November 1999

**Jane Korey
MATC Project Evaluator**

In four years the Dartmouth Mathematics Across the Curriculum project has created sixteen new courses: a six-course mathematics and physical sciences sequence (IMPS), two intermediate mathematics applications for science courses, and eight courses linking mathematics with a humanistic discipline. Additionally, it has influenced another thirteen, supporting the creation of new mathematics modules, case studies, or other interventions that add a mathematical lens. The evaluation team was charged with evaluating the effects of these courses on student learning, faculty development and institutional culture. Apportioning our resources for maximum impact, we focused on four categories of courses where the MATC influence was greatest: the IMPS sequence, the Introduction to Calculus featuring real-life applications, intermediate math applications for science, and the math and humanities courses. The evaluation results are based on extensive assessment–using attitude and achievement surveys, content tests, and in-depth interviews with students and faculty–of nineteen courses (a total of 36 course iterations) taught by 27 different faculty (a total of 44 "faculty-courses") and involving over 1800 "student-courses." [note 1. A faculty-course = one faculty teaching one course. A student-course = one student taking one course. The faculty/student-course total is higher than the number of students or faculty involved because some faculty and some students taught or took more than one course.] Looking at a broad range of courses over time allowed us to identify patterns in student learning and faculty experience that transcend a single course or discipline as well as to assess the effectiveness of MATC courses.

GENERAL FINDINGS•

Students' interest in mathematics is more important than their perceived math ability in determining whether they study more mathematics.Many students who view themselves as able mathematicians forego college mathematics because they see no career utility or intellectual reward.I have [AP] credit for calculus in high school but I don't want to go on. I've taken math every year of my life. But I'm not planning on doing something [that requires] higher levels of math. I'm decent at math, I usually get decent grades, but I don't enjoy math. (Mathematics and the New Universe student)

Since I came to Dartmouth the math courses I've taken, like multivariable calculus and differential equations, are really dry. You're always plugging and chugging numbers and doing proofs and stuff, and I really didn't like that so I stopped taking math courses altogether. Then I saw this course on the bulletin and it sounded really interesting because it encompassed a lot of different subjects, like infinity and the fourth dimension. I thought wow, this is so cool. (Science Fiction student)

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Real-life applications make mathematics more approachable and more interesting. Whether students' interests are pre-med, pre-engineering or pre-history, connecting mathematics to their existing interests transforms math from a "cut and dried" requirement to a relevant tool for advancing their learning.Taking a probabilty density function and finding out possible periods of incubation based on time of infection and things like that, I feel that's something a real doctor would do. You take this function and you find out when your patient was infected and you say, okay, you can possibly come down with AIDS in six years. That's very useful, for doctor and patient, you know? I thought it was really cool, to be frank. It was pretty cool. (Calculus Applications for Medicine and Biology student)

The chaos part of it made it seem a lot more interesting and colorful. I had thought of math as a black and white sort of thing, like pencil and paper and you write everything down. But just from pictures and stuff of strange attractors, they're so colorful and interesting. I guess it made it more colorful, sort of, having to put it to use. (Chaos student)

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Expanding the range of mathematics topics accessible to average college students increases their interest in mathematics.To most entering college students, calculusisupper level math. As a result, many who do not need calculus in their careers turn away from college math altogether. Students were excited by courses that offered non-calculus topics (combinatorics, number theory, group theory, probability, etc.), revealing unsuspected new worlds of mathematics to discover and enjoy.This course broke the calculus teaching level of math. I'd always associated calculus with knowing upper level math. Now I know calculus is just one section of a much broader mathematical area. Discrete math is big in its own right. (Discrete Mathematics for Computer Science student)

[The math] was basically just conceptual. Usually when you do math you don't really sit there and think about why it is the way it is; it's kind of a cut and dried type thing. So that was the best part of the course for me, to sit there and think about infinity and all that stuff. ( Matter of Time student)

