# Brainstorming

Ross Taylor should have been an ace in the classroom. He was, after all, a high school algebra teacher who literally wrote the textbook. Yet he'd be teaching from his very own book when kids would suddenly bark at him, "You don't know what you're talking about!'' "Just wait a minute,'' Taylor would rejoin, "I wrote this book!'' But it didn't make any difference; the students still didn't get it. They would take something like 4x-x, erase the x's, and say the answer was 4. Taylor would tell them that they were wrong, that they'd have to use the distributive property and do this and then that, but the students would just look at him as if he were nuts. And maybe he was, he now confesses, because he could have simply asked, "If you have four of these things, and you take one away, how many do you have left?'' That they would have understood.

Nothing about the teaching of mathematics, Taylor learned, ever went as planned. At Harvard University in the 1960s, Taylor was trained in the theories of the New Math. He then brought it back to the high school classroom, thinking he had discovered the instructional equivalent of a high wattage channel through which he could transmit his message. At first, everything seemed to be going fine. His students were quiet, and quiet was studious--or so he thought. Then, six weeks into using the new approach, a kid raised his hand and said, "What are you talking about?'' No one else, as it turned out, knew what he was talking about either. Taylor had been talking to himself.

Now retired from a 23-year career as director of the K-12 math curriculum in the Minneapolis public schools, Taylor claims that thousands of math teachers are teaching math the way he did. They stand at the board and toss out equations, axioms, and postulates--winged abstractions that flutter about the heads of oblivious students. As far as most teachers are concerned, he says, the student's role is "not to reason why, just convert and multiply.''

But simply converting and multiplying, Taylor and many other mathematics educators now agree, has not served the vast majority of American students very well. Each year, half of all high school students who take math do not go on to the next level, many undoubtedly sensing, as one education observer once put it, that "mathematics courses are chiefly designed to winnow out the weak and grind down the ungifted.'' Only 10 percent of high school students--most of them white males--eventually wend their way through calculus, the course considered indispensable for later work in scientific fields.

Just as bad, perhaps, is the fact that those who do take math do not take much from it. A University of Chicago study reveals, for instance, that only 30 percent of high school geometry students are adept enough at abstraction to understand the subject as it is currently taught. At the University of Minnesota's Institute of Technology, which requires a stellar SAT math score of 650 for admission, 55 percent of students fail first-year calculus.

In 1989, at a national convention of mathematics educators, Taylor discussed such problems with Sherry Fraser, a former math teacher and well-known innovator in mathematics education. Fraser believed that traditional approaches to math instruction simply were not working. Far too many students, she told Taylor, were never getting past rudimentary math classes. Taylor agreed, expressing frustration that there were no curricula available that managed to draw all students into higher-level mathematics, that managed to be both demanding and inclusive. "Actually,'' Fraser told him, "one is in the works.''

With several other Bay area mathematics educators, including two professors from San Francisco State University, Fraser was then working on what is now known as the Interactive Mathematics Pro-gram, a curriculum inspired by the new standards of the National Council of Teachers of Mathematics. The standards, which have generated a gale storm of controversy in a field long associated with choking chalk dust, aim to turn mathematics education into an intellectually demanding problem-solving process, emphasizing such things as inference, deduction, and logical analysis over the drill-and-skill learning that has characterized traditional math instruction.

Taylor wasn't the only one in Minneapolis having doubts about the efficacy of the old approaches to math education. Many high school math teachers were concerned about the increasing numbers of students abandoning the subject. In 1991, three Minneapolis algebra teachers--Jane Kostik, Judith MacAlpine, and Ed Anderson--journeyed to California to look at several new math programs that were in place in a number of Bay area schools. One of them, Sherry Fraser's Interactive Mathematics Program, stopped them cold. IMP was different, in actual classroom practice, than anything they'd ever seen. For one thing, algebra, geometry, and trigonometry were not taught as separate subjects but were folded into all aspects of the entire high school mathematics curriculum. In their attempts to solve a wide variety of real-life problems, students rolled dice, swung homemade pendulums, and cast shadows with flashlights. But it struck the teachers that these hands-on activities were much more than fun and games. Each had a mathematical purpose that meshed with the theme and other exercises of the given unit of study.

