Friday, February 4, 2011

The Sweet Spot:
Balancing Conceptual and Procedural Instruction in Grade-School Mathematics

The purely procedural approach to long division.
I stole these colorful illustrations from this website.

A few years ago, I was working in a charter school where the sixth-grade math teacher was having a hard time teaching long division. (I myself, let it be noted, was having a hard time teaching anything at all.) I sat in on some classes, and it was easy to spot the problem. The teacher had reduced the process of long division to a series of five steps: divide, multiply, subtract, bring down, repeat, with a mnemonic, which I forget, to help remember them. This must have seemed like an easy-to-follow script when they were planning the lesson, but it’s deceptively complicated: you have to know which numbers to divide, multiply, subtract, and bring down and what to do with all those quotients, products, and differences once you get them. (See Appendix A, for an idea of just how complex this gets.) The students didn’t understand why they were doing any of these steps, so they found all that information extremely difficult to keep track of. Long division, taught this way, became a dull and intricate labyrinth, riddled with small procedural booby-traps to derail the unsuspecting scholar. Yet, pure procedural methods like this one seem to be prevalent in contemporary instruction—try Googling “long division” and see what comes up.

More recently, my girlfriend asked me to teach her long division, a skill she’d never gotten her brain around back in elementary school. Eager to show her that mathematics is logical and comprehensible, not arbitrary and byzantine, I dove into a thorough explanation of the inner-workings of the long-division algorithm, complete with diagrams and concrete examples. After ten minutes, she was frustrated and I was worn out. My attempts to reveal the logic behind long division had made the process even more complicated than had my former colleague’s attempts to reduce it to a mechanistic sequence of discrete steps.

When I saw my error, I crossed out all those diagrams and did what, deep down, I knew I should’ve done from the start. I taught her short-division with remainders—an intuitive process that anyone who has ever had to share seven cupcakes with two friends has naturally engaged with. Then, when she’d mastered that, I gave her a very simple sequence of steps to memorize: start with the leftmost digit of the inside number; divide it by the outside number; put the answer on top and the remainder down below; then bring down the next digit and do it again. Within ten minutes, she was doing long division by herself.

In the teaching of mathematics, I believe that there’s a sweet-spot: a proper mix of conceptual understanding and procedural memorization that leads most quickly and naturally to mastery of the material. When we stray to either side of this sweet-spot, math becomes difficult to teach.

If we overemphasize procedural memorization, math appears arbitrary, byzantine, and dull—and in the long run, students progress more slowly than they would with a balanced method. Purely procedural methods, like the one used by my former colleague in Harlem to teach long division, are difficult to master, because children lack the intuitive understanding that would glue the different steps together in their minds. We invent rhymes and acronyms and other mnemonics, but they are never elaborate enough to capture the details of the procedure, and a great deal of minutia must be drilled into place. This website full of long-division mnemonics gives a nice sense of the pains proceduralist math teachers go to in trying to turn a complex procedure into a sequence of robotic steps; Appendix A gives a list of the dozens of steps and special cases that actually go into doing long division correctly. Even after all the memorizing and drilling, these kinds of procedures rarely stick in kids’ heads for more than a month or two, so teachers are usually forced to re-teach them, year after year. Even if quick mastery of state-test material were one's only goal, this hardly seems the best way to go about it.

The opposite scenario is more complicated. Progressive (or “constructivist”) math teachers are guided by two related principles: first, that students should never be doing work that they cannot understand; and second, that students will learn best and understand their work most clearly if they are led to discover mathematical principles on their own rather than having those principles handed down to them by the teacher. I want to clarify that this is not passive education: the progressive math teacher must guide the students with carefully chosen problems and puzzles that will lead them to the next discovery; in many ways, her task is subtler and more complex than that of the proceduralist—but are the results better?

In a real-world context, at least, constructivism has obvious drawbacks. Truly discovery-based instruction must move at the students’ pace. That makes it incompatible with any set grade-level expectations; for all kinds of reasons, it’s practically unfeasible to get away from such expectations in an urban school system (see my August 3rd post). Furthermore, since different students move at very different paces, this requires tremendous individualization of instruction, which is possible only with a student-teacher ratio far below that of regular public schools.

There are, however, deeper reasons to question the wisdom of orthodox constructivism. First of all, it tends to compound differences in student ability. There’s much more natural variation in ability in math than in other academic disciplines; when every student must discover and invent for herself, those who cannot see the patterns as quickly fall behind. There are kids who like to figure stuff out on their own and kids who like to have things explained to them: we should not assume that the latter are just being lazy; that may be how they learn best. The mathematically minded student will always want to discover the answer—no wonder, then, that the most mathematically minded teachers tend to favor progressive methods: they’re teaching the way they themselves like to be taught; but that’s not necessarily how most students should be taught.

This points to an even deeper issue—one which constitutes the heart of my misgivings about pure constructivism. The idea that students should understand a procedure before they use it arises, I think, from one of the great hobgoblins of pedagogy: the desire for our students’ understanding to mirror our own. For those of us with a strong grasp of the subject, the procedures and rules of math are deeply logical and interconnected; to use them without an understanding of this logic and interconnection seems to us barbaric—but this is a pedagogical fallacy.

In fact, it is the natural course of learning that understanding should be at first partial and functional, and only later broad and theoretical. Only when the student has become familiar with the practice of a discipline is she ready to see the patterns that lead to the theory. This is all the more true with young children, who—as theories of child-development tells us and as experience confirms—reside in a concrete world and do not think in purely abstract terms.

