Are elementary teachers Are elementary teachers ready to teach mathematics ready to teach mathematics

in the Information Age?in the Information Age?

Heréndira García TelloHeréndira García Tello

Instituto Latinoamericano de la Instituto Latinoamericano de la Comunicación EducativaComunicación Educativa

University of KentuckyUniversity of KentuckyLexington, KYLexington, KYFebruary 20, 2006February 20, 2006

AntecedentsAntecedents After 30 years of research on how children learn After 30 years of research on how children learn

mathematics, the literature reports activities, mathematics, the literature reports activities, teaching methods and educational tools that teaching methods and educational tools that may help students understand mathematics may help students understand mathematics

The mathematics textbooks written as part of the The mathematics textbooks written as part of the last curricular reform in Mexico (David Block, et last curricular reform in Mexico (David Block, et al. 2000) try to adopt a new approach to the al. 2000) try to adopt a new approach to the teaching of elementary mathematics, based on teaching of elementary mathematics, based on the research results on how children learn the research results on how children learn mathematics.mathematics.

The problemThe problem Elementary teachers do not understand many of Elementary teachers do not understand many of

the lessons in the new mathematics textbooks.the lessons in the new mathematics textbooks.

Elementary teachers need to ask for help to Elementary teachers need to ask for help to answer the exercises in the math textbooks, and answer the exercises in the math textbooks, and then, they read aloud the answers to students. then, they read aloud the answers to students.

Students only fill in the spaces in blank. Students only fill in the spaces in blank.

There is no teaching for understanding because There is no teaching for understanding because teachers cannot explain what they do not teachers cannot explain what they do not understand.understand.

Research questionsResearch questions

How to develop didactical experiences that How to develop didactical experiences that are meaningful to teachers and students?are meaningful to teachers and students?

How to help teachers to teach the lessons How to help teachers to teach the lessons they find difficult to teach?they find difficult to teach?

EnciclomediaEnciclomedia

Enciclomedia is a software that includes Enciclomedia is a software that includes an electronic version of the textbooks used an electronic version of the textbooks used in all subjects taught at elementary level. in all subjects taught at elementary level.

Almost all lessons have links to multimedia Almost all lessons have links to multimedia resources such as video clips, animations, resources such as video clips, animations, and interactive activities.and interactive activities.

The use of multimedia resources in The use of multimedia resources in the teaching of mathematicsthe teaching of mathematics

One goal for the multimedia resources One goal for the multimedia resources developed for the mathematics textbooks in developed for the mathematics textbooks in Enciclomedia was to provide teachers with Enciclomedia was to provide teachers with didactical scaffolds to help them with those didactical scaffolds to help them with those lessons that they find difficult to teach.lessons that they find difficult to teach.

Another goal was to use different Another goal was to use different representations for some mathematics concepts, representations for some mathematics concepts, to facilitate a conceptual understanding of to facilitate a conceptual understanding of mathematics.mathematics.

The use of imagesThe use of images

“ “Nothing can enter to the memory without Nothing can enter to the memory without passing through the doors of the imagination, passing through the doors of the imagination, without transforming itself into an image, and without transforming itself into an image, and this image should color itself with emotions … it this image should color itself with emotions … it is necessary to open our eyes throughout is necessary to open our eyes throughout images … we cannot understand if we do not images … we cannot understand if we do not speculate with images”speculate with images”

Giordano Bruno (1548-1576)Giordano Bruno (1548-1576)

Cognitive ScienceCognitive Science

Pictorial representations capture visual and Pictorial representations capture visual and spatial information in a much more usable form spatial information in a much more usable form than lengthy verbal descriptions.than lengthy verbal descriptions.

Imagery can aid learning, and some Imagery can aid learning, and some metaphorical aspects of language may have metaphorical aspects of language may have their roots in imagery.their roots in imagery.

Recent neurophysiologic results confirm a close Recent neurophysiologic results confirm a close physical link between reasoning with mental physical link between reasoning with mental imagery and perceptionimagery and perception

Thagard, P. (2004). “Cognitive ScienceThagard, P. (2004). “Cognitive Science.” The .” The

Stanford Encyclopedia of Philosophy. MIT PressStanford Encyclopedia of Philosophy. MIT Press

Representation and visualization Representation and visualization in mathematicsin mathematics

According to Duval (1998 and 1999), the According to Duval (1998 and 1999), the coordination of several representation coordination of several representation registers is fundamental for conceptual registers is fundamental for conceptual understanding as it helps learners to understanding as it helps learners to distinguish between the concept and its distinguish between the concept and its representation, and to recognize a concept representation, and to recognize a concept in any of its different representations.in any of its different representations.

Perception and experiencePerception and experience

Perception depends on experience: our Perception depends on experience: our perception is regulated by our experience.perception is regulated by our experience.

MethodologyMethodology

Data from teachers was gathered using a Data from teachers was gathered using a questionnaire.questionnaire.

Data from students was gathered videotaping a Data from students was gathered videotaping a class.class.

Two-hundred and eighty one teachers were Two-hundred and eighty one teachers were asked to answer the questionnaire. asked to answer the questionnaire.

Twenty four students participated in the study.Twenty four students participated in the study.

The responses of teachers and students to the The responses of teachers and students to the same exercise were compared.same exercise were compared.

WORKSHOPS FOR TRAINING ELEMENTARY TEACHERS IN THE USE OF THE MATHEMATICS INTERACTIVES LINKED

TO THE MATH TEXTBOOKS IN ENCICLOMEDIA

Date Duration in hours

Number of teachers

Mexico City: Instituto Latinoamericano de la Comunicación Educativa. Nov. 2004 8 10

Mexico City: Suburban elementary public school in Iztapalapa. Jan. 2005 8 20

Mexico City: Suburban elementary public school in Tlahuac. March 2005 8 25

Mexico City: Hotel Marriot (teachers from various states). Oct. 2, 2005 4 40

Mexico City: Hotel Marriot (teachers from various states). Oct. 11, 2005 4 40

Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación.

