Those Who Can’t… Teach?

May 8th, 2006 | View Comments

Nicholas Kristof of the New York Times published an editorial two Sundays ago (yes, I’m behind the times) about teacher qualifications. It is, unfortunately, hidden behind the beaded curtain that is TimesSelect, but thanks to the miracle of copyright infringement, you can read it in its entirety here.

The debate over teacher qualifications has been fueled by the No Child Left Behind Act, which mandates that all teachers be “highly qualified” by oh…about now. In order for teachers to be highly qualified, they must hold a bachelor’s degree, teacher certification, and “demonstrate competence in their subject area.”

I’m oversimplifying a bit here, but that’s the basic idea. Like everything else about No Child Left Behind, content is emphasized over pedagogy.

Kristof is basically parroting the arguments that have been trotted out in several Department of Education reports on teacher quality. Just go to www.ed.gov and do a search for “highly qualified teachers.”

The DoE basically argues that existing schools of education and teacher certification programs, which emphasize pedagogy over content, wind up producing a lot of teachers who can teach…something…but don’t really know anything. Thus, teacher prep should focus on content knowledge.

You can read a rebuttal by Linda Darling-Hammond, a well-respected education researcher, here (pdf, ~120K). The short version: the research doesn’t really support the arguments being advanced by the DoE.

I mean, you take Colin Powell, plunk him in a classroom of 40 urban high school students who have no idea who he is, in which there are no norms regarding manners, paying attention, doing your homework, or heck, even coming to class, and let him fend for himself. How much social studies do you think he can teach in an hour?

Or take me. I have a bachelor’s degree in math and have spent the last 4 years of my life studying how math can be taught and learned. I certainly have the content knowledge to teach K-12 math, though I have no meaningful teaching experience to speak of. Plunk me in a classroom of 7 students, aged 10-14, predominantly upper middle-class, and ask me to teach an hourlong lesson on surface area-to-volume ratio. How well do you think I do?

The answer? I spent half of my time doing classroom management, basically getting students to sit down, stop pestering each other, and pay attention to me. Remember, there were only seven students. In the remaining time I managed to work a few example problems.

By the end of the lesson, only one of the seven kids had any idea what I was talking about, and he didn’t even retain the knowledge for two days. He missed the same question he’d successfully answered in class when it appeared later on the quiz.

There is a lot more to effective teaching than content knowledge.

That said, I don’t think that concerns about teachers’ content knowledge are unwarranted when you’re talking about elementary math. An excellent book on this topic is Liping Ma’s Knowing and Teaching Elementary Mathematics.

Ma paints a very bleak picture of the state of American elementary math education, highlighting numerous weaknesses, the first and foremost of which is the American tendency to teach math as a string of computations that can be performed mechanically, without any reference to the underlying mathematical structure or how it relates to the real world.

Although the American teachers in Ma’s study were able to perform the computations themselves, they often lacked sufficient mathematical knowledge to answer student questions like, “Why do we have to put that zero there?” or “Why do we have to learn this anyways?”

Speaking of which, what is the purpose of math? Why do we do math? What is math good for? Why all the fuss about teaching math, when apparently no one but a select set of people ever uses it?

These are questions that math students ask all the time. The answers that adults give are typically along the lines of “well, if you’re in a supermarket and want to know how much four apples cost…” Answers of this type are wholly unsatisfying in the age of calculators. Furthermore, answers of this type completely fail to explain why anyone should need to know or do math beyond basic arithmetic.

So what is the purpose of math?

Math is a very precise way to describe things. It is a language, of sorts, with its own syntax and grammar. In math you have nouns–numbers and variables–and verbs–the various operators, including adding, subtracting, multiplying, and dividing.

You combine them to make equations–sentences–that describe how things in the real world relate to each other. Because math is so precise, it is a good way to describe things to another person so that they don’t misunderstand or misinterpret what you mean.

Math is abstract, which means I can use the same math to describe a large variety of things. When I’m writing math, I get to decide what numbers and variables stand for what real-world objects. This is like casting actors to play a particular character in a play.

For example, the equation y = mx could represent the area of a rectangle (A = lw), distance (d = rt), or Newton’s Second Law of Motion (f = ma). Because math is abstract, it can be used to describe and then solve any kind of real-world problem. The more math you know, the more real-world problems you can use it to describe and then solve.

