1. Copyright B. Buchberger 20031 White-Box / Black-Box Principle etc. Symposium Mathematics and New Technologies: What to Learn, How to Teach? Invited Talk Bruno Buchberger RISC, Kepler University, Linz, Austria Dec 10-11, 2003, Fondación Ramón Areces, Madrid

2. Copyright B. Buchberger 20032 Copyright Note: Copying is allowed under the following conditions: -The paper is kept unchanged. -The copyright note is included. -A brief message is sent to buchberger@risc.uni-linz.ac.at If you use the material, please, cite it appropriately.

3. Copyright B. Buchberger 20033 Contents The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples

4. Copyright B. Buchberger 20034 What are the “New Technologies”? Two (completely) different ingredients: –“technologies” like internet, web, graphics, laptops, tabletts etc. –“algorithmic mathematics” This distinction is crucial for discussing “what to learn, how to teach?”

5. Copyright B. Buchberger 20035 “Technologies” They are new. They are (useful) tools for all areas of learning and teaching. These technologies come (in a superficial view) from “outside of mathematics” and are applied to math learning and teaching.  Didactics of using these technologies is basically the same for all areas: Great chance and great challenge but not in the focus of my talk

6. Copyright B. Buchberger 20036 Algorithmic Mathematics is not new and new:

7. Copyright B. Buchberger 20037 Algorithmic Mathematics is not new Since early history, algorithms (“methods”) are the essential goal of mathematics. Algorithms come from within mathematics. Non-trivial algorithms are based on non-trivial theorems (i.e. non-trivial proofs). Math knowledge and math methods are only two sides of the same coin. Non-trivial algorithms trivialize an infinite class of problem instances.

8. Copyright B. Buchberger 20038 The efficiency of mathematical thinking: “Think once deeply and you need not think infinitely many times”. The ultimate goal of mathematics is to trivialize itself. This trivialization is never complete and is “not completable”. (By a version of Gödel’s incompleteness theorem.) The more is trivialized the more difficult (and interesting) it becomes to trivialize more.

9. Copyright B. Buchberger 20039 „Man“ trivialized

10. Copyright B. Buchberger 200310 Algorithmic mathematics is very new. In the past 40 years more algorithms have been invented than in the math history before.

11. Copyright B. Buchberger 200311 “The computer” (i.e. the universal, programmable automaton for executing any algorithm) is new. The computer is a mathematical invention. Its design has been given many years before the first physical realization was done. (Gödel, Turing, von Neumann, etc.) Its principal capabilities and limitations have been exactly clarified many years before the first physical computer was built. The logical design of the computer did not change over the past 60 years whereas its physical realization (the „natureware“) changes with increasing speed. („The computer: a thinking constant.“)

12. Copyright B. Buchberger 200312 The executability of mathematical algorithms by a mathematical machine („the computer“) is new. The execution of math algorithms on the computer, is one of the most exciting examples of application of mathematics to itself. (Self-application is the nature of intelligence and the intelligence of nature.)

13. Copyright B. Buchberger 200313 Executability of math algorithms on math machines have dramatically enhanced the invention capability in (algorithmic) mathematics. Executability of math algorithms on math machines have dramatically enhanced the application capability of mathematics.

14. Copyright B. Buchberger 200314 The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples

15. Copyright B. Buchberger 200315 Mathematical Invention: A Spiral The “Creativity Spiral” or “Invention Spiral” B. Buchberger. Mathematics on the Computer: The Next Overtaxation? Didactics-Series of the Austrian Math. Society, Vol.131, March 2000, pp. 37-56. (Used in talks since 1996, Derive Conference, Bonn.)

16. Copyright B. Buchberger 200316 Facts Results ….. Conjecture Insight …. Theorem Knowledge …. Algorithm Method ….

17. Copyright B. Buchberger 200317 A spiral is like a circle: It does not matter where you start. A spiral is more than a circle: Every round goes higher.

18. Copyright B. Buchberger 200318 Facts Results ….. Conjecture Insight …. Theorem Knowledge …. Algorithm Method …. “Seeing” (Observing) “Seeing” (Reasoning, Proving, Deriving, …) Extracting a Method Programming Applying Computing Experimenting

19. Copyright B. Buchberger 200319 Facts Results ….. Conjecture Insight …. Theorem Knowledge …. Algorithm Method …. more better

20. Copyright B. Buchberger 200320 some GCDs ….. GCD[m,n]= GCD[m-n,n] Euclid’s theorem Euclid’s algorithm GCD of large numbers First steps depend only on first digits Lehmer’s theorem Lehmer’s algorithm “better” = more efficient

21. Copyright B. Buchberger 200321 some linear systems triangula- rizable Gauß’ theorem Gauß’ algorithm some non- linear systems reducible to linear tangent systems Newton’s theorem Newton’s algorithm “better” = more general

22. Copyright B. Buchberger 200322 some linear systems triangula- rizable Gauß’ theorem Gauß’ algorithm some non- linear systems linear in the power products Gröbner bases theorem Gröbner bases algorithm “better” = more general

23. Copyright B. Buchberger 200323 some limits limit[f+g]= limit[f]+… limit[f*g] = … limit rules limit algorithm proofs for limit, derivative, … rules reducibility to constraint solving reduction theorem algorithmic prover for elem. analysis “better” = on the meta-level

24. Copyright B. Buchberger 200324 real world problem mathematical model … mathematical knowledge solution method “better” = more applicable Modeling Applying

25. Copyright B. Buchberger 200325 The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples

26. Copyright B. Buchberger 200326 Teaching Follows the Invention A (good) way of teaching: follow the path of invention. Allow the students to feel the pressure of an unsolved problem and the excitement of the invention. Don’t avoid all pitfalls and failures: –ideas don’t come from Kami (“God”) –but from Kami (“Paper”). Avoid some pitfalls and failures: Japanese “sensei”: the person who lived earlier. Master and teach all phases and aspects of the invention spiral.

