Is it time to kill calculus?

Math curricula are designed to shepherd students toward calculus. Some mathematicians think this path is outdated

Published September 26, 2020 2:00PM (EDT)

Math formulas are written on the school board  (Getty Images)
Math formulas are written on the school board (Getty Images)

Many parents relish reliving moments from our childhoods through our children, and doing homework with them is its own kind of madeleine. For Steve Levitt of "Freakonomics" fame — who is, in his own words, "someone who uses a lot of math in my everyday life" — a trip down memory lane vis-a-vis math homework became a moment of frustrated incredulity rather than gauzy reverie. "Perhaps the single most important development over the last 50 years has been the rise of data and computers, and yet the curriculum my children were learning seemed to have been air-dropped directly from my own childhood," he told me. "I couldn't see anything different about what they were learning than what I learned, even though the world had transformed completely. And that didn't make sense."

Levitt has made a career of questioning the received dogma. In this case, what he saw was that "A mathematical way of thinking, numeracy, data literacy, is far more important today than it has been; the ability to visualize data, the ability to make sense out of a pile of numbers, has never been more important, but you wouldn't know that from looking at the math curriculum." Data combined with the use of mathematical ideas had transformed the way he and others look at the world. Should data also change the way we teach mathematics?

* * *

In most schools, children are grounded in basic arithmetic in elementary school, and then, somewhere between middle school and high school, force-fed the "algebra-geometry-algebra sandwich". The first year of algebra ("Algebra I") continues to reinforce basic arithmetic, and then brings in fractions. The familiar starts to give way to the unfamiliar when variables and functions are introduced. That's when "x the unknown" makes its first appearance in word problems and linear equations, which for many marks a first sign of confusion rather than buried epistemological treasure.

Things then take a big turn, and math class time-travels to the days of ancient Greece for lessons in formal geometric proofs ("Geometry") that Euclid would have little trouble stepping in to substitute teach. Following that is a yearlong return to algebra ("Algebra II: The Sequel!"), which given the previous year's partial hiatus from x's and y's and numbers first requires a lengthy review and then finally a return to new functions (exponentials, logarithms, polynomials) that either amuse or irritate you, depending on your taste, predilections, and teacher.

For some math stops here. For others there is often an honors track that speeds things up. Increasingly, honors or not, students get to pre-calculus or calculus, which is often revisited in the first year of college, and is the last bit of formal math a person will ever taste. Apologies to the reader for any unhappy flashbacks – or indigestion — incurred.

The sandwich – and actually the entire mathematical meal — has had a long shelf-life. If Levitt felt like his kids were air-dropped into his childhood math classes, odds are this was true for his parents too. The origins of the curriculum go back to a famous 1892 "Committee of Ten" that met at the behest of the National Education Association to standardize public education. Like any good committee their first act was to create more committees – nine to be exact – each tasked with the consideration of a "principal subject which enters into the programmes of secondary schools in the United States and into the requirements for admission to college". Each of these subcommittees then considered "the proper limits of its subject, the best methods of instruction, the most desirable allotment of time for the subject, and the best methods of testing the pupils' attainments therein." Mathematics was one of them. So was Latin and Greek. 

Pre-college mathematics education at that time was like today, composed of arithmetic, geometry, and algebra. The committee had recommendations for each of these areas. The teaching of arithmetic should be "abridged by omitting entirely those subjects which perplex and exhaust the pupil without affording any really valuable mental discipline, and enriched by a greater number of exercises in simple calculation and in the solution of concrete problems." That could stand as a mission statement today for the teaching of any kind of mathematics. "Concrete geometry" would be part of grammar school mathematics and combined with drawing. The committee also recommended that all students, regardless of their aspirations have the same kind of mathematics education up through the first year of algebra. After that, distinctions start to emerge depending on your goals: trigonometry and more algebra for those going to "scientific schools", "commercial arithmetic" for those thinking of a business career.

While "the pupil who solves a difficult problem in brokerage" may still learn some good mathematics,  "The movements of a race horse afford a better model of improving exercise than those of the ox on a treadmill." Setting aside the confusing animal analogy (okay, an ox is slow, but it is really strong and why are you putting it on a treadmill?) the spirit of the metaphor reflects something of a general tension in education: how much do we teach for the world as it is today and how much for the unknown tomorrow? Specific applications or general principles?

This is a tension that is perhaps felt most keenly in the teaching of mathematics and has led to something of a back and forth in mathematics education. Among the most well-known attempts to revamp the mathematics classroom was the move in the 1960s to the "new math", which was a reaction on the part of mathematicians – and some math teachers – that mathematics teaching had become too utilitarian, a curricular decision made decades earlier, at least in part because it was observed that our soldiers in World War II were lacking in basic mathematical skills. The energy behind that revamping of mathematics teaching was the Space Race, initiated by the surprising launch of Sputnik and a perceived "math gap" that would have to be closed in order for the United States maintain international supremacy. Getting people into space and beyond the clouds would mean that we needed to start teaching a kind of math that was already in the clouds. The "New Math" would strip mathematics to its roots, going as far down as basic set theory – Venn diagrams – and rebuild the world of numbers from the ground up.

