do saber" (QUIJANO, 2005)

presente nos LD é que orientamos uma ríti a sobre a presença/ausên ia das relações étni o-ra iais no

LD e, através do diálogo om autores da área (D' AMBROSIO, 2010; KNIJNIK, 2010) omo isso pode,

ou não, ser solu ionado através de uma difusão dos ideais da etnomatemáti a no ensino institu ionalizado

brasileiro e, prin ipalmente, na onstrução de uma proposta de inserção onteudista nos novos LD.

Realizando uma análise expositiva em uma oleção de LD de Matemáti a (LDM) das séries �nais do

ensino fundamental brasileiro, pro essos de oordenação e avaliação do uso de imagem e onteúdos no

LDM es olhido, o diálogo deste om as políti as edu a ionais em prol a valorização ultural brasileira

e a implementação das Diretrizes Curri ulares Na ionais para a Edu ação das Relações Étni o-ra iais e

para o ensino de História e Cultura Afro-brasileira e Afri ana, dis utindo a ne essidade de uma re�exão

de olonial do urrí ulo e, onsequentemente, do livro didáti o brasileiro de Matemáti a.

Referên ias

D' AMBROSIO, Ubiratan. 2010. Etnomatemáti a e Edu ação. In: Knijnik, G; Wanderer, F

e Oliveira, C. J. (Orgs): Etnomatemáti a, urrí ulo e formação de professores. Santa Cruz dos

Sul/RS: EDUNISC.

D' AMBROSIO, Ubiratan. 2005. Etnomatemáti a � elo entre as tradições e a modernidade. Belo

Horizonte: Autênti a.

KNIJNIK, Gelsa. 2010. Itinerários da Etnomatemáti a: questões e desa�os sobre o ultural, o so ial

e o políti o na edu ação matemáti a. In: Knijnik, G; Wanderer, F e Oliveira, C. J. (Orgs). 2010.

Etnomatemáti a, urrí ulo e formação de professores. Santa Cruz dos Sul/RS: EDUNISC.

QUIJANO, Aníbal. 2005. Colonialidade do poder, euro entrismo e Améri a Latina. In: LANDER,

Edgardo. (Org.). 2005. A olonialidade do saber: euro entrismo e iên ias so iais. 1

a. ed. BuenosAires: Consejo Latinoameri ano de Ciên ias So iales - CLACSO.

SILVA, Ana Célia da. 2011. A representação do negro no livro didáti o: o que mudou? Salvador:

EDUFBA.

THOMPSON, John B. 2011. Ideologia e ultura moderna: teoria so ial ríti a na era dos meios de

omuni ação de massa. 9

a. ed. Rio de Janeiro: Vozes.

SILVA, Paulo Vini ius Baptista. 2008. Ra ismos em livros didáti os: estudo sobre negros e bran os

em livros de Língua Portuguesa. Belo Horizonte: Autênti a Editora.

90

WORKSHOP FOR TEACHERS I

ALTERNATIVAS COMPLEMENTARES AO LIVRO DIDÁTICO: DESENVOLVIMENTO

DE MATERIAIS PARADIDÁTICOS

Claudiomir Feustler Rodrigues de Siqueira

IFRS � Campus Canoas

laudiomir.siqueira� anoas.ifrs.edu.br

A onstrução de materiais didáti os omplementares ao livro didáti o é uma forma de desenvolvimento

do pro esso pedagógi o, possibilitando muitas vezes uma proposta didáti a mais estruturada dentro da

realidade dos alunos, bem omo a possibilidade de ontemplação de temas não explorados e o aprofunda-

mento on eitual de determinados onteúdos. Nesse sentido os materiais paradidáti os - livros, histórias

em quadrinhos, entre outros - onsistem numa alternativa na bus a de diferentes formas para despertar o

interesse e gosto pelo estudo de matemáti a e para a aprendizagem interdis iplinar (Furlan, 2002).

Os livros paradidáti os são materiais omplementares ao livro didáti o que podem ser impressos, digitais

ou disponibilizados rela ionando as duas formas. Apresentam aspe tos mais lúdi os do que o didáti o,

por isso são julgados mais e� ientes sob o ponto de vista pedagógi o (Menezes & Santos, 2001).

Nesses materiais riam-se enários para a investigação. Ao mesmo tempo, em que se desenvolve

uma estratégia para levar o leitor a motivar-se a seguir na leitura, se estabele e um roteiro orientado na

ompreensão dos on eitos/objetivos desta história estimulando a leitura, a es rita e a riatividade.

Desta a-se que a formação do leitor omeça bem edo, on�gurando o hábito pela leitura uma das

primeiras formas do livro didáti o �dar erto" (Laguna, 2012). No ensino bási o, na maioria das es olas,

obras além do livro didáti o, voltadas para a matemáti a são es assas. Nesse ontexto, podemos itar

omo exemplos mais onhe idos os livros �O homem que al ulava" - Malba Tahan e �O diabo dos

números" - Hans Magnus Ensensberger e o �lme: �Donald no país da matemági a", que muito pou os

alunos/professores onhe em.

Como forma de ampliar/estimular a aprendizagem matemáti a, esta o� ina tem por propósito dis utir

e apresentar a proposta didáti a da onstrução de histórias paradidáti as. Bem omo sua inserção nos

ambientes es olares omo forma de ensino e aprendizagem matemáti a. Ressaltando que ao entrelaçar-

mos uma história e um onteúdo matemáti o, vamos além da simples resolução de um exer í io ou do

domínio do algoritmo, possibilitamos trabalhar interdis iplinarmente na bus a de um ambiente de ensino

e aprendizagem mais fértil.

Referên ias

Furlan, S. A. 2002. A Geogra�a na sala de aula: a importân ia dos materiais didáti os. Brasília:

MEC/Seed.

Laguna, A. G. J. 2012. A ontribuição do livro paradidáti o na formação do aluno-leitor. Augusto

Guzzo Revista A adêmi a, 2: 43-52.

Menezes, E. T. e Santos, T. H. 2001. Verbete paradidáti os. Di ionário Interativo da Edu ação

Brasileira � Edu abrasil. São Paulo: Midiamix, 2001.

Disponível em: http://www.edu abrasil. om.br/paradidati os/. A esso em: 29 de nov. 2016.

91

WORKSHOP FOR TEACHERS J

GUIA DIGITAL DO PNLD: METARECURSO PARA A ESCOLHA DO LIVRO DIDÁTICO

Ri ardo Tibúr io, Rogério Igná io, Fran k Bellemain, Rosilângela Lu ena, Ver�ni a Gitirana

Universidade Federal de Pernambu o

ri o.tibur io�gmail. om, rogerioigna io�gmail. om, f.bellemain�gmail. om,

rosi.lu enas �gmail. om, veroni a.gitirana�gmail. om

Ao re onhe ermos a abrangên ia do uso de livros didáti os no apoio às atividades do entes, onvém

desta ar algumas de suas possíveis funções, dentre as quais: �auxiliar no planejamento didáti o-pedagógi o

anual e na gestão das aulas; favore er a formação didáti o-pedagógi a; auxiliar na avaliação da apren-

dizagem do aluno; favore er a aquisição de saberes pro�ssionais pertinentes, assumindo o papel de texto

de referên ia" (CARVALHO & LIMA, 2010). Portanto, a es olha e o uso do livro didáti o deve envolver,

dentre outros aspe tos, a a�nidade que o professor re onhe e entre a proposta apresentada e as suas

es olhas didáti o-pedagógi as além da sua utilidade para ampliar os onhe imentos que possui em sua

área de atuação.

Essa o� ina parte do trabalho desenvolvido pela equipe de avaliação do livro didáti o de Matemáti a

do MEC - Ministério da Edu ação e Cultura, bus ando dis utir a importân ia do Guia Digital do Livro

Didáti o omo metare urso destinado a orientar professores quanto à es olha das obras a serem utilizadas

em suas aulas. Ao onsiderar o termo metare urso per ebe-se a função do Guia Digital do PNLD 2016 e

2017 (Brazil, 2015; 2016) no sentido de ser um ambiente que possibilita a re�exão para a es olha, sobre

re urso, no aso o livro didáti o, que o professor utiliza em sua práti a do ente. Diversas pesquisas têm

se preo upado em analisar o trabalho de do umentação do professor, ompreendendo que esse pro�ssional

tem a responsabilidade de pro urar, sele ionar, adaptar e riar re ursos para sua práti a do ente (ROCHA

& TROUCHE, 2016; LUCENA & ASSIS, 2016). Segundo tal onstrução teóri a, o professor sele iona

re ursos e desenvolve esquemas de ação instrumentada, para a sua utilização em práti a, e vai pou o a

pou o ompondo sistemas de re ursos. Nesse pro esso, outros re ursos são utilizados pelo professor omo

auxiliar em tais es olhas e de isões, denominados de meta-re urso. O Guia do Livro Didáti o, impresso e

o digital, ompõem-se em um metare urso para o professor. Com isso, objetiva-se nesta o� ina apresentar

o Programa Na ional do Livro Didáti o, a omposição do Guia Digital, as razões que levam a ompor as

resenhas e omo as informações ontidas nelas podem ser a essadas através do Guia Digital. Os elementos

onstitutivos das resenhas e os relativos aos textos gerais que ompõem o Guia serão dis utidos à luz

de sua função informativa, mas também de outras funções que o Guia vem assumindo, tal omo a de

suporte ao uso do livro didáti o. Além disso, propomos dis utir a abordagem metado umental no intuito

de bus ar elementos sistemáti os que permitam melhor ompreender as práti as dos professores, por meio

da investigação-re�exiva sobre os re ursos que utiliza na práti a do ente e os metare ursos que utiliza

para a es olha do livro didáti o. Com base em tais dis ussões, os parti ipantes produzirão elementos que

expli item as suas práti as, o que fomentará as dis ussões no grande grupo. En erraremos a o� ina om

a so ialização de suas onstruções e om a dis ussão do que fora produzido.

Desse modo, � am estabele idos os seguintes momentos da o� ina: 1. Apresentação do pro esso

de avaliação do livro didáti o, do Guia digital e es olha pelo Guia (metare urso) - serão apresentadas

informações sobre a omposição do guia, os ritérios de avaliação, as � has de avaliação e omo as

informações são disponibilizadas no guia digital; 2. Instrumentalização dos ursistas om o Guia digital

- Pretende-se apresentar o Guia Digital, bem omo seus re ursos e possibilidades no sentido de orientar

na es olha das obras; 3. Apresentação da Abordagem Metado umental - Objetiva-se ompreender alguns

elementos da Abordagem Do umental e o que signi� a metare urso; 4. Dinâmi a: organização de grupos

- Esse momento tem por �nalidade onstruir e apresentar o mapa de re ursos pelos parti ipantes; 5.

En erramento - Momento destinado a avaliar a o� ina.

92

Referên ias

Brazil. Guia Digital do PNLD 2016. Brasília: MEC-SEB, 2015.

Brazil. Guia Digital do PNLD 2017. Brasília: MEC-SEB, 2016.

Carvalho, João. B. P. D, and Paulo F. Lima. 2010. �Es olha e uso do livro didáti o". In: Carvalho,

J.B.P.D. Coleção Explorando o Ensino. Brasília: Ministério da Edu ação, Se retaria de Edu ação

Bási a 17, 15- 30 .

Lu ena, Rosilângela, and Cibelle C. Assis. 2015. �Webdo Sistema de Re ursos e o Trabalho Coletivo

do Professor: Uma Via de Mão Dupla". Disponível em: http://lemate .net.br/webdo s/webdo 2/

( ódigo de a esso a ser soli itado aos autores).

Ro ha, Katiane, and Lu Trou he. 2016. �Da produção oletiva de livros didáti os digitais aos

usos feitos professores de Matemáti a: o aso do grupo fran ês Sésamath." EM TEIA - Revista de

Edu ação Matemáti a e Te nológi a Iberoameri ana. Re ife: Universidade Federal de Pernambu o

6, 1-22.

WORKSHOP FOR TEACHERS K

UMA ABORDAGEM DO CONCEITO DE HIPÉRBOLE COM BASE NA ANÁLISE DE

LIVROS DIDÁTICOS DO ENSINO MÉDIO

Nora Olinda Cabrera Zúñiga

Colégio Té ni o da Universidade Federal de Minas Gerais

no z�ufmg.br

Mariana Lima Vilela

Universidade Federal de Minas Gerais

marianalima�mat.grad.ufmg.br

Nayara Katherine Duarte Pinto

Universidade Federal de Minas Gerais

nayarak�ufmg.br

A proposta desta o� ina para professores é um desdobramento de uma experiên ia realizada om turmas

do ter eiro ano do ensino médio do Colégio Té ni o da Universidade Federal de Minas Gerais (Colte da

UFMG), no ano de 2015. Uma das autoras deste trabalho, que é professora efetiva de Matemáti a da

instituição, em onjunto om três estagiárias do urso de li en iatura em Matemáti a da UFMG, plane-

jaram e desenvolveram uma aula expositiva dialogada para introduzir e explorar o on eito de hipérbole

em sala de aula.

Para essa aula, um dos aspe tos ini ialmente onsiderados foi o da análise do onteúdo proposto no

livro didáti o adotado pelo Colte . Essa análise nos levou a per eber a ne essidade de adequarmos e

omplementarmos esse onteúdo, visando introduzirmos o on eito de hipérbole de maneira ompreensiva

aos alunos do ensino médio. Ressaltamos que o livro didáti o serviu de suporte na preparação e exe ução

da aula pelas do entes e de referên ia para os alunos.

Tendo em vista os retornos positivos que tivemos dos alunos e de do entes que presen iaram o relato de

experiên ia de Pinto et al. (2016), propomos essa o� ina, almejando familiarizar os professores do ensino

médio om uma abordagem em que se prioriza ompreender o on eito de hipérbole, pou o explorado

em livros didáti os desse nível de ensino. Como esperamos mostrar ao longo da o� ina, o uso riterioso

de material manipulável pode propi iar aos parti ipantes uma experiên ia na qual o on eito de hipérbole

seja uidadosamente desenvolvido.

Ini ialmente indagaremos, em diálogo om os professores parti ipantes, a respeito tanto da aprendiza-

gem das �ni as no tempo do ensino médio e da Graduação, quanto da do ên ia na es ola bási a. A

seguir, des reveremos e analisaremos abordagens do on eito de hipérbole propostas em diferentes livros

didáti os. Tentaremos indi ar al an es e possibilidades, em relação ao on eito almejado, sugeridos no

93

livro do aluno e nas orientações metodológi as no manual do professor. Na sequên ia, ministraremos

uma atividade para introduzir o on eito de hipérbole utilizando material manipulável, tomando omo

referên ia o trabalho de Pinto et al. (2016). Finalmente, realizaremos uma dis ussão em plenária.

Esperamos propi iar, mediante essa o� ina, uma re�exão pelos professores a respeito das suas práti as

pedagógi as e da importân ia de eles analisarem os livros didáti os utilizados na instituição de ensino e

motivá-los a adequarem abordagens de on eitos matemáti os propostas nos livros.

Referên ias

Ball, Deborah L., Thames, Mark H., and Phelps, Geo�rey. 2008. �Content Knowledge for Tea hing:

What Makes It Spe ial?� Journal of Tea her Edu ation 59: 389-407.

Brasil, 2014. �Guia de livros didáti os: PNLD 2015: matemáti a: ensino médio.� � Brasília: Min-

istério da Edu ação, Se retaria de Edu ação Bási a.

A essed November 09, 2016. http://www.fnde.gov.br/programas/livro-didati o/guias-dopnld/ item/5940-

guia-pnld-2015.

Pinto, Nayara K. D., Vilela, Mariana L., Santos, Fernanda G., and Zúñiga, Nora O. C. 2016. �Re-

�exões sobre a abordagem do on eito de hipérbole no ensino médio�. Experien e report presented at

the XII En ontro Na ional de Edu ação Matemáti a, São Paulo, São Paulo, July 13-16. A essed

November 09, 2016. http://sbempe. panel0179.hospedagemdesites.ws/enem2016/anais/autores-

N.html.

Shulman, Lee S. 1986. �Those who understand: knowledge growth in tea hing.� Edu ational

Resear her 15 (2): 4-14. A essed November 09, 2016. http://www.jstor.org/stable/1175860.

Sönnerhed, Wang W. 2011. Mathemati s textbooks for tea hing: An analysis of ontent knowledge

and pedagogi al ontent knowledge on erning algebra in mathemati s textbooks in Swedish upper

se ondary edu ation. Li entiate Thesis in dida ti s, in the framework of the Graduate S hool CUL.

Göteborgs Universitet: Inst för Pedagogik, Kommunikation o h Lärande. A essed November 13,

2016. https://gupea.ub.gu.se/bitstream/2077/27935/1/gupea_2077_27935_1.pdf.

94

Oral Communi ations

(in alphabeti al order of �rst author)

A SEMIOTIC INTERPRETATION OF THE CONCEPT OF DERIVATIVE IN A

TEXTBOOK

1

Lourdes Maria Werle de Almeida

State University of Londrina

lourdes�uel.br

Karina Alessandra Pessoa da Silva

Federal Te hnologi al University of Paraná

karinasilva�utfpr.edu.br

Considering the importan e of symboli language for on eptualization in mathemati s, our paper aims to

present a semioti analysis of the approa h related to the mathemati al obje t `derivative' in a textbook

on Integral and Di�erential Cal ulus. Our arguments with respe t to the derivative approa h are based

on the assumptions of Peir e�s semioti s theory (Peir e, 1972; Peir e, 1989; Peir e, 2005).

In this paper we onsider one of the Pier e�s tri hotomies, spe i� ally the one whi h des ribes the

signs from the point of view of the sign and dynami obje t relationship. In this relation the sign is

asso iated with the obje t via index, symbol or i on. Thus, a sign is not only a stru ture, but is also a

pro ess. In this ontext, a ording to Otte (2008), the symbol is the pro ess of the ir ular intera tion

between indexi al and i oni elements, realized as an endless pro ess of interpretation.

In the edu ational ontext, the on eptualization of the mathemati al obje t, in general, is an hored in

its presentation in the textbook. In this regard, the analysis of the book may indi ate how the mathemati al

obje t approa h ontemplates that sign relationship. That is, how the signs used by the authors an a t

as an i on, index or a symbol aiming at understanding the mathemati al obje t, parti ularly, the derivative

of a real fun tion of one independent variable.

To this end, we analyzed a textbook on Integral and Di�erential Cal ulus largely used, espe ially in

Brazilian universities. The results from this analysis indi ate that in this book the onstru tion of the

on ept of derivative omprises di�erent signs, and these signs may a t as i ons, indexes or symbols in

the student�s on eptual understanding of the obje t. In fa t, there are indi ations in the book that the

derivative symbol is the result of a pro ess whi h arti ulates indexes and i ons.

Referen es

Peir e, C. S. 1989. Es ritos Coligidos. São Paulo: Nova Cultural.

Peir e, C. S. 2005. Semióti a. 3. ed. São Paulo: Perspe tiva.

Peir e, C. S. 1972. Semióti a e Filoso�a: textos es olhidos. São Paulo: Cultrix.

Otte, M. 2008. The analyti /syntheti distin tion and Peir e's on eption of Mathemati s. In:

Semioti s and Philosophy in Charles Sanders Peir e. Ed. By: Fabbri hesi, R. and Marietti, S.

Cambridge S holars Publishing. 51-88.

TEXTBOOKS REVEALING THE DEVELOPMENT OF A CONCEPT - THE CASE OF

THE NUMBER LINE (1708-1829)

Mar ello Amadeo

Universidade do Estado do Rio de Janeiro (UERJ), Brazil

mar ello.amadeo�uerj.br

This work has the goal to analyse the development of a mathemati al on ept by means of textbooks.

Contrary to the usual historiography of mathemati s, where one reveals the emergen e of a new on ept

by some spe i� publi ation, say a resear h paper, the ase in question on erns a on ept where re ent

publi ations assert its unquestioned use over many enturies, but where no expli it re�e tions were pub-

lished by mathemati ians of former periods. The only viable a ess in su h a ase is the analysis of the

1

This resear h is related to a proje t funded by the CNPq.

97

pra ti e of mathemati ians and the best-adapted a ess is their pra ti e in dealing with the on ept in

textbooks. As a somewhat more expli it sour e of re�e tion, related entries in en y lopaedias have been

used.

The ase in question is the number line, whi h appears today as a natural, evident and unproblemati

notion. A loser analysis shows, however, that this notion is intimately onne ted to the on ept of

negative numbers; pra ti ing the number line reveals therefore ne essarily the ompli ated pro ess of

development and establishment of the on ept of negative numbers. An a ess to this pro ess is provided

by the analysis of mathemati s textbooks published during the eighteenth entury and the early nineteenth

entury, both at the se ondary level and the level of higher edu ation in various ountries. We make some

remarks about the di�eren es between this notion in the ontext of that period and this notion in the

present ontext, seeking to establish that a histori al reading of the development of a mathemati al notion

is needed and therefore the pra ti es of the time in whi h they were established should be respe ted. We

fo us on original sour es of Fren h and German mathemati s, whi h stand out in this period: the one

as the major mathemati al ommunity and the other as an emerging one. Our main result is that the

notion of number line, as we understand it as valid today, be ame established only in the �rst half of the

nineteenth entury, in the ontext of mathemati s tea hing at the se ondary level, as a result of a pro ess

of algebraization enabling a general on ept of negative numbers.

Referen es

Bézout, Étienne. 1770. Cours de mathématiques, à l'usage du orps royal de l'artillerie. Tome

Se ond, Contenant l'Algèbre & l'appli ation de l'Algèbre à la Géométrie. Paris: de l'Imprimerie

Royale.

Fis her, Ernst G. 1829. Lehrbu h der Elementar-Mathematik. Vierter Theil: Anfangsgründe; der

Algebra und der Lehre von den Kegels hnitten. Berlin/Leipzig: Nauk.

Prony, Gaspard Ri he de. 1795. �Cours d'Analyse appliquée à la mé anique.� In: Journal de l'É ole

Polyte hnique, an III (1795), tome 1, ahier 1, 92-119. Paris.

Reyneau, Charles-René. 1708. Analyse démontrée ou la Méthode de résoudre les problèmes des

mathématiques, et d'apprendre fa ilement es s ien es, tome 1, 1st ed. Paris: Quilliau.

STATISTICS EDUCATION IN TEXTBOOKS: BRAZIL'S NATIONAL TEXTBOOK

PROGRAM AND THE TEACHERS' MANUALS

Natália Amorim Gilda Guimarães


amorim_na�yahoo. om.br gilda.lguimaraes�gmail. om

This arti le is part of a dissertation resear h proje t whose main obje tive was to �nd eviden e of the

possible in�uen e of the Guide to Brazil's National Textbook Program (PNLD) on Mathemati s textbooks

used in the early years of primary s hool, spe i� ally in relation to tea hing statisti s.

The goal of Brazil's the Guide of PNLD, whi h began in 1996, is to evaluate, pur hase and distribute

textbooks. The Guide presents reviews of the olle tions, the result of an evaluation pro ess arried out by

tea hers in edu ational institutions all over Brazil. Its obje tive is to guide tea hers in hoosing textbooks.

Furthermore, it presents the riteria used in the evaluation and the guiding ompeten ies.

Sa ristán (1998) states that, when we de�ne urri ulum, we are des ribing the fun tions of the s hool

itself in a given histori al and so ial moment, its ontent and the ways in whi h said ontent is organized

and presented to tea hers and students. This author further proposes a model for urri ulum interpretation

organized in six developmental levels. In the study, our obje tive was to re�e t spe i� ally on the in�uen e

98

of the pres ribed urri ula (the PNLD Guides) and the urri ula presented to tea hers (tea hers' manuals

in textbook olle tions).

We analyzed the 5 (�ve) latest editions of the Guide to mathemati s textbooks, from 2004 and 4

(four) approved olle tions of mathemati s textbooks editions (2007 to 2016), for a total of 76 books.

Analyzing the obje tives proposed for the earliest years of primary s hool in the PNLD Guides on

the treatment of information (statisti s) axis, we found two de�ning moments. In the 2007 Guide, it is

stated that students need to re�e t on the fun tion of statisti s and not only on ertain types of graphi al

representations. It expresses a on ern that the data be real, onne ted to physi al or so ial reality,

with a resear h perspe tive, involving olle tion, sele tion, organization, representation, and inferential

analysis based on qualitative information or numeri al data. In the 2016 PNLD Guide, we observed a

se ond hange, whi h establishes resear h as the stru turing axis of statisti s edu ation, highlighting the

importan e of experien ing resear h and all its stages.

We observed that these obje tives are presented in the tea hers' manuals of all the olle tions analyzed

sin e 2007. However, analyses of the a tivities presented in the same olle tions are still being arried out

to see if, in fa t, the stated obje tives are borne out in the a tivities proposed. In the 2016 Guide, there's

an expansion of the ompeten ies to be developed in tea hing Statisti s, su h as re�e ting on a resear h

question, types of variables, types of graphs and tables, dire t numeri relation and s ales, measures of

entral tenden y, amplitude, and variability.

Referen es

SACRISTÀN, José. G. 1998. O urrí ulo: uma re�exão sobre a práti a. 3

a. Ed. Porto Alegre, RS:

Artmed.

TEXTBOOK FOR TEACHERS' LEARNING TO TEACH

Yuriko Yamamoto Baldin

Universidade Federal de São Carlos

yuriko�dm.ufs ar.br

Apare ida Fran is o da Silva

Universidade Estadual Júlio de Mesquita Filho

apare ida_fran is o57�hotmail. om

The aim of this proposal is to bring up the dis ussions about the ontemporary need of textbooks for

tea her edu ation ourses, fo used on tea hers� learning to tea h. We argue that along with the inves-

tigation of the textbooks for lassroom use and their manuals to tea hers, the advan es of mathemati s

edu ation resear h on the knowledge of tea hers to tea h imply the ne essity of textbooks for tea her

edu ation ourses. We dis uss the spe i� hara teristi s of books that are neither guiding manuals of

students� book, nor texts of re�e tions on edu ational theories and resear h arti les, but those that ould

support tea hers to study the pra ti es in lassrooms, grounded in theoreti al framework of Mathemat-

i s Knowledge for Tea hing (Ball, Bass, 2003), that en ompasses the on ept of Pedagogi al Content

Knowledge (Shulman, 1986). The ontent of su h book would omply with the on ept of Pra ti e-Based

Learning Task (Silver, 2009), orresponding to the dida ti al material meant to tea hers to understand

and to pra ti e e� iently the a tivities and resour es that are available nowadays in the literature, written

or in digital media. This proposal will present the book (Baldin, Silva, 2016), an out ome of a profes-

sional development program for mathemati s tea hers about �learning to tea h problem solving lessons

through inquiry, exploration, dis overy and extension", developed as pilot proje t within the proje t OB-

MEP (Brazilian Mathemati s Olympiad of Publi S hools, http://www.obmep.org.br). The methodology

of designing the ontent of the textbook follows losely the Lesson Study prin iples, and based in Mathe-

mati al Thinking skills in the lassrooms (Isoda, Katagiri, 2012). We onsider that the interest of tea her

edu ators that are re ognizing the book as potential tea hing material in tea her edu ation ourses is an

indi ation of the importan e to produ e textbooks for tea hers to learn how to tea h.

Referen es

99

Baldin, Yuriko Y and Silva, Apare ida F. 2016. OBMEP na Es ola, Resolução de Problemas na

Sala de Aula: uma proposta da OBMEP para apa itação de professores em estratégias de Ensino

da Matemáti a. Volume1. Rio de Janeiro, IMPA.

Ball, Deborah and Bass, Hyman. 2003. �Towards a pra ti e based theory of mathemati s knowledge

for tea hing". B. Davis and E. Simmt (Eds) Pro eedings of the 2002 annual meeting of the Canadian

Math. Edu ation Study Group, Edmonton AB

Isoda, Masami and Katagiri, Shigeo. 2012. Mathemati al Thinking: How to Develop it in the

Classroom. Singapore. World S ienti� .

Silver, Ed. 2009. �Toward a More Complete Understanding of Pra ti e-Based Professional Devel-

opment for Mathemati s Tea hers". R. Even and D. Ball (Eds) The Professional Edu ation and

Development of Tea hers. The 15th ICMI Study. New York: Springer.

Shulman, Lee. 1986. �Those Who Understand: Knowledge Growth in Tea hing". Edu ational

Resear her, Vol 5, n.2, pp 4-14.

AUTHORING ONLINE OPEN TEXTBOOKS WITH MATHBOOK XML

Robert A. Beezer

University of Puget Sound

beezer�pugetsound.edu

David Farmer

Ameri an Institute of Mathemati s

farmer�aimath.org

MathBook XML is an open sour e authoring system for textbooks that automati ally produ es both print

and online versions of a book from a single set of sour e �les. A key design prin iple is to rigorously

spe ify the stru ture of a book and then leanly separate the ontent from the presentation. On e an

author is provided the me hanism to do this, then it is possible for software to reate faithful versions as

print, PDF, HTML, EPUB, and other output formats, in luding those not yet imagined. As an extreme

example, onversion to Nemeth Braille has been su essful in an early prototype.

The web versions of MathBook XML textbooks o�er new opportunities for embedding intera tive

elements to assist students. Not only do these in lude standard elements like audio and video, but they

an also allow a reader to explore mathemati al ideas with tools like GeoGebra, JSXGraph, and Sage ells,

in addition to WeBWorK automated homework exer ises. There are also ex iting new ways to index,

ross-referen e, and link information from di�erent parts of a book, in addition to useful te hniques for

information-hiding. With MathBook XML sour e, the software reates output that onforms to many

a essibility guidelines. These books an all be read with a desktop or laptop omputer, but also work

well on small tablets and on smartphones that are smaller still.

We will demonstrate the nature of the online output with several textbooks made available with

open li enses. There are over twenty textbooks now in use in lassrooms, or in development, using

MathBook XML. Primarily these are undergraduate mathemati s textbooks, spanning the urri ulum

from developmental mathemati s, through al ulus, and up to more advan ed ourses su h as number

theory and abstra t algebra. However, other dis iplines are now being represented, in luding omputer

s ien e, poetry, and musi theory. More about the proje t an be found at mathbook.pugetsound.edu.

A WEBDOCUMENT STRUCTURE AS SUPPORT FOR THE ELABORATION OF

DIGITAL DIDACTIC MATERIAL FOR THE TEACHING OF MATHEMATICS

Fran k Bellemain

LEMATEC-EDUMATEC-UFPE, Brazil

f.bellemain�gmail. om

Ver�ni a Gitirana




LEMATEC-CAP-UFPE, Brazil

rogerioigna io�gmail. om

100

The produ tion of material for the tea hing of mathemati s in a te hnology-ri h environment by a tea her

is a very omplex, dynami and, in general, olle tive a tivity. It is omplex be ause it is hybrid and requires

the arti ulation of physi al as well as digital elements. It is dynami be ause it is a ontinuous pro ess in

onstant evolution and subje t to frequent modi� ations. It is olle tive be ause what is produ ed is at

least the result of tea hers' modi� ation of something produ ed by others, when it is not the result of an

e�e tive ollaboration.

