Notes and References

1. "The most visible force for change in the mathematics curriculum is the computer, a mathematics-speaking device that has totally transformed science and society." Steen, L. A., quoted in Ferrini-Mundy, J. and Graham, K. G., "An Overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and Curriculum Development." American Mathematical Monthly, 1991 98:7, 627-635.

   In 1943 the term "computer" was a job description, often applied to women hired by the U.S. military to help compute tables for artillery trajectories. For notes on the ACM video documentary that describes the shift of algorithmic processing from humans to machines, see http://ei.cs.vt.edu/~history/UNIVAC.Weston.html.
   Also, Palfreman, Jon, and Swade, Doron. The Dream Machine: Exploring the Computer Age, BBC Books, London, 1991, 208 pp.

2. See for example, the recent proceedings of the Conference on Electronic Communication in Mathematics, on-line at http://www.geom.umn.edu:80/docs/cecm/talks.html.

3. Wolfram Research is located at 100 Trade Center Drive, Champaign, Il. 61820. On the World-Wide Web, at http://www.wolfram.com. The Mathematica Book, 3rd Edition, is published by Cambridge University Press. For an excellent written introduction to Mathematica, see the review by Stan Wagon in The Mathematical Intelligencer, Vol. 13, No. 3, Summer 1997, pp 59-57.

   For technical details on Mathematica Version 3, including a history of the front end by its creator Theodore W. Gray, see Mathematica in Education and Research, special 3.0 issue, Vol. 5, No. 3.

4. The term "mathematics educator" is used here for anyone who teaches a mathematics class. Understandably, some mathematicians might not want to be called "educators," given attitudes such as those expressed in a recent editorial in the Notices of the American Mathematical Society. [Krantz, S., "Math for Sale". Notices of the AMS Vol. 42, No. 10,  1116.]

   Excerpt: "I know people, and probably you do too, who have made abrupt changes in their professional activities because they could not get funding from traditional sources. In one case a person who was formerly the world's greatest expert in a fairly narrowly defined branch of analysis suddenly became the world's greatest expert on teaching calculus using Mathematica."

   For examples of responses to Krantz, see Notices of the American Mathematical Society: Vol 43, No. 1,  1; No. 3,  285; No. 4, 406, and continuing.

5. Steen, L. A., ed., Calculus for a New Century: A Pump, Not a Filter, MAA Notes No. 8, 1987.

6. For a review and expansion of these themes see Ross, S. C. "Visions of Calculus," in Roberts, A. W. (Ed), Calculus. The Dynamics of Change. MAA Notes No. 39.

7. From the Calculus&Mathematica World-Wide Web site. Calculus&Mathematica has been professionally evaluated by Dr. Kyunmee Park and Dr. Kenneth Travers. Their study compares Illinois C&M students and Illinois students from the normal "book sections." Some of their findings:

   Generally the findings from an achievement test, concept maps, and interviews were all favorable to C&M students. The C&M group obtained a higher level of conceptual understanding than did the standard group without much loss of [hand-] computational proficiency. Furthermore, the C&M group's disposition toward mathemetics and the computer was far more positive than that of the standard group. . . .
   
   Generally, the C&M group seemed to more clearly understand the nature of the derivative and the integral than did the standard group. . . . A positive side effect of the [computer] lab was the rapport that was established among the students. When students gathered around the computer, worked together, and shared and developed ideas, a great deal of mathematics was learned.
   
   ...[Computer] capabilities helped students discover and test mathematical results in much the same way that a physics or chemistry student uses the laboratory to discover and test scientific laws. Those capablities provided the opportunities for the students to consider more open-ended questions and to encouter more realistic problems than often found in traditional calculus texts.

8. See "First In Computer-based Classes" (http://math.missouri.edu/~news/issue2/front2.html), and additional descriptions linked to that site: "By eliminating the mechanical and repetitive work from calculus, there is more time to devote to why things happen," Chair Elias Saab says. "When students see what is happening in calculus, instead of just imagining what is happening, they can develop a better sense of intuition and foresight. This has been especially helpful to physics, chemistry, and engineering students."

9. "Nothing which is taught at a secondary school was discovered later than the year 1800...Even among physicists, apart from those who work in quantum theory or relativity, I believe that those who do experimental work use hardly any more mathematics than was known to Masxwell in 1860." Dieudonné, J. Mathematics--The Music of Reason. New York: Springer-Verlag 1992, page 1.