By most accounts the program was a failure, sacrificing a direct inculcation of basic skills for the goal of exploring highly abstract general concepts, which while not completely disconnected from the day-to-day world of basic arithmetic and problem solving, was about as distant from it as the satellites it was supposed to help launch. Standards-based education and the Common Core followed soon after.

The uneasy relationship between applications and theory in the development of mathematics  curriculum is a reflection of  an ongoing – if slowly healing — rift within the discipline itself. There is pure mathematics and applied mathematics, the former a creation of mathematics for its own sake as opposed to in the service of solving a problem that is troubling someone in the real world. "Applied" is better than "impure," I guess, although it's an adjective that has ugly historical and elitist overtones. To the extent that engineering is a craft, it is a bias that one might trace back to the distinction between the scholar and the craftsperson, the university and the guild or technical school. In truth, a genetic family tree of math would show all kinds of connections and surprising worldly origins of even the "purest" mathematics.

Levitt's call for a mathematics curriculum centered around data is not born of anti-intellectualism. He is quick to point out that he is not "anti-math", rather that he is "anti-math-that-no-one-will-ever-use-in-the-first-place," at least in the classroom. As someone who loves math, he worries that a mathematics curriculum not connected to data and computers "runs the risk of being demoted." "If the best arguments  for mathematics is that it's part of the education of a well-rounded citizen and that it's good for brain development," that may very well be the undoing of math.

By his own admission, he is a Johnny-come-lately to the challenge of curriculum reform, but his academic star-power has enabled him to attract important and influential players to his mission of bringing data and computing to mathematics education. Levitt quickly stood up a small advisory committee of like-minded people that included statistics celebrity Nate Silver of, former Secretary of Education Arne Duncan, and former Google CEO Eric Schmidt.

Levitt's most important recruit may be Jo Boaler, the Nomellini & Olivier Professor of Education at Stanford. Boaler also directs YouCubed, a non-profit whose mission is to "inspire, educate and empower teachers of mathematics." Her work on mathematics education is widely cited and influential. The call she received from Levitt came at a perfect time, as she is currently working with California's Department of Education to revamp the mathematics curriculum around data.

For Boaler, the sclerotic nature of the mathematics curriculum is above all an equity issue, and for that she places calculus at the center. As Boaler points out, mathematics is usually the only subject in which kids – usually 6th graders – take a placement exam in middle school, the result of which sets them on their academic pathway through high school, on track – usually an "honors" track – to take calculus junior year of high school.

The curriculum as calculus funnel and an honors track to speed one's ride has downstream effects. My own home institution Dartmouth College is almost surely not an outlier among its peer group in that, while calculus is not required for admission, you would be hard-pressed to find a student here who didn't take calculus in high school. It's only the students who have made it through the initial placement – in sixth grade! – who have the ability to show colleges that they can succeed in calculus. For Boaler, calculus is a linchpin, not in and of itself, but because of the influence it has both on the curriculum that precedes it as well as its influence on students' college prospects moving forward.

Despite what some may think, calculus didn't end up as the last stop on the math track just to create a final hurdle for high schoolers. It was for many years, the most applicable math – outside of arithmetic – that you could find. It continues to be of great importance in all kinds of applied contexts, from medicine to engineering to finance, where modeling change – usually in the form of a "differential equation" – is crucial. It is mainly useful in continuous contexts (think fluids or asteroids in motion) and powerful for finding "analytic solutions" (formulas) that quantify a phenomenon indirectly encoded in a differential equation. All of this is still true, except what happened is that many new and interesting phenomena also started to be represented by numbers – i.e., the data revolution occurred. This didn't mean calculus became irrelevant, rather other important possibilities arose for mathematical thinking and learning and teaching.

Boaler calls calculus a "horrible and inequitable filter." Some of the inequity is around gender – placement testing preferences boys over girls, a finding that may be something of a surprise to many. Equity issues may also be redounding to the academy. A calculus-successful student body may very well be contributing to an over-representation in STEM in its entering classes, or rather that an underrepresentation or under-cultivation of humanities interests. Students have only so much time to take classes in high school and only so much energy. A history-interested student may very well be taking yet another difficult math course instead of another history or government or art course simply because she knows – or believes – that she has to wrap her head around calculus, which doesn't have great tangencies to her intellectual passions.

"Data science" is a name invented to distinguish the ideas and approaches used to analyze the new diversity and quantities of data that characterize modern data from those of classical statistics.  Done well, it is an integration of critical thinking and quantitative skills, storytelling with and through numbers, supported by evidence. It has strong connections to the humanities, both in spirit and practice, as many interesting kinds of data analyses are regularly performed on information derived from humanities subjects, often in digital humanities programs. The humanities context is reflexive, too: it's no coincidence that the new important work now being done in data bias came out of digital humanities programs. I'd wager that a student who excels at and enjoys data science is more likely to also have interest in and see beauty in the activity of close-reading a text, or image, or artifact, or working in the humanities more generally.

