Calculus at Crisis III: The Client Disciplines

In my last years at Penn State, I worked with faculty in the College of Engineering on issues of undergraduate education. They had two complaints about the mathematics department. First, we were failing too many of their students. Second, the ones we passed seemed incapable of using the mathematics we presumably had taught them when they got to their engineering classes. Chief among their specific gripes was an inability among their students to read a differential equation, to understand its assumptions of the relationships among the quantities being measured.

A decade later, in the Curriculum Foundations Project workshops with engineering faculty brought from across the country, we heard the same concerns about what their students should be learning from the mathematics department:

Students “should understand the reasons for selecting a particular technique develop an understanding of the range of applicability of the technique, acquire familiarity with the mechanics of the solution technique, and understand the limitations of the technique.” (from civil engineers, p. 59)

“There is often a disconnect between the knowledge that students gain in mathematics courses and their ability to apply such knowledge in engineering situations … We would like examples of mathematical techniques explained in terms of the reality they represent.” (from electrical engineers, p. 66)

“In an engineering discipline problem solving essentially mean mathematical modeling; the ability to take a physical problem, express it in mathematical terms, solve the equations, and then interpret the results.” (from mechanical engineers, p. 81)

From the current ABET (Accreditation Board for Engineering and Technology) Criteria for Accreditation, all of the references to mathematics under Curriculum talk about “creative applications,” building “a bridge between mathematics and the basic sciences on the one hand and engineering practice on the other,” and the use of mathematics in the “decision-making process.” As ABET moves into the criteria for specific programs, again the emphasis is entirely on the ability to apply knowledge of mathematics, not on any list of techniques or procedures.

In the biological sciences, the other big driver for calculus enrollments, the American Association of Medical College and the Howard Hughes Medical Institute have dropped the traditional lists of specific courses that students should take in preparation for medical and instead list the competencies that students will need. First among these is mathematics. Of the seven specific objectives within this competency, six speak of quantitative reasoning and the use of data, statistics, modeling, and logical reasoning. The seventh comes closest to calculus, but what they actually ask for is the ability to “quantify and interpret changes in dynamical systems,” a far cry from the usual calculus course. (For more on this report, see my column on The New Pre-Med Requirements.)

In the influential Vision and Change document crafted by the biological sciences with assistance from AAAS, six core competencies for undergraduate biology education are identified. Two of them are mathematical: quantitative reasoning and the ability to use modeling and simulation. The report goes on to specify that “all students should understand how mathematical and computational tools describe living systems.”

These examples can be multiplied in other client disciplines. What we see is a universal need for students to be able to use mathematical knowledge in the context of their own disciplines. In the case of calculus, the challenge is to understand it as a tool for modeling dynamical systems. This is why calculus is required by so many disciplines. But this is an understanding of calculus that is achieved by very few of our students because their focus has been narrowed down to learning how to solve the particular problems that will be on the next exam.

None of this disconnect between what we teach in calculus and the needs of the client disciplines is new. It now rises to the level of a force that is bringing us to crisis because these client disciplines are themselves under the same increased pressure to have their students succeed. There may have been a time when there was a sufficiently rich pool of potential engineers that we could afford the luxury of allowing the mathematics department to filter out all but the most talented, the ones who would succeed in spite of how we taught them. If it ever existed, that time has passed. Our client disciplines now have higher expectations for what and how we teach their students.

Nothing has driven this point home more clearly than Engage to Excel, the Report to the President from his Council of Advisors on Science and Technology (PCAST). (See my columns On Engaging to Excel, Response to PCAST, and JPBM Presentation to PCAST.) The frustration of the scientists in PCAST with calculus instruction that does not meet the needs of their disciplines is evident in their call for “a national experiment [that] should fund … college mathematics teaching and curricula developed and taught by faculty from mathematics-intensive disciplines other than mathematics, including physics, engineering, and computer science.” (Recommendation 3-1, p. vii)

While there was one particular physicist who was the driver behind this report, it did reflect the concerns of all of PCAST’s members. These are scientists and leaders in technology who deplored the fact that “many college students … often are left with the impression that the field [of mathematics] is dull and unimaginative.” (p. 28)

I have yet to find physicists, engineers, or computer scientists who want to take over our calculus instruction. They have better things to do. But some have been forced to do so, and others are contemplating undertaking it as a necessary correction to mathematical instruction that is not meeting their needs.

This completes my triad of forces that constitute the reason we are at crisis. It is the nature of a crisis that the solution is not readily apparent. Nevertheless, there are actions that can be taken to improve the situation. Next month, I will explore the first of these: drawing on knowledge of best practices for effective teaching and learning.