The general goal for this course will be to explore models of understanding, reasoning, and learning in mathematics and science and to explore their implications for the practices and problems of instruction and for curriculum at all levels -- from elementary through post-secondary.

For roughly the first half of the semester, we will examine models of understanding, reasoning, and learning in science and mathematics. We'll do this through readings from education research and cognitive science as well as by posing problems to people ("subjects") in experimental interviews and then analyzing what they say and do to try to understand their thinking.

In light of these various models and insights, we will then consider some practices and problems of instruction. We'll look at some current practices, watching videotapes of actual classes (K-12), trying some curricular software such as Microworlds Logo, Geometer’s Sketchpad and micro-computer-based labs (developed at Tufts’ Center for Science and Mathematics Teaching). We will also examine new curriculum frameworks and state standards. The final project will be to design some instruction in a particular mathematical or scientific topic area, ideally something you will be able to use in your teaching.

For this course, you will be required to do the following (subject, of course, to modification!):

• Read some articles (usually 1 or 2 per week) from the literature on mathematics and science education;
• Participate regularly in discussions;
• Engage in class mathematical and scientific activity and write short descriptions of your thinking and activity;
• Come up with some interesting mathematics or science problems/questions and conduct interviews about them with a number of subjects;
• Write up an analysis of the interviewees' understanding and reasoning in three of these interviews (3-5 pages each), and present these results in class;
• Read and assess other students' interview analyses;
• Develop plans and materials in preparation for student teaching and present them to the class, with a discussion of how these plans reflect your consideration of various aspects of understanding and reasoning in mathematics and/or science.

Grades

Grades will be based on written work and on participation in discussions.

Meeting Dates

Every Wednesday through December 16th with the exception of Sep. 30th (Yom Kippur) and Nov. 25th (Thanksgiving). One other class may need to be rescheduled: Wednesday October 14th. We will discuss this during our first meeting.

Email

every class member is expected to have an email account and to check for mail at least once a week and also on Wednesday before class meetings. There is a class list on the machine emerald. It is called mc@ccl.northwestern.edu. Please subscribe to this list as soon as possible. To subscribe send a message from your email account to "listproc@ccl.northwestern.edu". the body of the message should say: "Subscribe MC your name" (e.g., Subscribe MC Uri Wilensky).

Copying Fee

There is a $30 copying fee for the course materials. Please bring a check payable to Tufts University (or bring $30 cash) to next week’s class.

Note: We will not read all of these, but those we don't read serve as background support and are suggested auxiliary reading

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

Aiken, L. (1970). Attitudes towards mathematics. review of Educational research,, 40. pp. 551-596

Ball, D. (1990). With an eye on the Mathematical Horizon: Dilemmas of Teaching elementary school mathematics. Elementary School Journal, 93, 373-397.

Brown, A. L., & Campione, J. C. (1990). Communities of learning and thinking, or a context by any other name. Human Development. 21, 108-125.

Carey, S. (1988). Reorganization of knowledge in the course of acquisition. In S. Strauss (Ed.), Ontogeny, Phylogeny, and Historical Development (pp. 1-27). Norwood, NJ: Ablex.

Carey, S. (1986). Cognitive Science and Science Education. American Psychologist Vol 41, no 10.

Carraher, T.N. , Carraher, DW. & Schliemann, A. (1985). Mathematics in the Streets and in the Schools. British Journal of Developmental Psychology, 3, 21-29.

Chazan, D. Algebra for all students? National Center for Research on Teacher Learning.

Cobb, P. Wood T. & Yackel, E. (1993). Discourse mathematical thinking and classroom practice.. In Forman, Minick & Stone (Eds.) Contexts for Learning: Sociocultural Aspects of children's development. New York, Oxford University press.

Confrey, Jere. (1993). Learning to See Children’s Mathematics: Crucial Challenges in Constructivist Reform, In Kenneth Tobin (Ed.) The Practice of Constructivism in Science Education. Washington, D.C.: American Association for the Advancement of Science. pp. 299-321.

