Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
"There are many conversations, committee meetings, etc., today about the modernization of the undergraduate calculus course; but all too often the attack on the problem falls short of being comprehensive." So begins a 1958 article Bringing Calculus Up-to-date (AMM, 65 (1958), pp. 81-90) by M. E. Munroe. Too young at the time to be interested in Calculus, I first ran into the article while browsing the first volume of the collection A Century of Calculus. It's a remarkable two-volume set a small library in fact. A library of 310 papers selected from the American Mathematical Monthly, Mathematics Magazine and College Mathematics Journal.
40 years later, there is still no end to conversations and committee meetings, etc., but there appears to be a comprehensive effort under way to modernize instruction in undergraduate mathematics. In the November issue of the FOCUS newsletter, MAA has announced a two-year $900,000 grant from the NSF to construct the Mathematical Sciences Digital Library (MATHDL). MATHDL will comprise three related web-based components. The first part will be a new MAA publication - The Journal of Online Mathematics and its Applications (JOMA). JOMA is an official abbreviation. The journal will be available at www.joma.org. The initial focus is, of course, on Calculus.
In August, six mathematics professors went through 600 calculus mathlets small learning units found and tested by students. From them, they selected 18 for JOMA and another 50 for LOMA, where "L" stands for "Library." LOMA the second component of MATHDL will contain a select collection of online learning materials,including mathlets, that have been classroom tested and peer reviewed. The third part will review and list commercially available learning materials. Gene Klotz - a founder and director of Math Forum and a co-PI for the current grant - has suggested the latter be dubbed COMA to complete an easily memorizable triple of acronyms.
The articles for A Century of Calculus were selected through a similar process. For the first volume, the editorial committee read 750 papers. Abstracts were prepared, and each paper received a vote Yes, No or Undecided. Each paper was read by at least two committee members. Those papers that received at least one vote of Yes or Undecided were given a third, and in some cases a fourth and fifth reading until some agreement was reached whether to include the paper in the collection or not. For the second volume the editorial committee reviewed more than 500 articles.
The first volume was published in 1969 and then reprinted, along with the second volume, in 1992. As expected of items in a book set, the two 1992 volumes look exactly the same on the outside: the same cover, the same design, the same paper, they even have about the same number of pages. The two volumes also have very similar chapter structures. In both, the quality of articles and their relevance are excellent. However they read differently. If I were an grouchy old man, I would probably grumble about how, compared to the good old days, things have changed to the worse. On the inside, the two volumes differ remarkably. Using the modern terminology, I would say that the second volume suffers from a relatively poor user interface. And here is perhaps a lesson to learn for the JOMA project.
As a matter of fact, I've been skimming A Century of Calculus looking for some good ideas for JOMA. Munroe's quote in the opening paragraph came from the first volume (p. 70). The first volume was easy to browse and easy to relate to without making long stops at any one article. The following are a few items that caught my attention and thoughts that registered in my mind.
On page 48, R. G. Helsel and T. Radó pondered (MAA 55 1948, pp. 28-29) the question Can We Teach Good Mathematics To Undergraduates? They cited three ingredients of good mathematics: relevancy (calculus, for example, is overflowing with relevancy), rigor (all of the reasons must be given), and elegance (not to deprive the students of the very thing that affords us our greatest pleasure).
Page 111. Relating to Finding Derivatives of Trigonometrical
Functions by T. H. Hilderbrandt (AMM 25 (1918), pp. 125-126). Either
expanded by the formula for the sine of the sum of two angles and the
Page 114. Trigonometry From Differential Equations by
D. E. Richmond (AMM 61 (1954), pp. 337-340). This note shows how analytic
trigonometry may be developed in an elementary manner (with no use of
infinite series) from the differential equation:
Page 124. A Classroom Proof of lim sin(t)/t =
1 by S. Hoffman (AMM 67 (1960), pp. 671-672). The latter part of this
string of inequalities
Page 126. Derivative of sin() and cos() by C. S. Ogilvy (AMM 67 (1960), p. 673). The derivatives of the sine and cosine functions may be presented with the aid of vectors.
Page 153. A Peculiar Function by J. P. Ballantine (AMM 37 (1930), p. 250). A pie reposes on a plate of radius R. A piece of central angle is cut and put on a separate plate of radius r. How large must r be? Obviously r is a function of . It turns out to have the following formula:
The function has one discontinuity at = 0, and its second derivative has various discontinuities.
Page 251. Geometric Interpretations of Polynomial Approximations of the Cosine Function by E. R. Heineman (AMM 73 (1966), pp. 648-649). The polynomial approximations obtained from the power series expansion
converge to the cosine function... In one sense, it is clear that the
successive approximations become "better." The student, however, must be
warned against the pseudo-truth: "Every time you add another term, the
approximation gets better." The horizontal line
Page 287. On The Harmonic Mean of Two Numbers by G. Polya (MAA
57 (1950), pp. 26-28). x is in the interval
Problem I. Find
and the value of p for which it is attained.
Problem II. Find
and the value of p for which it is attained.
For Problem I, p = (a + b)/2. For Problem II, the solution is a little less obvious and may be new, but is certainly very little known: p = 2ab/(a + b).
This was my take on browsing the first volume of A Century of Mathematics. I also enjoyed just handling the book. Not so with the second volume. First, all articles in the first volume came with a footnote reference to their source, year of publication in particular. It piqued my interest to check the year just to confirm the obvious: the search for better ways is naturally an ongoing process. I expected the same quiet entertainment with the second volume. To my disappointment, the articles in that volume did not carry the footnote which I had come to expect. (To be fair, I later discovered that all sources have been listed at the end of the book. But having to peek there all the time was a great annoyance.)
Second, in the second volume the fonts changed from article to article. No, not very often, but often enough to cause a distraction. Third, in the second volume, the articles differ in style. Some are preceded by the author's picture and an introductory paragraph, some carry the author's affiliation seal, some come unadorned. I am aware of the reason why that is so the articles were photo reproduced from different journals but this did not help. Casual leafing through the volume was rather tiresome. And this is to be regretted, as the articles themselves are quite excellent.
In Testing Understanding and Understanding Testing (CMJ 16, pp. 178-185) J. Pederson and P. Ross offer a sample of problems that require for their solution an understanding of calculus concepts rather than formula memorization or mechanical symbol manipulation. The following applet, for example, helps observe the relations between a function and its derivative and integral with not a single formula involved.
(About the applet: Points on the graph below are draggable up and down. So that you can modify the function any other way you want. Simultaneously with the function, you may display its derivative and the integral as a function of its upper limit. If you drag the cursor away from the graph while the "Show tangent" box is checked, a short line will be displayed tangent to either graph of the function or the graph of the integral.)
Back to the MATHDL. To be useful, any software a stand-alone package or a library must, like a book, be coherent, reasonably comprehensive and consistent in its user interface. In this respect, learning materials in COMA will have a great advantage over freely available pieces from LOMA and even JOMA. The reviewing effort must be supplemented with purposeful development for the sake of consistency.
Fortunately, the JOMA proposal was written with proactive attitude in mind. Each topical area, like Calculus, will have a Table of Contents, and solicitations for missing pieces will be actively promoted through JOMA, now an online publication of the MAA. It will be hosted by the Math Forum. With Forum's experience at community building and publicity that comes from association with the MAA, the project has a very good chance for success.
Most certainly the many conversations, committee meetings, etc., devoted to the modernization of the undergraduate calculus course will not end with the planned inauguration of JOMA at the Joint Meeting in January. It would be to everyone's benefit, howerver, were JOMA to become the focus of many of those conversations and meetings.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1996-2000 Alexander Bogomolny