Are elementary teachers ready to teach mathematics in the Information Age?

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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
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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.

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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.

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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?

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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.

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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.

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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

Giordano Bruno (1548-1576)
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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

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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.

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Perception and experience

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

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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.

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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, 2005 15-19 hrs. 4 30
Hermosillo, Sonora XXI Simposio Internacional de Computación en la Educación. Oct. 3, 2005 10-14 hrs. 4 20
Hermosillo, Sonora XXI Simposio Internacional de Computación en la Educación. Oct. 3, 2005 15-19 hrs. 4 20
Acapulco, Guerrero XVIII Congreso Nacional de Enseñanza de las Matemáticas. Nov. 18, 2005 10-12 hrs. 2 10
Acapulco, Guerrero XVIII Congreso Nacional de Enseñanza de las Matemáticas. Nov. 18, 2005 16-18 hrs. 2 10
Mexico City Universidad Hebraica. Jan. 24, 2006 1630-1830 2 20
Mexico City Universidad Hebraica. Jan. 31, 2006 1630-1830 2 21
TOTAL 281
Data from teachers was collected from thirteen
workshops
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• 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.

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A lesson on fractions

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

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The Balance

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

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Students answers

• Strategy 1.
• 1 ½ 3/2 ½ ½ ½
• ¼ ¼ ¼ ¼ ¼ ¼ ¾ ¾
• Strategy 2.
• 1 ½ ¾ ¾
• Strategy 3.
• 1 ½ ½ ¼ ½ ¼

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Results

• Those teacherswhose answers to the problem were
  wrongcould not balance the interactive. All
  these teachers (N80) 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.

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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

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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.

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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.

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• 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

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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.

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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.