WhiteBox BlackBox Principle etc Symposium Mathematics and New

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 Copyright B. Buchberger 2003 1

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. Copyright B. Buchberger 2003 2

Contents The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples Copyright B. Buchberger 2003 3

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? ” Copyright B. Buchberger 2003 4

“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 Copyright B. Buchberger 2003 5

Algorithmic Mathematics is not new and new: Copyright B. Buchberger 2003 6

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. Copyright B. Buchberger 2003 7

• 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. Copyright B. Buchberger 2003 8

„Man“ trivialized Copyright B. Buchberger 2003 9

Algorithmic mathematics is very new. • In the past 40 years more algorithms have been invented than in the math history before. Copyright B. Buchberger 2003 10

• “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. “) Copyright B. Buchberger 2003 11

• 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. ) Copyright B. Buchberger 2003 12

• 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. Copyright B. Buchberger 2003 13

The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples Copyright B. Buchberger 2003 14

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. ) Copyright B. Buchberger 2003 15

Facts Results …. . Conjecture Insight …. Algorithm Method …. Theorem Knowledge …. Copyright B. Buchberger 2003 16

• A spiral is like a circle: It does not matter where you start. • A spiral is more than a circle: Every round goes higher. Copyright B. Buchberger 2003 17

Applying Computing Experimenting Facts Results …. . “Seeing” (Observing) Conjecture Insight …. Algorithm Method …. Extracting a Method Programming Theorem Knowledge …. “Seeing” (Reasoning, Proving, Deriving, …) Copyright B. Buchberger 2003 18

more Facts Results …. . better Conjecture Insight …. Algorithm Method …. better Theorem Knowledge …. better Copyright B. Buchberger 2003 19

“better” = more efficient GCD of large numbers some GCDs …. . Lehmer’s algorithm GCD[m, n]= GCD[m-n, n] Euclid’s algorithm First steps depend only on first digits Euclid’s theorem Lehmer’s Copyright B. Buchberger 2003 theorem 20

“better” = more general some nonlinear systems some linear systems Newton’s algorithm triangularizable Gauß’ algorithm reducible to linear tangent systems Gauß’ theorem Newton’s theorem Copyright B. Buchberger 2003 21

“better” = more general some nonlinear systems some linear systems triangularizable Gauß’ Gröbner algorithm bases algorithm linear in the power products Gauß’ theorem Gröbner bases theorem Copyright B. Buchberger 2003 22

“better” = on the meta-level proofs for limit, derivative, … rules some limits algorithmic prover for elem. analysis limit[f+g]= limit[f]+… limit[f*g] = … limit algorithm reducibility to constraint solving limit rules reduction theorem Copyright B. Buchberger 2003 23

“better” = more applicable Applying real world problem Modeling mathematical model … solution method mathematical knowledge Copyright B. Buchberger 2003 24

The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples Copyright B. Buchberger 2003 25

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. Copyright B. Buchberger 2003 26

• 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. Copyright B. Buchberger 2003 27

• 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. Copyright B. Buchberger 2003 28

• 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, … … Copyright B. Buchberger 2003 29

– 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 (and then proving) becomes possible. “reasoning” – Mathematics is the art of reasoning for gaining knowledge and solving problems. Copyright B. Buchberger 2003 30

The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples Copyright B. Buchberger 2003 31

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 …”? Copyright B. Buchberger 2003 32

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) Copyright B. Buchberger 2003 33

• 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. Copyright B. Buchberger 2003 34

The white-box phase of teaching linear systems some linear systems Gauß’ algorithm program arithmetics explore the problem triangularizable reason Gauß’ theorem Copyright B. Buchberger 2003 35

The black-box phase of teaching linear systems = some non-linear systems the white-box phase of teaching nonlinear systems Gröbner bases algorithm Gauß’ algorithm arithmetics explore the problem and observe non-linear = linear in the power products program prove Gröbner bases theorem Copyright B. Buchberger 2003 36

The black-box phase of teaching non-linear systems = some geo proofs the white-box phase of teaching geo theorem proving Geo theorem proving algorithm explore the problem and observe reducible to ideal membership program prove Rabinowitch theorem Copyright B. Buchberger 2003 37

• 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. Copyright B. Buchberger 2003 38

• 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”. Copyright B. Buchberger 2003 39

The “New Technologies” Mathematical Invention: A Spiral Teaching Follows the Invention The White-Box / Black-Box Principle RISC Research: Examples Copyright B. Buchberger 2003 40

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” Copyright B. Buchberger 2003 41

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. Copyright B. Buchberger 2003 42