1. 1 Trends in Mathematics: How could they Change Education? László Lovász Eötvös Loránd University Budapest

2. 2 General trends in mathematical activity The size of the community and of mathematical research activity increases exponentially. New areas of application, and their increasing significance. New tools: computers and information technology. New forms of mathematical activity.

3. 3 Size of the community and of research Mathematical literature doubles in every 25 years Impossible to keep up with new results: need of more efficient cooperation and better dissemination of new ideas. Larger and larger part of mathematical activity must be devoted to communication (conferences with expository talks only, survey volumes, internet encyclopedias, multiple authors of research papers...)

4. 4 Size of the community and of research Challenges in education: Difficult to identify ``core'' mathematics Two extreme solutions: - New results, theories, methods belong to Masters/PhD programs - Leave out those areas that are not in the center of math research today

5. 5 Size of the community and of research Challenges in education: Difficult to identify ``core'' mathematics Focus on mathematical competencies (problem solving, abstraction, generalization and specialization, logical reasoning, mathematical formalism)

6. 6 Size of the community and of research Challenges in education: Exposition style mathematics in education: teach students to explain mathematics to “outsiders” and to each other, to summarize results and methods,... teach some mathematical material “exposition style”?

7. 7 Applications: new areas Traditional areas of application: physics, astronomy and engineering. Use: analysis, differential equations.

8. 8 Biology: genetic code population dynamics protein folding Physics: elementary particles, quarks, etc. (Feynman graphs) statistical mechanics (graph theory, discrete probability) Economics: indivisibilities (integer programming, game theory) Computing: algorithms, complexity, databases, networks, VLSI,... Applications: new areas

9. 9 Traditional areas of application: physics, astronomy and engineering. Use: analysis, differential equations. New areas: computer science, economics, biology, chemistry,... Use: most areas (discrete mathematics, number theory, probability, algebra,...)

10. 10 -Internet Very large graphs: Applications: new areas @Stephen Coast

11. 11 -Internet -chip design -Social networks -Statistical physics -Ecological systems Very large graphs: -Brain -Does it have an even number of nodes? -Is it connected? -How dense is it (average degree)? What properties to study? Applications: new areas

12. 12 Applications: new areas and significance Challenges in education: - Explain new applications programming, modeling,... - Train for working with non-mathematicians interdisciplinary projects, modeling,...

13. 13 New tools: computers and IT Source of interesting and novel mathematical problems >new applications New tools for research (experimentation, collaboration, data bases, word processing, new publication tools)

14. 14 New tools: computers and IT Challenges in education: - Students are very good in using some of these tools. How to utilize this? nonstandard mathematical activities - How to make them learn those tools that they don’t know?

15. 15 New forms of mathematical activity Algorithms and programming Algorithm design is classical activity (Euclidean Alg, Newton's Method,...) but computers increased visibility and respectability.

16. 16 An example: diophantine approximation and continued fractions Givenfind rational approximation such that and continued fraction expansion

17. 17 New forms of mathematical activity Algorithms and programming Algorithm design is classical activity (Euclidean Alg, Newton's Method,...) but computers increased visibility and respectability. Algorithms are penetrating math and creating new paradigms.

18. 18 Mathematical notion of algorithms Church, Turing, Post recursive functions, Λ-calculus, Turing-machines Church, Gödel algorithmic and logical undecidability A mini-history of algorithms 1930’s

19. 19 Computers and the significance of running time simple and complex problems sorting searching arithmetic … Travelling Salesman matching network flows factoring … A mini-history of algorithms 1960’s

20. 20 Complexity theory P=NP? Time, space, information complexity Polynomial hierarchy Nondeterminism, good characteriztion, completeness Randomization, parallelism Classification of many real-life problems into P vs. NP-complete A mini-history of algorithms 70-80’s

21. 21 Increasing sophistication: upper and lower bounds on complexity algorithmsnegative results factoring volume computation semidefinite optimization topology algebraic geometry coding theory A mini-history of algorithms 90’s

22. 22 Approximation algorithms positive and negative results Probabilistic algorithms Markov chains, high concentration, phase transitions Pseudorandom number generators from art to science: theory and constructions A mini-history of algorithms 90’s Cryptography state of the art number theory

23. 23 New forms of mathematical activity Challenges in education: Balance of algorithms and theorems Algorithms and their implementation develop collections of examples, problems... No standard way to describe algorithms: informal? pseudocode? program? develop a smooth and unified style for describing and analyzing algorithms

24. 24 New forms of mathematical activity Problems and conjectures Paul Erdős: the art of raising conjectures Best teaching style of mathematics emphasizes discovery, good teachers challenge students to formulate conjectures. Challenges in education: Preserve this!!

25. 25 New forms of mathematical activity Mathematical experiments Computers turn mathematics into an experimental subject. Can be used in the teaching of analysis, number theory, optimization,... Challenges in education: Lot of room for good collection of problems and demo programs

26. 26 New forms of mathematical activity Modeling First step in successful application of mathematics. Challenges in education: Combine teaching of mathematical modeling with training in team work and professional interaction.

27. 27 New forms of mathematical activity Exposition and popularization Growing very fast in the research community. Notoriously difficult to talk about math to non-mathematicians. Challenges in education: Teach students at all levels to give presentations, to write about mathematics.