What is conceptual understanding?

A definition everyone agrees with, just you wait

I read a post by Greg Ashman, Conceptual Understanding is a Myth. I respect Greg not only because he is a serious, provocative writer about education but because he is Australian like me. So this got me thinking that I’d better check whether this thing that I’ve experienced all my life really exists. Let me try to describe that experience. If I’m trying to understand the proof of a theorem, there’s a stage where I am working through the steps, checking them carefully. If the proof is complex it’s easy to get lost, but you keep plodding. Then there’s a stage where you assemble all the pieces in your head, see how they fit together, understand the essential ideas. That’s conceptual understanding. Ashman dismisses it as “a feeling or impression.” I might call it a cognitive structure. But I don’t really care what you call it: it is consequential. Once you have it you don’t need to memorize the proof because you can reconstruct it yourself. And lest you think it’s something only mathematicians experience, I remind you of the fourth grade student Max, who grasped a general principle that had consequences for his ability to add and subtract.

I’ll get back to Ashman, but first let me take a tour through the literature, looking for definitions.

Relational and instrumental understanding

I found a 50-year old paper by Skemp which is strikingly relevant to today’s debates. Skemp distinguishes between relational understanding and instrumental understanding:

By the former is meant what I have always meant by understanding, and probably most readers of this article: knowing both what to do and why. Instrumental understanding I would until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’, without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was [what] they meant by ‘understanding’.

Skemp’s instrumental is our procedural, and his relational is our conceptual. He gives an analogy which is close to the description of my own experience above:

A person with a set of fixed plans [for navigating a town] can find his way from a certain set of starting points to a certain set of goals. The characteristic of a plan is that it tells him what to do at each choice point: turn right out of the door, go straight on past the church, and so on. . . .

In contrast, a person with a mental map of the town has something from which he can produce, when needed, an almost infinite number of plans by which he can guide his steps from any starting point to any finishing point, provided only that both can be imagined on his mental map.

This harmonizes with the experience of understanding a proof that I described above. For Skemp, relational understanding is a

conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point.

Skemp is candid in acknowledging his own biases, and in admitting that his earlier failure to appreciate instrumental understanding was a mistake. He makes an effort to play devil’s advocate for the notion, at the same time as arguing in the end for the superiority of relational understanding for the reason quoted above. He concedes that anything can be taught and understood in an instrumental way, and gives an amusing example from his own work in “modern mathematics,” the UK version of what was called the New Math in the US.

I was in a school which was using my own text, and noticed . . . that some of the pupils were writing answers like ‘the set of {flowers}’.¹ When I mentioned this to the teacher (he was head of mathematics) he asked the class to pay attention to him and said: “Some of you are not writing your answers properly. Look at the example in the book, at the beginning of the exercise, and be sure you write your answers exactly like that.”

This foreshadows a point Ashman makes 50 years later, citing examples of supposedly conceptual problems that can be answered procedurally. The strongest point in Ashman’s post is the difficulty of measuring conceptual understanding.

The five strands of mathematical proficiency

The National Research Council’s report Adding it Up describes five strands of mathematical proficiency: procedural fluency, conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. Their definition of conceptual understanding is “comprehension of mathematical concepts, operations, and relations.” They go on to say:

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. If students understand a method, they are unlikely to remember it incorrectly. They monitor what they remember and try to figure out whether it makes sense.

This echos Skemp’s notion of a schema. But notice a subtle shift here: Adding It Up has not made Skemp’s rich connected structure—knowing the layout of the town—part of the definition. Rather they have described that structure as a consequence of acquiring conceptual understanding. And they have given behavioral descriptions of conceptual understanding: integration, organization, ability to represent in multiple ways, ability to reconstruct when forgotten, ability to generate new knowledge. This is more than Ashman’s “feeling or impression.” And because these are behaviors, they are observable, and have the potential to answer the measurement question. The report hints at what such measurement might look like: “Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems.” Hold onto that thought about solving new and unfamiliar problems.

Procedural knowledge and conceptual knowledge

Ashman quotes a paper by Rittle-Johnson et al. to suggest that there is really no difference between procedural knowledge and conceptual knowledge, but he leaves out some context. Here is the full paragraph, with the bit Ashman quotes in bold.

There is a long-standing and ongoing debate about the relations between conceptual and procedural knowledge (i.e., knowledge of concepts and procedures). Although there is broad consensus that conceptual knowledge supports procedural knowledge, there is controversy over whether procedural knowledge supports conceptual knowledge and how instruction on the two types of knowledge should be sequenced. A review of the empirical evidence for mathematics learning indicates that procedural knowledge supports conceptual knowledge, as well as vice versa, and thus that the relations between the two types of knowledge are bidirectional.

Ashman takes that last sentence and runs with it.

Why would this be the case? Well, they would support each other if they were effectively the same thing. This could be an example of the jangle fallacy — the mistaken impression that two identical or near-identical things are different because they have been given different names.

But it is clear from the earlier sentences that the authors believe there is a difference between conceptual knowledge and procedural knowledge; indeed, they cite the Adding It Up definition approvingly. Two constructs can support each other without being identical.

Conceptual understanding and transfer

Ashman gives some examples of questions supposedly assessing conceptual understanding, pointing out that they could all be solved procedurally if students are given the right training. The examples are embarrassing and he is right to pillory them. Here he has rediscovered the point Skemp made 50 years ago. He makes the point that the real test of conceptual understanding is in the novelty of the questions. Here he has rediscovered the point that Adding It Up made 25 years ago, about “solving new and unfamiliar problems.” He then pivots to his own version of the jangle fallacy.

If you think that conceptual questions only work if students haven’t seen those question types before and if you think training students in methods to solve those specific questions is a form of cheating, you are really not grasping for the conceptual knowledge but for the transfer of learning.

Good point about transfer of learning being the behavior to observe. But what is the cognitive structure that would allow for transfer of learning? Ashman links approvingly to this chapter in the NRC report How People Learn, which has the choice quote, “Transfer is affected by the degree to which people learn with understanding rather than merely memorize sets of facts or follow a fixed set of procedures.” So it seems that conceptual understanding and transfer are, if not exactly two names for the same thing, two sides of the same jangling coin. Conceptual understanding is the cognitive structure that leads to transfer; transfer is the way you measure it. Ashman is right about that.

Assembling the sources, here is the definition everyone can agree with:² conceptual understanding is the connected cognitive structure—Skemp’s schema, Adding It Up’s “integrated and functional grasp”—that lets you handle problems you haven’t seen before. Ashman is right to point out that it’s hard to measure, but wrong to treat the measurement difficulty as evidence that it doesn’t exist.

Ashman declares conceptual understanding a myth and says we should measure transfer instead. But his own sources say that transfer is conceptual understanding measured. Ashman’s remedy detects the thing whose existence he denies. He hasn’t refuted it, he has renamed it.

1. The New Math was big on sets and set notation. I remember having to use those curly brackets to write the solution set of an equation in seventh year at North Sydney Boy’s High School.

2. Just kidding, there is nothing that everyone can agree with.