STATE UNIVERSITY OF NEW YORK AT OSWEGO
                 Computer Science Department


    I.  COURSE NUMBER AND CREDIT:

        MAT/CSC 125 - 3 semester hours

   II.  COURSE TITLE:

        Elements of Problem Solving, Mathematics, and
        Computation

  III.  COURSE DESCRIPTION:

        Introduction to elements of mathematics in a modern
        computing environment.  Exposure to geometry, number
        theory, combinatorics, and topology.  Programming in
        LOGO on the Macintosh.  Explicit reliance on general
        problem solving methodologies to drive mathematical
        explorations and guide computer programming
        activities.

   IV.  PREREQUISITES:

        none

    V.  JUSTIFICATION:

        There is a natural affinity among:

        (1)  The general problem-solving methodologies
             articulated by George Poyla, the team of Alan
             Newell and Herb Simon, and the current wave of
             cognitive scientists.

        (2)  The computational problem-solving metaphores
             championed by, most notably, Seymour Papert.

        (3)  The beautiful and accessible mathematics
             popularized by Martin Gardner, John Horton
             Conway, and others.

        To date there has been no course offering at SUNYCO
        dedicated to this precept.

        Wickelgren pioneered the notion that a course
        integrating (1) and (3) above would benefit the
        general student population.  The premise is that
        extensive practice in the conscious application of
        general problem-solving techniques to mathematical
        problems will result in the percolation of these
        strategies into the subconscious.  Moreover, an
        awareness of the general principles will enable
        students to avoid time spent blankly at formal
        problems with no idea of how to proceed in their
        solution, a common, irritating situation which is of
        no educational value.

        The failure of courses like Wickelgren's to take root
        in the educational system is at least partially
        attributable to an initial difficulty people often
        experience in relating the general principles to the
        mathematical problems.  The computer bridges this
        gap.  Papert goes so far as to suggest that the best
        way to explain Poyla to students is to let them learn
        Turtle Geometry in the LOGO context.  Graded
        exercises in Poyla's Principles provide the key to
        development of skill in their application.

        The computer animates problem-solving practice,
        promotes syntonic learning, encourages clarity of
        thought and expression, assists in the generation and
        organization of data for analysis, and most
        importantly, makes immediately plain the value of
        various problem-solving metaphores (e.g.,
        decomposition, debugging, "playing turtle").  With
        the advent of the Macintosh computer (and similar
        models), little time need be devoted to dreary
        technical preparations for computer use.  In short,
        the time is ripe for a course which tightly
        integrates (1), (2), and (3) above.

        This course will be submitted for General Education
        mathematics/computation certification.  Moreover, it
        should prove to be particularly appealing to (1) all
        education majors, and (2) anyone interested in
        exploring the idea of becoming a computer science
        major or a mathematics major.

        This course differs from MAT 102 and CSC 101 in that
        both are based upon a dissociated model of rote
        learning, which is antithetical to what is being
        proposed here.

        In the first year this course will be offered to a
        few sections of at most 30 students, perhaps two
        sections per semester.  This could be expanded with a
        substantial commitment to various resources.

   VI.  COURSE OBJECTIVES:

        (1)  Explicit knowledge of general problem-solving
             principles and reasoning methodologies,
             including:  representation, search,
             decomposition, state evaluation, hill climbing,
             action sequencing (control strategies), and
             inferencing, especially induction, deduction,
             and abduction.

        (2)  Exposure to mathematics, particularly geometry,
             number theory, combinatorics, and topology.

        (3)  Appreciation of the computer as a thinking tool.

  VII.  COURSE OUTLINE:

        NOTE:  Every third day will be "problems day."  These
        will be characterized by an emphasis on the theory
        and practice of problem solving, rather than on
        either mathematics or computing, per se.

        Prelude to Problem Solving
          Problems - Examples of Problems

        Introduction to the Computer and to Mathematics
          A Modern Computing Environment - The Macintosh
          What is Mathematics?
          Problems - General Problem-Solving Methods
          LOGO Fundamentals; Turtle Talk

        Geometry
          Turtle Geometry - Simple Designs
          Problems - Geometry
          Turtle Geometry - Simple Drawings
          Turtle Geometry - Polygonal Investigations
          Problems - General Problem-Solving Methods
          Turtle Geometry - Recursive Designs
          Turtle Geometry - An Interactive Game
          Problems - Geometry

        List Processing
          List Processing - Primatives; Recursive Processing
          List Processing - Implementation of a Sets
            Microworld
          Problems - List Processing and Set Manipulations

