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) 1 1 E-learn05

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

3. 3 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. 4 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. 5 5 Pépite software P ÉPI T EST P ÉPI P ROF Automatic Analysis of the whole test P ÉPI D IAG Automatic Coding of each exercice End-users: Students Teachers Researchers Checking by a human

6. 6 6 Example : Blandine

7. 7 7 Example : Aliou

8. 8 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. 9 9 Using Pépite : What teachers asked for ?  Remediation proposals according to profiles  A “cognitive geography” for the whole class  Grouping of students  …

10. 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. 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. 12 Blandine's Stereotype & Personal features UA3, T3, AC3

13. 13 Anissa'sStereotype & Personal features

14. 14 Cognitive geography of the class 2nd10 (10 th grade) UA3 T3 CA3 UA3 T3 CA2 UA3 T2 CA3 UA2 T3 CA3UA2 T3 CA1

15. 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. 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. 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!