A proposal for rethinking calculus class in the face of computational tools, originally developed in our work with XQ Math
Being a mathematician is no more definable as "knowing" a set of mathematical facts than being a poet is definable as knowing a set of linguistic facts. Some modern math ed reformers will give this statement a too easy assent with the comment: "Yes, they must understand, not merely know.” But this misses the capital point that being a mathematician, again like being a poet, or a composer or an engineer, means doing, rather than knowing or understanding. This essay is an attempt to explore some ways in which one might be able to put children in a better position to do mathematics rather than merely to learn about it.
— Seymour Papert, “Teaching Children to Be Mathematicians vs. Teaching About Mathematics” (1971)
In many schools today, the phrase "computer-aided instruction" means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer and, in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building.
— Seymour Papert, Mindstorms (1980)
Consider $\sqrt{x}$. You can think of a square root as “the number which, if multiplied by itself, yields $x$”, much as you can think of $i$ as being defined as “the number which, if multiplied by itself, yields $-1$”.
But that does not tell us how to compute a square root. There are a wide variety of algorithms for doing so, but one of the oldest is Heron’s method:
i.e. in general, let $g_{n+1}$ be the average of $g_n$ and $\frac{x}{g_n}$, computing successive $g_n$ until you get the level of numerical precision desired.
e.g. Let’s say you want to know the square root of 127, and your initial guess is that it’s near 20. This method would proceed something like this:
Almost immediately, the answer converges to $\sqrt{127} \approx 11.26942767$.
Needless to say, vanishingly few people who use square roots (even amongst those who say they understand square roots) know this (or any other) practical algorithm.
Why?
Because there are square root buttons within just a few seconds’ access of nearly everyone.
In other words, knowing how to calculate a quantity can be separated from understanding that quantity (and from using that quantity), provided convenient, accurate, computational representations.
This observation sits at the core of this essay, which aims to sketch out a case for translating this observation to calculus class.
Before getting ahead of ourselves though: Why does calculus class need help in the first place? Years of research and practical experience have established three, primary issues:
Most students who pass calculus don’t understand it and can’t apply it ⇒ From basic questions like “Is $\frac{dy}{dx}$ a fraction?” (Uygur & Ozdas, 2005) to the notion of limits (Tall & Katz, 2014) to the idea of integrals as the limit of a Riemann sum (Thompson & Silverman, 2008), most students leave their single variable calculus classes with a tenuous grasp on the mathematical fundamentals. And worse, even in introductory classes which are often the source of calculus problems (e.g. kinematics), students consistently fail to apply calculus concepts correctly (Muzangwa & Chifamba, 2012). People simply lack the conceptual understanding and ability to apply core calculus concepts, even when they might have demonstrated procedural fluency at one point in time.
Most calculus class content isn’t used by students in future coursework and careers ⇒ Most courses requiring calculus do not use the bulk of what’s covered in calculus class. They may use derivatives and integrals, but these applications almost certainly either focus on systems which are:
There are exceptions, but they are tiny—e.g. physics or mathematics—but they total barely 1% of all two- and four-year degree students. (Of course, it is reasonable to suggest that perhaps the under-enrollment in these programs is due in part to calculus’s issues!)
Calculus acts as a STEM gatekeeper, especially for the historically marginalized ⇒ Setting aside calculus’s efficacy and alignment with future coursework and careers, we also know that calculus acts as one of the most severe “filter courses” for STEM enrollment, with a pronounced effect on historically marginalized communities (Bressoud, 2021).
Taken together, this suggests that we are sacrificing swaths of potential STEM students in return for comparatively little: a weak grasp on knowledge that will go largely unused.
If we are to understand calculus’s problems (or develop possible solutions), it is important to get precise about what, exactly, we are trying to accomplish with calculus in our nation’s schools.
This requires that we differentiate calculus the class from calculus the domain from calculus the prerequisite.
Before anything else, calculus is a domain, i.e. it is a mathematical activity: the study of change. This domain has given rise to calculus the class, i.e. a collection of topics and techniques covered in high schools today, with little variation in scope and sequence. Which in turn has given rise to calculus the prerequisite, i.e. a link in a chain of progressions through mathematics. For most, this chain ends at calculus. For others, it continues on through physics or analysis or mechanical engineering.
For the vast majority of students in the United States, calculus the class looks something like this:
That is, you begin by exploring limits and continuity, you develop the derivative and apply it to more complex functions and optimization problems, and eventually you introduce integration and perhaps the basics of differential equations.