Data science is also highly collaborative, which a good deal of research shows is a working style where girls (and women) excel. Some of the same studies that show boys outperforming girls on timed tests, show girls outperforming boys when the tests have a collaborative framing. If data science were a part of a high school curriculum it could provide a mechanism for girls to show their quantitative skills and it also could be a boost to humanities programs as well. It would in short allow more students to showcase their talents to colleges in ways that could benefit both students and colleges.

* * *

Last March, Boaler and Levitt convened a Data Science Summit at Stanford. What was originally supposed to be a small working group soon mushroomed into a large meeting of more than fifty people that included well known mathematics educators, representatives from industry, and mathematicians. By the end of the Stanford meeting, rather than being energized by the day, Levitt was depressed. There were all these smart people in the room, committed to the idea of changing the curriculum, but all having different ideas. "It didn't seem like something that would happen in my lifetime," he says.

In fact, there already has been some substantial progress made in bringing data science to the schools. Notable is the Introduction to Data Science course that was co-developed by UCLA and the Los Angeles Unified School District and is already being rolled out in 15 southern California School Districts. In addition to a curriculum there is a professional development arm to help teachers acquire the skills to teach it.

In addition to explicit materials and courses like this, data is also making its way into the curriculum in more subtle ways. As part of thoughtful redesign twenty percent of the SAT now tests the ability of prospective college students to understand data, both in the quantitative and verbal parts of the test, the latter of which includes data in the reading comprehension piece. This is all part of concerted effort by the College Board CEO David Coleman to make the SAT more relevant to what is actually being taught in the "average" first year of college. Even more, from Coleman's perspective, if kids were going to be studying for the SAT, then that studying should be worthwhile even beyond its relevance to college admissions. Data literacy is a part of that mission.

The recognition of the centrality of data is also a part of a next generation of AP courses. The new AP Biology course has been redesigned to have a significant data analysis component. The AP footprint in computer science has been expanded to include an AP "Principles of Computer Science" course that focuses on data science and as such provides context for the a next programming course. Coleman is especially proud of the Principles course, as it has proved to be a gateway course for computing that is especially attractive to demographics that historically have been under-represented in computer science. Since its roll-out in 2016, the numbers of female, black, LatinX, and rural examinees have grown by 136%, 121%, 125% and 117% respectively. And in the first year after the Principles course was made available, enrollments in the well-known AP programming course doubled, with attendant and sustained increases in each of these populations in subsequent years. "the changes in who is doing computer science is something I'm really proud of," Coleman says. For Coleman, data science is a pathway to STEM diversity.

Another piece is a new initiative of Coleman's that the College Board is calling its "pre-AP curriculum". This fall "Pre-AP Geometry with Statistics" is being piloted around the country. It is a quarter the basics of data science with the rest basic geometry. The bridging conceit that both are contexts for deductive reasoning and the course joins the certainties of deduction with the probabilities of data science. A new Pre-AP Algebra II course will also have data analysis connections inserted through the appearance of functions with more than one variable.

Work like that being done at the College Board and other places does give Levitt some hope that math curricula will change — if not for his kids, then at least for his kids' kids. What Levitt, Boaler, and many others support – possibly as a short-term fix, but at least as a step forward – is the idea of streamlining the current curriculum. While it's not exactly as simple as "cutting two textbooks in half and gluing them together to make a new course," as Levitt says, there is something to that. It's the kind of thing that he and others could imagine organizing a group of mathematicians, and data scientists around to find a way to remove a year from the AGA sandwich.

A newly streamlined curriculum would then give space for a year of data exploration and integration of computing, maybe even more mathematics exploration – again, assisted by computing. A modification may also may put less strain on any requisite teacher training than a complete rewrite. It leaves open the possibility of a math curriculum that would be relevant for all the students, with a branchpoint that would depend upon interests: algebra, geometry/algebra+data followed either data science or calculus, or both! It would be better connected – maybe with the help of ideas from the new pre-AP courses – to the overall curriculum and in that, also possibly serve the purpose of getting more kids with a range of interests and abilities interested in mathematics. The devil is in the details, but this is the kind of near-term and seemingly achievable goal that Levitt, Boaler and others are now working toward.


By Daniel Rockmore

Dan Rockmore is the William H. Neukom 1964 Distinguished Professor of Computational Science at Dartmouth College, and an External Professor at the Santa Fe Institute.  In addition to his technical work he has a broad portfolio of science outreach work. He is the co-producer of the mathematics documentary The Math Life as well as three other  mathematics/computer science documentaries.  His writing for the general public has appeared in The New Yorker, Slate, and the Atlantic. His most recent books are two edited volumes, "What are the Arts and Sciences? A Guide for the Curious" (UPNE), and "Law as Data: Computation, Text, and the Future of Legal Analysis" (Santa Fe Institute Press), co-edited with Professor Michael Livermore of the UVA School of Law.

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