Common sense in mathematics

Too often students do not think when they look at math problems.

I proctored a test for a colleague recently.  It was a Calculus II class and the second part of the first problem involved finding the sum of an infinite series. There was a way to do the problem incorrectly that gave an answer of 1. The correct answer was 3. Several students gave the answer of 1, which would have been fine, because it was not a serious error except for one thing: the series had all positive terms and the series was

2/2 +3/4 + 4/8 + 5/16 + 6/32+ . . .

The first three terms add up to 2.25. Clearly the sum of the whole series was more than 2. I don’t object to the error, but once they made the error, they should have noticed it does not make sense.

Perhaps that is too subtle for some people, but consider the following problem. $5000 is invested at 4% interest, compounded annually. How much money is there at the end of 7 years? The correct answer is $6579.67.  A minor error in the calculations can give 1.03*10^26 or approximately $1030000000000000000000000000. I don’t object to the student making the mistake, I just object to them not recognizing that there is a mistake.

A few years ago a student did a Newton’s Law of cooling problem. Fudge is cooked to 234 degrees and then allowed to cool. Ten minutes after it is removed from the heat, its temperature is 200 degrees. What is its temperature after an additional ten minutes?

One student made two mistakes in his work and gave an answer of 4000 degrees. He then wrote, “The fudge spontaneously combusted.” He passed the common sense test.

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