In the usual lecture format classes devoted to a specific content area of mathematics the objectives are all related to the acquisition of new knowledge (mathematical facts) and procedures. These were a part, albeit a smaller part, of my course. In addition to specific content knowledge and techniquees which I refer to below as resources (following their description by A. Schoenfeld) I wanted to equip my students with psychological strategies for persisting in difficult situations, general strategies for solving problems, the ability to monitor their thinking so as to use their resources more efficiently and to socialize them to rules by which we judge the adequacy of solutions. These are described more precisely in the following.


Resources are the basis on which all problem solving is built. It consists of all the mathematical knowledge (facts and concepts), procedures, methods and techniques which an aspiring problem solver can access and apply to a given problem.

1. Students will obtain new content knowledge from the areas of logic, set theory, graph theory, combinatorial theory, number theory and abstract algebra.

2. Students' existing knowledge of logic, set theory, graph theory, combinatorial theory, number theory and abstract algebra will be strengthened.

3. Students will obtain new knowledge of proof and problem solving techniques including: pigeonhole principle, coloring, symmetry arguments, invariants, extremal principle.

Problem Solving Strategies/Heuristics

Heuristics are collections of loosely connected strategies for getting started and making progress in the problem solving process, that is, dealing with situations where there is not an obvious approach.

4. Students will learn to apply heuristics to novel (to the given student) mathematical problems, including: establish subgoals, try to relate the problem a previously solved problem, relax conditions, try an easier but similar problem, represent the problem in an alternate way.

Psychological Strategies and Mental Control

Psychological strategies are methods for persisting, developing confidence, and obtaining focus in problem solving. Control is the ability to monitor one's mental process so as to make efficient and effective use of resources.

5. Students will learn to use such psychological strategies as: engage in wishful thinking, apply creativity, use peripheral vision, bend the rules, break out of boxes, keep loose, be mentally touch.

6. Students will be learn to periodically judge the efficacy of their choice of methods and resources which they apply to a problem.

Mathematical Beliefs and Epistomology

Beliefs are the assumptions that an individual brings to his/her study of mathematics. It includes their notion of what the "rules of the game" are and how the "game is played." For example, what the student thinks of as a proof and the importance to the development of new knowledge and how knowledge is transformed is a part of their belief system.

7. Students will become acquainted with the accepted beliefs of the professional mathematical community.