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.
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
Students will obtain new content knowledge from the areas of logic, set
theory, graph theory, combinatorial theory, number theory and abstract algebra.
Students' existing knowledge of logic, set theory, graph theory, combinatorial
theory, number theory and abstract algebra will be strengthened.
Students will obtain new knowledge of proof and problem solving techniques
including: pigeonhole principle, coloring, symmetry arguments, invariants,
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.
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.
Strategies and Mental
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.
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.
Students will be learn to periodically judge the efficacy of their choice
of methods and resources which they apply to a problem.
Beliefs and Epistomology
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.
Students will become acquainted with the accepted beliefs of the professional