enhancing mathematical thinking in finnish elementary ... · enhancing mathematical thinking in...

www.helsinki.fi/yliopisto

Enhancing mathematical thinking in Finnish elementary school mathematics- pedagogical models (and modelling)

Dr. Heidi KrzywackiDepartment of Teacher EducationUniversity of Helsinki, Finland

Finland is located by the Arctic Circle…

Finland is in the North Europe, part of the Nordic countries

…and the home of the REAL Santa Claus is in the Finnish Lapland!

Where is Finland?

www.helsinki.fi/yliopisto 3

Finland in facts

• Population: 5.4 million, 17 inhabitants / km2

• Languages: Finnish (spoken by 91%)Swedish (~ 5%)Sámi (about 1,800 people)

• Religion: 79.9% Lutheran and about 1.1% Orthodox, other 1.7 % and none 17.7 %

• Independence: Declared on December 6, 1917. Previously a grand duchy in the Russian empire (1809-1917), and before that, a part of Sweden for 600 years

• Currency unit: Euro• Area: 338,424 km²• Capital: Helsinki (population ~600 000)

Department of Teacher EducationDr. Heidi Krzywacki Modelling Conference, January 10, 2013


1) Mathematics education aiming at development of mathematical thinking

2) Pedagogical models (and modelling)A. The use of manipulatives and illustrationsB. Inductive approach to elementary school

mathematicsC. Word problems and problem solving

3) Concluding remarks

4

Outline of the presentation



Aiming at development of mathematical thinking

5



• the significance of developing mathematical thinking skills... offering opportunities for the development of mathematical thinking and for the learning of mathematical concepts and the most widely used problem-solving methods.The discipline's concrete nature serves as an important aid in bringing together the pupil's experiences and systems of thought with the abstract system of mathematics.

Modelling is hardly mentioned in the curriculum!

6

National core curriculum: mathematics education (1/2)

(FNBE, 2004)Department of Teacher EducationDr. Heidi Krzywacki Modelling Conference, January 10, 2013


National core curriculum: mathematics education (2/2)

• Links with everyday life and practical problems• The use of concrete materials and illustrations

supporting conceptual and procedual understanding

• Various ways to process mathematical tasks and give reasons for solutions• oral and written communication, the use of

manipulatives, illustrations and symbolic expressions

7

(FNBE, 2004)Department of Teacher EducationDr. Heidi Krzywacki Modelling Conference, January 10, 2013


Teachers in Finnish elementaryschool

• is typically qualified to teach all 13 school subjects at grades 1 to 6

• has a Master’s level university degree(3+2 years, 300 credit points in total)• majoring in educational sciences / psychology• only 7 cp dedicated to mathematics education

• Teachers are considered as autonomous academic professionals who• choose learning materials and take full responsibility

for teaching and learning in the classroom• design and carry out assessment

8



Characteristics of Finnish school system (1/2)

• responsibility at the local level: neither school ranking, national exams nor pre-evaluation of learning materials

• Compulsory and intended instruction time of an elementary teacher is 667 lessons per year

NOTE: the number of teaching hours does not reflect explicitly the teachers’ total workload

9

(FNBE Kyrö, 2012)Department of Teacher EducationDr. Heidi Krzywacki Modelling Conference, January 10, 2013


Characteristics of Finnish school system (2/2)

• Total amount of lesson hours in 9-years comprehensive school is 5,750(cf. OECD average 6,500; max. Italy 8,300)• 32 weekly mathematics lessons

(~ 3.5 lessons per week)

• Schools are of various sizes and rather small in average• About 44% of all schools <100 pupils (12% of all pupils)• About 22% of all schools >300 pupils (54% of all pupils)• Average class size at elementary leve is 19.8 (OECD 21.4)

Department of Teacher EducationHeidi Krzywacki EDUVISIT, November 22, 2012

(FNBE Kyrö, 2012)


