August 1998

Not since the introduction of "New Math" in the 1960s has there been so much public controversy about the state of the nation’s mathematics education. It has been nearly a decade since the National Council of Teachers of Mathematics (NCTM) issued its groundbreaking standards calling for more emphasis on problem solving and student-initiated discovery. The standards have led to major changes in state education frameworks and curriculum, which, in turn, have sparked questions and criticisms. Some groups claim that reforms have downplayed traditional skills and content. For example, the Wall Street Journal has featured several editorials in the past year characterizing new curriculum based on the NCTM standards as watered-down mathematics which favors student creativity and group work over right answers.

Within EDC, mathematics experts cringe at the polemics of the media debate while simultaneously wrestling with the genuinely difficult issues at stake. Al Cuoco, director of EDC’s Mathematics Initiative, and EDC Vice President Wayne Harvey, a mathematics education researcher, emphasize that improving mathematics education goes beyond a simple choice between traditional mathematics and mathematics based on the NCTM standards. It requires a multifaceted approach focused on improving several things: curriculum materials; assessments of student understanding; the undergraduate and continuing education of teachers; and the relationships between K-12 teachers, mathematics faculty, and the general public. EDC currently manages projects focusing on all of these areas. Inherent in each project is a rich and robust definition of the word mathematics, as Cuoco and Harvey described in a recent interview.

Mathematical habits of mind

Q: When people at EDC talk about what mathematics students should learn, many use the phrase "mathematical habits of mind." What do we mean by that?

Al Cuoco: There are things kids need to understand about mathematics that are not at all the things we have traditionally taught them. The traditional focus has been on skills, but skills are just a means, not an end. So kids were learning the material in the book but missing the point of mathematics. One of the most useful things a kid can take away from class is a style of work¾a set of mathematical methods¾rather than a collection of results.

Take any mathematics fact, like the formula for the area of a triangle: one-half times the base times the height. Ninety-eight percent of kids will never use the actual formula in their lives. The reason to study it doesn’t lie in the fact itself but in the thinking that goes on behind that fact and what you can do with the fact. In our geometry curriculum, we ask students to find a method for cutting up a triangle in order to rearrange the pieces to get a rectangle. There are many ways to do this. Some students end up with a rectangle whose base is half the base of the triangle and whose height is the same as that of the triangle. Others end up with a rectangle whose base is the same as the triangle’s but whose height is half as big. Both methods lead to the "1/2 x b x h" formula, but that’s almost a side effect of the activity. What’s important, and what the class discusses, is the variety of methods for doing the cutting and rearranging, the proofs that the methods actually work, and the arguments that the different methods produce the same results. They then apply their findings to find the areas of other triangles—and to figure out area formulas for other shapes.

This example has all the ingredients of real mathematics: The problems are complex and nontrivial; students use abstraction to move their arguments from a particular triangle to the class of all triangles; and they end up with a beautiful mathematical theory in which a collection of results fit together to produce something much more satisfying than all the individual pieces.

Wayne Harvey: Al’s example illustrates that the mathematics doesn’t begin and end with a collection of facts. Traditionally, we’ve done a tremendous disservice to the public in how we’ve advertised mathematics. Many people believe mathematics and arithmetic are the same thing. That’s like saying that writing is the same as spelling. Parents tend to focus on how well their child is learning mathematical facts. Instead, I’d like to see them asking "How is my child’s understanding of mathematics helping him or her gain new understandings of the world?"

AC: Dealing with complexity at any level involves mathematics. Scheduling a school is a mathematical exercise. Using a word processor involves a lot of mathematical thinking. The most mathematical thing I’ve done in my life, outside of actually doing mathematics, is building a house¾not because of the calculations I had to do, though there was a lot of that. It was the mathematical thinking: doing thought experiments; predicting how something is going to look before it exists; trying to come up with an efficient process for carrying out certain tasks. Those kinds of things are the common currency of mathematics and it turns out they are also very useful in other parts of your life.

What about skills?

Q: Is there a tradeoff between teaching skills and teaching mathematical thinking?

WH: For some kids, teaching skills is enough. They can encounter the skills and, on their own, make connections and expand their thinking. But there’s so much more we could be teaching. We need to teach skills, but not to the exclusion of the rest of mathematics.

AC: One thing we can do is integrate the learning of skills with the learning of habits of mind. A mathematician friend of mine says that Chopin wrote a lot of etudes [practice exercises] that were also beautiful music. You can have exercises that are designed to build skills and have a point. They lead some place. There is a sequence to them.

WH: Music is a good example of how important practice is in developing skills and in building enjoyment in the learning. There are certain ingredients that come out of the practice, but it’s done in the service of something larger.

Q: Let’s turn to some specific examples of good mathematics activities. In some of your presentations, you use some nice examples of ways to "open up" traditional mathematics problems.

AC: One of the key points in these examples is that in order to solve the problems in the second column, you have to be able to do everything in the first column—plus more. You can solve the first problem by straight calculation. But what you have to do to solve the open version is imagine the calculation without actually carrying it out¾ because you can’t carry it out. You have to reason about the properties of the calculation. What does it mean for two numbers to add up to 11?

WH: Most of the challenges we face in our professions do not take the form of "Here’s the data, find some solution." Instead, it’s "Here’s some partial data and a partial solution. How can I determine the right questions to ask, the additional data I need?" Often, you need to conjecture about the answers even before all the data is available. And you need to work with others to develop those conjectures. That kind of thinking doesn’t come from practice solving closed problems where all the data is presented to you.

Q: Some critics of the NCTM standards view some open-ended problems as problems in which there are no wrong answers. Whatever students come up with is okay.

WH: That’s a legitimate concern. How can you tell whether a given problem really accomplishes the ends we’re discussing? We need better measurements. We’ve seen plenty of activities that purport to do deeper mathematics, but they don’t play out that way in classrooms.

Equity in math class

Q: Does the process of opening up problems provide a more equitable approach to mathematics by allowing for more ways into a problem and perhaps different levels of solutions?

WH: The issue of equity raises the question of what it means to be successful in mathematics. We need to realize that it’s okay to struggle sometimes. It doesn’t say anything bad about your talents if you work hard on a problem and you don’t find a way through it. There’s still learning to be had there. People will have different successes at different times with different problems. In mathematics, we tend to believe you’re successful if you get the right answers, and otherwise you’re doing poorly.

AC: We’ve also viewed the ability to do something quickly as an indication of being smart. It’s just not true. There are mathematicians who do things very quickly and some who have to crawl off and work on things for a while¾ sometimes for years.

WH: And along the way, they make mistakes and backtrack and start over . . .

AC: Computer science as a discipline has really capitalized on the value of learning from your mistakes. The whole notion of debugging a program is based on your ability to go back and analyze your mistakes so that you can go on to the next step.

I guess the larger point is that there isn’t one approach to mathematics that works for everyone. Still, I can point to three experiences I want every kid coming out of a middle or high school mathematics class to have: some experience solving difficult problems; some experience with abstraction; and some experience building a theory. Those kinds of experiences, to me, are the essence of mathematics.

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