**David Klein**

“There are several components to making a good math teacher,” says David Klein, professor of mathematics at California State University, Northridge.

“One of them, a very important one, is the kind of education training that they get in college, and another component is the quality of the textbooks that they’re required to use or are allowed to use in the schools where they teach. The reason I bring that up is even a highly competent, very knowledgeable, capable teacher, who is required to teach out of math books that don’t make any sense … is going to be handicapped by the low quality of the book. This was the focal point of the math wars of the second half of the 1990s and the first half of this decade. It overlaps in a very political way, and a very practical way too, with the quality of math teaching that goes on.”

In terms of university preparation, at the high school level, Klein says all teachers should have a bachelor’s degree in math, including coursework in geometry and advanced calculus, sometimes called analysis, which basically explains why calculus works. “For a teacher, especially a teacher who teaches AP calculus in high school, they need to know [for instance] what do we mean by a limit … What really is a derivative? … The teacher needs to understand the whys, not just the hows, because kids ask questions.”

And, while a major in math would be great for middle school teachers too, Klein says a minor in math is probably adequate for teaching at this level.

For aspiring elementary school teachers, Klein would like to see three specific courses in math with one-unit labs tied to the first two. First, since the main subject in elementary math is arithmetic, Klein suggests a class devoted to why arithmetic procedures work and ways that these concepts can easily be presented to children (for instance, how to teach the base-10 structure with money – pennies, dimes and dollars). Klein says there should be a filter to get into this course, to ensure that would-be teachers already have fluency in arithmetic and elementary algebra procedures before enrolling.

“Arithmetic should be taught in such a way that it gradually ramps up to algebra, so the danger – and we see it all the time – is that university students who want to become teachers sign up for a course like this and they don’t even know the arithmetic themselves. They don’t know how to add fractions. It’s pretty bad. What they get out of that class is they learn arithmetic. That’s all they learn; they don’t really learn how to teach it,” Klein says.

In the second of the three courses Klein describes, pre-service teachers would review geometry and learn relationships between geometrical ideas. Again, there would be a focus on explaining why things are true. “Being able to explain something requires a clearer understanding than just being able to get the right answer,” he says.

The last course, which wouldn’t have a lab attached, would be an algebra course, focused on how algebraic ideas are related to geometric and arithmetic ideas taught in elementary school. “Seeing what comes next helps a teacher know what to emphasize,” Klein says.

**Bruce R. Vogeli**

Bruce R. Vogeli, the Clifford Brewster Upton Professor of Mathematical Education, at Teachers College, Columbia University, has long focused on enhancing, extending and deepening the mathematical understanding of secondary school teachers. “In many, many instances, at institutions that certify secondary teachers, the mathematical training is superficial actually and so the instruction in the classroom is usually without very much mathematical depth or without the excitement that a good knowledge of mathematics can help to create in the classroom,” he says.

“Everybody thinks secondary school teachers ought to know a lot of mathematics but they do not put in place appropriate training programs. There’s still a lot of what were formerly called professional courses, the teaching of algebra, teaching of geometry, calculus for teachers. These have attached to them a reference to value in the classroom for prospective teachers, but every time you see that reference, you have to interpret this as: ‘This is not really hard mathematics. This is for teachers; they don’t need hard mathematics.’”

Yet, Vogeli says, “the tough stuff” is exactly what high school math teachers need. “They are not well enough informed in their field to prepare secondary school students to cope with the mathematical demands in the workplace, in colleges and universities. I don’t think we should try to prepare PhDs in mathematics beginning at the secondary school level, but we need to prepare people who are not afraid of mathematical challenges.”

**Hung-Hsi Wu**

“We have consistently taught our teachers mathematics of two kinds,” says Hung-Hsi Wu, professor emeritus of mathematics at the University of California, Berkeley and a former member of the National Mathematics Advisory Panel. On the one hand, departments are preparing their undergraduates for graduate-level study in mathematics, and, on the other, they’re offering courses designed just for prospective elementary or even middle school teachers. “The usual adage is that’s trivial mathematics” – and in some cases these math-for-teachers courses ignore or even vulgarize the actual mathematical principles at stake.

In short: “Neither is adequate,” Wu says.

Instead, Wu urges that we should think about teaching mathematics education – “school mathematics” – as we think about engineering.

“Engineering is the customization of abstract scientific principles to create processes or products that human beings want in their social or everyday use,” he says.

Likewise, in school mathematics, one should “tailor all those principles to the needs of human beings. In this case, the human beings are teenagers and young children; they have specific needs. Therefore, it’s our job to customize the mathematics we know into a form that they can use, that they can understand, without sacrificing the mathematical principles, of course.”

He continues: “Like all things, there are dual, opposite demands on what we’re trying to do. On the one hand, you want your product – you want school mathematics – to be user-friendly because if it is not user-friendly, our young kids cannot use it … On the other hand, yes it’s user-friendly; now, is it correct? Does it do any harm to our children to learn it?”

“Our job is to teach correct mathematics, not any mathematics,” Wu says.

Thinking about school mathematics in terms of engineering “lends clarity to the current debate in mathematics education,” Wu argued in a recent presentation. “There is no controversy in stating that engineering should not attempt to produce anything that caters to human wishes but defies scientific principles … There is also no controversy in stating that engineering should not waste time producing anything that is irrelevant to human needs no matter how scientifically sound.”