Your ATE project has as its goal to integrate the learning of mathematics with technical education through a joint effort by mathematics and technical faculty. What have been the main challenges in implementing an interdisciplinary project of this type? What have been the benefits?
The biggest challenge was getting the right people for the project. A total of 17 Wentworth faculty, including eight from mathematics, were involved in the first year's research and development work. The mathematics people were already committed to the philosophy and goals of the project, and we tried to bring in technical faculty (at least one from each department) who we thought would have much to contribute. While our choices usually worked out well, there were a few cases where faculty didn't "buy in" to the project as much as we'd hoped--they viewed their role as being more of a consultant rather than an active member of a team. Halfway through the first year, we realized that in some cases other faculty would have been better choices, and were able to make some personnel changes at that time, but it was tricky.
Scheduling was another difficulty, as it was hard to find times when all faculty could get together. The first prototype laboratory investigation was written essentially by the whole group over many meetings, and the varied input from many points of view was a big plus, but to make things workable and more efficient in writing about 30 labs over the summer, we split into teams of two or three, usually one mathematics person working with one technical person. While this made things more efficient, we lost some of the group synergy. Weekly meetings of the whole team allowed us to maintain some of that energy, but people became so focused on their own writing that it was sometimes hard to get really tuned in to what others were doing.
However, at those group meetings we had some of the most exciting experiences of the project. Getting 17 faculty from all disciplines to talk academic content and pedagogy was something none of us had seen before! When one group brought up the possibility of using the life of cutting tools to illustrate power functions with rational exponents, the environmental engineer in the group pointed out that a similar function described flow rate in a wastewater channel. We were able to develop a number of mathematical connections in this way.
The group meetings also revealed some inconsistencies in teaching similar subjects that were quite revealing. We realized that various courses (such as physics, mathematics, and strength of materials) often use different notations for the same quantities and relationships. We could probably lessen our students' confusion by communicating more about this.
Of course, a prime benefit of the interdisciplinary work, and the main reason we started our project in the first place, was to bring significant and non-contrived applied problems into our freshman mathematics classes (as well as to extend them to the high school level through another component of the project). As at many institutions in recent years, many of our mathematics faculty were eager to incorporate such rich content to replace the worn out "applications" that appear in mathematics texts, but didn't have the technical background to do it on their own. Pairing mathematics faculty with engineering and design faculty was our solution, which then allowed us to embed the mathematics (algebra, trigonometry, precalculus) in challenging and meaningful situations. "What can I use this for?" is a question we haven't heard this year.
Our initial strategy was to give mathematics faculty one course release from teaching so they could spend part of their time "living in" another department, talking to faculty and seeking out good problems on which to base lab investigations into mathematics topics. In retrospect, it would have been good to also provide time release for selected technical faculty in order to increase their "ownership" of the project. I think we could have avoided some of the problems I mentioned earlier regarding some technical faculty not being as involved as I had hoped.
Many mathematicians and mathematics educators worry that in most interdisciplinary programs mathematics exists to serve science and that the mathematics itself gets lost. How important is it that students see mathematics as a separate subject rather than just as a powerful tool in the service of other subjects?
I guess to a great extent I don't see mathematics (or physics, or other sciences) as separate subjects, so maybe I can't properly address this question. This might have something to do with the fact that my original background was in physics and biology, although I've been teaching primarily mathematics for twenty years now. It's partly because my students viewed mathematics as something they did up on the 3rd floor of Beatty Hall, then forgot about when they went to their other courses, that I got involved in things like this project.
I want my students to see mathematics as something that unifies all their technical subjects. I believe this enhances the stature of mathematics in technical students' minds, since most of my students have somewhat utilitarian views of education. Of course, when possible we still try to point out (or have students discover) the beauty of mathematics in its own right, as in a "Geometry of Design" investigation where exploration of proportion in architectural design leads to an examination of sequences and limits, and things like Fibonacci sequences and the golden mean.
Mathematics faculty also worry that the context-rich environment of an interdisciplinary course will impede rather than enhance learning since it will be harder for students to sort out the mathematics principles from the surrounding context. And they worry that students will not have the opportunity to take the mathematics they are learning to "the next level." What has been your experience in this regard?
I have to admit that we (the mathematics faculty) had to occasionally remind ourselves to focus on mathematics! The engineering and architecture problems were so much fun that we sometimes got carried away. Some mathematics folks with little exposure to the applications end started getting excited about the fields they had been investigating. But of course that's exactly what we want our students to do!
I'm not sure we want our students to "sort out" the mathematics from its context, as long as its not obscured in the process. Students at Wentworth are often quite singled-minded about their chosen majors, and context is important to their learning. I think a student who has waited and watched a voltmeter approach 6 volts very slowly while a capacitor charges, for example, will have a much more profound understanding of the concepts of asymptote and limiting behavior. Without the context, many of my students can't get beyond looking at mathematics as a bunch of isolated exercises in memorization.
I think contextual learning allows them to make more vivid connections that will enable better recall of previously studied topics when they do move to higher levels, and may make them more able and willing to move through those levels. I'm sure many of my students will immediately think of our work with building codes for stairway design any time they encounter inequalities in the future, and they'll be able to deal with them better because of having used them to determine appropriate dimensions for construction.
And I think the "standards-based" pedagogy we've incorporated, with group work and emphasis on modeling and writing about the mathematics involved in the lab investigations will pay significant dividends down the road. One instructor has remarked that her Calculus II students (who mostly had a "traditional" preparation) are unable to begin to approach a non-routine problem, whereas her freshmen will dive right in, assuming that they'll be able to find a workable strategy. It seems to me that our students are laying the groundwork for more confident use of mathematics in the future.
You give the example of students being exposed to the concepts of asymptote and limiting behavior while working with voltmeters and capacitors. Has anyone pointed out to students that these are examples of important concepts in mathematics and given names to those concepts? If so, what happens when the discussion shifts from engineering (or technology) to mathematics?
In this case, the exponential function lab is introduced at a point in the course where student have already studied asymptotes in connection with other function types (e.g., rationals), so the concept is not new and in fact the terminology is used in the lab. I think in all cases our lab investigations have fairly thorough explanations of the mathematics after the initial engineering or design problem is explored. Of course, a lot depends on how an instructor follows up in the subsequent class. Continuing to connect the notion of asymptote to real asymptotic behavior of such things as temperature equilibration or chemical reaction completion (curing of concrete is a good one for many technical students) will keep the concept alive, I think.
One of the paradoxes that can result from carefully planned professional development programs is that faculty get excited about exploring new areas and discovering ways to apply that knowledge to their own disciplines. So they create a neat instructional package for their students and bring it to the classroom. But those packages often lack spontaneity and rarely give the students the same opportunity to experience the delight of discovery that the faculty member experienced. Did you see that happening in your project?
Actually, we noticed a couple of related things. Faculty who used the lab projects in their classes sometimes weren't as excited as the authors of the lab projects, which probably had to do with the intense immersion of the authors in their allied fields during the writing phase. But the response of faculty was quite varied in this regard.
With students also, some were very enthusiastic and really "got into" particular labs while others didn't, and most had favorites among the labs. We think there are probably at least three variables that affect student response:
Last Update: June 17, 1997