In one unit, students examined the growth of an island rat population, learning, in the process, how to formulate a chi-square. In another, they sought to determine the best design for a honeycomb. Using the Pythagorean theorem and trigonometry to figure the area of polygons, they eventually determined that bees are "smart'' to choose hexagons.

There was in these classrooms a palatable seriousness of purpose. Students jotted down observations, compiled data on graphs and charts, and collaborated with classmates to come up with what were sometimes very novel solutions to difficult problems. They also--this virtually unprecedented in mathematics classrooms--wrote copious reports detailing what they had learned and how they had learned it.

One other thing made a powerful impression on the teachers: The students they observed were engaged in rigorous mathematics, and yet they didn't look like the typical high-level math kids; that is, most were not white middle-class males. In fact, they looked a lot like the students back at the teachers' own schools. It occurred to them that IMP might be the answer to the tracking that had so defined the history of mathematics education. Maybe here, at last, was a program rooted in "real'' math--rather than "general'' or "consumer'' math--that would allow the tracks of high and low achievers to finally converge.

Although Taylor had by this time retired from the Minneapolis school system, the teachers' enthusiasm for IMP upon their return from California motivated him to take a look at the program himself. The more he learned, the more he began to think that IMP just might be one of those rare instructional approaches that dovetail solid mathematical content with an understanding of how kids learn. "IMP was a radical departure from anything I'd seen,'' Taylor says. "It speaks to kids instead of mathematicians by starting with meaningful problems instead of a lot of abstract stuff teachers try to cram in.''

Taylor helped the teachers secure a National Science Foundation grant so they could receive training in how to teach IMP. Now, three years later, teachers at five Minneapolis public high schools have implemented the program; at one, Henry High School, all 10 math teachers use the curriculum. (Nationwide, IMP is in place in 140 schools in 13 states.)

For Henry High math teacher and Minneapolis IMP co-director Jane Kostik, giving up traditional teaching approaches for something so completely new was a bit like leaving behind the comforts of home. "You parent the way you were parented, and you teach the way you were taught,'' Kostik says. "We teachers learned in the old system, so it's natural for us to want to teach that way. You get up there, explain how something's done, and then assign problems one through 40. It may be boring to the kids, but you do it anyway; you're a teacher, and you like explaining things. But then you get to the point where you realize that there's got to be something better because all kinds of kids are falling through the cracks, and a lot of them are bright, creative kids who simply don't learn the way we did.''

Not surprisingly, many math educators don't see it this way. They argue that the open-ended problem-solving approaches favored by IMP and the National Council of Teachers of Mathematics don't provide students with essential basic skills. But these people, Kostik insists, miss an important point. "Our students learn the most important basic skill of all: how to use what they've learned to attack a problem.''

At Minneapolis' Edison High School, a dilapidated inner-city school so
security-conscious that I had to leave my driver's license to secure a
visitor's pass, an early morning first-year IMP class tackled this
problem: Al and Betty are playing a game with the spinner below. (Three
of the four segments on the spinner face are white; the fourth is
gray.) Each time the spinner comes up in the white area, Betty wins $1
from Al. Each time it comes up in the gray area, Al wins $4 from Betty.
In the long run, which of the players is likely to come out ahead in
this game?

The problem, which came near the middle of a unit on probability (units typically take four to eight weeks), struck me as disconcertingly simple. At a glance, I knew that they would come out even. I was wrong, of course, though it took me a while to see it. Thinking myself a bit of a fool, I felt somewhat better when I saw that none of the three students I was working with immediately got it either.

Teacher Becky Montgomery, who had a way of inspiring students with her own seemingly unrehearsed curiosity about the problems she presented, had pairs of us play the spinner game 25 times, using a pencil tip and paper clip as the spinner. Charting the results on the blackboard, we then saw that while Betty cumulatively won almost three times as many games as Al, Al still came out well ahead in terms of money won. In fact, it seemed apparent that he would always come out ahead.

"What's the probability that Betty will win?'' Montgomery asked. "Three to one,'' someone offered, and others agreed. "Is this game fair?'' No, most agreed, it was rigged in Al's favor. A boy explained why, demonstrating on the overhead projector what would happen if Betty and Al played 400 games. (The letter P in the following equations stands for probability.)

P (Betty) = 3/4

3/4 x 400/1 x $1 per game = $300 won

P (Al) = 1/4

1/4 x 400/1 x $4 per game = $400 won

The spinner game concluded with Montgomery assigning the Problem of the Week, or POW: Students, working with a partner, were to create games of their own that used strategy and probability. Their classmates would play the games and grade them.