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In-class group work helps college students learn mathematics.Provided that the problem is pitched at the right level–neither so difficult that no start can be made nor so easy that it can be quickly solved alone–students report that they learn better if they spend a portion of the class time working together on problems.They would give us a problem in class, and we’d have to do it right there in class, instead of just talking at us the whole time. That really helped me get out of just the note taking, write down whatever the professor says, but don’t really process it. Sometimes I get in [that] mode, but [group work] really helped me to internalize it, because you’d see it right away, before you’d actually go read about it on your own. [Then] you’d really see oh, this is why I’m trying to do it. This is what I’m trying to solve, plus helping me to learn to interact with other people, learn from other people, too. I thought it was just to be a great way of learning. (IMPS student)

Two or three minds are better than one. And you’re going to have a diversity of ways of thinking, a little more total creativity, and we’ll get some different ideas thrown out there, and we have more people to test them, and that works pretty well. Then you follow up on one idea that you think is good, and you keep working faster on it. It’s kind of like you have three minds linked in parallel. They can all work together, and then you can process things faster. (IMPS student)

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Interdisciplinary courses model the kind of creative, "out of the box" thinking needed in today's complex world.Linking quantitative to non-quantitative knowledge is particularly needed, but appears to be notably difficult for students. Courses where professors with widely differing perspectives make that connection introduce students to a critical intellectual process they are otherwise unlikely to encounter directly.[What I learned was] the interdisciplinary approach–just knowing how to integrate material that doesn't necessarily at the beginning seem like it would fit together. Finding ways, exploring ways and being creative in math which you wouldn't–you'd say creative and math don't go together. You're not creative in math. But you can be. And learning that you can be. And learning that when someone says, "Can you do these two things?" and you say, 'No," you probably can. You just need to figure out how. (Pattern student)

I always knew I was missing something, in that I didn't know exactly how I was supposed to put the math that I knew–all the math one side of my brain–into all the other stuff on the other side of my brain. I think this course helped me to kind of bridge the gap. That's probably the best thing it did for me. (Calculus Applications for Medicine and Biology student)

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Planning and collaboratively teaching interdisciplinary courses offer faculty a valued opportunity for intellectual, pedagogical and social exchange with their colleagues.Despite the extra work involved, faculty find that collaborative effort revitalizes their teaching.Chemist: It was fun because it was a challenge. Fitting it all together and making it appear to the students that it's all connected was a challenge, but it was fun to do that. It was also really rewarding to know that I could do a chemistry application of some physical principle, knowing they had just seen the physical principle, and seeing how[the physicist] presented it so I can adapt the lecture to say, here's what we learned in Physics. Here's the principle. Here's how it all works.

Humanist: I want to see what doing the mathematics does to my understanding of, say, Augustine's

Confessions. I want to see what happens when I go back toOne Hundred Years of Solitudeafter learning modern theories of time. In other words, I want to see what happens to thinking when you cross-pollinate like that. To me, that’s the real fun of doing it.Mathematician: When I make a breakthrough reading a humanities text, I feel less isolated (than when making a mathematical breakthrough). I feel that it has more connection with the rest of life. You read

One Hundred of Solitudeand you feel it says something about the way human nature is , and the way human beings behave. And when you can connect the math with that, I feel like I’m just more connected with humanity.

STUDENT OUTCOMES•

Integrated Mathematics and Physical Sciences.IMPS, which ran for two years as a three-term, six-course team-taught interdisciplinary math/physics/chemistry sequence for first-year students, is presently a two-term, four-course math/physics offering. Now in its fourth year, it has involved nine faculty members in teaching (five from math, four from science) and as many more in planning and support. 168 students (most of them pre-engineering) have accumulated 626 "student-courses." IMPS asks students to work hard, to work together, and to understand connections between math and science. Students believe it succeeds: "[The strength of the course] is the way it ties together math and physics. It's easier to learn, it's easier to understand, you understand it more thoroughly, and you understand applications. I think it speeds up the learning process, and it provided a lot of insight that is more difficult to grab on your own. They lead you through a lot of connections that I probably wouldn't have made until much later on." Through shared intellectual effort, "Impsters" (as they call themselves) emerge as a social unit as well–a community of scholars in the best liberal arts tradition.Because evaluation results are from the three years before IMPS stabilized its term length and teaching materials, we view present results as suggestive rather than conclusive.

