There are many articles written about math education that cover how it is being taught in schools, and most notably the difference between traditional methods and reform or progressive methods. In many of these articles, media coverage presents the debate about math teaching as both 1) a dichotomy between procedures and conceptual understanding, and 2) dismissed as “that math wars stuff” .

The experts interviewed in such articles often express frustration over the discourse that presents math teaching as a dichotomy between algorithms/procedures on one side (the traditional method) and problem-solving/exploration on the other (the progressive/reform side). Then somewhere within the article will be an observation that both sides agree that both procedural and conceptual understanding are important and mutually support each other. Therefore the two sides are in fact “saying the same things.”

Which makes a nice “can’t we all get along” type story, except for the inconvenient fact that we are not saying the same things. Ironically, while it may be true that both sides agree that procedures and conceptual understanding are important, there is a bit more “nuance” to the arguments raised. (My apologies for using the word “nuance” which is almost as overused as the word “narrative” these days.)

My characterization of traditional math is that we do teach for conceptual understanding; we just don’t obsess over it. The expectation that 100% of students will understand 100% of all concepts 100% of the time, is 100% unrealistic. If the expectation is that every student perfectly understand a concept before moving on — or delaying the teaching of a standard algorithm until students can “show understanding” — we would seldom move on. Students would be held back when ready to move on, which is often the case in classrooms where reform math practices are carried out.

Sometimes students understand the underlying concept before learning a particular procedure; and sometimes it comes after. And for some students and for some concepts and procedures, the understanding may never come. Some people when they hear me say that think I’ve committed blasphemy. So let me offer some “nuance”.

There are levels of understanding. An elementary school student’s understanding of a concept like fractional division is going to be at much lower level than say a math major’s understanding of it in college. Conceptual understanding for students in lower grades can be thought of as "contextual understanding"; i.e., what is it we're representing and doing when executing a mathematical procedure or algorithm, as opposed to doing it in isolation (as traditional math is accused of doing).

What is meant by isolated facts or procedures? There are some students who can be taught the procedure for how to divide two fractions like 3/4 ÷ 2/3 but not know what fractional division represents or what types of problems can be solved. For example, a student may know how to do the fraction division procedure to get an answer to the above problem but would not be able to apply it to solve a problem like this: How many 2/3 oz servings of yogurt are there in a 3/4 oz cup of yogurt? Teaching procedures in isolation is not an artifact of traditional math; it is just poor teaching.

The stage is set by the big picture, the context. By seeing what is represented by a type of problem — that is, the "context"— students know what type of problems can be solved by the procedure as well as how to use that procedure. Ultimately, students will end up using the memorized procedure — regardless of whether they know the derivation or whether they know only the context. It isn’t necessary for students to know every step and nuance of why the procedure works when they’re executing it. If I have two students, one who knows the derivation of the fractional division rule and the other who doesn’t—if both can solve the yogurt problem, I cannot tell who knows the derivation and who does not.

To wrap this all up, please allow me some shameless self-promotion. For those who want to see how conceptual understanding is taught and not obsessed over, you might wish to read “Traditional Math: An effective strategy that teachers feel guilty using.”

I really enjoyed reading this after finding you via typing key words that interest me into the Substack search bar.

I teach Calculus to high school seniors and juniors who, by virtue of their intense workloads and extracurricular commitments—our campus is fantastic, but many driven students mistake the buffet line of diverse participation options as a 39-course meal—often are pretty exhausted. Given the finite time I have with them and attention I can hold while presenting to them, my goals must include the necessary procedural fluency alongside the conceptual development that lets that have meaning to attach those procedures to.

Where I’ve landed in the last few years has been to diverge from that unrealistic goal of 100% understanding. My goal is to provide context, as you said. I want them to have a sense of how this new idea bridges old ones forward and of how a particular relationship confirms to intuition. I use many more metaphors—we analyzed functions via their derivatives’ graphs on Wednesday which I tied to figuring out if a person likes you from body language—and narrate the long proofs that centered so many college lectures, and then I walk them through problems while constantly pointing back to fundamental ideas, meaning, and the rich vocabulary being applied underneath the algorithmic procedure they follow. The instruction is richer—especially when Desmos lets me whip up an applet to make an idea or problem visual and interactive—and my students learn how to do the mechanics of the mathematics while developing fluency in the world. This strikes me as exactly what you’ve advocated.

Maybe I’ve gone astray from your piece; if so, I apologize. I appreciate your point, and I’d like to think that thoughtful design can accomplish sufficient learning on both fronts, but I am with you: if a student can solve that yogurt problem, they’ve bought time and opportunity to elevate their understanding later with prolonged exposure to it.

Happy Friday!