Mistrust the “must.”
That’s the word that characterizes so many aspects of the math major curriculum that it ought to be the subject of its own course. That course would probably be a required prerequisite of every other course in the program.
Every “must” in a curriculum erects barriers for students. Some of those ramparts are worth manning. Some are not. All of them restrain the flexibility of our programs and narrow the pipeline of potential talent in them. To get where TPSE envisions math programs are going, some of our “musts” will need to become “shoulds,” or even “coulds.”
Because every “must” in a curriculum is the result of a faculty choosing who its students are. To say that all math majors “must” be proficient in the methodologies of abstract algebra is to say “only those who are proficient in abstract algebra may be called math majors.” While I do believe that any person with the interest, time, and discipline to become proficient in abstract algebra can do so (it’s why I choose to make all my video lecture series public on my YouTube channel), the same could be said about a great many subjects that aren’t “musts” in the traditional curriculum. Mustn’t every math major be fluent in probability and statistics? Mustn’t they have proficiency with numerical methods? What about theories of voting and social choice? Technical writing? Who we imagine bachelor’s degree holders in mathematics will be after they graduate is clearly reflected by what our curriculum chooses to require, and the traditional curriculum leaves no doubt but that we imagine they all shall be as we are: advanced degree candidates and future faculty. As with many unreconstructed elements of the academy, the traditional curriculum serves primarily to perpetuate itself.
Which would have been defensible a century ago, when the road into and out of the academy was much narrower (not to mention more elite, far less diverse, and paradoxically far less expensive to travel). Today, the chances that a freshman math major will one day be a faculty member at a research-primary university are vanishingly slim — yet traditional programs remain structured so as to privilege that goal over the many others, the goals the vast majority of graduates ultimately reach instead.
That explains the absence of “musts” that would better address those other goals (such as statistics requirements). But it also explains the presence of “musts” that, by privileging the narrow goal, impair the ability to serve other goals. The requirements we have that crowd out the requirements we might need.
The sacred cows.
Can we have a math major program that doesn’t require abstract algebra? Real analysis? While the PhD-bound trainees will always benefit from both, many math major programs exist that require only one of these (or a choice of one). Likewise differential equations. Even vector calculus appears in some smaller programs as an elective.
These cows, once so sacred, are already less so. It doesn’t mean those courses (in particular) fade away, but it does mean fewer students taking them and more students taking other courses to fulfill their major. They have more flexibility to customize their pathway to their interests and plans. Math majors can come a la carte, not only prix fixe, so long as the meal still provides balanced nutrition.
Can there be a math major without calculus?
Can there be calculus without precalculus?
Looking through a precalculus textbook and realizing that *maybe* 1/5 of the content is actually used in calculus. pic.twitter.com/DkEUPOHTQc
— Robert Talbert (@RobertTalbert) June 15, 2018
Indeed, there are plenty of calculus courses across the academy that aren’t preceded by a course that bills itself as “precalculus.” Most often, these are calculus courses taught to nonmajors. The MAA’s “CRAFTY” subcommittee issued guidelines for college algebra in 2007 that position college algebra courses in the college curriculum as courses that
[…] can serve as a terminal course as well as a prerequisite to courses such as precalculus, statistics, business calculus, finite mathematics, and mathematics for elementary education.
That positions “precalculus” as a necessary prerequisite for “calculus,” presumably, but not for “business calculus.” What most determines whether a student succeeds in their experience with calculus evidently has less to do with precalculus per se than it does with coming equipped with the background that’s important to meet their calculus course’s goals, whatever those goals may be. (And, of course, fluent skills in algebra – which are more predictive of student success in calculus than even their prior exposure to calculus in high school!)
Contained in that fact is a recognition that not every student must learn limits, derivatives, and integrals in the same way, and not every course with “calculus” in its name is created equally. While a mathematician will recognize that calculus is one, the needs of physics majors and accounting majors from the calculus are many.
If calculus can be many, shouldn’t the ways of preparing for calculus be many?
And if calculus can be many, can’t a major in mathematics be many?
If you believe that disciplines are characterized by their ways of thinking and modes of inquiry, rather than by the specific nature of what they think and inquire about, then this answer is obvious. But even the staunchest traditionalist admits that the math major can have some flexibility in its content base, if only in the upper-division courses. Opting for an elective in partial-differential equations over an elective in number theory has long been viewed as a legitimate choice, and even if our primary goal as faculty members were to train the next generation of ourselves, we would want to preserve the ability for students to choose, as we did as undergraduates, to reach out for more breadth or reach down for more depth in the material that will most serve their future needs.
So differentiation in the math major is not a question of kind but of degree. All our programs draw a line between the courses and experiences that we decide all math majors need, and those that students decide they want. That line separates our perspectives as experts in our discipline from students’ perspectives as novices — which, sure, can be fickle and in flux. But it also separates our agency as faculty members who govern the institution and its curriculum, effectively deciding for them what students will learn in our programs, from students’ agency to decide for themselves what parts of our discipline best will serve their needs and plans. That line might just separate our sacred cows from our students’ Next Big Thing.
So, how do we better negotiate that line with everyone’s voices in the conversation — students, faculty, professional organizations, and employers? That’s the tricky part, but there are some promising practices that will be the subject of the next posts in this series.