As a thought experiment, let’s return to the square root with which we began this essay and attempt to bring a similar lens to calculus: Imagine you had a Differentiate
button which could take a derivative, and an Integrate
button which could evaluate both definite and indefinite integrals. Imagine these buttons:
What aspects of the calculus curriculum would go the way of Heron’s method? i.e. which would we drop as implementation details not core to people’s conceptual understanding or ability to apply those concepts?
Reasonable people can disagree on exactly which units might be wholly or partially elided, but here is one, informed suggestion for which might remain after such a transformation:
Notice nearly two-thirds of the units disappear; what remains is barely one-third of the total topics in AP Calculus AB’s original scope and sequence. At this point, whether it’s one-third or one-half seems like quibbling: Such Integrate
and Differentiate
buttons would dramatically open up the design space for approaching calculus.
If we could open up that design space, scrap the traditional calculus class, and begin anew, where should our design process begin? Presumably, the mathematical domain: What is the heart of “calculus”, and what tools and techniques are useful and accessible enough to merit inclusion in a broadly-targeted, introductory course?
Mathematically, there is one, fundamental idea in elementary calculus: the way a function behaves locally (e.g. its local curvature) and its global behavior (e.g. its overall shape, its long-term behavior, etc.) are intimately linked in both directions: i.e. the overall shape of a function is linked to its local curvature, the region enclosed by a function is linked to its overall shape, and therefore a function’s local curvature and the region it encloses are tied together.
This is a consequence of the astonishing fact that anti-differentiation is equivalent to the limit of approximating sums. This is of course the aptly named “fundamental theorem of calculus.” But how many calculus students see the mathematical import behind the strange fact that area and rates of change are so tightly tied together? How many could speak confidently to the local/global relationship, or even to anything deeper than the idea that “integration undoes differentiation” (or vice versa)?
Practically, the primary value of elementary calculus—the value that will show up reading epidemiology news or studying climate change—comes from becoming comfortable with thinking in terms of (a) local rates and total accumulations of change, and (b) extrema (e.g. minima, maxima, and optima).
As a technical matter, establishing (and then applying) these mathematical and practical tools requires some machinery, the crux of which are fundamental items in elementary calculus’s payload:
But despite the fact that the majority of the appeal (and the future use, such as it is) of calculus is its practical application, nearly everything else in the mathematics of elementary calculus concerns either:
Vanishingly few (less than one half of one percent) of two- and four year college students major in mathematics or statistics. Meaning something on that order will come to need those bona fides or clever calculation techniques. For everyone else, they will fall by the wayside.
Calculus is not only a domain and a class, but by virtue of being a class, it is also a prerequisite, a link in a chain leading onto further studies.
What are these studies?
This can be (and has been, e.g. Tyson, 2011) answered in terms of which classes people take, fields they major in, and jobs they pursue. But it is easy to miss the forest for the trees in doing so. In practice, for the vast majority of students, elementary calculus leads to one of three contexts for substantive application, in increasing order of incidence and applicability:
In mathematics, of course, the bona fides and computational techniques introduced in elementary calculus are valuable. In physics, the bona fides are slightly less valuable for most, but the computational techniques remain essential. And in domain specific courses, barely at all. In most fields, there is a small set of common patterns with whose solutions students quickly familiarize themselves, and the vast majority of real-world systems are either resistant to closed form solutions (e.g. the Navier-Stokes equations) or require numerical treatment to actually use (e.g. in PID controllers).
In other words, in elementary calculus we have a prerequisite where the bulk of material is targeted to the smallest slice of students’ future use cases.
Now we know—or at least have a rough idea of—what calculus is, and that elementary calculus class mostly doesn’t work (and incidentally, isn’t that useful to most people). But why doesn’t it work?
Over the years, researchers have carefully catalogued calculus students’ most common and fundamental challenges. The bulk of these challenges come back to the notion of a limit, and specifically ****seeing integrals as limits of [area] sums and derivatives as limits of ratios.
The challenge limits pose for novices hinges on the fact that though a limit is often talked about as a dynamic process (e.g. “As $x$ approaches $\infty$…”) they are not simply processes which “collapse” to a given value, in crucial ways (e.g. in the application of L'Hôpital's rule).
This challenge is a specific instance of a much more fundamental challenge as well, wherein students often have trouble seeing functions (and operators like derivatives and integrals) both as processes (i.e. a dynamic, mathematical object varying with dependent parameters, giving the local characteristics) and objects (i.e. monolithic objects having global characteristics).
