A. Remember the button math teachers used to wear--the one that said "Mathematics is not a spectator sport"? But is it a team sport? If so, who should be on the team? Is it true, as we have heard, that mathematicians hesitate to join wholeheartedly in interdisciplinary programs?
B. I taught an interdisciplinary college-level class this fall in which students were paired up in teams. Non-mathematicians were supposed to describe phenomena and the associated problems that they needed to solve. Mathematicians were supposed to build the models. I'm not sure how well it worked.
C. That's no surprise. It took me almost a year of working weekly with a biologist to begin speaking the same language. This collaboration came about because she was trying to understand biological papers with a lot of formulas. It took a long time to make any progress. Understanding another discipline is a long and difficult process.
D. Were you teaching each other, or negotiating?
C. She was providing me with biology papers from current projects in her lab. I was interested in the problems in these papers, and often could tell her how these things were working. We even published a few joint papers. Because this joint project went so well, I decided to teach a class in beginning mathematical modeling. Half the students were not from mathematics; they were paired with mathematics majors.
A. What value do you see in this pairing?
C. The non-mathematicians know what results they want and the mathematics students know how to get them. The teams worked reasonably well.
A. Do the non-mathematicians come with well-formulated questions?
C. I don't know about the well-formulated part ...
E. But tell us what happened when you took these ideas to freshman calculus.
C. Yes indeed! The biology faculty had been displeased with what the students were getting out of their calculus courses, and mathematicians complained about the mathematical abilities of the biology students that they got. A lot of mathematicians say "they cannot do algebra, so how can they do modeling"? I say, show students why they should want to learn a concept or a skill. Don't hit them over the head with the same stick.
F. It is interesting to me that what you want to accomplish seems very different from what others want to accomplish. Your vision is very different.
C. I know that my students know some biology. They are comfortable with science where observations lead to hypotheses, as opposed to mathematics which often starts with theory which is then illustrated with trivial examples. My conflict with my colleagues in mathematics is that I believe that without the right basis of intuition, students cannot learn mathematics using that approach--theory first, applications second. My approach rests on a very different philosophy in that nontrivial examples are introduced to show how the mathematics is used.
G. Could you have done this before your year-long discussions?
C. Probably, but that is because I had a double major in biology and mathematics. But the discussions really helped.
H. As a mathematician, how do you feel about what your students are learning in calculus?
C. They get what they need. They "blackbox" some of it, but that's OK. I teach them Maple--it gives them a crutch to move ahead. I feel good about what they are doing. Biology students don't have great algebra skills; should we let that stop them from moving ahead?
H. What do your students think of the course?
C. I push them hard. In the traditional course, students consistently ask why they are being forced to learn all that stuff. In my course they are still not comfortable, but they appreciate what they need. Now they complain that they only got three credit hours. When you start getting interdisciplinary you are asking a lot of students.
H. These same problems arise at the high school level, especially with students in vocational programs. Leaders in these fields, as well as recent federal legislation, advocate integration of academic and vocational study. But very few places seem to be able to make it happen.
E. I know one example where carpentry and algebra teachers got together to try to work to the same goal. But it is the only case I know.
A. Why are there not more?
C. It's so hard. Try mixing computers, biology, and mathematics all at once. It is too much to take on at the same time as one is also doing other things (teaching or research). Time is major factor.
I. And if you have time pressures at the college level, think about moving this challenge down to high school. Time is a real problem there, for both teachers and students. Teachers hardly have the opportunity to talk to others in their own discipline, let along take a year to be comfortable with the vocabulary of another field. Now with all the pressure of standards, teachers are feeling as if they are getting squeezed from every direction. And still not very much new gets accomplished.
A. One strategy to ease these separate pressures involves coordinated work on a single large project--for example, building a bridge. Students can get involved in issues of design, environmental impact, geography, history, economics, and so on. It is a good way to get students to look at broad-based projects.
J. But those kinds of projects require students to know something about many different subjects. That's unrealistic. I'm curious about whether students in the calculus class for biologists can answer the calculus questions without knowing biology.
