By Ben Blum-Smith, Contributing Editor
For reasons that will not be considered here, I recently learned this dance:
Although I have no background in any style of dance, I can now do the whole thing, start to finish. I am very proud.
My purpose in attaining this objective was unrelated to mathematics or teaching. Nonetheless, the experience put an eloquent fine point on a certain basic dialectic in math education.
Procedural vs. Conceptual
I spent a decade working in middle school and high school math classrooms before I trained as a research mathematician. Conversations regarding goals for students in elementary and secondary math education, and math education research, often distinguish between two types of knowledge: procedural and conceptual. These are fraught words, and you have your own ideas about the meanings. Nonetheless, for the sake of clarity (at least internal to this blog post), I will offer some definitions.
Conceptual knowledge: knowledge of what things really are, what they are all about, and how they are connected.
Procedural knowledge: knowledge of how to actually do things.
I hope with these definitions that I have not accidentally tripped any wires. If your background is anything like mine, the mere mention of this dichotomy may have already given you some unpleasant flashbacks. In one of my first teaching jobs, almost every department meeting eventually devolved, in a practically ritual way, into a bitter fight. And one of the perennial sticking points was which of these two knowledge types deserved priority. Those days were a high-intensity period in the Math Wars, and the “procedural vs. conceptual” dichotomy served, in my experience, as a kind of a “Math Wars bat signal”: once it came up in a conversation, powerful ideological fault lines showed up soon after, as though they had been summoned.
The terrain has shifted a bit since then. It eventually became fashionable, uncontroversial—indeed, obviously true—to assert that these two types of knowledge are both important, and are mutually reinforcing. Interest has grown in creative ways to serve both masters at once.
Nonetheless, educators still often have a propensity one way or the other at the level of educational values and aesthetics. For some, a calculus student who can differentiate elementary functions flawlessly, but doesn’t know what any of it means, ‘hasn’t actually learned any math.’ To others, ‘at least they can solve the problem!’ For some, it is distressing and concerning when a fourth grader can accurately identify a wide range of contexts modeled by subtraction, but can’t compute except by counting down on their fingers. Others feel this student has already learned the hard and important lesson, and believe that this will make learning a better computational method easy. These differences can persist even among educators who believe passionately in the joint value and mutual complementarity of the two types of knowledge.
For example, I fall on the conceptual side. Not intellectually: I believe strongly that mathematical knowledge comes in both types, that they’re both crucial, and that they’re mutually supportive. Every time I reflect on my own learning with this question in mind, it’s obvious how much my procedural knowledge has done for me. That said, I’m simply more passionate about teaching concepts than procedures. I am lit up by the challenge of getting students to perceive an unexpected connection or to understand the purpose of an important definition. I can also get excited about the how-to-do-this stuff when I know it will make my students feel powerful, or put them in a position to think about a particular interesting question or concept, but even in these cases it’s a means to an end. Meanwhile, my heart sinks a little when I read student work that evidences thoughtless application of a formula, even if the answer is correct.
These differences in taste can shape our curriculum design and our teaching choices even if we believe at the intellectual level in the importance of both types of knowledge. For example, my gut orientation toward conceptual knowledge means that when a student presents as stuck or lost and asks me what to do, my first instinct is always to pull their attention away from that question, down to the level of “what is this all about, and how is it connected to other things you know?”
I don’t think there’s anything wrong with these tendencies toward the conceptual or the procedural, and in any case, we have them whether we like them or not. But because they shape our teaching practice, I do think it’s useful to recognize them. Sometimes, the thing a student needs is conceptual; other times it’s procedural. I think both types of bias have their strengths, but each can also lead to teaching blunders caused by failing to recognize the needs of our students.
For example, my strong habit of assuming that the obstacle facing a student is conceptual, can make it hard for me to recognize when a student has a procedural need that’s not being met. I, and I think many conceptually-oriented educators, have a tendency to see the procedural knowledge—what to actually do—as a consequence or corollary of conceptual knowledge. So if a student presents with a difficulty doing something, I (we) take aim at the concept of which the desired action is (to us) a consequence.
This does actually work a lot of the time! And, there are plenty of times when it doesn’t, because it isn’t always reasonable or fair to assume that the student can get from “I know what’s going on” to “I know what to do” on their own.
Much to my surprise, learning a K-pop dance routine provided me with an incisive opportunity to reflect on both of these possibilities—from the student side.
Learning to hit you with that DDU-DU DDU-DU
When I set out to master the dance from BLACKPINK’s 2018 hit song “DDU-DU DDU-DU”, it was kind of like learning to walk. My lack of any kind of dance training, combined with my gender socialization, meant that half the stuff Jennie, Lisa, Jisoo and Rosé do in the dance practice video was missing entirely from my movement vocabulary. But I was up for a challenge.
