From Errors to Stereotypes: Different Levels of Cognitive Models in School Algebra

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From Errors to Stereotypes Different Levels of
Cognitive Models in School Algebra
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• Élisabeth DELOZANNE (CRIP5 Paris 5)
• Christian VINCENT (Math teacher)
• Brigitte GRUGEON-ALLYS, Lalina COULANGE,
  Jean-Michel GELIS (DIDIREM Paris 7),
• Jeannine ROGALSKI (Paris 8)

E-learn05
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Overview

• The Pépite project
• Studies on teachers diagnosis activity
• Stereotypes of students competence in algebra
• Monitoring learning activities
• Conclusion

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The Pépite Project

• Objective
• To help teachers diagnose student's competence in
  algebra in order to better regulate learning
• Multidisciplinary project
• 3 objectives
• Diagnostic
• Learning
• Dissemination of research results among teachers

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An educational study(Grugeon 1995, Artigue et
al. 2001)

• A set of exercises to test students
• 3 kinds of problems
• Technical, modeling and interpretation
• An analysis grid to characterize students' usual
  answers
• Students profiles
• A quantitative description
• Success rates on mastered skills
• A qualitative description
• Meaning of letters, algebraic calculus,
  translation between representations and types of
  justification
• A diagram to describe the flexibility between
  representations
• Algebraic, numerical, geometric, and natural
  language

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Pépite software
End-users Students
Teachers Researchers
PÉPIPROF
PÉPIDIAG
Checking by a human
Automatic Coding of each exercice
Automatic Analysis of the whole test
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Example Blandine
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Example Aliou
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Teachers diagnostic activity

• Diagnosis
• Focused on the class or a group of students
  rather than on individual students
• Often expressed in categories of students
• Overall achievement Good quick, average
  quick, average slow, students with many
  difficulties
• Well-known errors
• Closely linked with action
• Especially experienced teachers
• Pépite interesting but too demanding
• It reveals "hidden knowledge"
• Difficulty of computer access, too much time
  required to administer the test, not enough
  support for teachers to capitalize on the results
  of the diagnosis

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Using Pépite What teachers asked for ?

• Remediation proposals according to profiles
• A cognitive geography for the whole class
• Grouping of students

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Monitoring learning activities for the whole
class

• Problem an articulation between
• Catering for individual students
• Whole class management
• Objective Grouping of students
• Based on an accurate didactical analysis of the
  algebraic competence
• Supported by technology
• How to do it?
• Stereotypes of students profiles
• Learning situations linked with stereotypes

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Stereotypes

• Objective
• Conceptual tools to articulate personalization
  and whole class management
• Definition
• Stereotype a class of students personal
  cognitive profiles
• How are they used in the project?
• To present a single students diagnosis
• The stereotype it belongs to
• The students personal features
• To propose adapted learning situations
• Stereotypes
• To choose learning activities to work on in
  priority
• Student's personal features
• To tune parameters to activities

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Blandine's Stereotype Personal features
UA3, T3, AC3
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Anissa'sStereotype Personal features
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Cognitive geography of the class
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Stereotypes and teaching strategies

• Step 1 to destabilize erroneous rules
• Counter-examples with numbers (T3) or surface
  areas of geometrical figures (T2)
• Expressions
• CA3 a²2a 5 3a 8a etc.
• CA2 a²b² (ab)² etc.
• Step 2 to re-build knowledge
• UA3 task of generalization or proof in order to
  make sense of letters
• UA2 to write equations and to solve them
• Step 3 to use algebra in different contexts

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Conclusion

• Stereotypes
• Based on an accurate didactical analysis of the
  domain
• Supported by technology
• As a way for grouping students and to organize
  learning activities in a classroom that fit to
  the students' personal cognitive profiles
• Experiments are currently in progress to evaluate
  these teaching strategies
• We observe that teachers need time to appropriate
  these tools in their daily practices
• Future design of learning environments
• Automatically tune the parameters to the
  student's personal features

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Web sites

• Software download and research documents
• http//pepite.univ-lemans.fr
• Documents for teachers (in French)
• http//maths.creteil.iufm
• (formation continue, la compétence algébrique du
  collège au lycée)
• Thank you!