Learning and testing
Constructing a useful test can be very difficult.
Our tutoring consultant has had a very full schedule this past couple of weeks. He puts it down mostly to the first full-scale tests in classrooms, before which some students decided they needed extra preparation and after which other students found they should have had it. The first standardized-test dates of the school year added some additional pressure.
A few students have asked questions about how best to study, in particular how to memorize the necessary elements of trigonometry or the conjugations of a Spanish verb. Alas, there are no magic ways (that we know of). Repeated practice and self-testing over time constitute the basic technique. Later on, using the Unit Circle in problem after problem and writing proper Spanish sentences will cement the knowledge until it’s second nature. At least that’s the aim. Our tutor continues by pointing out that each person learns differently, and it’s valuable to discover one’s own best methods.
But recently he’s come to the conclusion that another bit of self-knowledge is equally important, at least in the short run: how much do I know, and how well do I know it? It’s possible to be quite mistaken about this. A student who can follow an example problem in class might decide she understands the method, but still be quite unable to work one on her own. One who works a problem by following a pattern may not understand how to apply the principles elsewhere (this is common in Physics). On the other hand, our tutor remembers one student in particular who was convinced she had no grasp of trigonometry. He made a few connections between the isolated ideas she did have, and suddenly she understood the subject. (He cherishes this memory, which is admittedly a bit simplified. Rarely has he gotten such a big return for such a small effort.)
Well, how does a student find out the state of his or her understanding? Mostly (in our subjects) through problem sets and tests (in the latter we include everything from weekly quizzes to final exams). If you’ve not been a teacher you may not understand just how time-consuming and tedious constructing these can be. You have to have variety, so the students are required to analyze each anew; and volume, to give plenty of practice.
It can also be difficult to test what you want to test. In Physics, for example, the teacher wants to see whether the student understands forces, combinations and components of forces, and how they produce acceleration. For this, the venerable inclined plane is trotted out again and again. Done correctly, an inclined-plane problem does indeed demonstrate these things. But there are limits to how much variety a teacher can include in such a problem. Many students simply memorize a pattern, involving mu cosine theta and m sine theta, and it works often enough for them to get by. Our astronomer tried to get around this by including short essay questions. They proved very difficult to grade. For students at the top and bottom, their understanding or lack of it was clear. However, to interpret a paragraph as either slightly or greatly confused was often a puzzle. When one student found he could store whole paragraphs out of the textbook in his calculator, our astronomer gave up the idea.
Of course the problem is not restricted to introductory Physics, and we would be surprised if something similar couldn’t be identified in subjects other than the hard sciences. The test of a really good teacher could be the ability to construct a good test.
