Wednesday, July 1, 2015

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.


  1. Wright State started a program (now spread to over 50 universities) in which engineering faculty teach a "math for engineering applications" course that students take before calculus. 89% of students who took that course got a C or better in Calculus compared to 60% who did not:
    They don't replace math department courses (or placement exams), but supplement/augment them.

    One root of the problem is transfer (or lack thereof). What the math students learn isn't transferring to physics, let alone engineering. Situated cognition/learning approaches help address this, by teaching concepts in context. This includes authentic learning techniques like problem-based learning, service learning, simulations, and the like.

  2. The following comments are a reaction to the three columns about Calculus at Crisis.

    a. How do you define marginally qualified and marginally prepared, and how are these attributes measured? One constant of my years (over 40 years) of teaching was the view of colleagues in disciplines from English, Economics, Physics, Mathematics, etc. that students in their classes were not "prepared."

    b. Mathematics has grown in importance as a tool for progress in so many disciplines, not only traditional ones such as Engineering, Physics, and Economics but others such as Business, Computer Science, and Biology. It is not clear to me that the mathematics college teaching community has kept apace of this reality. Not only does one hear that there is little room in the curriculum of courses in Calculus, discrete mathematics, and linear algebra for the diverse applications of these subjects, but there is even little "lip service" to putting the theory and applicability of mathematics on an equal footing in terms of the exposure that mathematics majors have to these two "poles" of mathematics. In recent years I know of examples where Computer Science departments are talking about or have taken back the teaching of discrete mathematics and linear algebra from the mathematics department at their school. For example, I have heard computer scientists complain that the linear algebra that their students see taught in the mathematics department makes no mention of finite fields or even the field of two elements, and emphasizes linear algebra over the complex numbers. Many mathematics courses are taught outside of mathematics departments (including those with Calculus content) because applicability of mathematics in mathematics department courses is rarely emphasized. The way people who get doctorates in mathematics are educated is such that they rarely see a broad range of applications of mathematics.

    c. Many mathophobic parents who want their children to go to elite/highly selective colleges encourage their kids to take Calculus in high school. And many high school students feel they would rather try to get AP credit for some mathematics course in the "protective" environment of high school rather than take mathematics in college.