MATH 506 Selected Topics: Mathematical Reasoning

Session 1 (January 28):

Sample Problem Illustrating Application of Polya’s Problem Solving Process

Find a rule for determining the n^th triangular number

Concepts/Terminology

recursion, recursive process, recurrence relation

difference equation

(explicit) functional equation, closed form solution, direct computation

iteration

Important Results

If s_n = 1 + 2 + 3 + … + n then s₁ = 1 and s_n = s_n-1 + n for n > 1.

If s_n = 1 + 2 + 3 + … + n then s_n = n(n+1)/2.

Session 2 (February 4):

Discussion of the relationship between the problem of Chapter 10 in 

[NCTM, 1991] and the problem about triangular numbers.

Concepts/Terminology

difference table

first, second, third, … differences

linear and quadratic growth patterns

inductive vs deductive thinking

Introduction of a counting problem – Counting the number of triangles in a

geometric pattern

Session 3 (February 11):

Discussion of the problem about the number of triangles in a geometric pattern

Concepts/Terminology

algorithm

pseudocode as a language for writing algorithms

mathematical representations

Consideration of some selected counting problems and associated algorithms

Brief introduction to Logo

Session 4 (February 18):

Discussion of sample middle school lessons

the lesson

relationship to “counting triangles” and “triangular numbers” problems

some possible extensions/generalizations

Reactions to “Strengthening a K-8 Mathematics Program with Discrete Mathematics

problem solving process

counting problems and mathematical representations

Venn Diagrams – suggest further study

graphs, trees, and networks – suggest further study

“Problem Solving with Discrete Mathematics” – Sample Problems

counting dominos – relate to triangular numbers

counting diagonals – relate to triangular numbers

recursion, difference equations, & Logo

direct computation, explict functional equations, closed form solutions

Session 5 (February 25):

Sets and Venn diagrams

interactive web sites

logic puzzles

attribute blocks

Two games

hex and sprouts

http://www.mazeworks.com/hex7/

http://www.math.utah.edu/~alfeld/Sprouts/

Graphs, Paths, and Circuits

http://www.utm.edu/departments/math/graph/

Konigsberg Bridges Problem

Floor Plan Walk

Logo Programming

Session 6 (March 4):(Independent Study Week)

Logo Programming and Recursion

Tower of Hanoi Puzzle

http://www.mazeworks.com/hanoi/index.htm

Session 7 (March 11):

Algorithms

PseudoCode

Spreadsheets & Recursion

Logo & Recursion

http://www.math.uic.edu/~burgiel/Mtht480/logo3.html

Session 8 (March 18): (Independent Study Week)

March 25 - Spring Break

Session 9 (April 1): (Independent Study Week)

Session 10 (April 8):

More Algorithms and Pseudo Code

Two Puzzles

A Little Graph Theory

http://www.utm.edu/departments/math/graph/

Session 11 (April 15):

Sample (graph theory) solutions for some puzzles.

Look at some lesson plans.

Introduction to Logic

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logicintro.html

Session 12 (April 22):

Graph Theory wrap-up

Look at some lesson plans

Introduction to Logic Continued

Conditional statements

Forward & backward thinking in crafting proofs

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logicintro.html

Session 13 (April 29):

Introduction to Logic Continued

Negations & Quantifiers

Valid arguments

Forward & backward thinking in crafting proofs

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logicintro.html

Sample lesson

Session 14 (May 6):

Introduction to Logic Continued

Some Proofs

Proof by Contradiction

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logicintro.html

Sample lesson

Session 15 (May 13): (Submit portfolios and evaluate course.)

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