Confrey, J. (1990). Powers of Ten. In E. Glaserfeld, (Ed.) Radical Constructivism in Mathematics Education. Kluwer Academic Press.

Cuoco, A. & Goldenberg, P. (1992). Reconnecting Geometry: A Role for Technology. Proceedings of Computers in the Geometry Classroom conference. St. Olaf College, Northfield, MN, June 24-27, 1992.

Damarin, S. (1995). Gender and mathematics from a feminist standpoint. In New Directions for Equity in Mathematics Education, Cambridge University Press.

Dubinsky, E. & Leron, U. (1994). An Abstract Algebra Story. American Mathematical Monthly.

Fischbein, E., Deri M. Nello, M. * Marino. M (1985). The role of implicit models in solving problems in multiplication and division. Journal for research in mathematics education, 16, 3-17.

Gay & Cole (1967). The New Mathematics and an old culture: A study of learning among the Kpelle of Liberia. New York: Holt, Rinehart and Winston.

Goldenberg, P. (in press). Habits of Mind as an organizer for the curriculum.Journal of Education. Boston University.

Greeno, J. G. (1991). Number Sense as Situated Knowing in a Conceptual Domain. Journal for Research on Mathematics Education, 22, 170-218.

Hammer, D. (1995). Epistemological considerations in teaching introductory physics. Science Education, 79 (4), 393-413

Hanna G. (1990). Some Pedagogical aspects of Proof. Interchange, 21 91), 6-13.

Harrison, A. G., & Treagust, D. F. (1996). Secondary students' mental

models of atoms and molecules: Implications for teaching chemistry.

Science Education, 80 (5), 509-534.

Hatano, G., & Inagaki, K. (1987). Everyday biology and school biology: how do they interact? The Quarterly Newsletter of Comparative Human Cognition, 9 (4), 120-128.

Hodson, D. (1988). Toward a philosophically more valid science curriculum.

Science Education, 72 (1), 19-40 --??

Hoyles, C. (1991). Computer-based learning environments for Mathematics. In A. Bishop, S. Mellin-Olson, and J. Van Dormolen (Eds.), Mathematical Knowledge: Its growth Through Teaching. Dordrecht: Kluwer, 147-172.

Hoyles, C. (1985). What is the point of group discussion in Mathematics? Educational Studies in Mathematics vol 16.

Kaput, J. (1995). Democratizing access to calculus: New routes to old roots. In A. Schoenfeld (Ed.) Mathematical Problem Solving and Thinking. Hillsdale, NJ.: Lawrence Erlbaum Associates.

Kuhn, D. (1989). Children and adults as intuitive scientists. Psychological Review, 96 (4), 674-689.

 Lampert, M. (1989). Choosing and using mathematical tools in classroom discourse. In J. Brophy (Ed.), Advances in Research on Teaching (pp. 223-264). Greenwich, CT: JAI Press.

Lampert, M. (1990.). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. In American Education Research Journal, spring, vol. 27, no. 1, pp. 29- 63.

Leron, U.: 1983, ‘Structuring Mathematical Proofs’, American Mathematical Monthly, Vol. 90, 3, 174-185.

Leron, U. & Dubinsky, E. (1993). An abstract algebra story.

Minstrell, J. (1989). Teaching science for understanding. In L. Resnick, & L. Klopfer (Ed.), Toward the Thinking Curriculum: Current Cognitive Research

National Council of Teachers of Mathematics. Commission on Standards for School Mathematics. (1989) Curriculum and evaluation standards for school mathematics. (Excerpt)

Noss, R.: 1988, ‘The Computer as a Cultural Influence in Mathematical Learning’, Educationasl Studies in Mathematics, 19, 2.

Noss, R. & Hoyles, C.: 1991, ‘Logo and the Learning of Mathematics: Looking Back and Looking Forward’, Hoyles, C. & Noss, R. (Eds.), Learning Mathematics and Logo, MIT Press, London, pp. 431-468.