I started with the chorus. I got as far as the first “Hit you with that DDU-DU DDU-DU,” but that little 4-beat bouncy lean thing that immediately follows it—
[The video is cued up at the exact point I’m talking about, but you lose the cueing once you play it. To rewatch, reload this webpage.]
I mean, I was lost. Right shoulder down, right hip up, lean back, left shoulder down, left hip up… while the hands are moving? How do you do the weight transfer smoothly while you’re bouncing? How do you bounce and lean at the same time? Where do I put my head this whole time?? Trying to assemble this strange little movement felt like trying to hold too many things in my hands at once: something was going to fall. If I got my hips in the right place, I’d forget about my shoulders. Get the shoulders? Mess up the bounce. The idea of doing all of it at once felt overwhelming. The idea of ever making it look cute felt way out of reach. I needed help.
My wife has an actual background in a highly relevant field, namely hip-hop dance. Also, as it turns out, she is a completely conceptually-oriented dance teacher. Her first move was to tell me to stop thinking about what to do with each body part. Instead, she said, focus on the attitude. She illustrated it with other, more familiar movements that differed in their details but shared the attitude. “It’s like, ‘Eyyyyyyyyyy!'” she said, demonstrating.
The parallel to how I respond to analogous situations as a math instructor was extremely apparent. There was a main idea here. My wife was pulling my attention away from the impossible-feeling task of assembling the whole out of a bunch of disconnected details, and toward a single main idea from which all those details would flow. She was elucidating that main idea through its connections to more familiar knowledge. The main idea was what was important. The details would work themselves out.
It worked! By focusing on the attitude, everything came together. The bounce was nothing more than feeling the music. The whole thing with the shoulders, the hips, and the lean, turned out to be nothing but a right-to-left weight shift shaped by the appropriate attitude. The hands were, like, I mean obviously, I just hit you with that DDU-DU DDU-DU—now I have to put the “guns” away, and where else would they go? The entire motion felt logical and coherent, and I could do it without even thinking too hard.
Score one for the conceptually-oriented lesson!
I kept going. Exactly 7 seconds deeper into the chorus, there is a second “Hit you with that DDU-DU DDU-DU,” and again the four beats that follow it threw me completely:
It’s just a turn. No fancy roll/lean/bounce stuff this time, just rotate 360 degrees over four beats, stepping on alternating feet, and end up in that same little shoulder-shimmy as before.
But I wasn’t getting it! I felt off-kilter, gangly, uncoordinated. I felt I had to keep lurching, yanking my weight in different directions—this did not feel cute at all. I kept being late to finish the turn and set up for the shoulder-shimmy. Furthermore, I didn’t understand how it was possible not to be late. I repeatedly watched my wife and all four members of BLACKPINK pull it off, but this seemed like magic.
Fresh off our previous success, my wife again took a conceptual approach. To her, the main idea of the turn is to feel the beat in the alternation of your steps. She had me practice those 4 beats without turning, just stepping right-left-right-left in place.
This was easy for me—but this time, it didn’t actually help. My problem wasn’t, as it turned out, a failure to feel the beat in my steps. I realized I had a more fundamental question: where should I put my feet?
When my wife responded with, “It doesn’t matter,” I had a little moment of acute empathy for every student I’ve ever driven up the wall by insisting they focus on an underlying concept when they want me to tell them what to do. In that moment, I was the student who needed some concrete steps to follow (pun intended), and I wasn’t getting them.
On the one hand, in saying “it doesn’t matter,” my wife was obviously telling me the truth. The four members of BLACKPINK are at that point in the song rotating their whole formation. They’re all turning, it’s all synchronized, but they’re not putting their feet in the same places at all. My wife’s own rendition involved turning in place, so that was different too. All five of them—Jennie, Jisoo, Lisa, Rosé, and my wife—were evidently successfully executing the same fundamental dance idea, while putting their feet in different places. It follows that this particular dance idea is not determined by the locations of the feet.
On the other hand, I understood the underlying concept, at least as my wife was presenting it to me, but this understanding was not clarifying for me how to actually do the turn. She saw the procedural knowledge as an immediate corollary of the conceptual knowledge, but to me it was apparent that she was using some additional, not-entirely-conscious prior knowledge to translate this underlying concept into actual steps to take, and this was knowledge that I didn’t have.
This elucidated a mistake I’ve made countless times in teaching. The student is stuck and asks me how to proceed. I assume it’s a conceptual problem and take aim at the underlying concept. The student seems to understand the concept and is frustrated I won’t just tell them what action to take. Because the appropriate action, AKA procedure, feels to me like an immediate corollary of the concept, I assume that there’s a subtler, undiscovered conceptual problem still lurking. Because, furthermore, I fear that I’ll short-circuit the student’s opportunity to address this underlying conceptual issue by revealing the appropriate action prematurely, I hesitate to answer the question about what to actually do.
