Learning Mathematics in College

Writing It Up

These pages are composed and maintained by Greg McColm, Department of Mathematics, University of South Florida.

Okay, you've worked out the homework exercise. What do you turn in? Many students work out plausible sounding solutions to problems on a piece of paper, and turn in that same piece of paper, rather like turning in a first draft of an essay. Students often get into the habit of doing homework this way when they take classes in which homework is not graded. But a moment's reflection suggests what is wrong with this approach.

• If tests are performances, then homework is rehearsal. While there are many things that one does during rehearsal that one does not do during performances (few recitals have people playing scales), the idea is to make onesself ready for the performance, which includes attempting to perform at performance level.

• In other courses, homework consists of answering essay questions (describe what will happen in a forest if all the foxes disappeared, and explain your reasoning). What you turn in is written in proper English, clearly written or printed, with diagrams if appropriate. Why should mathematics be any different?

• Mathematics homework tends to be graded by a upperclassman or graduate student who is being paid for three hours work for grading ninety or so papers (assuming thirty students turn in three assignments), or one every two minutes. This means that there is only time to grade perhaps two exercises per assignment. It also means that the grader has to go through each paper, rapidly find the problem to be graded, work out how correct the solution is, mark it, and go on.

Let's start with the materials. Buy a notebook just for doing homework, and work out the solutions in the notebook. Once you have worked out the solutions in the homework, and checked them over, get some clean sheets of paper and copy the homework solutions onto the clean paper.

• Modern word processors are capable of producing homework solutions, but if your handwriting is good enough -- and you take enough care to be readable and intelligible, you can do it by hand. You are learning to communicate by writing, so expect some of your grade to depend on intelligibility (in general, it is a fact of life that a happy grader is a lenient grader).
• Show all work. This means showing all the steps necessary to reach the conclusion if the problem is a computation, or all the steps necessary to convince a reader if the problem is a proof. There is a tendency to write almost all the work carefully except for a few steps that appear both obvious and not clear how to write out: that feeling of something being both obvious and hard to explain is a warning sign that you need to look at it more carefully.
• Give yourself lots of space, both for writing out computations and for pictures. (Most people find pictures useful, and bigger pictures are easier to see and work out than little ones.) Many students try to avoid wasting paper by writing in small letters, drawing small diagrams, and using two columns a page, and both sides of each piece of paper. All these are bad ideas: you should write in clear, legible writing, giving yourself space to write, and drawing large, easily read diagrams. And two columns per page, and using backs of pages, makes problems harder to find --- and easier to miss.

Two additional points.

• Scratch paper is very hard to grade, and it usually does not earn much partial credit. Working out the solution on scratch paper, or in your homework notebook (see above), but all that work is like scaffolding: just as scaffolding is necessary for building a building, but is unwanted after the building is built, so scratch work can just be in the way. Turn in exactly what is needed to show how the answer was obtained, or how the proof goes, but don't show your preliminary work. A solution should be as polished as possible because...
• Remember how rapidly a grader must grade: it is easier to grade a problem when the solution is in a nice format, with the steps clearly shown and the answer highlighted at the end. When your answer is set out this way, it is easier for the grader to find out exactly what you did. Remember: partial credit usually goes for that part of the problem that you did correctly, so if the grader can see what you did correctly, you should get more partial credit (in general, if a grader doesn't understand what is written, the grader assigns a very low grade).