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

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

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

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  1. 1 From Errors to Stereotypes: Different Levels of Cognitive Models in School Algebra • 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

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

  3. 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

  4. 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

  5. PÉPITEST Pépite software End-users: Students Teachers Researchers PÉPIPROF PÉPIDIAG Checking by a human Automatic Coding of each exercice Automatic Analysis of the whole test

  6. Example : Blandine

  7. Example : Aliou

  8. 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

  9. Using Pépite : What teachers asked for ? • Remediation proposals according to profiles • A “cognitive geography” for the whole class • Grouping of students • …

  10. 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

  11. 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 student’s diagnosis: • The stereotype it belongs to • The student’s 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

  12. Blandine's Stereotype & Personal features UA3, T3, AC3

  13. Anissa'sStereotype & Personal features

  14. 2nd10 (10thgrade) UA2 T3 CA1 UA2 T3 CA3 UA3 T2 CA3 UA3 T3 CA2 UA3 T3 CA3 Cognitive geography of the class

  15. Stereotypes and teaching strategies • Step 1 : to destabilize erroneous rules • Counter-examples with numbers (T3) or surface areas of geometrical figures (T2) • Expressions • CA3 : a²=2a ; 5 +3a = 8a etc. • CA2 : a²+b² = (a+b)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

  16. 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

  17. 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!

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