But sometimes, that’s what the student needs! The piece the student is missing may not actually lie in the concept, but instead in the way the concept entails the appropriate action—this is a kind of knowledge often not even visible to me, as focused as I am on the concept. In this situation, the student may need direct information about what to do. Seeing a complete solution demystifies this missing link, providing an opportunity to coordinate the underlying concept with the appropriate action.
This is what was happening to me with the turn. My only way forward was to directly mimic a correct example. I played the video back several times, focusing on Jennie—she’s the one in front at the beginning of the turn. Right foot steps out; turn 180 degrees on the right foot while swinging the left foot around the front; shift the weight; turn the other 180 degrees on the left foot, this time with the right foot moving backward; shift weight again; left foot behind right; step out with the right. The body is moving in the same absolute direction the whole time. Lemme try that…
It worked! Directly mimicking Jennie’s footwork gave me a structure to follow that solved the problem of how to turn around in exactly 4 beats without awkward direction changes. The abstract concept of feeling the rhythm in my feet could now inhabit the concrete set of motions I was following.
Score one for the procedurally-oriented lesson!
All of this is to say—we are hopefully on our way out of the false dilemma of procedural vs. conceptual knowledge, and toward a consensus that they are both critical, and are mutually reinforcing. Nonetheless, this wisdom can function as a bit of a platitude—preached, without always being lived. So I think it’s a worthwhile exercise to look, both in the classroom and outside of it, for opportunities to go beyond knowing it, to feeling it. And—who knew?—but learning a K-pop dance routine gave me the opportunity to feel it in my bones. Literally.
Notes and references
 Indeed, the lack of consensus about the meanings even extends to the possibility that by calling them knowledge types, I’m not being entirely faithful to the full range of their uses. See J. R. Star and G. J. Stylianides, Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality, Canadian Journal of Science, Mathematics, and Technology Education Vol. 13, No. 2 (2013), pp. 169–181 (link), which argues that while the terms refer to knowledge types among psychology researchers, they are better seen as referring to knowledge quality among math education researchers.
 An illustration: In 2015, in the Oxford Handbook of Numerical Cognition, Bethany Rittle-Johnson and Michael Schneider wrote, “Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative.” B. Rittle-Johnson and M. Schneider, Developing conceptual and procedural knowledge of mathematics, Oxford Handbook of Numerical Cognition (2015), pp. 1118–1134 (link).
 An example: M. Schumacher, Developing Conceptual Understanding and Procedural Fluency, on the Illustrative Mathematics Blog (link).
 While this and the next paragraph are focused on the situation in which I am wrong to withhold the “what to do” information, I hasten to add that this is, in general, a reasonable fear. If a student is in fact missing a conceptual piece of the puzzle, premature information about what to do may allow them to walk away from instruction with the belief that they have fully learned the concept when they actually did not. The student who applies a procedure in inappropriate contexts probably mis-learned it in this way. Judgement is required to determine what the student needs.
Nick Fortune and I wrote a blog about this in 2018 that may add to these body of research:
Thanks for adding this very relevant citation Karen! I see a connection between your work here and the blog post on the Illustrative Mathematics Blog linked in Note 3; both are aimed at pedagogy designed to serve conceptual and procedural goals simultaneously. I hope other readers add more links in this direction as well.
Ben, this was an amazing essay and I really enjoyed it. You make some very nice points. I actually see Conceptual versus Algorithm as part of a three step learning process beginning with Conceptual Understanding, proceeding to Development of Strategies, and Culminating with an Algorithm. In terms of Bloom’s taxonomy, the first stage is characterized by Comprehension and Application, the second by Analysis, Synthesis, and Evaluation, and the third is characterized by Knowledge. In the end, there is no longer a need to think about it because the ideas are encapsulated,
Thanks so much Amanda! Do you develop these ideas more fully in writing anywhere? (If so, might you add a link?)
May I suggest that people take a look at the 1999 article, Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education, in American Educator, Fall 1999, Vol. 23, No. 3, pp. 14–19, 50-52:
Thanks for adding this highly relevant citation! For readers interested in further investigation of the mutually supportive relationship between conceptual and procedural goals, I also suggest several pieces by Bethany Rittle-Johnson and her collaborators, for example the article in the Oxford Handbook of Numerical Computation linked in Note 2, and the citations therein.
Another area in which conceptual knowledge and procedural knowledge becomes extremely important is in society. Modern industry incorporates an enormous range of conceptual knowledge: how to make a car or a refrigerator or microprocessors which is held by a very small number of people. Each industry is designed around procedural knowledge: how to make 500 cars, refrigerators, or thousands of microprocessors per day to the exact same specification and almost no “reworking” or flaws. Those who work in manufacturing work with specialized fragments of procedural knowledge. Even cooking or making clothes is substantially procedural. Teaching procedural knowledge is done in polytechnics and trade schools across the world. Curiously this is an area which mathematicians have not studied, yet it is replete with algebraic and geometric structures.