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.