What Students Gain from Problem Solving
by Rudd Crawford and Steve Meiring
The point of studying mathematics is to become fluent at thinking mathematically, i.e., to acquire the ability to use the power of mathematics in personally meaningful ways to achieve some end. Solving problems – solving many of them – is the means for acquiring this know-how. We can say that problem solving is to learning mathematics content as essay writing is to learning grammar. Knowing mathematical formulas, say, is at the same level as understanding good sentence structure; they are both powerful tools, that shouldn't be thought of as ends in themselves.
We differentiate between problems and exercises. Exercises are simply practice at skills we wish to automize: solving equations, solving triangles by using the Pythagorean theorem or the law of cosines, so forth. Exercises are straightforward. Problems – "word problems" – as found in textbooks, also tend to be direct, straightforward, and sanitized applications of recently-learned skills to ostensibly real-world situations. Also we learn how to apply schemas for classes of problems often encountered – like time-rate-distance, percent, or mixture problems.
The problems in the Stella library, unlike typical textbook word problems, should best be defined as non-routine. These are the problems that are presented with no hints as to what methods or skills will be needed to solve them. They may seem to have insufficient or even contradictory given information. They are not straightforward, and it's often not clear that they have a solution at all. Thus, they are intended to move the solver into a space where, at first anyway, he doesn't know what to do. Note that the problems that society actually faces – providing a trustworthy voting process, feeding the world, getting out of Afghanistan, creating an environmentally sustainable civilization – are exactly of this sort – problems where we don't know, at the outset, how to go about solving them.
Problems like these kick us into a space that encourages heuristic thinking – that is, thinking about thinking. That is, pondering "what do we do when we don't know what to do?" The 50-cent word is meta-cognition. In the world outside of the mathematics class there are numerous ways to proceed – often hotly debated. Within the mathematics class, which is our concern here, there are also numerous ways to proceed, and we call these heuristics: actions to take that might help. 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", "make an easier version and solve that first”. The heuristics we recommend – twenty of them plus five extras – are listed and discussed elsewhere in this guide. The Introductory Problems which can be used to introduce the heuristics, giving the student (and you!) a chance to experience their power.
In beginning to work on a non-routine problem, the student must enter the space of the problem itself, and this is a difficult thing for her to do alone at first. If you think of solving standard textbook exercises as analogous to riding a bicycle with training wheels, non-routine problems are the bicycle with the training wheels removed. The student can veer left or right, go around in circles, speed up or slow down, or simply get off the bike and walk away.
Imagine a timeline, with three intervals on it. The first interval is the time in which a student meets the problem and begins to boot it up—to let the problem enter his space. (Some people say that when you read a book, the book is actually reading you. But we digress.). The third interval is the time after the problem is solved. The second interval, in the middle, is the one in which the serious work is done on understanding the problem, and grappling with it, seeking and hopefully finding a solution. It is typical that students want (or expect) this middle interval to be very short, or even of no length at all. "Just tell me the answer." "Can't I just google it?" (N. B., we do not consider googling to be a heuristic and we firmly discourage students from using it.)
There are several reasons for students' resistance at entering that middle zone. One is that it is clearly going to involve some hard work—maybe a lot. Students these days are typically overbooked, over-scheduled, and are caught up in the spell cast by emails, texting, social media, video games, and all the other engrossing ways of spending time. Who has time to sit and stare at a problem, waiting for an idea to hit? Another is that we simply don't like to be in situations where we feel frustrated and incompetent. And, alas, related to this discomfort for many students, is that being in the space, where hard thinking is required, lets other problems come crowding in, problems with considerable emotional weight – "Why didn't my dad come home last night?" "What if my mom loses her job?" "What if I'm pregnant?" For all of these reasons, and probably more that you can suggest, it can be difficult to encourage/entice the student to enter this middle space, where he will have to use his brain in ways he hasn’t yet explored, for an unknown length of time.
But this middle space is what education is all about. This middle space is where minds are trained to be imaginative, savvy, buoyant in the face of difficult situations, and resourceful – exactly the characteristics our leaders say that our students lack. We believe that traveling through mathematics textbooks, year after year, doing the routine problems contained therein, is not in itself going to help students become the skilled and savvy mathematicians, scientists, technicians, and craftsmen that we know our global economy requires. It is the ability to grapple with and solve non-routine problems, problems that aren't necessarily even clearly defined, that is what is needed.
Required is for students to have the opportunity to explore problem solving, which we believe is best done by having students wrestle optimistically with many, many non-routine problems and thereby to become fluent at using the powerful tools (the heuristics) which bring their personal (mastered) skills into play. The rewards are, first, the empowerment of students to accomplish more than they would ever have considered possible. Second is learning how to learn – how to focus one’s mind over an extended period, to work independently, to wrestle with technical material, and to write up technical presentations. These are the kind of skills useful in any context within an adaptive society in which workers can expect to have to be retrained numerous times.
This is, simply, thinking mathematically.
The problems in this library of Stella's Stunners are intended to be intriguing, so that a student will want to know the answer. We hope that they are intriguing enough that they will help entice the student, with the help of you, the teacher, into that middle space where the hard speculating, guessing, thinking, and checking can take place. We hope that students will come to feel comfortable in that space, comfortable in not knowing what to do, and confident that the tools available in that space, namely the tools of heuristic thinking, will help them master the problem, or at least empower a jolly good bash at it.
We hope that over time students will even come to enjoy living in that middle space, that they will see mathematical problem solving as a sport, a sport that is just as satisfying to their minds as physical sports are to their bodies.
It's a goal worth shooting for!