Are elementary teachers ready to teach mathematics in the Information Age? Heréndira García Tello Instituto Latinoamericano de la Comunicación Educativa University of Kentucky Lexington, KY February 20, 2006

Antecedents  After 30 years of research on how children learn mathematics, the literature reports activities, teaching methods and educational tools that may help students understand mathematics  The mathematics textbooks written as part of the last curricular reform in Mexico (David Block, et al. 2000) try to adopt a new approach to the teaching of elementary mathematics, based on the research results on how children learn mathematics.

The problem  Elementary teachers do not understand many of the lessons in the new mathematics textbooks.  Elementary teachers need to ask for help to answer the exercises in the math textbooks, and then, they read aloud the answers to students.  Students only fill in the spaces in blank.  There is no teaching for understanding because teachers cannot explain what they do not understand.

Research questions  How to develop didactical experiences that are meaningful to teachers and students?  How to help teachers to teach the lessons they find difficult to teach?

Enciclomedia  Enciclomedia is a software that includes an electronic version of the textbooks used in all subjects taught at elementary level.  Almost all lessons have links to multimedia resources such as video clips, animations, and interactive activities.

The use of multimedia resources in the teaching of mathematics  One goal for the multimedia resources developed for the mathematics textbooks in Enciclomedia was to provide teachers with didactical scaffolds to help them with those lessons that they find difficult to teach.  Another goal was to use different representations for some mathematics concepts, to facilitate a conceptual understanding of mathematics.

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

Cognitive Science  Pictorial representations capture visual and spatial information in a much more usable form than lengthy verbal descriptions.  Imagery can aid learning, and some metaphorical aspects of language may have their roots in imagery.  Recent neurophysiologic results confirm a close physical link between reasoning with mental imagery and perception Thagard, P. (2004). “Cognitive Science.” The Stanford Encyclopedia of Philosophy. MIT Press

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

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

Methodology  Data from teachers was gathered using a questionnaire.  Data from students was gathered videotaping a class.  Two-hundred and eighty one teachers were asked to answer the questionnaire.  Twenty four students participated in the study.  The responses of teachers and students to the same exercise were compared.

WORKSHOPS FOR TRAINING ELEMENTARY TEACHERS IN THE USE OF THE MATHEMATICS INTERACTIVES LINKED TO THE MATH TEXTBOOKS IN ENCICLOMEDIA DateDuration in hours Number of teachers Mexico City: Instituto Latinoamericano de la Comunicación Educativa.Nov Mexico City: Suburban elementary public school in Iztapalapa.Jan Mexico City: Suburban elementary public school in Tlahuac.March Mexico City: Hotel Marriot (teachers from various states).Oct. 2, Mexico City: Hotel Marriot (teachers from various states).Oct. 11, Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación. Oct. 2, hrs. 415 Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación. Oct. 2, hrs. 430 Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación. Oct. 3, hrs. 420 Hermosillo, Sonora: XXI Simposio Internacional de Computación en la Educación. Oct. 3, hrs. 420 Acapulco, Guerrero: XVIII Congreso Nacional de Enseñanza de las Matemáticas. Nov. 18, hrs. 210 Acapulco, Guerrero: XVIII Congreso Nacional de Enseñanza de las Matemáticas. Nov. 18, hrs. 210 Mexico City: Universidad Hebraica.Jan. 24, :30-18: Mexico City: Universidad Hebraica.Jan. 31, :30-18: TOTAL281 Data from teachers was collected from thirteen workshops

 A classroom equipped to use Enciclomedia was selected from five pilot schools to gather data from students.  A math class with twenty-four students was videotaped.

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

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

One-hundred and sixty teachers returned the questionnaire  Seventy teachers gave the correct answer, ¾.  Eighty teachers answered 1 in one box and ½ in the other empty box in the mobile.  Ten teachers made other types of mistakes.

Students´ answers  Strategy 1. 1 ½ = 3/2 = ½ + ½ + ½ 1 ½ = 3/2 = ½ + ½ + ½ = ¼ +¼ + ¼ + ¼ + ¼ + ¼ = ¾ + ¾ = ¼ +¼ + ¼ + ¼ + ¼ + ¼ = ¾ + ¾  Strategy 2. 1 ½ = = = ¾ + ¾ 1 ½ = = = ¾ + ¾  Strategy 3. 1 ½ = = ½ + ¼ + ½ + ¼ 1 ½ = = ½ + ¼ + ½ + ¼

Results  Those teachers—whose answers to the problem were wrong—could not balance the interactive. All these teachers (N=80) reported that the Balance does not work properly.  All students understood they had to divide 1 ½ into two equal parts. The group of students did balance the interactive.  Students also used the interactive to study equivalence of fractions; trying different fractions that also balanced the interactive.

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

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

Conclusions  If teachers do not know how to solve the exercises using The Balance, they will not use the interactive.  Indeed, they will avoid all the interactive activities that mark wrong the answers that they do consider correct.

 There is no teaching for understanding because teachers cannot explain what they do not understand.  The results only apply to the sample of teachers that participated in the study and cannot be generalized

Training teachers to teach with technology  Training teachers to use the interactives designed for the mathematics textbooks in Enciclomedia is not enough.  It is necessary to help teachers understand the mathematics they have to teach.  Teachers need to see the advantages of using multiple representations in the teaching of mathematics.

More work to do  How to help teachers and students to develop a conceptual understanding of mathematics?  We still need to develop a system within the interactives that help users to overcome their misconceptions, when they have them.  Use artificial intelligence to diagnose the level of understanding of the user.  Use techniques based on logic to guide the users towards different levels of understanding.