Math Teaching

One curious phenomenon is the way that some liberals oppose good math teaching for fear of being associated with conservatives. Take this article by Barry Garelick, identified as a mathematics analyst for a government agency:

I work for the federal government, which has a program that gives employees a chance to work on Capitol Hill to gain experience and knowledge of legislative and congressional procedures, which is valuable information when working in government. I applied for and received a six-month detail to work in a Democratic senator’s office. Senator X (so called, in keeping with mathematical convention to describe a class of variables, because, as I was also to learn, both the good intentions and the shortcomings of Congress are institutional) was interested in establishing a science project to nurture a “homegrown” breed of scientists and engineers who would then support that state’s burgeoning technology industry. Since I thought a likely place to start would be math education, the staffers working the education issue asked me to see what I could come up with.

I compiled a list of questions that I sent by e-mail to various mathematicians involved with the math education issue. The questions focused on the quality of textbooks and teaching, with emphasis on algebra and geometry. I also wanted to know whether K–6 texts taught arithmetic well enough to prepare students to learn algebra.

The nice thing about working on the Hill is that you almost always get responses to e-mails and phone calls. Fifteen minutes after I sent an e-mail to Harvard mathematics professor Wilfried Schmid, he called. I found out that his initiation into the world of K–12 math education was similar to mine—through his daughter. He explained how she was not being taught her multiplication tables. He was shocked at the math instruction she was receiving in the 3rd grade. Its substance was shallow, memorization was discouraged, students were kept dependent on mental crutches (her teacher made her work with blocks or count on her fingers), and the intellectual level was well below the capability of most of the kids in his daughter’s class.

Schmid’s reaction to the problems of math texts and teaching was similar to that of other mathematicians I talked to in the course of my Capitol Hill assignment, particularly those with children. Those dialogues led me to develop an ad hoc theory that I will postulate (this means I don’t have to prove it): Hell hath no fury like a mathematician whose child has been scorned by an education system that refuses to know better. In Schmid’s case, he talked to parents, school boards, and ultimately with the Massachusetts commissioner of education. Along with others, he succeeded in revamping Massachusetts’s math standards, much to the dislike of the education establishment and textbook publishers.

The controversy over K–12 math education has come to be known as the “math wars.” Like Schmid, mathematicians have been active in this debate, as has the “mathematics community” at large, including not only mathematicians at the university level, but teachers and others involved in the education establishment. They believe that students must master basic skills (the number facts, standard algorithms for adding, subtracting, multiplying, and dividing) in tandem with larger concepts about mathematics.

On the other side of the debate are the followers of an education theory that promotes discovery learning, minimization of both teacher instruction and repetitive drills, and a disdain for standard procedures (algorithms). The math being protested—by the mathematics community—is called a variety of things: “reform math,” “standards-based math,” “new new math,” and, most commonly, “fuzzy math.”

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Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I’ve seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student’s going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method—one that is held in disdain because it uses rote memorization and is not discovered by the student—the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.

So that depicts the situation. How do some liberals respond?

Though academic debate about mathematics curricula will no doubt continue, the field of argument is increasingly muddied by politics. It was in this context that I began my investigation into math education in 2002. I recall meeting with Senator X’s deputy chief of staff and two other staffers not long after completing my research on math curricula and the battles that had shaped—often, misshaped—them. “So what are your ideas on how math and science education can be enhanced?” they asked. My answer was something like, “You can enhance a car by painting it, but if the car has no engine, it’s not going to do much good.” This was not what they were expecting to hear. Nor were they expecting to hear that Lynne Cheney had also taken up the cause of anti-fuzzy math. At that point, the discussion took a decidedly troubling turn. These staffers—Democrats—now worried that they could not support policies that were also advocated by the wife of a powerful Republican.

I told them about the open letter from the two hundred mathematicians and urged them not to confuse the message with the messenger. “This is a real issue,” I said. “Kids aren’t learning the math they need to learn.”

I had discussions and sent e-mails in the hopes that I would at least get a chance to brief Senator X on the issue and, perhaps, persuade him to ask some tough questions of NSF when it came time to fund their programs. But I felt that at any moment everything was going to be whisked away.

And one day it was. The staffers in my office talked with other Democratic staffers on the Hill, who told them that it would be wise to stay away from the “fuzzy math/Lynne Cheney/Bush agenda” issue. Ultimately the staffers I was working with told me they couldn’t take a chance on having Senator X “come off like Lynne Cheney.”

This development was not surprising to any of the mathematicians with whom I had been working—most of them Democrats, like me. The senator was never briefed, and no investigation into NSF was launched. I was thanked for my hard work. I went back to my regular job and started tutoring middle school students in math at a school in D.C. while continuing to work with high school students in my neighborhood. That year a 9th-grade girl was having problems in geometry and came to me for help. “What seems to be the problem?” I asked. “I don’t know how to do proofs,” she said.

“I know,” I replied. “Don’t worry. It isn’t you.”

All politics is local, I decided.

Cheating kids out of an education in basic math is inestimably more harmful to their actual lives than anything that they might be taught about evolution.