Imagine Magazine - Johns Hopkins - March/April 2011 - (Page 6)

in my own words Street-Fighting Mathematics Sanjoy Mahajan, PhD associate Director, MIT Teaching and Learning Laboratory Lecturer, Department of Electrical Engineering and Computer Science, MIT Visiting associate Professor, applied Science and Engineering, olin College of Engineering do math, you need it. So much math is taught like this: “Here are all these results; now verify them,” or “Here’s a trig identity; check that it’s true.” It’s tedious, and I think students are right to run away from mathematics like that. There’s no adventure, no curiosity. There’s no fun in it. Teaching physics and engineering at Cambridge University and at MIT, Sanjoy Mahajan noticed a disturbing pattern among his students: they were afraid to make educated guesses. The popular course he designed in response, Street-Fighting Mathematics, gave students a set of practical tools for estimating solutions to realworld problems. His book of the same title was published in 2010 by MIT Press; two additional books, one for scientists and one for a lay audience, are forthcoming. An antidote to rigor mortis I had seen this problem everywhere I taught: students would just refuse to make guesses. Anything that they couldn’t logically deduce a priori ahead of time, they just wouldn’t do—which meant they couldn’t do anything because you never know what to look for until you have some kind of intuition that can guide you. I’d been really frustrated by how rigid people got with math. “Too much mathematical rigor teaches rigor mortis” is how I put this in the introduction to my book. I developed the Street-Fighting Mathematics course as an antidote. Getting past the labels I noticed that students were getting trapped in course labels. They would think, “Is this an analysis problem or a topology problem?” without realizing that there are common principles to these areas. In my course, I deliberately use problems from all different fields of mathematics. I organize the course around modes of reasoning: for example, reasoning by analogy, reasoning using extreme cases, using dimensional analysis. Several of the tools in Street-Fighting Mathematics come straight from physics and engineering, such as reasoning by extreme cases. In physics, you often use extreme cases. You might say, “Well, I don’t know how this thing is going to behave in this complex situation, but let me imagine what would happen if the friction went to zero.” Other tools came from finding names for things that I found I was using a lot, such as “lumping,” which I realized is the opposite of calculus. And from doing lots of math problems over the years, I figured out the tool of reasoning by analogy and realized that it has a distinguished history (for example, in Polya’s works). So the tools are really a mix of things that I noticed in my own thinking and that I picked up from learning physics and engineering. Too good at math A time to guess There’s a difference between thinking of new results and proving them. When you don’t know what’s true, you need to be able to say, “Well, I think there’s something likely to be true in this direction.” And as you move in that direction, you begin to see if it’s the correct direction. If it’s not, then you move in another direction. Educated guessing is what gives you a sense of where you might find something useful in solving a problem. Without it, you’re paralyzed. The way math is often presented makes it seem like you don’t need educated guessing, but to actually During my undergraduate work in physics, I was in some ways ill-served by being really skilled at mathematics. Because I could always get the right answer with mathematics, I was never forced to understand the physics. But there were three months during graduate school—I was a teaching assistant for a course called Order of Magnitude Physics and studying for my graduate physics qualifying exam—when I learned more physics than I had in my entire undergrad experience. It was then that I realized that there’s a lot of room for improvement in how we teach math and physics. 6 imagine Mar/Apr 2011

Table of Contents for the Digital Edition of Imagine Magazine - Johns Hopkins - March/April 2011

Imagine Magazine - Johns Hopkins - March/April 2011
Contents
Big Picture
In My Own Words
Problem Solving
Orange County Math Circle
Number Theory
Count Me In
National Mathematics Competitions
Math at the Science Fair
Just My Speed
MathPath
When Origami Meets Rocket Science
Selected Opportunities & Resources
Making a Difference
High School Options for Gifted Students
Off the Shelf
Word Wise
Exploring Career Options
One Step Ahead
Planning Ahead for College
Students Review
Mark Your Calendar
Knossos Games

Imagine Magazine - Johns Hopkins - March/April 2011