10. For a concise description, see pages 56 and 57 in Courant, R. (1957). Differential and Integral Calculus. New York: Interscience. From page 57:
    "In spite of all its defects intuition still remains the most important driving force for mathematical discovery, and intuition alone can bridge the gap between theory and application."

11. Douglas, R. G., (Ed)., Toward a Lean and Lively Calculus, MAA Notes No. 6. 1986.

12. See, for example, Borrelli, R. L, and Coleman, C. S., "New Directions for the Introductory Differential Equations Course." Also Carlson, D. and Roberts, W., "Changing Calculus: Its Impact on the Post-calculus Curriculum." Both references in Roberts, A. W. (Ed), Calculus. The Dynamics of Change. MAA Notes No. 39.

13. MathSource (http://www.mathsource.com)is a vast electronic library of Mathematica material, with over 100,000 pages of immediately accessible Mathematica programs, documents, examples, and more. You can either browse the archive or search by author, title, keyword, or item number.

14. Devlin, K., Notices of the American Mathematical Society, March 1991, 38, 3.

15. BYU Outcomes Study. For additional information, contact
    Gerald M. Armstrong
    Chair, Department of Mathematics
    gma@math.byu.edu

16. VanLehn, K., Mind Bugs : the Origins of Procedural Misconceptions.  Cambridge, Mass. : MIT Press 1990.

17. Cipra, B., Misteaks...and how to find them before the teacher does.... San Diego: Academic Press 1989.

18. Contributed by Terry Moore, Statistics Department, Massey University, New Zealand. Reply to article <5pcgg6$hko@crcnis3.unl.edu> USENET group sci.math.

19. Gordon, S. P., A Round-Table Discussion with the Client Disciplines. In  Roberts, A. W. (Ed), Calculus. The Dynamics of Change. MAA Notes No. 39.

    This discussion is just a starting point for a much deeper analysis of what students really need to know. For example, here is an excerpt from an electronic round-table discussion hosted by the AMS.
    [http://www.stolaf.edu/other/extend/Expectations/society.html]

    "Robert Meyer: 'We need hard data to see whether the skills cited in reports like SCANS and NCTM really matter in the world of work. It is not enough to just ask people what mathematical skills they use, or what they think their employees need. One should, instead, observe people at work and do research on this issue. There is some legitimate disagreement about which skills really matter.
    
    I think the ethos in education of testing beliefs about what skills really matter is not as strong as it should be. Education as a field is not disciplined sufficiently by the notion of validity. We need to conduct studies to see if what we believe to be valuable really matters in the lives of our students'."
    
    Robert H. Meyer is an assistant professor in the Irving B. Harris Graduate School of Public Policy Studies, University of Chicago.

20. Thomas, G. B., Calculus. Addison Wesley. Thomas' text, in its numerous reprintings, is often used as the canonical reference for college calculus exposition, 1953.

21. Abstracted from Resnick, L. B., Education and Learning to Think. National Research Council, 1998. See also, Perspectives on socially shared cognition, edited by Lauren B. Resnick, John M. Levine, and Stephanie D. Teasley. Washington, DC : American Psychological Association, c1991.

22. Schoenfeld, A. H., Mathematical Problem Solving. San Diego: Academic Press, 1985.

23. Asiala, M., Brown, A., DeVries, D. J., Dubinski, E., Mathews, D., and Thomas, K., A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II. American Mathematical Society (1996), 1--32.

24. Lakoff, G., "Embodied Minds and Meanings" in  Baumgartner, P. and Payr S. eds, Speaking Minds: Interviews with Twenty Eminent Cognitive Scientists. Princeton, 1995.

25. Penrose, R., "Mathematical Intelligence." in Khalfa, J. ed., What is Intelligence? Oxford, 1994.

26. Eisenberg, T. and Dreyfus, T., "On the Reluctance to Visualize in Mathematics." in Zimmerman, W. and Cunningham, S., Visualization in Teaching and Learning Mathematics. MAA Press.

27. Dehaene, S., The Number Sense.  Oxford Press. 1997. See also Changeux, Jean-Pierre and Dehaene, Stanislas, "Neuronal models of cognitive functions," Cognition, 33, 1--2, Nov. 1989, pp 63--109

28. Dunbar, S. and Fowler, D., Cinematic thinking and Mathematica notebooks. Mathematica in Education 1 (2) 1992, 3--8.

29. Ulam, S. M., A Collection of Mathematical Problems. Interscience, 1960.