The story so far
Last year I wrote about a kerfuffle of the mathematical portion of the Internet over a series of three articles by Keith Devlin that claimed not only that teaching multiplication as repeated addition was not only wrong, but outright harmful.
I mentioned I was working on an ultimate post that delivered a final verdict. This is still not that post, because the original got caught in a cycle known as “looking for more research”, “rewriting”, “reconsidering”, and then a touch of old fashioned “procrastination”.
However, Mr. Devlin has returned yet again to the topic in a January 2010 article, so to reduce my guilt over what will likely be an endless process, I’m going to put forth my main point.
My main contention
One common argument pattern (in education, politics, or anything really) is when two sides take different sets of premises but don’t argue about their premises; they instead argue about their ramifications. In an effort to get to the heart of the matter, in answering “is multiplication ever repeated addition?” I attest:
It depends on your philosophy of mathematics.
Two quotes from mathematicians
For newcomers, I’ll begin by saying it one more time: multiplication is not, repeat not, repeated addition. Not even for positive whole numbers. This is the one case where you do get the same answer, but getting the same answer to two procedures does not make them the same procedure. —Keith Devlin
Just as addition can be thought of as repeated counting, multiplication can be thought of as repeated addition. — Melving Fitting and Greer Fitting
So, one mathematician says multiplication is not repeated addition in any situation whatsoever, while another pair says it is exactly that! What’s going on? Consider the second quote in context…
Adding and multiplying defined from scratch: the executive summary
While there are many books on basic number theory, I’m going to discuss the one from the second quote, which happens to be online.
1. Start with the natural number system, by establishing counting numbers starting with 1, which have successors (eg the successor of 4 is 5). This is pre-addition, so the notation for “3 successors after x” is .
2. The addition x + y is defined as taking the yth successor after x, or as they put it, “We might think of 4 + 3 as a set of instructions to us, telling us to start with the number 4 and count off the next 3 numbers.” In their notation that would mean is the same as 4 + 3. In other words, addition can be thought of as repeated counting.
3. Now, multiplication.
Our informal guide is: to multiply 4 by 3 we write down 3 4’s and add them. But how can we express this without dragging in machinery to count things? We look for an alternative definition which will result in being the result of adding 4, three times, but won’t make it necessary to count 4’s.
This follows with an official definition of multiplication, notated by , as that operation such that
The first part the identity operation (5 times 1 is 5) but the second is considerably more clever. Rather than defining x times y, it defines x times y+1. This is best illustrated in steps:
Step 1. by the first definition.
Step 2. , adding 5 to the answer from Step 1.
Step 3. , adding 5 to the answer from Step 2.
Step 4. , adding 5 to the answer from Step 3.
and so on.
Stepping back, while the “recursive programming” nature of the definition is odd, it is clear the definition of multiplication is causing repeated addition.
4. Add 0 to get the whole number system. Addition is unproblematic: just define to be 1.
5. Multiplication, though, gets messier:
The idea we followed in the last chapter was that multiplication was repeated addition, and so told us to add 3 2’s together. This idea easy covers It tells us to add 3 0’s together . . . There is more a problem with however. If we apply our basic idea, we find ourselves adding together 0 3’s. We have no 3’s (or anything else) to work with, so we certainly can’t get a number for an answer.
This is neatly handled by special-case defining .
Note, however, that we are not strictly following the “repeated addition” algorithm anymore, although the special case is a “natural extension” of the idea.
Now, one can go on to fractions and so forth, but given that Mr. Devlin is arguing that multiplication is not repeated addition even in the case of positive whole numbers I’m going to stop here; just keep in mind each step can be a “natural extension” of the previous idea but the tweaks make them not formally identical.
If something is a “natural extension” (in this case, having to change things with 0) is it still “the same as” the original?
But more importantly: is the original formal method made to define addition, once changed, still “a part of” the fuller picture of mathematics? Is the formalism a seed still planted within the mathematical framework, or does the addition of an axiom (or two axioms, or five axioms) make any thinking about the “internal” model wrongheaded?
It is no accident many ardent defenders of multiplication as repeated addition come from computer science; in programming, old code has a framework built around it that does not change the nature of the original code. Whereas when mathematicians add to a theory, even a single extra axiom may radically shift the nature of a structure (take for example mathematics with or without the Axiom of Choice).
Want to mess with your brain further? Try the Doyle and Conway paper Division by three with the grand first line:
In this paper we show that it is possible to divide by three.
and the ominous:
There is more to division than repeated subtraction.
EDIT: Comments are closed for now. I will be posting more about this topic later at which point I will re-open for comments.
Filed under: Education