The study of this produ tion a tivity, whether to train tea hers or to elaborate support for this produ -

tion, is based on theoreti al and methodologi al prin iples related to the do umentary approa h (Gueudet

and Trou he, 2008), instrumental or hestration (Trou he, 2005), Planning and qualitative/formative eval-

uation (Gitirana, 2008)

Do umentation, at the same time pro ess (a tivity) and produ t (material), an bene�t signi� antly

from a omputational support. In this sense, we propose to approa h in this presentation the engineering

pro ess that allowed, through the integration of the theoreti al and methodologi al prin iples mentioned

above, to develop an instrumentation of the do umentation pro ess for the tea hing of mathemati s. We

will present in parti ular two notions, and their omputational modilizations in the engineering pro ess:

the notions of resour e and webdo umento, as well as the te hnologi al advan es produ ed from this en-

gineering, transdis iplinary result of the integration of issues related to edu ation and software engineering

itself.

Referen es

Adler, J. (2000). Con eptualising resour es as a theme for tea her edu ation. Journal of Mathe-

mati s Tea her Edu ation, 3, 205-224.

Gitirana, Ver�ni a. 2008, "Planejamento e avaliação em matemáti a". Silva, Janssen Felipe da,

Ho�mann, Jussara, Esteban, Maria Teresa..

Gueudet, G., & Trou he, L. 2008. Do Trabalho do umental dos professores: gênesis, oletivos,

omunidade. O aso da matemáti a (Trad. K. Ro ha, 2016). EM TEIA: Revista de Edu ação

Matemáti a e Te nológi a Iberoameri ana. 6, 1-43

Trou he, L. (2005), Constru tion et onduite des instruments dans les apprentissages mathématiques

: né essité des or hestrations, Re her hes en dida tique des mathématiques, 25, 91-138

ENGINEERING PROCESS IN THE DEVELOPMENT OF A DIGITAL GUIDE OF THE

BRAZILIAN PROGRAM OF TEXTBOOKS (PNLD)

Fran k Bellemain


f.bellemain�gmail. om

Ver�ni a Gitirana





rogerioigna io�gmail. om

Rosilângela Lu ena


rosi.lu enas �gmail. om

Pedro Alessio


pedro.alessio�gmail. om

PNLD - National Program of Textbooks is a Brazilian program whi h provides elementary and high state

s hools (from 6 to 18 years-old students) with textbook for students. Ea h three years, Brazilian Ministry

of Edu ation evaluates the textbook olle tions proposed by the publishers and elaborates a guide with

reviews of the approved olle tions on the basis of the evaluation pro ess. For ea h area of s hool

knowledge, tea hers' team of ea h state s hool sele ts a olle tion of textbooks among those approved

and listed in the guide, to use a ordingly their pedagogi al needs.

Until last year, the guide was provided in a digital and printed ri h text format with a volume per

subje t area. Sin e the PNLD 2016, an intera tive guide version with all areas was developed. The

101

intera tivity level of su h a guide for providing the best possible support for the tea her hoi e an be

high, but the needs and onstraints of all a tors and institutions involved limit the possibilities. Our team

(LEMATEC) had been in harge of on eiving and developing the digital guide the last two years. In

this presentation, we present how we built the digital guide of the PNLD (Brazil, 2015, 2016) onsidering

all the involved parameters, to providing an intera tive guide as useful as possible to support tea her's

hoi es of textbooks, onsidering their requirements to use in their pra ti e. The onsidered parameters

for the elaboration of the digital guide an be organized in the four following topi s related to the various

dimensions of the engineering pro ess of the digital guide elaboration:

• theoreti al ba kground whi h should s ienti� ally justify the ontent of the guide and the hosen

ontent explorations fun tionalities.

• spe i� ations whi h des ribe the main fun tionalities the guide should have.

• te hni al onstraints whi h hara terize the te hnologi al ontext in whi h the guide should be built

and later used.

• institutional onstraints whi h tell what an and annot be done in the guide.

We will present the engineering pro ess and its theoreti al and te hni al ba kground.

Referen es

Brazil, MEC,SEB. 2016. Guia Digital PNLD 2017, MEC/SEB: Brasília.

Brazil, MEC,SEB. 2017. Guia Digital PNLD 2018, MEC/SEB: Brasília

A METHODOLOGICAL TOOL FOR ANALYSING TEXTBOOKS

Marilena Bittar


marilenabittar�gmail. om

The textbook is a fundamental support for tea hing a tivity as eviden ed by resear h in di�erent ountries.

It an be said that the textbook represents the o� ial institutional knowledge to be worked by the tea hers

and, thus, its analysis, omplemented by the analysis of other sour es su h as urri ular dire tives, allows

an approximation with the institutional knowledge (Assude, 1996).

A textbook analysis reveals to the resear her diverse lands apes that an go from the study of the s hool

ulture in a given time to the identi� ation of possible reasons for learning di� ulties and the elaboration

of dida ti sequen es. A ording to the resear h aim, one theoreti al tool may be more pertinent than

another may. In my resear h, I use to work with the anthropologi al theory of dida ti s (ATD) (Chevallard,

1998). This theory onsiders that all human a tivity onsists in a omplishing a task t, of a ertain type

T, by means of a te hnique �, justi�ed by a te hnology è that permits, at the same time, to think about

it, to produ e it, and whi h itself is justi�able by a theory È. Therefore, ATD provides us a model named

praxeoly organization, whi h an be mathemati al or dida ti al, denoted as [T/�/è/È℄. This approa h

allows us to understand and to des ribe the mathemati al and dida ti hoi es presented in textbooks,

indi ates that to understand the hoi es of the authors of the textbooks allows to understand and question

the pra ti e of tea hers and to understand mu h of the reasons for students' learning di� ulties. On e the

importan e and analysis of textbooks has been eviden ed, it remains to re�e t on how to arry out su h an

analysis. In this text, I systematize the experien es and re�e tions experien ed with supervised students

in the announ ed perspe tive. The aim of this arti le is to present a methodologi al path omposed of

the following phases: the hoi e of material (book) to be analysed; the separation between Course and

Proposed A tivities elaboration/identi� ation of the mathemati al organization; elaboration/identi� ation

of the dida ti organization; analysis of the modelled organizations.

Referen es

102

Assude, T. (1996). De l'é ologie et de l'é onomie d'un système dida tique : une étude de as.

Re her hes en Dida tique des Mathématiques, 16/1, 47-72.

Chevallard, Yves. 1998. �Analyse des pratiques enseignantes et dida tique des mathematiques:

L'appro he anthropologique.� Re her hes em Dida tique des Mathématiques 19(2): 221-266.

A essed at http://yves. hevallard.free.fr/spip/spip/arti le.php3?id_arti le=27 on May 15th, 2013,

pages 1- 29.

RATIO AND PROPORTIONS 1716-2016

Kristín Bjarnadóttir

University of I eland � S hool of Edu ation

krisbj�hi.is

Arithmeti textbooks, written in the I elandi language, printed and in manus ript, have an unbroken,

although sporadi , history from 1716 until present. The �rst printed substantial arithmeti textbook

was published in 1780. Up to 1910s, textbooks began tea hing arithmeti from s rat h, adhering to the

tradition of libri d'abba o, des ribed by van Egmond (1980). During 1920s-1960s, the arithmeti textbook

tradition ontinued in two phases, intended for primary and lower se ondary edu ation. Sin e the 1970s,

beginning with a series highly in�uen ed by the New Math, �ve textbook series have been published for

the lower se ondary s hool level, all by a state textbook publishing house.

In this study we shall fo us on both ends of this sequen e. The ontent of the manus ript Arithmeti a

Islandi a of 1716 will be analysed with respe t to fa tors in�uen ing it and possible fa tors in�uen ed by

its text. Certain items point to that Arithmeti a Dani a, written in Latin by Geo Frommius (1649), the

Dane Jørgen From, published in Copenhagen, was a model for the I elandi Arithmeti a, whi h in turn

may be onsidered as an abridged version, omparing the sizes of the textbooks. The matter is however,

more ompli ated as Arithmeti a Islandi a may have more models, or it may be seen as a free and adapted

translation. Furthermore, exer ises, ontained in Arithmeti a Islandi a are also found in other younger

arithmeti manus ripts.

We shall in parti ular study the history of the tea hing of ratio and proportions in the form of the Rule

of Three and ompare its tea hing in Arithmeti a Islandi a and other books through the three enturies

to that of ratio and proportions in Skali 3B by Tofteberg et al. (in print), the last volume of the latest

textbook series for the lower se ondary level, to be published in early 2017. Skali is in turn an adapted

translation of a Norwegian textbook series, just as Arithmeti a Islandi a was a re�e tion of the Danish

Arithmeti a, three enturies earlier.

Referen es

Frommius, Geo. 1649. Arithmeti a Dani a Seu Brevis A Perspi ua, Institutio Arithmeti æ Vulgaris,

Astronomi æ, Geodæti e, In Usum Gymnasiorum Et S holarum Daniæ A Norvegiæ, Fussu Regio

Adornata À Georgio Frommio, In A ademiâ Hafniensi Mathematum, Professore Ordinario.

Lbs. 1694, 8vo. Arithmeti a Islandi a (1716).

Tofteberg, Grete N., Janneke Tangen, Ingvill M. Stedøy-Johansen and Bjørnar Alseth. (In print).

Skali 3B. Stærðfræði fyrir unglingastig. [Skali 3A. Mathemati s for lower se ondary level℄. Reykjavík:

Menntamálastofnun.

Van Egmond, Warren (1980). Pra ti al mathemati s in the Italian Renaissan e: a atalog of Italian

abba us manus ripts and printed books to 1600. In Annali dell'Istituto e Museo di Storia della

S ienza. Monogra�a N. 4, 3-36. Firenze: Istituto e Museo di Storia della S ienza.

103

HOW COMBINATORIAL SITUATIONS ARE REPRESENTED IN BRAZILIAN PRIMARY

AND MIDDLE SCHOOL TEXTBOOKS

2

Rute Borba


resrborba�gmail. om

Marilena Bittar

Universidade Federal de Mato Grosso do Sul, Brazil

marilenabittar�gmail. om

Juliana Montenegro


azevedo.juliana1987�gmail. om

Dara Silva


dara atarina�gmail. om

Textbooks in Brazil, as in many other ountries, play a entral role in Mathemati s tea hing and learning

and, in this sense, need to be onstantly assessed. The Brazilian National Textbook Programme (PNLD),

sin e the late 1990s, has been on erned with distin t aspe ts to be onsidered in textbook evaluations

and Carvalho and Lima (2010) observed advan es in textbook quality.

An important aspe t of assessment on erns how symboli representations are dealt with in text-

books as means to think about mathemati al on epts. To not onfuse a mathemati al on ept with

its representation, Duval (1995) emphasizes the importan e of semioti representation transformations:

internal transformations in whi h the system is the same (treatment) and transformations in whi h there

are pro essing hanges to the system ( onversion).

Con erning Combinatori s, Brazilian urri ulum orientations, BRASIL (1997), re ommend the tea hing

and learning of di�erent situations in all s hool levels (Primary, Middle and High S hool), i.e., Cartesian

produ ts (CP), arrangements (A), ombinations (C) and permutations (P).

This study analyses three textbook olle tions: lower Primary S hool (1st, 2nd and 3rd grades), upper

Primary S hool (4th and 5th grades) and Middle S hool (6th to 9th grades). The analysis on ern whi h

ombinatorial situations are presented and what symboli representations onversions are required.

In lower Primary S hool textbooks, 100% of the problems were Cartesian produ ts. Combinatorial

situations in upper Primary S hool textbooks were better distributed (54% were CP; 36% C; 10% P; but no

arrangements) and in Middle S hool textbooks presented an even better distribution (32% were CP; 22%

C; 24% P and 22% A). Presenting problems in only natural language and asking to onvert data to lists

or to numeri al expressions was not as frequent (only 27% of the a tivities) in Primary S hool textbooks

but was very frequent (88%) in Middle S hool textbooks. Early s hooling textbooks tend to use natural

language with drawings, tree diagrams or tables to aid the understanding of ombinatorial situations.

Positive aspe ts were observed in ombinatorial situations proposed, su h as onversions of varied symboli

representations, but a better distribution of problem types and more symboli representation onversions

still need to be present in textbooks.

Referen es

BRASIL. Ministério da Edu ação. 1997. Parâmetros Curri ulares Na ionais: Matemáti a. Brasília:

Se retaria de Edu ação Fundamental.

Carvalho, João, and Paulo Lima. 2010. Es olha e uso do livro didáti o. Volume 17, Brasília: MEC.

Duval, Raymond. 1995. Sémiosis et pensée humaine. Berna: Peter Lang.

MAKING DIGITAL MANIPULATIVES FOR MATH E-TEXTBOOKS: AN OVERVIEW OF

THE NOWADAYS MAJOR USER-FRIENDLY SYSTEMS

Humberto José Bortolossi

Fluminense Federal University, Brazil


2

Resear h �nan ed by CNPq (Conselho Na ional de Pesquisa), by Capes (Coordenação de Aperfeiçoamento de Pessoal

de Nível Superior) and by Fa epe (Fundação de Amparo à Ciên ia e Te nologia de Pernambu o).

104

Portable devi es like smartphones, tablets, and notebooks are in reasingly heap and, in onsequen e,

in reasingly popular. More than just a substitute vehi le for a ess to traditional paper textbooks ontents,

literature has pointed to how the use of e-textbooks through this type of te hnology an potentially

transform tea hing and learning of mathemati s (Pepin et al., 2016), in luding the design of learning

paths, expositions, tools, problems, exer ises, and essay tasks (Yerushalmy, 2015).

It is worth mentioning that the Brazilian government's largest program of free distribution of Basi

S hool textbooks (the PNLD program) requires ompeting publishers to deliver a digital version of the

textbook with intera tive resour es. Also, in addition to the questions with pen il-and-paper, the OECD's

Programme for International Student Assessment (PISA) now in ludes omputer-based assessments to

measure digital pro� ien y to solve mathemati al problems.

In this ontext, digital manipulatives are a key element! How to make su h digital manipulatives for

mathemati s and statisti s e-textbooks? In our presentation, we will make an overview of the nowadays

major user-friendly systems, that is, systems that use little or no dire t programming. Our des ription

will in lude power to express di�erent mathemati al ontents (e.g., mathemati al notation, symboli

omputations, 3D obje ts, et .), pri e, li ense of use of the system and the produ ed material, performan e

in less powerful equipment, ease of adaptation of the manipulative by the tea her and integration with

other authoring systems (e.g., iBooks and EPUB).

We hope that our analysis will be useful for professionals and publishers interested in the produ tion

of math e-textbooks.

Referen es

Pepin, Birgit et al. �E-Textbooks in/for Tea hing and Learning Mathemati s: A Potentially Transfor-

mative Edu ational Te hnology�. In Handbook of International Resear h in Mathemati s Edu ation,

edited by Lyn D. English, and David Kirshner, 229-247. New York: Routledge, 2016.

Yerushalmy, Mi hal. �E-Textbooks for Mathemati al Guided Inquiry: Design of Tasks and Task

Sequen es�. In Task Design In Mathemati s Edu ation: An ICMI Study, edited by Anne Watson,

and Minoru Ohtani, 229-247. Switzerland: Springer, 2015.

OPEN MATH BOOK PROJECT

Humberto Bortolossi

UFF, Brazil


Letí ia Rangel

UFRJ, Brazil

leti iarangel�ufrj.br

Cydara Ripoll

UFRGS, Brazil


Fábio Simas

UNIRIO, Brazil


Augusto Teixeira

IMPA, Brazil

augusto�impa.br

It is well known that the textbook plays a key role in tea hing and learning in s hools. (e.g., Haggarty

and Pepin, �International studies of a hievement in mathemati s,"). Moreover, Robitaille and Travers

(�International studies of a hievement in mathemati s," p. 706) argued that a great dependen e upon

textbooks is "perhaps more hara teristi of the tea hing of mathemati s than of any other subje t". In

the Brazilian ontext, Ma hado ("Sobre livros didáti os," p. 31) says that the textbook usually determines

what is taught and its logi al sequen e. We believe that, if tea hers were allowed to remix the textbooks

they use in an easy and simple way, as well as share their ontributions with the authors and other

tea hers, they would feel mu h more omfortable in adapting the textbook for their lasses on spe i�

ontexts, making the tea hing a reative pro ess as it should be. On the other hand, �at the heart

of the movement towards Open Edu ational Resour es is the simple and powerful idea that the world's

knowledge is a publi good and that te hnology in general and the Worldwide Web in parti ular provide

an opportunity for everyone to share, use, and reuse it." (Marshal and Catherine, "The promise of Open

105

Edu ational Resour es", p. 10). In this talk, we will present the LAM proje t (short for Open Math Book

in Portuguese, umlivroaberto. om) and its possible impa ts. This proje t is an e�ort of tea hers and

professors to develop a olle tion of mathemati s textbooks that are open (under the Creative Commons

BY-SA li ense) and ollaborative in a wider sense. First, it is being onstru ted in permanent onta t with

dozens of tea hers of regular s hools in Brazil, who are testing the material in their lassrooms. Moreover,

an open sour e platform is being developed, where any person or organization an easily ontribute to the

books by editing/ ommenting dire tly in the text (that has a referee pro ess) or fork whole books and

make their own version a ording to their spe i� purposes.

Referen es

David F. Robitaille and Kenneth J. Travers, �International studies of a hievement in mathemati s,"

in Handbook of resear h on mathemati s tea hing and learning, ed. Douglas A. Grouws (New York:

Ma millan, 1992), 687-709.

Haggarty, Linda and Pepin, Birgit. �An investigation of mathemati s textbooks and their use in En-

glish, Fren h and German lassrooms: who gets an opportunity to learn what?" British Edu ational

Resear h Journal, 28 (4) (2002), 567-590.

Marshall S. Smith and Catherine M. Casserly, �The promise of Open Edu ational Resour es."

Change, 38 (5) (2006) , 8-17.

Nilson J. Ma hado, �Sobre livros didáti os quatro pontos" Em Aberto 16 (1996) 31.

ON PREVALENCE OF IMAGES IN HIGH SCHOOL GEOMETRY TEXTBOOKS

Megan Cannon


mn annon�mail.usf.edu

Milé Kraj evski


mile�mail.usf.edu

Even though terminology used in mathemati s edu ation for the on ept of visualization is not uni�ed, the

importan e of visualization and graphi al representations of mathemati al obje ts and notions has been

well do umented in the resear h literature. Despite ample eviden e that images help guide our intuition

and help design a logi al hain of arguments needed in the proving pro ess, there is a la k of resear h

on how to expli itly formulate the hara teristi s of an image to foster this ognitive pro ess. What are

the attributes of an adequate illustration/image of a mathemati al obje t? What kind of illustrations

are urrently present in mathemati s textbooks? How an we design an e�e tive use of visualization to

support the reasoning and proving pro ess? How an we help students interpret visual information and

gain a ess to the information that is oded in every image? These are the questions we investigate and

for whi h we provide viable answers.

Our paper examines 14 geometry textbooks, eight of whi h were published by three major publish-

ers and six by non-pro�t organizations developing free and ustomizable textbooks aligned with state

standards. In ea h of these textbooks, we fo used on the visual representations in se tions dealing with

lassi� ations of triangles, parallel lines and transversals, parallelograms, and trapezoids. After de�ning a

typi al image for ea h on ept, we determined the per entage of typi al images among all images in the

lesson portion and exer ise portion of ea h textbook. Results show that the per entage of typi al images

among all images in ea h ategory of obje ts varies from 52% for parallel lines and transversals to 75% for

trapezoids. Potentially this may lead to additional hallenges imposed on students to orre tly on eptu-

alize a given mathemati al obje t, due to the restri tive nature of the image itself. We indi ate onfusing

hara teristi s stemming from typi al images and provide suggestions for over oming these obsta les. We

list possible venues for further resear h on evaluating images in mathemati s textbooks.

Referen es

106

Phillips, Linda M., Stephen P. Norris and John S. Ma nab. 2010. Visualization in Mathemati s,

Reading and S ien e Edu ation. New York: Springer.

Presmeg, Norma. 2006.�Resear h on visualization in learning and tea hing mathemati s�. In Hand-

book of resear h on the psy hology of mathemati s edu ation, edited by Angel

Gutierrez and Paolo Boero, 205-237. Rotterdam: Sen e. Vinner, Shlomo, and Rina Hershkowitz.

1980. �Con ept images and some ommon ognitive paths in the development of some simple

geometri al on epts�. In Pro eedings of the fourth international onferen e for the psy hology of

mathemati s edu ation (pp.177-184) Berkeley, California.

IDDM: INTERACTIVE DIGITAL DIDACTIC MATERIAL?

Apare ida Santana de Souza Chiari


ida h�gmail. om

Mar elo de Carvalho Borba

Universidade Estadual Paulista �Júlio de Mesquita Filho�

mborba�r .unesp.br

It is ommon knowledge that textbooks of mathemati s are one of the most (or the most) used resour e

by tea hers in preparing their lasses. Be ause it is the most used resour e, the way in whi h it is used

depends on several fa tors, su h as the ontent of the book, the pedagogi al obje tives of the tea her or

even the tea her's own knowledge about the subje ts that s/he intends to explore (Ribeiro and Amaral

2016).

In this sense, this paper aims to present and dis uss a very parti ular type of use of dida ti material

in the pre-servi e mathemati s tea hers online distan e edu ation. It is a �textbook� that is generated

ea h time by a olle tive of students-with-tea her-and-digital-te hnology. The result of it, IDDM, will be

presented in the next paragraph, and is based on the notion of humans-with-media (Borba 2012). The

resear h was ondu ted over a period of two years in online ourses o�ered by the Open University of

Brazil (UAB). The resear h was qualitative, entered on the strand alled Grounded Theory (Strauss and

Corbin 2008). Details an be found in Chiari (2015).

Two ategories were developed, whi h merged into a entral ategory alled IDDM (Intera tive Digital

Dida ti Material) from (a) the very parti ular type of use made in the Virtual Learning Environment

(VLE) by students and tea hers and b) the automati registration of the intera tions done automati ally

by the VLE. We believe that this type of use transforms the register of intera tions into Dida ti Material,

be ause it has be ome a sour e of study, Digital, be ause the developing in a VLE, and Intera tive,

be ause it is formed from the intera tions between students and tea hers. These elements ontribute to

the emergen e of the notion of IDDM.

We believe that IDDM introdu es new possibilities to the debate about textbooks be ause it is a very

parti ular type of "textbook" that arises from intera tions during the development of a ourse and, in this

sense, it be omes a �textbook� onstru ted by and for students, from their demands.

Referen es

Borba, M. C. 2012. �Humans-with-Media and Continuing Edu ation for Mathemati s Tea hers in

Online Environments.� ZDM, Berlim, 44: 802-14.

Chiari, A. S. S. 2015. �O Papel Das Te nologias Digitais Em Dis iplinas de Álgebra Linear a

Distân ia: Possibilidades, Limites E Desa�os.� Tese (Doutorado em Edu ação Matemáti a), Rio

Claro: Universidade Estadual Paulista �Júlio de Mesquita Filho.�

Ribeiro, Miguel, and Rúbia Bar elos Amaral. 2016. �Guia E Te nologia Dos/nos Livros Didáti os

de Matemáti a: Uma Primeira Dis ussão.� Edu ação Matemáti a Em Revista 1: 64-75.

Strauss, A, and J. Corbin. 2008. Pesquisa Qualitativa: Té ni as e Pro edimentos Para O Desen-

volvimento de Teoria Fundamentada. 2nd ed. Porto Alegre: Artmed.

107

PROJECT-BASED MATH-SCI INTERDISCIPLINARY CURRICULUM DESIGN AND

TEACHING TRAJECTORY : A PRELIMINARY CASE OF �EXPLORE THE

SUPER-EARTH�

Qi Chunxia

Beijing Normal University

qi hxia�126. om

Ma Fu

Nanjing Normal University

fuma30�sina. om

He Shengqing


Heshengqing99�163. om

Zhang Huying

This study hose �Explore the Super-Earth" as a topi and designed a mathemati s-s ien e interdis iplinary

proje t that is suitable to seventh grade students through a systemati analyzing of urri ulum and

textbooks.

Firstly we hose the topi of �s ienti� notation and its related operations" as the ase and analyzed

its urri ulum obje tives and litera y orientations a ording to Mathemati s Curri ulum Standards (2011

Edition) (China Ministry of Edu ation, 2011) as following �gure 1:

Figure 1 Expe tation of the proje t obje tives

Se ondly we analyzed the ontent organization in di�erent editions of China traditional mathemati s

textbooks (e.g Press, People's Edu ation Press.) to determine the basi knowledge and skills involved

in this proje t. Thirdly we hose a s ien e theme (that is, Explore the Super-Earth) that is relevant

to mathemati s ontent and suitable for students' ognition, and organized the �ve questions to guide

student to explore the knowledge that are in the Super-Earth.

Referen es

Berlin, D. F., & White, A. L. 1998.�Integrated s ien e and mathemati s edu ation: Evolution and

impli ations of a theoreti al model". In B. Fraser & K. Tobin (Eds.), International handbook of

s ien e edu ation. Dordre ht: Kluwer.

Chan, C. M. E. 2008. Using model-eli iting a tivities for primary mathemati s lassrooms. The

Mathemati s Edu ator, 11(1/2), 47-66

China Ministry of Edu ation. 2011. Curri ulum Standards for S hool Mathemati s in the Full-time

Nine-year Compulsory Edu ation (2011 Edition). Beijing: Beijing Normal University Press. [In

Chinese℄

English, L. D. (2010). Modeling with Complex Data in the Primary S hool. In R. Lesh et al. eds.

Modeling Students' Mathemati al Modeling Competen ies, 287-299?

108

Galbraith, P. L. 2007. Beyond the low-hanging fruit. In W. Blum, P. L. Galbraith, H. Henn, and

M. Niss Eds. Modelling and Appli ations in Mathemati s Edu ation: The 14th ICMI Study 79-88.

New York: Springer

M Graw, S. A. Ed. 2003. Integrated mathemati s. Choi es and hallenges. Reston: National

Coun il of Tea hers of Mathemati s?

Kraj ik, J., Blumenfeld, P., Marx, R. W., & Soloway, E. 1994. A ollaborative model for helping

s ien e tea hers learn proje t-based instru tion. Elementary S hool Journal, 5, 483-498

DOES A PROJECT-BASED LEARNING MATERIAL IMPROVE STUDENTS'

UNDERSTANDING OF RATIONAL NUMBER? A CASE STUDY

Qi Chunxia


qi hxia�126. om

Wang Ruilin

Capital Normal University

kk_jen�163. om

Lin Mengwei

Beijing Hepingli Middle S hool

18813043596�163. om

Rational number is the important ontent of the number and algebra for se ondary s hool. To �nd what

kind of material to help students' understanding the rational number, we design a unit of proje t-based

learning material and try to �nd whether it an impa t on students learning or not. The result shows

the impa t of proje t-based learning material on students' understanding the rational number and its

appli ation in real life. In addition, students' ability on ooperation with lassmate is also improved.

Referen es

Ala ap�nar, F. 2008. E�e tiveness of proje t-based learning. Eurasian Journal of Edu ational Re-

sear h, 32, 17-34.

Chinese Ministry of Edu ation .2011. Compulsory edu ation urri ulum standard of math, Beijing:

Beijing Normal University Publishing Group.

Ne³e Tertemiz. 2012. The E�e ts of Proje t- and A tivity-supported Pra ti es on Mathemati s

Edu ation A hievement and Student Views. Eurasian Journal of Edu ational Resear h, 46, 159-178

TEACHERS' PERCEPTIONS OF THE STANDARDS FOR MATHEMATICAL PRACTICE

BY TEXTBOOK TYPE

Jon D. Davis


jon.davis�wmi h.edu

Je�rey Choppin

University of Ro hester

j hoppin�Warner.Ro hester.edu

Corey Drake

Mi higan State University

drake�msu.edu

Amy Roth M Du�e

Washington State University Tri-Cities

m du�e�tri ity.wsu.edu

The majority of students in the U.S. are learning from the Common Core State Standards for Mathemati s

(CCSSM). An important omponent of the CCSSM are the Standards for Mathemati al Pra ti e (SMPs).

These SMPs are built on earlier standards su h as the Curri ulum and Evaluation Standards for S hool

Mathemati s (National Coun il of Tea hers of Mathemati s, 1989). Moreover, these standards were

used as a framework from whi h standards-based urri ula were developed (Senk & Thompson, 2003).

Given this onne tion between these urri ula and the SMPs we were interested in how middle s hool

mathemati s tea hers (MSMTs) using these programs and onventional mathemati s textbooks per eived

the SMPs. While various studies have examined tea hers' per eptions with regard to the CCSSM (e.g.,

109

Cogan, S hmidt, & Houang, 2013), none of these studies have used interviews to understand MSMTs'

per eptions of the SMPs. Interviews were ondu ted with 76 U.S. middle s hool mathemati s tea hers

(MSMTs) using onventional (N = 39) or standards-based (N = 37) urri ulum types to determine

their per eptions of the CCSSM and the SMPs. A mixed methods resear h design was used to explore

the themes in responses as well as to examine the di�eren es in theme frequen y by urri ulum type.

MSMTs using both urri ula types struggled in naming di�erent SMPs with less than two of eight SMPs

being mentioned on average. MSMTs using standards-based urri ula were more likely to identify the

instru tional philosophy of the CCSSM as aligned with the SMPs (Fisher's Exa t Test = 3.881, p = .05).Additionally, MSMTs using standards-based urri ula were statisti ally signi� antly more likely to see their

textbooks as aligned with the SMPs χ2(3) = 7.608, p = .026. The impli ations of these results will be

dis ussed.

Referen es

Cogan, Leland, William S hmidt, and Ri hard Houang. 2013. Implementing the Common Core State

Standards for Mathemati s: What We Know about Tea hers of Mathemati s in 41 States, a essed

November 10, 2016, https://edu ation.msu.edu/ep /publi ations/do uments/WP33Implementing

theCommonCoreStandardsforMathemati sWhatWeknowaboutTea herofMathemati sin41S.pdf.

National Coun il of Tea hers of Mathemati s. 1989. Curri ulum and Evaluation Standards for

S hool Mathemati s. Reston, VA: Author.

Senk, Sharon, L., and Denisse R. Thompson, eds. 2003. Standards-based S hool Mathemati s

Curri ula: What Are They? What Do Students Learn? Chi ago: University of Chi ago Press.