• There are no significant differences between IMPS students and traditional students in their subsequent total GPA, science GPA or engineering GPA or in their overall continuation into a science, math or engineering (SMET) major. In the first two years, 91% of IMPS completers declared SMET majors or minors, with the majority (63%) in engineering. 85% of traditional track (Physics 14) students chose a SMET major or minor. Performance of the two groups on various content tests is mixed.

• IMPS consistently recruits students into the engineering track: more students finish the course intending to be engineers than enter with that goal. Completion of the IMPS sequence itself is also good and getting better, from 68% in year 1, to 84% in year 2 and 88% in year 3.

• Professors in subsequent engineering courses rate the teamwork abilities of IMPS students significantly above that of their conventionally prepared peers.

• IMPS may advantage women. Women IMPS engineering majors posted the highest engineering GPA's, followed by traditional track men, traditional track women, and IMPS men. The GPA difference between traditionally prepared men and lower-performing IMPS men is statistically significant; the equivalent GPA difference between the smaller samples of IMPS women and lower-performing traditional track women is not. The perplexing performance difference between IMPS women and men raises questions about learning style that should be explored, particularly the role of collaboration and group support, the presence of female role models, and the possibility that new instructional pedagogies may have different consequences for men and women, increasing cognitive load and/or interfering with established learning strategies.

• Three years' worth of interview data indicate that IMPS offers an academic experience that is engaging, intense and deeply rewarding. Asked what he would take away from the course, one student wrote, "Math and science is scary and messy and wonderful and exciting. It's no longer about right and wrong answers; there are so many more things to think about. IMPS math and science isn't cameo, homogenized topics; it's the real thing at times–headaches and all."

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Intermediate Mathematics Applications Courses."Discrete Mathematics for Computer Science" and "Applications of Calculus to Medicine and Biology" were tightly focused courses which, like IMPS, targeted specific populations and tailored the mathematics directly to their interests. Using problems from the respective disciplines to motivate the mathematics (e.g., encryption, the growth of the AIDS virus) and involving students through group work and regular discussion, these highly successful courses stand as models for post-introductory mathematics offerings. The 58 students in these courses, most of whom entered with an introductory calculus background, left with their interest in mathematics heightened, their confidence buoyed, a new view of mathematics, and a lot of interesting and useful mathematics under their belts.•

Mathematics and Humanities Courses.417 students and fourteen different faculty members have participated in eight courses linking mathematics (mostly non-calculus) with art, music, philosophy, and literature. These courses have surmounted the daunting challenge of making mathematics accessible to a population ranging in preparation from high school algebra to differential equations, and in interest from the terrified to the wildly enthusiastic. As students perceive that mathematics can illuminate humanistic subjects (and vice versa), they also appreciate that mathematics can be a creative undertaking which is useful in all fields. Surveys reveal increased confidence and a new openness to mathematical thinking. Perhaps as important as their mathematical gains is their exposure to interdisciplinary dialogue. As professors actively construct a conversation between highly disparate disciplines, they model the kind of intellectual agility needed to negotiate an increasingly compartmentalized world. As one student concluded of her experience, "This course makes you think, and that's not a cliche."

