1. 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

2. Copyright Bruno Buchberger Copyright Bruno Buchberger: Copying, storing this file in data bases etc. is granted under the condition that the file is kept unchanged, this copyright note is included in the copy, a note is sent to buchberger@risc.uni-linz.ac.at If you use material contained in this talk, please, cite it appropriately.

3. Copyright Bruno Buchberger 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!

4. Copyright Bruno Buchberger 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!

5. Copyright Bruno Buchberger 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

6. Copyright Bruno Buchberger 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.

7. Copyright Bruno Buchberger 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.

8. Copyright Bruno Buchberger 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.

9. Copyright Bruno Buchberger 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.

10. Copyright Bruno Buchberger 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.

11. Copyright Bruno Buchberger Known Conjectured Proved Method The Invention Spiral of Mathematics More

12. Copyright Bruno Buchberger GCD[18,12]= GCD[6,12] GCD[x,y]= GCD[x-y,y] Euclid’s Theorem Euclid’s Algorithm The Invention Spiral of Mathematics GCD[ 2394830735890423089,…] Lehmer’s Conjecture ?

13. Copyright Bruno Buchberger Real Numbers integral[f]:=... integral[f+g]=…? integral[f+g]=... RR Integration Algorithm The Invention Spiral of Mathematics Thousands of Integrals Represen- tation Theorem ? Representation Theorem Risch Algorithm

14. Copyright Bruno Buchberger The math invention spiral also proceeds through meta-layers.

15. Copyright Bruno Buchberger Real Numbers integral[f]:=... integral[f+g]=…? integral[f+g]=... Proving

16. Copyright Bruno Buchberger Real Numbers limit[f]:=... limit[f+g]=…? limit[f+g]=... The Invention Spiral of Mathematics Proving

17. Copyright Bruno Buchberger Real Numbers is-continuous[f] : … is-continuous[f+g] <==…? is-continuous[f+g] <==... The Invention Spiral of Mathematics Proving

18. Copyright Bruno Buchberger Proofs in Analysis Conjecture on the reduction of proving to constraint solving Reduction Theorem Automated Theorem Prover for Analysis The Invention Spiral of Mathematics Thousands of Proofs

19. Copyright Bruno Buchberger 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.)

20. Copyright Bruno Buchberger 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

21. Copyright Bruno Buchberger 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.

22. Copyright Bruno Buchberger 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.

23. Copyright Bruno Buchberger 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.

24. Copyright Bruno Buchberger 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.

25. Copyright Bruno Buchberger 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.

26. Copyright Bruno Buchberger 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.

27. Copyright Bruno Buchberger 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.

28. Copyright Bruno Buchberger 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.