The Rochester Review, University of Rochester, Rochester, New York, USA
By Rose Ericson
Were they the daredevils on the luge? The bobsledders? The downhill skiers?
News reports were, confusingly, making different claims. So who were the real speed champs?
Borasi's objective was to entice these middle-school children into the realm of statistics--to get them to experiment with, and master, the use of statistical tools.
She could have just handed them a (ho hum) sheet of equations to solve, and got the predictable (ho hum) response.
But Borasi, a professor of teaching and curriculum at the Margaret Warner Graduate School of Education and Human Development, had another approach to try. She identified something these youngsters were passionate about: the thrilling Olympic competitions they were viewing every night on television. She pointed out an intriguing contradiction in the reporting on these events and guided the students as they were drawn into settling it.
"The discussion got us to the question of just what was being measured as 'speed,'" Borasi recalls. "And we realized that it could be average speed that was being cited, or it could be instantaneous speed, where you try to consider shorter and shorter intervals."
That the class, using average speed as their criterion, eventually calculated the women downhill skiers to be the swiftest Olympians was only an incidental part of the students' achievement. These middle-schoolers had begun to think like the budding statisticians they had become.
And they now knew that there was more than one correct answer to a mathematical question about speed, depending on how you define it.
Borasi uses the Nagano exercise to illustrate her premise that learning through inquiry--rather than by memorizing a rigid series of so-called "right" answers--is an important way in which math education needs to change if it is to bring out the best in American students. Too much emphasis on "right" vs. "wrong" interferes with the desire to learn, an essential first step, she points out, toward mastering the skills of reasoning and communicating mathematically.
Borasi's work is coming to the fore at a time when the pressure on schools to improve student performance has never been greater. Survey after discouraging survey shows American students lagging behind their peers around the world. (One recent 21-country survey of 12th-grade science and math students showed the United States ahead of only Cyprus and South Africa.)
"She's a star," says Warner School Dean Philip Wexler, putting it simply. In 1997, he tells you, Borasi was invited by the NSF to give its program research directors a seminar on the future of math education. "That's like going to the source of the river," Wexler says. "These folks set the agenda for the kind of research that gets funded in the coming years."
NSF officials are clear on the urgency of the need for reform:
"Too many students are being lost too early in mathematics," declares Diane Spresser, program director of Teacher Enhancement/Mathematics. "Clearly, if we are to guarantee that a first-rate mathematics education is available to every K-12 student in the U.S., we have to reform classrooms and schools."
Borasi's NSF-supported work, she adds, "has more than lived up to our expectations."
One of her collaborators, Marjorie Siegel, a reading specialist (more on that point later), speaks of her colleague's "deep understanding of mathematics that allows her to develop curricula that get at big concepts that other educators often miss." She also notes a "playful spirit" that enables Borasi to see things from a child's perspective and to tap into what naturally engages kids.
On the other hand, Siegel adds, "Raffaella has high expectations for herself and others. She's no milktoast."
After her college years, professional and financial considerations pushed her into leaving her home country. "I was lucky enough to get a Fulbright scholarship to come and study in America," she says. She earned her Ph.D. in 1986 from SUNY at Buffalo.
It was in the States that she discovered a world of interdisciplinary possibilities that she doubts she would have encountered back home.
When adding fractions, many beginners--doing what comes naturally--add numerators and denominators separately. For example, they add 1/4 and 1/2 and arrive at 2/6, rather than the correct answer of 3/4.
But when math educator Raffaella Borasi sees students making that mistake, rather than dismissing it summarily, she asks them to consider whether any circumstances justify adding fractions that way.
With baseball batting averages, for example, a player who hits 3 out of 4 times at bat and then 6 out of 7 times would have an average of 9 out of 11--an answer that clashes with the "correct" sum of 45/28, arrived at using the traditional mathematical computations.