WHEN IS AN EXPLORATION EXPLORATORY? A COMPARATIVE ANALYSIS OF

GEOMETRY LESSONS

Leslie Dietiker

Boston University

dietiker�bu.edu

Andrew Ri hman

Boston University

asri h�bu.edu

As the all for shifting the mathemati al experien es of students toward inquiry and problem solving

be omes in reasingly international, understanding how designs of written urri ula enable or onstrain

these experien es is needed. This study presents a omparison of two lessons on the same topi from U.S.

textbooks to explain how the sequen e of tasks in an a tivity an impa t the experien e of a learner. One

lesson, representative of a �reform-based� textbook (�A�), ontains investigations of onditions of triangle

ongruen e. The se ond is a �Te hnology Lab� on triangle ongruen e from a �traditional� textbook (�B�),

the design of whi h is atypi al for the that textbook. Comparing these lessons o�ers the opportunity to

distinguish between di�erently-designed �explorations� and allows us to re ognize how the stru ture of

ontent may enable or onstrain student exploration.

In order to ompare the unfolding mathemati al stru ture of these lessons, we use a narrative interpre-

tation of urri ulum. Spe i� ally, we interpret how mathemati al ontent unfolds sequentially for a reader

as a mathemati al story (Dietiker, 2014). As mathemati al questions are raised and ta kled, the hanging

moment-to-moment tension felt by a reader between what is known and what is desired to be known an

be interpreted as the mathemati al plot. In this study, we use odes developed from the literary theory

of Barthes (1974), marking how the mathemati al ontent emerged and sequentially hanged throughout

ea h lesson.

Our �ndings show how a lesson stru ture an prevent an �exploration� from being exploratory. The

mathemati al plot of lesson B is designed in a way that does not enable a reader to re ognize where the

investigation is going, preventing inquiry. In ontrast, the mathemati al plot of lesson A is sequen ed

110

to slowly reveal and withhold information in a way that ues the learner of what lay ahead, in reasing

student uriosity and sustaining interest. Lesson A raises and keeps open multiple questions through

signi� ant portions of the story. We also identi�ed the use of ambiguity and other narrative devi es that

an potentially in rease anti ipation and uriosity for what is to ome.

We propose that the design of lesson B may be onstrained by the extended traditional mathemat-

i al story within whi h it is pla ed, whi h likely produ es parti ular expe tations from readers for how

mathemati al questions will be asked and answered. In addition, we argue that this framework o�ers

new understanding of how designed textbook ontent in rementally emerges and hanges as a lesson

unfolds; an understanding that an support the design of lessons that are onsistent with the in reasingly

international urri ular reform movement.

Referen es

Barthes, Roland. 1974. S/Z. Translated by Ri hard Miller. New York, NY:

Dietiker, Leslie. 2014. �Telling New Stories: Re on eptualizing Textbook Reform in Mathemati s.�

In Pro eedings of the International Conferen e on Mathemati s Textbook Resear h and Development

2014 (ICMT-2014), edited by Keith Jones, Christian Bokhove, Geo�rey Howson, and Lianghuo Fan,

185-90. University of Southampton, UK.

COUNTING AND COMPUTATION IN MOZAMBICAN TEXTBOOKS FOR GRADE ONE

Jan Draisma

Pedagogi al University, Beira, Mozambique

jan.draisma�yahoo. om.br

In O tober 1970 I started working as a Dut h volunteer mathemati s tea her at Frelimo Se ondary S hool

in Bagamoyo, Tanzania. Sin e September 1964, Frelimo had been involved in an armed struggle for

independen e. In northern Mozambique large areas had been liberated and some basi so ial servi es had

been reated, in luding primary edu ation. In neighbouring Tanzania a se ond-dary s hool had been set

up, whi h moved to Bagamoyo in 1970.

The s hool also re eived the task of giving a tea her training ourse for primary s hool tea hers from

the liberated zones. I had to tea h mathemati s and its dida ti s. We had at our disposal a ma-themati s

textbook "Matemáti a - 1

a

lasse", written in Dar es Salaam for Frelimo by the East German tea her

Joa him Kindler, supported by William Minter (USA) and Mozambi an tea hers. In Grade 1, the most

di� ult topi is sums like 8 + 5 = 13 and di�eren es like 14 - 6 = 8. Fortunately, Kindler's book showed

the same strategy as I had learned as a hild in the Netherlands: 8 + 5 = ? 8 + 2 = 10 and 10 + 3 =

13, therefore 8 + 5 = 13.

On the 25th of April 1974, the Portuguese olonial (and fas ist) government fell during the "Revolution

of the Carnations", staged by aptains and the Portuguese army who didn't want to ontinue the olonial

wars in Angola, Guinea Bissau and Mozambique.

In January 1975, during the Transitional Government, the �rst Mozambi an onferen e on edu ation

in Beira de ided to abandon the Portuguese edu ational system, in luding urri ula, syllabi and textbooks.

I had the di� ult task to oordinate the �nal draft of the new mathemati s urri ulum from Kindergarten

to Grade 11, making use of several proposals presented to the onferen e, in luding ours from Bagamoyo.

Later in 1975 I suggested to the new Ministry of Edu ation to have the Grade 1 textbook of Kindler

and Minter reprinted, together with its tea hers' guide, mainly for use in Tea her Training Centres. During

a personal visit to the Tea her Training College in Lourenço Marques, I met the tea her trainer Fausto

de Sá. Among other things, we spoke about the strategy for �nding the sum of 8 and 5. His omment

on the 8 + 5 = (8 + 2) + 3 strategy was: why not do immediately 8 + 5 = 13, instead of using two

111

steps? Later I understood that his idea was that hildren ould �nd 8 + 5 = 13 by ounting, and that

they should memorise the basi fa ts from 1 + 1 = 2 till 9 + 9 = 18.

In my presentation I will show the ongoing history of the debate in Mozambique between ounting

strategies and omputation strategies, for problems like 8 + 5 = 13 and 14 - 6 = 8. Illustrations will

be given from textbooks published between 1971 and 2010, and from some resear h among Mozambi an

adults. Some omparisons with foreign traditions will be made. Unfortunately, the system urrently in

pla e does not en ourage the gradual improvement of textbooks.

Referen es

Draisma, Jan. 2013. Ensinar Matemáti a nas línguas Moçambi anas, om espe ial atenção para as

línguas do Niassa. Maputo, JV Editores.

Sá, Fausto Faria de. 1965: Didá ti a da aritméti a e da geometria. Lourenço Marques, Es ola do

Magistério Primário Cardeal Gouveia.

MATHS TEXTBOOK ANALYSIS IN THE TEACHERS TRAINING CURRICULUM � THE

EXPERIENCE AT UNICAMP

Henrique N. Sá Earp

University of Campinas, Brazil

henrique.saearp�ime.uni amp.br

The undergraduate tea hers training programme (Li en iatura) at Uni amp has re ently in orporated the

one-semester (60h) ourse on Analysis of Mathemati s textbooks and tea hing materials as a mandatory

urri ulum requirement. The ourse had existed before as an optional subje t and its has attra ted

substantial interest from students sin e its total restru turing in 2014.

The ourse is organised in sixteen weekly workshops of 3h30min ea h, divided in �ve three-week

modules in whi h teams of 4-5 students must omplete a on rete task, assessed by a written report and

an oral presentation of �ndings dis ussed with the whole lass. The tasks address the following abilities:

(i) to riti ise and improve on a textbook, (ii) to ompare two textbooks on the same ontent, (iii) to

extra t pedagogi al resour es from foreign textbooks, (iv) to produ e original ontent to a high standard

of ontent and layout, and (v) to take initiative and suggest an independent ontribution.

As to se ondary skill developments, tasks (i)-(iii) require the students to formulate and apply method-

ologi al proto ols and assessment grids and task (iv) requires advan ed layout and template management.

All the oursework must be submitted in LaTeX. Students are also en ouraged to oordinate asyn hronous

teamwork ombining hat apps, loud folders and shared editing alternatives.

About the author

I am adjoint professor of Mathemati s at Uni amp. I graduated (2001) and did a Masters degree (2002)

in Mathemati s at the Federal University of Rio de Janeiro, and hold a Masters degree in Theoreti al

Physi s from the University of Cambridge (2004) and a PhD in Mathemati s (2009) from Imperial College

London. My resear h areas are omplex di�erential geometry and topology.

I was harged with restru turing the ourse on Analysis of Mathemati s textbooks in 2014, and have

been tea hing it over the past three years. I have also been a member of the Department's Undergraduate

Studies ommitee for four years and the head of the Re ruitment and Outrea h ommitee for one year.

Referen es

Course website: http://www.ime.uni amp.br/∼ma225/Apresenta ao.htm.

Course manual: http://www.ime.uni amp.br/∼ma225/manual_da_dis iplina.pdf

112

NB.: both referen es (in Portuguese) have been suggested and produ ed by the students themselves as

instan es of task (v).

TABLES IN TEXTBOOKS FOR ELEMENTARYSCHOOL GRADES 4 AND 5

Betânia Evangelista Gilda Guimarães

EDUMATEC / UFPE, Brazil

mbevangelista�hotmail. om gilda.lguimaraes�gmail. om

In our daily lives, we deal with di�erent types of tables, be it in newspapers, magazines or textbook

a tivities. For Martí, Sedano, and la Cerda (2010), this is an indispensable tool in many ognitive

tasks, espe ially s ienti� ones. Despite this importan e, tea hing tables in s hools su�ers from many

inadequa ies. These an take pla e both in the way tables are on eived, lassi�ed, and/or presented, as

well as in the hoi e and management of tasks to be used with students in the lassroom, as is argued by

Estrella (2014) and Guimarães and Oliveira (2014).

The term �table� is often used to name a adre, a database or a list of fa ts that, among other things,

are presented within a frame, as observed by Guimarães and Oliveira (2014). Furthermore, for Estrella

(2014), tables di�er from other types of representations in their variety, stru ture, �exibility, notation,

representation and usage. It is not su� ient for it to be stru tured in rows and olumns, modeled in su h

a way to be onsidered a table; it must follow other aspe ts that are part of its representation.

To this end, the goal of this resear h proje t was to analyze the a tivities that involve tables in 6 (six)

olle tions of textbooks geared toward the 4th and 5th grades of primary s hool (ages 9-10), re ommended

by Brazil's 2016 National Textbook Program.

Based on our analysis of the a tivities, we lassi�ed the representations into adre, databases and tables.

We veri�ed the presen e of all of these in every major theme in mathemati s (numbers and operations,

spa e and shapes, size and measurements, treatment of information) in every book we analyzed. However,

we also found a la k of di�erentiation in the textbooks we studied in de�ning these representations, be ause

in many of these a tivities tables were used to name any representation with a re tangular stru ture,

divided into rows and olumns. It is important to note that in a database the information is presented

from individual re ords, ie it is possible to re ognize the hara teristi s of ea h resear h subje t. They are

raw data that have not re eived any statisti al treatment. From there, you an ategorize them, reate

lasses and sub lasses, quantify them and re ord in a table. It is noti ed that the data is presented in a

systemati way, an entire phenomenon study of one or more variables.

Given this, we believe it is important to pay attention to the nomen lature used in these a tivities,

sin e the way of presenting and treating data in ea h type of representation is di�erent and requires

distin t approa hes in distin t tea hing situations.

Referen es

Estrella, Soledad. 2014. �El formato tabular: Uma revisión de literatura.� Revista Ele tróni a

A tualidades Investigavas em Edu a ión. V. 14, Número 2.

Guimarães, Gilda, and Oliveira, Isabela. 2014. �Construção e interpretação de grá� os e tabelas.�

In: Carlos Roberto Vianna; Emerson Rolkouski. (Org.). Pa to na ional pela alfabetização na idade

erta: Edu ação Estatísti a. 1ed. Brasilia: MEC-SEB, v. 07, p. 21-38.

Martí, Eduard, Sedano, Esther Sedano, and la Cerda, Carla Aular. 2010. �Alfabetiza ión grá� a.

La apropia ión de las tablas omo instrumentos ognitivos.� Revista Contexto. Anos IX e X. No

10.

113

ON A TIMELINE FOR THE COMPLETE ACCEPTANCE OF THE SYMBOLS N, Z, Q, R

AND C IN TEXTBOOKS

Ri ardo Fajardo

Universidade Federal de Santa Maria, Brazil

rfaj�ufsm.br

This is an ongoing resear h. It is divided into three stages: 1. A sear h to �nd out whether or not

it has already been done, 2. A sear h for bibliographi al literature, and 3. A ontent analysis of the

sele ted bibliography. On the �rst item, we resear hed on international publi ations, as well as in the

internet. What we found was a website on �Earliest Uses of Symbols in Number Theory�, maintained by

Je� Miller. However, Miller (2012) presents only the earliest uses of su h symbols. In this paper, we are

going further and resear hing on textbooks to �nd out in what de ade (or year) these symbols where fully

a epted by the mathemati s ommunity, spe ially by textbook authors. On the se ond item, although we

have bene�tted from the literature presented by Miller in his website, we always investigated the original

sour e, but also have expanded onsiderably the sear h for textbooks on the subje t. We sear hed on

university`s libraries, virtual libraries, online used book stores, and on the internet in general. In the

following, we present a few preliminary results. So far, the �rst textbook mentioning all the �ve symbols

for the sets of natural numbers (N), the integers (Z), the rational numbers (Q) , the real numbers (R) ,

and the omplex numbers (C) was en ountered in the textbook entitled �Algebra and Number Systems�,

by Hunter, Monk, Bla kburn and Donald (1974). Although the book opy se ured was dated of 1974,

it is a reprint of the �rst edition published on 1971. Therefore, it is safe to assume that in the early

70`s these symbols were beginning to take hold. They also in luded the symbol for the whole numbers,

W = {0, 1, 2, 3, 4, 5, . . .} = {0} ∪ N . However, in the year of 1975, Dogde onsiders the letter apital

J to represent the integers in the textbook �Numbers & Mathemati s�. For the set of omplex numbers,

he explains that some authors onsider E2 (the Eu lidean spa e of two dimensions) to represent it, but

prefers the symbol C for this number set. Even almost ten years later, in the year of 1983, Coxford and

Payne published the textbook �Algebra 2 with trigonometry�. In it the symbol for the set of integers is a

apital I, and there is no symbol for the set of omplex numbers. Hen e, by su h date the symbol for the

integers and the omplex numbers was not yet settled. The resear h has not yet been �nished.

Referen es

Miller, Je�. 2012. Earliest Uses of Symbols in Number Theory. A essed on mar h 10, 2016,

http://je�560.tripod. om/nth.html.

Hunter, J., Monk, D., Bla kburn, W. T., and Donald, D. 1974. Algebra and Number Systems.

Glasgow, The University Press.

Dodge, Clayton W. 1975. Numbers & Mathemati s. Prindle, Weber &S hmidt, In .

Coxford, Arthur F., and Payne, Joseph N. 1983. Algebra 2 with trigonometry. Orlando: Har ourt

Bra e Jovanovi h, In .

COMPARING MATHEMATICS CURRICULUM AND TEXTBOOKS BETWEEN CHINA

AND THE UK: FINDINGS FROM THE SHANGHAI MATHS PROJECT

Lianghuo Fan

Univ. of Southampton

L.Fan�soton.a .uk

Ming Ni

East China Normal Univ.

niming�e nupress. om. n

Huiping Xu


xuhuiping�e nupress. om. n

Jing Xu


xujing�e nupress. om. n

114

In the urrent era of information te hnology and globalization, the international ex hange and ollaboration

between di�erent ountries in edu ation (Portnoi, 2016) in luding in the area of s hool urri ulum and

textbook resear h and development has made rapid progression in terms of both breadth and depth. A

re ent and highly visible example in this �eld is the Shanghai Maths Proje t between China and the

UK (e.g., see Farrington, 2015; Sun, 2015). The proje t is a ollaborative e�ort between the textbook

resear hers, developers and publishers in the two ountries. Under the proje t, an award-wining series of

learning resour e books, yi ke yi lian (or One Lesson One Exer ise), whi h overed all the primary and

se ondary years of mathemati s urri ulum and were originally published in Chinese in China for more

than two de ades with regular revision were translated and published in English after adaptation following

the English national urri ulum.

In this arti le, we shall �rst provide a ba kground about the Shanghai Maths Proje t, and then drawing

on our experien e in the proje t, present a omparison of mathemati s urri ulum in the two ountries

with fo us on the primary level. The omparison is made in a variety of aspe ts in luding mathemati s

ontent, stru ture and approa hes. Some on rete topi s su h as fra tions, time, money, temperature,

and statisti s are used to highlight the di�eren es between the two ountries and how these di�eren es

in�uen e the adaptation of the learning sour es based on the English national urri ulum.

Finally, using the Shanghai Maths Proje t books as examples, we will examine how di�erent ulture

and so ial ba kground play an important role in the development and formation of s hool mathemati s

textbooks and other learning resour es, whi h has been under-resear hed in the resear h on textbooks as

argued by resear hers (Fan, Zhu & Miao, 2013).

Referen es

Farrington, Joshua. 2015. �Shanghai Maths Textbooks from Collins". The Bookseller (Retrieved

from http://www.thebookseller. om/news/shanghai-maths-textbooks- ollins )

Fan, Lianghuo, Zhu Yan and Miao Zhenzhen. 2013. �Textbook resear h in mathemati s edu ation:

development status and dire tions". ZDM-International Journal on Mathemati s Edu ation, 2013,

45 (5): 633-646.

Portnoi, Laura M. 2016. �Globalization and edu ation." In Poli y Borrowing and Reform in Edu a-

tion, edited by Laura M Portnoi, 87-119. New York: Palgrave Ma millan.

Sun, Shuangjie. 2015. �Are we e� ient enough?" China Daily, 25 August 2015.

STUDENTS OF YOUTH AND ADULT EDUCATION: APPROPRIATING DISCOURSES

ABOUT LEARNING (AND TEACHING) MATH AND THE ROLE OF TEXTBOOKS

Maria da Con eição Ferreira Reis Fonse a

Universidade Federal de Minas Gerais, Brazil

m frfon�gmail. om

Cibelle Lana Forneas Lima

Prefeitura Muni ipal de Belo Horizonte, Brazil

ibelleforneas�edu.pbh.gov.br

The dis ussion about the ne essity and possibilities of di�erent mathemati al approa hes mediated by

di�erent textbooks for di�erent subje ts is an important issue for the �eld of Youth and Adult S hool

Edu ation (EJA), sin e the right of edu ation for young and older people is still under onstru tion. In

Brazil there has been a movement, stronger in the �rst de ade of the 21st entury, to in rease a ess

to edu ational programs that meet the demands of di�erent so ial groups previously ex luded from the

s hool system. This movement also addresses on erns about o�ering alternatives that are better suited

to desires and needs of di�erent so ial groups. However, we are still far from produ ing ideologi al and

operational onditions for the realization of a truly in lusive edu ational proje t in Brazil - a proje t that,

at the moment, is even more threatened by the ountry's politi al situation.

115

The dis ussions we propose in this work show lassroom situations in whi h the students of EJA

themselves evaluate the approa h of textbooks and their use in tea hing mathemati s to adults. These

situations were dete ted during the observan e of an EJA lass, whi h was equivalent to Middle S hool, in

an Extension Proje t of a Brazilian publi university. Su h observations involved audio and video re ording

of math lasses during a semester, interviews, and detailed notes in a �eld diary.

Our dis ussion is supported by the dis ursive analysis (Bakhtin, 1997a; b) of the intera tions that

onstitute two episodes in whi h students assume positions: regarding the infantilization of the s hool

ontents approa h for the adult publi , when they evaluate the in�uen e of the use of textbooks produ ed

for hildren in a s hool for adults; and stressing dis ourses on the role attributed to textbooks in s hool

edu ation, when they demand attention to the produ tion of a mathemati s textbook for the adult publi

of Basi Edu ation. The analysis fo uses on the ways of appropriating dis ourses on tea hing mathemati s,

dida ti material and learning mathemati s in EJA, relating the on ept of appropriation to the question

of meaning (Smolka, 2000).

Referen es

Bakthin, Mikhail 1997a. Estéti a da riação verbal. São Paulo: Martins Fontes.

Bakthin, Mikhail (Volo hinov) 1997b. Marxismo e Filoso�a da linguagem. São Paulo, Hu ite .

Smolka, Ana. L. B. 2000.�O (im)próprio e o (im)pertinente na apropriação das práti as so iais�.

Cadernos Cedes 50 (20): 26-40.

MULTIPLICATIVE SITUATIONS IN BRAZILIAN MATHEMATIC TEXTBOOKS

APPROACHES TO DECIMAL NUMBERS

Ver�ni a Gitirana



Paula Moreira Baltar Bellemain


pmbaltar�gmail. om

Ernani Martins dos Santos

Universidade de Pernambu o

ermasantos�gmail. om

Several studies have dedi ated many e�orts on studying multipli ative stru tures approa hes fo using

natural numbers �eld. Nonetheless, very few investigations addressed the multipli ative tasks in other

theme of s hool mathemati s. This paper dis usses some results of a resear h regarding the types of

situations presented in Brazilian Mathemati s Textbooks for Final Grades of Elementary S hools for

de imal numbers, onsidering the lassi� ation given by Vergnaud (1982) for multipli ative stru tures.

The study was part of the proje t "OBEDUC - A Study of Multipli ative Stru ture (Bahia, Ceará and

Pernambu o)" (Santana, 2012). The three olle tions of textbooks for �nal grades of elementary s hools

used in the s hools, part of the proje t, were analysed regarding the tasks proposed or solved involving

multipli ative stru ture in the se tions dedi ated to the study of de imal numbers. Ea h of the task

was identi�ed and lassi�ed per: simple or mixed stru ture; and lass and sub lass of the situation

by Vergnaud's lassi� ation. The results reveal that 37% of the tasks have mixed stru tures, and a

on entration of simple proportionality (80%) and a 14,5% of multipli ative omparison. Only one task

involves Cartesian produ t (Bilinear problems) and no one multiple proportions. The omplexity of the

tasks is given to the mixed problems, mainly those that arti ulate additive and multipli ative stru tures.

The large majority of mixed problems involves an additive situation on atenated with a multipli ative one

or vi e-versa. One to many, Partition, Quotation are well distributed among the problems involving simple

proportionality, and fourth proportionality is less presented. In this �eld, some situations whi h one of the

data is missing, and that students bring from his/her previous knowledge, an follow two di�erent kind

116

of lassi� ation: partition or quotations, depending on the data students hoose to add to the problem.

Thus, it be omes a problem with �exible lassi� ation or an ambiguity to be lassi�ed. It also reveals a

ontra sense on the term used to lassify situations alled by one to many, when for de imal numbers, the

problems require the orrespondent to a number less than one.

Referen es

Santana, Eurivalda Ribeiro dos Santos. Um estudo sobre o domínio das Estruturas Multipli ativas

no Ensino Fundamental. Projeto de Pesquisa �nan iado pelo OBEDUC-CAPES, 2012.

Vergnaud, Gérard. 1982. �Cognitive and Developmental Psy hology and Resear h in Mathemati s

Edu ation: Some Theoreti al and Methodologi al Issues.� For the Learning of Mathemati s 3, no.

2: 31-41. http://www.jstor.org/stable/40248130.

MATHEMATICS TEXTBOOKS AS DIFFERENT INSTRUMENTS FOR DIFFERENT

TEACHERS

Ver�ni a Gitirana Rosilângela Lu ena


veroni a.gitirana�gmail. om rosi.s lu ena�gmail. om

It is not new that textbooks play a entral role in mathemati s tea her pra ti e. Nonetheless, on helping

tea her training pro ess, we need to understand textbooks as instruments of tea hers' pra ti e. This

paper will dis uss partial results of a resear h that analysis tea hers' do umentary work, from on eption

to resour es exploitation, with spe ial interest on olle tive pro esses on in-servi e tea hers' training. Thus,

we use the do umentary approa h (Gueudet & Trou he, 2008) as a theoreti al framework to understand

the di�erent instruments mathemati s textbooks be ome for di�erent tea her do umentation. This paper

will fo us on the �rsts results on how mathemati s textbooks are onsidered in tea hers' pra ti e, during

the �rst stage of the tea her training, in a re�e tive interview and elaboration of a resour e map.

As a resear h methodology, a pro ess of in-servi e mathemati s tea her training to integrate digital

Te hnologies was designed on the basis of: re�exive methodology; valorization of tea her as author and

resear hers of their pra ti es. A �rst step was to olle t their �rst valise with an interview about their

do umentation. Four tea hers were involved in this training.

An videographi analysis of the interviews within the maps drawn revealed that all these tea hers have

mathemati s textbook as prin ipal resour e to their pra ti e, nonetheless, for ea h tea her, mathemati

textbook is an instrument di�erent to the other. One of the tea her, the textbook is in the onne tion

between him and the students representing instruments for both and an instrument whi h shape the

ma ro- urri ulum for the a ademi year. The se ond tea her analyzed, textbook represents his font of

tasks, mainly fo using on manipulative material annexed on them. For him, what shapes his urri ular

organization is students feedba k on mathemati s perspe tive. The third tea her shows to use textbook

as an instrument sour e of tasks given to students. And the fourth one, use textbooks also as sour e of

ontext within tasks, fo uses her urri ular hoi e on students' motivations.

Referen es

Gueudet, G., & Trou he, L. 2008. Do Trabalho do umental dos professores: gênesis, oletivos,

omunidade. O aso da matemáti a (Trad. K. Ro ha, 2016). EM TEIA: Revista de Edu ação

Matemáti a e Te nológi a Iberoameri ana. 6, 1-43

117

AUTHENTICITY OF CONTEXTUALIZED PROBLEMS RELATED TO ACCELERATED

MOTIONS: A CASE STUDY OF MEXICAN SECONDARY-SCHOOL MATHEMATICS

TEXTBOOKS

Gabriel Guzmán Josip Slisko Adrián Corona Lidia Aurora Hernández

Fa ultad de Cien ias Físi o Matemáti as, Benemérita Universidad Autónoma de Puebla, Puebla, Méxi o

ggbine�gmail. om jslisko�f fm.buap.mx a orona�f fm.buap.mx lhernan�f fm.buap.mx

The motion problems have been present in mathemati s tea hing sin e the time of Fibona i and his

famous Liber Aba i (Siegler, 2003). Many of Fibona i's motion problems have been formulated for rather

arti� ial situations (lion in a well, two snake on a tower,. . . ). Today authors of mathemati s textbooks,

for tea hing appli ations of quadrati equations, use a elerated motions that ould be reasonably well

modelled mathemati ally.

We tried to answer two resear h questions: (1) How often authors of Mexi an se ondary-s hool

mathemati s textbooks for third grade use a elerated motions in Earth's gravitation �eld for illustrating

quadrati equations? (2) How authenti are ontextualized learning tasks related to these motions,

a ording to Palm's riterions (Palm, 2009)?

We found that authors of 35 mathemati s textbooks (published by 21 editorial ompanies) present

236 learning tasks involving di�erent types of a elerated motions. Of these tasks, 109 tasks orrespond

to free fall, 107 tasks are related to laun hing obje ts upward and 20 tasks treat two-dimension paraboli

motions. Regarding authenti ity, various learning tasks violate Palm's one or more Palm's riterions.

Very alarming are fa ts that majority of authors don't dis uss physi al meaning of the oe� ient in

orresponding quadrati equations and that some of them give arbitrary value for the free-fall a eleration

(di�erent from 9.8 m/s2) or revel onfusion related to di�eren e between an abstra t parabola in position-

time plane (for verti al upward and downward straight-line motion) and real spatial paraboli path. These

and similar defe ts in physi s-related mathemati al tasks were observed in textbooks published in other

ountries, too (Korsunsky, 2002).

Referen es

Korsunsky, Boris. 2002. �Improper Use of Physi s?Related Context in High S hool Mathemati s

Problems: Impli ations for Learning and Tea hing.� S hool S ien e and Mathemati s, 102 (3):

107-113.

Palm, Torulf. 2009. �Theory of authenti task situations.� In Word and worlds: Modelling verbal

des riptions of situations edited by B. Greer, L. Vers ha�el, W. Van Dooren, & S. Mukhopadhyay.

Rotterdam: Sense Publishers.

Sigler, Lauren e. 2003. Fibona i's Liber aba i: a translation into modern English of Leonardo

Pisano's Book of al ulation. New York: Springer S ien e & Business Media.

THE CONCEPT OF SAMPLING IN MATHEMATICS TEXTBOOKS IN THE 5TH AND

9TH GRADES: AN ANALYSIS OF THE PROPOSED ACTIVITIES

Tâmara Marques da Silva Gomes

University Federal of Pernambu o, Brazil

tamara_msg�hotmail. om

The tea hing of statisti al on epts through resear h is in the ore of the re ommendations for the learning

of Statisti s, on e it is onsidered as an alternative to linking both theoreti al and pra ti al knowledges.

One of the steps of a resear h pro ess is the data olle tion, in whi h sampling pro edures are applied,

and where it is ne essary to onsider the kind and the number of individuals to be investigated, and

118

how to approa h and question them e�e tively to be as mu h representative as possible. At the time of

the analysis and interpretation of the olle ted data from a statisti al resear h, the resear her needs to

onsider how the data was sele ted, whi h methods were used in this sele tion and whi h variables are to

be analyzed, in the pursuit to understand the investigated ontext so that it will be possible to ompare

it to other situations (GOMES, 2013). Therefore, in all these previously mentioned stages it is possible

to observe the sampling impli itly and expli itly.

The aim of this study was to analyze how mathemati s textbooks in the 5th and 9th years of Basi

Edu ation propose working with resear h situations that involve sampling-related on epts. It was arried

out an analysis of the (10) ten most used textbooks in the 5th year of publi middle s hools in Brazil

and the (11) eleven textbooks approved by the PNLD for the 9th year. In Brazil, the National Textbook

Program - PNLD evaluates the dida ti on epts and approa hes of textbooks so that the government

may distribute them in publi s hools, a ording to the tea hers' demands.

It was noted that in the textbooks used in the 5th s hool year, the proposed a tivities work on

the on epts of sampling only impli itly, by bringing resear hes that involve samples or by requesting

that students olle t data from a determined sample. However, it is important to emphasize that the

word �sample" is not used. Added to this, the tea her�s manuals do not bring any suggestions for

the development of a tivities that address this topi . Hen e, it is up to the tea hers' will and their

previous knowledge to bring this issue to lass and develop re�e tions about it. As for the textbooks

used in the 9th s hool year, the on ept of �sample" is approa hed more expli itly, and they also bring up

de�nitions, some examples besides proposing dis ussions about the reasons for using samples. However,

it was possible to note that in most textbooks this dis ussion is still arried out super� ially, without more

detailed onsiderations and, in most ases, from � titious situations.

Su h observations have brought us to the on lusion that there is a palpable need for authors of

textbooks to propose a tivities that foster the experien e of �sampling" as one of the phases of an

investigative y le, in parallel to the development of resear hes in lasses, aiming at valuing the so ial

fun tionality of this s ienti� and edu ational pro ess.

Referen es

Gomes, T. 2013. O todo é a soma das partes, mas uma parte representa o todo?: Compreensão

de estudantes do 5

o. e 9

o. ano sobre amostragem. Master's Thesis. Brazil: University Federal of

Pernambu o, Pernambu o, 109 p.

