A young friend asked, why bother learning calculus, other than to get into college?
The answer is that calculus holds the keys to the secrets of the universe. If you don’t at least have an intuition for calculus, you’ll have a harder time building things that work (be they machines or organizations), and you’ll be prey to all kinds of crank theories. Of course, there are lots of other ways to go wrong in life too. Be grumpy. Don’t brush your teeth. Hang out in casinos. Wear white shoes after Labor Day. So, all is not lost if you don’t learn calculus. However, the world is less mystifying if you do.
The amazing thing is, calculus works. A couple of years ago, I found my kids busily engaged in a challenge, using a sheet of tinfoil of some fixed size to make a boat that would float as many marbles as possible. They’d managed to get 20 or 30 afloat so far. I surreptitiously went off and wrote down the equation for the volume of a rectangular prism, subject to the constraint that its area not exceed the size of the foil, and used calculus to maximize. They were flabbergasted when I managed to float over a hundred marbles on my first try.
The secrets of the universe come in two flavors. Mathematically, those are integration and differentiation, which are inverses of one another.
Technically, integration is figuring out the area under curves (or surfaces, or along paths). That’s of some interest if you need to figure out the volume of your elliptical kayak, but the real application of interest is in dynamics – the study of how things change over time. Dynamics is interesting because (a) lots of things change over time, and (b) our intuition about it sucks. In dynamics, integration is just the math behind accumulation – the way deficits build up in debt, CO2 builds up in the atmosphere, repeated small insults bring the anger of a sibling to the boiling point, or learning one thing enables learning more.
Derivatives are about figuring out the slope or gradient of curves or surfaces. This is interesting because it lets you predict which way things will change – if you drop a ball on a surface, which way will it roll? More importantly, you can use it to find points where things don’t change (the ball doesn’t roll), which means that you’ve reached a minimum or maximum. That’s handy to know if you want to maximize your bank balance, minimize your time in a 10k run, or fire a potato canon farther than ever before.
The trouble with calculus is that it’s taught backwards (or at least was, back in the old days). As I recall, we started with limits, then tackled derivatives, and finished up with integration. That’s a natural order of progression, because you need the infinitesimal stuff to develop the math for the derivatives, and once you know the derivatives of lots of things, it’s much easier to integrate. However, it wasn’t very motivating, except to the extent that some of the arcane word problems were intrinsically interesting to nerds like me.
If you learn the mechanics of calculus without the intuition, you end up somewhat empty-handed (or empty-headed) – but at least you get into college. However, if you learn the intuition behind calculus, you have a life long skill, even if you forget how to integrate x^2*sin(x), as I have, and you get into college.
An intuitive approach to calculus starts with science (with physics, specifically mechanics, being the most approachable and dynamic) and system dynamics. As with any modeling project, start with the question or problem to be solved. Why does conversation at the dinner table get louder and louder? What makes a pendulum swing? How does a sand pile maintain a consistent slope? How many rabbits can you breed in a year? Perhaps it’s best if the question emerges from observations – just play with the bathtub or sink.
The key question is, how do you represent what’s going on in these experiments or systems? Creating a representation is modeling, which you can do formally (e.g., build a simulation model) or informally (build a causal loop diagram, or talk about relationships and feedback). Once you have a model, you can integrate it (numerically). This lowers the barrier to doing calculus, because you only need algebra to build the model; software handles the rest. You still need to know what you’re doing, so it’s useful to practice graphical integration, which builds intuition without requiring abstract manipulation of symbols.
In a sense, once you have a dynamic model that you can integrate, you don’t need derivatives anymore, because you can find maxima (optimal solutions) fairly efficiently by trial and error in the model. Still, it’s useful to experiment with problems that combine integration and differentiation. Build a rubber band shooter. Model it (the trajectory of the projectile involves integration). Then figure out what launch angle maximizes flight distance (a derivative problem).
Some resources for proceeding this way:
- Business Dynamics
- Thinking in Systems
- The Systems Thinking Playbook
- The Creative Learning Exchange
- The Waters Foundation
- Road Maps
- All Models are Wrong: Reflections on becoming a systems scientist
The bible for system dynamics study, and not just for business. Good on graphical integration and thinking in general.
Donella Meadows’ gentler primer on systems, mostly non-mathematical and big on intuition and the big problems that confront humans.
Systems games that engage all the senses.
A gold mine of systems and learner-directed learning resources.
Another good systems thinking learning resource.
MIT’s online system dynamics courseware.
What are we trying to accomplish here, anyway?
Unfortunately, there’s a missing link. I don’t know of a good resource that bridges the gap from thinking about systems and physics to teaching the formal concepts of calculus (but maybe you’ll find something above, at the CLE or Waters sites).
Incidentally, I think it makes sense to approach statistics the same way, by experiment and example, maybe even at the same time.