Background:

This course is designed for middle/secondary teachers of mathematics and others who wish to enhance their capabilities in mathematical reasoning and problem solving. We consider topics from discrete mathematics such as sets, logic and proof, relations, graphs, recursion, and combinatorics and we emphasize the construction of well-organized arguments and justifications.

Objectives:

• To enhance the ability to think mathematically, 
• To enhance the ability to construct conjectures, arguments, and proofs.
• To employ a variety of problem-solving strategies, 
• To gain success in solving non-routine problems, and 
• To become more skillful in explaining and justifying conjectures, arguments, proofs, and problem solutions, both orally and in writing.
• To develop units/lessons addressing K-12 learning objectives related to mathematical reasoning.

The class will focus on content as well as teaching techniques that are aligned with NCTM recommendations.

Intended Audience:

Pre-service and in-service middle school and secondary teachers of mathematics and others wishing to strengthen their mathematical background. 

Prerequisite:

Approval of the department.

Texts:

National Council of Teachers of Mathematics (NCTM), Discrete Mathematics Across the Curriculum, K-12, 1991 Yearbook of the NCTM, 1991.

Briggs, W., Ants, Bikes, & Clocks: Problem Solving for Undergraduates, Society for Industrial and Applied Mathematics (SIAM), 2005.

Schedule:

Technology:

June 21, 22, 23, 24, 27, 28, 29, 30. 8:30-11:30 & 12:30-2:00 daily.

(Some sessions will be focused "on your own work" or "group work." 

Excel, Graphing Calculator

 Tentative Schedule

Days 1-3:  Introduction to Mathematical Reasoning and Problem Solving.  The nature of mathematical reasoning. Problem solving approaches and strategies, mathematical representations, inductive and deductive reasoning, generalizations, closed-form solutions, recursive thinking, difference equations, algorithms, and use of spreadsheets.

Days 4-5:  Problem Solving & Topics in Discrete Math.  Further investigation of topics in set theory, the study of functions and relations, graph theory, and combinatorics. 

Days 6-7: Logic, Reasoning, and Proof.   Direct and indirect proofs, proof by contradiction, and proof by mathematical induction.  Sample theorems derived from discrete structures and number systems.

Day 8:  Presentations.  Project presentations based on units/lessons developed, or adapted, to address K-12 learning objectives.

In-Class Work

25%

Assignments

25%

Portfolio

25%

Project

25%

Total

100%