Calculus Without Fear

Artificial intelligence is becoming too important for people not to appreciate its core mathematical foundations in calculus, probability theory, and linear algebra. Most advanced countries are regearing their secondary schools to introduce these topics to teenagers. The US is not, largely because its educators are obsessed with remedial arithmetic. This needs to change. Calculus and scientific coding (e.g., Python with NumPy and Matplotlib) should be standard fare for American 10th graders, just after they complete Algebra 1. Early or fast learners can tackle it in 9th grade. Late or slow learners might wait until 11th grade and need two years instead of one to complete the core sequence.

Why focus on secondary school? Because that is where American math education is particularly deficient. Too few graduates can handle STEM majors in colleges because the math or math-like reasoning is too intimidating. While most colleges offer remedial math, this often delays graduation for so long that students give up on STEM so as to avoid stigmatization and extra costs. Since STEM jobs are substantially better-paying on average than non-STEM jobs— reflecting both the increasingly sophisticated needs of the economy and the relative dearth of STEM-competent graduates—math deficiencies in high school drag down US economic growth and stoke further inequality.

Why calculus? Because it is the Swiss Army knife of higher math. It is key to analyzing gradual change and limits of random processes. Calculus makes it easy to solve a host of problems. But it takes time to get comfortable with the general approach, namely, that of decomposing complex phenomena into an infinite host of simpler parts, analyzing each of the parts, and then reaggregating these analyses for insights into the whole.

Calculus has a bad reputation in American schools, largely because of the triple-D way in which it is taught: dry, dull, and daunting. Trying to defer the pain, schools typically insert two or three years of courses between first-year algebra and calculus: e.g., geometry, second-year algebra, and pre-calculus. But that isn’t necessary. Integrals are areas under curves; Archimedes figured that out over two millennia ago without needing algebra. Derivatives are slopes of curves, which does require some algebra. Yet even simple quadratics provide a decent foundation.

The only conceptual challenge is to grasp the notions of sums or ratios of miniscule quantities. But that’s not inherently more difficult than the challenge of imagining a whole number that is distinct from what it counts, or a zero or negative number that can’t be counted, or fractions as ratios of two numbers, or real numbers that aren’t ratios, or imaginary numbers that aren’t real. In fact, most of what we call precalculus—infinite series, exponentials, logarithms, and trigonometric functions—is better grasped by teaching calculus first.

Why don’t we reverse the order already? Inertia. Trigonometry greatly facilitates surveying and navigation. Logarithms simplify long error-prone arithmetic. The calculus connections, which were not discovered until later, had little bearing on the speed and accuracy of the human computers we needed to train. In contrast, now that we rely on electronic computation, developing an intuitive grasp of the methodology—why to take that approach, how it works, and when it can fail—is far more important than our proficiency in emulation. Here calculus shines, as few subjects offer such profound insights so cheaply.

Granted, without attempts at emulation, most people find it hard to grasp the concepts. That’s where scientific coding comes in. Have students teach computers to do their calculus homework for them. Have the computer calculate discrete approximations to slopes and areas over a broad grid and then narrow the grid spacing to make the approximations converge. Have the computer model movements of basketballs or satellites subject to gravity and track the parabolas, ellipses, and hyperbolas that emerge. Program computers to differentiate and integrate without calling on the built-in functions. In the process, students can master programming skills that help them understand how computers solve problems, improve their own problem-solving skills, and expand their options for STEM careers.

Best of all, it is easy to teach this material in triple-E ways: engaging, exciting, and entertaining. There are many ways to do so and there’s no need to declare one of them best. Instead, give schools, teachers, parents, and students more options between vendors, or mix and match. Monitor the results, publicize them, and let competition select for the best. That works for sports, where many American secondary schools excel. It can work for math too.

Image: Adobe Stock


  • Kent Osband

    Kent Osband holds a Ph.D. in economics from the University of California, Berkeley, worked in international finance, and is the author of "Calculus for the Curious."

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10 thoughts on “Calculus Without Fear

  1. As a professional scientist, I can attest that most of the value of calculus stems from its concepts: rates of change of a variable and sums of a quantity as it changes. These concepts are simpler than solving second-year algebra problems and far more important the trig identities. By the way, the simple concept of cosine and sine can be taught earlier too, and are so valuable. All of this can be taught to young people 13 to 15 years old.

  2. I couldn’t agree with your position more. In fact, I would strongly urge to start even earlier than high-school. As your book “Calculus for the Curious” illustrates, you can “dress” complex concepts such as breaking down curves into smaller and smaller bits of straight lines at kindergarten level, and build on it step by step such as adding the bits (Integration and differentiation); it’s all nothing more than counting. Honestly, it is not until Group theory that math becomes anything other than clever methods of counting and keeping track of otherwise complicated addition problems.

    Keep up the good work. The country needs self-less, optimistic advocates like yourself to get past the disillusionment created by the idiocracy that rules the current day, and educate the youth despite forces of new-age religions fiercely exerting against numeracy and reason.