Pedagogical models(and modelling) in mathematics education

11



Learning mathematics:students have an active roleA child is seen as an active and social learner who is

basically willing and motivated to learn as well as capable for self-regulated process

• thinking at concrete level (Piaget, Galperin)• embodied cognition (tactile and kinestetic approach)• verbalization and oral communication

(also performing initiatives)

• support to building up self-efficacy (Erikson)• positive learning experiences, assessment and

feedback

12



Learning arithmetic mathematical concepts1. Concrete strategies• Conceptual understanding based on concrete models

and illustration• manipulatives• illustrations and figures• images of the use of the concrete

2. Mental strategies• conceptual understanding and processes without

images of the concrete• One or more steps are embodied in the arithmetic

processes

3. Automatizised conceptual understanding

VERBALIZATION



1. Concrete strategies• Conceptual understanding based on concrete models

and illustration• manipulatives• illustrations and figures• images of the use of the concrete

2. Mental strategies• conceptual understanding and processes without

images of the concrete• One or more steps are embodied in the arithmetic

processes

3. Automatizised conceptual understanding © Hellevi Putkonen

•demonstration in frontal teaching

• manipulatives used by individual students (or by pairs)

•together with teacher guidance•students in their own pace

8 + 5 = 8 + 2 + 3 = 13

127 + 38 = 127 + 30 + 8 = 157 + 8 = 165or = 127 + 3 + 35 = 130 + 35 = 165

etc.

Learning arithmetic mathematical concepts

VERBALIZATION



15


A. The use of manipulatives and illustrations

B. Inductive approach to elementaryschool mathematics

C. Word problems and problem solving


A . The use of manipulatives and illustrations• manipulatives and illustrations play a helpful role in

learning mathematics, especially in enhancing conceptual understanding and also in problem solvinge.g., attribute blocks, geometric shapes of different colours

and sizes; counting cubes, base ten blocks, fraction pieces, number line, surface model etc.

• BUT these are potentially confusing if their presentation is haphazard, disorganized, or lacking appropriate guidance and instruction from the teacher

teacher’s training and pedagogical thinking (!)16


Getting to know different kinds of concrete materials (manipulatives)...

17

A . The use of manipulatives and illustrations

0,03

... and learning how to use materials and models

18


http://youtu.be/Fu5djaxLJak


In teacher education student teachers• acquire knowledge and skills (cognitive potential) to

use manipulatives and illustrations• learn how to make (abstract) and formal mathematics

approachable• get to understand what school mathematics is about

and the meaning of well-structured and meaningful instruction

EXPERIENCE makes a difference!

19




20






• Individual cases as a basis for understanding mathematical rules and facts• investigating phenomena through student work

e.g., commutativeness (a + b = b + a), the sum of angles of a triangle, calculating the area of parallelogram etc.

A teacher needs to guide students to find general notions based on their investigationsstudents’ previous knowledge and skills form a

starting-point for investigations

21

B. Inductive approach to mathematics



• Previous knowledge about triangles and angles

each student draws a triangle(or a teacher gives triangles of different shapes and size)

setting the aim for investigation

22


1. Introductory phase

2. Examining and finding the mathematical rule

3. Rehearsing4. Limitations of

the rule

B. Inductive approach:The sum of angles of a triangle






the rule


• Examining the triangles: measuring angles and guiding students to make notes about their measurement

listing outcomes of student work and letting students to figure out that

the overall sum of angles is 180°






the rule


• Applying new knowledge and rehearsingvarious tasksmaking sure that all students understand






the rule


• Summing up the investigative session and its outcome

the sum of angles of a triangle applies with all triangles but not with other shapes


In teacher education student teachers• get familiar with inductive approach as a method in

mathematics classroom

discuss which contents can be approached especially through inductive approach and student investigations

a special challenge is to have an impact on student teachers’ views of school mathematics

26

B. Inductive approach to mathematics




27






• Writing down and drawing1. All information given in the initial task2. Drawing a picture or other illustration (using

manipulatives)3. All steps of the solving process4. Finding the solution(s) and reasons behind it

• Making the problem solving process visible support to the process and understandingcommunicating mathematics with othersbecoming aware of personal problem solving

process and reflecting on the process

28




Altogether 20 witches are to travel to magic mountains by 8 brooms. Each broom can fit 4 witches at the most but each broom must have at least 2 witches. Find out how all witches can travel to the mountain.