INTERACTIVE TEXTBOOKS: THE CASE OF FRACTIONS

Stefan Ho h, Kristina Reiss, Frank Reinhold, Bernhard Werner, Jürgen Ri hter-Gebert

Te hni al University of Muni h

stefan.ho h�tum.de, kristina.reiss�tum.de, frank.reinhold�tum.de, werner�ma.tum.de, ri hter�ma.tum.de

The proje t �ALICE:fra tions� aims at designing and evaluating an intera tive mathemati s textbook for

the introdu tion of fra tions in the lassroom. Sin e young students are regarded as �digital natives�

(Prenzky 2001), the transition from lassi al to ele troni textbooks seems to be a logi al step. However,

developing an e-book is not only digitizing text as key features of omputer-based learning environments

like intera tive exer ises, adaptive demands, or automati feedba k may be integrated.

ALICE en ompasses �rstly the design of an intera tive e-book on fra tions. This e-book follows a

onstru tivist point of view. In parti ular, the design takes into a ount that learning experien es are

supported by students' a tions as well as by visual and symboli representations (Bruner, 1966). The e-

book o�ers intera tive diagrams and manipulatives, thus a�ording opportunities for exploring mathemati al

on epts and operations. Feedba k is provided on most a tions performed by the students. It is task-

related, immediate, and tailored to the individual answer. Self-regulated learning is en ouraged and

119

supported by graded assistan e for the solution of problems. Moreover, the e-book adapts to a student's

pro� ien y by suggesting new problems based on the di� ulty levels of problems already solved. Inserting

data is fa ilitated by an automati re ognition of handwriting.

ALICE en ompasses se ondly the evaluation in grade 6 lassrooms. In a regular pre-test and post-test

design, 29 lassrooms were split into an experimental group and a ontrol group; six lassrooms were

equipped with tablet-PCs, twelve used a print version of the ALICE e-book and had thus no opportunity

for hands-on experien es, and eleven lassrooms worked with their regular textbooks. In all lassrooms,

fra tions and their spe i� properties were introdu ed. This study does not only provide data on students'

performan e but gives also insight into their problem-solving behavior. In parti ular, time on task is

measured by the e-book and may be used as an indi ator of the di� ulty of a spe i� type of task.

In our presentation, we will on entrate on the learning environment and will provide data on how

students use the di�erent omponents.

Referen es

Bruner, Jerome S. 1966. Toward a Theory of Instru tion. Cambridge: Harvard University Press.

Prensky, Mar . 2001. �Digital Natives, Digital Immigrants Part 1�, On the Horizon 9, no. 5: 1-6.

doi: 10.1108/10748120110424816.

MATHEMATICS AND ITS USEFULNESS: A BRIEF ANALYSIS OF TEXTBOOKS IN

MOZAMBIQUE

Abdul arimo Ismael

Universidade Lúrio, Mozambique

aismael�unilurio.a .mz

With the new dynami in re ent years in the textbooks elaboration, distribution and utilization pro esses

in Mozambique, arisen new and more authors and a tors in evaluation pro ess of the quality of the books.

If, on the one hand, the urri ulum of mathemati s in Mozambique re ommend the in orporation of

Mathemati al skills fo used on the development of students' thinking, showing the need for the learning

of mathemati s in solving the problems of life, something useful, on the other hand, it is known that the

ontent of a good textbook should be useful and meaningful, using lear language and ontaining many

examples (Karling, 1991). This aspe t orrespond, in some sense, the ethnomathemati al starting point,

whi h is bringing the world into the lassroom (Keitel et all, 1989; Gerdes, 2005). For more experien ed the

authors are, textbooks have sometimes numerous di�erent problems, espe ially the overhead of theorizing,

arti� iality examples, gratuity illustrations, among others (Neves, 2002). This study had as its main fo us

mathemati s textbooks of the primary and se ondary levels in use or that have been used in Mozambique,

produ ed by publishers based in Mozambique and was based on written work produ ed by master's

and undergraduate students in tea hing mathemati s at the Pedagogi al University in the subje t alled

Ethnomathemati s and Edu ation. It intended to �nd out the extent to whi h textbooks in orporate

a tivities, exer ises or realisti examples that an provide an insight into the usefulness of mathemati s

in general and parti ularly in Mozambi an so iety. Results show some obvious on ern of the authors,

however, as it seems, with limited spa e and reativity. The results also point to some theorizing and

abstra tion of ontent, and ertain inadequa ies in lo al ontextual terms, examples, exer ises, problems

and a tivities.

ARGUMENTATION AND MATHEMATICAL PROOF: SOME PROBLEMS IN BRAZILIAN

TEXTBOOKS

Ana Paula Jahn

USP, Brazil

anajahn�ime.usp.br

Aline dos Reis Matheus

CAEM-IME-USP, Brazil

aline.matheus�gmail. om

120

The mathemati al proof is a entral element in a ademi mathemati s and, among resear hers of Math-

emati s Edu ation; there is onsensus about its importan e also in s hool mathemati s that is pra ti ed

in Basi Edu ation. This onsensus does not represent a return to the formalist tenden y that guided

the tea hing of Mathemati s in Brazil until the 1950s, whi h is guided by the dedu tive organization of

mathemati al knowledge, within the Eu lidean model of exposition. In this tenden y, the tea hing was

entered in the presentation of the tea her, leaving the students a passive role before the knowledge.

Nowadays, on the ontrary, resear h ontributions in Mathemati s Edu ation show that the importan e

of working with argumentation and proof is pre isely in stimulating the a tive posture of students to-

wards mathemati al knowledge and that, rather than establishing the pra ti e of rigorous proof, it must

be an argumentative pra ti e that simultaneously respe ts and stimulates the development of students'

rationality (MATHEUS, 2016). However, the work with mathemati al argumentation and proof in Basi

Edu ation imposes a series of hallenges and questions. Here, we will fo us on the following question:

taking as referen e the urri ular re ommendations and those oming from resear h in Mathemati s Ed-

u ation, what problems are ommonly found in Brazilian textbooks regarding mathemati al proof? It is

a matter of great importan e, for two reasons. First, it is well known that the on epts onveyed by

textbooks have an important impa t on the on eptions and pra ti es of tea hers (SILVA, 2010). Se ond,

the adequate reintegration of mathemati al proof in Brazilian Basi Edu ation is a hallenge to fa e. In

this arti le, we will dis uss some of the main problems found in textbooks with respe t to the theme,

subdividing them into two ategories: spe i� problems and stru tural problems. The spe i� problems

- hasty generalizations; inadequate equivalen e between a proposition and its re ipro al; omission of jus-

ti� ation et . - have already been pointed out in several studies and have been the subje t of on ern

of the PNLD expert ommittee, the Brazilian National Textbook Program. Although it is important to

over ome this type of problem, it is ne essary analyse what we all here stru tural problems, su h as: the

presen e of demonstrations only from the �nal grades of Elementary S hool, without adequate preparation

for this; absen e of proposals that lead students to onje ture; ex essive emphasis on routine tasks that

do not require pra ti e of argumentation and proof et . Over oming these stru tural problems seems to

be essential if textbooks are to e�e tively ollaborate in the development of students' ability to argue and

prove in Mathemati s.

Referen es

Matheus, Aline dos Reis.�Argumentação e prova na matemáti a es olar". Dissertação (Mestrado

em Ensino de Matemáti a) � IME-USP, São Paulo, 2016.

Silva, Daniel Romão da. �Livro Didáti o de Matemáti a: lugar históri o e perspe tivas". Dissertação

(Mestrado em Edu ação) � FE/USP, São Paulo, 2010.

A COMPARISON OF DIFFICULTY LEVELS OF PROBLEMS INCLUDED IN

MATHEMATICS TEXTBOOKS AND PROBLEMS USED IN REAL CLASSROOM

Chunlian Jiang

University of Ma au, Ma ao

ljiang�uma .mo

Qin Li

Binhai Primary S hool, Shenzhen, China

liqin7720�126. om

Chunxia Qi


hunxia qi hxia�126. om

Mathemati s textbooks, as an important omponent of intended urri ulum, are the main sour es of

tea hers' mathemati s tea hing in real lassrooms. Chinese mathemati s tea hers always look for tea hing

materials from other resour es to supplement the ine� ien ies of problems in luded in the textbooks. In

this study, we ompare the di� ulty levels presented in mathemati s textbooks and the problems that

three tea hers in a se ondary s hool used in real lassrooms in a onse utive lessons over a topi in plane

geometry using the framework Bao (2004) developed for omparing problems presented in Chinese and UK

mathemati s textbooks. We also used Item Response Theory to analyze the di� ulty levels of problems

121

used in one lesson. We used a questionnaire to ask from what re ourses that the three tea hers sele ted

problems for their use.

The following results were obtained: (1) The number of tea hers used in real lassrooms are 3-5

time that were in luded in mathemati s textbooks they used and about half of the exemplary problems

used in real lassrooms are sele ted from other resour es. (2) The problems used by tea hers used in

real lassrooms are more di� ult than the problems in luded in mathemati s textbooks from the �ve

dimensions of Bao's framework (i.e., investigation, ontext, omputation, reasoning, and topi overage).

(3) The problems used in real lassrooms were arranged from the least to the most di� ult, however, the

problems assigned for homework are not arranged in similar ways. (4) Tea hers sele ted from syn hronized

tea hing supplementary materials but not from tea her's guide for the textbook series.

Keywords: Mathemai s textbooks; di� ulty levels; se ondary mathemati s; Pythagorean Theorem.

Referen es

Bao, J. (2002). A omparative study on omposite di� ulty of Chinese and British intended

mathemati s urri ula. Global Edu ation, 31(9), 48-52.

NORWEGIAN TEACHERS USE OF RESOURCES FOR PLANNING INSTRUCTION

Camilla Normann Justnes Olaug Ellen Lona Svingen

Norwegian Centre for Mathemati s Edu ation

amilla.justnes�matematikksenteret.no olaug.svingen�matematikksenteret.no

In Norway tea hers are free to de ide whether and how mu h they use urri ulum materials and other

resour es in tea hing mathemati s. There is a strong tradition of using urri ulum materials published

by ommer ial publishers. This material is often a pa kage that typi ally onsists of a student textbook

(in luding exer ises), a tea her guide, and web-resour es of various kinds (in luding, but not limited to,

exer ises, games, tests and �lms). This material thus has potentially a great impa t on instru tion in

Norwegian lassrooms. The la k of national quality assuran e of published urri ulum material led us to

investigate what kind of resour es tea hers use and how they use them.

We wish to jointly report from two ase studies in luding seven tea hers in upper primary s hool in

Norway. In both studies the tea hers were interviewed about whi h resour es they use, how they use them,

and the rationale for their hoi e of resour es. One of the studies then went on to analyse the ontent of

tea her guides in two frequently used urri ulum material pa kages, whereas the other study onsidered

the impa t of a professional development ourse on the tea hers� use of re ourses. This ourse was a lo al

PD-programme for in-servi e tea hers with the aim to implement inquiry-based learning in mathemati s.

The study followed the tea hers in the �rst four months of a one-year programme.

Although our studies revealed that a range of resour es seem to in�uen e these Norwegian tea hers'

planning, both studies on�rmed that the tea hers depend heavily on the urri ulum material pa kages used

at their s hool. Espe ially in the pro ess of planning their lessons, the analysis revealed an intensive use of

the student textbook and/or the tea her guide to de ide both mathemati al ontent, tasks and progression

for their instru tion. Moreover, we found that the tea hers who parti ipated in professional development

in luded more resour es and per eived themselves as less dependent on the urri ulum material.

Even though we found that the professional development ourse under onsideration had an impa t

on the tea hers� �exibility in hoosing and using re ourses, it should be noted that parti ipation in PD

is not systemati in Norway and varies between s hools and distri ts. Hen e, many tea hers are left with

the urri ulum material as their main resour e for tea hing. This puts the urri ulum material in position

to provide professional development for tea hers in Norway.

Referen es

122

Justnes, Camilla Normann. �Mathemati s Tea hers Working with Resour es : A Case Study of Four

Tea hers' Development.� Høgskolen i Sør-Trøndelag Avdeling for lærer- og, tolkeutdanning, 2013.

Svingen, Olaug Ellen Lona. � Analysis of Two Tea her's Guides - Chara teristi s and Tea hers Use

of Them.� Høgskolen i Sør-Trøndelag Avdeling for lærer- og, tolkeutdanning, 2014.

Further referen es in the two referen es above.

RUSSIAN POST-REVOLUTIONARY MATHEMATICS TEXTBOOKS: A SHORT LIVED

HISTORY

Alexander Karp

Tea hers College, Columbia University

apk16� olumbia.edu

The proposed presentation is devoted to Russian textbooks from 1918-1931 and onstitutes a ontinuation

of related studies by Karp, 2009, 2010, 2012.

After the revolution of 1917, an attempt was made to establish a new so ialist s hool, whi h reje ted

the traditions of drills and rote memorization, hara teristi of the old pre-revolutionary approa h. The

theoreti ally �new� approa h was in fa t a strange blend of Ameri an progressive edu ation; ideas developed

by the international reform movement, whi h arose from the International Commission on Mathemati s

Instru tion; ertain traditions of Russian pre-revolutionary demo rati pedagogy; and Soviet phraseology,

often in a rather primitive form. In the early 1930s, s hools were redire ted toward the old ways, and all of

these experiments were reje ted as left-wing perversions, and their authors were denoun ed as s hemers.

Surviving textbooks from this brief period are of onsiderable interest, if only as re ords of an attempt

at a new approa h to the problem of edu ational literature. In fa t, the very appearan e of these textbooks

was already a kind of ompromise � initially, it was believed that no textbooks should exist. And textbooks

that were published later until 1931 were indeed not always and in every respe t in tune with reality, but

the adaptation of a methodologi al idea to reality (espe ially su h a multifa eted reality as Russia's during

the �rst third of the twentieth entury) ordinarily requires more time than the experimenters were allowed.

A number of the ideas voi ed at the time, however, turned out to be � de ades later � on e more popular

and mu h in demand. The history of the attempt to plant approa hes borrowed from abroad on new soil

is also of interest in and of itself.

The presentation will analyze ertain textbooks, omparing them as far as possible with textbooks

used before them and after them, and also juxtaposing the development of mathemati s edu ation with

so ial-e onomi and politi al development. The presentation will also dis uss the authors of the analyzed

textbooks, their developments as �gures in mathemati s edu ation, and their lives.

Referen es

Karp, Alexander. 2009. �Ba k to the future: The onservative reform of mathemati s edu ation

in the Soviet Union during the 1930s-1940s." International Journal for the History of Mathemati s

Edu ation 4 (1): 65-80.

Karp, Alexander. 2010. �Reforms and ounter-reforms: S hools between 1917 and the 1950s." In

Alexander Karp & Bru e Vogeli, Eds., Russian mathemati s edu ation: History and world signi�-

an e, (pp. 43-85). London-New Jersey-Singapore: World S ienti� .

Karp Alexander. 2012. �Soviet mathemati s edu ation between 1918 and 1931: a time of radi al

reforms." ZDM/International Mathemati s Edu ation 44(4): 551-561.

123

TEACHER FIDELITY DECISIONS AND THE QUALITY OF ENACTED LESSONS

Ok-Kyeong Kim


ok-kyeong.kim�wmi h.edu

Tea hers make various de isions when they use urri ulum to plan and ena t a lesson. They de ide

whether to use the task (lesson or unit) in the urri ulum and, if so, how to use it. The urri ulum usually

in ludes various kinds of resour es regarding how to ena t the task (lesson or unit), su h as questions to

ask, and representations, models, and strategies to use. Tea hers de ide whether to use, modify, or omit

ea h of these elements provided in the urri ulum. I all su h de isions �delity de isions (FDs), whi h

indi ate various possible adaptations tea hers make as they use written lessons to design instru tion. One

important question to ask is how su h FDs impa t the quality of ena ted lessons, or the quality of the

transformation from the written to the ena ted.

This study examined the kinds of FDs and their impa t on the quality of the ena ted lesson, espe ially

those that support or hinder the a omplishment of the goals of the written lesson and those that promote

students' engagement at a high or low level of ognitive demand. Mathemati al points of the lesson and

ognitive demand are two important aspe ts that an indi ate the quality of instru tions and opportunities

for students to learn in the lesson. Resear h questions are: What �delity de isions do tea hers make

within individual lessons? What impa t do su h �delity de isions have on the ena ted lesson in terms of

mathemati al points of the lesson and ognitive demand?

The data used for this study were drawn from a larger proje t investigating tea her- urri ulum re-

lationship and tea her urri ulum use. In order to identify a general pattern, I analysed �ve tea hers'

ases, ea h of them using a di�erent urri ulum. Three onse utive lessons that ea h tea her ena ted

were examined along with the orresponding written lessons. First, I identi�ed tea her FDs-whether ea h

of them was per eived as a use (i.e., following the guidan e in the urri ulum), hange, or omission, or

whether any new elements were added. Then, I examined whether ea h FD supported the a omplishment

of the mathemati al points ( on eptual and pro edural aspe ts) of the written lesson and whether the

FD promoted high ognitive demand. Finally, I ompared the oding summary of individual tea her ases

to see the pattern and followed up on interesting aspe ts of the results.

The �ve tea hers revealed di�erent patterns of making FDs. For example, one tea her not only followed

the urri ulum guidan e, but also added many new elements to instru tion; another tea her added a lot

of new elements rather than using the provided guidan e. Overall, using the guidan e provided in the

urri ulum tended to a�e t the quality of the lesson positively. However, the impa t of using the provided

guidan e on ognitive demand varied a ross �ve urri ulums.

This study illuminates tea her apa ities that are needed to make appropriate FDs to transform the

written to ena ted lessons produ tively, su h as re ognizing important mathemati al points and addressing

them in instru tion, and noti ing and bridging gaps in the resour es provided by the written lessons. Also,

this study suggests urri ulum designers make the goals and intentions of tasks, a tivities, and lessons as

transparent as possible to tea hers. Simply listing goals at the beginning of the lesson and providing general

guidan e do not seem su� ient for tea hers to arti ulate mathemati al points and design instru tion toward

the points.

ENEM AND MATHEMATICS TEXTBOOKS FOR HIGH SCHOOL: AN ANALYSIS OF

THE VOLUME OF GEOMETRIC SOLIDS

Katy Wellen Meneses Leão Rosilângela Lu ena


katywellen�gmail. om rosi.lu enas �gmail. om veroni a.gitirana�gmail. om

124

Thousands of Brazilian High S hool students undertake ea h year the test of ENEM - National Exams

of High S hool, aiming to enter university. Textbooks, one of the most used resour es by tea hers and

students, should play an important role for students' development. We onsider that it should exist

ompatibility between what is approa hed in mathemati s textbook and the mathemati s knowledge

required in the main Brazilian evaluation of high s hool. After mapping ENEM questions regarding

Magnitude, we delimitated the study to volume. Thus, the study used the Theory Anthropologi of

Dida ti (CHEVALLARD, 1999) as its theoreti al framework. This proposed paper dis usses some of the

results of a resear h whi h aims to analyze ontributions given by mathemati s textbooks approa hes to

high s hool students to solve questions of volume of geometri solids of ENEM tests. By mapping the

questions, tasks and te hni s involved to solve them orre tly were identi�ed for ea h question. Then,

we analyzed one Brazilian Mathemati s textbook olle tion of high s hool, aiming to analyze whether

the textbook approa h to volume of geometri solids support those kind of tasks, exploring any of the

required te hni s and te hnologies whi h justify them. The six last editions of ENEM and one olle tion

of Brazilian Mathemati s Textbook for high s hool, approved in the textbook national evaluation program

- PNLD 2015. The tasks involved in ENEM questions were lassi�ed into six groups, among whi h those

involving de omposing and re omposing a solid to al ulate the volume. The results show that most

ontents, abilities, and ompeten es required to solve ENEM questions of volume were found in textbooks

approa h. However, the textbook approa h rarely orrelate it to other �eld of knowledge, as well as

to others blo ks of mathemati s, su h as proportion and per entage. Thus, its approa h shows to be

insu� ient. Questions of the textbook analyzed use largely dire t al ulation of the volume, using its

formula. They also show to be rarely ontextualized. Then, they show not to be su� ient to enable a

good support to ENEM test.

Referen es

Brasil, FNDE 2015. �PNLD 2015 - Coleções mais distribuídas por omponente urri ular - En-

sino médio�. A essed November 15, 2016, http://www.fnde.gov.br/programas/livro-didati o/livro-

didati o-dados-estatisti os.

Chevallard, Yves. 1999. �Analyse des pratiques enseignantes et dida tique des mathématiques:

l'appro he anthropologique. Re her hes em Dida tique dês Mathématiques�. Grenoble: La Pensée

Sauvage Éditions 19 (2): 221 - 266

HOW BOOKS FROM THE 6TH TO THE 9TH GRADE PROPOSE HORIZONTAL

TREATMENT OF COMBINATORICS

3

Ana Paula Lima Rute Borba

Universidade Federal de Pernambu o � UFPE, Brazil

lima.apb�gmail. om resrborba�gmail. om

Combinatori s is usually treated less formally in the early years of Primary S hool as guided by the National

Curri ulum Parameters (BRASIL, 1997) and in High S hool this ontent omes to full formalization, being

expe ted in Middle S hool (6th to 9th grade) an intermediate stage of formalization. In this sense, the

aim of the present study is to analyse how Horizon Content Knowledge, one of the six dimensions of

tea her knowledge proposed by Ball, Thames and Phelps (2008), is en ouraged in textbooks, so that it

may help tea hers in a gradual tea hing of Combinatori s along basi edu ation. Thus, it is expe ted that

textbooks, being instruments widely used by tea hers for the tea hing of Mathemati s, should propose

a gradual deepening of on epts, in parti ular those present in ombinatorial situations, from one s hool

grade to another and from one level of s hooling to another.

3

Proje t �nan ed by Fundação de Amparo à Ciên ia e Te nologia de Pernambu o (Fa epe).

125

In the present study, part of a do toral resear h in progress

4

, it is proposed an investigation on how

Combinatori s is treated in textbooks used by Mathemati s tea hers of the Colégio de Apli ação (CAp),

a Middle and High S hool maintained by the Universidade Federal de Pernambu o - UFPE. In this study,

analysis on erning the olle tion used at Middle S hool, in the hapters and topi s about Data Pro essing,

are presented. In the analysed olle tion, 56 a tivities were identi�ed involving ombinatorial situations,

being 13 a tivities in the 6th year, 20 in the 7th year, only 4 in the 8th year and 19 in the 9th year.

Throughout the a tivities, in the textbooks there are guidelines for the tea her on how to deal with

di�erent situations during ea h grade of Middle S hool. It is suggested to use strategies of gradual

formalization for the resolution of the situations proposed, starting with drawings, lists and multipli ative

reasoning in 6th grade and, in the following grades, that the use of trees of possibilities should be

in orporated, so students an observe patterns in di�erent and more omplex ombinatorial situations.

This type of presentation and of orientation of ombinatorial situations an support the development of

ollaborative a tivities among Mathemati s tea hers in this level of s hooling and also with the tea hers

who tea h in High S hool, who are responsible for the formal tea hing of Combinatori s.

Referen es

Ball, D.;Thames, M. and Phelps, G. 2008. �Content Knowledge for Tea hing: What Makes It

Spe ial?� Journal of Tea her Edu ation 59 (5): 389-407.

Brasil. 1997. Se retaria de Edu ação Fundamental. Parâmetros Curri ulares Na ionais (PCN):

Matemáti a. Ensino de primeira à quarta série.

FRACTIONS IN TEXTBOOKS AND MATHEMATICS TEACHERS' SPECIALIZED

KNOWLEDGE

Beatriz F. Litoldo, Marieli V. R. Almeida, Pris ila K. A. Costa, Miguel Ribeiro

Universidade Estadual de Campinas (UNICAMP), Brazil

beatrizfernanda_r �hotmail. om, marieli.almeida�outlook. om, prikabbaz�hotmail. om, mribas78�gmail. om

The entrality (and permeability) of fra tions in understanding many other mathemati al subje ts makes

their understanding important. Van de Walle (2009) states that the fra tions represent a great hallenge

for the students, sin e the in omprehension of this on ept results in varied di� ulties in the al ulations

involving them. This understanding, by the students, will only be possible if the tea her owns and

dominates. The textbook appears in the s hool s ene as an instrument that helps the tea her to share

his knowledge with his students, assuming the role of tea her tea hing aid, mediating the onstru tion of

knowledge. In this dire tion, it be omes essential a dis ussion of the ontent of this resour e asso iated

with the spe ialized knowledge of the tea her to approa h the theme.

Thus, we aim to analyze the spe ialized knowledge of the mathemati s tea her involved in under-

standing the tasks proposed in the textbook on fra tions. For this we use the ase study dis ussing the

approa h of the subje t in the textbook EJA Moderna (2013). Su h material is used in the 7th grade of

Elementary S hool, in the Youth and Adult Edu ation (EJA) modality SP. The textbook in question ad-

dresses fra tions in a hapter se tion, overing the on epts of fra tion, equivalent fra tions, simpli� ation

and omparison of fra tions, as well as addition and subtra tion of fra tions and per entage.

The main mathemati al fo us observed in the treatment of fra tions is part-whole, although the

on ept of reason in some exer ises is mentioned. When dealing with addition and subtra tion operations,

the material does not present the on ept of least ommon multiple (MMC), hoosing to transform the

fra tions with di�erent denominators into equivalent fra tions until rea hing a ommon denominator.

4

The general aim of the resear h is to investigate ollaborative a tivities among Mathemati tea hers of the CAp-UFPE

and how, from these a tivities, the tea hers mobilize di�erent forms of knowledge for the tea hing of Combinatori s.

126

Another option that ould repla e the use of the algorithm of resolution of the operations is the geometri

approa h, whi h ould fa ilitate the understanding of the students.

Regarding the mathemati al knowledge needed by the tea her to work with fra tions in the EJA, we

onsider important, besides knowing what a fra tion, its types and relations and knowing the algorithms,

knowing the best ways to approa h the subje t.

Referen es

Veridiano, Maria C. da S.; Lima, Diana M.; Miguel, José C.; Almeida, José J. P.; Veronese, Paula

C. de F.; Votre, Mayara M. 2013. EJA Moderna: Edu ação de Jovens e Adultos: anos �nais do

ensino fundamental. São Paulo: Moderna.

Van de Walle, John A. 2009. Matemáti a no Ensino Fundamental: Formação de professors e

apli ação em sala de aula. Porto Alegre: Artmed.

REDESIGNING OPEN TASKS IN MATHEMATICS TEXTBOOKS

Ant�nio José Lopes

Centro de Edu ação Matemáti a (SP), Brazil

bigode�pentaminos.mat.br

Joaquin Giménez

Universtat de Bar elona, Spain

quimgimenez�ub.edu

Yuly Marsela Vanegas

Universitat Autònoma de Bar elona, Spain

yulymarsela.vanegas�uab. at

The a tivities regularly presented in mathemati s textbooks for Primary Edu ation are losed, and without

hallenges (Boaler, 2015). Many authors state that the role of tea hers is to enlarge the textbooks

proposals by opening and hallenging them. In our theoreti al ba kground, we assume that tasks do not

exist separately from the pedagogies and dida ti s asso iated with their proposals. In these tasks, learners

an gain a ess to the mathemati s, and the transformation of the subje t (Jaworski, 2014).

The redesign pro ess, in the ase of textbooks, should be in reformulating the tasks, after being

hearing the students' voi es to learn from the lassroom experien e. We stablish a blog to transform the

onventional onne tions among tea hers, urri ulum developers, and author-resear hers in mathemati s

edu ation to improve the produ ts itself. Let us see an example, whi h introdu es the need of ollaborative

framework. It is a task for 10-11 years-old hildren: �Design the basis of a building a ording to the

following measurements. Use a sheet of paper to do ea h pla e. Cut ea h of the sites, and paste to the

building. 3 rooms (3m x 3m ea h); 1 dining room (4m x 6m); 1 dining room (3m x 6m); 1 orridor (5m x

2m); 1 kit hen (7m x 3m); 2 bathrooms (2m x 2m ea h) and 1 garden (3m x 12m)� (Lopes & Giménez,

2015). During the s hool experien e, one tea her noti ed that the task ould be more hallenging, by

observing hildren's drawings: Where is the orridor in your answers? Did you a hieve to have a re tangle?

The textbook's authors immediately talk about next edition, we will put a new senten e: �Dis uss about

your answers, with your olleagues� More examples will appear in the large presentation, in order to dis uss

redesign a tion pro esses with tea her-resear hers helping redesign pro ess.

Referen es

Boaler, J. 2009. �The Elephant in the Classroom: Helping Children Learn and Love Maths�, London.

Souvenir Press.

Jaworski, B. 2014. Mathemati s edu ation development. Resear h in tea hing and Learning in

pra ti e. In J. Anderson, M. Cavanagh, & A. Pres ott (Eds.), Curri ulum in fo us: Resear h guided

pra ti e (37th Annual Conferen e of the Mathemati s Edu ation Resear h Group of Australasia, pp.

2-23). Sydney: MERGA.

Lopes, A. and Giménez, J., 2015.Matemati a do otidiano 5. São Paulo. S ipione.

127

THE CHARACTERISTICS OF JUNIOR HIGH SCHOOL MATHEMATICS TEXTBOOKS

IN JAPAN RIGHT AFTER WWII

Naomi hi Makinae

Institute of Human S ien e, University of Tsukuba

makinae�human.tsukuba.a .jp

In 1949, the Textbook `Chugakusei-no-Sugaku [Mathemati s for Junior High S hool Students℄' written by

Ministry of Edu ation was published. In previous histori al resear h, this textbook has been re ognized

as eviden e that Ameri an progressive edu ation was imported for post-war mathemati s edu ation. And

the �rst unit `Jutaku [The House℄' has been pi ked up as typi al ase of this textbook and riti ized as a

poor tea hing material for mathemati s. O� ers of the Ministry did not agree with Ameri an progressive

edu ation and wanted to keep the Japanese way of mathemati s tea hing. Su h an intention of the

writers has been ompletely ignored in previous histori al riti ism. In order to onsider this point, I make

omparison between the textbook `Chugakusei-no-Sugaku' and the Ameri an original textbook `Everyday

Junior Mathemati s' written by William Betz. Through the omparison, I revaluate the unit `Jutaku' in

`Chugakusei-no- Sugaku'

Both textbooks have similar units `Jutaku' and `The House' that start from houses as everyday things,

and fo us on geometri al �gures from their shapes. But the order of mathemati al ontents di�ers. In

`The House', de�nitions of �gures are given �rst, and then properties and relationships are onsidered.

In ontrast, in `Jutaku' the onsideration of hara teristi s and properties of �gures from the houses

shapes is �rst, then de�nitions are made based on foregoing onsiderations. Furthermore, the role of

everyday things is di�erent. In `The House', everyday things are used as an introdu tion for mathemati al

ontents. In ontrast, in `Jutaku' the fun tion of mathemati s is pointed out from everyday things, and

then mathemati al ontents are re�ned. In summary, the tea hing based on everyday matters was in

ommon in both textbooks. But in "Chugakusei-no-Sugaku", we an see the intention of writers that

students reate mathemati s through mathemati al a tivities with everyday things.