  3. Great article! I feel a bit cheated by existing US k-12 math teaching paradigms. I went through a significant portion of my educational endeavors assuming that I just didn’t have “it”, an innate intuitive understanding of math. I learned later on that with a bit of hard work math intuition can be built, and proud to say I love math now – on my path to taking higher level mathematic courses on the side.

    But, I’ll always question the “what if” of my academic pursuits had I not been failed in k-13 math education. Definitely plan to do better by my children, whether or not the schools finally do a better job.

  4. I’m not naturally wired for mathematics & I was too stupid to realize that I should give up trying. I’m also a visual learner & not a reader learner which is required for success in university since there is a finite time for the courses. So for the simple minded like me this is calculus for a visual learner. Calculus is the only mathematics of motion between limits or two vertical strings, rolled up space or just plain old numbers as well as a function (f(x)). Calculus is really motion along a path, wave or string between two limits expressed as numbers or a function f(x). Think of calculus like a car that is approaching a vertical string called a wall which calculus, as it moves along a path, wave, string, calls a limit. If the car hits the wall the car hits its’ limit or vertical string & is destroyed. If calculus hits its’ limit or rolled up space, calculus stops moving & is destroyed. Most mathematics like arithmetic, algebra or trigonometry only calculates the limit or value of the rolled up space of the expression which they call the answer or answers. Some people like to drive their car as close to the wall, limit or rolled up space as they can without hitting it. If you like competitions you can see if you can drive a car closer to the wall then your competition. Calculus is competitive too. Calculus tries to get as close to the wall as it can so like you in your car you approach as close to the wall as you can like calculus in fractions of measurement or ratios that is really slicing a path, wave or string into sections.

    1. Thanks for your comment. I commend your emphasis on motion; that’s how Newton thought of calculus. However, a wall is an awkward metaphor for a limit in calculus, since hitting a wall is an abrupt change and calculus limits are smooth. A better metaphor imo is a regular polygon that keeps doubling the number of sides. Even though each vertex is pointed, the limit is a smooth circle.

      Viewing circles as regular polygons with infinite sides is a very calculus-type notion. Computer graphics apply this notion by drawing all curves as chains of very short straight line segments meeting at shallow angles. Conversely, a calculus-related technique can convert straight line segments into smooth “Bezier” curves; it is visually striking. Imo every high school student should learn these relations. Among other things, it undermines the false dichotomy between STEM and art.

  5. I’m definitely in favor of trying to get more of this material going in K-12, and I think Python’s a good choice for it – both “friendly” (as far as computer languages go) and also the de facto language for AI/ML.

  6. Once you believe learning should be “entertaining” you have already lost. I made it through four quarters of calculus, a quarter of differential equations, a quarter of linear algebra and at the graduate level a quarter of graph theory and a quarter of complex variables. None of these courses were “entertaining”. They were hard work and the professors stressed its importance and relevance. Bottom line was I was interested and wanted to learn the material.

    Any student who has to be entertained in order to master a math subject will never master that subject. Learning takes discipline. It takes dedication. It takes hard work. Sorry, but making it into some video game or computer programming exercise is a waste of time, money, effort and the everybody’s time—including that of the student. Instead, focus on the students who really wants to master the subject matter.

    1. Thanks for your comment on my article. I share your disgust with “entertainment” used as a pretext for dumbed-down content and half-hearted effort. However, sports contests show that entertainment can also motivate serious quest for mastery, and that’s what I have in mind. Consider this playful proof of the Pythagorean Theorem – it strengthens intuition with no loss of rigor. What’s dumbed-down about proving Kepler’s Three Laws — as my short book Calculus for the Curious does, unlike the standard Calculus BC syllabus — and using that to inspire awe of the universe and hunger to learn more? What’s half-hearted about teaching computers to simulate planetary motion as my coding exercises do?

      Bear in mind too that I am describing an introductory course intended for all high school students. If we follow your advice to “focus on the students who really want to master the subject”, we would likely reach less than 10% of the cohort, and hardly any of them can really master it in one year.

    2. Excellent point.
      I think teaching is like catching fish with your bare hands.
      As one who has taught these introductory level classes I know direct entertainment-value has its place: breaking down resistance to change innate to all humans. Yet, as you pointed out, that won’t carry the day.

      However, there is another type of entertainment: the induced kind. Regardless of the subject, when RELEVANCE is made the center of teaching effort the, a connection to the student’s mind can be made like hooks into a very wet, slippery bar of soap. When the student finds the relevance of a topic to his/her own interests, then the said brain will generate its own entertainment. And that WILL carry the day, every day.

      Evolution has programmed the urge to survival over a billion years. Both the resistance to change, and the value of relevance come from the same evolutionary programming. As products nature we will never be smarter than nature (California politicians notwithstanding). Work WITH nature; you’ll lose every time going against it.

  7. What a wonderful essay. Congratulations. In addition to making a pragmatic and perceptive case for calculus unencumbered, at least in part, by a traditional series of preparatory fields, and able to yield insight and application, sui generis, the essay may also be pointing to a more generalizable argument relevant to many other disciplines that are set in a field of extended linear prerequisites, and is thereby addressing institutional efficiency. Law education may be an example. Regards.

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