As an educator, I have strong concerns when teachers fail to provide clear and simple explanations to students struggling with concepts in math and science. The rare, gifted student will always be able to connect the dots, but less talented students are often shortchanged. The concern is not just that the student may fail to understand a particular problem that is poorly explained, but that a chronic pattern of weak, inadequate, or inappropriate explanations may have the effect of turning students off to math and science, a very big problem for the United States.
I see poor math and science explanations all the time, not just in the classroom from inexperienced or disinterested teachers, but in places you would least expect it. An example came from an article I was reading in Scientific American, a popular and highly regarded science magazine. The author was discussing the Monty Hall problem, in which a game show contestant is shown three doors. Two doors conceal donkeys, while one door hides a brand new car. The writer went on to explain that the contestant picks door number one, but before that door is opened, door number two is opened to reveal a donkey. Then, the contestant is asked if he would like to switch to door number three. To most of us, it seems that there would be a 50-50 chance of getting the car with either of the two doors that remain closed, so a switch would not improve ones chances of winning. It turns out, against all intuition, that your chances improve if you switch. Originally, with three doors to choose from, there was a probability of 1/3 that door number one concealed the new car. By switching to door number three after the second door is opened (revealing a donkey), you will have a 2/3 chance of getting the car, twice as favorable as when the game began. How is this possible?
At this point, most students are intrigued, but it is the explanation that will either turn them on to the subject of probability or turn them off, perhaps for good. The author begins his explanation by saying that there are three possible configurations of doors:
1. donkey, donkey, new car
2. donkey, new car, donkey
3. new car, donkey, donkey
Blundering forward, he says that in the first two cases, switching takes you to a favorable outcome, whereas switching in the third case leads to an unfavorable outcome, therefore switching provides a 2/3 chance of winning.
Based on this explanation, most students look at the second configuration and wonder how switching to door number three results in a favorable outcome, as there is a donkey behind door number three. Moreover, most students will say that the second configuration is impossible because a donkey was shown to be behind the second door, so the configuration "donkey, new car, donkey" is illegitimate. Between the legitimate first and third configurations, one is good for switching and one is bad for switching, therefore it should not matter whether the contestant switches or not.
The student is not to blame for the confusion here. Instead, the failure lies squarely on the author who did not explain things clearly enough. In fact, all the author needed to say was that for the second configuration,the contestant would have been shown the donkey behind door number three, leaving the contestant the choice of switching to door number two, and thus to a favorable outcome. By omitting this one simple statement, the problem is virtually impossible to understand, with the result that student interest vanishes, and light turns to dark.
We need to think more carefully about how we explain the intricacies of math and science to students, who incidentally make up a significant proportion of readers of Scientific American. Doing it well opens the minds of young people to the beauty of these subjects, but doing it poorly destroys their interest. Like it or not, the future of the United States is tied to the intimate exchange of ideas between teacher and student taking place every day in classrooms across the nation. If students dislike math and science, we will not have an adequate supply of mathematicians and scientists, which in turn leads to the inability of our country to maintain a competitive position in the world. It also leads to a society that stops trying to unfold the secrets of the universe, which I have always regarded as the best part of being human.