copyright bruno buchberger teaching without teachers? bruno buchberger research institute for...

Copyright Bruno Buchberger

Teaching Without Teachers?

Bruno Buchberger

Research Institute for Symbolic Computation

University of Linz, Austria

Talk at VISIT-ME 2002, Vienna, July 10, 2002


Copyright Bruno Buchberger:

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The Simple Message

• There are two poles:

– populist view:

by recent advances, (math) teaching, can be automated;

thus, let’s automate it and dismiss the teachers!

– purist view:

(math) teaching is an art;

thus, let’s protect it against automation!


• My view:

– Don’t marry to the populist view.– Don’t marry to the purist view.– Don’t be satisfied with a compromise between the two views.

– Rather, let’s fully expand and enjoy the tension between the two poles!


The Two Poles from a Different Perspective

• Liberal access to knowledge:

– a web full of interactive courses– the student finds her personal way trough the material – the criterion of success is success in life

• Regulated access to knowledge:

– institutions / teachers who offer a canon of courses– institutions / teachers who prescribe curricula – the criterion of success is a certificate


My View

• Let’s do all we can for making the liberal access possible

and, at the same time,

let’s cultivate the role of the personal teacher.

• Both– the liberal access– and the cultivation of personal teaching

will be possible to an unprecedented extent.

• The reason for both advances will be the same: the current and future increase in automation.


The Liberal Access and Automation

• The computer allows the creation of teacher-less interactive courses.

• The web (the “global” computer) offers easy access to a huge course base.

• In principle, the web (with the appropriate “middleware”) also allows to create students - courses - teachers magmas.

• In mathematics, the sophistication of computer-based interactive courses is driven by the advances of computer mathematics.

• Automation of mathematics ==> automation of math teaching.


Personal Teaching and Automation

• The creation of interactive web-based courses is a didactically demanding task for teachers.

• Personal teaching has to be deployed on higher and higher levels of knowledge.

• Thus, personal teaching becomes more and more sophisticated.

• In mathematics, increased sophistication of teaching goes hand in hand with increased sophistication in the automation of mathematics.


Mathematics and Automation

• The goal of mathematics is automation.

• The goal of mathematics is to trivialize mathematics.

• The goal of mathematics is explanation (= making things of high dimension plane = making complicated things simple).

• Mathematics is didactics.

• The process of trivialization is completely non-trivial.

• Think nontrivially once and act trivially infinitely often.


• The process of trivialization (automation) in math, in the past 30 years, has seen enormous advances (“symbolic computation”, “computer algebra”, math software systems, …)

• The process of trivialization is never finished.

• The process is a spiral that arrives at higher and higher levels.


Known

Conjectured

Proved

Method

The Invention Spiral of Mathematics

More

More


GCD[18,12]=GCD[6,12]

GCD[x,y]=GCD[x-y,y]

Euclid’sTheorem

Euclid’sAlgorithm


GCD[2394830735890423089,…]

Lehmer’sConjecture ?


Real Numbersintegral[f]:=...

integral[f+g]=…?

integral[f+g]=...

RR IntegrationAlgorithm


Thousands ofIntegrals

Represen-tationTheorem ?

RepresentationTheorem

RischAlgorithm


• The math invention spiral also proceeds through meta-layers.


Real Numbersintegral[f]:=...

integral[f+g]=…?

integral[f+g]=...

Proving


Real Numberslimit[f]:=...

limit[f+g]=…?

limit[f+g]=...


Proving


Real Numbersis-continuous[f] :<=>…

is-continuous[f+g] <==…?

is-continuous[f+g] <==...


Proving


Proofs inAnalysis

Conjecture onthe reduction of provingto constraint solving

ReductionTheorem

Automated Theorem Proverfor Analysis


Thousands ofProofs


Example of Automated Theorem Proving: the Theorema system.

See the accompanying file .ps file containing

- a knowledge base formalized in Theorema

- a call of the PCS prover of Theorema

- the proof generated completely automatically by the

PCS prover

(Proof of the proposition that the limit of a sum is the sum of the limits.)


• Meta-layers are ubiquitous in the advancement of mathematics, for example:

– compute with numbers– observe and prove laws on numbers expressed by terms with

variables– compute with polynomials: the domain of terms with variables– observe and prove laws on polynomial sets– compute with polynomial sets (e.g. Groebner bases)– automated geometry theorem proving by polynomial set

computing– automated proving applied to the expansion of Groebner bases

theory


The Global Math Process

• The process of researching / applying / teaching / studying mathematics is an expanding process in a global magma (= unstructured powerful mass with the potential of becoming structured).

• In this process, mathematics is applied to itself.

• Self-application is the nature of intelligence and a natural phenomenon.

• The expansion is directed towards more and more automation and, at the same time, towards more and more individualization on higher and higher levels.


• Always, this process is “just at the beginning”.

• In this process, everyone is (should be) a student, a teacher, a researcher, a contributor, a user.

• The growing global math magma consists of

– knowledge bases (to be organized in a new way)– algorithm libraries / “math software systems” (to be purified)– teaching agents– organizational tools (e.g. “semantic” search engines, to be expanded into

math knowledge management tools)

– and people.


Global Math Teaching

• Teaching will never be obsolete in this global magma.

• Rather, teaching will be more and more important and challenging.

• Everyone should be a teacher and nobody should be a teacher only.

• Math teaching is a paradigm: If we manage math teaching, we master teaching in (all, some, many?) other areas.


• Teach all phases of the invention spiral in dependence on the teaching situation (the “White-Box / Black-Box Principle”).

• There is nothing like “obsolete mathematics” and nothing like “absolutely necessary mathematics”.

• Creating “teacher-less” interactive courses is one important field of teaching.

• Whatever level of automated teaching will be achieved, personal teaching will be necessary on the next higher level.


• Free yourself from routine (by being creative on the next higher level) and enjoy the next higher level.

• The notion of routine is relative. It depends on the particular phase in the evolutionary process of teaching.


The Education of Math Teachers

• Educate them to be first class citizens in the global math magma.

• Educate them to be ahead of the current needs.

• Educate them to be able to stay ahead of the current needs:

– master the depth, don’t try to master the breadth– thinking is the essence of mathematics– thus, grasp the accumulated thinking culture of mathematics– don’t separate mathematics and computers.


• Math Teaching is the most subtle aspect of doing mathematics:

– Teaching math for a changing audience is like living again through the various stages of the evolution of mathematics.

– This needs full penetration of the current stage of mathematics and, at the same time, full understanding of and dedication to the individual situation of the learner.


Teaching Without (Math) Teachers?

• Math teaching is one of the most exciting and most substantial professions today:

– mathematical thinking technology is the essence of science / technology based society

– contribute to teacher-less teaching for well-defined user-communities

– contribute to teacher-based individual teaching.

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