Solve the problem above and reflect on the problem solving process and the strategies you used in finding the solution.

NOTE: illustrate and communicate your problem solving process as accurately as possible.

29


C. Word problems and problem solving:Word problem ’Flying witches’



C. Word problems and problem solving:Word problem ’Flying witches’

Illustration

Describing the process

Mathematical formulation

Reasoning


Class 4A sells packages of rolls at the school ‘s Xmas market. A package of rolls includes 4 whole wheat rolls and 3 small breads made out of organic rye. Altogether 4 packages are sold before noon. How many rolls are then sold?

Start solving the problem by drawing a picture of the situation and then continue with formulating an equation describing the situation based on your picture.

31


C. Word problems and problem solving:Word problem ’Rolls for sale’

4 + 3


Class 4A sells packages of rolls at the school ‘s Xmas market. A package of rolls includes 4 whole wheat rolls and 3 small breads made out of organic rye. Altogether 4 packages are sold before noon. How many rolls are then sold?

Start solving the problem by drawing a picture of the situation and then continue with formulating an equation describing the situation based on your picture.

32


C. Word problems and problem solving:Word problem ’Rolls for sale’

4 · (4 + 3)


In teacher education student teachers• …learn the basics about problem solving• … have an experience of solving problems and

analysing process

a special challenge is to help improving teachers’ self-confidence and willingness to carry out problem solving activities in their classroom

subject matter knowledgeposing problems and guidance

33




Concluding remarks

34



1. Reachable aims and differentiation

• Developing activities suitable for different kind of learners• What is(are) the objective(s) of the activity?• Reflecting on how well the chosen tasks worked in

the classroom and how to modify them if needed?

• Assessment procedures should reflect the nature of teaching and learning



2. Paying attention to affective aspects

• becoming aware of personal strategies and qualities as a mathematics learner

• commucating (sharing) with others verbalization• positive experiences and encouragement

avoiding categorisation in the classroom based on mathematical performance

’everyone can do mathematics’routine tasks are also needed



3. Approaching mathematics through pupils’ eyes

• Concrete abstract• The contextualisation of mathematical tasks should

be close to children’s everyday life or something they find easy to approach

• Mathematical performance (thinking) is not only performed through correct symbolic expressions• Manipulatives and illustrations• sensitiveness for understanding pupils’ ways of

thinking



4. Resources and materials supporting school teaching

• Teachers need support in enhancing their practice in mathematics classroom, for example, suitable tasks and pedagogical ideas

learning materials (textbooks and worksheets)teacher guide books (pedagogical support for

teachers)computer-aided teaching and learning

Not only materials but also education that supports teachers (and schools) to enhance their practice


39

4. Resources and materials...

Kiitos!


Finnish National Board of EducationFNBE, (2004). National Core Curriculum for Basic Education.

www.oph.fi/englishKyrö, M. (2012). International comparisons of some features of

Finnish education and training. FNBE. http://www.oph.fi/english/publications/2012/international_comparisons_of_some_features_of_finnish_education_and_training_2011

Ministry of Education and Culture http://www.minedu.fi/OPM/?lang=en

Department of Teacher Education (UH)http://www.helsinki.fi/teachereducation/

E-mail: [email protected] of Teacher EducationDr. Heidi Krzywacki Modelling Conference, January 10, 2013

Further information

enhancing mathematical thinking in finnish elementary ... · enhancing mathematical thinking in...

Documents