In June, the National Council on Teacher Quality released an important study finding that most education schools in the U.S. are weak in their training of teachers in mathematics. Not surprisingly, American students don’t, on average, learn math very well.

In this week’s Pope Center Clarion Call, I comment on the NCTQ study.

A math professor wrote me about the piece, and I reproduce his letter after the jump.

This is a good article highlighting an important aspect of problems with mathematical preparation. I think the problem begins in elementary school. What I have found in 40-plus years of teaching college math is that students’ difficulties in calculus stem from not really understanding the most basic mathematical concepts.
The students’ problems with addition and division of algebraic fractions stems from their lack of understanding of the same operations on fractions formed from integers. They have weak understanding of geometry, and their understanding of percentages is quite poor. On top of that, they have not internalized ninth-grade algebra to the point of second nature.
What is more important is that they don’t seem to grasp what it means to understand mathematics. They mostly think it is all recipes — formulas to be plugged into without understanding the derivations of the formulas themselves, theorems to be used without understanding their proofs.
When I teach the quadratic formula in college (it is amazing that so many of our students are still taking courses covering ninth-grade algebra), I always spend time on its derivation. Fortunately, I don’t often have to teach it, but I had to a couple of years ago, and when I derived the quadratic formula I had students openly rebelling during class time, telling me I didn’t know how to teach because I was doing so. Similar reactions occurred when I tried to explain how solving algebraic inequalities is based on such simple rules as the product of a positive and a negative is negative. When I would use numerical examples to explain this principle and show how it applied in solving quadratic inequalities, they would say my course was too theoretical. When I got my student evaluations from the course, they claimed I spent too much time on theory. Last year when I got my annual evaluation from my department, I got criticized for being too theoretical.
I have been teaching math since 1965, and I have a pretty good idea of what students are missing when they are having trouble. You can make an honest attempt to correct their misconceptions so that they will genuinely understand, or you can streamline the exams and homework so that they can memorize templates without understanding what they are doing.
A colleague of mine described the situation like this. If you need to smooth out a rough piece of wood you can take a piece of sandpaper and press, or you can take glue and cover up the rough spots.
The one part of your article I am not sure of is that I am considerably less sanguine about the ability of mathematicians to alleviate the situation. We have very little experience dealing with children, at least as it comes to teaching math. We have little practical experience about knowing the limits of their concentration. As parents we may have been involved in teaching our own children (which is a valuable experience), but it is not the same as explaining something like division of fractions to a room full of kids with widely different abilities and knowledge.
One problem that confronts elementary schools is that the wide differences in abilities in math present far more difficulties than the wide differences in reading. With reading, one can allow students to choose their own books. It is tougher to do this with math. When you have some students who could easily understand algebra in fifth grade and others who still don’t understand borrowing and carrying, it is hard to devise a one-size-fits-all curriculum.
I think what is needed is better cooperation between math departments and schools of education. When I was chairman of the math department at DePaul, I was involved in discussions concerning the creation of a master’s degree in math for middle-school teachers. The project floundered over issues about how much math would be in the curriculum; beyond the usual arguments about methods courses verses content courses, we also had disputes about courses the school of ed wanted on multiculturalism and on political maneuvering within a school building. As we were unable to agree on what to teach, no program was created.
Ten years later, some math faculty from area colleges developed a program with a more limited focus. They are giving a one-year course in ninth-grade algebra to middle-school teachers so they will be able to teach the subject to eighth graders. The courses are taught at the schools. At the end of the year the teachers take an exam, and if they pass they receive some kind of certification. This is a far more limited but more effective use of everyone’s time.
Instead of spending a lot of money on a master’s degree for the purpose of improving their salary, they spend no money on a course that will actually result in improved education for our children. I don’t want to exaggerate the importance of this program. It is quite limited in scope, but I brought it up because I think it comes closer to something that might actually work.
As for the training of elementary-school teachers, we do teach a two-course sequence in math. Whether it does the job, I don’t know. We’ve had some good people teach the courses, but I don’t know if it really prepares the students to teach math in elementary school. I hope that it helps.
I like your overall way of looking at education problems. You are asking if all this stuff does any good, and if so, is it worth the cost. It is difficult for academics to question their own value, but I think it is reasonable to ask them to do so.
Jon Cohen
DePaul University