Post on 07-Jul-2018
transcript
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.