The strange interconnectedness of mathematics muddies this essential pedagogical truth. Math is a castle built of math: math is the tool, math the substance, and math the product. The theory and the practice are therefore one, since the theory created the practice and the practice the theory. It therefore seems reductive, even barbaric, to separate the two, to teach the practice as if it were mere practice and not a gleaming mechanism of interlocking theory—but only the most talented students can see the patterns and interconnections from the outset. Most, I think, benefit from exploring mathematics first in a muted form, through a narrower lens. As Ms. Dickinson put it, “The truth must dazzle gradually.”

But it has to dazzle. We have to provide the necessary foundation of conceptual understanding to allow deeper inquiry in later years. How do we find that necessary balance of procedures and concepts, that sweet spot? I think that the best math teachers are always searching for that proper balance, and no doubt the experienced ones know where it lies far better than I. I hope, however, that the taxonomy of math knowledge that I mentioned in my last post will provide some added insight. I said I was going to discuss that taxonomy this week, but here we are, Friday Afternoon, 1,400 words in (not to mention three discarded drafts and an appendix), and I haven’t begun to write about it. So, once again, it will have to wait for next week. By now I know better than to make any promises, though. You’ll just have to read and see. It’s like a cliff-hanger, only much, much dryer.

Appendix A   ^ ^

The step-by-step procedure for long division presented in the sixth grade math class described at the beginning of this post leaves out a lot of important details and is much simpler than the actual process of long division. Below is a more complete script, one which provides all the details that a robot-child would need to do long division. I provide this to illustrate how absurd it is to try to memorize the long division procedure without understanding the short-division procedure of which it is composed.

Step 1:  Find how many times the divisor goes into the leftmost digit of the dividend.
·         If it goes in zero times, try again with the two leftmost digits of the dividend; if the result is still zero, then try it with three, and so on.
·         Otherwise, go to step 2.

Step 2:  Put the result above the digit of the dividend you just used. (If you just used more than one digit of the dividend, put the result above the rightmost of the digits you just used.)

Step 3:  Multiply the number you just wrote down by the divisor.

Step 4:  Subtract the result from the leftmost digit of the divisor.

Step 5:  Bring down the next digit of the dividend so it is next to the result of step 4, to form a two-digit number. (If the result of step 4 was zero, this will only be a one-digit number.)

Step 6:  Find how many times the divisor goes into the number formed in step 5.

Step 7:  Repeat steps 2-6 until you have reached a step 5 but have no more digits in the dividend to bring down.
·         If the result of the last time you did step 4 was zero, then you’re done, and the answer is the number written above the dividend.
·         Otherwise got to step 8.

Step 8:  Form a fraction by placing the result of the last time you did step 4 above the divisor. (Don’t forget to reduce!) The answer is the number written above the dividend plus this fraction.

Obviously, this would have to be more complicated if you wanted to get a decimal result rather than a fractional one.


  1. In some sense, the division between concepts and procedures is superficial, since the way in which higher mathematical knowledge is understood is itself procedural. Understanding math means understanding what operations (procedures) defined over numerical symbols are valid, and composing these valid operations into more complex ones. I don't think you can "understand" the concept of a number independently of the operations that manipulate it. The reason I bring this up is that perhaps the question is not how to balance concepts vs. procedures but what particular procedures are useful for someone to know. I definitely think that long division is not a useful procedure (not since the invention of the calculator... or even the abacus!), but it may be harder to come up with positive examples without discerning how a student will use math in the future.

  2. Sam’s comment pushes on some deep questions about what conceptual knowledge is and why it matters. I maintain that it does exist and does matter, but what and why are complicated questions.

    It’s true that most of math can be viewed as a set of rules for manipulating symbols, but this is not how human beings interact with the subject—either mathematicians or students. Viewed as symbol-manipulation rules (as pure procedures, in other words), mathematics would be incomprehensible to the human mind. It is the patterns and symmetries within mathematics and the metaphorical relationships between mathematics and the concrete world that allow us to understand math, to give it form in our minds, to have intuition about it, and to use it to solve problems. No doubt, you could relate many of these concepts to some procedural rule, but that is not how students and mathematicians interact with those concepts, and the ability to perform the relevant procedure does not necessarily imply a clear understanding of the concept. I hope to give a more detailed discussion of this issue in my next post.

    Sam also raises the question of the relevance of long division to modern life. Reexamining which topics are worth teaching is certainly worthwhile, and long division would probably be high on many people’s list for topics to get the axe. The reality is, however, that the content of the math curriculum in almost any school, and certainly at any public school, is tightly constrained by the SAT and by the State Departments of Education. Without such external constraints, we might well reexamine the entire curriculum—why, for example, do we teach trigonometry, as opposed to say, number theory, linear algebra, set and group theory, topology, and so on? Trigonometry underlies none of these fields, nor is it strictly necessary for the study of calculus—you could easily cut the trig out of calc—but why teach calc anyway? And most of algebra 2 seems unnecessary, once you drop calc and trig. A lot of the standard highschool math curriculum is arbitrary; much of it arises out of historical accident; and rarely does it cover the most interesting topics. Questioning this is a valuable exercise, but at the end of the day, this is the reality we have to live with, as educators, as students, and as citizens.

  3. I totally agree with Sam's comment. I really makes you think about the objectivity of your action.