by Tom

Here’s a fun math problem to solve:

Let N be an odd number.

a. Prove that N squared is odd.

b. Prove that when N squared is divided by 4, the remainder is 1.

c. Prove that when N squared is divided by 8, the remainder is 1.

d. Find an odd N such that N squared divided by 16 leaves a remainder that is not 1.

If you’re like me, you probably remember doing this kind of work at some point in your education. Given enough time and motivation, you could probably figure it out. But if you’ve taught third grade for the past 24 years (like me) you probably haven’t done a problem like this in a long while.

So what? Well, a new study released by the National Council on Teacher Quality (NCTQ) concludes that colleges of education are under-preparing elementary school math teachers. As a teacher who considers himself reasonably prepared to teach math to third graders, my initial reaction is to defend my colleagues and the institutions that prepared us. However, I think we do have to own some of this. I have sat next to teachers during math trainings who appear to be learning (or not learning) some very basic math for the first time. And I think it’s a fair generalization to acknowledge that college undergraduates who are strong in math do not gravitate towards elementary education. We tend to get a lot of people who are competent in math, but not obsessed by it. I think it’s reasonable to expect teacher education programs to ensure that their graduates know how to do college-level math.

On the other hand, most people would agree that with many math skills you either “use-it-or-lose-it.” The minute you stop solving algebraic equations is the minute you stop knowing how to solve them. I teach third grade. I was once pretty good at algebra; I’m still very good at subtraction and multiplication. In their report, NCTQ includes five different math problems that they feel elementary teachers should be able to solve. The problem above is one of them. Although there was a time in my life when I could have solved each of these problems, I have to admit that now is not one of those times. Why not? Simple: I do not work with this material on a day-to-day basis. It is not currently critical that I know how to do this type of math. That’s how people forget how to do things. (By the way, this is exactly why the American Red Cross makes us take CPR classes every two years to retain our certification. They assume we haven’t had to save anyone’s life lately, and they know we’ll forget how to do so.)

The NCTQ and its panel of experts (none of whom teach elementary school) believe that grade-school teachers should have more rigorous coursework in math content. Perhaps. But my concern is that a large portion of this “rigor” will be left behind once these teachers begin working with their students. I think their time in college would be better spent on methods courses and content-level math classes in which they focus on developing concrete understanding of the math with which they will have day-to-day interactions.

Let me give you an example. In my school we use a professional development tool called Lesson Study. Small teams of teachers collaborate on high-quality lessons designed to meet the specific needs of our student population. Then we observe the lesson as it’s taught to a class of students, analyze what we notice, and refine the lesson for further use.

A few years ago, we noticed that our fourth graders were having trouble learning how to multiply two-digit numbers. The curriculum we were using wasn’t working for our students, so we came up with a way to introduce the concept using base-ten blocks. It built on their experience of using these manipulatives to multiply single-digit numbers, and our lesson was designed to facilitate a seamless transition to the standard multiplication algorithm.

Working as a part of this team deepened my own understanding of the underlying concept of multiplication. At the same time, it increased my pedagogical skills. Since I use this learning day-to-day, year in and year out, I am a better math teacher because of the experience.

Beginning teachers have a lot of important things to learn. Things that they’ll use everyday. Learning how to “find an odd N such that N squared divided by 16 leaves a remainder that is not 1” is not one of them.