People don’t like losing points for errors in their solution of a problem. So, what can you do to insure you won’t lose points again the next time you are given a similar problem? Most folks seem to look through the corrections, then perhaps ask the teacher to solve the problem for the entire class, and perhaps even make a note to study that type of problem before the next test or quiz. Few do more than that… and the next time they are given a similar problem the probability they will lose points unfortunately remains greater than zero.
The solution I advocate to this dilemma involves a combination of repetition and self-awareness.
Every problem you lose points on, whether on homework, a quiz, or a test, should be copied neatly onto a new piece of paper, and put aside for a day or two before being solved again. When you sit down to solve it again, pay careful attention to how you feeling while you are working, to the pace of your work, and to your train of thought. Are you feeling frustrated or hesitating anywhere during the problem? Are you uncertain about which step to take next at any point? Do you have any doubt that your answer is correct before you compare it to the known correct answer? If you answered “yes” to any of these questions, then copy the problem onto a fresh piece of paper again, and put it aside for a day or two… then repeat.
When you can solve the problem with confidence and efficiency, then you have probably learned what you needed to know and you are now ready to try solving a similar problem… one you have not solved previously, as a test of your newly improved skills.
Why does this process work? By repeatedly solving the same problem, you can focus on learning only the things that were bugging you about the problem. If you were to use a new problem for practice, it could be different in one or more ways that would distract you from focusing on just the parts of the problem you are having trouble with. By waiting a few days between solution attempts, you give your mind a chance to ponder the parts of the problem that were bugging you, to transfer things from short term memory to long term memory, and to establish connections between related ideas.
Points are often lost on topics that were covered long ago (distributive property, fractions, notation, setting up simultaneous equations, etc.), not necessarily on topics that were introduced recently. Yet many people seem to assume that it is the newer topics that are giving them difficulty. Solving the same problem repeatedly until you fully understand how to work all the parts of the problem that were giving you trouble allows you to focus on “sticking points”, whether they are of recent or ancient vintage, without introducing new issues.
If you were to skip re-solving the problem you lost points on, and instead do a bunch of problems “just like it”, there is a chance they could all lack the item(s) that gave you trouble on the original problem. You would also find that after you figured out the process for solving the first problem, you tend to go into “auto-pilot” mode for the rest of the problems because you know you can use the same solution approach to all of them. This “wastes” perfectly good practice problems. It is more useful to work on a variety of problem types in a sitting, with no two quite the same, since figuring out what approach you wish to take when solving a problem is half the battle.
So, repeatedly re-solve the problem you lost points on, with some time between each re-solving, until you have mastered it. Then take a break for a day or two before trying a couple of problems of differing types (including one of the type you lost points on) to verify that your new-found mastery is sticking with you.
Update 2/9/10: You may also be interested in George Woodbury’s posting on roughly the same topic.