Bonnie Gold, Monmouth University, Monmouth, NJ, firstname.lastname@example.org
Several years ago, after struggling with supplementing traditional calculus texts with labs and trying to squeeze everything into three meetings per week, I decided to look for a calculus reform text. When I read the Ostebee/Zorn text (which I'll refer to as O/Z), I was excited -- here at last was a text which (usually) said exactly what I told my class because it was left out of traditional texts. I immediately decided to try using the text, but was faced with two problems. First, if the text says what I usually lecture over, what do I do in class? The answer was not hard: get the students actively working on the ideas in the section by having them do problems, labs, worksheets, etc. However, that was predicated on their having read the text before class. This led to the second problem: how to get them to read this text, which, unlike most, actually addressed the (reasonably good) student, rather than the professor.
The first year I had students keep journals. I used the preliminary edition of O/Z, which had wide margins with many questions directed towards students to make sure they were reading carefully, as well as humorous, historical, and philosophical comments. (It still has marginal notes, but fewer, with most of the humor removed.) I required students to answer these questions and invited them to include any further responses to the text -- difficult-to-understand sections, errors, etc. Each week I collected, semi-randomly, about 1/3 of the students' journals, read them, made comments, and gave a grade. The journal counted for 15% of their grade, as it involved a substantial amount of work. While this was successful in getting the students to read the text, they very much resented the time it took. They also resented the humorous comments in the margins. I think they felt left out of the jokes, just as my children do when looking at cartoons in the New Yorker -- they didn't have the sophistication to understand them. However, a few students responded very well to the journals, using them as course notes helpful in reviewing for tests. The journals also involved a massive amount of work for me -- I wanted to ensure students responded to a substantial number of the questions -- this took me about 6 hours each week. I started dreading weekends.
Fortunately, during the year, I attended a minicourse at the Winter AMS/MAA Meetings on question-based courses. While I've not implemented that kind of course, I did modify the idea, in place of the journals. I now require students to come to class each day with questions over that day's reading. My instructions to them are: "To insure that you do a thoughtful job of reading the text, you are to come to class every day having read the section assigned for that day, and to bring a sheet of paper with at least three important questions which reading the section has answered for you, or which the reading has raised, either because it was unclear or because you are wondering about extensions or special cases. Please make it clear which is the case by writing 'answered' or 'left unanswered' next to each question. These questions should not be simply 'What does the word . . . mean,' nor should they be questions which appear in the text. The questions should show that you've read the section AND thought about what you read. Questions 'answered' should be questions the authors were trying to answer by writing that piece of text. The questions will count for 10% of your grade (and will be graded 'good' (= A, roughly), 'OK' (= B), 'needs improvement' (C-) )." I also give a bonus point (one point added to their test grade sum) for finding misprints in the text, which some students get very good at. I warn them that I will NOT lecture over the reading, although I will answer any questions they have on it.
I've been VERY pleased with the student response to this requirement. It usually takes students a week or two to understand what sort of questions I'm looking for. After the first few days, I hand out examples of good questions from a few class members, to help those who aren't getting the idea. It's clear that virtually all students are reading the text, not just skimming the section titles, once they learn that questions such as "what is an application of derivatives" get a comment from me, "This question can be asked about any section without reading the section. You can do better." Students who have a hard time writing interesting 'answered' questions will often write a less interesting question, but follow it with the answer to the question, in their own words. But many students ask really interesting questions which involve extending the discussion in the text. Here are some examples, from a section, in second semester calculus (Section 7.3 of O/Z), on the midpoint and trapezoidal approximations and their error bounds. [I have copied the questions literally from the students' papers , including grammatical errors.]
1. Since Ln is an exact approximation if f´(x) = 0, and Tn is exact when f´´(x) = 0, is there any sort of approximation that is exact when f´´´(x) = 0?
2. It seems that, in some cases, Tn would be a better approximation than Mn. But in Theorem 3, it shows that |I - Mn| 2 and |I - Tn| 2. [I represents the integral; C is not the text's notation, but the student's summary.] Is Mn always twice as good of an approximation than Tn? (Half the error?)
[Here I should mention, in the only example for which the error is actually computed in the text, the error for the trapezoidal sum is exactly twice that for the midpoint sum.]
3. For a function with changing concavity, is it possible to know if the function is overestimated or underestimated by Mn and Tn?
4. Although K2 can be any upper bound on |f´´(x)|, isn't it to our advantage to pick the smallest possible upper bound?
These questions show some confusions, as they should, but also a lot of serious engagement with the text. I use the questions in several ways. First, I begin each class by asking whether there are questions on the reading for that day. Usually only the bolder and better students are willing to ask their questions publicly, but those provide me with a springboard to emphasize the main points of the section and, in the process, clear up the questions some of the weaker students have. Then, after I have read their questions that evening, I will comment, at the next class meeting, on any confusions or concerns that several students have expressed. I respond to all unanswered questions, either in class, on the paper, or by suggesting that they come see me about it in office hours. Reading and commenting on the questions takes me about an hour for a class of 30. I usually to do it before the next class period, so students get a response while they're still interested.
Because, for most students, this is the first time they have been expected to read a mathematics book, I also spend part of a couple of class periods helping them learn to read the book. I usually model the process by reading with them about a page of the text during the second class meeting. Then, during the second week, I use a classroom assessment technique from Angelo and Cross's book, called Content, Form, and Function Outlines, in which one takes a small portion (say, a page) of the text and analyzes each piece for its content (what it says), form (how it says it: e.g., graph, example, theorem, etc.), and function (why it's being said: what purpose does that example, or narrative, have in the section). Asking why authors have put information in the text is very hard for many students, but doing this exercise helps them write better questions later.
The continual flow of questions and answers leads to a nice dialogue, especially with many of the shy students -- it is clear to them that I am interested in what they are thinking and what concerns they have. They still don't like having to read the book -- they'd prefer to be spoon-fed, as they have been in high school -- but they do appreciate the individual attention their questions get them, and I do see substantial progress in their ability to read, and make sense of their reading.
Copyright ©1998 The Mathematical Association of America