Thinking About Thinking - 2017


The SAT’s Problems Are Easier than Life’s Problems

By Balaji Prasad

“It is hard to convince a high-school student
that he will encounter a lot of problems more
difficult than those of algebra and geometry."
-- E.W. Howe

I wish you a wonderful and Happy New Year filled with problems that yield gracefully to you!

Life is full of problems. My SAT students have to deal with about 155 problems during the test. Outside the test, they and the rest of us have way more than 155 problems to deal with, many of which are likely to be vexing and perplexing.

Tests like the SAT have problems similar to the day-to-day problems of life we all face. Except, the SAT problems are simpler. I think a lot about problems and about solving them, especially the ones that my students care about. It helps me demonstrate that the SAT test can be easier than some students think.

The nature of problems

One way to understand the nature of problems is to reflect on an interesting subset of problems: primal instinct problems. We would get little argument that problems that help us survive are valuable. They fill some need. When we are hungry, we seek to satisfy the hunger. That search results in foraging and finding food. The hunger is satisfied. The problem is “solved". This example suggests that a problem is, quite simply, just a gap between what there is, and what we would like something to be. The realization that there is a gap provides a stimulus to close the gap, through action. So maybe feeling that something is a problem is not so bad after all? It seems to get things done. You feel hunger, and you address the hunger. In this particular example of survival, our involuntary functions kick in, realize the existence of a gap, marshal the resources available, and “solve" an important problem.

But not everything is as simple. In fact, for some people, fixing hunger itself is a difficult problem. There are people in this world who can't get enough to eat! Some of us may not face food shortages, but we do deal with many problems that don't seem to have solutions, or at least easy solutions. It seems fair to say that there are problems that yield, and those that don't. But even this may be too simple a view.

Problems are probabilistic

The problem with most problems is that they are not tagged as being solvable or unsolvable. Also, “solvability" needs to be defined in a very personal way. You don't want to just know if a problem is solvable in general. You want to know if you can solve the problem, given your capabilities and constraints. Problems are definitely not tagged with any such indicator. So what we are left with are problems that may be solvable, but then again may not be. This is a real problem! How can we be motivated to put in the effort to solve something, if we don't know that it can be solved? If we don't put in the effort to solve a problem, that problem is, for all practical purposes, unsolvable. If we do, it may be solvable. So what we have is a set of problems, some of which are solvable, some that may be solvable, and some that are unsolvable. The last category includes those problems that are actually solvable, but seem unsolvable.

There are too many problems!

Time and resources are always limited. We don't have the time to address every problem that we can dream up. So we have to find some way to be efficient about all this. Somehow, we have to be able to focus attention on those problems that are worth solving. And, they should be ones that have a good chance of being solved with a reasonable amount of effort.

Here is a practical exercise. Just pick up a pen and a sheet of paper, and start writing out your problems – where you are relative to something of interest, and where you would like to be. One item per line. This is the start of your personal “problem solver's portfolio".

Now, go over every line item, and make a notation about the importance to you of solving the problem – whether it is a “High", “Medium" or “Low". The list now distinguishes between the problems that must be solved, the problems that should be solved, and those that are a less critical to your life.

Step 2. Go over the list once again. This time, put an assessment of the probability that a given problem can be solved, using your capabilities and a reasonable amount of resources.

It is quite possible that this exercise revealed some things that you may not have realized. You may discover that there are problems that are both important and solvable, that you are not doing anything about. You may also find that the problems that you thought were important are not that important when you look at them in the broader context of all your problems. And, you may find some problems that are just not worth going after, because solving such problems may be Pyrrhic victories – you lose more by solving them, than by letting them be.

Problems may be poorly defined

Some of the more important problems can be difficult to solve. But this may have less to do with the actual problem itself, and more do with how the problem is framed. Intuitively, we often define problems in ways that can be addressed with the resources we know we have. For example, if you have a calculator, you may think of certain ways of solving the problem, but you may not think of other ways of solving the problem that may actually be more elegant and faster. We could go back over the list we created to check if we are letting “solutions" creep into how we framed our problems. Our problems may not be quite what we thought they might be. Some may be easier than we thought. And, some of them may be actually quite different from the ones that we should actually be looking to solve.

Life is hard, but the SAT is easy

The SAT problems my students face pale in comparison to life's problems. They come neatly tagged as “solvable"; otherwise, the examiners would not have put them on the test. They also come framed in a reasonable way – you are given some things, a starting point, and a target to get to. You don't have to worry about whether the problem is important, whether it is solvable, and whether it is defined properly. That's half the battle won even before you start, unlike life's problems. The SAT is easier!