I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6, and AC = 7, what is the smallest angle? *(Edit for clarity: The question is simply asking for a name of the angle, not its measure.)*

One of my students said he didn’t understand how we could figure that out. I didn’t answer him, since it’s a final exam. But as I thought about how I might have answered it, it occurred to me that this is a case in which the conceptual answer is so much simpler and straightforward than the procedural one, even as students cling to procedures.

Here’s the conceptual answer: The shortest side of a triangle is always opposite the smallest angle. That’s the way I try to teach this. It does rely on a concept that students consistently struggle with, of “opposite angle”, but that’s an important one regardless.

There’s a more important related concept from which this one follows: As an angle grows, the opposite side length grows, and vice versa. From here, we can use an isosceles triangle to demonstrate the short side concept with a scalene triangle. Specifically, an isosceles triangle has two congruent sides and two congruent angles (another, related concept). If angles A and C of triangle ABC are congruent, then sides AB and BC are likewise congruent. It follows, from the general concept (angles and sides grow together), that if AC gets longer, angle B gets larger. Hence if BC > AB, then angle A has a greater measure than angle C.

This longer explanation allows for quite a few important concepts to be connected together. Above all of these is the most important concept of mathematics: Everything coalesces, everything is connected. That is the beauty of mathematics.

However, even the short, specific conceptual answer shows a useful understanding.

Now let’s look at the procedural answer, which I found myself reverting to when teaching this topic earlier this year.

- Find the shortest side
- Identify the opposite angle
- If you’re having trouble with this, the opposite angle is the letter that isn’t listed when you list the side (some students needed this prompting)

- This is the smallest angle

This, to me, is far more mental work. And yet, more students were comfortable with it. There are clear steps, even if there’s little if any evidence of understanding of the key point. Procedures are far more difficult to generalize than concepts because there’s no real depth to a procedure.

Imagine learning to cook like this. Gather the ingredients in the recipe. Mix some of them together. Melt some of them. And so on. Why are we adding chocolate chips in step six? Because the recipe says so. What happens if we’re out of chocolate chips? We hit a brick wall and have to stop. What happens if we misread “tbsp butter” as “stick butter”? Well, we have a mess, and we don’t know why. What happens if the cookbook has “stick butter” as a misprint? We have no way of judging that. We have to trust the cookbook, because we don’t know what all the stuff does anyway.

What happens if we put \(-1^2\) into the calculator and get \(-1\)? Well, that’s what we must get when we square \(-1\). Of course, the calculator isn’t lying, it’s just (as Skemp might put it) playing by rugby rules when we’re playing soccer. The calculator has been taught, as is the standard mathematical convention, that \(-1^2 = -(1^2)\), while my high school students tend to think that \(-1^2 = (-1)^2\). Nobody’s “right”, since this is convention, not mathematics, but we can’t play rugby by soccer’s rules.

If there are no concepts, if there are just procedures, then there’s little if anything to flag a student’s mind that \(-1^2 = -1\) isn’t what they wanted when they went to square negative one. This is another case where procedure blocks concept. My students know, with a fairly solid degree of confidence, that the product of two negatives is a positive. That can be understood either procedurally or conceptually, but the conceptual understanding there relies on approaches that are typically beyond most high school math students (one explanation: when we multiply complex numbers, we multiply their vector length and add their rotations; a negative number has a rotation of 180°; ergo, the product of two negatives has a rotation of 360°).

However, what my students struggle with is the concept that \(a^2 = a \times a\). They have a separate procedural rule that a number squared will always be positive, and they haven’t mapped it onto this. And, to be fair, once we get into non-integer exponents, it stops being appropriate to think of exponents as repeated multiplication. It’s not that \(a^2\) is the same exact thing as \(a\times a\), but rather that it is always true that they return the same value. *Side note: There’s one integer power where this is also problematic. In general, we can argue that \(x^0 = 1\) because that’s the empty product (that is, we’re multiplying x no times at all), but \(0^0\) is indeterminate, not universally defined as 1.*

The point being, if students don’t see any clear relationship between \(a^2\) and \(a \times a\), they’ll have less reason to question \(-1^2 = -1\) when their calculator tells them so.

Sometimes, as with the product of two negatives, the concept is indeed tricky, and the topic might be better taught procedurally. Other times, as with the relative relationship between side lengths and angle measures, the concept could be simpler than the procedure, and lead to deeper overall understanding.