Referen es

Betz, Willam 1944. Everyday Junior Mathemati s, New York: Ginn and Company.

Mathemati al Asso iation of Ameri a & National Coun il of Tea hers of Mathemati s 1940. The

Pla e of Mathemati s in Se ondary Edu ation The 15th Yearbook of the National Coun il of Tea h-

ers of Mathemati s, New York: Bureau of Publi ations Tea hers College Columbia University.

Monbusho 1949. Chugakusei-no-Sugaku [Mathemati s for Junior High S hool Students℄, Tokyo:

Tyutou-gakkou-kyoukasyo-kabushiki-kaisya, (in Japanese).

Progressive Edu ation Asso iation 1940. Mathemati s in General Edu ation, New York: D. Appleton-

Century Company.

BRAZIL'S MATHEMATICS TEXTBOOKS: AN OVERVIEW OF THE GOVERNMENT

POLITICIES

Lu as Carato Mazzi

University of Campinas (UNICAMP)

lu as arato12�gmail. om

Rúbia Bar elos Amaral

São Paulo State University (UNESP)

rubiaba�r .unesp.br

Brazil's government sin e the 20's has an interest in textbooks. In 1929, the �rst poli y measure was

developed in a sense to think the role and the spa e of the textbooks in Brazil's edu ation. In that year was

reated the National Textbook Institution (INL), whi h had the goal of being the spe i� organ to legislate

128

on the poli ies of the textbooks (Guimarães, 2011). However, this institute had its �rst attributions only in

1934, after Gustavo Capanema assumes his pla e as the edu ation minister. He stated the attributions of

INL were to help the edition of some literary works; to elaborate a national di tionary and en y lopaedia;

and to expand the number of publi libraries in the ountry (Tavares, 2014). Although they were great

propositions, it fails in a omplish its goals.

Two de ades after, in 1966 it was reated within the INL the Committee of the te hni al book and

textbook (COLTED), whi h aimed to oordinate the textbooks' produ tion, edition and distribution. This

ommittee made a partnership between the Ministry of Edu ation of Brazil (MEC) and the United States

Agen y International for Development (USAID). At this moment, on e more, Brazil's edu ation had an

in�uen e of other ountries. In 1971, COLTED was extinguished and the INL along with the publishers

started to promote the o-edition of the textbooks. Finally, in 1985, the government established a National

Program of Textbooks (PNLD). This program parti ipates from the analysis of textbooks itself until the

distribution to the s hools (tea hers and all the students of the publi s hool in Brazil), based on the

tea hers' hosen (Amaral, Ribeiro and Godoy, 2014).

Base on the history, the laws and the resolutions reated by the Ministry of Edu ation, this paper

aims to present these government poli ies sin e the 20's until the development of the urrently program

of textbooks (PNLD), omprehending their prin ipal ideas and goals, and identifying what those mea-

sures in�uen ed in Brazil's textbooks development. This Program have been ensuring the quality of the

textbooks, be ause the deep riteria to approve the textbook that an the used at most of the s hools.

Referen es

Amaral, Rúbia B.; and Ribeiro, Carlos M.; and Godoy, Juliana S. 2014. �Choosing textbooks without

looking at the textbooks: the role of the other's interpretations�. Pro eedings of the International

Conferen e of Mathemati s Textbook Resear h and Development 1, 154-158.

Guimarães, Fernanda M. 2011. �Como os professores de 6

o. ao 9

o. anos usam o livro didáti o de

iên ias�. Dissertação de Mestrado da Universidade Estadual de Campinas, 109p.

Tavares, Mariana R. 2014. �Editando a nação e es revendo sua história: O Instituto Na ional do

Livro e as disputas editoriais entre 1937-1991�. Aedos, 15 (6): 164-180.

ASSESSING A NEW EL SALVADOR MATHEMATICS TEXTBOOK: CASE STUDY �

HOW DOES IT CONTRIBUTE TO IMPROVE LEARNING?

Fran is o Mejía

Ministry of Edu ation of El Salvador

Fran_Ramos93�yahoo.es

Norihiro Nishikata

Japan International Coorporation Agen y

Nishikata.Norihiro�friends.ji a.go.jp

El Salvador is one of the ountries su�ering from low mathemati s a hievement. It is ommon to be

observed during mathemati s lasses that students parti ipate very passively, just listing to what tea her

says and opying what tea her indi ates. It is said that one of the reasons, above mentioned, ould

be shortage of learning materials to support students' learning pro ess. However, even there are some

materials at s hool level, usually they are not well used be ause of inadequate textbook ontents and

also ine� ient tea her's orientation. Therefore, Ministry of Edu ation of El Salvador (MINED) laun hed

ESMATE (El Salvador Mathemati s) proje t in 2015, re eiving te hni al support by Japan International

Corporation Agen y (JICA), with the aim of improving mathemati s a hievement. This proje t onsists

of three major omponents, new textbook development, implementation of National Plan of Continuous

Tea her Training (Plan National de Forma ión Contínua) and also reforming program of mathemati s

edu ation for the National University of El Salvador.

ESMATE proje t adopted hypotheti al strategies, fo using on 3 major fa tors whi h ould have signif-

i ant orrelation with the improvement of mathemati s a hievement, su h as providing learning material,

in reasing learning time during lass and tea her's learning assistan e.

129

1. New textbook development, also student workbook and tea her guide in a ordan e with appropriate

sequen e onsidering a tual a ademi level of student.

2. To guarantee su� ient students' learning time (approximately 20 minutes ea h math lass) so that

ea h student ould study a tively during lass, adding another learning opportunity at home with

student workbook.

3. To hange tea her's paradigm from talking a lot to se ure student learning opportunity to study

a tively by their own utilizing new textbook.

New mathemati s textbook has been developed from �rst to ninth grade in 2016, and it is expe ted

to be validated during 2017 s hool year, and to be distributed nationwide in 2018. Currently ESMATE

proje t is resear hing new textbook e�e tiveness in 8 pilot s hools. ESMATE proje t attempts to de�ne

textbook editing strategy following the next steps:

1. Analyzing the results of baseline survey regarding a tual student learning a hievement level.

2. Revising urri ulum sequen e and redistributed number of lasses per study topi in a ordan e with

student understanding level.

3. De�ning editing strategy as learning material not tea hing material.

4. Proposing learning pro ess to tea hers to se ure su� ient a tive learning time solving appropriate

problems whi h new textbook provides.

In this session, we would like to expose our textbook editing strategy sharing some examples, and also

intermediate term results of the new textbook e�e tiveness and appropriateness of the Proje t hypothesis

strategy.

AN ANALYTICAL FRAMEWORK FOR STUDYING THE IMPACT OF TECHNOLOGY

ON THE USE OF MATHEMATICS RESOURCES IN TEACHING AND LEARNING

Ida Mok

The University of Hong Kong

L.Fan�southampton.a .uk

Lianghuo Fan

The University of Southampton

ia mok�hku.hk

The development of te hnology in the �eld of mathemati s edu ation has been rapid and fast over the last

few de ades. There are signi� ant hanges with respe t to the tools of demonstration, devi es and display,

tools for al ulation, drawing and graphing, omputer-assisted learning platforms and internet. Te hnology

is obviously having an impa t on di�erent levels of urri ulum, in luding textbooks and a ting as an agent

of hange (Hut hinson and Torres, 1994). A ordingly, we believe it is imperative to advan e theoreti al

underpinnings, whi h have been largely under-developed, for resear h on the impa t of te hnology on the

development and use of mathemati s resour es.

In 2010 Fan proposed to in lude the te hnology prin iple in the �Six Prin iples for Textbook Develop-

ment�. Later, in his study of the impa t of te hnology on mathemati s textbooks in China and Singapore

for the last 15 years, Fan (2011) o�ered an operational de�nition for ICT for studying textbooks that in-

luded: al ulator, omputer, internet and software, and argued that in studying the impa t of te hnology

on mathemati s urri ulum one should in lude three questions: What to tea h? How to tea h? Why to

tea h? Applying Fan's framework, Mok analysed how te hnology was re�e ted in the junior se ondary

mathemati s textbooks in Hong Kong (Mok, 2104). The results showed the impa t of te hnology in

Hong Kong textbooks was a result of top-down urri ulum reforms in an Asian ontext, mostly guided by

the mathemati al ontent stipulated in the urri ulum and the development lustered around three major

ategories: self-learning platforms, IT a tivities making use of di�erent software and proje ts making use

of internet resour es.

130

Drawing on our earlier resear h as well as related literature in this area, in this paper, we propose a

revised analyti al framework for investigating the impa t of te hnology in the use of mathemati s resour es

in tea hing and learning and dis uss its impli ations in this �eld of resear h on textbooks.

Referen es

Fan, L. 2010. �Prin iples and pro esses for publishing textbooks and alignment with standards: A

ase in Singapore.� Paper presented at the APEC Conferen e on Repli ating Exemplary Pra ti es

in Mathemati s Edu ation. Koh Sumai: Thailand.

Fan, L. 2011. �Re-looking at the Impa t of Te hnology on the Development of Mathemati s

Curri ulum.� Presentation at the 10th International Conferen e on Te hnology in Mathemati s

Tea hing. Portsmouth, UK.

Hut hinson, T., & Torres, E. 1994. �The textbook as agent of hange.� ELT journal 48 (4):

315-328.

Mok, I. A. C. 2014. �How te hnology use is being re�e ted in junior se ondary mathemati s text-

books in Hong Kong.� In Pro eedings of the International Conferen e on Mathemati s Textbook

Resear h and Development (ICMT-2014), edited by K. Jones, C. Bokhove, G. Howson & L. Fan,

339-344. Southampton, UK: University of Southampton.

ADAPTATION OF MATHEMATICS TEXTBOOK TO DAILY SITUATIONS OF

STUDENTS IN MOZAMBIQUE: CASE OF 8TH CLASS

Saide Issufo Momade

Universidade Pedagógi a de Moçambique

saidinhomomade�hotmail. om

The textbook is one of the most important resour es for tea hing - learning mathemati s. For this reason,

a number of studies have examined the use of mathemati s textbooks by tea hers. This study examines the

extent to whi h tea hers adapt textbooks suggested exer ises to real life appli ation. That is be ause the

mathemati s tea hing program in Mozambique suggests that tea hing - learning in mathemati s should

equip students with the basi skills needed to solve problems and explore situations based on experien es

of everyday life. The present empiri al investigation results from analysis of the adequa y between the

program ontents, textbook exer ises and students' daily problems. The literature review fo uses on

the tea hing of mathemati s program of 8th grade, in relation to the proposed methodology and the

onsisten y with the exer ises presented in the textbook. In order to raise questions related to the daily

exer ises of the students, this study evaluates the exer ises in relation to their usefulness in solving the

problems of the day to day. Then, the results of the study are outlined in this arti le fo using on the

ommon and divergent points between the tea hing program and the ontents presented in the textbook

of 8th grade mathemati s tea hing in Mozambique. The analysis has shown opportunity for tea hers to

use mathemati games in order to bring the ontents into students' real life.

Referen es

BORIN, J. 1998. Jogos e resolução de problemas: uma estratégia para as aulas de matemáti a.

3.ed. São Paulo: IME/USP,.

CHEVALLARD, Y.1985. La Transposition Dida tique. Du savoir savant au savoir enseigné. Greno-

ble: Pensées sauvages

MEC/INDE. 2007. Plano urri ular do ensino se undário geral. (PCESG), Maputo.

131

INDE/MINED. 2010. Programa, Matemáti a 8

a. lasse, ed.

© INDE/MINED - Moçambique,

Maputo.

GANDRO, R.C. 2000. O onhe imento matemáti o e o uso de jogos na sala de aula. Tese.

Doutorado. Universidade de Campinas. Campinas: Uni amp.

RÊGO, R.G.; RÊGO, R.M.2000. Matemáti a Ativa. João Pessoa: Universitária/UFPB, INEP,

Comped.

SMOLE, K.S.; DINIZ, M.I.; MILANI, E.2007. Jogos de matemáti a do 6

o. ao 9

o. ano. Cadernos doMathema. Porto Alegre: Artmed.

LINGUISTIC, CULTURAL AND PEDAGOGIC DIMENSIONS OF GEOMETRY:

NAVIGATING TEXTBOOK DEVELOPMENT IN A CROSS-NATIONAL PROJECT

Candia Morgan

University College London Institute of Edu ation

andia.morgan�u l.a .uk

Teresa Smart

University College London Institute of Edu ation

teresa.smart�u l.a .uk

Natalya Panikarskaya

The Nazarbayev Intelle tual S hools

panikarskaya_n� .nis.edu.kz

Arman Sultanov

The Nazarbayev Intelle tual S hools

sultanov_a� ep.nis.edu.kz

To support the introdu tion of new urri ula a ross ten se ondary s hool subje ts, in luding mathemati s,

a ademi s from UCL Institute of Edu ation (IOE) have been working with the Nazarbayev Intelle tual

S hools (NIS) in Kazakhstan to develop textbooks. A entral omponent of this work has been to support

the development of lo al Kazakhstani tea hers to be ome independent textbook authors. During the

�rst three years of the proje t, we have worked with an author team of six experien ed and su essful

mathemati s tea hers - and an interpreter. In this paper we will explore some of the issues that we have

fa ed and lessons we have learnt about international ollaboration of this type. See (Bridges, 2014) for

ba kground to edu ational reform in Kazakhstan and the role of NIS.

The original proje t brief suggested that the major fo us of the IOE �experts� should be on guiding the

authoring of textbooks that would in orporate the �best pra ti e� pedagogy desired by the NIS leadership,

aiming to develop problem solving, reativity, independent learning and language awareness, supporting

the national tri-lingual edu ation poli y. The author team shared these aims but we soon dis overed

fundamental di�eren es in our understandings of tea hing, learning and mathemati s that have aused

us to problematize the brief. As might be expe ted, ommuni ating mainly through an interpreter has

been hallenging (the authors all speak Russian as their �rst language and most have very limited English

� while the IOE team has even more limited Russian). However, as we have struggled to understand

ea h other, we have also ome to learn more about linguisti and ultural di�eren es that annot be

resolved simply by � orre t� translation. In ontradi tion of the ommon assumption that the language

of mathemati s is universal, we have found ategories of mathemati al obje t that are distinguished and

named in one language but not in the other. This phenomenon may in some ases lead to di�eren es

in the stru ture and emphasis of the urri ulum and hen e di�eren es in the ontent and organisation of

textbook material.

While linguisti and ultural di�eren es and di�eren es in pedagogi al beliefs have arisen a ross all

areas of the mathemati s urri ulum, geometry has proved to be espe ially hallenging. In our presentation

we will fo us on two major issues:

• relationships between the axiomati stru ture of Eu lidean geometry and beliefs about tea hing and

learning

• the nature of de�nitions of geometri obje ts (su h as angle) and how these relate to the learning

experien es that may be o�ered to students and to their experien e of the physi al world.

132

Referen es

Bridges, David, ed. 2014. Edu ational Reform and Internationalisation: The Case of S hool Reform

in Kazakhstan. Cambridge: Cambridge University Press.

TEXT BOOK ANALYIS: A CASE FOR UGANDA TEXTBOOKS

Imma ulate Kizito Namukasa

Western University

inamukas�uwo. a

The resear h questions of the inquiry reported in the presentation were: What is the hara ter of s hool

mathemati s, as evin ed a ountry's approved textbooks? And what does this hara ter say about the

nature of s hool mathemati s taught in the ountry. I take the ase for Ugandan mathemati s text-

books. Spe i� ally, I onsider the nature of ontent emphasized, the pedagogi al approa h depi ted, the

assessment pra ti es shown, the real-life ontexts used as well as an illary materials mentioned in the

textbooks. As a result of this analysis, three broad hara teristi s of Ugandan s hool mathemati s are

revealed in luding: the esoteri , intelle tually oriented and formal nature of the ontent taught, tea hing

pra ti e dominated by written, national examinations, and, pedagogi al presentation that uses minimal

lo al and urrent ontexts.

With de entralization and universalization of edu ation in the 1990s and privatization of tertiary

edu ation in 2000, the system of edu ation in Uganda, Afri a is undergoing rapid hanges that in lude

in reased demand for se ondary edu ation and growth of private s hools and universities. Competitive

examinations for entran e to se ondary grades, as eviden ed in edu ational frameworks of the olonial

era persist. The language of instru tion is English. The reported analysis is based on textual analyses of

textbooks on erning primary and se ondary s hool urri ula. Textbook analyses have been ommonly

used to get to the ta it hara ter of urri ula (Mau h & M Dermot; 2007). Textbooks often in�uen e

the nature of ontent-what is learned and lassroom a tivities - how, when and in what order the ontent

is learnt (Robitaille & Travers, 1992). Analysis of the visible elements of textbooks reveals aspe ts of the

invisible elements su h as the lived urri ulum and instru tion.

In the presentation, I share �ndings from the analysis on: (i) the nature of the ontent and the

presentation style, (ii) the textbook pedagogy, (iii) in orporation of an illary materials and assessment,

and (iv) the nature of real-life ontexts used in the textbooks. It should be noted that an empiri al

study of the examination system, of tea her edu ation programs, and other mathemati s tea hing and

learning resour es as well as �eld work in s hools and edu ation o� es might reveal more or even some

ounter hara teristi s about the urri ulum and pedagogy of Ugandan s hool mathemati s. In general,

three broad hara teristi s about Ugandan s hool mathemati s an be identi�ed: (a) It is mainly esoteri ,

intelle tually oriented and formal; (b) Its tea hing pra ti e is dominated by written, tests; and ( ) It is

presented in a way foreign to learners and uses minimal lo al and urrent ontexts.

Referen es

Robitaille, David. F and Travers, Kenneth (1992). �International studies of a hievement in mathe-

mati s.� In Grouws, Douglas A., and National Coun il of Tea hers of Mathemati s. 1992. Handbook

of resear h on mathemati s tea hing and learning. Toronto: Ma millan.

Mau h, Elizabeth and Megan M Dermott. 2007. �Can elementary mathemati s textbooks be im-

proved to fa ilitate student understanding of mathemati s?� Mathemati s and Computer Edu ation

41 (2): 127-136.

133

LEARNING MATHEMATICS WITH VIDEOS

Liliane Xavier Neves

Universidade Estadual de Santa Cruz, Brazil

lxneves�ues .br

Mar elo de C. Borba

UNESP - IGCE - Rio Claro, Brazil

mborba�r .unesp.br

Hannah Dora Gar ia La erda

UNESP - IGCE - Rio Claro, Brazil

hannahdorala erda�gmail. om

In this work we dis uss issues related to the in�uen e of digital te hnologies on students' daily lives in

s hool and how the intera tion with digital videos has hanged the types of edu ational materials used

to learn. Videos an be understood as part of the transformation pro ess of textbooks. Elements that

hara terize audiovisual materials take edu ational tools to a di�erent level, making videos a sour e of

edu ational information permeated by visual elements, orality, gestures, body expressions and sounds,

given their multimodal nature (Walsh, 2011).

These ideas are entral to the resear h entitled �Digital videos in distan e learning mathemati s",

developed by members of the Resear h Group in Informati s, other Media and Mathemati s Edu ation.

This resear h proposes the joint produ tion of videos by students and tea hers expressing mathemati al

ontent dis ussed in the lassroom, from a Freirean perspe tive (Freire, 2003). Videos will play multiple

roles: they are textbook as they may introdu e a topi , at the same time tea hers may use them as a

tool to evaluate how students see and understand mathemati s. Video is one of the most, if not the

most ommon, means of ommuni ation among university students. Videos produ ed by a olle tive of

humans-with-media may be ame a digital obje t for others to learn. Video is produ ed by humans-and-

te hnologies and it shapes humans as well. Digital te hnologies intera t with the human being in the

produ tion of knowledge, shaping not only the way knowledge is produ ed, but also helping to onstitute

what humans may be ome (Borba, 2012).

In these times when our sense distin tion is being transformed, Boll and Axt (2011) laim that edu ation

must be onne ted to a digital wisdom. In this way, this resear h fo uses on an annual video festival that

brings mathemati s students together. With this festival we seek to intervene in the virtual environment

by stimulating the produ tion of videos with mathemati al ontent by students and tea hers together

and reate a lo us for the sharing of these digital edu ational materials. We also intend to motivate the

students to a hange of posture allowing a more a tive role in the universe that involves their learning.

Referen es

Boll, Cíntia. I. and Axt, Margarete. 2011. �Fen�menos de sentidos éti os e estéti os em suspensão".

RETEME 1 (1): 45-52.

Borba, Mar elo de C. 2012. �Humans-with-media and ontinuing edu ation for mathemati s tea h-

ers in online environments". ZDM - The International Journal on Mathemati s Edu ation. 44:

802-814.

Freire, Ana Maria Araújo. 2003. Cartas a Cristina: Re�exões sobre minha vida e minha práxis,

edited by Paz e Terra.

Walsh, Maureen. 2011. Multimodal Litera y: Resear hing lassroom pra ti e, edited by Primary

English Tea hing Asso iation (e:lit).

THE PROCESS OF CHOOSING AND USING THE TEACHING BOOK BY TEACHERS

OF MIDDLE SCHOOL IN ITAITUBA WEST OF PARÁ

José Ant�nio de Oliveira Júnior

Federal Institute of S ien e and Te hnology of Pará, Campus Itaituba

jose.antonio�ifpa.edu.br

134

The textbook assumes the role of great importan e within the s hool ontext, to understand its use,

it requires to re ognize several aspe ts from its on eptualization, produ tion and hoi e. A ording to

Batista(2000) many s holars like Oliveira (1968), Lajolo (1998), Soares (1968) and Zilberman (1998)

defend this in lusion in the ategory of textbooks as a set of texts well before the invention of the press.

Regarding mathemati s the resear her Magda Soares ites that the dida ti book had its origin with the

Greeks and that the work of Eu lides �The Elements� had ir ulation like s hool text from 300 a.C.

Thus mathemati s must be understood throughout history as a pro ess of produ tion and onstru -

tion of human knowledge, where we emphasize its aspe t of human a tivities, its pure part of the very

onstru tion of mathemati s, its aestheti s and also emphasized the dedu tive method.

Furthermore, people are at the same time subje ted to information that must be understood in a riti al

way that requires mathemati al knowledge, su h as notions of al ulation, treatment of information,

magnitudes and measures, spa e and forms, and others.

Hen e this work envisages a perspe tive of a tea hing of mathemati s that an not leave aside, nor

negle t all these aspe ts mentioned previously and spe i� ally in what on erns the approa h of the

ontents of the textbooks of mathemati s in the last stage of basi edu ation, that is, as Do tea hers

hoose and use the math textbook to tea h their lasses for e�e tive and meaningful math tea hing and

learning?

The proposal of this work lists how the high s hool tea hers of the muni ipality of Itaituba do Pará

hoose and use the textbooks of mathemati s, that is, what are the riteria adopted in the hoi e? What

is the privileged approa h? And what literature relevant to the tea hing and learning of mathemati s?

The on eptual aspe ts? The use of problem solving?

Within this panorama the analysis will be done with questionnaires applied to mathemati s tea hers

who tea h in high s hool in all the s hools of the muni ipality with the obje tive of answering the questions

listed previously, in a future perspe tive of understanding the hoi es of the mathemati al olle tions

adopted For use in the s hool and how they develop the ontents with the students, and more, with the

analysis of the results of the questionnaires, we will have as the main spring of this work to laun h a

proposal for the next hoi e of the textbook in high s hool, taking as a parameter the re ommendations

Of o� ial do uments su h as PNLD, PCNS, among others, a ording to the lo al needs of tea hers, in

line with how students learn the ontents signi� antly.

Referen es

BATISTA A. A. G., Re ommendations for a Publi Poli y on Dida ti Books.Brasil: Ministry of

Edu ation, Se retariat for Basi Edu ation, 2001

LAJOLO, M. Textbook: one (almost manual) user. In: In Open, year 16, n.69.

OLIVEIRA, A. L. From The textbook. Belo Horizonte: Ed. Bernardo Alvares, 1968.

TEXTBOOKS ON DESCRIPTIVE GEOMETRY TRANSMITTED TO BRAZIL:

RECEPTION AND DIFFUSION

Thiago Ma iel de Oliveira

Universidade do Estado do Rio de Janeiro

thiagomoliveira�gmail. om

Viní ius Mendes Couto Pereira


vini iusmendes pereira�hotmail. om

The arti le presents the re eption of textbooks on des riptive geometry in Brazil. The historiography

of this dis ipline fo us on the Brazilian translation of Gaspard Monge's book and its use at the Royal

Military A ademy of Rio de Janeiro. Do uments re ently found in publi ar hives show the presen e of

other textbooks, like Silvestre-Françoix La roix's �Des riptive Geometry�. The arti le will also presents

Brazilian textbooks on this subje t, published during the 19th and 20th enturies by professors of several

135

s ienti� institutions (Polyte hni S hool of Rio de Janeiro, S hool of Mines at Ouro Preto, Polyte hni

S hool of São Paulo and National S hool of Fine Arts), like the Brazilian translation of the Éléments de

Géométrie Des riptive ave nombreux éxer i es par FIC. Brazilian publi ations about des riptive geometry

show the re eption of new developments on the subje t developed within the Fren h ommunity. The

arti le will highlight Alvaro José Rodrigues' work. He was a former professor at the National S hool of Fine

Arts and published three volumes on des riptive and proje tive geometry during the �rst half of the 20th

entury. His work is hara terized by the relations that the author establishes between the onstru tive

methods of the dis ipline and its history.

Referen es

Gabaglia, Eugênio de Barros Raja. 1946. Elementos de Geometria Des ritiva om Numerosos

Exer í ios. Rio de Janeiro: F. Briguiet.

Monge, Gaspard. 1799. Géométrie Des riptive, Leçons donées aux é oles normales l'an 3 de la

Republique. Paris: Baudoin.

Sakarovit h, Joel. 1998. Epures d'ar hite ture: De la oupe des pierres a la géométrie des riptive

XVI - XIX sié les. Basel: Birkhäuser.

TEXT ANALYSIS ON FRACTIONAL CALCULUS CONCEPT

Adrian Muñoz Oroz o Flor Monserrat Rodríguez Vásquez Martín Patri io Ár iga Alejandre

Universidad Autónoma de Guerrero

adrianmunoz1993�hotmail. om �ormonr�hotmail. om mpar iga�gmail. om

Sometimes mathemati al on epts are presented in urrent tea hing as �nished obje ts and we forgot its

histori al pre edent, hen e the de ontextualization of its own genesis is one problemati in mathemati s

edu ation. (Cantoral and Farfán, 2014; Ana ona, 2003). In this regard, histori al and epistemologi al

resear h is good resour e both to strengthen the learning pro esses of students, as for initial tea her train-

ing in mathemati s (Sierra, 2000; Gua aneme, 2008; Ar avi, 1991). On the other hand, in mathemati s

edu ation �eld, fra tional al ulus is a mathemati al on ept developed little and it has a high potential

in the present time. For example, this on ept allows the modelling of population growth phenomena,

whi h do not �t ideally to the usual models su h as exponential fun tion, but it allows explain better

the behaviour of population. So, attend the de ontextualitation of mathemati al on epts, a ontribution

towards edu ational resear h is the histori al analysis of the Fra tional Cal ulus.

Therefore, in this resear h we show some histori al and epistemologi al aspe ts that allowed the

development of the on ept of fra tional al ulus. Therefore, we analysed text histori al books using

qualitative texts analysis methodology (Ku kartz, 2014) and using software QDA.

Referen es

Ana ona, Maribel. 2003. �La Historia de las Matemáti as en la Edu a ión Matemáti a.� EMA 8

(1): 30-46.

Ar avi, Abrahan. 1991. �The experien e of history in mathemati s edu ation: Two bene�ts of using

history.� For the Learning of Mathemati s Revista 11 (2): 11.

Cantoral, Ri ardo and Farfán, Rosa. 2004. Desarrollo on eptual del ál ulo. Méxi o: Interna ional

Thomson Editores.

Gua aneme, Edgar. Una aproxima ión a la rela ión Historia de las Matemáti as-Cono imiento del

profesor. Conferen ia presentada en el �Ter er en uentro de Programas de forma ión ini ial de

profesores de matemáti as�, 24 a 25 de Abril 2008 en Bogotá.

136

Ku kartz, Udo. 2014. Qualitative Text Analysis: A Guide to Methods, Pra ti e and Using Software.

London: Sage publi ations.

Sierra, Modesto. 2000. �El Papel de la Historia de la Matemáti a en la Enseñanza.� Números 43

(1): 93-96.

A STUDY ABOUT TRANSMISSIONS OF CALCULUS TEXTBOOKS TO BRAZIL

Vini ius Mendes Couto Pereira


vini iusmendes�id.u�.br

The present study relates of the resear h to analyze how the Brazilian mathemati al ommunity be ame

established. Thus, the pro ess of the emergen e of a mathemati al ommunity in a ountry that initially

o upied a peripheral position implies a hange in the fo us from tea hing to resear h - that is from the

use of textbooks imported from a metropolis until the moment in whi h proper produ tions were a hieved.

In this sense, we have studied how the pro ess of transmitting the basi on epts of al ulus o urred,

analysing whi h on eptions of analysis, prevailing in one of the metropolis ountries, were transmitted to

Brazil and how the transmitted knowledge was transformed and reworked in order to onstitute a proper

produ tion. In this intention we have investigated the pro ess to transmission of al ulus textbooks

to Brazil starting with the �Traité Eléméntaire de Cal ul di�érentiel et de Cal ul intégral� (1802), by

Sylvestre-François La roix already translated into Portuguese as Tratado Elementar de Cal ulo Di�eren ial

e de Cál ulo Integral, the �rst al ulus textbook used in Brazil, from 1812, and the version published in

1842 by José Saturnino da Costa Pereira, Elementos de Cal ulo Di�eren ial e de Cal ulo Integral, segundo

o sistema de La roix, being the �rst work of proper produ tion in Brazil. In the following de ades, one

ontinued, however, to transmit foreign textbooks, mainly Fren h ones, so that for a long time the work

of Saturnino remained the only al ulus textbook being an own produ tion in Brazil.

In this way we have identi�ed the main hara teristi s of the al ulus textbooks transmitted to Brazil,

as well as, the di�erent generations transmitted, in the sense established by Zerner (1994).

We intend to present a study of various aspe ts of al ulus textbooks transmitted, omprising the

period from 1810 - the year of founding the �rst institution of higher edu ation - to 1934, the year in

whi h the Italian mathemati al s hool arrived at the Universidade de São Paulo, as well as to reveal, whi h

were the relevant proper produ tions in analysis.

Referen es

La roix, Silvestre François, 1812. Tratado Elementar de Cál ulo Diferen ial e Cál ulo Integral.

