Selection and Construction

Okay, you have the ideas written up on scraps of paper or index cards or sheets of newsprint or (heaven forbid) buzzing around in your head. What do you do with them?

Doing homework is a creative act: that is not accidental, for one of the most important things you learn to do in school is how to create things. But creation is something that takes time and effort.

• Because creation is so important -- civilization consists of a vast matrix of things people have created -- we teach young people how to create. That is what homework is supposed to do: give you practice creating things.

• But creativity is hard. Writing is easy, says Donald Maass, just sit down and open a vein. It isn't quite like that, for that implies that all you have to do is cut yourself open and watch the ink pour out. In fact, it requires effort and attention.

1. Things don't assemble themselves: there may be many ways to assemble the solution, some better than others.
   
2. The materials collected for assembling the solution may not be the materials you will want. Some things may be missing, and you will have to look for them; some will be unnecessary, and can become a time-consuming distraction.

Short homework problems and projects are relatively straightforward: you figure out how to do it, and then you do it. Longer problems, which require a lot of work to find a solution, or a lot of work to construct a solution of many parts, are more problematic. You may get an idea of how to solve the problem, but then ... there is a lot of detail work to do before you write it up. It may be helpful to construct an outline:

• An outline usually is a system of nested lists.
  • Start with a list of parts of the problem.
  • Each part should have its own list of sub-parts, or of routes to attack it, or similar problems, etc.
• One list-like routine popular in computer science is modular programming.
  • Start with a list of parts, and an idea for solving each part.
    • Each part is divided into a list of subparts, each with its own idea for a solution.
      • Repeatedly divide parts until the ideas for solution are actually miniature computer programs.
    • Then assemble all these parts into a vast computer program.
  • The result is a program which can be readily analyzed -- and debugged -- part by part.

• Drift. A homework problem or project has a specific goal, and if the problem or project takes time to resolve, it is easy to forget what the goal is. (This is a common joke about term papers: half way through, one asks, now what is this about?) Working off the outline, you can find where you are in the outline; if you can't, that's a sign that either you are adrift, or that the outline is inadequate and needs adjustment.

• Schizophrenia. There is a story about a donkey that starved to death because it was between two bales of hay, and it could not decide which to go to. Frequently, there are, or appear to be, several different ways to solve a problem. Many students are tempted to take both routes, either partway down both, or jumping back and forth, or doing all of one and pieces of the other, etc. Perhaps subconsciously, there is the feeling that two solutions are better than one. But as a practical matter, a single working solution is better than two non-working ones. It is sometimes wise to remember Descartes' dictum that when in the woods, to leave, it may not matter which direction you go, as long as you keep going that direction. An outline that says "this is part of solution X" and "that is part of solution Y" should help you develop a single working solution to the problem.

• Obsession. The ancient Greeks allegedly believed in moderation in all things, and certainly a stubborn resistance to schizophrenia can lead to an equally serious problem: a determination to keep going the direction one is going, even though it doesn't seem to be going anywhere. Sometimes when there are several ways to attack a problem, some approaches are better than others: solution is more readily at hand, the solution is clearer, the solution is nicer to deal with, etc. Notice that what you will really need is judgement: "avoid schizophrenia" and "avoid obsession" are both mere slogans, and countered by the slogans "keep an open mind" and "stick to it," respectively. We all know the sad story of James Maxwell, obsessed with his quaternions while vectors were just under his nose; but we also know the happy story of Albert Einstein, who devoted seven years to constructing a single relativistic framework for electromagnetism and gravity. A lot depends on what it is you want to do, and what sort of person you are.

• Incompleteness. But one problem is not controversial: a long, winding "solution" fading into a swamp is little help. Here the outline with criteria for success can be helpful: a solution should actually solve the problem, and be checkable against the criteria. So a computation should conclude with a number, vector, formula, or some other object, and that computation should follow a procedure that works in general for that kind of problem (few teachers give credit for psychic powers). A proof should be a mechanical and convincing argument.

• Ooops. There is a story of a university that asked professors to account for each day on the job: one professor reported, "Monday, tried to prove theorem; Tuesday, tried to prove theorem; Wednesday, tried to prove theorem; Thursday, tried to prove theorem; Friday, theorem is false." First, you have to be able to check a solution. There are several ways to do this: you can ask if seems reasonable; you can go back and check each step; or if is a general problem (verify a formula) you can check it with specific examples. You will find that sometimes you know you have an error but you just can't find it. This is a common problem: in the computer software industry, the software engineer who detects, locates, and corrects errors --- the debugger --- is a high status position. You have several options: you can take examples or logic and go through the faulty solution step by step, or you can start over.