This process/object distinction is fundamental to the pedagogy of mathematics (cf. Sfard, 1991). Repeatedly throughout human development, we learn to take a process (like counting) and abstract and package this process as a manipulable object (like numbers) in an act called reification. This ongoing cycle is central to building mathematics’ deeply nested conceptual structures.
Given how central limits are to calculus students’ challenges, it would be natural to turn to limits and examine them more closely, trying to understand how to better introduce them and to help students understand them as mathematical objects in their own right.
But that’s a red herring.
To see why, look at the historical sequence of the development of the four big ideas of calculus,
In doing so, we find something surprising: The historical development of ideas in calculus seems to be nearly the reverse of the order of introduction of concepts in modern, elementary calculus (which typically sees students being introduced to limits, then differentiation, then integration, and finally series).
Why?
Modern elementary calculus emulated the wrong period in history: In 1821, Cauchy proved that the limit of a sequence of continuous functions is continuous. In 1826, Abel produced a counterexample using Fourier series. Research mathematicians like Cauchy and Weierstrass then prototyped the (ε, δ)-definition of a limit and related machinery required to resolve this crisis and put elementary calculus on solid, theoretical foundations. This led to modern analysis, which was the kernel from which modern elementary calculus grew. While this machinery is certainly suited to establishing the mathematical rigor of calculus, it has little to do with the conceptual underpinnings or practical applications of elementary calculus.
This is not just a matter of historical pedantry: We know from work of those like Piaget (cf. Piaget, 1989) that the historical growth of ideas in mathematics often roughly parallels the developmental growth of ideas within an individual, and we see those conclusions mirrored in modern research on misunderstanding and threshold concepts in calculus (e.g. Scheja & Pettersson, 2010; Meyer & Land, 2006).
If the order in which concepts is introduced is part of the problem, why not just focus on that? Some have (e.g. Bressoud, 2019), and we’re confident that this approach offers an improvement to the median curriculum. But there is a much bigger opportunity for transforming how people learn calculus.
Limits are specific to calculus, of course. But elementary calculus’s problems run deeper than that because calculus is a math class, and math class has a few, structural problems too, the primary symptoms of which are:
We see the deeper causes of this as a twofold failure of authenticity:
These failures are interrelated: Without real, motivating work, we have to create artificial work. That artificial work, invariably, lacks the necessary subtlety, complexity, and texture to support the multiple perspectives required by real understanding. Bill Thurston, the famed low-dimensional topologist wrote (Thurston, 1994):
People have very different ways of understanding particular pieces of mathematics. To illustrate this, it is best to take an example that practicing mathematicians understand in multiple ways, but that we see our students struggling with. The derivative of a function fits well. The derivative can be thought of as:
- Infinitesimal ⇒ the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
- Symbolic ⇒ the derivative of $x^n$ is $nx^{n-1}$, the derivative of $\sin(x)$ is $\cos(x)$, the derivative of $f \circ g$ is $f^{\prime} \circ g * g^{\prime}$, etc.
- Logical ⇒ $f^\prime(x) = d$ if and only if for every $\epsilon$ there is a $\delta$ such that when $0 < |\Delta x| < \delta$,
$\left |\frac{f(x+\Delta x)-f(x)}{\Delta x} - d \right| < \delta$
- Geometric ⇒ the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
- Rate ⇒ the instantaneous speed of $f(t)$, when $t$ is time.
- Approximation ⇒ The derivative of a function is the best linear approximation to the function near a point.
- Microscopic ⇒ The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.
This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions. Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions.
I can remember absorbing each of these concepts as something new and interesting, and spending a good deal of mental time and effort digesting and practicing with each, reconciling it with the others. I also remember coming back to revisit these different concepts later with added meaning and understanding.
The list continues; there is no reason for it ever to stop. A sample entry further down the list may help illustrate this. We may think we know all there is to say about a certain subject, but new insights are around the corner. Furthermore, one person's clear mental image is another person's intimidation:
- The derivative of a real-valued function $f$ in a domain $D$ is the Lagrangian section of the cotangent bundle $T^{*}(D)$ that gives the connection form for the unique flat connection on the trivial $\mathbb{R}$-bundle $D \times \mathbb{R}$ for which the graph of $f$ is parallel.
These differences are not just a curiosity.
Most word problems and worksheets which most people spend their time differentiating and integrating are simply not real, meaningful, or complex enough to cultivate the multiple ways of understanding and problem solving judgment required for deep conceptual understanding or fluid capacity for application.
