The Appeal to Common Practice

Folks are still kicking around the repeated-addition-vs.-multiplication stuff, and clearly there are more than two sides to this debate, which, to the extent that it represents a development at all, is a positive development.

Deserving of more scrutiny, though, is this widespread notion among commenters--a keystone belief supporting many different arguments--that students' collective intuitions about mathematics serve as a kind of useful, or even necessary, foundation on which curricula should be built. This is, essentially, a warped version of the notion (or perhaps mantra) of what's known as "teaching from the known to the unknown."

Known to Unknown

At any point in a child's life or schooling, he or she presents with a number of things he or she can do and a number--which could be 0--of things he or she knows. We can refer to these collectively as the "knowns." And, of course, the "unknowns" are all those things a child does not know or cannot do at any of the same points. The problem of teaching from the known to the unknown involves making some kind of connection from a student's knowns to a very restricted set of unknowns, which, taken together at any point, form a kind of immediate curriculum.

Now, of course, it is impossible to teach without going from the known to the unknown in some way. On the one hand, a student can't learn anything if s/he has absolutely no knowledge or skills (because then s/he wouldn't exist), and on the other hand, nothing can be described purely in terms of itself (although God takes a good crack at it in Exodus when he says to Moses, "I am that am"). The inevitable connection from known to unknown itself is not at issue. What is at issue is the way this connection is made. What knowns are connected to what unknowns?

The Best "Known"

Over a wide variety of topics, educators will often argue about the quality of the knowns to be connected to specific unknowns. The ongoing debate about whether to teach fractions first or decimals first is an area where this argument pops up, with some making the case that place value is the better "known" to be connected to the unknown of rational numbers (decimals first) while others argue that equal shares is the better known (fractions first). Similarly, one can argue that, for the unknown of improper fractions, proper fractions serve as the best "known," whereas another can argue that, because improper and proper fractions are used in such diverse situations (e.g., "no one says that they have 8/5 dollars"), we must scrap the use of proper fractions as the "known" in introducing improper fractions and come back to the connection later.

Proponents of the equivalence between repeated addition and multiplication--for brevity's sake, I'll just call them "repeaters"--often argue that repeated addition is the best "known" (or, in most cases, the only known) that can be connected to the unknown of multiplication. And there are several problems with these arguments, given the various ways they are presented and the premises on which they rest. One of those is called the appeal to common practice.

Appeal to Common Practice

This is a fallacy. And it works like this: Such and such an action is justified because it is what everyone else is doing or what we've always done. Now, virtually no repeaters that I've read actually commit this fallacy so nakedly. But it does creep up somewhat, um, "un-nakedly." Here's Mr. Mark, falling into the fallacy with repeated multiplication:

Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication. Find me a beginners textbook or teachers class plans that explains exponentiation to kids without at least starting with something like "5² = 5 × 5, 5³ = 5 × 5 × 5."

The second of those sentences is pretty clearly the fallacy of appealing to common practice, to the extent that it is used in any way to justify or excuse the teaching of exponentiation as repeated multiplication. But notice what is said in the first sentence: "Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication." This, too, is an appeal to common practice, but the practice in this case is not necessarily the teaching of exponentiation as repeated multiplication to fifth or sixth graders but, rather, the teaching of everything before that. The argument is that repeated multiplication is the best "known" because currently the 8 to 10 years of schooling prior to teaching the "unknown" of exponentiation don't prepare students for learning exponentiation any other way (or any better way).

But these circumstances do not make repeated multiplication the best "known," just the most expedient "known." The same goes for repeated addition as a "known" connected to the unknown of multiplication. Expedience is, of course, criterial when one is a classroom teacher forced to get Johnny to grok multiplication any way she possibly can, but it is not a convincing justification for repeated addition as the best foundation for learning multiplication.

The idea that multiplication and repeated addition are not the same has implications for the reorganization of our mathematics curricula. It cannot be judged convincingly one way or the other by how well or how badly it fits into those curricula.

As always, more later.