Experience Before Instruction, Examples Before Theorems

I’ve spent a lot of my time in the past few days working around the house, constructing bookshelves (from store-bought Ikea-like kits), tearing down some interior walls in our basement, and whatnot. I found myself thinking about the physics courses I’ve taught in the past, and how some key points in the courses were perfectly illustrated by what I was experiencing with my own hands.

For example, some of the finishing nails used to secure baseboards are extremely thin, and to remove them is just like pulling weeds, which I discussed in a previous post. If you try to jerk them out, they just snap. One must pull gradually if one wishes to pull the nail out whole.

Large spiral nails come out with a creaky sound, and require a lot of effort. As a result they are a little warm upon removal … the work-energy theorem in action. (Sure, some of the work done in pulling the nail out is converted to the creaky sound and to the kinetic energy of the nail being pulled, but a lot of the work goes to thermal energy.)

Breaking the dry wall with my hands is just like pulling weeds, too. Pulling too sharply leads to a handful of dry wall “crumbs” breaking off, but pulling more gradually is more likely to break off a larger chunk, which is then easier to transport out to the garage. (By the way, dry wall is pretty hard, which makes this video clip of an NFL player pretty amazing, if it is legitimate. I played it frame by frame, and it looks real.)

One of the shelves has facings on the front edge of each shelf and the uprights, which I had to slide onto the rough edges. They were a tight fit, and I found that if I got the facing going, then I could slide it on relatively easily, but if I stopped it would be a real pain to get it started again. Yes, the coefficient of static friction is greater than the coefficient of kinetic friction, as anyone who has ever dragged a refrigerator across a floor knows.

My point is that what we learn in elementary physics classes is somewhat abstract, yet it is so apparent in our world. That is, it is taught abstractly, not that it is intrinsically abstract. And it appears to be abstract and somewhat meaningless to students, because they either have no experience of the concepts or they do not connect the concepts to their experiences.

For example, a fellow was about to back out of the driveway this afternoon, when I reminded him that his trunk lid was still up, blocking his rear window. He told me to “watch this,” then started to back out and then abruptly stopped, at which point the lid slammed shut. Newton’s first law in action. (And yes, I watched carefully to make sure that nobody was nearby as he was backing out.)

It is said that children who spend a lot of time moving about on jungle gyms (“monkey bars”) have an easier time visualizing in three dimensions when they grow into university students. More generally, students who have a rich experience of working with their hands have an easier time understanding basic physics, particularly if their teachers help them connect the concepts with their experiences.

It’s true in mathematics as well. One of the worst aspects of mathematics education is the definition-theorem-proof-example style that I suffered with so much as a student, and which is still very common, particularly in upper-level undergraduate and graduate courses. Much better for the student is lots of examples first, especially if they include concrete calculations, so that they have a repertoire of experience to draw upon. This sets the stage for abstraction, which is thereby much more meaningful.

This is how it has worked historically, time after time. Only after intensive work with concrete problems have mathematicians been able to invent the right definitions that allowed them to state and prove interesting theorems. This is highly creative work, and we do our students a terrible disservice to hide this natural sequence from them in the interest of “efficiency.”  It might allow instructors to cover a lot of material efficiently, but it certainly does not lead to deep understanding very efficiently. On the contrary, the poor students are left to figure things out on their own, and good luck to them. Most students have no idea that they ought to seek out lots of concrete experience before tackling the lecture material, or they just don’t have the time because they can’t keep up with the pace.

There are a number of very nice books that are leading the way towards a much more humane way of teaching, but we still have a long way to go to bring the pedagogy of our upper-level courses in line with a natural way of learning. Elementary and high-school teachers are doing a much better job of this nowadays then when I was a student, and more power to them.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 21 January 2021.)