Instructors in reformed courses across the country are hearing the same complaints from their students: "This isn't what we did in high school." "You're making us work too hard." While students intend these remarks as criticisms, we see them as high praise. In courses that emphasize conceptual understanding and de-emphasize symbol manipulation, students complain that they are not learning what they need for engineering or physics or the next math course. To the extent that the rest of the curriculum remains unreformed, there may be some truth in that, but that merely tells us the job is not finished yet -- not that it can't be done. Of course, it was the engineers and physicists and mathematicians teaching higher level courses who let us know that the traditional calculus course was not preparing students for their courses. The same folks, along with biologists, chemists, economists, and others, are applauding our new focus on problem solving, modeling, concepts, exposition, and technology. (As Larry Evans, chair of the Duke physics department put it, "There's nowhere to go but up.") But it takes time for this message to filter into student consciousness.
Deb Hughes Hallett has spoken often of how scary it is for students to be thrust into a totally new environment with totally new demands, when they were secure in the knowledge of what "math" is and how it works -- even if they saw themselves as people who could never "do math" very well. Once we get them through that scary period -- which may vary from about four weeks to a semester and a half -- many of our students find that they are having fun, and we see them doing things that calculus students never did before, such as writing intelligent papers about solutions of real problems. Furthermore, they are finding meaning in the modeling and interpretation phases of problem solving, contradicting their prior beliefs that math has nothing to do with anything else and that it has no intrinsic meaning either. Janet Ray notes the new phenomena -- which may occur more frequently in community colleges than in elite universities -- of grades rising from semester to semester and of students taking more mathematics courses "for fun."
Most of us did not start out consciously intending to solve problems of access to mathematics and science by under-represented minorities and women. However, we have stumbled onto some solutions to those problems as well. By downplaying the roles of symbol manipulation and of traditional testing, we have created courses in which students with weak algebra skills (disproportionately, female and/or minority students) can achieve success in other ways. In so doing, they build confidence in their thinking abilities, and they often learn from their peers the necessary symbol skills they were lacking. In traditional courses, a grade of C or D is a clear indicator of worse grades to come, eventually leading to failure. In a reformed course, students can and do get stronger over time. It is no accident that courses developed by "elite" institutions such as Harvard and Duke are finding widespread acceptance at schools with large non-traditional student populations.
A key factor in expanding opportunities for women and minorities is small group work that is required, structured, and supervised. The work of Uri Treisman (University of Texas) and others has shown the importance of minority students working in groups to learn from each other and to support each other. Various reform groups are finding that this works well whether or not minority students are the majority in their groups. We are also finding, in the language of Belenky et al. , that women working in small groups "find their own voice" in ways that are not likely to happen in a classroom-size group. In time, as they discover that they have good ideas and can express them, women students often become vocal leaders in the larger group as well. As teachers, we have all known for a long time that the students who speak up most in class are likely to be among those who achieve the most success. Now, by starting with small-group activities, we are finding ways to make that success-through-participation accessible to almost all students.
The other side of this coin is that we have frustrated some students (disproportionately, white male students) who have been very successful at symbol manipulation in high school, and who really believe that (a) this is what mathematics is all about and (b) they want to major in mathematics. Typical question: "I can do the math -- why do I have write about it too?" Answer: "The 'math' you can do so well can be done by a machine. If you don't understand what you are doing and why you are doing it, you're not going to be a very good math major, let alone mathematician." Another typical question: "I can solve these problems by myself -- why do I have to be saddled with a partner?" Answer: "Real problems are too big to be solved by one person working in isolation. That partner you may be carrying algebraically may also be carrying you conceptually. An important part of learning is learning to work with others, to bring out each person's strengths and contributions, so that the final product is greater than the sum of the parts." Many of our hot-shot symbol pushers eventually buy into this, but some do not. I am not disappointed when the latter students, frustrated by B's when they think they should be getting A's, announce that they are never going to take another math course. Better they should be frustrated as freshmen, when there is still time for them to find something they are really good at (and at which they haven't been displaced by a machine), than to have them major in mathematics and frustrate themselves and us as juniors, still rejecting the idea of "understanding" as part of mathematics.
Closely related to the problems of understanding mathematics is a fundamental problem we overlook at our peril: Most of our students are functionally illiterate, at least in mathematics. That is, they don't know how to read, and they don't know how to write. It is one of the great scandals of our time that students can actually "succeed" in an academic endeavor (I'm not talking about social promotion in school) without being able to read or write. We did that to ourselves and to them: We created courses in which writing was not expected, and we created "textbooks" for which reading was unnecessary. Now, as we focus on understanding, we are confronted with the folly of our past practice. Most of our institutions have discovered the concept of "Writing Across the Curriculum" -- and have implemented some form of writing program, with varying degrees of success. But where are we going to find "Reading Across the Curriculum"? Do we have to invent it ourselves? If we don't, how can we expect students to actually read books about mathematics when they have never had to do that before?
As reformed courses become the "mainstream," no longer "experimental" sections that students can avoid by taking some other track, we can expect to see increased levels of frustration, bitterness, even hostility. That was already evident a year ago when Elizabeth Culotta wrote about Project CALC , and the situation has gotten worse at Duke, now that every student who starts with Calculus I is in a laboratory section. Hostility levels seem to be lower at schools that are not as selective as Duke or Harvard (where the students are absolutely certain that they know better than the faculty what a calculus course is supposed to be), but this is likely to become a real issue at every school that attempts serious reform.
There's an important message here for department chairs and deans about rating and rewarding the teaching effectiveness of the faculty. In a time of transition, teaching must not be evaluated solely by students -- who always have a vested interest in the status quo -- and it may even be necessary to consider negative evaluations as a positive sign and vice versa. Of course, that means numerical scores on five-point scales are not enough. One must listen to or read student comments very carefully to know whether their complaints are about positive or negative aspects of teaching. Better yet, samples of student reports should become part of the process of deciding whether students are in fact learning -- which is the most important issue in deciding whether the faculty are in fact teaching. This is a tough issue: Several conference participants noted the relatively high median age of those in attendance, reflecting the risk for young faculty members in "getting involved" with reform. David Lomen (University of Arizona) observed that, to get that median age down, we have to learn to value teaching in ways other than student evaluation forms.
At Duke, we have evidence that offsets the negative attitudes displayed by our students at the end of Calculus I (and sometimes later): Jack Bookman, evaluator for Project CALC, has conducted a series of tests with sophomores and juniors, half of each group having had the Project CALC course and half the traditional course in their freshman year. Once they have had time to reflect, the same students who wrote very negative course evaluations about Project CALC have much more positive attitudes than their peers toward mathematics in general and its role in their lives .
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