This post has its genesis at work in the past week (March 19–23, 2018), it has its underpinnings though in 25 years of thought about the number system(s) we get to play in.

For those who know me, you know I work in two places, one a university where I help teachers struggle with math and STEM learning; the other a non-profit organization dedicated to the same goal. So it will not surprise you that a question about dividing by fractions would come up in my daily work. This week it did, and I have been reveling in it for a few days. Let me put the context on this for you oh gentle reader… On the Green Wall at work, from time to time, I post problems to pester the minds of my co-workers and remind them that there is a vast and wonderful world of mathematics out there to think on. But this week, one of those co-workers placed a sequence much like what you see below with the single word, “INSIGHTS?” written in Purple pen:

So just to clarify, first off this is not the exact problem on the wall, but it is illustrative of the conflict, second most of the dialogue I will relate here is enhanced for dramatic effect, but the gist of my buttheadedness is probably accurate.

What is happening at work is a number of people are reading Liping Ma’s book from the 1999, “Knowing and Teaching Elementary Mathematics: Teachers’ understanding of fundamental mathematics in China and the US.” This book has had a profound effect upon the psyche of American Mathematics teachers for nearly two decades now. It outlines, repeatedly, how shallow American understandings of mathematics is. Dr. Ma refers throughout the book to what came to be known as PUFM or the Profound Understanding of Fundamental Mathematics, something her study demonstrated was in great lack in the US.

So powerful in fact was this lesson that now many teachers who do possess PUFM have an inferiority complex about coming from the US and what I now consider an undue reverence for those from other countries. This undue reverence is the topic of this essay, which now has a way too long introduction.

The main issue at hand is the line of the problem where my purple callout points, the one where the parentheses from the line above magically drop off and convert the “divide by 4” into a “multiply by 4.” This magic step LOOKS like a DISTRIBUTION of division BUT IT ISN’T, do not be fooled. Take a moment and go back through those six lines and verify that what looks right is indeed mathematically sound.

The problem that this episode stems from comes up in the book to illustrate how a Chinese teacher describes an “informal proof that division by a fraction is in fact multiplication by a reciprocal.” To which I say HOGWASH!

First off, there is NO PROOF that division by a fraction is the SAME as multiplication by its reciprocal. Not in a mathematical sense anyway, perhaps in some other more ambiguous communication, but not mathematics. You cannot prove that which is an axiom or definition these stand, as it were unprovable. That dividing 1 by 3/4 is the same as multiplying 1 by 4/3 is a consequence of the very nature of the Real Numbers. These wonderful concoctions behave precisely the way we want them to, consistently and coherently.

I spent quite a bit of time attempting to persuade my co-workers to this fact, but yet they persisted in the need to understand, nay demonstrate concretely, just how the rule of thumb of (a/(b/c)) = a/b*c works. As it was stated in the Ma text that this rule of thumb is well known among children and teachers in China. To which I replied, “Oh like the wonderful rule of thumb “minus a minus is a plus!” (e.g. 7- -5 = 7+5) “how wonderfully understandable that one is?” yeah I am a butthead.

I am sorry to sound obnoxious, but not sorry that I am not wrong. 🙂

The Axioms that call into existence the Real Numbers and turn them into a commutative ring require only two operations, addition and multiplication, not four. The other two, subtraction and division, are metaphors, phantoms really, that obscure reality. It should not be surprising that occasionally these phantoms create ghostly problems. By ghostly I mean they appear to be there, but if you look closer they disappear into thin air.

This is the case with the divide a divide becomes a divide them multiply above. I think, you will see it better if you look closely at (a/(b/c)) = a/b*c for a minute or two. My point in this is that while some cultures have stronger metaphors for some truths, we still need to examine the metaphors used. This “trick” relies upon a circular bit of logic and as such obscures what really is happening. If you look close that blurry haze will dissipate, and the truth of how the trick uses what it is trying to prove reveals itself. The multiplication at the end is in fact making use of the need for a multiplicative inverse of an element, not an operation. If could have one thing change in all of the teaching of mathematics it would be the cessation of the idea that “subtraction is the opposite of addition” and “division undoes multiplication,” and all their cousins.

The truth is, division is a dream, felt to be real until you wake from it and discover that it isn’t actually there (like that spider who spins webs made of fishing line to ensnare you).

I must say that I was about this with my good friend James Tanton and feel somewhat vindicated by his responses to me. I won’t fill the pages with them, they are copious and a delight though, but they are not mine to share. In these James reminds me that each new set of numbers is created to extend the last: The Naturals beget the Integers because we can’t solve problems like 4–7 =n, the Integers beget the Rationals because we can’t solve equations like 5x=3 in the Integers…and so on we need the Irrationals for things like sqrt(2), and the Complex for things like sqrt(-2) right on up until we discover that when we define the polynomials we have a complete collection that has answers to all the questions you can ask within itself, YAY!

So, in closing, I want to say that while we in the US have a ways to go till more of us who teach math truly do possess PUFM, we can get there without having to rely upon other peoples’ phantoms.

POSTSCRIPT: I want to say that I did in fact love the chance to have these chats with my co-workers, they challenged me to be better at thinking about these things. Furthermore I want to also say that in my conversations with James he reminded me that fractions are NOT an easy thing to understand and that expecting 6th graders to get them in that one time they are taught, and have them forever is kind of crazy. Rather it would be best to continue the fraction talk clear through all of secondary schooling, allowing the context to grow in sophistication over time. When an 11th grade class is struggling with the idea, do not belittle them and say, “You should have got this in such and such a class,” no this is the best time to have that discussion, dive deep with them into the joys that are the numbers we have made up to suit our purposes. Enjoy the dialogue. I sure have this week. Thank you.

PPS: It is my plan that by June of 2018 I will be posting some of my Green Wall problems to the web for others to play along with. If you would like to know when this begins, please send me a quick email.