The math book you didn't know you needed to read

 At least, I didn't know I needed to read it. The book is Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma. I came across the title in a discussion of teaching math in homeschooling, and on a whim I put it on hold at the library. I didn't have huge expectations for it as math is not usually what I read about. Not only that, but it is actually Ms. Ma's doctoral dissertation. Dissertations are not usually exciting reading. (My apologies to anyone who has ever written one or is writing one, but you'll probably agree with me.) This had all the markings of a book I skimmed a bit and then returned to the library. 

Given all that, this was a surprisingly compelling book. Really.

Much of the book centers around four math problems... a subtraction problem involving regrouping, multiplying a three-digit number by a three-digit number, dividing a mixed number by a fraction, and a geometry problem involving area and perimeter. The two groups of teachers, one group is from the US and the other is from China, are asked to solved each math problem and then discuss (in interviews) how they solved it and how they would teach it to students. 

What I found fascinating was the difference in approach and understanding by the different teachers, and the differences were striking. The US teachers in general did not come off so well in all of this. Some couldn't figure the more advanced problems, and many didn't understand what they were doing in others even if they got the correct answer. The Chinese teachers on the other hand had an amazing grasp on what was going on with each calculation. They knew why they worked, could prove it, and offer alternative ways of solving the problems. 

And as I read it, much of my elementary math education came to mind. Math was a series of tricks you did to get the correct answer. Why these tricks worked was mysterious and if there was a reason for doing them it was never shared. I ended up doing the math but never really understanding why or what I was doing. Reading the interviews with the US math teachers made some sense of what I experienced. There is a good chance that they didn't understand the actual math behind it all, either. 

The most egregious example of this was in discussing the multiplication problem. The teachers were asked how they would go about explaining to students what went wrong if they answered a problem like this:

  123

x456

  738

  615

  492

1845

The problem here is that the partial products are not placed in the correct columns to indicate their actual place value of 615-10's and 492-100's. What was horribly disturbing was that many of the US math teachers Ms. Ma interviewed could give no reason why the partial products should be moved over on each line. Even worse, when talking about how sometimes zeros are added, they had no idea what the justification for them would be, with some teachers using stars or animals as a place filler. 

Now, since I was a student in the same system that these teachers were a part of, I'm not sure I could have answered these questions any better than they could when I first started teaching math to my children. But, after 20+ years of having children ask questions, of having to come up with different ways to help them understand what we were doing, of using quite a few math curricula, I figured it out. More than once on this homeschooling journey I have been helping a child only to discover something about math that no one ever taught me. I can say without a doubt that my elementary math skills are better now than they have ever been. So, I'm afraid that I don't have much patience for other teachers who have been teaching more children for longer periods of time. There's just very little excuse for not knowing when you are the teacher.

In comparison, the Chinese teachers knew exactly what was going on and had real solutions for helping students understand where the error in the problem lay. In contrast to the US teachers who just repeated how to 'do the trick' over and over, they presented the problem in different ways so that they student could discover what was going wrong. The Chinese teachers were so clear in their explanations that I now feel as though I finally understand why, when dividing fractions, you change the division sign to multiplication and flip the second fraction. That had been the one 'trick' that I could never quite figure out the why behind it all. It was kind of a relief to finally understand it. (I'll probably need to read it a few times to be sure it's all cemented in my head.) 

The book was published twenty years ago, and I know things have changed in the elementary math world since then. There is a lot more emphasis on the why behind it all, and this is a good thing, I think. Tricks only get you so far, and unless you really understand what is going on, they cannot help you solve something new. This is part of my beef about a video I've seen going around. On one side is someone solving a multiplication problem the traditional way, with carrying and partial products. The other side of the screen is a teacher showing a class how the problem can be solved using expanded notation so that the understanding of place value is there. As the person on the right side of the screen is fixing a sandwich and eating lunch having solved the problem, the teacher is still explaining. The point of the video is to try to convince people that just learning to solve the problem by doing the trick is far superior to actually understanding what is going on. This is ridiculous and I grind my teeth every time I see it. Teaching is more than just showing a trick and being done.

So if you are teaching children math or if you are just interested in education in general, read this book. I bet you'll learn something new as well.