Tradução de Fran is o Cordeiro da Silva Torres. Impressão Régia. Rio de Janeiro.

Pereira, José Saturnino da Costa, 1842. Elementos de Cál ulo Di�eren ial e de Cál ulo Integral,

segundo o sistema de La roix, para uso da Es ola Militar. Typogra�a Na ional. Rio de Janeiro.

S hubring, Gert, 2002. �A Framework for Comparing Transmission Pro esses of Mathemati s to the

Ameri as�, Revista Brasileira de História da Matemáti a, vol. 2, no. 3: 45-63.

Zerner, Martin, 1994. La Transformation dês Traités Français d'Analyse (1870-1914). Laboratoire

J-A. Dieudonné. Université de Ni e-Sophia-Antipolis. Ni e.

MATHEMATICS TEXTBOOKS IN EAD: CHATEREZING THEM AS DIFFERENT

TEACHERS' INSTRUMENT

José Wilson Pereira Ver�ni a Gitirana


wilsonquinho�gmail. om veroni a.gitirana�gmail. om

137

Textbook is one of the main resour e that Distan e Learning Institutions makes available in the virtual

learning environment (VLE) for tea hers and students. A ording to Brazilian Referen es for Distan e

Edu ation (Brazil, 2007), it should be organized to enable students to learn. The ontent-tea her, as

known textbook authors at Distan e Learning ourses, to produ e his/her work material, must follow

some methodologi , epistemologi and politi prin iples that also should be expli it in its Institutional

Pedagogi Proje t. It re ommends that textbook should allow students and tea hers (trainer and tutor)

to onstru t knowledge in intera ting themselves. This paper presents some results of a resear h that

investigates Distan e Learning tea hers' on eptions of the textbook made available by the institution and

s hemes of instrumental a tions they plain. The theoreti al framework was the Theory of Instrumental

Or hestration (Trou he, 2005), fo using on understanding dida ti on�guration. Three tea hers of

Analyti Geometry ourses from two Distan e Learning Courses of Mathemati s tea her training under-

graduations omprised our subje ts. As regards the method, we adopted some te hni s used in the Dida ti

Do umentary Approa h (Guedeut & Trou he, 2008). We visit tea her's virtual lass in the VLE, then an

interview with ea h tea her was undertaken, �nally we request them to draw a map of the resour es used

by them in lasses. Our results reveal the need to in rease in the textbooks, during mediation, regarding

the ontents, solved tasks, worksheets, hyperlinks towards video external to the VLE. It is also lear the

need to think about a olle tive onstru tion of the textbook, dis ussing in parallel with the main authors

of distan e edu ation, tea hers and exe utors tutors.

Referen es

Brazil, SED-MEC. 2003. Referen iais de Qualidade para Cursos a Distân ia. SED/MEC: Brasilia.

Gueudet, G., & Trou he, L. 2008. �Do Trabalho do umental dos professores: gênesis, oletivos,

omunidade. O aso da matemáti a� (Trad. K. Ro ha, 2016). EM TEIA: Revista de Edu ação

Matemáti a e Te nológi a Iberoameri ana. 6, 1-43, 2016.

Trou he, L. 2005. �Constru tion et onduite des instruments dans les apprentissages mathématiques:

né essité des or hestrations�. Re her hes en Dida tique des Mathématiques. v..25, pp. 91-138.

SOCIAL REPRESENTATIONS OF LEARNING MATHEMATICS BY TEACHERS OF

EARLY YEARS OF ELEMENTARY SCHOOL AND THE CHOICE OF THE TEXTBOOK

Patri ia Bastos Fosse Peres

UNESA

patri iaperes.mat�gmail. om

The tea her's so ial representations on the mathemati s learning pro esses of their students onstitute

a kind of inter onne tion element between what students learn and how the tea her tea hes. From

the perspe tive of mathemati s edu ation, thinking about how the student learns is essential to develop

the edu ational pro esses that will result in meaningful learning, in whi h the mathemati al obje ts are

onstru ted from their meanings in a given ontext.

It is in the �s hool� psy hoso ial area that on epts, ideas, beliefs and values are onstru ted about

obje ts that run among subje ts, easing to be individual opinions and beginning to have meaning among

them. To the �set of on epts, propositions and explanations not subordinated to reason and logi � (FR,

2007, p. 590), Mos ovi i gave the name of so ial representations. A ording to the theory of so ial

representations, designed by Serge Mos ovi i in 1961, so ial a tors a t as a fun tion of ea h other when

guided by ommon sense purposes.

In the so ial representations' pro ess of formation about an obje t, the subje t organises their own

system of referen es from the intera tion with the other and assigns meaning to their a tions. Knowing

the meanings of Mathemati s learning by the tea her an help to understand the reasons for ertain lines

138

and hoi es. For Mos ovi i (1976/2012, p. 46), �so ial representation is the 'preparation for a tion', not

only be ause it guides behavior, but above all be ause it remodels and re onstitutes the elements of the

environment in whi h the ondu t must take pla e�.

Among these elements, we believe the textbook's hoi e should be made from the tea hers' on eptions

about what supports their pra ti es. A ording to Mazzotti (2008, p. 136), the textbook hosen by a

ertain group of tea hers gives lues as to what is �seizing its so ial representation. Moreover, tea hers of

a subje t an organize themselves into groups that refer to some textbook�. Thus, it is intended to show

that the so ial representations of learning result in the sele tion of textbooks and the way they are used

in the lassroom.

Referen es

Mazzotti, Tarso. 2008. �For a Pedagogy in So ial Representations�. Edu ation and Contemporary

Culture 6 (11): 121-142.

Mos ovi i, Serge. 2012. Psy hoanalysis, its image and its audien e. Petrópolis: Vozes.

Ponte, João Pedro. 2008. Studying the knowledge and professional development of the mathemati s

tea her. Lisboa: IE-ULISBOA.

Sá, Celso Pereira de. 2007. �So ial Representations in Re ent History and Current So ial Psy hology.

History of Psy hology: Dire tions and Paths. Rio de Janeiro: Nau. 587-606.

HOW TO ANALYZE HISTORICAL TEXTBOOKS FROM A DIDACTIC PERSPECTIVE?

THE DIDACTIC ANALYSIS AS A METHODOLOGICAL APPROACH

Miguel Pi ado

Universidad Na ional of Costa Ri a

miguepi ado�hotmail. om

Luis Ri o

Universidad of Granada, Spain

lri o�ugr.es

In the edu ational �eld, we understand the textbooks as dida ti al resour es that provide, to both tea hers

and students, stru tured and relevant information for a spe i� level, or systematized subje t matter, that

answers to an established urri ulum. These textbooks in lude spe ialized information related to a spe i�

subje t matter, whi h respond to a determined edu ational fun tion. We onsider the histori al textbooks

as the literary text designed and used in di�erent periods of the edu ational history of a ountry or region,

that are not part of the urrent edu ational urri ulum.

This ontribution proposes the dida ti al analysis as a te hnique to study the on eptual and dida ti al

spe i� ities that one may identify in mathemati s histori al textbooks (Pi ado, Gómez y Ri o, 2013).

Spe i� ally, from the partial analyses that ompose the dida ti analysis, we propose the analysis of the

ontent to re ognize the formal mathemati aspe ts from the arti ulation of mathemati al on epts,

their representations, and the phenomena that onne t it to di�erent ontexts. In addition, we propose

the ognitive and instru tive analysis as te hniques to highlight the mathemati al tasks, the expe ted

a hievements regarding these tasks, the possible errors and di� ulties that may ome up among the

students and the dida ti materials proposed to tea h established on epts spe i� mathemati al on epts.

We exemplify this proposal by means of a study arried on to hara terize the in orporation of the

metri al-de imal units in the arithmeti textbooks in Spain during the se ond half of the XIX entury

due to the implementation of the De imal Metri System (DMS) in 1849 and its reper ussion in the

Spanish edu ational system at that time. This study allowed the re ognition of on epts and mathemati al

pro edures that, at that time, people linked on the implementation of the DMS a ording to the a ademi

level (elementary, se ondary, or tea hers' training edu ation). It also allowed identify some trends related

to the use of representations to illustrate these units and about situations to show their usefulness in the

everyday living, and the teleologi al aspe ts and tea hing strategies proposed. This has also hara terized

139

the tea hing of these units during a histori al edu ational period in Spain. Likewise, we are using this

te hnique to hara terize the training of tea hers in the Normal S hool to tea h arithmeti in Costa Ri a

during the �rst half of the XX entury from the analysis of textbooks.

Referen es

Pi ado, Miguel, Gómez, Bernardo, and Ri o, Luis. 2013. �El Análisis Didá ti o en el estudio del

Sistema Métri o De imal en un libro de texto históri o de matemáti as.� In Análisis Didá ti o

en Edu a ión Matemáti a, edited by Luis Ri o, José Luis Lupiáñez, and Marta Molina, 403-414.

Granada, Spain: Comares.

THE COMPARISON OF MATHEMATICAL COMPETENCIES IN TEXTBOOKS FROM

BRAZIL AND SINGAPORE

Weverton Ataide Pinheiro Kin Hang Lei Feng-Lin Lu, Tai-Yih Tso

National Taiwan Normal University

wevmat�gmail. om misslei2001�hotmail. om fenglin.lu�gmail. om tsoty�ntnu.edu.tw

Textbooks have a strong in�uen e on how mu h students learn and what they learn (Törnroos 2005).

Results in international assessments have shown that students from di�erent ountries have performed

di�erently in mathemati s, what motivates resear hers to understand why some students have su h a

good performan e while others have it poorly.

Students mathemati al performan e on exams is related to students' mathemati al litera y, whi h

in turn is related to mathemati al ompeten e. Mathemati al ompeten e is de�ned by Morgan Niss

(Niss 2003) as the ability to understand, judge, do, and use mathemati s in a variety of intra and extra-

mathemati al ontexts and situations in whi h mathemati s plays or ould play a role. Niss also de�ned

mathemati al ompeten ies as the elements of mathemati al ompeten e in the same work. For example,

mathematizing (the apa ity to link an extra-mathemati al ontext with an intra-mathemati al on ept)

is one of the mathemati al ompeten ies whi h form what she so alled mathemati al ompeten e. Pisa

framework is designed through the use of Niss' ompeten ies.

This paper ompares the eight Niss' mathemati al ompeten ies in Pythagorean theorem related

exer ises, within four di�erent levels from a Brazilian and a Singaporean textbook, in order to omprehend

to whi h extent the levels of the ompeten ies required by ea h textbook's exer ises are related to the

performan e of students from both ountries on the OECD PISA 2012 assessment. The reason for

hoosing these two ountries' textbook is that both ountries have ompletely opposite results on PISA,

Singapore on the top and Brazil on the bottom.

Singaporean textbooks have a great emphasis on the mathemati al ompeten y mathematizing, that is

the key ompeten y for PISA exam. Thus, results have shown that the level of mathemati al ompeten ies

required by Singaporean textbooks exer ises is statisti al more related to the tasks on PISA exam, what

shows a possible eviden e for the dis repan y on the a hievements of students from these ountries.

Further details about this resear h are going to be presented in the onferen e.

Referen es

Niss, Mogens. 2003. "Mathemati al ompeten ies and the learning of mathemati s: The Danish

KOM proje t." In 3rd Mediterranean onferen e on mathemati al edu ation, 115-124. 2003.

Törnroos, Jukka. 2005. "Mathemati s textbooks, opportunity to learn and student a hievement."

In Studies in Edu ational Evaluation 31, no. 4, 315-327. 2005.

140

THE CONCEPTION AND DEVELOPMENT OF TEXTBOOKS FOR DISTANCE

LEARNING COURSES: THE CASE STUDY OF A TEACHER EDUCATION COURSE


Universidade Federal do Rio de Janeiro

mar ia�im.ufrj.br

Here the aim is to re�e t on the on eption and development of mathemati s textbooks for undergraduate

distan e learning ourses (EAD). In parti ular, we a knowledge the EAD undergraduate ourses as an

important governmental initiative for tea her edu ation in Brazil, although being aware of negative resear h

reports on the e� ien y of this modality of mathemati s edu ation in other ountries (see e.g., Safavi

et al, 2013) in ontrast to supportive evaluations in Brazil, whi h is elaborating its own EAD model

(see Moran, 2008). One amongst other spe i� ities of the model onsists of writing proper material and

textbooks, raising a need to investigate the pro esses of their development and use. We report on a ase

study adopting Shulman's (1987) notions on tea her knowledge though alerted on the di�erent modes of

linguisti ommuni ation (see Reddy, 1979) when analysing the written textbooks developed by a group

of professors. They implemented the task simultaneously elaborating a notion of lessons on paper, as

they per eived it demanded a pro ess that is di�erent from those when writing textbooks for fa e-to-fa e

le ture ourses and that it also does not oin ide with the wordings of the intera tions in a lassroom

le ture. Our analysis fa es the hallenge of problematizing the e�e t of the linguisti form of the textbook

produ ed for distan e learning on the mathemati s s ienti� ontent to be ommuni ated.

Referen es

Moran, José Manuel. 2016. Avaliação do Ensino Superior a Distân ia no Brasil. Disponível em:

www.e a.usp. br/prof/moran/avalia ao.htm. A esso: 15 nov. 2016.

Reddy, M. 1979. The Conduit Metaphor � A Case of Frame Con�i t in Our Language about

Language. In Ortony (1979): 284-324.

Safavi, Abdollah; Rostamy-Malkhalifeh, Mohsen; Behzadi, Mohammad-Hassan, and Shahvarani,

Ahmad. 2013. �Study on the E� ien y of Mathemati s Distan e Edu ation.� Mathemati s Ed-

u ation Trends and Resear h. Available online at www.ispa s. om/metr Volume 2013, Year 2013

Arti le ID metr-00017, 6 Pages doi:10.5899/2013/metr-00017

Shulman, L. S. (1986). Those who understand: Knowledge growth in tea hing. Edu ational

Resear her, 15(2), 4-14.

ONE-STEP MULTIPLICATION AND DIVISION WORD PROBLEMS IN THE 3RD

GRADE TEXTBOOKS IN BOSNIA AND HERZEGOVINA

Karmelita Pjani¢

University of Biha¢, Fa ulty of Pedagogy, Bosnia and Herzegovina

kpjani �gmail. om

A ording to the urri ulum for the 3rd grade in Bosnia and Herzegovina, on epts of multipli ation and

division are taught by modeling real world situations expressed in words and (or) a pi ture.

The main goal of this paper is to analyze the types of one-step multipli ation and division word problems

represented in mathemati s textbooks for the third grade of primary s hool in Bosnia and Herzegovina. We

examine distribution of one-step multipli ation and division word problems that in ludes 5 multipli ation

stru tures (equal groups, rate, multipli ative omparison, re tangular array and Cartesian produ t) in four

3rd grade textbooks approved by relevant ministries of edu ation.

141

The resear h results show that all types of one-step multipli ation and division word problems are not

adequately represented nor distributed in the third grade mathemati s textbooks. Results of the qualitative

analysis point to in onsisten ies in division problems formulations and, in some ases, non- omplian e of

word problem formulation with the a ompanying illustrations.

Sin e the textbook is the basi sour e of information to students as well as the basi , and sometimes

the only sour e of ideas to tea hers in lesson preparation, it is ne essary to revise word problems given in

textbooks and ensure that all types of multipli ation and division word problems are presented in textbooks,

whi h would ontribute to a better understanding of multipli ation and division on epts among the third

grade pupils.

Keywords: one-step word problems, multipli ation, division, textbook, Bosnia and Herzegovina.

Referen es

Anghileri, J. 1989. �An investigation of young hildren's understanding of multipli ation.� Edu a-

tional Studies in Mathemati s 20, 367-385.

Brown, S. 1992. �Se ond-grade hildren's understanding of the division pro ess.� S hool S ien e

and Mathemati s, 92(2), 92-95.

Chapin, S. H., Johnson, A. 2006. Math matters. Understanding the Math You Tea h, Grades K-8,

se ond edition, Sausalito, CA: Math Solutions Publi ations.

Greer, B. 1992. �Multipli ation and division as models of situations.� In Handbook for resear h on

mathemati s tea hing and learning, edited by D.A. Grouws, 276-295. New York: Ma millan.

Nesher, P. 1988. �Multipli ative s hool word problems. Theoreti al approa hes and empiri al

�ndings.� In Number on epts and operations in the middle grades, edited by J. Hiebert & M.

Behr, 19-40, Hillsdale, NJ: Lawren e Erlbaum.

S hmidt, S., Weiser, W. 1995. �Semanti stru tures of one-step word problems involving multipli-

ation or division� Edu ational Studies in Mathemati s 28, 55-72.

Rea hing out to lassrooms with Learning Materials through the textbooks of Mathemati s

Anup Rajput

Professor of Mathemati s and Head, Department of Elementary Edu ation,

National Coun il of Edu ational Resear h and Training, New Delhi, India

anupn ert�yahoo. o.in

Dharam Parkash

Ex. Professor of Edu ational Te hnology,

National Coun il of Edu ational Resear h and Training, New Delhi, India

dharamp�gmail. om

This paper outlines the e�orts being made and tried out to over ome the systemi bottlene ks of getting

the learning materials (LM) to every hild so as to provide her on rete learning experien es for qualitatively

enhan ing the understanding of mathemati s through Textbooks.

In order to improve the quality of Mathemati s tea hing/learning, quite a lot of e�orts have to be put

in to provide on rete learning experien es to learners at the primary stage. The fo us should be to reate

and provide LM for young learners so that along with the tea hers the learners ould explore and learn

by using on rete learning experien es. It was therefore, planned that ea h lassroom gets enough LM to

a hieve this goal. But due to systemi bottlene ks and problems, the LM somehow ould not rea h the

lassroom.

Questions in the minds of the team members were very many like �Is there a way to over ome the

systemi bottlene ks? How to rea h out to every hild in the lassroom?� et . All su h questions

142

prompted us to analyse the situation around and �nd out what is it that is able to rea h every learner and

every lassroom despite systemi problems. The answer was very obvious �the Mathemati s Textbook�. It

was then felt that the mathemati s textbook may be used to provide LM to ea h learner and lassroom,

espe ially in grades I and II.

A publishing house en ouraged the authors to initiate the development of su h textbooks and materials.

It took quite some time to develop textual material whi h integrated the use of LM along with various

pedagogi al strategies. Learning materials were reated using all kind of materials like ard board, plasti

sheets, and wooden blo ks and so on. Then the question ame up �where to sta k up the LMalong with

the textbooks?� A pou h of LM was reated and it was provided along with textbooks with appropriate

mention in the text.

Despite having an odd shaped pou h along with textbooks and di� ulties in storing and transportation,

the book was adopted by many s hools and tried out. But this experien e further prompted us to reate

a mathemati s textbook with limited LM being the integral part of the textbook. Based on the riterion

of suitability, material used, ost of the textbook, ease in using it by tea hers, learners and parents et .

and so on, the LM was put in the end portion of the books using material like thi k ard board papers

and perforated sheets. An orientation programme to tea hers for use of su h books was also reated.

The books are used in around twenty s hools and the systemati feedba k is obtained. The feedba k is

en ouraging and the assessment of learning out omes showed positive hange in ea h lass.

All this experimentation has made us realise that at primary stage e�e tiveness of mathemati s books

an be further enhan ed if LM is integrated with it. The lassrooms a ross twenty s hools are transforming

slowly to make us realise that it ould be right strategy. Survey done with tea hers, parents and s hool

administrators helped us learn more about use of su h textbooks.

NOT MY KIND OF ALGORITHM: COMPARING USE OF SHARED LANGUAGE IN

MATHEMATICS TEXTBOOKS AND COMPUTER SCIENCE INSTRUCTIONAL

MATERIALS

Kathryn Ri h

UChi ago STEM Edu ation, University of Chi ago, USA

kmri h�u hi ago.edu

Carla Stri kland

UChi ago STEM Edu ation, University of Chi ago, USA

astri kland�u hi ago.edu

T. Andrew Binkowski

Computation Institute, University of Chi ago, USA

abinkowski�u hi ago.edu

Bolstered by President Obama's CS4All initiative, interest in bringing omputer s ien e instru tion to all

K-12 students in the United States ontinues to grow. Dis ussions of what su h instru tion might entail

for elementary students, and how it might be in luded in an already- rowded urri ulum, have led naturally

to explorations of how omputer s ien e might be integrated with existing subje ts (Israel et al., 2015).

In parti ular, mathemati s seems to be a logi al integration point, given the shared attention to logi al

systems of thinking in the two dis iplines.

What is the extent of the overlap between mathemati s and omputer s ien e, and what are the key

di�eren es to be addressed in an integrated ourse? In an attempt to answer this question, we ondu ted

a systemati review of a set of resear h-based K-5 mathemati s textbooks, Everyday Mathemati s (Bell et

al., 2016), with attention to four key terms used in both mathemati s and omputer s ien e � algorithm,

iteration, abstra tion, and variable. We ondu ted the same review of two sets of popular instru tional

materials for omputer s ien e in K-5, ode.org's Computer S ien e Fundamentals (2016) and CS Un-

plugged (Bell, Witten, and Fellows, 2015). Using standard meanings of the terms used in the resear h

143

literature of ea h �eld as a lens, we analysed the meanings of ea h term ommuni ated to students and

tea hers via the materials.

In this paper we dis uss the results of this analysis, highlighting both similarities and di�eren es in the

manner in whi h the terms are used in ea h set of instru tional materials. We dis uss the impli ation of

these results for the design of integrated urri ula, with �rst re ommendations for how the �eld might

leverage the similarities and attend to the di�eren es.

Referen es

Bell, Max, Jean Bell, John Bretzlauf, Amy Dillard, Robert Hart�eld, Andy Isaa s, . . . Peter Sae ker.

2016. Everyday Mathemati s (4th Ed.). Chi ago, IL: M Graw-Hill Edu ation.

Bell, Tim, Ian H. Witten, and Mike Fellows. 2015. CS Unplugged: An enri hment and extension pro-

gramme for primary-aged students. Adapted by Robyn Adams and Jane M Kenzie, 2015 revision by

Sam Jarman. Christ hur h, New Zealand: Computer S ien e Unplugged. http:// sunplugged.org/books/

�Computer S ien e Fundamentals for elementary s hool.� ode.org, a essed September 28, 2016,

https:// ode.org/edu ate/ urri ulum/elementary-s hool.

Israel, Maya, Jamie N. Pearson, Tanya Tapia, Quentin M. Wherfel, and George Reese. 2015.

�Supporting all learners in s hool-wide omputational thinking: A ross- ase qualitative analysis.�

Computers and Edu ation 82: 263-279.

THE TEACHING OF LOGARITHM: METHODOLOGY DIFFERENCES BETWEEN

BRAZILIAN AND AMERICAN TEXTBOOKS

Crístiam Wallao Rosa Ri ardo Fajardo

Universidade Federal de Santa Maria

ristiamwallaorosa�gmail. om rfaj�ufsm.br

This paper is one result of the Master's dissertation �The Mathemati s in Ameri an and Brazilian text-

books: a study on the tea hing of logarithms�. Aiming to resear h on the methodologi al di�eren es

existing on the presentation between Brazilian and Ameri an textbooks, thirty three mathemati s text-

books were analyzed. In most of these textbooks the de�nition of logarithm is presented as, y , of a

positive number, x , to a base b (positive and di�erent from 1), is the se ond number of an ordered

number pair (x, y), su h that x equals b raised to y; or, algebrai ally: logbx = y ⇔ x = by, whereb, x > 0, and b 6= 1. This de�nition was the most pe uliar ontent observed in the study. However, others

were also identi�ed su h as: ontextualized exer ises, the use of a s ienti� al ulator, and the appli ation

of omputer language programs like Fortran and Basi . Two textbooks, one Brazilian and the other one

Ameri an, aught the attention due to their very di�erent way of presenting the ontents. They are,

respe tively, �The mathemati s in the renewed s hool�, by Netto (1972), and �Algebra and Trigonometry:

stru ture and method, Book 2�, by Dol iani (1977). The Brazilian one presented the material in a very

short way, with no ontext, and with a set of exer ises requiring only the use of formulas. On the other

hand, the Ameri an textbook presented the material in a very detailed manner, divided into sequential

hapters and exploring well the subje t. It also presents the s ienti� al ulator, a very important tool

at that time. The number of exer ises between the two textbooks is a very important di�eren e, sin e

onsidering a tivities, examples and exer ises, it was found a total of 1,673 in the Brazilian textbook,

and a total of 4,929 in the Ameri an ounterpart. The history of mathemati s was also en ountered,

mostly in Ameri an textbooks, being used as an introdu tion in the hapter and assisting the student to

be pla ed loser to the subje t at hand. In our point of view, this fa t be ame a signi� ant di�eren e to

be investigated. As a general analysis, the Ameri an textbooks are more well produ ed fo using on the

learning, sin e they present a variety of ways to explore the same subje t. The Brazilian textbooks, mainly

144

the ones published until the 1980's, are engaged in the traditional way of tea hing with straightforward

examples and no appli ations.

Referen es

Dol iani, M. P., Sorgenfrey, R. H., Wooton, W., Kane, R. B. 1977. Algebra and Trigonometry:

stru ture and method. Book 2. Boston: Houghton Mi�in Company.

Netto, S. P., and Goés, C. C. 1972. Matemáti a na es ola renovada. São Paulo: Saraiva.

TEACHERS' PERSPECTIVE OF TEXTBOOK CHOICE: THE DIGITAL GUIDE AS A

META-RESOURCE

Ri ardo Tibur io dos Santos

UFPE, Brazil

ri o.tibur io�gmail. om

The textbook is onsidered an indispensable resour e in support of tea hing a tivities. Gerard & Roegiers

(1998) list the most important roles of this resour e for tea her as: to aid in planning the lasses; to

further the a quisition of knowledge; to foster dida ti -pedagogi al tea her's training; to help in assessing

student's learning. By onsidering the roles of this resour e, a resear h to understand what hara teristi s

of mathemati s textbooks are onsidered indispensable by mathemati s tea hers, onsidering those from

Elementary Edu ation in Brazil. A resear h methodology was designed to olle t data within a workshop

on the use of the Digital Guide of PNLD to hoose the textbook. A questionnaire was reated and

undertaken with the tea hers who were doing the ourse. The questionnaire aimed to observed some

models of tea hers' work with textbooks, some expe tations and hara teristi s that these professionals

onsider indispensable in the works. Within those data olle ted, the Digital Guide of PNLD (Brasil,

2015) was presented, as a resour e built to help tea hers to hoose the best textbook for his/her use

in lassrooms: a meta-resour e. It is onsidered a meta-resour e be ause it is a resour e that guides

the hoi e of another resour e. We justify the guidelines on the use of the Guide when we realize the

importan e of the National Textbook Program � PNLD. The �rsts results of this study show that tea hers

use the textbook as a fundamental resour e in their lasses and for their lass planning. It is also observed

that the fa ility of a essing information in the Digital Guide helped tea hers in their hoi e.

Referen es

BRASIL. SEB-MEC. 2015. Guia Digital PNLD 2016. Brasília: MEC / SEB, 2016.

GERARD, F. M.; ROEGIERS, X. 1998. Con eber e avaliar manuais es olares. Porto � Portugal.

Porto Editora..

INTERBREEDING PARADIGMS IN RESEARCH ON MATHEMATICS KNOWING AND

LEARNING

Thorsten S heiner

University of Hamburg, Germany / Ma quarie University, Australia

thorsten.s heiner�uni-hamburg.de


Federal University of Rio de Janeiro, Brazil

mar ia�im-ufrj.br

145

Textbooks are often onstrued as the primary vehi les for making subje t matter a essible to students,

a view in whi h textbook designers turn to the stru ture of the dis ipline to se ure a foundation for

developing textbooks. This viewpoint, whi h is prin ipally stru turalist in orientation, has been hallenged

by s holars taking onstru tivist perspe tives on mathemati s knowing and learning. Often stru turalism

and onstru tivism are framed as ompeting perspe tives between one seems to have to hoose. In this

presentation, we present re ent advan es in resear h on mathemati s knowing and learning that aims to

put in dialogue these seemingly on�i ting paradigms produ tively (Pinto & S heiner, 2016; S heiner,

2016). A ase study of one student's knowing and learning the limit on ept of a sequen e is dis ussed

(S heiner & Pinto, 2014), in whi h the student a tively intera ts with the subje t matter in omplex

dynami ways: his knowledge system is shaped by the meaning of the mathemati al on ept, but his

knowledge system also shapes the meaning of the mathemati al on ept. The ase study reveals riti al

insights that lead to interbreeding traditional paradigms. One onsequen e is to a knowledge a omplex,

dynami view on mathemati s knowing in whi h knowledge is onstrued as a pro ess, rather than taking up

a stati view in whi h knowledge is interpreted as an obje t that an be onstru ted ( onstru tivist view)

and de onstru ted (stru turalist view). In a knowledging rather than denying students in the reation of

the meaning of a mathemati al on ept, some impli ations for textbook resear h and development are

drawn out. The fun tion of textbooks as `emerging orientations' is outlined.

Referen es

Pinto, Már ia M.F., and Thorsten S heiner. �Making sense of students' sense making through

the lens of the stru tural abstra tion framework.� Pro eedings of the First Conferen e of the

International Network for Dida ti Resear h in University Mathemati s. 474-83. 2016.

S heiner, Thorsten. �New light on old horizon: Constru ting mathemati al on epts, underlying

abstra tion pro esses, and sense making strategies.� Edu ational Studies in Mathemati s 91.2

(2016): 165-183.

S heiner, Thorsten, and Már ia M. F. Pinto. �Cognitive pro esses underlying mathemati al on ept

onstru tion: The missing pro ess of stru tural abstra tion.� Pro eedings of the 38th Conferen e of

the International Group for the Psy hology of Mathemati s Edu ation and the 36th Conferen e of

the North Ameri an Chapter of the Psy hology of Mathemati s Edu ation. Vol. 5, 105-12. 2013.

SOME REMARKS ON TEXTBOOKS ANALYSIS: HERMENEUTICS OF DEPTH AND

EDITORIAL PARATEXTS

Leandro Josué de Souza

UNESP/Bauru, Brazil

leandrojosue�gmail. om

Mirian Maria Andrade

UFTFPR/CP, Brazil

andrade.mirian�gmail. om

Bruno Alves Dassie

UFF, Brazil

badassie�gmail. om

Hermeneuti s of Depth (HD) is a methodologi al referential theory used to analyze symboli forms.

The symboli forms are produ ed, or an be assumed to be produ ed, intending to �say� something.

Thus, we an a�rm that they are full of intentions human produ tions. Written texts, for instan e, an

be onsidered symboli forms and, therefore, may be analyzed by HD. HD is proposed, as a method,

through three di�erent and inter onne ted movements of analysis, whi h, a ording to Thompson, an be

onsidered separate stages of a sequential method. These movements or dimensions, or yet phases, an be

brie�y des ribed as: So ial-histori al Analyses � that aims to understand the psy hologi al-so ial-histori al

ontexts that surround the symboli form and that ontribute to its onstitution; Formal or Dis ursive

Analyses � that intents to understand the materiality of the symboli form, or, in other words, its fo us

is on the internal details of the subje t being studied; Interpretation/Reinterpretation � is the movement

that synthesizes, or �bond� together, the past movements, thus produ ing the re ord of an interpretation

(Thompson 2011).