• Introduction to Calculus with Real-life Applications.The Fall Mathematics 3 course uses real open-ended case studies (such as Milankovich's theory about glacier formation) as the vehicle for teaching calculus concepts. By recapitulating the analysis on computers, many students appreciate that the calculus they are learning can be used to answer questions that are real, practical, and interesting. As one related, "Instead of just learning some nonsense you're not going to use, Math 3 brought it more into the real world, where you see you are actually going to use it sometime in the future." Working with case studies also gave students a deeper understanding of the mathematics: "It showed me that math wasn't as clean cut as everyone perceived it to be. You have to understand what's going on." The value of the case study concept has been confirmed through students like these (a majority of the class in 1997), but it has been harder to extend those successes to the full range of Math 3 students. 1998 revisions of the case studies intended to make them more broadly relevant were not successful, but the effort usefully emphasized lessons which will guide ongoing modifications: applications must be varied, relevant to student interests, keyed to the level of student learning, and require doing real mathematics.Comparison of the 449 Fall calculus students with their 202 peers in the standard Winter course has been complicated for the last two years by Winter personnel changes and Fall curricular changes. One result has been consistent: Fall students have scored better than Winter students on a math test taken ten weeks after the course, although that difference has not always been statistically significant.

FACULTY OUTCOMESFaculty benefit from involvement in interdisciplinary courses, and for many of the same reasons as students. Like students, they discover that interdisciplinary courses are more work than traditional courses, but well worth the investment. Working with colleagues from other departments allows them to be students again, exploring other fields and approaching their own from a fresh perspective. Pedagogical issues are forced into the open and strategies shared as collaborating instructors confront novel teaching challenges. One of the most productive interchanges occurs between novice and established faculty: the experienced teacher mentors the novice, while the junior member introduces the senior to new pedagogical techniques which are incorporated into his or her tool kit. Similarly, faculty are introduced to new pedagogy as they join the teaching rotation for these courses. Designing and teaching an interdisciplinary course exemplifies the project's applications-based philosophy that we learn better in the context of problems that are real and compelling. Pedagogical issues that would draw professorial yawns in a teaching methods workshop are riveting when their own material and students are directly at stake.

Collaborative teaching also counteracts debilitating departmental isolation. Repeatedly, faculty (especially junior faculty) regretted that they had few contacts outside their departments and enjoyed little of the collegial interchange that ought to pervade a liberal arts campus. While interdisciplinary courses represent some additional cost to the College, the payoff in professional development is substantial. Faculty are intellectually energized, establish complex relationships with their colleagues, acquire new teaching skills and, of course, teach their students better. Many professors report that collaborative teaching has been such a rewarding intellectual, personal, and pedagogical experience they wish they could co-teach all their courses.

INSTITUTIONAL OUTCOMESThe heart of a college is its faculty, and as faculty pursue opportunities to grow as scholars and teachers, the institution gains. Through its interdisciplinary course development, MATC has offered a mechanism for professional growth that complements and extends the College's existing interdisciplinary teaching structure. The number of regular faculty who wish to teach interdisciplinary courses alongside mathematicians continues to increase. The project has also reinforced on campus and beyond the importance and feasibility of interdisciplinary learning as preparation for the real work of the world. The success of courses like IMPS and the College's support for MATC encouraged a group of College and Medical School faculty to establish the interdisciplinary Human Biology Program. Likewise, MATC provided an example for faculty seeking to create an interdisciplinary materials science graduate program. The enhanced opportunities for students and faculty to engage in a mode of thinking widely believed to be critical to twenty-first century success places the College in the vanguard of higher education institutions. Extending the interdisciplinary and numeracy message well beyond the Hanover campus, MATC authors have produced nine manuscripts scheduled for publication by Key College Publishing in the next eighteen months (with more in preparation), hosted 137 faculty at five workshops, mounted an art exhibit in collaboration with the Hood Museum, created math videos, and established a website–to name only some of its varied activities. Finally, and perhaps most important, MATC has raised the issue of quantitative competence with administrators and, one by one, with participating faculty. MATC's efforts to improve mathematics instruction for all students addresses a serious national issue whose solution will require the cooperation of educational institutions at all levels.

© Jane S. Korey, January 2000. All rights reserved. Dartmouth College