Math: Does the U.S. Teach It Well?

Elizabeth Boulton

I love math. Is that a weird thing to say? I love algebra, calculus, problem-solving math. I love working with equations and making them say what I want them to say. I like looking at the world in terms of math. When I’m driving on the highway, I try to guess how fast the cars around me are driving. Given that I am going 65 mph and I have gotten 5 car lengths (about 50 feet) closer to the blue Honda in front of me over the last 5 minutes, how fast is the blue Honda moving?

Unfortunately, there are many people in this world who don’t feel the same. Not everyone has to like math, but everyone should at least be given the tools to be comfortable with math. I have tutored many people in physics, and since physics has so much algebra in it, in reality, we spend a lot of time talking about math. It is always frustrating for me, when my tutees cannot rearrange an equation in order to find the variable they need, when they can’t figure out what equation to use, or when they can’t see how to make an equation out of the words in the problem. Let me be clear: I’m not frustrated with them. I’m frustrated because I can’t articulately explain how they can know to do those things. I can lead them through rearranging equations until I am blue in the face, but I can’t communicate any general rules that they can use for every problem.

A few months ago, a friend of mine sent me an article, “Teaching the Conceptual Structure of Mathematics.” She described herself as having hated math in school and we have discussed her frustrations many times. When she ran across this article, she sent it to me saying, “When I was a kid in math I would ask, every time, ‘but why would you use that formula there?’ and my teachers would (without fail) tell me I didn't need to know why—I just needed to memorize it and apply it on the exam.”

After I read the article, everything made sense: why so many of my students have trouble using algebra in physics, why my friend suffered through math classes, why I was frustrated at my own inability to teach my students the patterns of math. Could the answer to all of my questions be that math is very poorly taught in the United States public K-12 system?

According to the article, schools in the U.S. teach math by teaching the rules of math and having students apply these rules over and over. On the surface, this doesn’t seem like a bad way to teach, except when you consider that this doesn’t give anyone a framework to approach new math problems that maybe don’t look exactly like the ones for which they were taught the rules. The article lays out two main features of instruction that would allow students to form a framework for math. First, teachers and students need to explicitly discuss mathematical concepts. Second, students must struggle with the concepts in order to fit them into their emerging framework.

The authors of the article note two studies that corroborate these findings. The two studies looked at about a hundred teachers in each of the following countries: Germany, Japan, the U.S., Australia, the Czech Republic, Hong Kong, the Netherlands, and Switzerland. The conductors of the study found that what separated the high-achieving countries (all of the above except the U.S. and Australia) from the low-achieving ones wasn’t the class size, the kinds of problems used, nor the teaching style, “but the kinds of learning opportunities teachers created for students, namely, making explicit connections in the lesson among mathematics procedures, problems, and concepts and finding ways to engage students in the kind of productive struggle that is required to understand these connections in a deep way.” The teachers in each country gave their students two main types of problems: problems that can be solved using a procedure and problems that must be solved by making connection between previously solved problems. In every country except Japan, many more procedure problems were given. However, in every classroom, the teacher transformed some of the connection problems into procedure problems by giving extra instruction. For example, a teacher might present a connection problem, and then solve a sample problem that was exactly like the connection problem, which would transform it into a procedure problem. Figure 1 shows the percentage of connection problems the teachers in each country transformed into procedure problems.

The compelling evidence from this articled supported by my personal experiences makes me firmly believe that the U.S. is teaching math the wrong way. What can we do about this? How can the system be changed in order to provide better math education? How can we in higher education correct for all the damage that has been done in lower level math? I don’t have any of these answers, but I’m hoping that together we can come up with some.

If you found this post interested, I urge you to read the full article that I discussed, “Teaching the conceptual Structure of Mathematics” by L.E Richland, J.W. Stigler and K.J. Holyoak. Another interesting read on the same topic is “A Mathematician’s Lament” by Paul Lockhart.

Figure 1