Background

Project MOSAIC is rooted in two basic questions. What mathematics should we teach? How do we create an environment where change is possible?

To start, note that we have used the word mathematics. To many people, this has come to refer to an established curriculum, particularly the geometry, algebra, trigonometry, and calculus found in secondary schools and the continuation of this curriculum with the various levels of calculus, differential equations, linear algebra, discrete math, etc. at the college or university level.

We use mathematics more broadly to refer to the theory and skills that support the study and description of quantity, pattern, variation, and change. Included in this are topics that were not included in the traditional curriculum as it developed historically. These non-traditional topics include statistics and modern computation.

What mathematics should we teach?

What mathematics should we teach? In considering this question, we have been motivated by the remarkable series of reports commissioned in 1999-2001 by the Mathematical Association of America as part of the CRAFTY project: Curriculum Renewal Across the First Two Years. The purpose of the workshops was summarized clearly by Susan Ganter and William Barker in the project's final report:

Mathematics can and should play an important role in the education of undergraduate students. In fact, few educators would dispute that students who can think mathematically and reason through problems are better able to face the challenges of careers in other disciplines - including those in non-scientific areas. Add to these skills the appropriate use of technology, the ability to model complex situations, and an understanding and appreciation of the specific mathematics appropriate to their chosen fields, and students are then equipped with powerful tools for the future.
Unfortunately, many mathematics courses are not successful in achieving these goals. Students do not see the connections between mathematics and their chosen disciplines; instead, they leave mathematics courses with a set of skills that they are unable to apply in non-routine settings and whose importance to their future careers is not appreciated. Indeed, the mathematics many students are taught often is not the most relevant to their chosen fields. For these reasons, faculty members outside mathematics often perceive the mathematics community as uninterested in the needs of non-mathematics majors, especially those in introductory courses.
The mathematics community ignores this situation at its own peril since approximately 95% of the students in first-year mathematics courses go on to major in other disciplines. The challenge, therefore, is to provide mathematical experiences that are true to the spirit of mathematics yet also relevant to students' futures in other fields. The question then is not whether they need mathematics, but what mathematics is needed and in what context.

Even though the CRAFTY workshops covered a broad range of fields -- biology, business and management, chemistry, computer science, engineering in various flavors, mathematics, statistics, physics -- the conclusions reached are remarkably consistent across all disciplines. They call for, among other things, much greater emphasis on

  • Mathematical modeling, the process of constructing a representation of an object, system, or process that can be manipulated using mathematical operations.
  • Statistics and data analysis.
  • Multivariate topics. The reports refer specifically to two- and three-dimensional topics. Many of the topics mentioned are related to the traditional calculus sequence (including linear algebra, differential equations, and multivariable calculus) --- we'll refer to these topics as calculus.
  • The appropriate use of computers.

Similar recommendations are found in the highly regarded Bio2010 report. Bio2010 lists an extensive set of topics for biology students to learn: Calculus (including multidimensional calculus), Linear Algebra, Dynamical Systems (including the phase plane, feedback, limit cycles), Probability and Statistics, Information and Computation, Data Structures. Each of these is subdivided into several topics. In addition, the committee recommends that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes.

How to Make Change Possible?

There are many critiques of contemporary mathematics education, and many attempts to improve it. Ironically, these critiques often stem from the success that the mathematics community has had in creating high expectations. It's widely appreciated and accepted that the development of mathematics skills is an important aspect of every student's education. Society has invested very substantial educational resources into mathematics: students in primary education spend a substantial proportion of their time with mathematics; college-bound secondary students will almost always spend at least three out of four years studying mathematics; standardized tests for college admission are largely about mathematical reasoning; a large fraction of college and university students will take some mathematics.

