What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

Learning isn't possible without unlearning whatever temporary construct we used as a crutch, but it doesn't mean using that temporary construct is wrong even if it's not technically correct.

More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:

From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.

From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.

Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.

Repeated addition that ain't...

It boggles my mind that you've been studying Galois theory yet somehow try to reduce the algebraic structures associated to two binary operations to playing with one of them.

I think it's okay to say: "It's useful to visualize (or think of) multiplication as repeated addition, in contexts where that makes sense." Properly qualified the statement seems true and unbojectionable, and I think it's a useful way to help students understand multiplication. (It's not the only way.)

The reservation I'd have with the statement "Multiplication is repeated addition" is that the use of the word "is" implies an identification, or a necessary derivation of one from the other. The operations aren't identical, and one is not necessarily derived from the other. As operations, they're distinct.

(The issue also has nothing per se to do with units. When you introduce units of length or area, you're using numbers in a particular applied context. Outside of that context, numbers don't have units.)

In everyday usage "addition" and "multiplication" are conventional designations. Conventions always have ambiguities and edge cases.

The most common formal structure in which you have both addition and multiplication is in a ring. A ring is a set with two operations, which we call "addition" and "multiplication" --- but those are just names. We could just as well have used "foo" and "bar". We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention/convenience. (Note also that those operations are not unique.)

Rings include the integers, rationals, reals, complex numbers - but also (for instance) polynomials with real coefficients, 42 x 42 matrices with real entries, quaternions, or finite fields.

Now consider what happens as you list the axioms for a ring. You say that addition is associative, addition has an identity element (conventionally denoted "0"), every element has an additive inverse, and addition is commutative. That's addition.

Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted "1").

At the moment, you have two independently defined operations, which have nothing to do with one another. "Multiplication" is therefore not identical to (or defined as) repeated addition. But it's not too useful to have two independent unrelated operations. You connect the two operations by introducting the distributive axiom: For all a, b, c in your ring,

  a * (b + c) = a * b + a * c   and   (a + b) * c = a * c + b * c.

(scythe pointed out the importance of the distributive law in another reply.) The distributve axiom is huge! Consider

  a * (b + c) = a * b + a * c.

From left to right, it says that "you can multiply out"; from right to left, it says that "you can take out a common factor", which are both standard operations in algebra.

Once the distributive property is available to connect the operations, it explains why you can "think of" multiplication as repeated addition: For instance,

  3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1 = 3 + 3.

So you say "3 times 2" can be thought of as "3 added to itself 2 times". It does not say that multiplication is repeated addition, if "is" means "defined as" or "derived from",

Or consider one of the standard illustrations given to kids in grade school:

  3 * 2 is   $ $ $   which is   [ $ $ $ ]
             $ $ $              [ $ $ $ ]

This is 2 groups of 3, i.e. 3 + 3. So 3 * 2 = 3 + 3.

But what happened here? We relied on our physical intuition to "know" that putting the original 6 dollars into 2 bags in the second step didn't change the number of dollars. But formally, it is

  3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1,

which is the distributive law again.

(BTW regrouping the dollars again into 3 groups of 2 is a standard way of motivating commutativity of multiplication, since 3 groups of 2 is visibly "the same as" 2 groups of 3.)

(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don't know math history well enough to say, but historical derivation doesn't imply identity or logical derivation.)

So I think saying "Multiplication is repeated addition" is a little sloppy in the use of the word "is", but we can agree to disagree about how much sloppiness is okay. The statement is fine as a way of giving students one (of many) ways to think about multiplication. (Pictures are important, too!)

Here's something to think about. (I don't have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as "7 i added to itself (3 + 4 i) times". If you don't like the sound of that ... why? You might say "You can't have '3 + 4 i' things." Well, in the real world discrete "things" come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that "-3 things" means "I'm missing 3 things" or "you owe me 3 things". Nothing stops us from making agreements about the use of words. So maybe we could "agree" that "(3 + 4 i) times" means exactly an occurrence of the expression "3 + 4 i" in a context like this one. Is there any harm in that? :-)

The author is speaking from personal experiences teaching children K-10 (ages 5-16). It sounds like she is documenting her experience teaching these various temporary constructs, and that her experience is that teaching multiplication as repeated addition is a less effective way of teaching it.

> Ironically, this article wants to be very normative about which mathematical intuition is better…

No, the article is not being normative about “which intuition is better,” this is an incorrect reading of the article.

The article is giving advice about how to teach multiplication. Advice that is apparently based on years of experience teaching multiplication.

It’s not uncommon for people to experience teachers who prescribe specific intuitions about math rather than accommodate different intuitions—but the author is not doing that. What she’s doing here is outlining the various ways in which one particular intuition may fail you.

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may.

I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach multiplication when we got to that point (which we did about a year later).

I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.)

I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her.

With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller.

It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?