**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.

*To add your voice to this discussion, e-mail comments, letters, and op-ed
articles to: extend@stolaf.edu or click here.*

*Last Update: *April 7, 1997