The Nature of Mathematics

This course is organized around a series of intertwined themes each of which has aspects that are pertinent to primary, secondary and post-secondary mathematics as well as to mathematics as it is employed by those who use it in diverse ways in the pursuit of their livelihoods. These themes are

· mathematical objects and actions,

· mathematics as a tool for understanding one's world,

· mathematics as an intellectual undertaking in its own right,

· learning mathematics and/or making mathematics,

· meaningful assessment of mathematical attainment,

· mathematics and culture - how universal is the subject?

There will be both assigned readings and problems in the course. Students will be evaluated on the basis of participation in class discussion and problem-solving sessions as well as a final presentation. This course meets the requirement for a graduate level course in mathematics for purposes of certification. The course is limited to math students in the Midcareer Math and Science Program and math concentrators in the Teaching and Curriculum Program. Mathematics students in MCMS and TAC are expected to enroll in the course. The permission of the instructor is required.

This course is not offered in the 2001-02 academic year

Course Outline

1) mathematical objects and actions (1 week)

__Suggested readings for Part 1__

· *Mathematics - Our Invisible Culture*, Allen L. Hammond in __Mathematics Today__, Lynn Arthur Steen, ed., New York, Vintage Books, 1978, pp.15-34

· __Can Technology Help Us Make The Mathematics Curriculum Intellectually Stimulating and Socially Responsible?__ - Schwartz (paper on reserve in library and available for purchase from ETC secretary in Nichols House).

2) mathematics as a tool for understanding one's world (3 weeks)

Suggested readings for Part 2

· *Biological Modeling: Population Dynamics* - John Macqueen, in __New Applications of Mathematics__ C. Bondi, ed., Penguin 1991

· *Scaling Methods in Physics and Biology* - John Rallison, in __New Applications of Mathematics__ C. Bondi, ed., Penguin 1991

· __Semantic Aspects of Quantity__ - Schwartz (paper on reserve in library and available for purchase from ETC secretary in Nichols House).

3) mathematics as an intellectual undertaking in its own right (3 weeks)

Suggested readings for Part 3

· *The Essence of Mathematics* - C.S. Pierce, in __The World of Mathematics__, Vol. 3, James R. Newman, ed., NY Simon & Schuster, 1956, pp. 1773-1783

· __The Problems of Mathematics__, 2nd Edition, Ian Stewart, NY, Oxford University Press, 1992

· *The Mathematical Way of Thinking*, Herman Weyl, in __The World of Mathematics__, Vol. 3, James R. Newman, ed., NY Simon & Schuster, 1956, pp. 1832-1849

4) learning mathematics and/or making mathematics (3 weeks)

Suggested readings for Part 4

· *Mathematics & Plausible Reasoning* - Princeton, Princeton University Press, 1954

Vol. I - Induction and Analogy in Mathematics,

Vol. II - Patterns of Plausible Inference,

· *How to Solve It*, G. Polya, New York, Doubleday, 1957

· *Intellectual Mirrors: A step in the direction of making schools knowledge-making places*, J.L. Schwartz, Harvard Educational Review, February 1989, pp. 51-61.-

5) meaningful assessment of mathematical attainment (2 weeks)

Suggested readings for Part 5

Selected reports -

National Council of Teachers of Mathematics

New Standards Project

Balanced Assessment in Mathematics Project

6) mathematics and culture - how universal is the subject? (1 week)

Suggested readings for Part 6

· *Africa counts; number and pattern in African culture,*. Claudia Zaslavsky, Boston, Prindle, Weber & Schmidt [1973]

· *Ethnomathematics : a multicultural view of mathematical ideas*, Marcia Ascher, , Pacific Grove, Calif. : Brooks/Cole Pub. Co., 1991.

· *Ethnomathematics : challenging eurocentrism in mathematics education*, Arthur B. Powell and Marilyn Frankenstein, eds., Albany : State University of New York Press, 1997.

Bibliography

This is a collection of books and articles that I have found interesting and helpful. It is not comprehensive and I welcome all suggestions for additions to the list. [JLS]

Ethnomathematics : a multicultural view of mathematical ideas, Marcia Ascher, , Pacific Grove, Calif. : Brooks/Cole Pub. Co., 1991

__Philosophy of Mathematics__, P. Benacerraf & H. Putnam, eds., Englewood Cliff, Prentice-Hall, 1964

__New Applications of Mathematics__ C. Bondi, ed., Penguin 1991

__What is mathematics? : an elementary approach to ideas and methods__, Richard Courant and Herbert Robbins. 2nd ed. revised by Ian Stewart,. New York : Oxford University Press, 1996.

