## Calculus in context

I’ve taught calculus for a number of years and have largely been dissatisfied with the texts. I love mathematics in its own right.  I don’t think mathematics needs to justify its existence or its questions.  However, a calculus class is an opportunity to train the minds of future scientists and engineers.

I’ve grown tired of the long lists of functions to do A and B to or the psuedo-context problems (to use the phrasing of Dan Meyer) that are supposed to represent applications.  Students often want to know when they’ll run into these specific functions, how will calculus show up in their major, how does the now ubiquitous graphing calculator come into play or services like Wolfram Alpha?  The applications seldom have references for the student or instructor to follow up on.

I’m trying to design a class where if you forget all the rules for integration and differentiation you’d still look back and say “yeah, that was a great class, I really grew as a problem solver.”  This is a tall order in a class of biology, ecology, engineering, physics, chemistry, computer science, and premed students.  Moreover, we teachers are all too aware of the problem of transfer.  The student seems to know how to do calculus class but doesn’t seem to know how to  do calculus anywhere else.

My department recently changed books to “Calclulus” by Hughes-Hallett, Gleason, and McCallum published by Wiley which is a great start.  It often asks students about units, interpretations of the derivative, graphical skills, technology applications etc.  It treats limits quickly and gets on with the process of thinking about derivatives, but spends a whole chapter on derivatives before learning any of the ‘short cuts’ to calculating them.  It is rich in qualitative reasoning.

This has led me to look for more problems of this sort.  Strogatz’ book “Nonlinear Dynamics and Chaos” is a good start.  This has led me to think about how early you can introduce differential equations to students.  Once you have the concept of the derivative, even before you calculate any you can talk about a reaction equation such as:

$latex \frac{dy}{dt}=k(a-y)(b-y)$

Where $a and $a,b,k$ are all constant.  $y$ is measured in grams and $t$ is measured in seconds.  ‘a’ and ‘b’ represent chemicals that combine in some way to produce chemical ‘y’ and this differential equation describes the process.  Questions you could ask:

(a) what are the units on dy/dt, a, b, and k?

(b) Make a graph of dy/dt versus y.

(this is a really good one.  From this graph you can obtain huge amounts of information about what happens to y(t) depending on the initial conditions. You can also talk the range of applicability of the model, that is, does it make sense for $y(0)>b$?

(c) What is the rate of change if $y(0)=0$?

(d) What is the limit of $y(t)$ as $t\to\infty$ if $0?

Those are just a couple of questions that only use the basic idea of a derivative and a graph to answer.  They will be hard though they will require a lot of work on the part of the student.  Then there are some other more challenging questions you can ask such as:

If you mess around with the differential equation:

$latex \frac{dy}{dt}=kab(1-y/a)(1-y/b)$

Then you can ask about  conditions such as $0 or $y< but $y and look at how to approximate the differential equation.

It’s a lot of work, but you’ll be able to construct really clear pictures of what $y(t)$ looks like from this sort of reasoning and it doesn’t require any fancy mathematics, just fancy thinking which is exactly what we need our students to develop.  I’ll post more in the near future!