I was recently asked about calculus instruction: Which is easier, reforming pedagogy or curriculum? The answer is easy: pedagogy. This is not to say that it is easy to change how we teach this course, but it is far easier than trying to change what we teach. The order and emphasis of topics that emerged in the 1950s has proven extremely hard to shift.
Not that we have not tried. During the Calculus Reform movement of the late 1980s and early 1990s, NSF encouraged curricular innovation. Several of these efforts adopted an emphasis on modeling dynamical systems, introducing calculus via differential equations and developing the tools of calculus in service to this vision. This provides wonderful motivation, and this approach survives in a few pockets. It is how we teach calculus at Macalester, and the U.S. Military Academy at West Point has successfully used this route into calculus for over a quarter century. But despite its appeal, this curriculum necessitates modifying the entire year of single variable calculus, raising problems for institutions that must accommodate students who are transferring in or out. It also is a difficult sell to those who worry about “coverage” since it requires devoting considerable time to topics—modeling with differential equations, functions of several variables, and partial differential equations—that receive little or no attention in the traditional course.
This month, I want to talk about a promising curricular innovation that Pat Thompson at Arizona State University has been developing in collaboration with Fabio Milner and Mark Ashbrook, Project DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus). It has the advantage that it fits more easily into what is expected from each semester. It has been under development since 2010 and is slated to be the curriculum used for all Calculus I sections for mathematics or science majors at ASU beginning in fall, 2018.
Thompson began with research into the misconceptions that students carry into calculus and that impede their ability to understand it. I quote these common misconceptions from his website (patthompson.net/ThompsonCalc/About.html).
- Calculus, like the school mathematics, is about rules and procedures. Students think that calculus is difficult primarily because there are so many rules and procedures.
- Variables do not vary. Therefore rate of change is not about change.
- Integrals are areas under a curve. Students wonder, "How can an area represent a distance or an amount of work?"
- Average rate of change has little to do with rate of change. It is about the direction of a line that passes through two points on a graph.
- A tangent is a line that "just touches" a curve.
- Derivative is a slope of a tangent. The net result is that, in students' understandings, derivatives are not about rates of change.
f(x) = x3 – 3x + 2,
many students see the expression f(x) as nothing more than a lengthy way of writing the dependent variable, and functions are seen as static objects that prescribe how to turn the input x into the output f(x). With this mindset, differentiation and integration are nothing more than arcane rules for turning one static object into another.
Choosing to define integrals as areas and derivatives as slopes, as is common in the standard curriculum, is equally problematic. It reinforces the notion that calculus is about computing values associated with geometric objects. To complicate matters, while area is a familiar concept, slope is far less real or meaningful to our students. Too many students never come to the realization that the real power of differentiation and integration arises from their interpretation as rate of change and as accumulation.
While the earliest uses of accumulation were for determining areas, those Hellenistic philosophers who mastered it also recognized its equal applicability to questions of volumes and moments. By the 14th century, European philosophers were applying techniques of accumulation to the problem of determining distance from knowledge of instantaneous velocity. None of these come easily to students who are fixated on integrals as areas.
Seeing the derivative as a slope is even more problematic, a static value of an obscure parameter. Differentiation arose from problems of interpolation for the purpose of approximating values of trigonometric functions in first millenium India, in understanding the sensitivity of one variable to changes in another in the work of Fermat and Descartes, and in relating rates of change as in Napier’s analysis of the logarithm and Newton’s Principia. Derivative as slope came quite late in the historical development of calculus precisely because its application to interesting questions is not intuitive.
These insights provide the starting point for Thompson’s reformation of Calculus I. His textbook, which is still a work in progress, can be accessed at patthompson.net/ThompsonCalc. See Thompson & Silverman (2008), Thompson et al. (2013), and Thompson & Dreyfus (2016) for additional background. I find it deliciously ironic that one of the first topics he tackles is the distinction among constants, parameters, and variables. If you look at the calculus textbooks of the late 18 th century through the middle of the 19 th century, this is exactly where they started. Somehow, we lost recognition of the importance of elevating this distinction for our students. Thompson goes on to spend considerable effort to clarify the role of a function as a bridge between two co-varying quantities. And then, he really breaks with tradition by first tackling integration, which he enters via problems in accumulation.
This accomplishes several desiderata. First, it ensures that students do not begin with an understanding of the integral as area, but as an accumulator. Second, it makes it much easier to recognize this accumulator as a function in its own right. Students struggle with recognizing the definite integral from a to the variable x as a function of x (see the section of last month’s column, Conceptual Understanding, that addresses Integration as Accumulation). Thompson begins by viewing the integrand as a rate of change function. The variable upper limit arises naturally. Third, and perhaps most important, it gives meaning to the Fundamental Theorem of Integral Calculus, that the derivative of an accumulator function is the rate of change function.
Differentiation can then be introduced in precisely the way Newton first understood it: Given a closed expression for the accumulator function, how can we find the corresponding rate of change function?
Next month, I will expound on exactly how Thompson introduces these steps, but for now I would like to conclude with a comparison of the two curricular innovations, that of Thompson and the approach described at the start of this column that emphasizes calculus as a tool for modeling dynamical systems. The latter does overcome the problem of student belief that the derivative is to be understood as the slope of the tangent. It brings to the fore the derivative as describing a rate of change. The problem is that it does nothing to clarify the role of the integral as an accumulator. In some sense, it makes it more difficult. As we teach calculus at Macalester, the integral is introduced purely as an anti-derivative, making it extremely difficult to give meaning to the Fundamental Theorem of Integral Calculus. I have to work very hard in the second semester to help students understand the integral as an accumulator and so justify that this theorem has meaning. In a very real sense, Thompson approach begins with this fundamental theorem.
Thompson, P.W. and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (MAA Notes Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America.
Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools. 30:124–147.
Thompson, P.W. and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientitific Discipline (pp. 355–359 ) Hannover, Germany: KHDM.