__The Mathematical Universe__ - William Dunham, NY, John Wiley & Sons, Inc. 1994

__Descartes' Dream__, P.J.Davis & R. Hersh, Boston, Houghton Mifflin, 1986

Mathematics: The new golden age, K. Devlin, Penguin, 1988

__Chaos: Making a new science__, New York, Viking, 1987

__A Mathematician's Apology__, G.H. Hardy, Cambridge University Press, 1940

__Mathematics and the Imagination__ - E. Kasner & J. Newman, New York : Simon and Schuster, 1940.

__Mathsemantics: Making Numbers Talk Sense__ - MacNeal

__The World of Mathematics__, (4 volumes) James R. Newman, ed., NY Simon & Schuster, 1956

__Islands of Truth: A mathematical mystery cruise__, I. Peterson, New York, W.H. Freeman, 1990

__The Mathematical Tourist: Snapshots of modern mathematics__, I. Peterson, New York, W.H. Freeman, 1988

__Mathematics & Plausible Reasoning __- Princeton, Princeton University Press, 1954

Vol. I - Induction and Analogy in Mathematics,

Vol. II - Patterns of Plausible Inference,

__How to Solve It__, G. Polya, New York, Doubleday, 1957

__Ethnomathematics : challenging eurocentrism in mathematics education__, Arthur B. Powell and Marilyn Frankenstein, eds., Albany : State University of New York Press, 1997.

__Vision in Elementary Mathematics__, W.W.Sawyer, Penguin, 1964

__Can Technology Help Us Make The Mathematics Curriculum Intellectually Stimulating and Socially Responsible? __- J.L. Schwartz

__Semantic Aspects of Quantity__ - J.L. Schwartz

__Intellectual Mirrors: A step in the direction of making schools knowledge-making places__, J.L. Schwartz, Harvard Educational Review, February 1989, pp. 51-61.-

__Shuttling between the Particular and the General: Reflections on the Role of Conjecture & Hypothesis in the Generation of Knowledge in Science and Mathematics__, J.L.Schwartz, in __Software Goes to School__, Perkins, Schwartz, West & Wiske, eds. Oxford University Press, 1995

__Mathematics Today__, Lynn Arthur Steen, ed., New York, Vintage Books, 1978,

__Does God Play Dice__,: The mathematics of chaos, Ian Stewart, Penguin 1989

__The Problems of Mathematics__, 2nd Edition, Ian Stewart, NY, Oxford University Press, 1992

__The Mathematics of Physical Quantity__ - H. Whitney, 1968, American Mathematical Monthly, Vol. 75.

__Evolution of Mathematical Concepts__, R.L. Wilder, NY, John Wiley & Sons, 1968

__Africa counts; number and pattern in African culture__,. Claudia Zaslavsky, Boston, Prindle, Weber & Schmidt [1973]

A Brief Bibliography on Assessment

Baxter, Gail P., Richard J. Shavelson, Sally J. Herman, Katherine A. Brown, and James R. Valadez. "Mathematics Performance Assessment: Technical Quality & Diverse Student Impact." Journal for Research in Mathematics Education, 24, 3, (1993): pp. 190-216.

Cain, Ralph W. & P.A. Kenney, "A Joint Vision for Classroom Assessment." The Mathematics Teacher 85, 8, (1992): pp. 612-615.

Charles, Randall I., and Edward A. Silver, eds. Research Agenda for Mathematics Education: The Teaching and Assessing of Mathematical Problem Solving. Hillsdale NJ: Lawrence Erlbaum Associates, 1988.

Kulm, Gerald. Assessing Higher Order Thinking in Mathematics. Washington DC: AAAS, 1990.

Kulm, Gerald. Mathematics Assessment: What Works in the Classroom. San Francisco CA: Jossey-Bass, 1994.

Leder, Gilah, ed. Assessment and Learning of Mathematics. Victoria, Australian Council for Educational Research, 1992.

Lesh, Richard and Susan J. Lamon, eds. Assessment of Authentic Performance in School Mathematics. Washington DC: American Association for the Advancement of Science, 1992.

Lester, Frank K. and Diana L. Kroll. "Evaluation: A New Vision." The Mathematics Teacher 84, 4, (1991): pp. 276-283.

Mathematical Sciences Education Board. Measuring What Counts: A Conceptual Guide for Mathematics Assessment. Washington DC: National Academy Press, 1993.

Mathematical Sciences Education Board. For Good Measure: Principles and Goals for Mathematics Assessment. Washington, DC: National Academy Press, 1991.

Office of Technology Assessment. Testing in American Schools: Asking the Right Questions. Washington DC: OTA, 1992.

Petit, Marge. Getting Started: Vermont Mathematics Portfolio — Learning How to Show Your Best!. Cabot VT: Cabot School, 1992.

Resnick, Lauren B. and Daniel P. Resnick. "Assessing the Thinking Curriculum: New Tools for Educational Reform" in Changing Assessments: Alternative Views of Aptitude, Achievement and Instruction. edited by Bernard R. Gifford and Mary C. O’Connor, Boston MA: Kluwer Academic Publishers, 1992.

Romberg, Thomas A., ed. Mathematics Assessment and Evaluation: Imperatives for Mathematics Educators. Albany NY: State University of New York Press, 1992.

Schwartz, Judah L. and Katherine A. Viator, eds. The Prices of Secrecy: The Social, Intellectual and Psychological Costs of Current Assessment Practice, Cambridge MA: Educational Technology Center, Harvard University, .1990.

Webb, Norman L. "Assessment of Students’ Knowledge of Mathematics: Steps Toward a Theory" in Handbook of Research on Mathematics and Learning, edited by Doulas A. Grouws, New York NY: Macmillan, 1992.

Webb, Norman L. and Arthur F. Coxford, eds. Assessment in the Mathematics Classroom. 1993 Yearbook, Reston VA: NCTM, 1993.