146

Thompson laims that HD alone may not always be able to answer some questions and that, in the

interpretative pro ess, other methods an be added by the interpreter to ollaborate with the analyses of

the symboli form (Thompson 2011). As a result, HD an be added to other methodologi al referential

theories that an somehow ontribute to answering prior questions depending on the type of questions

and the resear h subje t. In Brazil, History of Mathemati s Edu ation resear hers have been using HD

with the on ept of Editorial Paratexts, among others, as introdu ed by Gérard Genette. This author

explains that a text only be ome a book by various omplements or addends (the so alled paratexts) that

are used in order to ompose a book and spread some messages about that book. A ording to Genette,

these omplements are the author's name, the book title, the prefa e, the footnotes, the endnotes, the

releases, the advertisements, et ., that say something about the book or help in rease its appeal, as well

as its omprehension (Genette 2009).

Therefore, this arti le intends to present HD, as given by Thompson, Editorial Paratexts, as pre-

sented by Genette, and dis uss how they have been on urrently used as a methodologi al approa h or

methodologi al referential theories to analyze textbooks and books from a histori al perspe tive in History

of Mathemati s Edu ation resear hes. Furthermore, we are going to present some of the works devel-

oped by Brazilian resear h groups that study the History of Mathemati s Edu ation and mobilize these

methodologi al referential theories to analyze su h symboli forms.

Referen es

Genette, Ge?rard. Paratextos Editoriais. 2009. Cotia, SP: Ateliê Editorial.

Thompson, John B. Ideologia e Cultura Moderna: Teoria so ial ríti a na era dos meios de omu-

ni ação de massa. 2011. Petrópolis, RJ: Editora Vozes.

THE TEXTBOOK IN MATHEMATICS: FINDINGS FROM A SYSTEMATIC REVIEW

Natasja Steen

Aarhus University

nat�zun.dk

Matilde Stenhøj Madsen

VIA Læreruddannelsen i Aarhus

masm�via.dk

In this oral presentation we investigate the question: �Whi h potentials and limitations of using textbook

in mathemati s tea hing an we identify?� We answer this through a qualitative systemati review in two

parts. First we ondu ted a des riptive review of 451 referen es that resulted in nine analyti themes, sorted

in four ategories: `the textbook', `the textbook and the students', `the textbook and the urri ulum' and

`the textbook and the tea her'. These ategories express relations that the textbook is part of.

Se ond we ondu ted a qualitative synthesis of three of the analyti themes. The �rst theme was

hosen from the ategory `the textbook' and fo used on the on ept of the textbook. The �ndings

in lude a suggestion for a de�nition of the on ept `textbook'.

An analyti theme from the ategory `the textbook and the students' was also hosen for an extended

qualitative synthesis. This synthesis explored the on ept of `Opportunity to Learn' as it is des ribed in the

literature. The �ndings are summed up in a model for Opportunity to Learn in relation to the textbook.

The third synthesis was ondu ted with one of the analyti themes in the ategory `the textbook and

the tea her' and fo used on the potential in�uen es from the textbook on tea her learning. The review

explores if the textbook an make a di�eren e in tea her learning and in what ways and under whi h

ir umstan es the textbook an have an impa t. The �ndings are transformed into a proposed model of

the relation between the textbook and the tea her that is des ribed here as `Opportunity to Tea h'.

Referen es

147

Steen, Natasja, and Matilde Stenhøj Madsen. 2016. Lærebogen i matematik. Et systematisk review

af lærebogen i matematikundervisningen. Master thesis in Edu ational Theory and Curri ulum

Studies (mathemati s). Emdrup, Denmark: Aarhus University. Available from http://library.au.dk/

or by sending a request to the authors at nat�zun.dk.

TEXTBOOK ANALYSIS IN UNIVERSITY TEACHER EDUCATION

Ysette Weiss

Johannes Gutenberg-University of Mainz

weisspid�uni-mainz.de

In German middle and se ondary s hools, mathemati al textbooks are extensively used in lesson prepara-

tion, in lassroom as well as for students' homework. In the last two de ades, the mathemati s tea hers

at ea h s hool in parti ular were asked to prepare on retised s hool urri ula based on very general

edu ational standards and ompeten y models. Lesson planning for mathemati s was generally done by

mathemati s tea hers at s hool based on the mathemati s textbook series used in that parti ular s hool.

Also in preparation of the exer ises for homework, tests and tasks for individual learning, mathemati s

textbooks still serve as an essential basis (Rezat, 2009). An internal ommission at s hool, onsisting of

all mathemati s tea hers, a parent and a student representing their respe tive groups, makes the sele tion

of the mathemati s s hoolbook. Taking into onsideration the long German tradition of textbook develop-

ment and the use of s hoolbooks in mathemati s lasses ( f. Otte, 1981), it is even more astonishing that

there are hardly any anoni al subje ts in mathemati s tea her edu ation at university related to mathe-

mati s textbook analysis. The paper presents the on ept of a seminar given on mathemati s s hoolbook

analysis, whi h has been held and developed over six years in 15 di�erent groups. The aim of the seminar

is the get a quainted with di�erent mathemati s textbook designs with regard to impli it or expli it intro-

du tion of mathemati al on epts, di�erent ontextualisations, systematisation and formalisation of the

ontextualised on epts, di�erentiation in exer ise tasks, as well as the omparative analyses on spe i�

topi s. A major problem of tea her training is the often ited double dis ontinuity (Klein, 1933, S.2). In

this ontext, textbook analysis is an ex ellent way to hange the student's perspe tive (solving tasks) to

the tea her's perspe tive (lesson planning with tasks) to the author's perspe tive: What is the on eptual

understanding of the notion? Whi h tasks are in the zone of proximal development of a possible student?

Working with a textbook also provides tools to deal with di�erent levels of awareness (Mason, 1998). It

takes a longer time and a ompanying guidan e until the students themselves ask questions on erning

not only the solution of a task but also about the edu ational value, the development of meaning and

about the existen e of mathemati al obje ts. The paper presents the development of the seminar and

shows exemplary problems, whi h led to the present on ept of the seminar.

Referen es

Klein, Felix. 1933. Elementarmathematik vom höheren Standpunkte aus, Bd. I: Arithmetik, Algebra,

Analysis. 4. Au�age. Berlin: Julius Springer 1933.

Mason, John. 1998. �Enabling tea hers to be real tea hers: Ne essary levels of awareness and

stru ture of attention.� Journal of Mathemati s Tea her Edu ation 1 (3): 243-267.

Otte, Mi hael. 1981. �Das S hulbu h im Mathematikunterri ht.� Mathematiklehrer 3: 22-27.

Rezat, S. 2009. �The utilization of mathemati s textbooks as instruments for learning.� Pro eedings

of CERME 6, Lyon, Frankrei h: INRP, 6: 1260-1269.

148

INTEGRATING THE CONCLUSIONS OF TEACHERS' FEEDBACK INTO THE NEW


Gergely Wints he, Dániel Katona

Eötvös Loránd University (ELTE), Hungarian Institute for Edu ational Resear h and Development (OFI),

Veres Péter Gymnasium

wints he� s.elte.hu, danikatona�gmail. om

Wints he has had the opportunity to develop new learning tools within the framework of the So ial

Renewal Operational Programme (SROP) 3.1.2/ B-13 funded by the EU. The budget of the proje t was

more than 7 million euros. The proje t involved not only the writing and printing of new textbooks and

onstru ting their digital ba kground, but also the exploration of tea hing pra ti e in onne tion to the

usage of the new textbooks, olle ting and analyzing feedba k, and monitoring the tea hing pro ess.

The present paper fo uses on the o� ial feedba k olle ted from tea hers, and a short summary will be

presented about the hanges in the revised textbooks in�uen ed by the feedba k. (Even & Olsher, 2014)

The main universities in Hungary ondu ted studies in order to lay down the fundamental guidelines for

our programme before the start of the proje t. Most of these studies emphasized the ru ial importan e

of olle tive thinking of tea hers and students, as well as the signi� an e of the ooperative development

of digital support. These studies reinfor ed our belief that tea hers and students should play a leading

role in the development.

The main milestones of the proje t were:

First year: Planning and reating the �rst version of the textbooks and workbooks for grades 1-2, 5-6,

and 9-10, a ording to the Hungarian s hool system.

Se ond year: Testing the �rst version of issues by 50 tea hers for ea h subje t and olle ting feedba k

and reating textbooks for grades 3, 7, and 11.

Third year: Rewriting and re-editing the pilot textbooks written in the �rst year, based on the tea hers'

feedba k, and reating the pilot books of the series for grades 4, 8 and 12. (Wints he, 2015)

During the proje t, more than 20.000 pie es of feedba k per textbooks were olle ted. The users of

the pilot textbooks �lled out a survey before and after the s hool year and the o� ial users had to �ll

out a web-based questionnaire after every le ture. We also ondu ted some interviews at the end of the

s hool year. We lassi�ed the feedba k using ategories (su h as misprints, errors, theoreti al problems,

dida ti al problems, onstru tive suggestions, good pra ti es, et .). Some of the proposals were useful,

reative and de�nitely worthy to be integrated into the re-edited textbooks during the revision pro ess,

and we got an overall pi ture about the expe tations and wishes of the tea hers.

Referen es

Wints he G. 2015. �The Prin iples of the New Mathemati s Textbooks in Hungary� A ta Mathe-

mati a Nitriensia 1 (1): 183-187. http://www.amn.fpv.ukf.sk/papers.php

Even R., and Olsher S. 2014 Tea hers Editing Textbooks: Changes Suggested by Tea hers to the

Mathemati s Textbook They Use in Class Southampton Pro . ICMT-2014: 43-48.

THE EMBODIMENT OF CORE COMPETENCY IN TEXTBOOK COMPILING

Liu Xiaomei

Capital Normal University

xiaomeiliu2013�163. om

Ma Fu

Nanjing Normal University

fuma30�aliyun. om

Zhang Huiying

Institute of Shijiazhuang Edu ation S ien e

jkszhying�sina. om

Con epts like number sense, symbol sense, spatial sense, geometri intuition, on ept of data analysis,

al ulating ability, reasoning ability and model thinking are learly presented in the Curri ulum Standard

149

of Mathemati s (2011) as ore on epts that re�e t nature of Mathemati s urri ulum and embody

urri ulum obje tives.

These ore on epts have strong asso iation with urri ulum ontent, while surpassing urri ulum

ontent. How to embody ore on epts mentioned above in textbook ompiling? How to develop om-

prehensive ompeten y of students in pro ess of knowledge presentation and students' problem solving?

These are signi� ant questions for us textbook ompilers to do further resear h. This arti le presents

some ore on epts mentioned above as examples(su h as symbol sense, model thinking and reasoning

ability) to analyse embodiment of these ore on epts in urrent textbooks, as well as a hievements in

tea hing when tea hers use textbook reatively .

Case: Intuitive Per eption and Reasoning

Suppose the terrestrial equator is surrounded by an iron wire one mile longer than the length of

terrestrial equator, how mu h is the spa e between iron wire and the terrestrial equator? Is it big enough

to pla e an apple? Please pi ture this in your mind �rst and then al ulate spe i� ally to he k your

answer. Try to ommuni ate with your partner.

The aim of this ase is to make students realize that intuition is untrusted a ording to the on�i t

between result a essed by imagination or intuition and by al ulation. However, renovation of questions

of the same style is arried out during spe i� tea hing pro edure. They raise questions after students'

estimation of the spa e between basketball and terrestrial equator instead of starting al ulating immedi-

ately.

How mu h is the spa e between iron wire and basketball �terrestrial equator� when the earth is repla ed

by basketball and the basketball �terrestrial equator� is surrounded by an iron wire one mile longer than

the length of basketball �terrestrial equator�? What if basketball is repla ed by ping-pang?

What an students gain during this pro ess after tea hers' renovation of questions in textbook? Can

they better understand meaning of reasoning and fun tion of representing by symbols?

Tea hers' reative pro ess during textbook using not only in reases students' a hievements, but also

provides rare materials and experimental experiment for development of textbooks.

Referen es

Fuma. 2012. �Textbook for Compulsory Edu ation: Mathemati s� (grade 8). Beijing: Beijing

Normal University Press.

MAPPING THE TERRAIN � MAKING THE TEXTBOOK'S CONTENT & PEDAGOGY

TANGIBLE

Mi hal Yerushalmy Shai Olsher

University of Haifa

mi halyr�edu.haifa.a .il olshers�edu.haifa.a .il

Textbooks are a vital ingredient for mathemati s tea hers' lesson preparations and their pedagogi pra ti e

and are seen as vehi les for tea her learning and professional development. In many ountries, in luding

Israel, textbooks are per eived to re�e t the views expressed in national urri ular do uments. Hen e,

textbook represents what the edu ational system views as appropriate for students to learn and how

it should be taught (Chazan & Yerushalmy, 2014). The proposed talk fo uses on new-mode digital

mathemati s textbooks that an be ustomized by all of their users: urri ulum developers, tea hers, and

students. As tea hers have expressed a desire to personalize digital textbooks, we have identi�ed a riti al

need for them to make informed de isions that re�e t their own ideas and views, while understanding the

impli ations of their de isions on the approa h originally set by the textbook's author. In response, we study

the design of organizational tools whi h will provide tea hers with a window into the underlying stru ture

of the digital textbook, by giving them a multi-dimensional intera tive mapping display of its embedded

150

mathemati al, pedagogi al, and urri ular omponents. We study the modi� ation of intera tive textbook

(part of the VisualMath urri ulum Yerushalmy, Katriel, & Shternberg, 2002/2014) by following hanges

guided by an awareness of the balan e between the learning obje ts and an awareness of the rationale of

their sequen ing.

Initial testing with experien ed tea hers led us to understand that though important to the resear h

ommunity, some of the mapped aspe ts meant little to the tea hers, while other aspe ts that were not

in luded were addressed. These �ndings are aligned with the previous �ndings by Olsher & Even (2014)

that show that tea hers have substantial ideas when it omes to making textbook more a essible. Thus,

to meet the hallenge of bridging the author and the user aspe ts an organizing tool should be �exible

in terms of the mapped aspe ts � the latter ould be determined by the tea hers as well as resear hers,

and not only limited to the initial aspe ts proposed by the developer. In our talk we present relevant

hara teristi s of urri ular materials in textbooks that ould be tagged, thus reating data sets that an

represent tea hing sequen es that are linked to the representation of the balan e between the learning

obje ts, and a omparative view of sequen es that visually emphasizes the di�eren es between data sets

of learning obje ts.

Referen es

Chazan, D. & Yerushalmy, M. 2014. �The Future of Mathemati s Textbooks: Rami� ations of Te h-

nologi al Change�. In Media and Edu ation in the Digital Age Con epts, Assessments, Subversions,

edited by Sto hetti, M, 63-76.

Olsher, S., & Even, R. 2014. �Tea hers editing textbooks: Changes suggested by tea hers to the

math textbook they use in lass�. In Pro eedings of the International Conferen e on Mathemati s

Textbook Resear h and Development (ICMT-2014) edited by K. Jones, C. Bokhove, G. Howson, &

L. Fan. Southampton, GB: University of Southampton.

Yerushalmy, M., Katriel, H., & Shternberg, B. 2014. The VisualMath fun tions and algebra e-

textbook. Israel: CET �The Centre of Edu ational Te hnology.

Available from: http://visualmath.haifa.a .il/, a essed O tober 31, 2016).

OPPORTUNITY TO LEARN PROBLEM SOLVING IN DUTCH PRIMARY SCHOOL


Mar van Zanten

SLO / Utre ht University

m.vanzanten�slo.nl

Marja van den Heuvel-Panhuizen

Utre ht University

m.vandenheuvel-panhuizen�uu.nl

Problem solving is an important part of the mathemati s urri ulum. Whether students develop problem

solving skills, depends substantially on whether they are o�ered the opportunity to gain experien e in

problem solving tasks. Sin e it is known that in the Netherlands tea hers follow largely their mathemati s

textbook (Hop, 2012; Meelissen et al., 2012), these textbooks are of de isive in�uen e on the Dut h

ena ted urri ulum. This means that the degree and the way in whi h tasks on problem solving are

in luded in the textbooks determine for a great part the opportunity to learn problem solving.

About a de ade ago, problem solving tasks were only minimal present in Dut h textbooks (Kolovou,

Van den Heuvel-Panhuizen, & Bakker, 2009). Sin e the textbooks then in use all have been repla ed by

new editions and other textbooks, the question arises whether the opportunity to learn has in reased with

the new generation of textbook series. To answer this question, we analyzed three Dut h textbooks having

together the largest market share, as well as one new textbook series that has only a small ir ulation but

that expli itly laims to enhan e students' problem solving skills.

In our presentation the following questions will be addressed:

151

• To what degree ontain the new generation Dut h textbooks mathemati al problem solving tasks?

• In what way do these textbooks fa ilitate the opportunity to learn problem solving?

• How in lusive are these textbooks with respe t to o�ering opportunities to learn problem solving for

students with varying mathemati al abilities?

Also, we give thoughts to the on eptualization of problem solving and methodologi al hallenges of

textbook analyses.

Referen es

Hop, Mi hel. 2012. Balans van het reken-wiskundeonderwijs halverwege de basiss hool 5 [Balan e

of mathemati s edu ation halfway primary s hool 5℄. Arnhem, the Netherlands: Cito.

Kolovou, Angeliki, Marja van den Heuvel-Panhuizen, and Arthur Bakker. 2009. "Non-routine prob-

lem solving tasks in primary s hool mathemati s textbooks � A needle in a haysta k." Mediterranean

Journal for Resear h in Mathemati s Edu ation, 8(2), 31-68.

Meelissen, Martina, et al. 2012. PIRLS en TIMSS 2011. Trends in leerprestaties in lezen, rekenen

en natuuronderwijs [PIRLS and TIMSS 2011. Trends in a hievement in reading, mathemati s

and s ien e℄. Ens hede / Nijmegen, the Netherlands: Twente University & Radboud University

Nijmegen.

152

Poster Presentations

(in alphabeti al order of �rst author)

THE MATHEMATICAL PROBLEM SOLVING IN TEXTBOOKS

Maria Madalena Dullius Geovana Luiza Kliemann

Centro Universitário Univates

madalena�univates.br geovanakliemann�universo.univates.br

This paper emphasizes the problem solving as a methodology for Mathemati s tea hing. In order to

in rease the knowledge about the study, the theoreti al support of several authors was fundamental. On

the PCNs, we highlight authors su h as Dante (2010), Diniz (2001), Polya (1995) and Lopes (2009).

This resear h aimed to diagnose how and what for tea hers of the 1st year of High S hool, from six

publi s hools in Vale do Taquari use Mathemati s textbooks, and to understand how these materials

feature the problem solving. In addition, the resear h aimed to reate a tea hing material for tea hers to

approa h Mathemati s through problem solving. In order to a hieve the proposed obje tives, we ondu ted

a qualitative study in whi h the resear hers were ontinually involved in the pro ess. Initially, an interview

was ondu ted with ten Mathemati s tea hers to realize how mu h importan e these professionals attribute

to textbooks during the planning and development of their lasses, and to identify their per eptions of

the approa h to problem solving in the urrent ontext. Subsequently, an analysis of the textbooks of the

series involved that were adopted in s hools in luded in the initial interviews was held. Finally, the material

for edu ational intervention was produ ed to help tea hers to use this methodology in di�erent ways. As

a result, it is possible to point out that textbooks are used by tea hers under di�erent aspe ts and that

they present some innovations, but the problems are, in most ases, linked to some spe i� ontent. It is

onsidered, therefore, that this resear h allowed re�e tions and even hanges in the per eption of tea hers

as to methodology of problem solving.

Referen es

Dante, L. R. 2010. Matemáti a: ontexto e apli ações. 1

a. Ed. São Paulo: Áti a.

Diniz, M. I. 2001. Os problemas onven ionais nos livros didáti os. In: Smole, K.S.; Diniz, M.I.

(org.). Ler, es rever e resolver problemas: habilidades bási as para aprender matemáti a. Porto

Alegre: Artmed, p. 99 - 102.

Kliemann, G. L. 2015. Poten ialidades e limitações de material didáti o para explorar resolução de

problemas matemáti o. Dissertação (Mestrado) � Curso de Ensino de Ciên ias Exatas da Univates,

Lajeado. Disponível em: http://hdl.handle.net/10737/802.

Lopes, J. de A. 2009. O livro didáti o, o autor e as tendên ias em edu ação Matemáti a. In:

Na aratto A. M.; Lopes, C. E. (Org.). Es ritas e leituras na edu ação Matemáti a. Belo Horizonte:

Autenti a, p. 35 - 62.

Polya, G. 1995. A arte de resolver problemas. Um novo aspe to do método

THE COLLECTIVE PRODUCTION OF DIGITAL DIDACTIC MATERIALS AS PART OF

INITIAL FORMATION OF MATHEMATICS TEACHERS: A CASE STUDY INVOLVING

DIGITAL TEXTBOOK PRODUCTION


UNIAN-SP, Brazil

igna io�ufpe.br

This paper presents a part of a do toral resear h that aims to investigate evolutionary and evaluative

aspe ts of the initial formation of mathemati tea hers in produ ing and experimenting with. We use

the do umentary approa h, Guedet and Trou he (2009), to guide the study be ause it provides means

155

investigate inter onne tion between the professional development of a tea her and the resour es that

he/she uses, onsidering evolutionary aspe ts of both. We also used Instrumental Or hestration Theory

(Trou he, 2005) to support the identi� ation of pre-servi e tea hers' hoi e elements for lasses exe ution

using the textbook produ ed. We present a partial analyzes of the �rst (of two) data olle tion obtained

from the lessons planning pro ess, prototyping, room use of the digital textbook versions and the preliminar

auto-evaluation the subje ts made. The data were obtained from video-re ords of olle tive interviews

and video-re ordings of meetings between the pre-servi e tea hers and the pre eptor tea her, who is the

tea her of the lass in whi h the traineeship will be developed.The partial results suggest that, while

the pre-servi e tea hers, based on their own experien es as students, propose the planning to introdu e

fun tions starting with a problem to be explored using dynami resour es, the pre eptor tea her suggests

using a tivities of the textbook to be expanded with digital resour es that interns dominate. This �nding

is relevant to the resear h be ause they are di�erent on eptions of digital textbook involving di�erent

resour es, what by its turn is an element to be investigated during the produ tion of the material.

Referen es

Gitirana, Ver�ni a, Marilena Bittar and Rogério Igná io. 2014. �Objeto Edu a ionais Digitais:

políti as e perspe tivas". Anais do II Fórum GT 6 SBEM - Edu ação Matemáti a: novas te nologias

e edu ação a distân ia. Rio de Janeiro : SBEM: 1-8.

Trou he, Lu . 2005. �Constru tion et onduit des instruments dans les apprentissages mathéma-

tiques: né essité des or hestrations". Re her hes en Dida tique des Mathématiques, 25(1), 91-138

Gueudet, Ghislaine, Birgit Pepin, and Lu Trou he. 2009. "Towards new do umentation systems

for mathemati s tea hers?" Edu ational Studies in Mathemati s, 71(3), 199-218.

GEOMETRIC CONSTRUCTIONS WITH RULER AND COMPASSES IN BRAZILIAN

TEXTBOOKS

Ana Paula Jahn

IME-USP

anajahn�ime.usp.br

Mar os Alves dos Santos

CAEM-IME-USP

mar osime�gmail. om

From the middle of the nineteenth entury, the geometri onstru tions of Eu lidean Geometry had a

tortuous traje tory in the urri ula of the Brazilian Basi Edu ation. A ording to Zuin (2001), with the

ena tment of LDB 5692/71, Geometri Design (DG) is no longer a ompulsory subje t, whi h ontributed

to the ex lusion of many subje ts from s hool institutions. From the 1990s onwards, mathemati al

dida ti texts again in luded a tivities or hapters devoted entirely to geometri onstru tions with ruler

and ompass, and in 1998, the National Curri ular Parameters of Mathemati s (NCPs) suggested that

geometri onstru tions be addressed within the Mathemati al dis ipline (BRASIL, 1998). Even if their

tea hing is advo ated in the NCPs and that several studies in the �eld of Mathemati s Edu ation emphasize

their importan e in the integral formation of students of Basi Edu ation, based on our experien e in the

formation of tea hers, we observe that an e�e tive work with geometri onstru tions has not yet been

fully integrated into the Mathemati s lasses of Brazilian s hools.

For almost two de ades, Wagner (1998) stated that this subje t should be res ued, given its potential

as an auxiliary tool in the learning of Geometry, sin e the onstru tions with ruler and ompass already

appear in the �fth entury BC and were of great importan e in the development of Greek mathemati s.

We agree with this author and raise the following question: to what extent are the mathemati al textbooks

ontributing to this res ue? Or, more spe i� ally, how do geometri onstru tions appear in the urrent

Mathemati s textbooks of the �nal grades of Elementary S hool? In this ontext, we arried out a

study aimed at des ribing and analyzing how the geometri onstru tions � whether in the onventional

156

paper and pen il environment or in Dynami Geometry environments � are integrated in the textbooks

that make up the PNLD (Programa Na ional do Livro Didáti o) 2017 Mathemati al Textbooks Guide

(BRASIL, 2016). We believe that su h a survey is important, sin e it is known that the ontent of

textbooks has a strong in�uen e on tea hers' pra ti es.

Referen es

BRASIL. Se retaria de Edu ação Fundamental. 1998. Parâmetros Curri ulares Na ionais: Matemáti a

� 3

o. e 4o. i los do Ensino Fundamental. Brasília: MEC/SEF.

BRASIL, Se retaria de Edu ação Bási a. 2016. Guia de Livros Didáti os do PNLD 2017 �

Matemáti a. Brasília: MEC/SEB.

WAGNER, Eduardo. 1998. Construções Geométri as. Rio de Janeiro: So iedade Brasileira de

Matemáti a.

ZUIN, Eleni e de Souza Lodron. 2001. �Da Régua e do Compasso: as onstruções geométri as omo

um saber es olar no Brasil�. Dissertação (Mestrado em Edu ação) � FE/UFMG, Belo Horizonte.

MATHEMATICS TEXTBOOK STRUCTURE IN EL SALVADOR WITH THE AIM OF

INCREASING LEARNING TIME AS MUCH AS POSSIBLE

Félix Abraham Guevara Menjívar Wendy Stefanía Rodríguez Argueta

Ministry of Edu ation of El Salvador

abrahamguev�gmail. om wendyrodriguez19�hotmail. om

Currently, the Ministry of Edu ation in El Salvador is elaborating new mathemati s textbooks and work-

books for students and tea her's guide from grade 1 to 11 to distribute nationwide in 2018.The ministry

adopted hypotheti al strategies, fo using on 3 major fa tors whi h ould have signi� ant orrelation with

the improvement of mathemati s a hievement, su h as providing learning material se uring appropriate

a ademi level, in reasing learning time during lass and tea her's learning assistan e as follows?

1. New textbook development, also student's workbook and tea her's guide in a ordan e with appro-

priate sequen e, onsidering a tual a ademi level of students.

2. To guarantee su� ient students' learning time (approximately 20 minutes ea h math lass) so that

ea h student ould study a tively during lass, adding another learning opportunity at home with

student's workbook.

3. To hange tea her's paradigm from "talking a lot" to se ure students' learning opportunity to study

a tively by their own, utilizing the new textbook.

In this poster session, we try to larify our proposed stru ture of these three types of materials

on retely.

One of the editing strategies is �one page per lass" to maintain appropriate lesson progress so that

the tea her an over all grade ontents until �nishing s hool year.

a. Review b. Problem . Solution d. Con lusion e. Exer ise f. Challenge

• Supplied to s hool not to student and expe ted to be used more than 3 years.

• Textbook won't be allowed to bring home, but to be utilized just during math lass by students.

• Sin e textbook for �rst and se ond grade is adopted "workbook style" allowed to bring home, just

these two grades textbooks will be supplied to ea h student every s hool year.

157

• Workbook will be supplied to ea h student, allowed to bring home.

Referen es

Ministerio de Edu a ión, 2008. Programas de estudio de matemáti a: Primer, Segundo y Ter er

C Ci lo/ Ministerio de Edu a ión 1a. ed.

Ministerio de Edu a ión, 2008. Matemáti a 3, Cole ión Cipotas y Cipotes / Ministerio de CCEd-

u a ión 1a. ed.

Ministerio de Edu a ión, 2008. Currí ulo al servi io del aprendizaje: Aprendizaje por ompeten ias/

CCMinisterio de Edu a ión 2da. ed.

HISTORICAL APPROACH OF THE TOPICS MATRIZES, DETERMINANTS AND

LINEAR SYSTEMS PRESENT IN THE TEXTBOOKS

Daniel Martins Nunes

Federal Institute of the North of Minas Gerais

daniel.nunes�ifnmg.edu.br

This study aims at presenting data obtained after the analysis of three olle tions of textbooks available for

hoi e through National Textbook Program, 2015, to high s hool lasses: Conne tions with Mathemati s

- Vol. 2, Fábio Martins de Leonardo; Mathemati s: Context & Appli ations - Vol. 2, Luiz Roberto Dante;

Mathemati s - Paiva - Vol 2, Manoel Paiva. The analysis onsisted of observation of histori al aspe ts

present in the topi s of Matri es, Determinants and Linear Systems in the mentioned textbooks, having

su h dire tion the guidelines of Brazil (2006) and Miguel and Miorim (2011). In addition, we analyzed the

presen e of errors omparing the information olle ted with the works developed by Boyer (1974); Eves

(2011) and Smith (1958).

In general, the History of Mathemati s is represented shyly in these olle tions. Only one of the

olle tions presents a dida ti sequen e based on the histori al ontext (Matri es - Linear Systems -

Determinants). In the others olle tions there is no su h organization and the History of Mathemati s

arises only as a resour e to present the biography of mathemati ians. Moreover, in one of the books we

dete ted an error in the informations, whi h leads us to re�e t on the need to he k su h fa ts before you

play them wrongly.

Thus, we on lude that we must be areful when hoosing a textbook as referen e for our work.

Moreover, it is ne essary to give value the works that bring problematization and dida ti sequen e in

a ordan e with the History of Mathemati s, be ause, as highlighted in Brazil (2006); Miguel and Miorim

(2011), this approa h brings more bene�ts to student learning show them the relationships between the

mathemati al theories learned and that therefore they are not dis onne ted.

Referen es

Boyer, Carl Benjamim. 1974. History of Mathemati s. Translate by Elza F. Gomide. São Paulo:

Edgard Blu her, Ed. University of São Paulo.