To make better use of the resources that society already invests in mathematics, CRAFTY reports called for mathematicians to form collaborations with the partner disciplines. But, for several reasons, establishing interdisciplinary partnerships is difficult. One important reason is that many partner disciplines are not yet prepared to engage the mathematics curriculum. In potential major partnership areas such as biology, the mainstream curriculum involves college mathematics only weakly if at all. Since mathematics is hardly used, there is little force for alignment. The result is that innovation is a matter of faith. As in the movie Field of Dreams, the whispered call from on high is, If you build it, they will come. But it's unreasonable to expect mathematics faculty to plow under their fields in anticipation of the future arrival of partners. It's understandable that the mathematics curriculum is shaped by the more immediate and tangible forces of the internal structure and priorities within mathematics and, to a lesser extent, by the STEM disciplines most likely traditionally to engage the mathematics curriculum: physics, chemistry, and engineering.

Another reason why establishing partnerships is difficult is that potential STEM partners are understandably hesitant to invest the energy that's needed until they have a solid idea of what the result will be. Such support takes the form, for instance, of requiring specific courses. For example, examination of requirements for the biology major at the top 10 US News liberal arts colleges show that only four require any calculus-level mathematics course and four require a statistics course. (Two of the schools require both; four require neither.) Such requirements can be satisfied with any of a number of courses, including Advanced Placement high-school courses. As a result, contact between biology departments and mathematics departments is often unfocused. A requirement for specific courses creates a site for the catalysis of curricular alignment, but instituting such a requirement requires faith that it would prove successful and relevant. In other words, the partners can't be expected to come until it's built!

At the level of individuals seeking to form interdisciplinary partnerships, consider the plight of a calculus instructor making partnership with an ecologist. The ecologist would sensibly be interested in improving student understanding of fluctuations in populations and how they relate to interactions among species. Or, she might want to have students use field data to study certain relationships while mathematically adjusting for variation in other measured quantities. Calculus, broadly speaking, is highly relevant here: models of growth and interaction, partial derivatives for analyzing models while holding some quantities constant. Yet the early calculus curriculum is not so relevant; its first-year is devoted to functions of a single variable and their algebraic transformation to form derivatives and integrals. The calculus instructor, however masterful at teaching calculus, is likely to be unaware of statistical approaches to approximating data involving multiple variables; the ecologist is likely unaware of the natural relationship of calculus to those statistical approaches. So the potential for calculus instruction to inform the ecology student is largely lost.

The MOSAIC community aims to bring together disciplinary partners who are well positioned to engage the mathematics curriculum and to help align it with the recommendations of the CRAFTY reports. These partners are the faculty already teaching statistics, computation, and the various facets of mathematical modeling.

Using the MOSAIC topics to initiate alignment will build interest in the partner disciplines and encourage their participation; a curricular MOSAIC gives partner disciplines many more points of contact with the mathematics curriculum. It also serves an internal curricular purpose; each MOSAIC subject provides a setting in which the concepts and techniques of the other subjects can be explored.

For example, consider again the calculus instructor seeking to make a partnership with an ecologist. Incorporating modeling concepts in a calculus course lets the calculus instructor illustrate important ideas in calculus, and provides the opportunity for a direct link with ecology. When computation is used in calculus, it becomes possible to simulate dynamics (e.g., solving differential equations that are relevant to ecology and showing integration as an accumulation of change) and to use data as the basis for approximating the relationship between variables.

The idea of uniting MOSAIC subjects is completely consistent with the mainstream mathematics community. The MAA Committee on the Undergraduate Program in Mathematics (CUPM) \cite{cupm-2004} recommended that courses designed for mathematical sciences majors should ensure that students gain experience in careful analysis of data. It goes on to say, A variety of courses can contribute to this experience, including calculus, differential equations and mathematical modeling, as well as courses with a more explicit emphasis on data analysis. For example, students in a calculus course can fit logistic and exponential models to real data or encounter finding the best-fitting line for data as an optimization problem.

Announcements

Upcoming M-CASTs

Feb 10, 1pm (CST)
Identifying Modeling Misconceptions


JMM January 2012 Presentations:
MAA PREP Workshop
July 9-13

Videos of Recent M-CASTS


The MOSAIC Wiki


Talks from the July 2010 Kick-Off Workshop