You can use math to describe the gravitational pull of the earth. This is important if you want to send a satellite into space so we can have good weather reports, satellite TV, and GPS.

You can use math to describe how the price of a product relates to how many people will buy it. This is important if you want to sell things and make money.

You can use math to describe how students in different classrooms are performing on a test. This is important if you want to know if one group of students is doing better than another.

In short, math is useful for engineers, anybody who wants to understand the flow of money, including economists, financial planners, retailers, and other businesspeople, and anybody who may want to do statistics, which includes scientists, actuaries, pollsters, policymakers, and sports enthusiasts, just to name a few groups. Oh, and mathematicians.

People in these careers may or may not do a lot of number crunching themselves, but when you get handed a page full of numbers, you’d better know enough math to understand what it means. Math is a gateway to a large number of careers.

How many elementary math teachers can give answers like that?

Granted, 6th graders can’t exactly be sending satellites into space. However, there are plenty of real-life statistical problems that are both interesting and tractable for all age groups.

And there are also worthwhile scientific questions as well. Although they may seem trivial for adults, they can be extremely engaging for students. The following example is adapted from a series of actual lessons described in a book chapter by Rich Lehrer and Leona Schauble (2001).

Take these rectangles:

How would you describe the difference between the rectangles in the top group and the rectangles in the bottom group? You might say that the ones in the top group are wider or the ones in the bottom group are taller, but that’s not really true.

Students generate ideas for a while, but eventually they hit upon the idea that the short sides of the rectangles in the top group “go into” their long sides two times, while the short sides of the rectangles in the bottom group go into their long sides three times.

In other words, they have hit upon the idea that it’s the ratio between the two sides that defines each “family” of rectangles. Once students have the idea of ratios, you can have them explore other ratios like density, which is the ratio of mass to volume. You could give the students a bunch of objects and ask them to identify which will sink and which will float in water. Eventually students will generate enough data to create the following plot, in which the slopes of the lines represent density:

The blue line represents the density of water. It has a slope of 1. Anything with a density less than that (slopes flatter than the blue line) will float, anything more (slopes steeper than the blue line) will sink. Thus the green objects will float while the orange objects will sink.

These lessons took students many weeks to learn, but those lessons covered concepts like measurement, area, multiplication, shape similarity, ratio, division, mass, volume, density, graphing, and slope, all in a way that reflects how math is actually used to solve problems. As a student, would you rather do a unit like that? Or would you rather do worksheets with times tables and story problems about apples and oranges?

This brings me back to my original question about teacher qualifications. The unfortunate truth is that many elementary math teachers lack the sort of deep and flexible understanding of math that would enable them to create meaningful units like this and to really answer questions like, “What is this good for?”

Currently, elementary teachers do not need to have a math degree or any specific math training in order to teach math, and the tests of math competence that teachers take predominantly test low-level computational skills. Think of the kinds of problems you’d see on a test like the SAT or the ACT. You may need to use some trig, but that’s all.

What is true of students is also true of teachers: although we can expect people with math competence to pass the standardized math competence exams, we can’t expect that people who pass the math competence exams to actually be mathematically competent.

As in the case of Liping Ma’s American teachers, it is possible, and even common, to be able to perform computations proficiently and yet not know enough math to teach it competently.

It is not uncommon to hear of teachers who go into elementary ed precisely because they don’t want to have to teach “hard” math like algebra. Unfortunately, since algebra is but a slight extension of arithmetic, this probably means these teachers are also ill-equipped to teach arithmetic. And even more unfortunately, students can easily pick up on teachers’ distaste, confusion, or fear about math, and carry these attitudes with them for the rest of their school careers.

Thus we have a situation in which those teachers who are entrusted to start building a fundamental skill are often those who are least equipped to do it properly. No wonder so many students wind up hating math.

The solution is not as simple as making the content exams harder. I think we, as a nation, really need to step back and re-learn what it means to do math competently. Only then will we be able to devise tests that truly reflect mathematical ability rather than human calculator ability. Fortunately, there are good people leading the charge.

Reference

Lehrer, R., & Schauble, L. (2001). Similarity of form and substance: From inscriptions to models. In D. Klahr & S. Carver (Eds.). Cognition and instruction: 25 years of progress (pp. 39-74). Mahwah, NJ: Lawrence Erlbam Associates.