The raison d'etre of calculus is differential equations. (One might counter that optimization is a good reason to study differential calculus. But max/min is now a button on every calculator, so optimization problems no longer justify a semester course.) Never mind that most calculus students never get there -- all the really interesting problems involve ODE's. Traditionally, to understand ODE's one needed lots of technique. And, to get to that point, one needed (in reverse order) power series, integration techniques, the Fundamental Theorem, definite and indefinite integrals, derivatives, limits, functions, trigonometry, and algebra. The traditional curriculum often had a whole semester or more intervening between power series and differential equations, so one really had to be persistent to get to the good stuff.
Now we can pose the problem embodied in a differential equation on Day 1 of a calculus course: The time-rate of change of some important quantity has a certain form -- what can we say about the time-evolution of the quantity? Furthermore, we can draw a picture of the problem: a slope field. The meaning of solution is then clear: We seek a function whose graph ''fits'' the slope field. Even the essential content of the existence-uniqueness theorem is intuitively clear -- the details can wait for that course in ODE's. By that time, the survivors will have a clear idea of what that course is going to be about and why the details matter.
To be more specific, suppose our initial question is ''What can we say about growth of the human population, past, present, and future?'' Students recognize that this is an important question, and they start to get engaged with ideas. (This is an example of Harel's Necessity Principle [10].) Students can make conjectures about growth rates, such as proportionality to the population, and explore where they lead. We can trace solutions using the same technique by which we drew the slope field -- that's Euler's Method. When we observe that human population is changing more or less ''continuously'' -- one-year or 10-year intervals may not be good enough -- we are led naturally to the derivative concept. Technology enables experiments that lead to discovery of what's ''natural'' about the natural exponential function.
There are many models students might pose for population growth, but we don't have to keep guessing. We have about 1000 years of more or less reliable data that we can try to fit to a model. Using log-log and semilog graphing, we can find that the historic data are not exponential. Rather, the growth rate is proportional to the square of the population, so the data fit a hyperbola with a vertical asymptote. Students are initially shocked to find that the asymptote occurs within their lifetime (about 2030). Then they really have to think about what all this means. (See [16], Chapter 7 Lab Reading.)
The details of the preceding paragraph involve substantial mathematics -- numerical, symbolic, and graphical. Note also the echoes of the Kolb cycle: concrete experience with data plots, reflective observation about what the plots mean, abstraction in the symbolic models and their solutions, and active testing of the symbolic solutions against the reality of the data. Then the cycle starts again with the vertical asymptote: What does it mean? How can we fit it into an abstract scheme? How can we test whether our scheme fits with reality?
| Title page |
| Reform or Renewal?
| Students
| Problems |
| Cognitive Psychology
| Brain Research
| Technology and Learning |
| Renewal in Calculus Courses
| References |