Richard Askey, Professor of Math at the University of Wisconsin, Madison, has given me permission to share his article, "What do we do about calculus? First, do no harm," with this group.
David Klein Math Dept. Calif. State University, Northridge ____________________ What do we do about calculus? First, do no harm.
In the spring of 1994, the Dean of our Engineering School paid the expenses of four speakers to tell the Mathematics Department how to teach calculus in a modern way. To him, a modern way was intensive use of computers. The real goal was to have us teach the same amount of calculus but with fewer credits, so that more of the time of engineering students could be spent taking engineering courses. A joint committee was set up to look at what has been taught and what needed to be taught. The conclusion was that with heavy use of computers it would take more time to teach the same material rather than less, so nothing came of the push to cut the number of credits. I asked the first speaker about proofs. He replied that calculus was not the place to do proofs. Proofs should start in the junior year of college, primarily for students who are mathematics majors. There was an education meeting held at the University of Chicago. A Japanese education official said, "About half of the ninth-graders could express quantitative relations using letters (variables) and could write geometrical proofs" . I asked if ninth grade Japanese students could learn how to do proofs, why couldn't our calculus students also do this? While we have a higher percentage of students taking calculus now than we did forty five years ago when I was a young college student, we do not begin to have 50% of the age cohort taking calculus. Saying that we have so many more students taking calculus that we cannot possibly expect them to be able to do this is looking at the wrong comparison group. In addition to proofs in geometry, there are other proofs in Japanese middle school books, such as a proof that the square root of 2 is irrational. See . The second speaker talked about differential equations, and began with this equation:
x' = x^2 - t
with an initial condition. Once this was put up on an overhead, I worked out the solution. The speaker said this was a differential equation that could not be solved [exactly], and I let this go by without saying anything. After talking on other topics, the speaker came back to this, her favorite equation, and put up an overhead, which seemed to show a pole. She said that a pole was there and that a colleague had shown this. This equation is just a Riccati equation, so can be linearized, and the linear equation solved. In the present case, the linear equation is the Airy equation, so a solution is easy to find. Riccati equations are important in control theory, and there were electrical engineers in the room, so I did not want them to think that the mathematicians did not know what was happening. I asked if they then solved the equation to explain where the pole comes from. She repeated what she had said earlier about a colleague having proven that the equation could not be solved. I said that it depends on what is meant by solved. In the present case, the solution is easily found in terms of Airy functions, which are Bessel functions in a slight disguise. The pole that was claimed to exist comes from the smallest zero in the denominator. There is a tendency to downplay the role of infinite series in calculus and in differential equations. The usual argument for differential equations is that it is hard or impossible to see the long range behavior from a power series. In the present case, it is the smallest zero that is in question. The other argument given against power series solutions of differential equations is that few differential equations have solutions that can be written in the form of a nice series. It is one of those miracles of nature that many very important problems lead to just those differential equations that can be solved in series with nice coefficients. These series have the property that the term ratio of the coefficients is a rational function of n, and are called hypergeometric series. A course in calculus is not the place to study hypergeometric series in detail, but the most important one, the binomial theorem, should be there. The ratio test for convergence is as popular with students as it is because it is easy to compute the limit of a term ratio that is a rational function of n, and many of the elementary functions studied in calculus have power series of this type. Students should start to be led in the direction of seeing that this class of functions is important. The third speaker was someone I have known for years, so I asked some questions in an e-mail before he arrived. One was about differentiating x^n. This can be done in several different ways. The traditional one in our texts was to quote the binomial theorem to get started. This used to be a standard topic in algebra. One of the new calculus books does it this way, and refers the reader to any high school algebra book for a proof of the finite binomial theorem. I called and asked one of the authors if he had looked at any high school algebra or precalculus books recently. He said no. He should, for the binomial theorem is no longer the staple it once was. For example, the precalculus book written by the faculty of The North Carolina School of Science and Mathematics  does not have either the binomial theorem or the geometric series. In response to my question to readers of the e-mail discussion group calc-reform, someone replied that most of his students had taken calculus in high school. If there is anything students remember from high school, it is the formula for the derivative of x^n, so he does not give a derivation. Many people who are supporting the current reform efforts do not like formulas, and so do not want to use the binomial theorem to differentiate x^n. One solution to this problem is to use another formula. Instead of writing (x+h)^n - x^n -------------- h it is possible to write y^n - x^n ----------- y - x or even (qx)^n - x^n q^n-1 -------------- = x^(n-1)* ----- qx - x q-1 and use the sum of a finite geometric series. This arguemnt also works when n is rational. In the course of changing variables to see this, you give an introduction to the chain rule and to the simple form of l'Hospital's rule. However, some of those who do not like formulas even object to the formula for the sum of a geometric series, so there is a way to differentiate x^n without using any formula. Just observe that (x+h)^n=(x+h)(x+h)...