Thus, the "wrong" answer in certain instances can be correct--reflecting the need, Borasi and other math reformers say, to question long-held assumptions.
"In this country I learned about things like classroom dynamics--how the relationship between the students and the teacher, and factors like anxiety, confidence, expectations, and beliefs, make up an integral part of the learning process. These are important variables to take into consideration when you're thinking of changing curriculum or methods."
When in the mid-'80s Borasi joined the Warner School and found herself its only math educator, it was a natural step for her to start working with others outside her own field. "If I wanted to collaborate with anyone, and by nature I don't like to work by myself, I needed to start hearing what other people were doing in disciplines other than math," Borasi recalls.
One important collaboration has been with her former Warner School colleague Marjorie Siegel, a reading specialist now on the faculty at Columbia University. The pair is engaged in preparing a book based on findings from an NSF project examining the place of reading in mathematics.
"It was interesting to combine two areas like that, to see the parallels and how reading could become a tool for learning in math," Borasi says.
Siegel adds: "Most people see reading a text in math just as a vehicle to get to the 'real' work." But that narrow view misses opportunities. Integrating reading with math, she goes on, creates students who "live mathematics, who are math learners and thinkers and inquirers"--not simply memorizers.
To show how it can work, Siegel and Borasi cite the case of the ovoid monument. As part of their study, the researchers asked a middle-school class to read an article about a Canadian mounted police unit that, for whatever reason, wanted to build a huge, egg-shaped monument.
If the students were to be responsible for designing such a structure, how would they proceed? Intrigued, they took notes on their reading, worked in groups to decide what questions needed to be answered, ranked the questions in terms of importance--and eventually came up with their recommendations for how the Mounties should proceed.
Following a more traditional classroom approach, the students would simply have been given a set of formulas from which to design an egg. The inquiry approach, on the other hand, sparks their curiosity and invites them to explore possibilities that, Borasi and her proponents say, lead to more profound learning and "ownership" of the problem-solving processes.
Both Borasi and Siegel say that in their reform efforts they have endeavored to avoid criticizing current systems and to respect the people and institutions that have gone before them.
When you are proposing changes, "there are political forces, historical and sociological forces, that you need to consider," Borasi says. "If you want to be a reform teacher, you'd better know how schools work, because otherwise you'll get squashed before you can make any changes.
Too many teachers fall prey to ideologies that she considers counter- productive, a prime example of which, she says, is the traditional focus on finding the "correct" answer and treating "wrong" answers as problems.
Talking about how errors can open up new and productive areas of inquiry, she offers an analogy.
If you're rushing to a business appointment, you know you can't afford to make a wrong turn and waste time.
But suppose you are visiting Paris, and you're out looking for a grocery store. If you get lost and find yourself near the Eiffel Tower, that could be a great thing, she suggests. You could pop up in the tower, chat with the locals (that is, if you'd been paying attention in French class), and explore unfamiliar neighborhoods. Your arrival at a suitable épicerie will happen naturally during the course of events.
If you're in an exploring mode, it's hard to pin down just what constitutes a mistake, Borasi says. "You're much more open to consider errors as opportunities for learning, or for more exploration."
"The first and easiest step," Borasi says, "was helping the teachers realize that students could do a lot more than they thought they were capable of." And that happened, she adds, as soon as they stopped expecting total accuracy "and instead let the kids discuss and argue and conjecture."
"Clearly, our program changed the way these teachers taught in their classes," Borasi says. More than one of them, she reports, has said, "I just can't go back to teaching the old way."
Cindy Callard, a seventh- and eighth-grade math teacher at Brighton's Twelve Corners School, agrees. Since she's been applying Borasi's methods, she says, "kids are much more involved in their own learning. They come up with ideas and discuss them, rather than just listen and take notes."
