Conceptual understanding is a myth
Transfer is a better focus
For the reasonable price of just $5 AUD per month or $50 per year, you can be a full subscriber to Filling the Pail with complete access to my extensive archive on a range of topics, as well as access to the unabridged version of my weekly Curios posts.
If you begin to participate in the debate about mathematics teaching, two things quickly become apparent. The first is gaslighting. This is where people will claim there really is nothing to debate and other people are creating false binaries. These arguments are similar in form to claiming the existence of Birmingham proves there is no such thing as Northern or Southern England.
The second feature is a focus on the hocus pocus of conceptual understanding. This is a dodge. Advocates of inquiry learning will sometimes concede that explicit teaching is effective at teaching ‘rote’ procedural knowledge, but inquiry learning better delivers on more nebulous aims. One of the commonly-stated aims that inquiry learning purportedly supports is the development of conceptual understanding.
Conveniently, this is less likely to show up on standardised assessments.
When researchers try to assess conceptual understanding, a strange transformation takes place. Suddenly, it becomes conceptual knowledge. When looking at the evidence from such research, one response is to highlight the fact that procedural knowledge and conceptual knowledge are mutually supportive. As Bethany Rittle-Johnson and colleagues write, “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.”
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.
The reason understanding becomes transformed into knowledge in these studies is that understanding is a feeling or impression. You may feel you understand linear algebra and you may sense someone else understands fraction addition. However, this is hard to measure. What does ‘understanding’ look like beyond being able to do and say certain things? Therefore, it is the doing and saying that are measured and these look a lot like knowledge. How, exactly, is this different to procedural knowledge?
I recently tweeted about a paper in this tradition by Michael D’Erchie and colleagues that lists examples of the fractions and algebra questions they asked German middle-school students to solve in order to assess their conceptual and procedural knowledge.
One of the questions asks how many fractions sit between these two fractions:
The answer is that there is an infinite number, but students could be tricked into listing only the eighths, including or excluding the end-points. The underlying idea can, and often is, explicitly taught to students and so students who answer this correctly may simply have remembered that there are an infinite number of fractions.
Another question shows a shape made up of two equally-sized segments that is two-thirds of the whole and students are asked to add to it to make a whole. This is a question that could be taught procedurally. One of the questions asks which of two fractions is larger and then asks for an explanation. Middle-school students can and should be taught a procedure for determining which of two fractions is larger. So, is it the explanation that is the critical point that assessing conceptual knowledge? Maybe not. Speak to a colleague who has explicitly taught students how to write explanations in examined courses in science, economics or business management and you will realise that this is their daily work.
You may think teaching these procedures sounds like cheating. Surely, these questions work as a test of conceptual knowledge if, and only if, students have not already been shown how to solve them. OK, so how do we control for that? Perhaps we need to make the questions really weird and unusual to help ensure they haven’t seen anything like them before? Maybe, but the questions in the D’Erchie et al. paper are standard.
I don’t think anyone would claim that ‘conceptual knowledge’ questions should be so weird and unusual that students need to invent new mathematical concepts to answer them, so there need to be some elements that can migrate from question types that students have seen before to the new questions we want them to answer. If so, we are now approaching something similar to Thorndike’s theory of transfer of learning.
Transfer is the ability to apply what we have learned in one context to a different context and it is notoriously difficult to achieve in educational psychology research. Thorndike proposed that the ability to transfer learning depends on the extent to which different contexts share common elements. We cannot simply learn to think better. Learning Latin or chess does not make a person smarter in some general way; it makes them more able to complete tasks and solve problems that share common elements with Latin or with chess.
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. There is nothing structural about a question that makes it conceptual rather than procedural because a procedure—and sometimes even the recitation of a definition—may be used to solve both. No, it is the fact students have not seen this question type before that is important.
Crooks and Alibali published a paper in 2014 that examined the ways that maths education researchers had, like D’Erchie and colleagues, attempted to measure conceptual understanding. They found that the most common conceptual question students were asked was to explain what the ‘=’ sign means. Vital as an appreciation of the principle of equivalence is in mathematics, answering such a task requires little more than reciting a definition—a little like the ‘infinite fractions’ question above. Ironically, therefore, advocates of conceptual understanding who rage against the recitation of rote knowledge are perhaps attempting to measure their own success by requiring the recitation of rote knowledge.
