GeoGebra with an IHS generates abductive argumentation during proving process

1. The 2ndInternational GeoGebra Conference GeoGebra International Conference GeoGebra with an IHS generates abductive argumentation during proving process Nam NGUYEN-DANH University of Würzburg, Germany • Hagenberg, Austria, 30/08/2011

2. Difficulties in the Proving Process • Realizing geometric invariants • The gap between argumentation and proof • Writing a formal proof Nam NGUYEN-DANH, University of Würzburg

3. An Interactive Help-System Level 0: Information Level 1: Construction Level 2: Invariance Level 3: Conjecture Level 4: Argumentation Level 5: Proof Level 6: Delving Nam NGUYEN-DANH, University of Würzburg

4. Delving Proof Argumentation Conjecture Invariance Construction Information Information Level The students read the information of the problem such as: unknown, data, condition and requirement Nam NGUYEN-DANH, University of Würzburg

5. Delving Proof Argumentation Conjecture Invariance Construction Information Construction Level The students construct drawing satisfying the problem, sometimes including auxiliary figures Nam NGUYEN-DANH, University of Würzburg

6. Delving Proof Argumentation Conjecture Invariance Construction Information Invariance Level The students search for static invariants, moving invariants and determine the a geometric transformation Nam NGUYEN-DANH, University of Würzburg

7. Delving Proof Argumentation Conjecture Invariance Construction Information Conjecture Level Based on the recognized invariants, the students formulate their conjectures Nam NGUYEN-DANH, University of Würzburg

8. Delving Proof Argumentation Conjecture Invariance Construction Information Argumentation Level The students produce arguments in order to find more invariants, make more conjectures and validate their conjectures Nam NGUYEN-DANH, University of Würzburg

9. Delving Proof Argumentation Conjecture Invariance Construction Information Proof Level The students ’organize’ (select, combine) plausible arguments in a chain of logical reasoning Nam NGUYEN-DANH, University of Würzburg

10. Delving Proof Argumentation Conjecture Invariance Construction Information Delving Level The students use some thinking strategies to delve into the problem: generalization, specialization, analogy,… Nam NGUYEN-DANH, University of Würzburg

11. The Bridge Problem • A river has straight parallel sides and cities A and B lie on opposite sides of the river. Where should we build a bridge in order to minimize the traveling distance between A and B (a bridge must be perpendicular to the sides of the river)? Nam NGUYEN-DANH, University of Würzburg

12. Basic TOULMIN’s Model of Argumentation C (Claim): A statement D (Data): Data justifying the claim C W (Warrant): The inference rule that allows data to be connected to the claim Nam NGUYEN-DANH, University of Würzburg

13. Abductive Argumentation in Toulmin’s Model Nam NGUYEN-DANH, University of Würzburg

14. Abductive Arg.  Deductive Prf. Nam NGUYEN-DANH, University of Würzburg

15. Abductive Arg.  Deductive Prf. Nam NGUYEN-DANH, University of Würzburg

16. Nam NGUYEN-DANH, University of Würzburg

17. Thank you for your attention!