(x+h) and observe that x^n appears once. The next term, h*x^(n-1), appears once for each factor, so n times. Every other term in the expansion has at least two factors of h. This way the student can understand why (x+h)^n = x^n + n*h*x^(n-1) + terms that involve h^2 or higher powers of h. Contrast this with the treatment in . The formula for the expansion of (x+h)^n is stated when n=2,3,4,5. Then the authors write "we can say that (x+h)^n = x^n + n x^(n-1)*h + terms involving h^2 and higher powers of h". There is a big difference between "we can say that" and "we see why". Mathematics should be an open subject, where students do not take such simple facts because "we can say that" or because the computer algebra system gives such a formula. Another of my concerns can be illustrated by a problem in the same book, but other books could have been used equally well. This deals with when something has been shown to be true. Consider problem 48 on page 365 of . This has three parts. The first is to use Riemann sums to evalute the integral from 1 to 2 of ln x . The second is to evaluate this integral using anti-derivatives. The same integral had been done in the text, but from 2 to 3. Both of these parts are fine, except it would have been better not to use "evaluate" in the first part, but "approximate", and it would have been better for the students to have been asked to do an integral that had not been done in the text in the second part. However, it is the third part that bothers me. The students are asked to "Explain in words why your answers verify the Fundamental Theorem of Calculus". This has not "verified" the Fundamental Theorem of Calculus, but has illustrated that the approximation in the first part gives an approximation to the exact value obtained in the second. The first definition of "verify" in the dictionary at my desk is: "To prove to be true". Words mean things and they are important. Meanings should not be changed without very good reasons. The fourth speaker tried to show us how a computer algebra system could be used in a lecture setting. One of his main examples was Simpson's rule. He set up the problem, got three linear equations in three variables, and said that these were far too complicated to solve at the board. He then displayed the solution via a computer algebra system. I asked him why he felt it was necessary to do the interpolation at the points a, a+h, and a+2h, when you could do the interpolation at -h, 0, and h, or even -1, 0, and 1. Then the equations fall apart and it is very easy to do the algebra by hand. There is an important mathematical lesson taught when doing this: you can adapt the coordinate system to the problem at hand. That lesson needs to be learned whether you do calculations by hand or by computer algebra.
In talks on mathematics education, I frequently start with four guidelines that should be considered when teaching, writing a book, or developing a curriculum:
* Do not lie to your students but don't tell them the full truth. * Some results in mathematics are more important than others and this should be reflected in texts and in class. * Mathematics is not a secret guild where something is true because I say it is or because a computer algebra system says it is. When something simple and important is studied, reasons should be given. * Words are important and their meanings should not be changed without very good reasons.
Examples where these have not been followed have already been given. There are many more in newer books. Since first mentioning these, I have decided to add one, which is very important for textbook writers and curriculum developers to observe.
* Be careful that what you are doing does not lead others to make changes that will hurt the long-term education of students.
In his book  on textbooks written for the TIMSS study, Geoffrey Howson makes the following point: "The passing of the 1960s emphasis on algebraic structure need not be regretted. What is sad is that it has not been replaced by some other clear philosophical or pedagogical structure more appropriate to school mathematics." He ends this paragraph with "A first attempt to establish such a framework of `recurring themes' has been made by Gardiner . It is an idea which deserves further consideration, development, and elaboration." In the absence of such guidelines, textbook writers, curriculum developers and test writers will look at the current curriculum and try to provide material that will get students ready for later courses. Thus, one frequently overlooked point is how changes being made for one reason will impact in other ways. The newer calculus books tend to be more qualitative, and this is starting to show up on the AP Calculus exams. For various reasons, which will not be listed here, the knowledge of arithmetic and algebra that students starting calculus have has fallen. As a response to this poorer knowledge of algebra, the Harvard Consortium has tried to finesse the problem by emphasizing the use of graphing calculators. Other reasons are given for this, but a quotation from Tony Phillips suggests that this was a major factor. After saying that students' manipulative skills have become much weaker, Phillips continued with: "And the HCC curriculum makes a great virtue out of this necessity. By eliminating some of the symbolic manipulation from calculus, they were able to make the course more accessible to students." This was written in a newsletter from the Harvard Calculus Consortium. The report