The fallacy of teaching by derivation

A large chunk of my research time this week was spent trying to teach myself a couple of relevant background concepts.  Whenever I find myself in this situation, the first thing that happens is that I feel really dense.  I’m an Nth-year PhD student, how could I not understand that topic?  The second thing that happens is that I get frustrated about the way the subject is presented in most references.

Without giving away too much about my field of study, let me just say that it can involve a lot of math.  But, for the most part, the math is applied: it’s used to describe real things that exist and happen.

The topics I was looking up this week were almost universally presented in texts and notes in this fashion:

  1. Brief introduction to topic.  We’re talking 2 or 3 sentences.
  2. Extensive, step-by-step derivation of the key equation(s) used to describe the phenomenon.  Usually these equations are named after people.
  3. [optional] Another handful of sentences explaining the results from part 2.

And, ta-da!  You know the math, so now you know the concept!

Except… my brain doesn’t work that way.  It doesn’t easily convert gammas and rhos and plus signs to a mental picture of real physical things.

It’s not that I’m bad at math.  I’m good at math, in fact.  Very good.  I don’t mean to brag or nuthin’, but my partial differential equations professor in college asked me for my notes at the end of the semester.  It can be time-consuming to work through derivations, but I can absolutely do it.

And it’s not that the math isn’t important.  Mathematics is a critical tool for describing, interpreting, and predicting the world.  If you’re in a highly applied field, like, say, architectural engineering, being able to do the math correctly could make the difference between a building staying up or collapsing on the people inside of it.

But knowing the math is not the same thing as knowing the concept.  Teaching the former does not automatically confer upon your students an understanding of the latter.

My brain isn’t the only one that struggles to convert equations to conceptual understanding.  During my time as a PhD student, I’ve been a teaching assistant (read: lab instructor, grader, substitute teacher, and cell-phone patroller) for a number of introductory classes.  Some of those classes were geared toward students hoping to major in the subject, while others were designed for folks trying to check a box on their list of Gen Ed requirements.

It turns out that there are exactly two differences between these groups of students.

One, the majors are significantly more motivated than the non-majors, on average.

And two, the majors are better at math.

You know what wasn’t a difference?  Their conceptual understanding.  The majors could rearrange equations and calculate numerical answers like nobody’s business, but they made the same basic errors as the Gen Ed students when asked to explain how things work.  All that math we were showing them didn’t help them build a better mental framework of the fundamentals.

So why do we do this?  Why is “teaching by derivation” the default in science and engineering?

Is it that there’s a subset of people whose brains do readily make the connection between math and concepts, and those are the people who go on to be STEM professors?

Is it that (and I suspect this is most likely) that our professors are just presenting the material the same way they were taught?

Or is it that (and I really do wonder if this might be a little true, too) that they don’t fully grasp the underlying concepts themselves?

Whatever the case, if you are teaching science or engineering, or putting together lecture notes, or writing a textbook, I beg of you: please, please, please explain the concepts before you go through all of the derivations.  If the derivations reveal new concepts, explain those too!  Use words!  Use pictures!  Be different!

Your students will thank you.