copyright b. buchberger 20031 white-box / black-box principle etc. symposium mathematics and new...

Copyright B. Buchberger 2003 1

White-Box / Black-Box Principle etc.

SymposiumMathematics 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


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Contents

The “New Technologies”

Mathematical Invention: A Spiral

Teaching Follows the Invention

The White-Box / Black-Box Principle

RISC Research: Examples


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?”


“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


Algorithmic Mathematics is not new and new:


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.


• 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.


„Man“

trivialized


Algorithmic mathematics is very new.

• In the past 40 years more algorithms have been invented than in the math history before.


• “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.“)


• 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.)


• 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.



• 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.)


FactsResults

…..

ConjectureInsight

….

TheoremKnowledge

….

AlgorithmMethod

….


• A spiral is like a circle: It does not matter where you start.

• A spiral is more than a circle: Every round goes higher.


FactsResults

…..

ConjectureInsight

….

TheoremKnowledge

….

AlgorithmMethod

….

“Seeing”(Observing)

“Seeing”(Reasoning,

Proving,Deriving, …)

Extracting a MethodProgramming

ApplyingComputing

Experimenting


FactsResults

…..

ConjectureInsight

….

TheoremKnowledge

….

AlgorithmMethod

….

more

better

better

better


some GCDs…..

GCD[m,n]=GCD[m-n,n]

Euclid’stheorem

Euclid’salgorithm

GCD oflarge numbers

First stepsdepend onlyon first digits

Lehmer’stheorem

Lehmer’salgorithm

“better” = more efficient


some linearsystems

triangula-rizable

Gauß’theorem

Gauß’algorithm

some non-linear systems

reducibleto lineartangentsystems

Newton’stheorem

Newton’salgorithm

“better” = more general


some linearsystems

triangula-rizable

Gauß’theorem

Gauß’algorithm

some non-linear systems

linear inthe powerproducts

Gröbner basestheorem

Gröbnerbases

algorithm

“better” = more general


some limits

limit[f+g]=limit[f]+…limit[f*g] =

…

limit rules

limit algorithm

proofsfor limit, derivative, …

rules

reducibilityto

constraintsolving

reductiontheorem

algorithmicprover

for elem.analysis

“better” = on the meta-level


real worldproblem

mathematicalmodel

…

mathematicalknowledge

solutionmethod

“better” = more applicable

ModelingApplying



• 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.


• 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.


• 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.


• 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, …– …


– 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.



• 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 …”?


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)


„Man“

trivialized

• 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.


some linearsystems

triangula-rizable

Gauß’theorem

Gauß’algorithm

The white-box phaseof teaching linearsystems

arith-metics

explorethe problem

reasonprogram


Gauß’algorithm

The black-box phaseof teaching linearsystems =

the white-box phaseof teaching non-linear systems

arith-metics

explorethe problemand observe

proveprogram

some non-linearsystems

non-linear =linear in the

power products

Gröbner basestheorem

Gröbner bases

algorithm


33

Gauß’algorithm

arith-metics

Gröbner bases

algorithm

explorethe problemand observe

proveprogram

some geoproofs

reducibleto ideal membership

Rabinowitchtheorem

Geo theoremproving

algorithm

The black-box phaseof teaching non-linearsystems =

the white-box phaseof teaching geotheorem proving


• 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.


• 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”.


8

„Man“

trivialized

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”


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 20031 white-box / black-box principle etc. symposium mathematics and new...

Documents

copyright note

talk slide

algorithmic mathematics

madrid slide

symposium mathematics

nontrivial algorithms

talk bruno buchberger

algorithms methods