by Rudd Crawford
One reason we study mathematics is to become fluent at thinking mathematically, that is, to harness the power of mathematics to achieve some end. Solving problems — many problems — is the best means for acquiring mathematical fluency. Problem solving is to learning mathematics content as essay writing is to learning grammar. Knowing mathematical formulas is like understanding good sentence structure; they are both powerful tools that should not be thought of as ends in themselves.
In the Stella Project we differentiate between problems and exercises. Exercises provide practice at skills like solving equations, solving triangles using the Pythagorean Theorem or the Law of Cosines, and so forth. Exercises are straightforward and algorithmic. The word problems found in textbooks tend to be straightforward — direct applications of recently learned skills to ostensibly real-world situations.
We call the problems in the Stella Project non-routine problems. These problems are presented with no hints as to what methods or skills will be needed to solve them. Some may seem to have insufficient given information or even contradictory information. They are not straightforward, and it is not always clear that they even have a solution. Thus they are intended to leave the solver not knowing, at first anyway, what to do. Note that the problems society faces — providing a trustworthy voting process, feeding the world, getting out of Afghanistan, or cleaning up oil spills — reflect exactly this sort of quandary, problems where we do not know, at the outset, how to go about solving them, or whether they can be solved at all.
Non-routine problems can lead us into heuristic thinking, or pondering what to do when we don’t know what to do. In the world outside of mathematics class there are numerous ways to proceed — often hotly debated. Within the mathematics class, there are also numerous ways to proceed, and we call these heuristics things to try. Examples of heuristics are “guess, and use the feedback from your probably wrong answer to guide you to a better one,” “draw a good picture,” “break the problem up into bite-sized chunks,” and “write a good equation.” The heuristics we recommend — 25 of them — are borrowed from Problem Solving — A Basic Mathematics Goal by Steven P. Meiring, published in 1980 by the Ohio Department of Education. The Introductory Problems lend themselves to a variety of these heuristics and thus can be used to introduce the heuristics and display their power.
In beginning work on a non-routine problem, the student must enter the space of the problem, which is a difficult thing to do at first. Imagine a time line with three intervals on it. The first interval is the time during which a problem is presented and the solver becomes clear about what is being asked for. Some people call this “booting up the problem in your head”; Polya called it “understanding the problem.” The third interval is the time after the problem is solved, during which the student may enjoy a feeling of success, reflect on the solution and how it was obtained, and perhaps even discover a more elegant solution. The second, middle interval, is the woodshed — the one in which the student actually grapples with the problem, seeking and usually finding a solution. It is typical that students want both the initial and middle intervals to be very, very short, or even of no length at all (“just tell me the answer”).
There are several reasons for students’ resistance to problem solving. One is that some hard work may well be involved — how much is unknown. Students these days are typically overbooked, overscheduled, and caught up in the spell cast by e-mails, texting, Facebook, video games, and all of the other engrossing ways of spending time. Who wants to sit and stare at a problem, waiting for an idea to hit? Another inhibitor is that we simply do not like to be in situations where we feel frustrated and incompetent. And related to this discomfort, for some students, is the fact that hard thinking evokes other problems with considerably more emotional weight: “Why didn’t Dad come home last night?” “What if Mom loses her job?” “What if I’m pregnant?” It can be difficult in such circumstances to entice students to engage in problem solving, thinking in ways not previously experienced, for an unknown length of time, and with no certainty of success.
But education is all about training minds to be imaginative, savvy, persistent, and resourceful — exactly the characteristics our public leaders say our students lack. Trudging through mathematics textbooks, year after year, is not in itself going to help students become the skilled mathematicians, scientists, technicians, or even literate citizens our global economy requires. It is the opportunity to grapple with and solve non-routine problems, problems that are not necessarily clearly defined, that provides students, our prospective adults, with the intellectual robustness that our country needs in its citizens.
Stella’s hope is that each problem in her collection of Stella’s Stunners is intriguing enough that students will want to know the answer and be willing to engage in the hard thinking necessary to find a solution. We hope students will come to feel comfortable in not knowing what to do, but confident that the tools of heuristic thinking will help them master the problem they face, or at least have a jolly good run at it. We hope that over time students will come to see mathematical problem solving as a sport that is just as satisfying to their minds as physical sports are to their bodies. Our country needs such students.