Andy Liu—University of Alberta
There are two stages in solving a problem. The first stage is to convince yourself that you have solved the problem. The second stage is to convince other people that you have solved the problem. The first stage is the creative one and is indicative of how mathematics is actually done. The second stage is more formal and often has little in common with the first stage, but ironically it is how mathematics is usually communicated and taught.
Let us illustrate with a simple geometry problem.
P is any point inside an equilateral triangle ABC. Perpendiculars are dropped from P to BC at D, CA at E, and AB at F. Which has the greater total area: triangles PAF, PBD, and PCE, or triangles PAE, PBF, and PCD?
The symmetry of this problem compels us to jump in with both feet and say, “They are the same!” However, the proposer of the problem may be having fun with us, so let’s test our hypothesis with some special positions for P. Putting P at the center of ABC, then at the midpoint of BC, and then coincident with A, we see that in each case our hypothesis holds true. (See figure 1.) So we are confident that our conclusion is correct.
A good way to solve problems is to make use of special cases. The first two attempts at incorporating the special positions into the general diagram are not particularly fruitful. The shaded regions do not correspond exactly. (See figure 2.)
The third special position—the simplest one—delivers! The shaded regions correspond exactly. Moreover, instead of having P coincide with A, we can have it coincide with B and C as well, yielding the diagram in figure 3.
Since PUV, PWX, and PYZ are all equilateral triangles, PZ and PW are both parallel to BC, so that P lies on ZW. Similarly, P lies on UX and VY. Moreover, PXAY, PZBU, and PVCW are parallelograms, and a diagonal of a parallelogram bisects its area. Everything falls into place!
We are now at the end of the first stage—which is where the trouble begins. If you read the solution to this problem in some problem book, you are likely to get the following:
Construct lines ZW, UX, and VY through P, parallel to BC, CA, and AB respectively. Then PUV, PWX, and PYZ are equilateral triangles while PXAY, PZBU, and PVCW are parallelograms. Denoting the area of a polygon Q by [Q], we have
This is very neat and very impressive. The key step is in the first line—the construction of ZW, UX, and VY. But notice that in our work on the first stage, this construction emerges as practically the final step. The two stages run in essentially opposite directions!
Emphasizing the second stage over the first has significantly negative consequences. Performing an excess of exercises may lead students to look at a problem and say, “This one I can do,” and start their write-up before thinking through their own reasoning. Alternatively, they may say, “I have never seen anything like this before,” and move on for lack of any ideas.
The most difficult thing about solving a problem is getting started. Where do ideas come from? There are many techniques, but no sure-fire method because problem solving is not a science. It is an art. Progress comes only with regular practice and sustained effort on the first stage of the process. Unfortunately, only reading solutions by other people is ineffective because other people tend to write only about the second stage of problem solving.
Perhaps this is to be expected. In the world of research mathematics, only the second stage is required to get a paper published. The paradoxical result of this is that we disseminate new theorems of mathematics in a fashion that offers little insight into how they were created.
Bringing this back to the world of the classroom, the upshot is that more attention needs to be paid to the first stage of problem solving. Perfecting the expository skills is certainly important; however, there will not be anything to write about if students cannot conduct informal explorations. They must learn to make something out of nothing, which is the essence of research.
About the author: Andy Liu teaches at the University of Alberta, from which he obtained a Ph.D. in mathematics and a graduate diploma in elementary education, thus becoming perhaps the only person officially qualified to teach mathematics from kindergarten to graduate school. Email: firstname.lastname@example.org
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