8/18/2019 Common Core State

1/26

8/18/2019 Common Core State

2/26

!ize of the community and of research

"athematical literature dou#les in every $years

%mpossi#le to &eep up with new results:need of more efficient cooperation and#etter dissemination of new ideas.

'arger and larger part of mathematical activity

must #e devoted to communication (conferenceswith expository tal&s only, survey volumes,internet encyclopedias, multiple authors ofresearch papers...)

8/18/2019 Common Core State

3/26

*

!ize of the community and of research

Challenges in education:

+ifficult to identify core-- mathematics

Two extreme solutions:

New results, theories, methods #elong to"asters/0h+ programs

'eave out those areas that are not inthe center of math research today

8/18/2019 Common Core State

4/26

!ize of the community and of research

Challenges in education:

+ifficult to identify core-- mathematics

2ocus on mathematical competencies(pro#lem solving, a#straction, generalization

and specialization, logical reasoning,mathematical formalism)

8/18/2019 Common Core State

5/26

$

!ize of the community and of research

Challenges in education:

3xposition style mathematics in education:

teach students to explain mathematics to4outsiders5 and to each other, to summarizeresults and methods,...

teach some mathematical material4exposition style56

8/18/2019 Common Core State

6/26

7

8pplications: new areas

Traditional areas of application: physics,astronomy and engineering.

9se: analysis, differential e uations.

8/18/2019 Common Core State

7/26

;

'!%, ...

8pplications: new areas

8/18/2019 Common Core State

8/26

?

8pplications: new areas

Traditional areas of application: physics,astronomy and engineering.

9se: analysis, differential e uations.

New areas: computer science, economics,#iology, chemistry, ...

9se: most areas (discrete mathematics,num#er theory, pro#a#ility, alge#ra,...)

8/18/2019 Common Core State

9/26

@

%nternet

>ery large graphs:

8pplications: new areas

A!tephen =oast

8/18/2019 Common Core State

10/26

1B

%nternet

chip design

!ocial networ&s

!tatistical physics

3cological systems

>ery large graphs:

8/18/2019 Common Core State

11/26

11

8pplications: new areas and significance

Challenges in education:

3xplain new applicationsprogramming, modeling,...

Train for wor&ing with non mathematiciansinterdisciplinary proDects, modeling,...

8/18/2019 Common Core State

12/26

1

New tools: computers and %T

!ource of interesting and novelmathematical pro#lems Enew applications

New tools for research (experimentation,colla#oration, data #ases, wordprocessing, new pu#lication tools)

8/18/2019 Common Core State

13/26

1*

New tools: computers and %T

Challenges in education:

!tudents are very good in using some of

these tools. Cow to utilize this6nonstandard mathematical activities

Cow to ma&e them learn those tools thatthey donFt &now6

8/18/2019 Common Core State

14/26

1

New forms of mathematical activity

8lgorithms and programming

8lgorithm design is classical activity

(3uclidean 8lg, Newton-s "ethod,...) #utcomputers increased visi#ility andrespecta#ility.

8/18/2019 Common Core State

15/26

1$

8n example: diophantine approximation and continued fractions

Given ,α ∈ ¡ find rational approximation / p q

such that | / | / p q qα ε − < and 1/ .q ε ≤

α

ε <m α

n α

| | | |m n p q pα α α ε − − = − <

0aα α ≈ = 0 ?aα −

1

0 0

1 1a

a aα α

≈ = − −

0

1

1a

a

α ≈ +

continued fraction expansion1/q ε ≤

8/18/2019 Common Core State

16/26

17

New forms of mathematical activity

8lgorithms and programming

8lgorithm design is classical activity

(3uclidean 8lg, Newton-s "ethod,...) #utcomputers increased visi#ility andrespecta#ility.

8lgorithms are penetrating math andcreating new paradigms.

8/18/2019 Common Core State

17/26

1;

"athematical notion of algorithms

=hurch, Turing, 0ost

recursive functions, calculus, Turing machines

=hurch, GHdel

algorithmic and logical undecida#ility

8 mini history of algorithms 1@*BFs

8/18/2019 Common Core State

18/26

1?

=omputers and the significance of running time

simple and complex pro#lems

sortingsearchingarithmetic I

Travelling !alesmanmatchingnetwor& flowsfactoring I

8 mini history of algorithms 1@7BFs

8/18/2019 Common Core State

19/26

1@

=omplexity theory

0JN06

Time, space, information complexity

0olynomial hierarchy

Nondeterminism, good characteriztion, completeness

Kandomization, parallelism

=lassification of many real life pro#lems into 0 vs. N0 complete

8 mini history of algorithms ;B ?BFs

8/18/2019 Common Core State

20/26

B

%ncreasing sophistication:upper and lower #ounds on complexity

algorithms negative results

factoringvolume computationsemidefinite optimization

topologyalge#raic geometrycoding theory

8 mini history of algorithms @BFs

8/18/2019 Common Core State

21/26

1

8pproximation algorithms positive and negative results

0ro#a#ilistic algorithms

"ar&ov chains, high concentration,phase transitions

0seudorandom num#er generators from art to science: theory and constructions

8 mini history of algorithms @BFs

=ryptography state of the art num#er theory

8/18/2019 Common Core State

22/26

New forms of mathematical activity

Challenges in education:

8/18/2019 Common Core State

23/26

*

New forms of mathematical activity

0ro#lems and conDectures0aul 3rdLs: the art of raising conDectures

8/18/2019 Common Core State

24/26

New forms of mathematical activity

"athematical experiments

=omputers turn mathematics into an

experimental su#Dect.=an #e used in the teaching of analysis,

num#er theory, optimization, ...

Challenges in education:'ot of room for good collection ofpro#lems and demo programs

8/18/2019 Common Core State

25/26

$

New forms of mathematical activity

"odeling

2irst step in successful application of

mathematics.

Challenges in education:

=om#ine teaching of mathematicalmodeling with training in team wor& andprofessional interaction.

8/18/2019 Common Core State

26/26

7

New forms of mathematical activity

3xposition and popularization

Growing very fast in the research community.

Notoriously difficult to tal& a#out math to nonmathematicians.

Challenges in education:Teach students at all levels to give

presentations, to write a#out mathematics.