So far, we’ve spent a lot of time understanding what’s the matter with calculus. This is because it is important to be precise about the problem to understand whether a possible solution is viable.
We believe computational tools and perspectives offer such an opportunity. To understand that opportunity, we need to return to the conversation about square roots with which we began this essay and its moral: understanding and application can be separated from knowing how to calculate. (This is the converse of our earlier observation: one can—and too many do—calculate a quantity without understanding or being able to apply it.) This separation is only practical when you have access to convenient, accurate representations.
When we talk about computational tools, we have in mind three varieties:
A1
is always twice A2
”)These distinctions (declarative v. imperative v. function and numeric v. symbolic) are rarely absolute. They represent dimensions of design which different tools afford and emphasize to different degrees. e.g. One can write functional (and with some work, symbolic) code in Python just as Mathematica supports numerical solutions. These distinctions are highlighted here because they are the dimensions which are most salient to understanding theses tools’ possible contribution to learning elementary calculus.
These families of tools offer complementary strengths uniquely suited to the skills and concepts of elementary calculus.
Declarative tools allow people to begin working with the differences and sums which are numerical precursors to the derivative and integral, but with real numbers, unobscured by variables. How many times have you written a spreadsheet formula where a cell is a constant delta relative to one above it? And linked that to a change in another quantity? In doing so, you just constructed the rudiments of $\frac{f(x+\Delta x)-f(x)}{\Delta x}$, thinking about a quantity $x$, and how some other quantity changes when $x$ changes "just a little bit", creating discrete derivatives and integrals as simply as this:
Imperative tools allow people to repeatedly cross the boundary between process and object, writing (or taking apart) a function and alternately seeing it as a process (executed line-by-line) and an object (an abstraction which can be composed with other such abstractions). Imagine basic integration of Newtonian mechanics (for, e.g., a simple video game). One might naively implement something like leapfrog integration, e.g. in one dimension:
Or imagine a PID controller in a robotics project controlling the depth of a submarine with a ballast:
And symbolic tools allow people to:
Collectively, these tools offer us exactly the Differentiate
and Integrate
buttons we imagined earlier while going directly to the heart of calculus’s challenges in two, primary ways:
These tools’ strengths go deeper as well, opening opportunities to tackle some of the more fundamental challenges elementary calculus [class] has inherited from math classes writ large.
In practice, the appeal of these kinds of tools and approaches is simple: They make it possible to spend a great deal of time and energy developing a conceptual and practical, application-oriented understanding of differentiation and integration while minimizing the symbolic manipulation and bookkeeping required to do so. But realizing this possibility will require dedicated work creating (not simply retrofitting) tools, materials, and curricula (and transforming both the preparation of educators and the evaluation of students).
It is worth pausing to note that neither this opportunity nor its recognition and exploration are new. There is a long history of work recognizing and prototyping the possibilities of these kinds of computational approaches ranging from SimCalc (Roschelle, 2012) to Boxer (diSessa, 1995) to LOGO (Papert, 1979) to Mathpert (Beeson, 1998). Spreadsheets, programming, and computer algebra systems have made regular appearances in calculus education research for almost a half-century. None of these will be novel to any researcher’s ears. What has changed is the degree of availability, affordability, and capacity of computational tools and the concomitant increase in the widely-acknowledged importance of computational fluencies. There is now an opportunity to revisit these conversations with an unprecedented degree of concreteness, cultural openness, and political buy-in.
More ambitious interpretations of this situation lead to visions redefining and rethinking calculus entirely. The renaissance in machine learning naturally raises the possibility of centering linear algebra and gradient descent: What would prevent an introduction to calculus contextualized by deep learning? Or perhaps differentiation and integration should become simply tools in a simulation toolbox in a course featuring techniques ranging from the Monte Carlo to finite difference methods. Perhaps the core, conceptual ideas of calculus should be distributed across advanced science courses, aided by computational tools. There’s nothing preventing a calculus-based physics, mathematical biology, or physical chemistry course from supplanting calculus’s role as a terminal mathematics course. Or maybe a serious robotics course emphasizing control theory could take its place?
Less ambitious interpretations of this situation might lead to simply redesigning and reconsidering the on-ramp to calculus. What if we were to replace precalculus and Algebra II with a computational calculus course? One where learners had the opportunity to spend an entire year using and exploring and grasping the conceptual core of calculus before digging into its mathematical bona fides and specific techniques for calculation? Or what if each unit within a traditional calculus class were redesigned to reflect that progression, beginning with computational explorations which were then enriched by more comprehensive mathematical treatments?