Oct. 2, 2005 10-14 hrs.

4 15

Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación.

Oct. 2, 200515-19 hrs.

4 30

Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación.

Oct. 3, 200510-14 hrs.

4 20

Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación.

Oct. 3, 200515-19 hrs.

4 20

Acapulco, Guerrero: XVIII Congreso Nacional de Enseñanza de las Matemáticas.

Nov. 18, 200510-12 hrs.

2 10

Acapulco, Guerrero: XVIII Congreso Nacional de Enseñanza de las Matemáticas.

Nov. 18, 200516-18 hrs.

2 10

Mexico City: Universidad Hebraica. Jan. 24, 200616:30-18:30

2 20

Mexico City: Universidad Hebraica. Jan. 31, 200616:30-18:30

2 21

TOTAL 281

Data from teachers was collected from thirteen workshops

A classroom equipped to use Enciclomedia was A classroom equipped to use Enciclomedia was selected from five pilot schools to gather data from selected from five pilot schools to gather data from students. students.

A math class with twenty-four students was videotaped.A math class with twenty-four students was videotaped.

A lesson on fractionsA lesson on fractions

Teachers have pointed out this lesson as one Teachers have pointed out this lesson as one that is difficult to understand and to teach.that is difficult to understand and to teach.

The BalanceThe Balance

          

                            

The Balance was designed to facilitate the teaching of a lesson 39 on fractions

One-hundred and sixty teachers One-hundred and sixty teachers returned the questionnairereturned the questionnaire

Seventy teachers gave the correct Seventy teachers gave the correct answer, ¾. answer, ¾.

Eighty teachers answered 1 in one box Eighty teachers answered 1 in one box and ½ in the other empty box in the and ½ in the other empty box in the mobile.mobile.

Ten teachers made other types of Ten teachers made other types of mistakes.mistakes.

Students´ answersStudents´ answers

Strategy 1.Strategy 1.

1 ½ = 1 ½ = 3/23/2 = ½ + ½ + ½ = ½ + ½ + ½

= ¼ +¼ + ¼ + ¼ + ¼ + ¼ = ¾ + ¾ = ¼ +¼ + ¼ + ¼ + ¼ + ¼ = ¾ + ¾ Strategy 2.Strategy 2.

1 ½ = = = ¾ + ¾ 1 ½ = = = ¾ + ¾ Strategy 3.Strategy 3.

1 ½ = = ½ + ¼ + ½ + ¼ 1 ½ = = ½ + ¼ + ½ + ¼

ResultsResults

Those teachers—whose answers to the problem Those teachers—whose answers to the problem were wrong—could not balance the interactive. were wrong—could not balance the interactive. All these teachers (N=80) reported that the All these teachers (N=80) reported that the Balance does not work properly.Balance does not work properly.

All students understood they had to divide 1 ½ All students understood they had to divide 1 ½ into two equal parts. The group of students did into two equal parts. The group of students did balance the interactive. balance the interactive.

Students also used the interactive to study Students also used the interactive to study equivalence of fractions; trying different equivalence of fractions; trying different fractions that also balanced the interactive.fractions that also balanced the interactive.

Teachers’ and students’ Teachers’ and students’ perceptions and experiencesperceptions and experiences

We observed that teachers and students We observed that teachers and students did not interpret in the same way the did not interpret in the same way the exercise and the registers used in the exercise and the registers used in the study: the mobile and the balance study: the mobile and the balance

Perception and experiencePerception and experience

The meaning attached to a representation The meaning attached to a representation (or register) depends on the knowledge (or register) depends on the knowledge that the subject has about the concept that the subject has about the concept being represented, it also depends on the being represented, it also depends on the previous experience the subject has had previous experience the subject has had with the object or concept being with the object or concept being represented as well as the level of the represented as well as the level of the development of the cognitive structures of development of the cognitive structures of the person.the person.

ConclusionsConclusions

If teachers do not know how to solve the If teachers do not know how to solve the exercises using The Balance, they will not exercises using The Balance, they will not use the interactive.use the interactive.

Indeed, they will avoid all the interactive Indeed, they will avoid all the interactive activities that mark wrong the answers that activities that mark wrong the answers that they do consider correct.they do consider correct.

There is no teaching for understanding There is no teaching for understanding because teachers cannot explain what because teachers cannot explain what they do not understand.they do not understand.

The results only apply to the sample of The results only apply to the sample of teachers that participated in the study and teachers that participated in the study and cannot be generalizedcannot be generalized

Training teachers to teach with Training teachers to teach with technologytechnology

Training teachers to use the interactives Training teachers to use the interactives designed for the mathematics textbooks in designed for the mathematics textbooks in Enciclomedia is not enough.Enciclomedia is not enough.

It is necessary to help teachers It is necessary to help teachers understand the mathematics they have to understand the mathematics they have to teach.teach.

Teachers need to see the advantages of Teachers need to see the advantages of using multiple representations in the using multiple representations in the teaching of mathematics.teaching of mathematics.

More work to doMore work to do

How to help teachers and students to develop a How to help teachers and students to develop a conceptual understanding of mathematics?conceptual understanding of mathematics?

We still need to develop a system within the We still need to develop a system within the interactives that help users to overcome their interactives that help users to overcome their misconceptions, when they have them.misconceptions, when they have them.

Use artificial intelligence to diagnose the level of Use artificial intelligence to diagnose the level of understanding of the user.understanding of the user.

Use techniques based on logic to guide the users Use techniques based on logic to guide the users towards different levels of understanding.towards different levels of understanding.