1 trends in mathematics: how could they change education? lászló lovász eötvös loránd...

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

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1

Trends in Mathematics: How could they Change

Education?

László Lovász

Eötvös Loránd University

Budapest

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

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

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

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

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

Applications: new areas

Traditional areas of application: physics, astronomy and engineering.

Use: analysis, differential equations.

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

Applications: new areas

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

-Internet

Very large graphs:

Applications: new areas

@Stephen Coast

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

Applications: new areas and significance

Challenges in education:

- Explain new applications

programming, modeling,...

- Train for working with non-mathematicians

interdisciplinary projects, modeling,...

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

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

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

An example: diophantine approximation

and continued fractions

Given , find rational approximation /p q

such that | / | /p q q and 1/ .q

m

n

| | | |m n p q p

0a 0 ?a

10 0

1 1a

a a

01

1a

a

continued fraction expansion1/q

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

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

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

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

Increasing sophistication: upper and lower bounds on complexity

algorithms negative results

factoringvolume computationsemidefinite optimization

topologyalgebraic geometrycoding theory

A mini-history of algorithms 90’s

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

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 stylefor describing and analyzing algorithms

24

New forms of mathematical activity

Problems and conjecturesPaul 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

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

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

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.