Wherever one falls on the spectrum of ambition, it is worth keeping the complaints of Gilbert Strang—author of one of the more famous calculus textbooks—in mind. All the way back in 2001 (Strang, 2001), Strang wrote:
Calculus I, Calculus II, Calculus III—what an imbalance in our teaching! All the rest of mathematics is overwhelmed by calculus. The next course might be differential equations (more derivatives), and the previous course is probably precalculus. I really think it is our job to adjust this balance, we cannot expect others to do it.
Strang’s preferred alternative focus is linear algebra, but the broader point remains: calculus probably takes up too much space in the curriculum (both in absolute terms, and relative to the uses and purposes which motivated its inclusion), and curricular innovation can reclaim that space. Retaining the spirit—if not the letter—of elementary calculus means returning to the domain and asking afresh, “What does the mathematical study of continuous change entail?” and secondarily, “What might learners comfortable with algebra and geometry be able to do with the powerful ideas of that study?”
Seymour Papert is best known for his role in developing the programming language LOGO and the associated educational philosophy and movement, constructionism. Papert spent his life exploring the potential for the computer to transform how we learn and think (and our understanding of how we learn and think). Papert did this both in the lab and in practice, working across research, education, and policy over his career.
This essay is much less ambitious than Papert’s work, but has been written in a similar spirit. We hope we have made the beginnings of a constructive case for the opportunity for computational tools and thinking to transform calculus.
Our thesis is simple. Calculus the domain is full of powerful ideas. Calculus the class is burdened by historical detritus preoccupied with calculation. By dropping this baggage (and similar distractions in the traditional Algebra II/Precalculus sequence), we can secure a precious opportunity to rethink calculus (or at least the on-ramp to calculus). Computational perspectives allow us to drop this baggage by delegating most of the work of calculation while enabling a narrower focus on the conceptual underpinning and practical applications of calculus’s most powerful ideas, all while centering increasingly salient computational tools and perspectives.
But however simple that thesis may be, implementing it is a separate matter entirely. In place of an all-encompassing conclusion, we’d like to end with a warning, also due to Papert. Reacting to the growing salience of computers in education, in 1996, he wrote a parable (Papert, 1996).
Imagine that writing has just been invented in Foobar, a country that has managed to develop a highly sophisticated culture of poetry, philosophy and science using entirely oral means of expression. It occurs to imaginative educators that the new technology of pencils, paper and printing could have a beneficial effect on the schools of the country. Many suggestions are made. The most radical is to provide all teachers and children with pencils, paper and books and suspend regular classes for six months while everyone learns the new art of reading and writing. The more cautious plans propose starting slowly and seeing how "pencil-learning" works on a small scale before doing anything really drastic. In the end, Foobarian politicians being what they are, a cautious plan is announced with radical fanfare: Within four years a pencil and a pad of paper will be placed in every single classroom of the country so that every child, rich or poor, will have access to the new knowledge technology. Meantime the educational psychologists stand by to measure the impact of pencils on learning.
Reflecting on having deployed this parable in talks and papers and advocacy over the years, Papert goes on to say,
I first used this parable in the early days of computers to warn against basing negative conclusions about computers on observations about what happens when computers are used in a manner analogous to that pencil experiment. At that time I ended the story with something like "And not surprisingly, the Foobarians concluded that pencils do not contribute to better learning." Subsequent events have indeed shown my fears to be well-founded: Conclusions of a Foobarian kind have in fact slipped into the accepted wisdom of American educators. For example, educational experiments in which children's access to computers and to computer culture was far short of what would be needed to learn programming have been accepted as proof that programming computers is not an educationally valuable experience for children. But in telling the Foobar story today I would give it another, even more insidious, ending.
In fact what I now understand that the Foobarian educators would actually do is not reject the pencil but appropriate it by finding trivial uses of the pencil that could be carried out within their meager resources and that would require minimal change in their old ways of doing things. For example they might continue their oral methods of doing chemistry but use the pencils to keep grade sheets. Or they might develop a course in "pencil literacy" which would include learning what pencils are made of, how to sharpen them and perhaps how to sign one's name.
We should be careful not to make Foobar’s mistakes. There are good reasons to introduce changes incrementally in many educational environments. But that does not mean there should be no place where more dramatic experiments are undertaken. The computer is a powerful, intellectual tool which can transform the character of intellectual work both students and teachers undertake, if we let it.
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