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With all of the problems in the world today, I thought I'd editorialize a little bit on our collective, and yet to the greater part unexamined decision to label the mathematical exercises we perform every week for homework as "problems."
My primary confusion and question on this matter is the pejorative nature of this label. When I think to myself that I have problems to do for homework, I start to go in one of two directions psychologically. I may begin to self aggrandize myself, thinking that I am a problem solver and a true go-getter and what have you. On the other hand, I may become overwhelmed with the dreary bulk of the problemi-ness of the problems. In both cases, the word problem
lends the exercises the feeling of a task, of something to be overcome.
In the case of "problem" the dictionary gives the derivation from the Greek prefix "pro," meaning forward, and the Greek word ballien, meaning "to throw or to drive." Thus the essential nature of the word is something of a proposition, anything put forward at all. This would not exclude the left leg in the dance the "Hokey-Pokey." You put your left leg in, you take your left leg out, you put your left leg in and you shake it all about. According to the Greeks, during the period of time your leg was in the circle, it became a problem.
The dictionary has three definitions for the word in modern usage. First, it is "a question proposed for solution." Second it is "a question, matter, situation or person that is perplexing or difficult." Third, in Mathematics, it is "anything required to be done or requiring the doing of something." Thus, in terms of the mathematical definition of the word, my original sense of a task to be done or something to be overcome was not far off the mark. It is this sense of the word I like the least, and also the least representative of better mathematical exercises I've experienced.
The best Math problems I have had have fit the first two definitions of the word far better than the
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third. When made correctly, a problem is far more than something merely to be done. It is an event, or even a ritual or a rite of passage. One begins without comprehension even of what the problems asks. This understanding comes in levels, with many false ideas and red herrings along the way. The bulk of the work is in understanding the question. Once the question is fully understood, the answer is at hand. If it is not, the question is not yet fully understood.
These days we have come to be able to say things like: "I'm having a problem with problem number 5." In this case we are using the world problem in totally different ways within the same sentence. In the first sense the word is used to express frustration and impasse. In the second it is used to represent a question, a question that, if it is good, is an opportunity to extend the limits of your comprehension. It is important that we recognize the distinction between these usages for two reasons: that we may value our opportunities, and that we might value our difficulties.
I had an art teacher who was fond of telling stories where the difficulties we faced and the bruises we got on the way turned into gold for us later on. Math problems can be this way too. Things I didn't understand that were driving me crazy one day, a week later would suddenly crystallize and be clearer and more understood than anything I had glossed over thinking I had comprehended perfectly.
Math problems are essentially opportunities for growth of the mind and the imagination. Simply classifying them as tasks or as chores is a disservice to people since it negates their true nature as questions put forward to perplex and thereby inspire growth. Problems are not merely things to be done but an avenue to internal possibility and to more and more interesting layers of the unknown.
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