## Devlin's Angle |

Conceptual understanding is one of the five strands of *mathematical proficiency*, the overall goal of K-12 mathematics education as set out by the National Research Council's 1999-2000 Mathematics Learning Study Committee in their report titled *Adding It Up: Helping Children Learn Mathematics*, published by the National Academy Press in 2001.

I'm a great fan of that book, so let me say up front that I think achieving conceptual understanding is an important component of mathematics education. That appears to pit me against one of the two opposing camps in the math wars - the skills brigade - so let me even things up a bit by adding that I think many mathematical concepts can be understood only after the learner has acquired procedural skill in using the concept. In such cases, learning can take place only by first learning to follow symbolic rules, with understanding emerging later, sometimes considerably later. That probably makes me an enemy of the other camp, the conceptual-understanding-first proponents.

I do agree with practically everyone that procedural skills that are not eventually accompanied by some form of understanding are brittle and easily lost. I believe that the need for rule-based skill acquisition before conceptual understanding can develop is in fact the norm for more advanced parts of mathematics (calculus and beyond), and I'm not convinced that it is possible to proceed otherwise in all of the more elementary parts of the subject.

[LONG ASIDE: Theoretically, I think it probably is possible to achieve understanding along with skill mastery for any mathematical topic, but it would take far too long, with a likely result that the student would simply lose heart and give up long before achieving sufficient understanding. But as language creatures - the "symbolic species" to use Terrence Deacon's term [Terrence Deacon, *The Symbolic Species: The Co-Evolution of Language and the Brain*, W.W. Norton, 1997] - we have a powerful ability to learn to follow symbolic rules without understanding what they mean; and as pattern recognizers with an instinct to perceive meaning in the world, once we have learned to play such a "symbol game" it generally does not take us long to figure out what it means. Chess playing is an excellent example of how learning to play by simply following the rules eventually leads to an understanding of the game. Thus, while the idea that students should "understand before they do" has a lot of appeal, it ignores that fact that nature has equipped us with a far more efficient method of learning.]

But that is not my focus here. Rather the question I am asking is, what exactly *is* conceptual understanding?

My problems are, I don't really know what others mean by the term; I suspect that they often mean something different from me (though I believe that what I mean by it is the same as other professional mathematicians); and I do not know how to tell if a student really has what I mean by it.

*Adding It Up* defines conceptual understanding as "the comprehension of mathematical concepts, operations, and relations," which elaborates the question but does not really answer it.

"*Suppose that you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractable claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying 'kitty' each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience.*"

This idea is appealing, but not without its difficulties, the primary one being that the learner may end up with a concept different from the one the instructor intended! The difficulty arises because an abstract mathematical concept generally has fundamental features different from some or even all of the examples the learner meets. (That, after all, is one of the goals of abstraction!)

An important illustration of this that has been much studied is the modern mathematical concept of a function. For instance, the Israeli mathematician and mathematics educationalist Uri Leron, in his article "Mathematical Thinking and Human Nature: Consonance and Conflict"
[*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004.* (3) 217-224] wrote:

"*According to the algebraic image of functions, an operation is acting on an object. The agent performing the operation takes an object and does something to it. For example, a child playing with a toy may move it, squeeze it, or color it. The object before the action is the input and the object after the action is the output. The operation is thus transforming the input into the output. The proposed origin of the algebraic image of functions is the child's experience of acting on objects in the physical world. . . . Inherent to this image is the experience that an operation changes its input - after all, that's why we engage it in the first place:
you move something to change its place, squeeze it to change its shape, color it to change its look. *

*
But this is not what happens in modern mathematics or in functional programming. In the modern formalism of functions, nothing really changes!
The function is a "mapping between two fixed sets" or even, in its most extreme form, a set of ordered pairs. As is the universal trend in modern mathematics, an algebraic formalism has been adopted that completely suppresses the images of process, time, and change.*"

Leron and others have carried out several studies demonstrating that many mathematics and computer science students at universities have formed erroneous concepts of functions; assuming in particular that applying a function to an argument changes the argument. Such is the power of the original examples, that even when presented with the correct formal definition of the general, abstract concept, the learners assume features suggested by the examples that are not part of the abstract concept. It can take an experienced instructor some time to uncover such a misconception, let alone correct it.