After class, I told Montgomery that I had gotten the spinner game wrong, that I had thought Al and Betty would come out even.

"Who benefits from that kind of thinking?'' Montgomery asked me. "Well, the lottery, of course,'' she said, answering her own question. "Most people don't get past what their intuition tells them. In fact, a lot of them will look at the spinner game and immediately say, 'Betty's going to win.' But these kids will understand odds and how the lottery works. They'll know that Betty has a much better chance to win at the spinner game than they ever will at the lottery.''

Montgomery and her IMP colleagues believe that numerical sagacity is a kind of inoculation against various social and political deceits. Unlike most citizens, IMP graduates would never have to take a politician's interpretation of charts, facts, and figures at face value. They would have the skills to interpret the statistics for themselves.

Over the next two days, the students in Montgomery's class worked to solve other probability games and puzzlers--all permutations of the spinner game. They predicted the winner of games in which darts were randomly dropped onto a "rug'' with colored patterns of variously assigned values. And they calculated the odds with which certain sums and products would turn up on a series of dice rolls. With almost every activity, new concepts were introduced.

During one session, the students were given a deck of playing cards and the following problem: Mia gets two points for a heart, one point for a diamond, and no points for a club or spade. What would her average score be each time she cuts the deck?

This turned out to be a sticky one. The students saw right away that there are 13 of each suit in a standard deck of 52 playing cards and that the probability of turning up any one suit on a single cut was one in four. This understood, they easily calculated that if they played 100 games, the odds were that each kind of card would be drawn, on average, 25 times. What they had a much harder time understanding was the seemingly elementary concept of "average score.'' They seemed to grasp on an intuitive level what a baseball average indicated or that an annual income of $100,000 was well above average, yet when Montgomery asked them to calculate an average she was met with blank stares.

For a moment, then, it was back to basics. Montgomery wrote on the blackboard the numbers 5, 2, 10, 1, 7, saying that the average was five. In a flash, an inspired student deduced the principle behind calculating averages: "You take the total and then divide it by the number of things you've got.'' From here the card problem crept toward a resolution. One of the students I was working with eventually jotted down something like this after a number of false starts:

P (heart) = 1/4

1/4 x 100 = 25 games won

P (diamond) = 1/4

1/4 x 100 = 25 games won

P (clubs and spades) = 2/4

2/4 x 100 = 50 games won (no points awarded for club or spades)

hearts 25 x 2 = 50 points

diamonds 25 x 1 = 25 points

clubs and spades 50 x 0 = 0

75 total points

But members of my group still had trouble coming up with the elusive "average score per cut.'' They were rescued by a girl from another group who reminded them, perhaps with a hint of exasperation, that to find the average you had to divide total points by the number of times played: 75 points divided by 100 games equals .75, the average score per cut.

After this first-year class, I visited two different third-year IMP
classes at nearby Henry High School, where the students were also
working on probability as part of a unit called "Pennant Fever.'' They
had spent two weeks using concepts such as combinations and
permutations, trying to find out if "The Good Guys'' were likely to
maintain their three-game lead over "The Bad Guys.''

Third-year IMP was markedly different from first-year in terms of mood. While first-year students were clearly absorbed by the problems at hand and highly responsive to Montgomery's questions, they were also somewhat wary of working in groups and sometimes hesitant to offer to the class solutions of which they were less than certain. By the third year, this had all changed. Without the least bit of reserve, the students exchanged ideas, argued over varying approaches to problems, and exchanged quips as they worked together. There was plenty of humor, too, though it was confined to the margins, like the "math wonderwoman'' whose wisecracking caricature, offering up slangy nuggets of mathematical wisdom, kept popping up in the corner of each student's transparency the teacher projected.

Of course, the problems here were also considerably more challenging than those the first-year students tackled. In an exercise called "Fair Spoons,'' for example, students were asked to calculate how many red and blue spoons must be placed in a bag if someone randomly drawing them were to have a 50 percent chance of a match. The first volunteer, working through the problem on the overhead projector, concluded, "If the number of red spoons equals the number of blue spoons, then the probability of getting a match equals 50 percent.'' Gleefully, if good-naturedly, her classmates pointed out the error in her thinking. She had failed to realize that drawing, say, an initial red spoon from a bag of three red and three blue spoons would drastically alter the probabilities of a red on the next draw: it would be well below 50 percent.