Brazil. 2006. Curri ulum Guidelines for Se ondary Edu ation: Natural S ien es, Mathemati s and

Te hnologies. Brasilia: Department of Basi Edu ation.

Eves, Howard. 2011. Introdu tion to the History of Mathemati s. Translation by H. Hygino

Domingues. Campinas: Editora Uni amp.

Miguel, Antonio, and Maria Angela Miorim. 2011. History in the mathemati s edu ation: proposals

and hallenges. Belo Horizonte: Editora Autênti a.

158

Smith, David Eugene. 1958. History of Mathemati s. New York: Dover Publi ations

ANALYSIS OF BRAZILIAN TEXTBOOKS IN TWO STAGES: PRE-MODERN TIMES

(1950) AND MODERN (1960 AND 1970)

Alexandre Souza de Oliveira

UNINOVE-SP/ PUC-SP

professor.oliveira�yahoo. om.br

This study seeks to identify how di�erent authors of textbooks dealt with the role of edu ation from the

transformations stemming from the Modern Mathemati s Movement. Separate analysis of textbooks in

two stages: Pre-Modern Times and Modern. The �rst orresponds to the 1950s, during whi h the urrent

edu ational legislation was the Ministerial De ree of 1951. The se ond is the name we use for the 1960s

and 1970s, during whi h were published by GEEM in 1962 A�airs minima for a Modern Mathemati s

Program Gym, Tips for Program s ript to the hair of mathemati s in 1965; and the Laws of Edu ation

Guidelines and Bases, 1961 and 1971. The results indi ate that there is a ertain standardization in

relation to: fun tion as a spe ial ase of relationship; representation ratio / fun tion by arrows diagram;

domain on ept, ounter-domain and image. The aspe ts that di�erentiate the analyzed olle tions are:

the emphasis on symboli language, rigorous approa h to the subje t, the on ern with abstra tion,

ontextualization, the use of exer ises / a tivities for ontent approa h.

Referen es

BLOCH, M. 2002. Apology of history or the o� e of the historian. Translation André Telles. Rio

de Janeiro: Jorge Zahar.

BRAZIL. 1961. Ministry of Edu ation and Sports. Law No. 4,024 / 61 of 20 De ember 1961.

Establishes Guidelines and Bases of National Edu ation.

CERTEAU, Mi hel. 2007. The writing of history. Translation: Maria de Lourdes Menezes. 2nd ed,

Rio de Janeiro, RJ: University Forensi .

CHARTIER, R. 1991. The world as representation. Translation: Andrea Daher and Zenir Campos

Reis. Advan ed Studies, 11 (5): p. 173-191.

CHERVEL, A. 1990. Stories of s hool subje ts: re�e tions on a sear h �eld. Theory & Edu ation,

n.2. Porto Alegre: Pannoni a. P. 177-229.

CHOPPIN, A. 2004. History of books and edu ational issues: State of the art. Magazine: Edu ation

and Resear h. São Paulo, v. 30, n.3, p.549-566.

GEEM. 1962. Minimum A�airs for a modern math program to the gym. modern mathemati s for

se ondary edu ation. Grade Tea her n. 1, 1st edition, São Paulo, SP: GEEM.

GEEM. 1965. Suggestions for a program s ript for the hair of mathemati s. In: modern mathe-

mati s for se ondary edu ation. Grade Tea her n. 1, 2nd edition, São Paulo, SP: GEEM 1965.

VALENTE, W. R. 2008. Osvaldo Sangiorgi and the Modern Mathemati s Movement in Brazil. Rev.

Dialogue Edu . Curitiba, v. 8, no. 25: p. 583-613.

THE TYPES OF PROBLEMS ADDRESSED FROM THE CONTENT OF FUNCTIONS IN

THE DIDACTIC BOOK: MATH CONTEXT AND APPLICATIONS

José Ant�nio de Oliveira Júnior

Federal Institute of S ien e and Te hnology of Pará, Campus Itaituba, Brazil

jose.antonio�ifpa.edu.br

159

The use of problem resolution in the lassroom is a topi that deserves attention from tea hers who tea h

mathemati s. This is very important be ause it is in problem resolution that students develop skills that

they an use to solve problems in the s hool environment and in situations that go beyond the boundaries

of the lassroom.

We an point out that problem resolution has gained a good in entive in the events held on mathemati s

edu ation and the publi ation of the National Curri ulum Parameters (1998) that propose, together

with mathemati al edu ators, the possibility of students to mobilize, develop the apa ity to manage

information,

Onu hi and Allevato (2004) argue that working in the perspe tive of problem resolution is not easy,

requiring of the tea her a vast knowledge mathemati al and opening the �exibility of new ideas. Dante

(2009) also points out the di� ulty of tea hing problem solving on e, that this skill involves a variety of

thought pro esses developed that need to be losely monitored by tea hers.

In the present work we will investigate how to lassify the problems proposed by the dida ti book

�Mathemati s ontext and appli ations" regarding the ontent of fun tions whi h the types of problems

that the book addresses for the development of the learning of this ontent? If the problems addressed

are of re ognition, algorithms, standard, heuristi , appli ation or puzzle, among others, where we will form

ategories of problem typi� ation and analyze this approa h with volume 1 of high s hool in a perspe tive

of how problems are organized in to promote the tea hing and learning of students of a ording with the

previously mentioned literature.

Referen es

DANTE, L.R. Formulation and resolution of problems: theory and pra ti e.1.ed.São Paulo: Áti a,

2009.

ONUCHIC, L. de La R. Tea hing-Mathemati s Learning through Problem Solving. In: BICUDO

Maria A.V. (org) Resear h in Mathemati al Edu ation: Con eptions and perspe tives. São Paulo:

UNESP 1999.p.199-218.

POYA, G. The art of solving problems. A new aspe t of the mathemati al method. Rio de Janeiro:

Inter iên ia, 1995.1196p.

BOOKS TASKS' ADAPTATION TO DIFFERENTIAL AND INTEGRAL CALCULUS

CLASSES

Nélvia Santana Ramos

UTFPR � CP

nelvia_ramos�hotmail. om

William José Gonçalves

FATEC/UNIVALE

williamboatemati a�gmail. om

André Luis Trevisan

UTFPR - LD

andreluistrevisan�gmail. om

Based on Palha, Dekker, Gravemeijer e Van Hout-Woltersa (2013) e Palha, Dekker, Gravemeijer (2015) is

presented here a proposal to Di�erential and Integral Cal ulus lasses (DIC) based in the organization of

tasks resolution episodes. It's broa hed the "learning arrangement" organized by tasks proposal, adapted

from dida ti al books (Gafanhoto; Canavarro, 2014) whose resolution does not demand a previously

explanation about the subje t to itself onne ted, amenable to students' resolution in groups and working

in a ollaborative way.

Su h tasks, whi h are entral elements from this type of approa h, are planned to be proposal in regular

lasses, in di�erent moments during the DIC ourse, whi h emerge DIC's fundamental on epts, taking in

onsideration that general and rigorous de�nitions do not dispense with intuitive and reative phase from

mathemati al ideas (Reis, 2009). These tasks allow to emerge and deepen fundamental on epts from

math by a progressive pro ess of mathematization (Freudenthal, 1991) and take, in their formulation and

proposal of routing, the real environment from a DIC lassroom.

160

In this paper, it's presented a hara terization from these tasks, as example from instru tional materials

to DCI lasses. In spe ial, it is presented an example whi h explores related aspe ts to the on ept of

in�nite limit from a real fun tion from a real variable. It is also given data from a real tea hing episode,

in intuition to analyze the dis ussion and the mathemati al on epts whi h were systematizes based on

this environment.

Referen es

Freudenthal, Hans. 1973. Mathemati s as an edu ational task. Dordre ht: D. Reidel Publishing

Company.

Gafanhoto, Ana Patrí ia and Canavarro, Ana Paula. 2014. �A adaptação das tarefas matemáti as:

omo promover o uso de múltiplas representações". In Práti as Pro�ssionais dos Professores de

Matemáti a edited by João Pedro da Ponte, 113 - 134. Lisboa: Universidade de Lisboa.

Palha, Sonia P. A. G., Dekker, Rijkje, Gravemeijer, Koeno, and Van Hout-Woltersa, Bernadette.

2013. "Developing shift problems to foster geometri al proof and understanding". The Journal of

Mathemati al Behavior 32 (1): 141-159.

Palha, Sonia P. A. G., Dekker, Rijkje and Gravemeijer, Koeno, and Van Hout-Woltersa, B. 2015.

"The e�e t of shift-problem lessons in the mathemati s lassroom". Interna ional Journal of S ien e

and Mathemati s Edu ation 13 (1): 1589-1623.

Reis, Frederi o. 2009. �Rigor e Intuição no Ensino de Cál ulo e Análise". In Edu ação Matemáti a

no Ensino Superior: Pesquisas e Debates edited by Lilian Nasser and Maria Clara Rezende Frota,

81 - 97. Re ife: SBEM.

TEACHING PROBABILITY IN EARLY SCHOOL YEARS: THE APPROACH IN

BRAZILIAN TEXTBOOKS

Mi haelle Santana Rute Borba


mikarmoraes�hotmail. om resrborba�gmail. om

Tea hing probability on epts in s hool is highly essential for today's itizen that has to onstantly deal

with ertainties and un ertainties. Therefore, it is riti al for tea hers to promote the development of

a wide range of probability on epts through experien es, enabling students to observe and then draw

on lusions, awakening s ienti� thinking riti al to their edu ation. In this sense, textbooks are very

important be ause they may dire t tea hers to ontents to be taught in ea h s hool level and year

and show how to broaden the understanding of ea h on ept, in parti ular those on erning probability.

A ording to Carvalho and Lima (2010) textbooks ontain hoi es on: the ontents to be studied; the

methods to be used for students better understanding; and the urri ulum organization to be adopted

throughout s hooling years.

The present study aimed to analyze how Brazilian Elementary S hool mathemati s textbooks approa h

the on ept of probability and how authors onsider the dimensions pointed out by Vergnaud (1986) in

his Theory of Con eptual Fields: situations that bring meanings to a on ept; properties and relationships

of the on ept; and symboli representations used to represent and work on the on ept. With this aim,

11 5th grade textbooks of the Brazilian National Textbook Program were analyzed. Analyses onsidered

the way the on ept of probability was introdu ed; notions dealt with; the types of a tivities provided and

symboli representations used.

It was observed that the ommonly used ways to introdu e probability are linked with the ideas

of per entage, fra tions or ombinations. Furthermore, it was noti ed that �ve notions, namely han e,

probability, experiment, random, predi tion and trial, were largely overed, with han e the most frequent,

161

although no textbook dealt with all these notions. Regarding the types of a tivities, word problems were the

most frequent with 50% of them only with instru tions and no type of auxiliary symboli representation.

In general, the textbooks analyzed do not deeply explore the on ept of probability and the proposed

tea hing is fragmented. Thus, an improvement is ne essary to better motivate early s hool students in

their development of probabilisti thinking.

Referen es

Carvalho, J. B. P., Lima, P. F. 2010. �Choi e and use of the textbook.� e. 17: 15-30.

Vergnaud, G. 1986. �Psy hology of ognitive and dida ti development of mathemati s. An example:

additive stru tures.� Psy hologi al Analysis 1: 75-90.

COLLIGATIVE PROPERTIES AND MATHEMATICAL SIGNS PRESENTED IN THE

DIDACTIC BOOKS OF CHEMISTRY: A SEMIOTIC ANALYSIS

5

Karina Alessandra Pessoa da Silva

Federal Te hnologi al University of Paraná, Brazil

lourdes�uel.br

Lourdes Maria Werle de Almeida

State University of Londrina

karinasilva�utfpr.edu.br

Our ondition of relationship with the world is mediated by a multiple network of languages and represen-

tations. We ommuni ate through readings or produ tion of texts, gestures, fa ial expressions, through

images, graphs, signs, numbers, sensations, so that we an onsider ourselves to be beings of language.

The diversity of uses of language and representations refers us to referen es that investigate signs and

their representative fun tions.

In the edu ational ontext, the textbook an be onsidered an entity of language and it presents a

multipli ity of signs that represent obje ts. Our understanding of the sign and the obje t is based on

Peir ean semioti s (Peir e, 1972; Peir e, 1989; Peir e, 2005). In his writings, Peir e de�nes sign as

anything that o upies pla e of the obje t and that is apable of giving rise to other signs.

Considering the importan e of sign language, our resear h proposes to present a semioti analysis of

the mathemati al signs approa hing Colligative Properties presented in textbooks of Chemistry for High

S hool, approved by the National Program of Dida ti Book (PNLD

6

) of the year 2015. The obje tive

of our analysis is to investigate if the mathemati al signs used by the authors of the textbooks have the

potential to generate other signs and to provide the mathemati al on eptualization for the phenomenon

under study.

The results of our analysis indi ate that the mathemati al signs, more spe i� ally the graphs, on-

erning the approa h of olligative properties in Chemistry textbooks, an suggest mathemati al obje ts

and provide their understanding by the student.

Referen es

Peir e, C. S. 1972. Semióti a e Filoso�a: textos es olhidos. São Paulo: Cultrix.

Peir e, C. S. 1989. Es ritos Coligidos. São Paulo: Nova Cultural.

Peir e, C. S. 2005. Semióti a. 3. ed. São Paulo: Perspe tiva.

5

The resear h is related to a proje t funded by CNPq.

6

Program implemented by the Brazilian Ministry of Edu ation (MEC), in triennial y les, with the obje tive of evaluating

and providing publi s hools of Basi Edu ation with textbooks.

162

COGNITIVE DEMANDS TO COMPARE MATHEMATICS TEXTBOOKS IN DIFFERENT

COUNTRIES

Xhevdet Thaqi

University of Gjilan

xhevdetthaqi�yahoo. a

Joaquin Giménez

Centro de Edu ação Matemáti a (SP)

quimgimenez�ub.edu

Antonio José Lopes

Universtat de Bar elona

bigode�pentaminos.mat.br

The rationale behind this study on erns the issues s hool administrators and tea hers/prospe tive tea hers

fa e over the progress of these students in mathemati s lassrooms in two di�erent ountries (Spain/Cata-

lonia, Brasil and Kosovo). This study aimed to present a method of examining primary s hool mathemati s

textbooks with the purpose of evaluating students' expe ted past learning and omparing students' ex-

pe ted geometri al transformations learning a ross the di�erent urri ula. The analyti al pro edure of the

examination of text ontent as presented in this study is itself a diagnosti te hnique for assessment of the

students' past learning, whi h addressed the main obje tive of the study. The results were analysed and

ompared among the textbooks of two ountries in terms of ognitive domains that these ontent required.

The results were interpreted in the ontext of the textbooks and relevant urri ulum. This interpretation is

des riptive in nature and serves as a means to identifying the nature of similarities and di�eren es a ross

the textbooks from Spain/Catalonia and Kosovo in terms of the geometri al transformations on ept

image (Thaqi, 2009).

The textbooks analysed of any of the three ountries do not re�e t the urri ulum obje tives and

tea hers do not over these absen es. Consider the example in Catalonia urri ulum is required that

pupils in primary �... re ognize turns, and make shadows, symmetries and turns ...� and � re ognize

obje ts that have generated a parti ular shade �or � identify the transformation that relates two given

�gures ...�et ., but there is no a tivity that would develop it.

As a main result, we found that the organization of ontent is based upon a dis onne ted way of

presentation about transformations, and a few attentions paid to variability issues At one level, the study

provides information about ognitive requirements of geometri al transform a tions ontent in elementary

textbooks and this information an be used to improve mathemati s urri ulum and tea hing in other

ountries. We see that the a tivities posed math textbooks for primary edu ation do not give the possibility

to students understanding and building a lear on ept of geometri transformation. There is ne essary

a good prepared professional knowledge tea her of primary s hool. The ontent presented in textbooks

may be useful in the orre t and omplete onstru tion of the on ept of geometri transformation only

when the tea her is well prepared to develop a tivities and to tea h them.

Referen es

Jagoda, E., & Swoboda, E. 2011, Stati and dynami approa h to forming: the on ept of rotation,

Pro edings of CERME7, (pp, 558, 567) University of Rseszow.

Mar hini, C. & Vighi, P. 2009. Can we develop geometri al understanding by fo using on isometries?

A tea hing experiment by the means of geometri al artefa ts. Pro eedings of SEMT '09, (pp. 169-

176)

Thaqi, Xh. (2009). Aprender a enseñar transforma iones geométri as en Primaria desde una

perspe tiva ultural. (phD dissertation, Bar elona University. Spain).

A VIRTUAL SPACE AS AN ALTERNATIVE TO DIFFERENCIAL AND INTEGRAL

CALCULUS BOOK

André Luis Trevisan Alan Fran o do Couto Mar ele Tavares Mendes

UTFPR � LD, Brazil

andreluistrevisan�gmail. om alanf outo�gmail. om mar eletavares�utfpr.edu.br

163

Di�erential and Integral Cal ulus lasses (DIC) used to be summarized by the y le, de�nition - theorems -

proprieties - examples - new de�nition, in a rigorous approa h whi h fo us on algebrai aspe ts and leaves

out the students' intuition. In sequen e, it is proposed tasks, in general from the dida ti al books, whi h

resolutions ontribute only to develop in the students a reasoning knowledge based on the identi� ation of

similarities (Lithner, 2000), supported by some key-word usage and from examples previously presented.

Meeting to this perspe tive, proposes from DIC lasses based on tasks resolutions episodes (or, some

alled for Palha, Dekkes, Gravemeijer e Van Hout-Woltersa, 2013; Palha, Dekker, Gravemeijer, 2015, shift

problem lessons), whi h mobilize the use of di�erent representations and utilize omputational resour es

(Gadanidis; Borba; Silva, 2015) ontribute to the development of a type of reative reasoning and not

only an imitative one (Lithner, 2008).

In this paper, that is a fragment from a tivities developed in a resear h proje t alled �Investigation of

an edu ational environment to Di�erential and Integral Cal ulus in real tea hing onditions�, it is presented

and analyzed a virtual spa e proposal as support to DIC. Besides its des riptions, it is presented an analysis

of elements, whi h distinguish itself from usual materials of tea hing, su h as: the organization of topi s

with subje ts presented in spiral format, dynami al hara ter - by applets and videos usage, the strategy

to make available the subje ts and the proposition of investigative tasks at the end in ea h topi .

Referen es

Gadanidis, George; Borba, Mar elo de Carvalho and Silva, Ri ardo S u uglia Rodrigues. 2015.

Fases das te nologias digitais em Edu ação Matemáti a: Sala de aula e internet em movimento.

Autên ia: Coleção Tendên ias em Edu ação Matemáti a.

Lithner. Johan. 2000. �Mathemati al Reasoning in Task Solving�. Edu ational Studies in Mathe-

mati s, 41 (2): 165 - 190.

Lithner. Johan. 2008. �A Resear h Framework for Creative and Imitative Reasoning�. Edu ational

Studies in Mathemati s, 67 (3): 255 - 276.

Palha, Sonia P. A. G.; Dekker, Rijkje,; Gravemeijer, Koeno, and Van Hout-Woltersa, Bernadette.

2013. �Developing shift problems to foster geometri al proof and understanding�. The Journal of


Palha, Sonia P. A. G.; Dekker, Rijkje and Gravemeijer, Koeno, and Van Hout-Woltersa, B. 2015.

�The e�e t of shift-problem lessons in the mathemati s lassroom�. Interna ional Journal of S ien e

and Mathemati s Edu ation 13 (1): 1589-1623.

THE PRESCRIBED CURRICULUM FOR COMBINATORICS AND WHAT IS

PRESENTED IN 5TH GRADE BRAZILIAN TEXTBOOKS

Glau e Vilela Rute Borba


glau evilela_�hotmail. om resrborba�gmail. om

Fa ed with all the variety of material o�ered to the tea her, the textbook still o upies a prominent pla e

in tea hing and learning pro esses in the s hool ontext. In the last de ades in Brazil, this resour e is

be oming more and more the target of resear h, espe ially in what on erns the approa h of spe i�

ontents from the o� ial urri ular guidelines. For Sa ristán (2000), the o� ial urri ular guidelines,

de�ned by him as pres ribed urri ulum, a ts as a starting point in ordering the urri ular system, and

be omes a referen e in the elaboration of dida ti materials. In this sense, Sa ristán (2000) de�nes

textbooks as presented urri ulum that translate and interpret the meaning and ontents presented in the

pres ribed urri ulum.

164

The present study aims to analyze the guidelines given on erning Combinatori s in the o� ial do u-

ments for Elementary Mathemati s Edu ation, the relations of these orientations with the approa h to the

problems involving ombinatorial reasoning in 5th year textbooks and the guidelines presented in tea hers'

manuals, taking into a ount varieties in the three dimensions of on epts proposed by Vergnaud (1986)

(meanings, invariants and symboli representations). For the analysis, we used as o� ial do uments the

National Curri ular Parameters � PCN (BRASIL, 1997), eight 5th grade Mathemati s textbooks and their

respe tive manuals. These textbooks were hosen randomly and the 5th grade was hosen be ause previ-

ous studies showed that, in Elementary S hool, more ombinatorial situations are presented in this s hool

year.

The pres ribed urri ulum analyzed presented indi ations of paths to be followed for the tea hing

of Combinatori s. This approa h be omes expli it when the do ument emphasizes the need of a tivities

with di�erent ideas of multipli ation, dealing expli itly with one of the ombinatorial situations, that is the

Cartesian produ t, and pointing out possible strategies su h as drawings and tree diagrams. We observed

the absen e of spe i� orientations for the tea hing in the �rst s hool years with di�erent ombinatorial

situations (arrangements, permutations and ombinations).

Regarding the results obtained in textbooks, we observed that the meanings with the highest total

per entages of presentation were ombination and Cartesian produ t. There was little variation of the

symboli representations used in the presentation and in the request for solving the problems. It was

also veri�ed that there was no on ern, on the part of the authors, to present the invariant properties of

Combinatori s, nor did they draw the tea her's attention on the di�erent meanings involved. The results

show little orientation in textbooks to the approa h of this ontent in the early years of s hooling.

Referen es

Brasil. SEF. Parâmetros Curri ulares Na ionais (PCN): Matemáti a. Ensino de primeira a quarta

série. Brasília: MEC, 1997.

Sa ristán, J.G. O Currí ulo: uma re�exão sobre a práti a. Porto Alegre: ArtMed, 2000.

Vergnaud, G. 1986. Psy hology of ognitive and dida ti development of mathemati s. An example:

additive stru tures. Psy hologi al Analysis 1: 75-90.

THE COMPARATIVE STUDY OF STATISTICAL COURSE DIFFICULTY DEGREE IN

JUNIOR HIGH SCHOOL MATHEMATICS TEXTBOOKS OF CHINA, THE US AND

AUSTRALIA

Jianbo Wang

Beijing Normal University Publishing Group

11050238�qq. om

The paper sele ts from China, the US and Australia respe tively two types of junior high s hool math-

emati s textbooks that are examined and approved by edu ational administrative organs or widely used,

and ondu ts a omparative study of the statisti al ontent in these textbooks in light of ontent width,

ontent depth, and depth of statisti al exer ises. The paper ranks the six types of textbooks in terms of

statisti al ourse di� ulty degree as follows: Mathemati s textbooks and Conne ted textbooks rank �rst

and se ond respe tively, Heinemann textbooks and ICE-EM textbooks rank third and fourth respe tively,

PEP textbooks and BNUP textbooks rank �fth and sixth respe tively. The resear h �ndings of this study

o�er thoughts and enlightenment about the ourse reform of the statisti al ontent in China's junior high

s hool mathemati s ourse and the ompiling of statisti s in textbooks.

Keywords: mathemati s textbooks; statisti s; ourse di� ulty; omparative study.

165

THE EFFECT OF LITERACY IN THE NEW HUNGARIAN MATHEMATICS

TEXTBOOKS

Gergely Wints he Gergely Szmerka

Eötvös Loránd University (ELTE), Hungarian Institute for Edu ational Resear h and Development (OFI),

Veres Péter Gymnasium

wints he� s.elte.hu szmerka.gergely�gmail. om

Everyone needs high-level litera y skills in our knowledge-based so iety. The modern on ept of litera y

in ludes not only the reading apability (in a narrow sense), but also ultural, information and digital

understanding, as well as a lot of other ompeten ies that are essential in the 21st entury. Nowadays

people have a ess to an extensive amount of information and one has to understand, reorganize and

mobilise the appropriate part of the information �ood. (Varga & all 2015)

The knowledge of mathemati s, similarly to a lot of other subje ts, is partly a fun tion of the litera y

level of the pupils. We 'lost' some of our students be ause they have no han e to understand mathemati s

as they have problems with the meaning of the words and the senten es. One of our goals is to support the

e�orts of tea hers of other subje ts by raising the reading and understanding apability of the students.

That is why we built long stories, and a lot of textual based games and problems in our new textbooks.

In this part I will fo us on our main aims at the beginning of the textbook development. Our target

group was that of the 'average students' with a basi level mathemati al knowledge. Our goals were

grouped a ording to the age groups (level) of the students. On the �rst level, in grade 5 and 6, we fo use

mainly on the general litera y skills and on so ial ompeten y, as well as on some basi math information.

On the se ond level, in the textbooks for grades 7 and 8 we lay more emphasis on mathemati s, and we

tried to make use of the inner onne tions with other subje ts too.

Here I quote a tea her opinion from an early feedba k. �The long and ompli ated textual based

problems are not as motivating for the students as one ould wish. They �ght with the text, they have

problems even with the literature based text, and here there are mathemati al ontexts in luded as well. It

is a hallenging task for them, and for us too. My 5th grade lass is a good example. Normally they have

no problems with the (more pure) math tasks in the test, but they omit the long text based problems,

although it would be easy as a math problem. I suppose it was too long as a text.�

Results

In our poster we show how we applied textual problems, games and stories, and we give a lot of

examples from our new textbooks. We will present the detailed answers of 64 tea hers after using the

textbooks for more than two years. Their attitude had ompletely hanged as a group. We an ite some

relevant answers and the statisti s of their hanges. In this regard, we are approximately satis�ed. Even

the relu tant tea hers proved that the level of the litera y ompeten y and the so ial ompeten y of their

student groups has been raised, so we a omplished our goal.

Referen es

Varga, K., Kojanitz, L., Dobszay, A. Wints he, G., 2015. Textbooks in a Knowledge-based So iety.

CIDREE yearbook, 46-61. http://www. idree.org/publi ations/yearbook-2015/

USE OF THE BALANCE MODEL IN LEARNING TASK BY AUTHORS OF MEXICAN

SECONDARY-SCHOOL MATHEMATICS TEXTBOOKS: ITS PRESENCE AND

AUTHENTICITY

Yolanda Zamora Josip Slisko Adrián Corona María Ara eli Juárez

Fa ultad de Cien ias Físi o Matemáti as, Benemérita Universidad Autónoma de Puebla, Puebla, Méxi o

y_z orona�hotmail. om jslisko�f fm.buap.mx a orona�f fm.buap.mx arjuarez�f fm.buap.mx

166

It is known that students reveal big on eptual and pro edural di� ulties in their early algebrai thinking

(Sleeman, 1986). One way to help students over ome di� ulties in solving linear equations is to start

with the balan e model (Vlassis, 2002; Waren & Cooper, 2005). Mexi an mathemati s urri ulum for

se ondary s hool ( orresponding to junior high s hool in the USA) suggests expli itly the use of that

model.

We tried to answer two resear h questions: (1) How present is this urri ular suggestion in published

mathemati s textbook for se ondary s hool? (2) How authenti are learning tasks involving the balan e

method, a ording to Palm's riterions (Palm, 2009)?

Of 30 mathemati s textbooks for the �rst grade of se ondary s hool, 10 of them do not follow the

urri ular suggestion, by not mentioning the balan e model for linear equations. The resting 20 textbooks

present 86 learning tasks related to di�erent forms of balan e model. Nevertheless, the presen es vary

greatly, going from only one to eleven ases. The absen e of the model in 10 of 30 textbooks is strange

be ause all of them had to pass a review pro ess by the Ministry of Publi Edu ation, who designed the

urri ulum and is in harge of its implementation in s hools.

Many of learning tasks, involving the balan e model, violate one or more riterions for authenti s hool

problems. Some of them even have ontradi tory data and an't be solved.

Referen es

Palm, Torulf. 2009. �Theory of authenti task situations.� In Word and worlds: Modelling verbal

des riptions of situations edited by B. Greer, L. Vers ha�el, W. Van Dooren, & S. Mukhopadhyay.

Rotterdam: Sense Publishers.

Sleeman, D. 1986. �Introdu tory algebra: A ase study of student mis on eptions.� Journal of


Vlassis, Joëlle. 2002. �The balan e model: Hindran e or support for the solving of linear equations

with one unknown.� Edu ational Studies in Mathemati s 49 (3): 341-359.

Warren, Elizabeth, and Tom J. Cooper. 2005. �Young hildren's ability to use the balan e strategy

to solve for unknowns.� Mathemati s edu ation resear h journal 17 (1): 58-72.

PROCEDURAL KNOWLEDGE DEVELOPMENT MODEL IN CHINESE CLASSROOMS �

EXAMPLES BEYOND THE SCHOOL TEXTBOOK

Jinyu Zhang

Department of Mathemati s, East China Normal University

Jinyuzhang1984�163. om

Pro edural knowledge is a omponent of mathemati s that learners need to handle every day. Even

though it only took a small part during the whole ir le of mathemati s a tivity, this type of �knowing

how� knowledge still be the ne essary part of mathemati s textbook. There is an impression that Chinese

students drill in this highly repeat problems during their daily learning life. But as a high s hool math-

emati s tea her in China, it seems useless if only put �pressure and time� in students learning pro ess.

In this arti le, a pro edural knowledge development model is built to larify the fun tion of a traditional

tea hing method whi h used widely in China. Instead of giving students a bun h of homework full of

highly repeat pra ti es, experien e tea hers in China use some lassi but arefully designed examples

beyond the textbook in the lass to help students.

Referen es

Clark, J. M., Cordero, F., Cottrill, J., Czarno ha, B., DeVries, D. J., St. John, D., et al. (1997).

�Constru ting a s hema: The ase of the hain rule.� The Journal of Mathemati al Behavior, 16,

345-364.

167

Dubinsky, E. (1991). �Re�e tive abstra tion in advan ed mathemati al thinking.� In D. Tall (Ed.),

Advan ed mathemati al thinking (pp. 95-123). Dordre ht, The Netherlands: Kluwer.

Dubinsky, E., & M Donald, M. (2001). �APOS: A onstru tivist theory of learning in undergrad

mathemati s edu ation.� In D. Holton (Ed.), The tea hing and learning of mathemati s at university

level: An ICMI study (pp. 273-280). Dordre ht, The Netherlands: Kluwer.

Gu, L., Huang, R., & Marton, F. (2004) �Tea hing with variation: A Chinese way of promoting

e�e tive mathemati s learning.� In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn

mathemati s: Perspe tives from insiders (pp. 309-347). Singapore: World S ienti� .

168

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