Transcript Slide 1

Perspectives on
Transition Courses
Alex M. McAllister
[email protected]
Mathematics Department, Centre College
MAT 290: Foundations of
Mathematics
Foundations of Mathematics develops the abstract thinking
and writing skills necessary for proof-oriented
mathematics courses and surveys various areas of
mathematics.
Important mathematics concepts and questions are
studied from several areas, including mathematical
logic, abstract algebra, number theory, and real analysis.
Further topics include complex analysis, statistics, graph
theory, and/or other areas of mathematics according to
the interests of instructor and students.
Two Motivating Questions
 How
can we help our majors succeed?
Particularly in their upper-level courses?
 What
should every math major “know”
when they graduate?
How can we help ???
Transition from computationally-oriented lower
level courses to the more theoretical upper level
courses.
The hallmarks of a mathematician:
 Sound reasoning
 Communicating with precise language
 Asking probing questions about mathematics
Two Motivating Questions
 How
can we help our majors succeed?
Particularly in their upper-level courses?
A Transition-Bridge course…
in what context???
 What
should every math major “know”
when they graduate?
What are the fundamental ideas and
questions of mathematics ???
What are the fundamental ideas and
questions of mathematics ???
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Pythagorean Theorem
Fermat’s Last Theorem
Division Algorithm
Fundamental Theorem of
Arithmetic
Uniqueness of
objects/factorization
There exist infinitely many primes
Goldbach’s Conjecture
Riemann Hypothesis
The square root of two is
irrational
Fundamental Theorem of
Algebra
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Graphs of Complex Functions
Mean Value Theorem
The definition of the derivative
Basic properties of functions
Basic properties of derivatives
The definition of the integral
Fundamental Theorem of
Calculus
Taylor’s Theorem
Fourier Series
Proofs by Induction
What are the fundamental ideas and
questions of mathematics ???
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The definition of
number/cardinality
|Z| = |Q|
Cantor’s Theorem
Pascal’s Triangle
The Binomial Theorem
Basic Combinatorics
Basic Probability
Euclidean geometry
Non-Euclidean geometries
The Gödel Incompleteness
Theorems
Basics of Set Theory
Russell’s Paradox
Graph Theory
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Attributes/adjectives of
functions:
 one-to-one
 onto
 increasing/decreasing
 continuous
 differentiable
 integrable
 Equal
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Basics of logic
 connectives
 truth tables
 arguments
 quantifiers
Two Motivating Questions
 How
can we help our majors succeed?
Particularly in their upper-level courses?
A Transition-Bridge course…
in what context???
 What
should every math major “know”
when the graduate from Centre College?
A Survey course…
The Book…
A Survey of Advanced Mathematics
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Chapter 1: Mathematical Logic
Chapter 2: Abstract Algebra
Chapter 3: Number Theory
Chapter 4: Real Analysis
Chapter 5: Probability and Statistics
Chapter 6: Graph Theory
Chapter 7: Complex Analysis
Lead up to Infinitude of the Primes
Embedded Question 9 on pp 145
(a) Determine if the following integers are prime; if not,
give a nontrivial divisor.
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2+1
2•3+1
2•3•5+1
2•3•5•7+1
(b) Formulate a conjecture about the number
p1 • p2 •…• pn + 1.
Lead up to Infinitude of the Primes
Embedded Question 9 on pp 145
(a) Determine if the following integers are prime; if not,
give a nontrivial divisor.
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2+1 = 3
2•3+1 = 7
2 • 3 • 5 + 1 = 31
2 • 3 • 5 • 7 + 1 = 211
(b) Formulate a conjecture about the number
p1 • p2 •…• pn + 1.
Lead up to Infinitude of the Primes
Embedded Question 9 on pp 145
(a) Determine if the following integers are prime; if not,
give a nontrivial divisor.
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2+1 =3
2•3+1 = 7
2 • 3 • 5 + 1 = 31
2 • 3 • 5 • 7 + 1 = 211
(b) Formulate a conjecture about the number
p1 • p2 •…• pn + 1.
(c) Find two primes greater than 50 that divide
2 • 3 • 5 • 7 • 11 • 13 + 1 ( = 30031 = 59 • 509).
(d) Reformulate your conjecture. What can you say about
the primes p1, …, pn (not) dividing p1 • p2 •…• pn + 1?
Prime Number Theorem
Exercise #62 on pp 150
Complete the following table. The following data
suggests lim (n->∞) π(n) / [n/ln(n)] = ??
n
10
1,000
100,000
10,000,000
1,000,000,000
π(n)
4
168
9,592
664,579
50,847,534
n/ln(n)
π(n) / [n/ln(n)]
Prime Number Theorem
Exercise #62 on pp 150
Complete the following table. The following data
suggests lim (n->∞) π(n) / [n/ln(n)] = ??
n
10
1,000
100,000
10,000,000
1,000,000,000
π(n)
n/ln(n) π(n) / [n/ln(n)]
4
4.343
0.921
168
145
1.158
9,592
8,686
1.104
664,579
629,420
1.056
50,847,534 48,254,942
1.053
Some Success
Student
Body
MAT 290
Majors
Minors
2006
1129
16
11
7
2007
1145
20
21
11
2008
1186
28
20
7