By Keith Devlin @profkeithdevlin

I’ve written on this blog before about the false impression math educators can leave their graduates with, by focusing entirely on tightly constrained and carefully crafted problems, devoid of context, that yield a single right or wrong answer within a pre-determined length of time. For sure, such problems are ideal for developing specific math skills. But what they do not do is prepare their students for using mathematics effectively to solve real-world problems — or even classroom problems outside the math class (such as in the physics class, the biology class, or the economics class).

Kind problems is the term often used nowadays to refer to such problems. What we need – in addition – I have argued, is to spend more time working with our students on how to set about solving wicked problems.

I wrote about this most recently just two posts ago, in October. Before that, I raised the same issue in my November, 2019 post. In both posts you will find a definition of “wicked problems”.

What prompted me to return to this theme this time was a super little video that was just published by science educator Dr. Derek Muller on his Veritasium platform. 

Muller got an Engineering Physics degree from Queen's University in Canada and went on to earn a PhD in physics education research at the University of Sydney in 2008, where his thesis topic was Designing Effective Multimedia for Physics Education.

The video in question was bound to catch my eye, being about the physics (and hence the mathematics) of bicycle dynamics, which nicely combines two of my greatest life passions.

I’ve written about connections between mathematics and bicycles here before. There was a 2018 post about applications of mathematics to bicycle aerodynamics, a 2016 post that used the problems riders often have changing modern road-bike tires as an allegory for effective math teaching, and a pair of posts in February and March of 2014 showing how technically difficult mountain-bike riding closely resembles effective math-problem solving (à la Polya, see also here).

So, there was nothing in the new Veritasium video that was new to me. What impressed me was Muller’s presentation. Visual explanations of science is what he studied for his doctorate, and what he is good at.

Watching Muller makes it clear that, whereas we may think of the bicycle as a simple mechanical device, its design and its physics (and hence mathematics) are far from simple. Check out this 2016 article from Nature for a fairly short, nice discussion of one aspect of bicycle dynamics. And, for a deep dive into the mathematics of the bicycle, grab a large mug of coffee and settle down for a long session with the Wikipedia entry on the subject. It’s messy!

Bicycle dynamics is not a wicked problem; it’s a tightly constrained engineering issue. Everything follows the laws of physics. But it’s a complicated problem, where the gap between being able to ride a bicycle and being able to describe, with mathematics, how we can do what we do when we sit on the saddle and ride, is a vast chasm that is beyond most people. It’s every bit as wide a gap as between flying in an airplane, or even piloting a plane, and understanding the mathematics of airplane flight. (And no, it’s not “just Bernoulli’s equation.” The common, two-dimensional-side-view description you see on many Websites isn’t even close. Beginners should start here, on NASA’s educational site. For beginners!)

And this is the point I want to add to my ongoing math-ed theme. It's not just wicked problems that highlight the limitations of math education that focuses exclusively on tightly constrained, essentially artificial problems with relatively simple, clear, unique answers a good student can find in at most twenty minutes. Simple-seeming physics problems of our everyday world such as “What keeps a bicycle upright when it’s in motion?” also stretch mathematics way beyond the familiar classroom experience. “Kind problems” are not always kind.

We need to take three educational bags to the math classroom.

Bag 1. Teaching a collection of basic math skills is important.

Bag 2. So too is trying to help our students see and appreciate the beauty and elegance of the context-free, pure mathematics canon. That is worth our time as educators for a variety of reasons. 

Bag 3. But we do our students a disservice if we allow them to graduate without giving them a glimpse (ideally an experiential glimpse) of using math to solve complex real-world problems, both wicked problems, where math is usually just part of the answer, and tricky applications in, say, physics and engineering, where the math still “rules” but it’s complicated and messy. Most uses of math outside the math classroom are in this third bag.

Further reading: The Veritasium webpage on Muller’s bicycle segment has a lot of references to the math and physics of bicycles.