|Ivars Peterson's MathTrek|
September 22, 1997
That's the sort of brainteaser that greeted my students as they entered the classroom back in the days many years ago when I was a high school math teacher. Such puzzles tended to bring the students to the room for the start of class somewhat more promptly than normal and generally got them thinking mathematically as soon as they entered.
At the school where I taught, students were placed in math classes according to their level of achievement and ability, and I often had students who were on the lower end of the math scale. What always struck me was the amazing ingenuity and perseverance the students displayed in solving many of the puzzles I posed, even as they struggled with the demands of a fairly standard curriculum in algebra or beginning calculus.
Math teachers who love their subject, understand it well, and care about conveying the sheer joy of mathematics have many different ways of communicating their enthusiasm. They create memorable experiences that are long remembered by their students, even as they build skills and encourage clear thinking. Different math teachers may use different styles and methods, and they may favor different topics. However, they all share the goal of building an appreciation of both the usefulness and the intellectual wonder of mathematics.
Much of the current debate about the quality and direction of math education has focused on the usual suspects: curriculum, standards, and technology in the classroom. Largely missing from many of these discussions has been any serious consideration of the crucial role played by the men and women charged with using such guidelines and tools in the classroom.
The current generation of math teachers includes a large number who, as students, survived the so-called new math of the 1960s. Some may even have thrived on it, inspired by a few dedicated math teachers in their past. Unfortunately, many of their colleagues who teach other subjects and many of their neighbors and friends recall only the bad times in math class. Math teachers work in an environment in which many people routinely dismiss math as largely irrelevant to anything they do, take pride in being innumerate, or display a palpable fear of mathematics.
One unfortunate by-product is the polarization of school communities into the math types and the English types. Some students have told me that they ended up majoring in math and science because they didn't like writing or had trouble in English class. Yet, as future scientists, engineers, and mathematicians, they will undoubtedly face writing reports and grant proposals, preparing evaluations, and performing other duties that require effective communication. Similarly, students who shy away from math will face difficulties in understanding and appreciating a significant fraction of what happens around them.
Much of what students remember from the mathematics classroom has to do not with textbooks or computers but with people -- the encouragement and interest of teachers, parents, and peers. We need a greater focus on helping math teachers do their jobs more effectively, and there are many ways in which that can be accomplished. One way is to encourage teachers to express their own mathematical interests and enthusiasms in the context of a given curriculum.
Several years ago, I attended a show-and-tell session sponsored by the National Science Foundation, at which various groups displayed innovative uses of technology in teaching a variety of topics. In many instances, participants proudly exhibited videos of their systems in use: Typically, they would should show an enthusiastic instructor (with computer) interacting with a group of eager, engaged students. The subjects were wildly different, from fractals and random walks to music from around the world. The student response was uniformly encouraging.
As I observed these demonstrations, it occurred to me that the specific topics didn't really matter and the common factor wasn't the technology. What made these experiences rewarding for the students was the presence of the instructor -- someone who cared about the subject and used whatever tools were at hand to illuminate that passion.
With the right teacher, it's possible to build an entire curriculum, from English and history to math and science, around a theme such as bridge building, folk music, or sand. The students will learn something of value. However, no matter how well developed and thoroughly tested a given program is, what works for one teacher will not necessarily work for another. The teacher, not the topic, makes the difference.
In looking to the future of math education, it's important to examine what today's students are learning and how they are learning it. It is from this group that the math teachers of tomorrow will come. We will need the dedication of focused, enthusiastic, engaged, well-informed math teachers. Whether enough of them will emerge from a generation that has become comfortable with the flashy snippets of Sesame Street, MTV, and the World Wide Web remains to be seen.
Answer to problem: A domino covers exactly one black square and one white square. Modifying the checkerboard as specified removes two black squares or two white squares. Hence, there is no way to place the dominoes to cover the entire board.
Copyright © 1997 by Ivars Peterson.
Cheney, Lynne. 1997. Once again, basic skills fall prey to a fad. New York Times (Aug. 11).
Hubler, Shawn. 1997. Teachers who make magic. Los Angeles Times (Sept. 12).
Lewis, Ricki. 1997. Study highlights need for more scientists in classroom. The Scientist 11(Sept. 1):1.
Mackenzie, Dana. 1997. Project NExT helps new Ph.D.s in the classroom -- and beyond. Science 277(Aug. 22):1031-1032.
Mitchell, Russ. 1997. Building a learning society. Wired (October):136.
Perelman, Y. I. 1984. Fun with Maths and Physics: Brain Teasers, Tricks, Illusions. Moscow: Mir Publishers.
Ratnesar, Romesh. 1997. This Is Math? Time (Aug. 25):66-67.
Romberg, Thomas. 1997. Mediocre is not good enough. New York Times (Aug. 11).