In fact, as several students went on to demonstrate, the correct answer was strongly counter-intuitive: To have a 50 percent chance of a match, you'd need to begin not with even numbers of red and blue spoons, but with one red and three blue, among an infinite number of other possibilities (three red and six blue; six red and 10 blue).

An even more difficult problem involved calculating the chances of 10 workers who eat at one of two lunch counters each day all randomly deciding to eat at one of the establishments on a particular day. The students were to write a short report informing one of the counter's owners how many stools they need in order to be 95 percent certain of having enough stools for any one day in a month.

This problem lost me in a hurry, but the students attacked it with zeal. What mattered, they quickly agreed, was "combinations, not permutations''; that is, what mattered was simply the number of people showing up at the counter, not the order in which they showed up. And the odds of all 10 showing up at the same place were, they figured, 1 x (.5)10 = < 1. (The actual figure is .000976.) One girl, with amazing speed and clarity, launched her classmates through a series of calculations to demonstrate that the number of stools required was in fact 8.

Critics of the IMP approach say that while these problems may be
interesting, they don't demand the kind of rigor associated with the
systematic study of algebra, geometry, trigonometry, and
precalculus--the traditional sequence in American mathematics
education. And there are many math traditionalists, particularly in
ever-polarized California, who condemn the NCTM standards and programs
based on them for diluting full-strength mathematics with a fuzzy
emphasis on process, cooperative learning, and entertaining brain
teasers that kids might best solve on a long car trip. These critics
generally believe that math is first and foremost about getting right
answers, not the joy of thinking through problems. Particularly cynical
about NCTM-based programs is the voluble textbook publisher John Saxon,
who has long claimed that the so-called "progressive'' approaches of
teaching math entail a covert but nevertheless lethal lowering of
standards.

"The leaders of the NCTM,'' Saxon charges in a stinging two-page ad that recently appeared in this magazine and Education Week, "want to raise the grades and the self-esteem of minorities . . . instead of ensuring that the students know what they should know.''

Of course, skepticism about supposedly new and improved ways of teaching math is shared by many math teachers--particularly long-tenured veterans who have seen an array of fads come and go. One such veteran is Edison High School's David Kotz, whose geometry class I sat in on. His students were working through a chapter titled "Medians, Altitudes, and Perpendicular Bisections.'' I could have been in a time warp, so similar was the material to what I remembered studying more than 20 years ago. ("If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the end points of the segments.'')

While his students worked on a series of problems, Kotz told me that he had tried teaching the IMP way but quickly decided it wasn't for him. He simply was too accustomed to "the old way'' to change. "I'm happy [the IMP teachers] are doing what they're doing,'' he said. "But I'm more comfortable with telling kids how to do something.''

As he talked, Kotz echoed a common criticism of the proponents of the NCTM standards and associated curricula: They mean well, he said, but their emphasis on cooperative problem-based learning and authentic assessment--most IMP students submit portfolios as opposed to taking traditional tests--has weakened the rigorous standards that have always so defined math. "The students take good tough tests here,'' Kotz said of his classroom. "The new standards, on the other hand, get away from long-established testing procedures that really stretch students. I believe that students need to get cranked up for good tough tests, so when they get the results, they'll know where they stand.''

Preliminary research, though, demonstrates that IMP students are holding their own, at least as far as their performances on standardized tests are concerned. In a limited study (researchers at the University of Wisconsin are now conducting a more thorough one), control groups of IMP and non-IMP students achieved similar results on the SAT, even though IMP students had much less exposure in class to the standardized-test format.

Kotz encouraged me to talk with his students, so I asked them about their perceptions of IMP. One said, "In IMP, you swing things from the ceiling and stuff like that.'' Another said, "You get a lot of weird, confusing problems that you're somehow supposed to solve.'' One boy, who had dropped out of IMP after an unhappy year, said, "I like problems from a book, not a bunch of packets like IMP has.'' I told him I wasn't sure what he meant. "A textbook,'' he said with some irritation. "I like problems from a textbook.''

Still, several others said they had wanted to take IMP but had been steered away from it by counselors and former teachers. One girl explained that her counselor had told her that she was too smart for IMP, that it wasn't the right thing for her. "That's the problem I've always had with math,'' the girl said, "teachers who say there's only one way of doing something when I know that's not true.''