Instead of using the traditional "spiral" approach, in which students cover a variety of topics every year, with a little bit more about each of them added every time around, Callard's students now get more focused lessons. Using the "inquiry" method, Callard may take six weeks, instead of the customary two-week block, to teach an area such as formulas. But she is confident that her students learn formulas so thoroughly this way that they won't need to be taught them ever again --sparing them the deadly "been there, done that" syndrome.
"How much did they learn just by sitting there anyway?" Callard says of the traditional approach. Simply "covering" a topic, she notes, doesn't ensure that students have been learning.
Getting willing teachers excited about, trained in, and contributing to Borasi's concepts has been only part of the picture.
"The reform we're getting into asks people really to change their goals," Borasi says. "What they need to teach and what the students need to learn is not what everyone thought it was 10 or 20 years ago."
Probability and statistics, for example, are now so embedded in everyday life (think finance and sports) that if you want to be a literate citizen, says Borasi, "you need to know about things that previously weren't taught until college."
As with any change, resistance is to be expected, and reform comes slowly. She concedes that her thinking represents a radical departure from normal methods of instruction, and that it may be hard for some teachers and schools to show the flexibility and willingness to take the risks that her views imply.
But adopting a new attitude about inquiry and exploration may be the most direct path teachers can take to help their students learn to appreciate, and, yep, even enjoy, their math lessons.
And that may in this instance, Borasi believes, provide a clear-cut "right answer" to a longstanding problem about math.
Compute its area so that we can determine how much paint will be needed.
Warner School professor Raffaella Borasi--who believes that the traditional focus on "correct" answers to math questions closes off useful areas of inquiry--uses this exercise to illustrate her point that seemingly straightforward math problems do not necessarily have a single "correct" answer, or a "correct" path to reach it.
Borasi says there are a number of valid approaches to solving a problem based on a complex diagram such as the above. You might divide the figure into smaller components and use known formulas for determining the area of those pieces. You could cut your work in half by using symmetry in calculating your answer. Or you could just start counting squares.
The catch here is the number of partially covered squares in the diagram. Given no further information, the problem-solver is left to estimate the area of the incomplete squares--including a decision as to whether the painted area is defined by the inside or the outside of the boldly drawn outline. Depending on the guesses upon which further (valid) calculations are based, the "correct" answers may vary widely.
When Borasi posed this problem to math teachers in her professional development program, their arguably "correct" answers ranged from 305 square units to 403 square units.
(On the other hand, knowing the mishaps that can befall even simple paint jobs, perhaps the real answer to "How much paint?" is "More than you would think.")
Sometimes, getting a wrong answer to a math problem might be better than getting it right, says Warner School professor Raffaella Borasi. Her revolutionary opinions about revitalizing mathematics education are causing U.S. teachers to sit up and pay attention.
uring last year's Winter Olympics, Raffaella Borasi challenged a middle-school math class to figure out which were the fastest athletes at the Nagano competition.
vidence of Borasi's growing clout in math-education circles is perhaps best illustrated by the multiple National Science Foundation grants she has won during the last 10 years.
orn in Italy, Borasi graduated from the University of Torino in 1981. Long before that time she had already fallen in love with mathematics and has since committed herself to rousing a similar passion in others--or, at the very least, to making it more appealing to reluctant learners.
"Education is organized differently in different countries," she says. "I've always been fascinated with the teaching and learning of math as a science." But in Italy math education was "very philosophical and anecdotal," with teachers relying on instinct rather than research to evaluate and improve their teaching methods.
When Do 1/4 and 1/2 Add Up to 2/6? Maybe You Should Ask Ted Williams
ow could I educate teachers if I don't take a stand about all the constraints that don't let them do a good job?"
nder another of her NSF grants, Borasi and her team have been working with teachers in four Rochester-area school districts.
Freelancer Rose Ericson wrote the article about the Simon School's Clifford Smith in the Winter 1998-99 issue of Rochester Review.
How Much Paint to Order?
An enlarged replica (scale 1:100) of this fish is to be painted red in the middle of the floor of a new zoo aquarium.
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