Didactical practices of computer algebra in mathematics education

1. Science Teaching Department Didactical practices of computer algebra in mathematics education Nurit Zehavi Weizmann Institute of Science, Israel TIME 2008 TIME 2008

2. The MathComp Project(Mathematics on Computers) The MathComp is a R&D project that began in the early 1990s, with the aim of integrating CAS into teaching to broaden learning opportunities and to promote greater mathematical understanding. Webs of Meaning Noss & Hoyles, 1996 Windows on Mathematical Meanings: Learning Cultures and Computers TIME 2008

3. Issues discussed in CAME symposia (1999) • Chevallard’s anthropological approach • Instrumentation • Bridging the gap between techniques and theory • CAS & DGS • Teachers in transit (professional development) (2003) http://www.lkl.ac.uk/research/came/ TIME 2008

4. Chevallard’s anthropological approach to the didactics of mathematics • Didactics of mathematics as a domain of research is part of ‘anthropology of mathematical knowledge’ that is concerned with the emergence and growth of the mathematical knowledge in educational settings (“institutions”). • An educational setting can be characterized by its theory of practices, i. e. a praxeology. TIME 2008

5. The components of praxeology of a given topic: • (Types of) Tasks • Techniques The constructed knowledge is strongly dependent on the perspectives of the partners within a specific educational setting on the above components. • “Technology” (Discourse of the techniques) • Theory TIME 2008

6. Two Topics: • From a word problem to a family of word problems:In what ways can CAS help in solving word problems? (Zehavi & Mann, IJCAME, 1999) • Justifications of geometric results using CAS: Can an integration of features of DGS and CAS be useful for approaching proof? TIME 2008

7. Study I: In what ways can CAS help in solving word problems? The School of Pythagoras Pythagoras, who lived in the sixth century BC, ran a school. He was once asked how many students are in his school. After thinking awhile, he said: 1/2 of the students are now participating in a math class. 1/4 of the students are now in a science class. 1/7 of the students are now silently exercising their minds. In addition to all the above, three students are in the garden. How many students were in the school? TIME 2008

8. Debugging the model TIME 2008

9. 0 PD - Teachers’ contribution: Tutorials for making explicit the implicit restrictions of the model, for example: Change the problem so that the number of students in the School of Pythagoras will be 280 instead of 28. TIME 2008

10. How long did Diophantus live? Diophantus, known as the 'father of algebra'– hisepitaph“This tomb hold Diophantus Ah, what a marvel! And the tomb tells scientifically the measure of his life….” (Newman (ed.) The World of Mathematics,1956) Invent a similar problem A new type of tasks: The equation is an object to explore; not just a tool for finding an answer TIME 2008

11. x Math & Music 1/9 of my life I solved math exercises by hand. Then I was introduced to Derive and used it for 1/3 of my life. Later, I knew everything! so I did math in my head for another 1/3 of my life. In the last 26 years I did not learn any math, I just played music. How long have I lived? TIME 2008

12. Students’ contribution Why do you want us to deal with ancient people?! Let’s invent problems on someone more interesting like … ET – The Extra Terrestrial!!! TIME 2008

13. Student and teacher contribution:A Basketball Game Sport journalist: Eitan scored 40% Tom scored 25% Eyal scored 15% And Motti, as usual, made 5 three-point Shots. Readers: The report must be wrong! TIME 2008

14. Expanding the praxeology of learningword problemsin aCASenvironment Didactical perspectives: • Transferring the control over the process of modeling to the student. • By inventing problems the students share responsibility, with the teachers, of the techniques and the validity of the established knowledge in class. • Enabling students to see any given problem as a member of a family of problems through its model. TIME 2008

15. A new practice: resource e-book for teaching analytic geometry with CAS A classical task: “Show that the locus of points through which two perpendicular tangents are drawn to a given ellipse/hyperbola is ….." The director circle TIME 2008

16. New perspectives on conic sections Classical textbook solution: Viète's formula Users with basic CAS experience: Utilizing CAS for traditional techniques Users with good mastering of CAS: New instrumented techniques further exploring the solution and the problem Discourse TIME 2008

17. di rect r ix 1350 1200 600 450 A new task: viewing the parabola under α or 180-α An unfamiliar relationship TIME 2008

18. Illustration of tangents to hyperbola when sliding a point on the director circle the pair of tangents switch from touching one branch to touching both The relation between visual clues and formal proof TIME 2008

