A Common Mistake in Teaching Math and Science

The following is a typical approach to teaching math and science and then testing the students on what they were taught. Textbooks which have questions at the end of a unit or chapter will sometimes also make the same error, which tends to show up then in students' homework grades. I will give two examples; one from math, and one from science. (It has turned out, as you will see below, that the science example I chose is even much more difficult than I had thought when I chose it for being unfairly difficult in the first place.)

Students are taught that distance equals rate times the time, the formula being d = rt. Some examples are given to show that the longer you drive at some rate or the faster you drive for some time, the further you will go. Students are also shown that if you know the distance and the rate, you can use the formula to find the time; and vice versa, if you know the distance and the time, you can find the average rate. So then some problems are worked that are pretty straightforward, where the time and rate are given and the student needs to compute the distance, or where the distance and either the time or rate is given and students need to compute the other.

Then some examples or some problems might be given where the rate varies during a trip, and the student needs to compute the average rate. Typically, the problem is something like: John drives for one hour at 60 mph and then drives for an hour at 30 mph. What is his average rate of speed for the total distance? The answer is essentially that since he drives one hour at 60 mph, he goes 60 miles in that time, and since he drives the next hour at 30 mph, he goes 30 miles in that time. Therefore he has driven 90 miles in two hours, so he averaged 45 mph, which is the average of his two speeds. One can work it out using the formulas because you need to find the total distance divided by the total time, so you first figure out the distance he drives at 60 and the distance he drives at 30, add them together to get the total distance, and then divide that by the total time, 2 hours, to get the average rate he drove. Some students may have some difficulty with that when it is first presented, but after it is explained and some more problems of that sort are given, many or most students will be able to do them. So they practice on things like: a plane flies 200 mph for an hour with the wind, and then it flies 100 mph against the wind for an hour. What is its average speed. The answer will be 150 mph.

So far so good. But then on the test or in the chapter questions in a textbook, the following problem might be given: John drives 100 miles at 60 miles an hour but then comes to a long patch of highway construction and drives the next 100 miles at 30 mph. What is his average rate of speed. Students then go through the same process they have "learned" and they figure the answer to be 45 mph because he has gone the same distance at the two different rates, so the average rate for the whole distance must be the average of the two rates, just as it has been in all the problems they have worked in the book and in class. Or there might be a variation to the problem, such as: John needs to average 60 mph over a two mile course in order to qualify for a race. After the first mile, he develops some sort of engine problem that allowed him only to average 30 mph for the second mile. How fast would he have had to drive the first mile in order to qualify at the average of 60 mph? The answer seems fairly obviously to be 90 mph.

Both those answers are wrong, however. I will explain that in a minute, but the typical rationale for the book's or the teacher's changing the form of the problem is to see whether, or how many, students really understand the formula d = rt and understand the concept of average rates, etc. The view of the teacher or the text is that they have only added a slightly different "wrinkle" in order to see how the students adapt what they have learned to take it into account and solve the problem. Normally, few students, if any can adapt.

There will sometimes be some students who will get the answers right, but most will get them wrong and be very surprised and confused. Moreover, many of those students will then become discouraged, figuring they just don't understand math and are no good at it. That is a bad deduction on their part. What is really true is that they were taught incorrectly and they were the victim of a trick that the teacher does not realize is an unfair trick. The teacher and textbook writer will know the question is a trick question, but they will think it is fair because they mistakenly think they are only testing for what ought to be understood. I will come back to this. But first the science example................