Image: Integration by parts. Remember?
Nearly every university science student takes 1st-year calculus. The content is fairly standard: functions, limits, derivatives, max/min problems, working up to integration and often capped off by the powerful but counterintuitive trick of integration by parts*. I think 1st-year calculus is widely seen (both by the Math departments that teach it, and the other science programs whose students take it, as the sine qua non of mathematical training for all scientists.
Until about a month ago, I would have said so without hesitation. In fact, I did. I teach a population biology course** that’s cross-listed with our Forestry program, and when Forestry proposed cutting their 1st-year calculus requirement, I screeched as loud as anyone (and a good deal louder than some). Surely you can’t do population biology without calculus! But I’ve been thinking about it, and I may have been wrong. Maybe biologists don’t need calculus.
I need to be careful about my argument here. Biology certainly needs calculus – in my own field, for example, we’ll always have theoretical ecologists using calculus to build and solve population-dynamic models. But that’s not the same thing as every biology student needing calculus. What do my population biology students need to know, mathematically? And what parts of calculus are on the list? A full answer would be too much, and too boring, for a blog post, but I think some attempt is important.
I’ll start with a claim that I would have considered, just as a month ago, heretical: I don’t think my students need to know how to take a derivative or to integrate a function. Whoa – there goes virtually all of calculus right there, you’re thinking! Not quite. What my students definitely do need to know is that dN/dt is a population growth rate, and more generally, that d(something)/d(something else) is a rate of change of the “something” with respect to the “something else”. We use such derivatives to specify the basic models of population growth (exponential growth, logistic growth), of metapopulation dynamics, of interspecific interactions (the Lotka-Volterra models of competition and predation), and much, much else.
So if we specify population models as derivatives, why am I exploring the notion that my students don’t need calculus? Here’s the thing: once we’ve written down the derivative, we do (at least in my course) the rest of what we need to do with it using only basic algebra. Take the Levins model for metapopulations, for example:
dp/dt = mp (1-p) – ep
where p is the proportion of patches occupied by a focal species, m is its migration rate among patches, and e is the per-patch risk of extinction. (If you aren’t familiar with this model, don’t worry; it’s just an example).
Having taken calculus myself, my first instinct is to integrate this to get an expression for p(t). But I suppress this urge, because actually, I don’t care about p(t). What I care about is whether there’s a stable equilibrium p*, and to find this all I have to do is set dp/dt = 0 to find p* = 1-e/m (technically, this alone doesn’t assure stability, but I can verify that graphically). The two-species Lotka-Volterra competition questions are a bit more complicated, but the approach is the same. Here we specify dN1/dt and dN2/dt, but our real interest is finding conditions under which species 1 and 2 can coexist, and this question can be answered by setting dN1/dt = dN2/dt = 0 and plotting the resulting “zero net growth isoclines”.
This is getting a bit technical, so I won’t offer any more examples. What’s important is the bottom line: I teach an entire semester of population biology without integrating or taking the derivative of anything. My students don’t need calculus – at least, not beyond the fact that d(something)/d(something else) is a rate of change, which might be zero.
This doesn’t mean that my students don’t need math. The “biologists don’t need math” trope drives me crazy***. I’ve heard it from well-meaning parents, high-school guidance counsellors, university admissions offices, physicists, you name it; and it’s just plain wrong. My students do need math; but I’m suspecting more and more that what they need is not what 1st-year calculus delivers. My students need statistics (and they do take it, although the way we teach it is another issue). They need basic algebra, logarithms and exponentiation, simple matrix manipulations, and a few more things – but even as I type out this list, I realize it’s boring, that your list might differ from mine, and that it misses the point. What my students really need (mathematically) is just one thing: comfort.
What do I mean by “comfort”? I mean that my students need to see a page of quantitative material – an equation, a graph, a statistical table, an ANOVA table – without panicking. They need to be willing to write down an idea about how something works in the form of an equation, or to look at an equation and ask themselves what it suggests about nature. They need to understand that math isn’t a thing distinct from biology; it’s just a language we use to think precisely and to deduce the consequences of what we’re thinking. In short, they need to think of math as something quite ordinary, and something that’s their friend rather than their enemy.
I strongly suspect that for most students, 1st-year calculus doesn’t deliver mathematical comfort. If anything, it may take it away. How can we deliver that comfort? I don’t know. I’d hope to see students arriving from their K-12 education with it – but while some do, many obviously don’t. One thought I’m toying with: maybe trying to build on top of high-school math just takes students higher when they’re already afraid of heights. Maybe we should focus instead on knitting the math they already know into the questions they care about in their discipline – so they stop noticing the height in the first place. Later in their careers, of course they can learn more math – when they have some mathematical comfort, and when they’re motivated by a problem in their discipline. (That, after all, is how I learn.)
Speaking of comfort, the conclusion I’ve come to makes me uncomfortable. That’s partly because if I’m right, then I’ve been wrong for a long time. And it’s partly because no matter how clearly I think I’ve laid this out, someone is sure to assume I’m just rolling over on “biologists don’t need math”. So I’d be happy to be (re)convinced that biologists do need calculus. Bring it on.
UPDATE: I like the look of this newish text for teaching calculus: Biocalculus, by Stewart and Day. I’m not sure it’s a first-year text; but the approach, thoroughly grounded in biological problems, would help my students a lot. Thanks to Amy Hurford for suggesting it in the Replies.
© Stephen Heard (email@example.com) February 9, 2016
Related post: Why do we make statistics so hard for our students?
*^When I was an undergraduate, I was pretty good at calculus. My Engineering friends, though, thought I was brilliant at it. Here’s why. When they were stumped on an integration problem, they’d talk about in the residence breakfast line. I’d say something like “Stuck, eh? What do you need to integrate?”, and then pretend to listen while they told me. I’d gaze thoughtfully upward for a moment, then say “I think you need to integrate by parts”. They’d go off and try, and it would always work, and they’d assume I had solved the problem in my head. In fact, I knew just two things: engineers are really good at integrating, and engineers really hate integrating by parts. If they were stuck, they must have tried everything else. But the problem wouldn’t be on an assignment if it weren’t soluble, so it had to be one that requires integration by parts! There – I’ve pulled aside the curtain. There never was a wizard.
**^I would argue that population biology needs at least as much math as any other subdivision of biology, so it’s a good test case: if any biology student needs calculus, surely it’s a population biologist.
***^My mother, my aunt, two uncles, and my grandfather all taught math. In many ways, I teach math. I like math. It pains me to even consider reducing the world’s total teaching of it.