I. Number and Title of Course

Mathematics 183-Problem Solving in Mathematics

A. The course is an introduction to the process of problem solving. At present, there is no course in the Mathematics Department which focuses on process rather than the mathematical content.

B. The course is intended to provide a foundation in the processes of problem solving which will be useful as a preparation for other courses in mathematics.

C. The course is needed to provide a forum for students to develop their own problem solving style, and to assume a leadership role in problem solving.

D. The course is needed to meet the program requirements to prepare mathematics teachers for secondary schools.

E. The course is a requirement in the K-9 math extension program.

A. To concentrate on the process of problem solving rather than the content and to use and internalize problem solving processes.

B. To develop the student's confidence and originality in problem solving in mathematics.

C. To prepare students to handle the problem solving of more advanced courses involving mathematics.

D. To improve the teaching skills of prospective secondary mathematics teachers.

E. To provide more depth in applying and understanding how problem solving processes can be used to increase mathematical learning.

F. To provide a means for assessment of the strengths, weaknesses, and preferences in the mathematical problem solving of the undergraduate.

A. Problem solving processes applied to mathematics

1. Creative problem solving

a. Data-finding

b. Problem-finding

c. Idea-finding

d. Solution-finding

e. Acceptance-finding

2. Polya's approach

a. Diagrams, tables, and charts

b. Special cases

c. Reduction in size

d. Identifying cases

e. Relaxing restrictions

f. Separating various parts of the conditions

g. Trial and error, approximation and verification

h. Generalization

3. Wickelgren's approach

a. Implicit information and inference

b. Stating subgoals

c. Contradiction

d. Working backwards

e. Relations between problems

4. Pattern search and recognition, estimation, making models, restatement of a problem, and problem posing.

5. The problem solving approaches of Adams, Descartes, Euler, Gauss, Hadamard, Poincare, and others,chosen for interest or appropriateness.

B. Suitable problems selected from algebra, geometry, analysis, and the applications related to any of these areas.

C. Problems selected from recreational mathematics.

Adams, James L., Conceptual Blockbusting, San Francisco, CA, Freeman, 1974.

Ball, W.W., Rouse,rev. by H.S.M.,Coxeter, Mathematical Recreations and Essays, New York, NY, Macmillan Co., 1960.

Barbeau, E.F., Klamkin, M., and Moser, W. (eds.), 1001 Problems in High School Mathematics, Book I, Problems 1-100, Montreal, Quebec, Canadian Mathematical Congress,

Brown, Stephen I, and Walter, Marion, I., The Art of Problem Posing. Hillsdale, NJ, Lawrence Erlbaum Associates, Inc., 1983.

Debono, Edward, Practical Thinking. New York, NY, Penguin Books, 1976.

Debono, Edward, Lateral Thinking, New York, NY Penguin Books, 1977.

Dudeney, H.E., Amusements in Mathematics, Mineola, NY, Dover Publishing, 1958.

Dudeney, H.E., 536 Puzzles and Curious Problems, New York, NY, Schribner's

Gardner, M.,Mathematical Carnival, New York, NY, Knopf, 1975.

Gardner, M.,Mathematical Magic Show, New York, NY, Knopf, 1977.

Gardner, M.,More Mathematical Puzzles of Sam Loyd (selected and edited by M. Gardner), Mineola, NY, Dover Publications, 1960.

Graham, L.A., The Suprise Attack in Mathematical Problems, Mineola, NY, Dover Publications, 1968.

Hadamard, J., The Psychology of Invention in the Mathematical Field, Mineola, NY. Dover Publications, 1954.

Honsberger, Ross, Mathematical Games, Washington,~D.C., Mathematical Association of America, 1973.

Journal articles from the Mathematics Teacher, Arithmetic Teacher,

Mathematics Teaching, Mathematics Magazine, School Science and Mathematics, American Mathematical Monthly.

Mathematical Challenges: Selected Problems from the Mathematics Student Journal, Washington, D.C. NCTM, 1965.

Mathematical Challenges II,Plus Six: Selected Problems from the Mathematics Student Journal, Washington, D.C., NCTM, 1974.

MeschLowski, H. Ways of Thought of Great Mathematicians, San Francisco,CA, Holden Day, 1984.

NCTM, 1980 Yearbook,Problem Solving in Mathematics,Reston, VA, NCTM, 1980.

Noller, R., Heintz, R.,and Blaeuer, D.,Scratching the Surface, Buffalo, NY, D.O.K., 1977.

Polya, G. How to Solve it, 2nd Edition, Garden City, N.Y., Doubleday, 1957.

Polya, George, Mathematical Discovery, Volumes I and II, New York,?NY, John~Wiley and Sons, 1962.

Polya, George, Patterns of Plausible Inference, Volumes I and II, Princeton, NJ, Princeton University Press, 1954.

Raudsepp, Eugene, More Creative Growth Games , New York, NY, G.P. Putnam and Sons, 1980.? ?

Rubinstein, Mosha, F.,Patterns of Problem Solving, Englewood Cliffs, NJ, Prentice Hall, 1975.

Salkind, C. The Contest Book, Problems from the Annual High School Contests of the M.A.A., New York, NY, Random House, 1961.

Schoenfeld, Alan, Mathematical Problem Solving, Orlando, FL, Academic Press. , 1985.

Whimby, Arthur,and Lochhead, Jack, Problem Solving and Comprehension, 3rd. ea., Hillsdale, NJ, Lawrence Erlbaum Associates Inc., 1983.

Whimby, Arthur,and Lochhead, Jack, Beyond Problem Solving and Comprehension An Exploration of Quantitative Reasoning, Hillsdale, NJ, Lawrence Erlbaum Associates, Inc., 1984.

Wickelgren, W. How to Solve Problems, San Francisco, CA, Freeman, 1974.

Lectures, class discussion, small group problem solving, class presentations and write-ups, written examinations, one?to?one discussion with instructor, problem solving projects.

Four years high?school math or equivalent

3 semester hours

Mathematics 183-PROBLEM SOLVING IN MATHEMATICS. An introduction to the basic techniques of problem solving, creative problem solving,the methods of Polya, Wickelgren, and others, and applications to the areas of algebra, geometry, analysis and recreational mathematics. The emphasis is on process rather than on content.

A master's degree in mathematics is the minimum formal education required. All members of the Mathematics Department meet this requirement. Beyond this, a person teaching this course should have an interest in exploring a variety of problem solving techniques and applying them to mathematics.

This course was examined in accordance with established procedures, and approved by the Department of Mathematics on May 15, 1986. ______________________________________Department Chairperson

Present classroom facilities are adequate.