Noss, R.: 1994, ‘Structure and Ideology in the Mathematics Curriculum’, For the Learning of Mathematics, 14, 1.

Papert, S. (1972). Teaching Children to be Mathematicians Versus Teaching About Mathematics. Int. J. Math Educ. Sci Technology, vol 3, 249-262

Peterson, P. L., Fennema, E., & Carpenter, T. (1992). Using children's mathematical knowledge. In B. Means, C. Chelemer, & M. S. Knapp (Ed.), Teaching advanced skills to at-risk children. (pp. 68-111). San Francisco: Jossey-Bass.

Schoenfeld, A. (1988). When good teaching leads to bad results: Disasters of well taught mathematics classes. In T. P. Carpenter, & P. L. Peterson (Ed.), Learning mathematics from instruction (pp. 145).

Schoenfeld, A. (1991). On Mathematics as Sense-Making: An Informal Attack on the Unfortunate Divorce of Formal and Informal Mathematics. In Perkins, Segal, & Voss (Eds.) Informal Reasoning and Education.

Schwartz, J. & Yerushalmy, M. (1987). The geometric supposer: an intellectual prosthesis for making conjectures. The College Mathematics Journal, 18 (1): 58-65.

Sfard, A. (1994). Mathematical Practices, Anomalies and Classroom Communication Problems. in Ernest, P. (ed). Constructing Mathematical Knowledge: Epistemology and Mathematics Education. Falmer Press.

Simon, M. (1989). Intuitive understanding in geometry: the third leg. School science and Mathematics, 89, 373-379.

Smith, J.P., diSessa, A.A., & Roschelle, J. (1994). Reconceiving Misconceptions: A Constructivist Analysis of Knowledge in Transition. Journal of the Learning Science, 3, pp. 115-163.

Smith, J. P. (1994). Competent Reasoning with Rational Numbers. in Cognition and Instruction.

Steele, C. (in press). A Burden of Suspicion: How stereotypes shape the intellectual identities and performance of women and african-americans.

Tate, W. (1994). Race, Retrenchment and the Reform of School Mathematics. Phi Delta Kappan v75 n6.

Thurston, William. (1994). On Proof and Progress in Mathematics. Bulletin of the American Mathematical Society. Volume 30, Number 2, April, 1994.

Treisman, P. U. (1988). A Study of the Mathematics Performance of Black Students at the University of California, Berkeley. In N. Fisher, H. Keynes, & P. Wagreich (Ed.), Mathematicians and Education Reform (pp. 33-46). American Mathematical Society.

Treisman, U. (1992). Studying students studying calculus: A look at the lives of minority mathematics students in college. College Mathematics Journal, 23 (5), 362-372.

Turkle S., & Papert, S.. (1991). Epistemological Pluralism and Revaluation of the Concrete. In Idit Harel & Seymour Papert (Eds.) Constructionism. Norwood N.J. Ablex Publishing Corp.

van Hiele, P. (1958). A method of Initiation into Geometry. In H. Freudenthal (Ed.) Methods of Initiation into Geometry.

van Hiele, P. (1976). How can one account for the mental levels of thinking in math class? Educational Studies in Mathematics, 7.

von Glaserfeld, E. (1987). Learning as a Constructive Activity. In Janvier, C. (Ed.) Problems of Representation in the teaching and learning of Mathematics.

Warren, B., & Rosebery, A. S. (1996). "This question is just too, too easy!" Perspectives from the classroom on accountability in science. In

L. Schauble, & R. Glaser (Ed.), Innovations in learning: New environments for education. (pp. 97-126). Mahwah, NJ: Lawrence Erlbaum Associates.

Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism. Norwood, NJ: Ablex Publishing Corporation.

Wilensky, U. (1995). Paradox, Programming and Learning Probability. Journal of Mathematical Behaviour, 14, June.