By Marion Deutsche Cohen
As extra-credit at the end of most tests, I include a "preview problem," meaning a problem about the next topic to be covered. For example, for the latest Elementary Statistics quiz, which covered hypothesis testing about two means, I included a problem about two proportions. (I also included the TI83+ keystrokes).
This practice has worked wonders for me, but it¹s probably not really a time-saver in the same sense that most of the articles for this column provide. Rather than saving preparation or grading time for the teacher, it saves time with respect to keeping up with, or getting ahead of, the syllabus.
I give an in-class quiz every Friday, as well as a test at least every four weeks, so the idea of teaching through the quizzes and tests can go a long way. In actuality, I usually do finish the syllabus at least a week ahead of time, and am often (to my knowledge) farther along than many of my colleagues teaching other sections of the same course. (This semester, this happened despite the fact that I had to miss two classes, and also lectured on clarifying sub-topics not included in the syllabus.)
Some might feel that, as far as the teacher's time is concerned, this idea is more of a time sink than a "swim." After all, one does have to make up and grade the problem. However (and this might be because I'm a poet and writer), I find that such "extra credit preview problems" simply occur to me; I don't need to put time into making them up. And grading doesn¹t seem to me to be a "time sink," since it¹s only one problem (sometimes two), and also since those students who choose to do the problem usually get it right. (As we know, it's a lot easier to grade when the student gets it right.)
Most students in my classes do, or partially do, or try to do, the extra credit problems, so what I'm saying applies to many. This kind of extra-credit can entail relaxed "quiet time" with the new material, since they've already finished the test. (The lecture format doesn't provide quite the same atmosphere.) Students know that it¹s extra credit so they have nothing to lose. Also, they get the chance to try it out themselves, without being taught or guided, and many find out that they're up to, or almost up to, the task.
I'm always on the lookout for ways in which testing can be used as an aid in the learning process; at any rate, it's a shame when testing is detrimental to the learning process. The extra-credit preview problems seem to me a good way to make testing become learning, to "kill two birds with one stone."
One thing to be careful about is that the problem isn't so difficult, or so far afield from the familiar, that they wind up doing it wrong or getting too bogged down on it, thus practicing and cementing their misconceptions (along, perhaps, with negative feelings). To this end, I make these problems not-hard ? meaning, problems that the students are ready for. I'm very careful to word the problems as strategically as I can, and I realize that such problems aren¹t advisable for all topics.
Also, I make sure that the test itself (without the extra-credit) is short enough so that there's time left over to do the extra-credit. (Also, for classes containing more than their usual share of math-anxious students, I make the extra-credit problem a separate handout, so that students who don¹t want to think about them don¹t have to. (Some students have, admittedly, remarked, "They make me nervous." Many of these same students, though, have eventually come around.)
That this "preview extra credit" constitutes a "syllabus-moving-along-er" becomes evident when we get to the class after the test. The "new" material (upon which the extra-credit was based) is now next on the agenda, and since the students have already experienced it, they're more comfortable with it ? even if they didn't make any progress with it on the test, and even if they got it wrong (although I usually give a small amount of extra-credit just for trying).
During this class I soon see that many students have completely learned the topic via the "preview extra credit," and are ready for the topic after that. (When that happens, though, I make sure to go over ? or have them go over ? one other example of the topic.)
Moreover, I find that often, when some of the students in the class know a topic that moves things along for the rest of the class (though how and why is subtle; a lot of it is probably psychological, and student interaction must have a lot to do with it). I do, however, need to be wary that some students don't get lost in the drift, or feel pressured to pretend to understand that topic. In my experience, this rarely happens, and can be easily dealt with.
A variation: Sometimes I do "preview problems" as in-class "seatwork," rather than on tests. That is, I give the students a problem to do at their seats (not to be collected) before showing them how, or before actually teaching the topic. In general, students seem to be more likely to do seatwork than they are to do oral recitation, and sometimes they're more likely to do seatwork that they are to do extra-credit problems on a test!
It all depends, sometimes on the personalities of the students in the class, or on how long the class period is, or on the particular topic involved. In-class seatwork might not offer the same relaxed "quiet time" aspect as test-taking; on the other hand, working together offers other advantages. There are pros and cons to each; it's not hard to gauge, either is a change of pace from the other, and from the straight lecture style ? and the two practices save time and energy in different ways.
This "preview extra-credit" is not only a time-saver (in whatever sense); it's also an energy-saver, and a stress-saver. Students simply don't have to work as hard at it, and I myself feel more relaxed, knowing that there's already been progress made in this new topic. Also, I believe that it's possible for a topic to be "over-taught" ? that is, it sometimes happens that all a student might need, in order to learn something, is a problem or two clearly worked out, rather than explanation or even motivation (or with minimal explanation, perhaps simply a formula). The extra-credit preview-problems often prevent "over-teaching," while simultaneously allowing for as much "over-teaching" as necessary. Some might suggest that I simply have the students read something in advance before every class (or before some classes). My first-approximation answer would be: that's just not my style. If pressed further, I'd say that the textbook's handling of the new topic is often not to my liking, or to the students' liking. (Of course, that depends.) Also, the spirit is different; my "extra-credit previews" seem less invasive, and they involve the element of choice, which sits well with me and probably with students.
Certainly reading portions of the book beforehand isn¹t the same thing as doing an extra-credit challenge problem, nor is it the same thing as learning by doing. Besides, when I think of the term "time-saver," I'm also thinking of saving the students' time. Moreover, my extra-credit previews often convey to students the (true) information that a particular topic is not "as big of a deal" (meaning as difficult) as they might think. A teacher could, of course, do both ? ask the students to read ahead, with the incentive that there will be an extra-credit problem on the new material. (In fact, that might be an idea for me to consider for the future in select situations.) In general, there are many ways to do things, and the ways can be combined.
Some students choose not to touch extra-credit problems (although they probably do look at them, which can be beneficial). At the other extreme, some students find them fun. (Students have asked me, "How come there weren¹t any extra-credit problems on that test? I was looking forward to them!") In between are those who are motivated by the "extra credit" aspect of the extra credit. All in all, students who choose to can feel very positive when they see what they're capable of doing on their own, at the same time knowing that they have the option not to.
Time Spent: Zero to five to fifteen minutes (depending) making up the extra-credit preview problems, plus five to ten minutes grading them.
Time Saved: 15 minutes to an entire class or more than one class. It¹s not always possible to measure.
Aggravation Saved: in my experience, a lot!
Marion Cohen is the author of Crossing the Equal Sign (Plain View Press, TX), a collection of poetry about the experience of mathematics. She teaches part time at Arcadia University.