Catching Mistakes:

The goal, when doing mathematics, is not to make any mistakes. And mistakes can be reduced to a minimum by adopting and following good work habits for doing mathematics. Tips on what constitutes good work habits were given earlier in this part (see “Study Tips” in the section entitled “How to Study Technical Calculus”). You may wish to go back and review them once again. Mistakes can also be reduced by being aware of where they are most likely to occur in our work and , then, to be most attentive in these areas to see that they don’t occur. We have tried to call to your attention areas in which many students frequently make mistakes so that you will alertly avoid them.

Analyze all mistakes that you make, whether they occur on an exam or as part of your own informal work. You may notice a pattern to your mistakes; these patterns may suggest other areas to heighten your attentiveness when doing mathematics.

Although we may reduce mistakes from occurring in our work, it is not realistic to expect that we can eliminate mistakes completely from our work. We can’t keep them from occurring, but we can be observant of techniques and signposts which help us to catch them before they become a permanent part of our work.

One tip-off that a mistake may be present in your work is when the mathematics turns really ugly. The computations suddenly become more burdensome, the numbers become irrational numbers, not easily represented by radicals only by truncated decimals, or the algebra and its expressions suddenly become overly long with no hope of getting rid of terms by cancellation and other simplifications. When this occurs, and you will know when it does, look for a mistake in the line or two before the sudden turn for worse. If you find one, great. Correct it and proceed from that point. If you don’t find one, well, return to the ugly stuff with added resolve and intensity to fight through it to the solution of the problem.

Another technique for finding possible errors in your work is to get in the habit of asking, at the conclusion of any problem, the question “Is my answer reasonable?” This strategy is particularly useful when you are solving word problems. Judge the answer obtained in your own intuitive understanding of what the outcome ought to be. Does the answer make sense in the setting of the problem? If the problem has units like feet or ft/sec, are the units consistent with the answer sought. Examine the arithmetic of units, i.e., how they add, subtract, multiply, and divide, how they factor and combine, in order to gain insights about the correctness of your work.