The following is an excerpt from an article I wrote a few years ago: “Problem Solving By Analogy/ Problem Solving As Analogy,” The Mathematics Educator (2007), Vol. 17, No. 2, 2–6
Click here to download a PDF of the article
There are at least three different ways to go hiking in the woods.
- One is to be led down a previously created path, often by an expert who’s taken the path before.
- Another is to follow a path with which you are already familiar, perhaps after being led along the path several times.
- The third is to be willing to leave a familiar path to try a completely new trail when the need arises (or just for the heck of it).
Being led on the hike is efficient, if your goal is to get to the end of the trail. You can see some sights but only those you are led to and only if you’re paying close attention. It is probably the most comfortable method for novice hikers, because it isn’t necessary for them to keep track of where they are. However, this also doesn’t help them learn to get to places that are not on the trail.
Traveling alone on a path with which you are familiar is less efficient, since now you don’t have an expert to keep you on the trail. However, it’s definitely more interesting, since you can choose when and where to stop and how quickly to walk. Of course, in order get to a location you haven’t already been to, you must be willing to stray off the path and possibly blaze a new trail every now and then.
Some of these new locations might be just off the path, while others may be far away from your comfort zone, but these less-traveled sights are often the most interesting (and educational). And each new trail you forge provides you with new locations you know how to get to (and return to later).
However, many hikers aren’t natural explorers. It isn’t likely that someone who has always been led through the woods will stray far from the known path. It takes a rare person to feel confident enough to take over leading the group to the end of a trail (or even back to where they started) or to choose to lead the group entirely off the path just to explore, unless they had been given the chance to explore in the past. Unless it is your responsibility to get everyone back to the trailhead, your mind is usually focused on following the leader.
The second method (taking a path with which you are familiar but without a leader) is more work, since you need to pay closer attention to where you are and which branches of the path you could choose. You might feel limited to taking the particular path you are on, but the real fun comes when you veer from the known path and explore, knowing that if you choose the left fork and arrive at a dead end, you can always find your way back and choose the other fork.
When you’re the leader of the hike (or at least an active participant in the decision making), whenever you break off a familiar path and look for new sights to see, you must be aware of where the familiar path is (in relation to your current location) so you don’t get lost.
Learning to be comfortable with straying off the path (or becoming a trailblazer) comes from experience, but that experience need not be a solo effort. A good hike leader will point out trail markers and share his or her decisions with fellow hikers, letting them in on the thought processes being used as options are chosen and decisions are made. And novices could be encouraged to take charge under the watchful eye of the hike leader, who allows them to make the decisions. Even if the novices get lost, the leader can keep track of their location and bring them back to familiar territory (or help them solve the problem of finding their way back themselves).
As novices become more comfortable (and experienced) with making decisions and realizing that every decision is reversible, they are more willing (and able) to explore on their own. If novice hikers ever find themselves in unfamiliar territory, they will have a very difficult time finding their way out if their only hiking experiences involved having been led or traveling on familiar paths.
Therein lies the key connection with problem solving. If we are to help students learn to solve new problems that they haven’t seen before, they need experiences—guided and otherwise—that allow them to try and fail, try something else, and eventually arrive at a solution to the problem.
They aren’t alone, though, because the teacher/hike leader is there as a safety net—not to solve the problem for them, but to serve as a mirror, reflecting their strategies and progress, asking probing questions that encourage the novices to think through their options.
Experienced hikers, like expert problem solvers, are able to keep track of where they are, where they need to be, and the options available to them at any given moment. By explicitly discussing these options and decisions with novices (hikers and problem solvers), the novices gain an understanding of the process and are more likely to be able to navigate the paths themselves.
George Pòlya, in Mathematics and Plausible Reasoning, spoke of the importance of intellectual courage—being ready to revise ones beliefs. I like to expand this notion a little further to include a willingness to persevere even when you don’t know whether your strategy will lead to a solution or to a dead end. Isn’t that the goal we have for all our students?