Instructors

Changes in Instructors

We're just like the students: Changing the way we have always "done math" in the classroom is scary. But once we get past that, we are having fun, working harder than we ever did before, and finding meaning in what we are doing. As Janet Ray notes, more faculty are talking to each other than ever before -- both about mathematics and about learning. Where faculty are aware of the damage being done by traditional teaching methods and are open to possibilities for change, the idea spreads rapidly.

One of the critical ways in which we are like our students is in our avoidance of responsibility -- and even those who wear the "reform" label proudly have problems with responsibility on occasion. Our students believe we are responsible for "educating" them; we know that one can educate oneself but not another. Yet, we toss the responsibility back to them without necessarily taking responsibility for providing what they need for effective learning. In particular, we resist the idea that it is our job (if nobody has done it yet) to help our students learn to read a mathematics book or to write coherent sentences and paragraphs about mathematics. It's our job (if nobody has done it yet) to help students learn how to work together to achieve a common goal. The list of important new jobs is much longer, but before we can begin on those, we have to educate ourselves about how to do these things. And then we have to take responsibility for helping our colleagues educate themselves as well. Frank Demana (Ohio State University) says that the biggest mistake they made in their projects was to underestimate the importance and difficulty of educating the faculty for real change.

I am often asked how we convinced our mathematics department to accept the laboratory approach as the only way calculus would be taught at Duke. (We're only half way to that "only way" -- the other half our calculus students start beyond Calculus I, and most of them are not in Project CALC sections -- but the principle has been accepted.) In fact, the Duke faculty convinced themselves by volunteering to teach the laboratory course. Many of them did so under terrible conditions, when we were still experimenting with the workload and other aspects of the course. All had to cope with the problems of becoming novices again; in addition to the damage that does to one's ego, they re-learned that anything you do for the first time is a lot more work than is repetition of a comparable task. Most of our faculty experienced the rejection of negative student evaluations, even when there was no hint of any problem throughout a semester of cordial collegiality with students who were achieving at a high level. Nevertheless, almost every individual who taught a section of the course decided that, even though there were still things that needed work, we were headed in the right direction.

How much work does it take to teach a reformed course? Too much -- so far. Most of the instructors I have asked this question are working much too hard to justify the effort for one course. But most are also doing the course for the first time. Those of us who have taught such courses two or more times after achieving stability of materials and syllabus have come to the conclusion that the effort required in the steady state is about the same as that required for a junior-senior level majors course. And that's exactly as it should be -- our freshman and sophomores are just as important as our majors, no more and no less.

The dominant metaphor for what calculus reformers are doing in the classroom is "coaching." Our students can understand that no one learns to play basketball or violin by listening to lectures about dribbling or bowing. They can also understand that people who lack the talent to be a varsity athlete or a concertmaster can nevertheless learn to play well enough to achieve personal satisfaction. To do so, they have to start playing -- preferably under the guidance and direction of a good coach. (Violin coaches are usually called "teachers" -- there's probably a message there.) Furthermore, most athletic and artistic endeavors are more satisfying when practiced with other people, not in isolation. Why should intellectual endeavors be any different?

In [7] I wrote of a personal metaphor for my role in helping students through the scariness of giving up "math" and learning to do mathematics: that of hospice nurse (my wife's profession). I frequently see in my Calculus I students the classical stages of dying described by Elizabeth Kubler-Ross: denial, anger, bargaining, depression, and finally acceptance. Not everyone goes through all the stages or in the same order, and some never make it to acceptance -- at least, not before the course is over and the evaluation form filled out. But this pattern is what I see in my students, and an important part of my job is to nurse them through the stages of dying in their anti-intellectual life, easing their pain and respecting their dignity, in the hope of their reaching joyful acceptance and new intellectual life.

We're just like the students: As more and more faculty and graduate students become involved in teaching reformed courses, whether by choice or because of departmental decisions, we see more and more who are not ready to give up their investment in the status quo. We must recognize that there are pains to be eased and dignities to be respected among our colleagues, just as among our students. Unfortunately, it is not clear who will be the nurses in a highly competitive academic environment that traditionally has little respect for time and effort devoted to making sure that students learn. Supportive and sensitive deans and department chairs can play the role of nurse, but most administrators were not chosen for those qualities. With or without nurses, there is an important role for "support groups" -- groups of three or four faculty and graduate students who share responsibilities for a given course and who meet regularly (perhaps over lunch) to share experiences, horror stories, and solutions. Notice that this is exactly what we are recommending to, even requiring of, our students: We're just like the students.

We're not at all like our students: Simple arithmetic tells us that the percentage of potential mathematics majors among the 600,000 or more students starting calculus each year is vanishingly small -- never mind the percentage of potential mathematicians and mathematics educators. So, if we ask ourselves how we learned mathematics -- that is, if we use ourselves as model learners -- we're going to get it wrong. Even if every student came to us fully prepared to study calculus, they would not be like us. The fact that we learned in a system dominated by lectures -- and did wonderfully well -- does not prove that that system "works." Indeed, those of us who "made it" know that we didn't walk out of the lecture hall knowing more mathematics. Rather, we learned there what we had to go home and work on so it would become part of our repertoire, so we could understand the concepts and master the techniques. Most of the students we teach today have been taught, both overtly and subtly, not to do what we did to learn mathematics. They use textbooks that don't have to be read, and they learn "algorithms" for mimicking steps in worked-out examples so they can "do" homework and tests. It's widely considered "unfair" to ask students on a test to do a problem they have not done already. Our students, at least as they come to us, are not at all like us.

Nevertheless, they are our students, and we are responsible for seeing that they learn: “Teach the students you have, not the ones you wish you had" [6]. In time we can expect the preparation of college students to change for the better. As the NCTM Standards take hold and become part of a nationwide reform effort, many of the problems we see in getting lower division students to understand mathematics will either be dealt with in elementary and secondary schools or not be created in the first place. But we can't wait around for that change and still be responsible to the students we have now. Nor can we declare the problem unsolvable by blaming the victims -- among whom I include not only the students, but also their teachers and their parents. We now have constructive proof from a large number of reform projects, being replicated on a much larger number of campuses, that we can reach students where they are. In addition to confronting them with appropriate challenges to their imperfect understanding of mathematics, we can teach them to read and write -- or support the efforts elsewhere on our campuses to do that. We can gently and supportively turn students around, thereby encouraging them to become thinking and working adults, actively engaged in constructing their own knowledge of mathematics and of its power as a tool in whatever intellectual discipline they choose to pursue.