But it was not always this way! As a college math major filling out applications to masters & secondary credential programs, I definitely saw myself as a high school teacher, much more interested in the complexity and rich structure of Algebra II and trigonometry and calculus than in the usual middle school topics. And I *certainly* had never gone back to closely examine my own conceptual understanding of the foundational mathematics we learn in elementary school. Who wants to teach fractions and decimals when you could be initiating kids into the wonders of trigonometric functions??
So, I got into a secondary program, started student teaching Algebra I, & learning all the magical things they teach you in Curriculum & Instruction (ie, "methods") class about how kids make sense of ideas like variables and functions and data analysis over time and what it really means to understand all these things anyway. It was mystifying and terrifying and amazing, and in addition to learning how to teach, those experiences also unlocked for me an entirely new dimension of understanding. It was exhilarating ("Who knew math could be even MORE AMAZING??) but also a bit panic-inducing ("How the HECK am I supposed to get kids to understand it THIS way?!?"). All in all, though, I was starting to feel pretty darn good about my content knowledge.
And then, one day, Cathy Humphreys came to class.
She came to teach us about fractions.
I still remember a bunch of us thinking it was a bit odd; I mean, as secondary math teacher candidates, nearly all of us had undergraduate degrees in math, and we were certainly all quite strong in the subject by any reasonable measure. Shouldn't we be learning how to teach kids about systems of equations or rate of change? Still, we were also reasonably open minded and willing to try just about anything, and mostly approached the session with an attitude of, "What the heck, let's see what Cathy Humphreys has to teach us about fractions."
She brought pattern blocks, and one of the first things she told us was that as we thought about fraction problems today, we were not to use any procedures or algorithms that we knew for performing calculations with fractions; instead, we were to reason our way through the problems, performing only operations that we could explain in the context of the problem. (For example, "There's 1/3 of each pizza leftover and 6 pizzas, so to find out how much is left, we can multiply 6 pizzas times 1/3 of a pizza, ie, add six 1/3 's together" = OK; "We need to divide 6 5/8 meters of ribbon into strips of 1/4 meters, so invert the 1/4 and multiply straight across" = not OK unless you can explain why on earth that makes sense.)
Well, we were all of course believers in deep conceptual understanding, and we weren't sure why people like us who were obviously very good at math would need to fall back on memorized procedures, but we were happy to oblige.
I wish I could remember more about the specifics of the session--which operations and problems we did, and in which order, and what the conversations around them were. I do remember talking about and modeling problems like 3 ÷ 1/4, where the divisor fits evenly into the dividend, which didn't trouble any of us too much, and we were even fine with problems like 1/2 ÷ 1/4 or 6/8 ÷ 1/4. (In fact, I might go so far as to say we were actually pretty proud of our ability to explain the solutions in a completely conceptual way and even illustrate our methods using the pattern blocks.)
But something I remember very clearly was how the mood in the room shifted when we began working on a fraction division problem using the pattern blocks where the divisor did not fit so neatly. I think that most (or all?) of us had thought we'd breeze through it, but it quickly became obvious that that wasn't the case.
Since I don't remember the first real stumper Cathy asked us to think about, let's pretend it was 1 ÷ 2/3 (partly because this video exists as an illustration). It was easy enough to take my 1 (the yellow hexagon) and find two of the 1/3 blocks (the blue trapezoids) and place them on top; what I was having trouble with was completing the mapping of the equation in my mind, 1 ÷ 2/3 = x, onto the concrete representation.
Like my classmates, I remember staring blankly at my pattern blocks, completely paralyzed, unexpectedly humbled by elementary school math that for years I'd considered too boring to even consider beyond its procedural, symbolic role in formal algebra. I knew 101 ways to integrate a function and could rattle off epsilon-delta proofs with the best of them, but somehow couldn't solve a fifth grade arithmetic problem without falling back on a memorized procedure.
I could see the 1 and I could see the 2/3, but when I thought about division, my brain yelled "Equal parts!!" and I could not knock into place a) the leftover 1/3 or b) where in the representation the division was happening. When I tried to think concretely about 1 ÷ 2/3, what I came up with was, "Split 1 into equal groups of 2/3," which was just clearly not possible. Abstractly, I knew the answer was 3/2 or 1 1/2, but I could not for the life of me see where that 1 1/2 was, or make sense of that "equal groups" construct.
There was quite a lot of tentative murmuring and later full-on discussion about this among our group of secondary candidates. We shared our representations, made observations, and explained the thoughts and questions that were tripping us up. The light bulb came on for a lot of us, I think, when someone asked, "What if instead of interpreting 1 ÷ 2/3 as "Divide 1 into 2/3 equal groups," we thought about it as, "How many 2/3 fit into 1?"
And with that observation, suddenly a switch in my thinking flipped. I looked at my model, at the 2/3 of the yellow hexagon covered with the two blue trapezoids, and at the troublesome 1/3 that was left, and suddenly there it was, plain as day, 1 1/2, right in front of my face.
Of course, putting it into words was something else all together. "There it is!" we squealed, pointing wildly to our blocks. "There's 3/2!"
"THERE!" More exuberant pointing.
But Cathy didn't only want us to see the 3/2; she wanted us to put it into words, to explain it, mathematically and precisely. She pressed on our burgeoning understanding with questions like, "But if we said the yellow hexagon is 1, how can we have an answer that's more than 1?" and "Wouldn't 1 1/2 be a yellow hexagon and half of another one?" and "If the answer is 1 1/2, but not 1 1/2 hexagons, then 1 1/2 what?"
(This was 12 years ago, but I can't think back on the discussion now without MP 3 and MP 6--and realistically, a whole host of others--jumping out at me.)
It was through these questions that we finally got to the idea of the changing unit. The answer was not 1 1/2 "ones" (yellow hexagons); it was 1 1/2 two-thirds. Inside our "one" (the yellow hexagon), we could fit one complete 2/3 (two blue trapezoids), plus another 1/3 (one blue trapezoid)--ie, another half of a 2/3.
So if we thought about the problem as, "How many 2/3 fit into 1?," it was easier to see that the answer was, "One (complete 2/3), plus another half (of a 2/3), or 1 1/2 (two-thirds)."
Friends, I don't know about you, but I was *definitely* not taught fraction division this way; mine was definitely, definitely not to reason why. Flip it and multiply straight across you say? DONE AND DONE. It never occurred to me that there could be a reason; it was just the rule.
I know for a fact that this was a huge part of why I hated and struggled with math so much in school. I don't memorize well, at all. If there's some procedure I need to carry out, even as an adult, I need to understand why, to completely make sense of each step, if it's going to be any good to me. And when the focus is on getting an answer, quickly and consistently and at all costs, true understanding--understanding that can be enriched and expanded and built on over time--falls by the wayside. Teaching procedures without helping students make sense of them in the context of the mathematics they already know is a mathematical dead-end.
Next time, how I made sense of even CRAZIER problems, ie, ones where the we must divide a SMALL fraction by a BIGGER fraction. (Crazy talk, I know.)
***(P.S., Cathy, if you ever read this and have a better memory of this day than I do--or if I've totally bolloxed what actually happened--please let me know so that I can make update this post! And, thank you again for teaching me how to divide fractions. :) )***