On the number line at the right, the tick marks are all evenly spaced and the values for the tick marks increase from left to right. One can perform repeated addition of 2 from the rectangle to arrive at the value for the circle or repeatedly subtract 2 from the circle to get to the rectangle. In other words, you can determine the additive relationship between the values for the circle and rectangle.

What you cannot do, however, is determine the multiplicative relationship between the values. If the rectangle is at 4, then the circle is at 18, and $$\mathtt{\frac{18}{4} = 4.5}$$, which means nothing in the context of this number line.

We can also determine a multiplicative relationship without being able to identify an additive one. Given the same assumptions as above, we know that the value associated with the square is, without a doubt, $$\mathtt{\frac{1}{4}}$$ the value for the rectangle. But there’s no way of telling what the distance between them is.

Although getting into the math is not the reason for my writing this post, I want to stick with the above contrast briefly. The point is simple, but (believe me) hard to swallow: no matter how tightly connected the two operations are, no matter how many years you have taught it this way or how fine you turned out as an adult after learning it this way, no matter Peano or extensionality, multiplication is not repeated addition. It is not that it is “not just” repeated addition. And it’s not “just semantics.” Multiplication is not repeated addition. The two are not the same.

One important reason for the distinction, in my view, is that it forces us as educators to “level up” to the multiplicative and treat it as basic and fundamental. We are forced to connect multiplication to more intuitive operations, like scaling and stretching, which in turn means that students will have more direct psychological access to ratio, scale, unit rate, slope, and on and on, rather than having to build everything up from the numerical-additive every time a new concept is introduced.

In one of his famous classic lectures (39:42), Richard Feynman perfectly explains why multiplication is to be chosen over repeated addition, even if they lead to the same intuitive conclusions in every context in which they are applied. Except, of course, Feynman is talking about physics. He says, referring to three ‘different’ laws of motion:

These theories are exactly equivalent. The mathematical consequences in every one of the different formulations of the three formulations—Newton’s laws, the local field method, and this minimum principle—give exactly the same consequences. What do we do then? . . . They are equivalent. Scientifically, it is impossible to make a decision. . . . Psychologically, they are different because they are completely unequivalent when you go to guess at a new law . . . they become not equivalent in psychologically suggesting to us the guess as to what the laws might look like in a wider situation.

So it is with repeated addition and multiplication. Absorbing multiplicative reasoning into our bones, treating multiplication as fundamental rather than derived, allows us to “guess at” what’s going on in a lot of middle-school and later mathematics. For most of these topics, like slope and scale, a repeated-addition intuition will look like no intuition at all.

Insanity: Repeatedly Adding and Expecting Multiplication

The difficulties we have moving away from repeated addition and toward multiplication in mathematics are reflected in how we think about more everyday things too.

I know I’m not alone in having been exposed to the idea that simply adding on more opinions, more voices, more collaboration, is an absolute good—one that, if we just add enough, can accomplish anything. And I’m not alone in having watched that fail more often than not. Post-mortem analysis tends to reveal (surprise, surprise) that we didn’t add on enough, we didn’t work hard enough, we weren’t good enough.

In education, it looks like this (to me): Whatever good thing we can do for students or whatever bad thing we can avoid doing, the message about this thing is delivered in the exact same way to the whole of education, no matter the grade level. If the message is that X is good, the message is to do it in first grade, then again in second grade, and third grade, add on, add on, repeatedly.

This way of delivering messages betrays an assumption about the purpose of schooling: that it is not designed to systematically scale up student knowledge and ability, but to provide practice for the same general “skills” over and over. Not only that, but it paints a picture of an education system that is not really a system at all—just a series of stations, manned by employees with no responsibility to each other.

The multiculturalists of the 1980s and 1990s accepted a too romantic, essentialist view of language that helped fragment the school curriculum. They seemed to believe that Americans could transcend particularity, that we did not need communal knowledge shared by all but could happily exist as a universe of separate cells: out of many, many. These cells could then all function together if students achieved critical-thinking skills. But neither the critical-thinking idea nor curricular fragmentation has worked out for the social groups that these ideas were supposed to help. Gap closing has stagnated; the achievement gap persists.

–E.D. Hirsch, Jr.