In contrast, the procedures these same advocates disparage are far harder to reproduce. At the very least, they involve recalling the procedure and then applying it to a new set of numbers and most of the time, there will be considerably more transfer of learning required than that. This means successfully applying procedural knowledge will be much harder to teach students than performing some tasks classified as assessing conceptual knowledge.
Rather than asking students to recall definitions or fooling ourselves that some kinds of procedural questions really are procedural, whereas others are conceptual, we should focus instead on measuring transfer of learning.
Conceptual understanding is a myth.
Excellent piece!
The belief that teaching procedures will result in “math zombies” is entrenched in educational culture. The people pushing these ideas view the world through an adult lens which they’ve acquired through the very practices that they feel do not work. They become angry that their teachers (supposedly) didn’t explain all these things to them and are certain that they would have liked math more and done better if only their teachers would have focused on the nebulous "conceptual understanding". Their views and philosophies are taken as faith by school administrations, school districts and many teachers — teachers who have been indoctrinated in schools of education that teach an "understanding uber alles" approach to math.
These ideas are so entrenched that even teachers who oppose such views feel guilty when teaching in the traditional manner so reviled by well-intentioned reformers. Given that today’s employers are complaining over the lack of basic math skills their recent college graduate employees possess, the fixation on conceptual understanding that prevails in the early grades has created a poster child in which “understanding” foundational math is often not even “doing” math at all.
I agree that mathematics education researchers tend to get stuck on conceptual understanding, viewing it as a prerequisite to learning any kind of procedure, despite the fact that procedural knowledge and conceptual understanding often develop concurrently, and that one does not necessarily have to come before the other. They even sometimes seem to believe that learning procedural knowledge too soon is harmful, which is something I don't think there is any serious evidence for. And of course, as you point out, measuring conceptual understanding is flawed, since so-called demonstrations of conceptual understanding can be learned by rote just as well as procedures. But I don't think this amounts to conceptual understanding being a "myth", rather than it being hard to measure. It is something real and something we want our students to develop. In this sense I am sympathetic to mathematics education researchers, even when I think they unnecessarily place conceptual understanding in opposition to procedural knowledge.
I'm reminded of a post that I read on, I believe, Reddit, a few years ago. In it the author claimed that students in some Asian country (I don't remember which) tended to be procedurally fluent, but to lack conceptual understanding. The example they gave was that if you asked the students to compute the least common multiple of 15 and 18, they could do so without any problem. (I assume they used Euclid's algorithm to find their greatest common divisor, and then computed the least common multiple as the product of 15 and 18 divided by their greatest common divisor, 3, which gives 90.) But if you gave them the arithmetic sequences 15, 30, 45, ... and 18, 36, 54, ..., and asked them which is the first number found on both sequences, they had no idea. Now I don't know if these students really exist (though it seems possible). But if they do, I assume their lack of ability to answer the question comes from the arithmetic sequence context being new to them and something they cannot connect to the context in which they learned about least common multiples. Which you would probably call a lack of transferability.
I also want to push back against the claim of gaslighting. I'm not saying educationists never engage in gaslighting (they sometimes do), but I don't think claiming that the traditionalist/progressive dichotomy in mathematics teaching is a false binary is an instance of this; I think it's a sincerely held belief. You, Greg, view so-called "traditional" teaching methods such as explicit teaching (though I think your form of effective explicit teaching isn't exactly traditional) as an objectively superior teaching method, while inquiry learning is objectively inferior, and you identify the goal of schools to be teaching as efficiently as possible. In this view, you either teach explicitly all the time, or you don't. You advocate the former, your opponents advocate the latter. There is obviously a debate there. But your opponents view explicit teaching, inquiry learning, project-based learning, and so on, as tools in an effective teacher's arsenal. Sometimes one is more appropriate than the other, but not always. (The fact that they probably think of "explicit teaching" as only meaning lecturing is probably salient here. See, they use "explicit teaching", but also exercises, problems, etc.) So there is no debate: a skilled teacher must be fluent in all teaching methods and know when to implement them.