Thus, whereas conceptual understanding is a goal that educators should definitely strive for, we need to accept that it cannot be guaranteed, and accordingly we should allow for the learner to make progress without fully understand the concepts.

The authors of *Adding It Up* seem to accept this problem. Rather than insist on full understanding of the concepts, the committee explained further what they meant by "conceptual understanding" this way (p.141), "... conceptual understanding refers to an integrated and functional grasp of the mathematical ideas."

The key term here, as I see it, is "integrated and functional grasp." This suggests an acceptance that a realistic goal is that the learner has sufficient understanding to work intelligently and productively with the concept and to continue to make progress, while allowing for future refinement or even correction of the learner's concept-as-understood, in the light of further experience. (It is possible I am reading something into the NRC Committee's words that the committee did not intend. In which case I suggest that in the light of further considerations I am refining the NRC Committee's concept of conceptual understanding!)

Since the distinction I am making is somewhat subtle, let me provide a dramatic example. As the person who invented calculus, it would clearly be absurd to say that Newton did not understand what he was doing. Nevertheless, he did not have (conceptual) understanding of the concepts that underlay calculus as we do today - for the simple reason that those concepts were not fully worked out until late in the nineteenth century, two-hundred-and-fifty years later. Newton's understanding, which was surely profound, would be one of functional understanding. Euler demonstrated similar functional understanding of infinite sums, though the concepts that underpin his work were also not developed until later.

One of the principal reason why mathematics majors students progress far, far more slowly in learning new mathematical techniques at university than do their colleagues in physics and engineering, is that the mathematics faculty seek to achieve full conceptual understanding in mathematics majors, whereas what future physicists and engineers need is (at most) functional understanding. (Arguably most of them don't really need that either; rather what they require is another of the five strands of mathematical proficiency, *procedural fluency.*) I have taught at universities where the engineering faculty insisted on teaching their own mathematics, precisely because they wanted their students to progress much faster (and more superficially) through the material than the mathematicians were prepared to do.

Teaching with functional understanding as a goal carries the responsibility of leaving open the possibility of future refinement or revision of the learner's concept as and when they progress further. This means that the instructor should have a good grasp of the concept *as mathematicians understand and use it.* Sadly, many studies have shown that teachers often do not have such understanding, and nor do many writers of school textbooks.

I'll give you one example of just how bad school textbooks can be. I was visiting some leading math ed specialists in Vancouver a few months ago, and we got to talking about elementary school textbooks. One of the math ed folks explained to me that teachers often explain whole number equations by asking the pupils to imagine objects placed on either side of a balance. Add equal numbers to both sides of an already balanced pairing and the balance is maintained, she explained. The problem then is how do you handle subtraction, including cases where the result is negative? I jumped in with what I thought was an amusing quip. "Well," I said with a huge grin, "you could always ask the children to imagine helium balloons attached to either side!" At which point my math ed colleagues told me the awful truth. "That's exactly how many elementary school textbooks do it," one said. Seeing my incredulity, another added, "They actually have diagrams with colored helium balloons gaily floating above balances." "Now you know what we are up against," chimed in a third. I did indeed.

I suspect that I am not alone among MAA members in my ignorance of what goes on at the elementary school level. My professional interest in mathematics education stretches from graduate level down to the top end of the middle school range, with my level of experience and expertise decreasing as I follow that path. Sure, I can see how the helium balloon metaphor can work for the immediate task in hand of explaining how subtraction is the opposite of addition. But talk about a brittle metaphor! It not only breaks down at the very next step, it actually establishes a mental concept that simply has to be unlearned. This is surely a perfect example of using a metaphor that is not consistent with the true concept, and hence very definitely does not lead to anything that can be called conceptual understanding.

Devlin's Angle is updated at the beginning of each month.