"I regret that I didn't take it,'' she said, "because most of my friends who take it love it.''

Suggestions that IMP somehow represents a lower track and diminished
standards, that it's an interesting diversion from "real'' math, drive
the Minneapolis IMP teachers to distraction. As far as they're
concerned, the fact that some counselors, teachers, parents, and
students see it as a remedial program in disguise demonstrates the lock
that the school-as-a-sorting-mechanism concept still has on people. As
one teacher said, "Parents still want to know if their kid is going to
be in the redbirds or bluebirds.''

Almost all of the IMP teachers I met in Minneapolis had been traditional algebra teachers who said that if it weren't for IMP, as many as two out of three students they now teach would be out of math altogether. But this, they emphatically pointed out, did not mean that IMP was easier or inferior math. In fact, they insisted that IMP was structured so that problems were "layered'' in terms of complexity, presenting students of all abilities with challenges.

One afternoon, I asked a group of IMP teachers what they thought of John Saxon, who attacks the NCTM for its emphasis on problem solving as opposed to teaching skills. ("This is a horrible mistake!'' he says.) There was a moment of silence, as if mentioning Saxon's name had broken some taboo; his drill-oriented textbooks are anathema to everything the IMP teachers are trying to accomplish.

Finally, Jean Stilwell, a teacher at Henry High, spoke up. "If you want to teach students to think, then you have to do all the things John Saxon attacks,'' she said. "If you want your students to do well on multiple-choice skill-based tests, then use Saxon's texts. But for the world of work, what difference does it make how you do on a multiple-choice test? No one hires you to solve algebra equations--they hire you to solve problems. So Saxon does a great job of teaching you obsolete things. His students learn, review, learn, review, but to what purpose?

"We find that when people use lots of drill and skill, lots of repetition, they can do well for a while, but that it won't last. A few years, and it's gone. That's how I learned calculus, and when I opened the book a few years later, I could remember nothing. I got A's, but did I really learn anything?''

At this point, several students from the third-year IMP classes joined the conversation. Uncowed by the presence of their teachers, they rather giddily extolled the virtues of their mathematics classes, tossed a few playful barbs at their non-IMP classmates, and swapped stories about their favorite IMP units. The winners were "The Overland Trail,'' a unit that asks students to calculate, among other things, which group of pioneers will run out of water first, and "All About Alice,'' in which Alice's weight is doubled or halved by eating or drinking certain magical items. In the first unit, students learn to represent situations in terms of algebraic equations. In the second, they work with exponents and are introduced to logarithms.

"A lot of the regular math students put us down for being in 'easy math,' '' one girl said. "But we know we've learned things that they haven't, and they feel threatened by that.''

Elaborating, a classmate said, "They're in geometry, and yet, they don't know how geometry works. It's like they take geometry and then, when they're done with it, throw it out of the window. And then they get into analysis, and when that's done, they throw that out of the window, too. They'll look at a problem they haven't seen for a while, and the first thing that comes out of their mouth is, 'I can't remember how to do it.' But that doesn't happen to us because we know there's no one way to solve something--you could try this, or this, or this.''

This was the same student who had so lucidly explained the lunch-counter problem to her classmates. She was obviously a gifted mathematics student, as articulate as she was quick, and so I asked her if she ever felt hampered by "slower'' students. "Never,'' she said without hesitation. "Sometimes the best way to figure something out is to explain it to others.''

Before leaving Minneapolis, I again visited with Edison High's Becky Montgomery, this time at the end of the school day. Although committed to the IMP way, Montgomery was still teaching one class of traditional algebra, which, she said, was driving her "nuts.'' I asked her why, and she grumbled about how constrictive it is. Then she caught her breath and, in a burst, summarized just what this new approach to mathematics was trying to accomplish.

"It seems as if we've too often forgotten how we've learned something, how the knowledge we possess has come about,'' she said. "We just look at the back end, the answer, and then say, 'I'm going to tell you how to get it, and you're going to know it.' But we've forgotten that people learn by going through the experience of finding something out. If they don't have that experience, it doesn't matter how you show them the answer--that answer will have no eloquence, no richness. IMP brings that experience back.''

Vol. 07, Issue 04, Page 1-24

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