Sometimes, getting a wrong answer to a math problem might be better than getting it right, says a University of Rochester professor. Her startling opinions about revitalizing mathematics education are causing leading U.S. educators to sit up and pay attention.
Raffaella Borasi, associate professor of education at the University's Margaret Warner Graduate School of Education and Human Development, addressed the National Council of Teachers of Mathematics at their annual meeting, April 25-28, in San Diego. Last year, NCTM judged Borasi's paper on changing attitudes toward errors the best one published in 1994 in the Journal for Research in Mathematics Education, the leading journal in its field. Borasi has also received five National Science Foundation grants underwriting her work during the past 10 years.
This summer, under terms of her most recent NSF grant, Borasi will lead an initiative to improve mathematics instruction in local middle schools. She will work intensively with lead teachers from the Brighton, Spencerport, Hilton, and Honeoye Falls/Lima school districts, showing them how they can improve their mathematics programs.
Borasi's new book, Reconceiving Mathematics Instruction: A Focus on Errors, (Ablex, $26.50) has just been published.
How can a new attitude about a wrong answer lead to a deeper understanding of mathematics? Borasi compares it to getting "lost" on your way to a destination. If you're in a hurry to make an appointment, you don't want to take any risks that might make you late; getting lost is a nuisance and a frustration. But if you've moved to a new neighborhood and are exploring it, getting "lost" becomes the occasion to get to know the whole area better -- to find a new route, or discover shops you might want to revisit.
In this spirit, Borasi conducted a teaching experiment in the late 1980s at School Without Walls, an alternative high school in Rochester, NY. Her students were poor at math, but she taught them to use errors as "springboards to inquiry."
Take the problem of adding fractions, for example. Many students solve the problem by adding numerators and denominators separately: They add 1/4 and 1/2 and arrive at an answer of 2/6, rather than the correct answer, 3/4.
Borasi suggests exploring with students whether there are some instances in which it makes sense to add in this way. Baseball batting averages are one instance. A player who gets 3 hits out of 4 times at bat in one game and 6 hits out of 7 times at bat in another, has an average of 9 out of 11 -- not 45/28, which is the result of adding the fractions 3/4 and 6/7.
By end of the term in which Borasi's students explored the significance of errors, they were performing much better and were enjoying learning math.
The problem with traditional styles of instruction
Generations of students have come to regard math with fear and loathing because of the rigid, authoritarian way it was taught, says Borasi. Too many of us learned there was only one right answer to the question, what does 2+2 equal? We also learned that if your answer wasn't "4," the teacher marked it wrong. The laws of mathematics governed grades, too: As your wrong answers increased, the value of your grade decreased, in inverse proportion.
But mathematicians themselves, she says, understand that making errors and learning from them are essential to growth in understanding. She points out that important discoveries in mathematics -- such as the development of a non-Euclidean geometry and the early development of calculus -- had their beginnings in the failure of a mathematician to get an answer to a problem that seemed correct.
The real problem with so much emphasis on right answers and applying formulas correctly, Borasi says, is that it gets in the way of students learning something more important. It is better that students learn to pose and solve math-related problems, to appreciate the value and potential applications of mathematics, and to reason and communicate mathematically, she says.
Current thinking in the field backs up Borasi. For the past two decades, leading mathematical educators have agreed that traditional attitudes and instructional techniques are deeply flawed. Reformers have called for new classroom environments that
Show many math questions can have MORE than one right answer. Help students articulate mathematical ideas. Provide experiences that teach students careful reasoning and disciplined understanding. Encourage exploration.
Borasi concedes that her thinking represents a radical departure from normal methods of instruction, and that it may be hard for some teachers to show the flexibility and willingness to take risks that her views imply. But adopting a new attitude about errors may be the most direct path teachers can take to helping their students enjoying math and gain a deeper appreciation of mathematical reasoning.