"New" and "Old" Calculus:
Student Reactions and Comments

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that would never occur in real life, with repeated reminders of how to use the chain rule from Calculus I. In the other class, almost the entire period was devoted to the students, who repeatedly and pointedly asked that I explain to them what f_xy tells them about the shape of a surface; the significance of f_xx and f_yy were clear to them, but what does the mixed partial mean in terms of the behavior of the function? While not an especially important mathematical point, it was one that these students were not willing to pass on until they fully understood it!

I also had assigned routine differentiation problems similar to the ones above to reinforce the mechanics and had tested my students on them, but my emphasis in the reform course is on much more than merely performing the routine mechanics-and that is reflected in what my students come to find important.

In turn, I have come to expect incredibly insightful and mathematically sophisticated questions and written comments from these students. For example, prior to the first class test in Calculus I, based on Chapters 1 and 2 of the CCH text, one of my students asked, "Let me see if I have this straight:" Then, in a single breath at an incredibly rapid speed, he said, "A function is concave up if it is increasing at an increasing rate or decreasing at a decreasing rate and it is concave down if it is increasing at a decreasing rate or decreasing at an increasing rate. Right?" Sure! But, if this were something he had memorized, he would not be asking the question; instead, he was clearly verbalizing a solid graphical image.

About a month into Calculus I, I briefly introduced the notion of an implicit function, commented that in general it is extremely difficult to graph such functions, introduced the idea of implicit differentiation, gave two dumb examples, and then a third example of finding the equation of the tangent line to an implicitly defined curve. A student immediately asked, "Couldn't you do the same thing at many different points, draw the tangent lines, and trace the curve as it touches each of the tangent lines?" Yes, we certainly could do that! The only question is whether the idea should be attributed originally to Euler or Poincaré?

On the first test based on the same two chapters, where students had seen families of functions and the concept of the derivative without any formal differentiation formulas, I included a problem giving the graph of the derivative of a function which is positive across almost all of an interval, but which dips below the x-axis for a relatively short time. The students were asked to indicate the points where the unseen function achieves its maximum and minimum. I expected that they would reason as follows: The derivative is mostly positive, so the function is mostly increasing, and therefore it has its minimum at the left endpoint and its maximum at the right. Of the 28 students in the class, nine produced that line of reasoning for a problem they had never seen before. Of the remainder, 14 came up independently with the idea of using the graph of the derivative to sketch a graph of the actual function (reversing the process of graphical differentiation that they had seen). 13 of them actually drew reasonable sketches for the function and used it appropriately to answer the question. More significantly, under the pressure of an exam, these students created the concept of the antiderivative, a notion which had not previously been mentioned in class!

As part of each of my courses, I assign the students a series of individualized projects often based on their social security numbers. Each student is required to submit a formal written report describing the results of his/her investigations on each project. In Calculus I, one of the projects I typically ask them to perform is a complex max-min analysis of their social security polynomial, which is produced by using the digits of their social security number as the coefficients of an eighth degree polynomial. Although the results are usually surprisingly good, some are really incredible. For instance, in one recent report, I found the following gem: "Although it cannot be seen on a graph, even by zooming in, there is a change in concavity between the local maxima and minima. I decided to create a visual aid by using the third derivative. This local maximum of F'' corresponds to the lowest point of the concave up segment on the graph..." Ideally, when teaching any course, we would like our students to be able to take the ideas they have learned and adapt them to other situations. Having a Calculus I student develop a third derivative test for possible points of inflection and apply it intelligently certainly fits that goal.

Over the last few years, I have come to expect a high level of intellectual performance of my students in reform courses at all levels ranging from college algebra up through Calculus III. Perhaps the most telling indictment of the traditional calculus course is the following comment from a calculus student, overheard during a visit to another major university that was in transition to reform calculus: "The old calculus was much easier. You didn't have to understand what you were doing to get the right answers."

Editor's Note
This article has been adapted from portions of a more extensive article published in Primus. The original article contains additional examples of student questions and comments in reform courses.