from a committee looking at the future of the AP Calculus exam reads like a description of the Harvard Calculus book. This is a very poor idea. Let me explain why with an analogy. When my son was in high school, the precalculus book they used was . This was a very nice book, and he learned a lot from it. We could not use it for the corresponding course at the University of Wisconsin. Our course went about twice as fast, and the students who took it were in general not as good at mathematics as those who took this in high school. Many of the students who will use mathematics seriously are now taking calculus in high school. They need to develop technical skills beyond those of students who take calculus for the qualitative ideas there. The past AP Calculus exams were reasonable exams. My finals in first and second semester calculus were in general a bit harder than the AB and BC exams. A few years ago, there were a couple of students in calculus lectures who wanted to transfer to California Institute of Technology. Exams were given to these students, in chemistry, mathematics, and physics. The mathematics exams over a three year period were sent to me, to share with the students or for them to take. They were a bit harder than the final exam in my second semester calculus course, as is appropriate. They should be at the level of exams for an honor section. The MAA has published translations of university entrance exams for some Japanese universities . The one for students who want to study humanities at Tokyo University is harder than the sophomore placement exam for Cal. Tech. Thus, the level of the traditional AB and BC advanced placement exams was not too high. It is likely that the newer one will continue at an equally high level for a couple of years, but from then on it needs to be watched carefully. There are experienced high school teachers who feel the earlier exams were harder than those given in the last few years, and with a more qualitative exam it will be very easy to have the level slip as students who take the exam have less technical skill. The message from the Calculus Reform programs that is being heard is that students do not need to be able to do algebra well. The message from the NCTM Reform is that students do not need to know how to do arithmetical calculations well. Both of these messages are different from those sent by the countries that did best on TIMSS. See  and . Technology has a place in mathematics instruction, but it needs to be used carefully. Until much more is learned about the drawbacks, it should not be pushed too much. In England, the results on the age 11 exams in the summer of 1995 were so poor that six months later a decision was made that calculators were no longer to be used in one of the two maths exams. A teacher in Milwaukee talked at the Wisconsin Mathematics Council meeting in May 1995. He said that he was probably the first one to use calculators in high school in the Milwaukee area, and also the first to use graphing calculators. However, he now has some serious doubts about the wisdom of using them as much as he had. First, students do not know enough yet to profit from their use to the same extent that teachers could, with their greater knowledge. Second, he looked at books from 30 years before and found that many topics now in second year algebra were in first year algebra then, topics from second year algebra then are frequently done in precalculus now, and quite a few things that were once done in precalculus are not done in high school now. He did not mentioned specific things, but conic sections comes to mind as something that is frequently not done now, either in precalculus or in calculus. The binomial theorem is another. Both belong somewhere, and high school seems the right place. Unfortunately, NCTM put conic sections down for decreased emphasis. Other countries do them and many other things that we do not do. We need to look seriously at what is being done in the rest of the world, to see what our students could learn if they had a good mathematics program. Then we need to develop one. The most important part of this is not calculus, but elementary school. However, what we do in calculus has an impact on the rest of our mathematics program, so we need to be very careful about what we do and how we talk about what we are doing. The medical advice of "Do no harm" is good advice for us as well.
 B. B. Barrett et al, Contemporary Precalculus Through Applications: Functions, data analysis and matrices, Janson, Dedham, MA, 1992.  Albert Beaton et al, Mathematics Achievement in the Middle School Years, TIMSS International Study Center, Boston College, Chestnut Hill, 1996.  Robert Fisher and Allen Ziebur, Integrated Algebra and Trigonometry With Analytic Geometry, Second Edition, Prentice-Hall, Englewood Cliffs, N.J., 1967.  Anthony Gardiner, Recurring Themes in School Mathematics, Birmingham Univ., UK Mathematics Foundation, Birmingham, 1992.  Geoffrey Howson, Mathematics Textbooks: A Comparative Study of Grade 8 Texts, TIMSS Monograph No. 3, Pacific Educational Press, Univ. of British Columbia, Vancouver, 1995.  Deborah Hughes-Hallett, Andrew M. Gleason, et al, Calculus, Wiles, New York, 1994.  Kunihiko Kodaira, editor, Japanese Grade 9 Mathematics, 1984, translation published by Univ. of Chicago School Mathematics Project, Chicago, 1992.  Tatsuro Miwa, Mathematics in Junior and Senior High School in Japan: Present State and Prospects, in , 172-190.  William Schmidt, Curtis McKnight and Senta Raizen, A Splintered Vision: An Investigation of U.S. Science and Mathematics Education, Kluwer, Dordrecht, Boston, London, 1997.  Izaak Wirszup and Robert Streit, Developments in School Mathematics Education Around the World, Proc. of UXSMP International Conference on Mathematics Education, Univ. Chicago, March 1985, NCTM, Reston 1987.  Ling-Erl Eileen T. Wu, Japanese University Entrance Examination Papers in Mathematics, Mathematical Association of America, Washington, D.C., 1993.
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