        Production Systems
          Production Systems - Fundamentals
          Production Systems - Expert Systems
          Problems - General Problem Solving

        Number Theory
          Number Theory - History and Lore
          Number Theory - Generating Number Sequences in LOGO
          Problems - Number Theory
          Number Theory - Investigating Conjectures in LOGO
          Number Theory - Applications
          Problems - Number Theory

        Counting and Arranging
          Combinatorics - Combinations and Permutations
          Combinatorics - Counting in a Combinatorial
            Microworld
          Problems - Combinatorics
          Topology - Topological Games
          Topology - Topological Theorems
          Problems - Topology

        Inference
          Deductive Inference - Deductive Puzzles
          Deductive Inference - Automating Deduction
          Problems - Deductive Inference
          Inductive Inference - Inductive Games
          Inductive Inference - Automating Induction
          Problems - Inductive Inference

        Computer Models of the Real World
          Object Oriented Systems Design
          The Shapes World
          Problems - Robot Planning and Problem Solving

 VIII.  METHODS OF INSTRUCTION:

        (1) Lectures
        (2) Readings
        (3) Problem Sets (which will include some
              programming

        NOTE:  The Macintosh Laboratory will permit a tight
        coupling of lectures with student computer
        interaction.

   IX.  COURSE REQUIREMENTS:

        LOGO Programming, readings on general problem
        solving, problem sets drawn from various fields of
        accessible mathematics.

    X.  MEANS OF EVALUATION:

        (1)  Written Problem Sets
        (2)  Written Examinations

   XI.  RESOURCES:

        (1)  Macintosh computers, including one or two for
             course development.
        (2)  Software, LOGO for the students and a variety of
             tools for courseware development.

  XII.  BIBLIOGRAPHY:

        Abelson, H. & diSessa, A. Turtle Geometry: The
           Computer as a Medium for Exploring Mathematics.
           The MIT Press, 1980.

        Brown, S. & Walter, M. The Art of Problem Posing.
           The Franklin Institute Press, 1983.

        Brualdi, M. & Genise, L. Logo and Models of
           Computation. Addison-Wesley, 1987.

        Coral Software. Object Logo. Coral Software Corp.,
           1986.

        Gardner, M. Mathematical Magic Show. Knopf, 1977.

        Gardner, M. Wheels, Life and Other Mathematical
           Amusements. Freeman, 1983.

        Gardner, M. Mathematical Puzzles and Diversions.
           Simon and Schuster, 1961.

        Gardner, M. New Mathematical Diversions from
           Scientific American. Simon and Schuster, 1966.

        Harvey, B. Computer Science Logo Style. The MIT
           Press, 1985.

        Hofstadter, D. Godel, Escher, Bach. Vintage Books,
           1979.

        Jocobs, H. Mathematics, A Human Endeavor. Freeman,
           1970.

        Klarner, D., Ed. The Mathematical Gardner. Wadsworth,
           1981.

        Minsky, M. The Society of Mind. Simon and Schuster,
           1986.

        Newell, A. & Simon, H. Human Problem Solving.
           Prentice-Hall, 1972.

        Nosal, M. Basic Probability and Applications.
           Saunders, 1977.

        Papert, S. MINDSTORMS: Children, computers and
           powerful ideas. Basic Books, 1980.

        Poyla, G. How to Solve It. Doubleday, 1957.

        Shafer, D. Artificial Intelligence Programming on
           the Macintosh. SAMS, 1986.

        Schroeder, M. Number Theory in Science and
           Communication. Springer-Verlag, 1984.

        Silver, A. Editor. Teaching and Learning Mathematical
           Problem Solving: Multiple Research Perspectives.
           Lea, 1985.

        Sobel, M. & Maletsky, M. Teaching Mathematics: A
           Sourcebook of Aids, Activities, and Strategies,
           2nd Ed., Prentice-Hall, 1987.

        Townsend, M. Discrete Mathematics: Applied
           Combinatorics and Graph Theory. Benjamin/
           Cummings, 1987.

        Watt, D. Learning with Logo. McGraw-Hill, 1983.

        Wenger, E. Artificial Intelligence and Tutoring
           Systems. Morgan Kaufman, 1987.

        Weir, S. Cultivating Minds: A Logo Casebook.  Harper
           & Row, 1987.

        Wickelgren. W. How to Solve Problems. Freeman, 1974.

        Winston, P. Artificial Intelligence, 2nd Ed. Addison-
           Wesley, 1984.