C. They have good intuition on the biology side, but they do not have reliable intuition about mathematics.
J. I can't imagine an interdisciplinary program like this because I have never seen a successful example. In looking at musical scales, for example, you can talk about interval ratios and even-tempered scales without ever having seen a keyboard. If you require that students need to know an area of application in order to do a problem, you--and they--are in trouble. Sure, there are lots of applications of calculus in chemistry and biology for example, the action potential of nerves. But if you can pose problems so that students can get the answers from the mathematics itself, then you are on safer ground than if the students need to know both biology and mathematics. I can't imagine a truly interdisciplinary course. It is unrealistic, perhaps impossible.
K. I agree, but for a somewhat different reason. In the interdisciplinary programs I have seen, the mathematics is in there to serve the science. Often the mathematics itself gets lost. There is no opportunity for the students to pursue mathematical concepts. The mathematics is not clearly visible, except as a collection of skills needed to accomplish the matter at hand. It appears "as needed," so students get no experience grappling with mathematics itself. I feel uneasy because what I have seen lets the mathematics go.
C. When I first gave my biology students a graph of the growth of a puppy vs. time, using m and t for mass and time, they asked me whether they could put x and y there. Biologists complain that students don't know straight lines when they use m and t instead of x and y. Students compartmentalize their learning so that mathematics is one thing, science something else. "If it's math, it must be x and y, and can't have anything to do with life or society." Mathematics is not relevant to what they do, so they forget it the moment they walk out the classroom door.
R. I agree. Interdisciplinary issues do not motivate mathematics faculty. Other teachers may be motivated to do this, but mathematics is not pushing the envelope. Mathematics teachers are used to traditional goals-- numbers, graphs, equations. For these reasons, the interdisciplinary projects in this state rarely engage mathematicians.
K. I worry that some mathematics I want students to know might be missed in interdisciplinary study.
H. This discussion helps explain why mathematicians are less enthusiastic than other teachers to join these programs. But the dominant view in vocational education and tech-prep programs is that because real problems are holistic, discipline boundaries don't really matter all that much. Can mathematicians feel that the study of mathematics is complete if it is centered in rich applications?
G. Maybe one of the difficulties is that the kinds of things that people will encounter at work are really very varied. When prospective teachers take a mathematics class focused on their role as future teachers they can see the relevance of that class for their job. But making a similar connection to other areas of work is much more difficult.
M. And this has consequences for students. Since teachers know so little about the role of mathematics in careers, I find quite often that students' sense of what is gong to be relevant to their future careers is way off base.
N. I think you can convince an eleven-year-old that mathematics is relevant. The new middle school mathematics curricula that have been inspired by the NCTM Standards get into rich mathematics from the ground up.
D. Yet even in scientific endeavors, as likely as not, if there's some mathematics involved, it is put off to an appendix. For most of life, you can get along fine without mathematics.
L. I'm hearing a dichotomy. We may want to have these large-scale problems and have everyone get involved, but we also want to put on our mathematicians' hats and be sure the discipline gets its due.
O. If all your knowledge is skill-based or procedure-based, you are not going to recognize mathematics in other situations.
L. I worry about using rich problems for students. If you don't have the mathematical background, how would someone know what kind of problem it is?
P. You absolutely need both the discipline-based knowledge and the integrated "big problem" approach.
J. Teachers in other subjects are trying to implement changes. How can we help teachers understand the mathematics they need?
C. Just using the word "mathematics" can create a psychological block.
Q. I teach programming to students who do not know much mathematics. This makes me wonder how much mathematics you need to function in a classroom that's not itself about mathematics. Do we know what people really need? Maybe it is not what we thought, or what mathematicians believe.
A. That opens up a whole other range of issues, of great
importance but well beyond the agenda of this discussion. It does
seem clear from what we have heard that in addition to the many
ordinary impediments to innovation, even the desirability
for mathematics of integrated curricula is not at all clear. That
gives us all room for much continued discussion.
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Last Update: April 7, 1997