19. Proofs and dynamic geometry Hanna, (ESM, 2000) • Experimental work with dynamic geometry could lead educators to question the need for analytical proofs. • The key role of proofs in the classroom is to promote mathematical understanding. • Exploration of a problem can lead one to grasp its structure and its ramification, but cannot yield an explicit understanding of every link. TIME 2008

20. Proofs and dynamic geometry Laborde et al. ( PME Handbook, 2006) Teaching and Learning Geometry with Technology: When students were asked to justify conjectures the teachers did not mention the possibility of using Cabri to find a reason or elaborate a proof. TIME 2008

21. A significant difference between DGS and CAS in DGS – The algebraic infrastructure that enables construction and animation is hidden. in CAS – Users need to develop the algebraic expressions for producing constructions and insert slider bars related to the parameters for animation. TIME 2008

22. Proof and CAS Sliding… The algebraic expressions that create the animation can be used for justifying the visual results. Mann, Dana-Picard and Zehavi (2007) and Proving…. TIME 2008

23. a=3, b=2 Symbolic computation in CAS Implicit plotting Solve Simplify TIME 2008

24. Symbol Sense in the messy expressions explicit understanding of every link. Towards Didactical Transposition... TIME 2008

25. Study II:Teachers’ justifications of geometric results using CAS(n = 43) • (a) teachers' views on the need for an algebraic proof of unfamiliar geometric results obtained by experimenting with slider bars in CAS, and • (b) the types of proofs they produced using the expressions that generated the animation. TIME 2008

26. Goals • To construct, with the teachers, elements of a praxeology for integrating the new perspectives on conic sections into teaching. • To determine how a CAS, which enables both symbolic manipulation and animation, can promote mathematical understanding by bridging experimental mathematics with deduction. TIME 2008

27. The tasks Part (a) Is it possible to draw two tangents to the hyperbola from every point in the plane, which is not on the asymptotes? Do two tangents to the hyperbola from specific points touch the same branch? Justify your answers x•y = 1 interior exterior TIME 2008

28. Implement two slider bars to animate a pair of tangents drawn from P(X, Y) to hyperbolax•y = 1and identify the loci of points from which: (The expressions were defined, but not displayed) No tangent can be drawn; A single tangent can be drawn; Two tangents to the same branch can be drawn; Two tangents can be drawn, one to each branch. Rate the need for students to prove algebraically the results; explain your pedagogical arguments. TIME 2008

29. Part b: Here are the expressions obtained by the CAS while designing the animation of tangents through a general point P(X, Y). (X, Y ≠ 0) TIME 2008

30. Follow through the derivation of the coordinates of the tangency points, and use these expressions to prove your findings. Please rate (and explain) again (from 1 to 6), the need for students to provide algebraic proof of the partition of the plane into four loci. (X, Y ≠ 0) TIME 2008

31. Distribution of teachers’ rating (n = 43) H 22 20 TIME 2008

32. Teacher H: Answer (part a) Intersection ofsame/different colortangents Rating: 2 Explanation TIME 2008

33. Teacher H: Answer (part b) Reflection: Symbol Sense Rating: 6 Reflection Explanation respectively explanation of every link TIME 2008

34. PDPractice: Distribution of proofs and arguments given by the teachers (H) Discourse Techniques Relevance to classroom practice TIME 2008

35. The study focused on: • The coordination of algebraic, graphical and geometrical representations to deal with problems related to tangents to conic sections using a CAS. • The possibility of building a praxeology for learning about proofs based on these problems. • The particular place of actions with slider bars in this praxeology. TIME 2008

36. Discourse • Slider bars demonstrate, in a dynamic way, the effect of a parameter, in an algebraic expression, on the shape of the related graph. • The algebraic expressions encapsulate the relationships between the different parameters. For approaching proof these relationships need to be unfolded by means ofsymbol sense. TIME 2008

37. Discourse • Suchsymbol sense motivates not only qualitative exploration of the effect of changing the value of the parameter of the geometric representation; • but also quantitative explanation of the cause of the change. TIME 2008

38. Conclusion Teacher professional development within curricular R&D(of story problems, tangents to conics, and other topics)can help in the didacticaltransposition of new practices that incorporate technology. TIME 2008

39. A new practice: Viewing hyperbolas k = 1.7 2 No tangent Acute angles 1 1 Obtuse angles 2 Thank you! TIME 2008