27. Copyright B. Buchberger 200327 The teaching of math in application fields (economy, engineering, medicine etc.) is different: –The application of methods is in the focus. –This is a very important part of math teaching, which of course today profits tremendously from the availability of algorithmic mathematics in the form of “mathematical systems” like Mathematica etc. –The other phases of the spiral, e.g. “proving”, cannot be trained extensively. –This type of teaching is not in the focus of this talk.

28. Copyright B. Buchberger 200328 For “complete math teaching”: –Master and teach all phases and aspects of the invention spiral. –What to teach? This question has not the same importance as the question of teaching the math invention technology. –One can never be complete in terms of “what to teach” but one should be complete in terms of the phases and aspects of the mathematical invention process. –The “what to teach” is the more standardized the younger the students (children) are.

29. Copyright B. Buchberger 200329 Aspects of the invention process: –modeling, representing, … –inventing, analyzing, specifying problems –decomposing into subproblems –retrieving knowledge, check applicability, using existing “technologies” –conjecturing knowledge, inventing methods –arguing, discussing, reasoning, proving, verifying, comparing, generalizing, cooperating, … –programming “in the small and in the large” –assessing programs and systems –documenting, presenting, storing, … –applying, assessing results, … –…

30. Copyright B. Buchberger 200330 –For young children, the phases of the invention process are indistinguishable: “Touch, play, see, and memorize”. –For adults: the efficiency of mathematics stems from the distinction between observing, reasoning, and acting. –Somewhen between the age of 14 and “reasoning” (and then proving) becomes possible. –Mathematics is the art of reasoning for gaining knowledge and solving problems.

31. Copyright B. Buchberger 200331 The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples

32. Copyright B. Buchberger 200332 The White-Box / Black-Box Principle When should we apply “technology” in math teaching? (Remember: In this talk, “technology” = algorithms.) Example: Should we teach “integration rules” when we have systems that can “do integrals”? Example: Should we teach “linear systems” when we have systems that can “do linear systems”? Example: Should we teach … when we have systems that can “do …”?

33. Copyright B. Buchberger 200333 B. Buchberger. Should Students Learn Integration Rules? ACM SIGSAM Bulletin Vol.24/1, pp. 10-17, January 1990. (However, introduced already in talk at ICME 1984, Adelaide)

34. Copyright B. Buchberger 200334 When should we apply “technology” in math teaching? The Populists’ Answer: Stop teaching things “the computer” can do! The Purists’ Answer: Ban the computer from math teaching! The White-Box Black-Box Principle: Absolute answer is not possible, Answer depends on the phase of teaching.

35. Copyright B. Buchberger 200335 some linear systems triangula- rizable Gauß’ theorem Gauß’ algorithm The white-box phase of teaching linear systems arith- metics explore the problem reason program

36. Copyright B. Buchberger 200336 Gauß’ algorithm The black-box phase of teaching linear systems = the white-box phase of teaching non- linear systems arith- metics explore the problem and observe prove program some non-linear systems non-linear = linear in the power products Gröbner bases theorem Gröbner bases algorithm

37. Copyright B. Buchberger 200337 explore the problem and observe prove program some geo proofs reducible to ideal membership Rabinowitch theorem Geo theorem proving algorithm The black-box phase of teaching non-linear systems = the white-box phase of teaching geo theorem proving

38. Copyright B. Buchberger 200338 The white-box black-box principle is recursive. You may start at any round in the spiral. The black-box phase is exactly the moment for applying “technology”, i.e. the current math systems. This moment is relative and not absolute. There is nothing like “absolutely necessary” and “absolutely obsolete math content”. There is nothing like “absolutely creative” and “absolutely technical” topics in math.

39. Copyright B. Buchberger 200339 You may want to walk in the reverse direction through the spiral (black-box / white-box). “Program” may also mean “train to apply in examples”.

40. Copyright B. Buchberger 200340 The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples

41. Copyright B. Buchberger 200341 From the RISC Kitchen We don’t want to be just users of the technology. We don’t want to be just implementers of the technology. We want to be creators of the technology. See Mathematica Notebook “RISC Research”

42. Copyright B. Buchberger 200342 Conclusion The technology is permanently expanding through the global invention spiral. The algorithmic result of one invention round is tool for the next round. Math teaching should teach the “thinking technology of mathematical invention” in well-chosen white-box / black-box invention rounds whose contents depend on many factors. The contents of mathematics are the accumulated and condensed experience of mankind in gaining knowledge and solving problems by reasoning.