1 trends in mathematics: how could they change education? lászló lovász eötvös loránd...
Post on 22-Dec-2015
217 Views
Preview:
TRANSCRIPT
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,...)
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.
top related