I have a complicated relationship with calculus: it reveals the elegance of mathematics, yet highlights the struggles within math education.
Calculus beautifully connects mathematical concepts in a way that expands your understanding. A close parallel is Darwin’s Theory of Evolution: once grasped, you begin to interpret the natural world through the lens of survival. You understand why antibiotic resistance occurs (survival of the fittest). You realize why we crave sugary and fatty foods (an evolutionary drive to consume high-calorie resources during scarcity). Everything interlocks.
Calculus offers a similar enlightening experience. Do these formulas not appear inherently related?
They are indeed connected. However, many of us encounter these formulas in isolation. Calculus empowers us to start with $text{circumference} = 2 pi r$ and derive the others – a concept that would have resonated with the ancient Greeks.
Unfortunately, calculus can also represent the flaws in mathematics education. Many lessons are filled with artificial problems, obscure proofs, and rote memorization that crush our natural curiosity and enthusiasm.
It shouldn’t be this way.
Math, Art, and the Power of Ideas
School taught me a crucial lesson: The true challenge in math isn’t the mathematics itself, but sustaining motivation. Specifically, staying encouraged despite:
- Educators prioritizing research over effective teaching.
- Self-fulfilling prophecies that math is difficult, uninteresting, unpopular, or “not for everyone.”
- Textbooks and curricula more focused on profits and standardized test scores than genuine understanding.
‘A Mathematician’s Lament’ [pdf] is a compelling essay addressing this issue, and it has resonated with many people:
“…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”
Imagine teaching art in this manner: Children, no finger painting in kindergarten. Instead, we will study pigment chemistry, the physics of light, and ocular anatomy. After twelve years of this, if students haven’t developed a deep aversion to art, they might consider coloring independently. After all, they possess the “rigorous, testable” fundamentals to begin appreciating art. Right?
Poetry shares a similar fate. Consider this quote (formula):
This above all: to thine own self be true, And it must follow, as the night the day, Thou canst not then be false to any man. — William Shakespeare, HamletIt’s an elegant expression of “be authentic” (and if that means writing unconventionally about math, so be it). However, in a math-class approach to poetry, we would be counting syllables, dissecting iambic pentameter, and diagramming subjects, verbs, and objects.
Math and poetry are like fingers pointing towards the moon. Don’t mistake the finger for the moon itself. Formulas are a means to an end, a language to articulate mathematical truths.
We’ve lost sight of the fact that mathematics is about ideas, not just mechanically manipulating the formulas that express them.
Okay, What’s Your Approach to Learning Calculus?
Fair enough. Here’s what I won’t do: recreate existing textbooks. If you need immediate answers for an upcoming exam, numerous websites, lecture videos, and quick study guides are readily available to assist you.
Instead, let’s uncover the fundamental insights of calculus. Equations alone are insufficient – I aim to deliver the “aha!” moments that make everything click into place.
Formal mathematical language is just one form of communication. Diagrams, animations, and straightforward explanations can often provide deeper understanding than pages filled with proofs.
But Isn’t Calculus Difficult to Learn?
I believe anyone can grasp the core concepts of calculus. You don’t need to be a novelist to appreciate Shakespeare.
It’s achievable for you if you have a foundation in algebra and a general interest in mathematics. Historically, reading and writing were skills reserved for trained scribes. Yet, today, a ten-year-old can master them. Why?
Because of expectation. Expectations significantly influence what is possible. So, expect calculus to be just another subject to learn. Some individuals delve into the intricate details (the writers/mathematicians). But the rest of us can still admire its workings and expand our minds in the process.
It’s about how far you choose to explore. My goal is for everyone to understand the fundamental principles of calculus and experience that “whoa” moment of realization.
So, What is Calculus Really About?
Some define calculus as “the branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables.” This definition is accurate but not particularly helpful for beginners seeking to Learn Calculus.
Here’s my perspective: Calculus does for algebra what algebra did for arithmetic.
- Arithmetic focuses on manipulating numbers (addition, multiplication, etc.).
- Algebra identifies relationships between numbers: $a^2 + b^2 = c^2$ is a well-known relationship describing the sides of a right triangle. Algebra uncovers entire sets of numbers – knowing ‘a’ and ‘b’ allows you to determine ‘c’.
- Calculus reveals relationships between equations: you can observe how one equation ($text{circumference} = 2 pi r$) connects to a related one ($text{area} = pi r^2$).
Calculus empowers us to investigate various questions:
- How does an equation change and evolve? How does it accumulate over time?
- When does it reach its maximum or minimum point?
- How do we work with variables that are constantly in flux? (Think of heat, motion, populations, and more).
- And much more!
Algebra and calculus are a problem-solving team: calculus generates new equations, and algebra solves them. Similar to evolution, learning calculus broadens your understanding of how the world functions.
Let’s See an Example of Calculus in Action
Let’s put this into practice. Suppose we know the formula for circumference ($2 pi r$) and want to find the area. How do we proceed?
Imagine a solid disc as being composed of concentric Russian nesting dolls.
There are two ways to visualize a disc:
- Draw a circle and fill in the interior.
- Draw a series of rings using a thick marker.
The total “space” occupied (area) should be equivalent in both cases, correct? And how much area does a single ring encompass?
The outermost ring has a radius “r” and a circumference of $2 pi r$. As the rings become smaller, their circumference decreases, but they maintain the pattern of $2 pi cdot text{current radius}$. The innermost ring is essentially a point, with virtually no circumference.
Now, here’s where the calculus concept becomes apparent. Let’s unroll these rings and arrange them in a line. What emerges?
- We obtain a series of lines, forming a jagged triangle. However, as we use thinner and thinner rings, this triangle becomes smoother and less jagged (we’ll explore this further in subsequent articles).
- One side of this shape represents the smallest ring (0), and the other side represents the largest ring ($2 pi r$).
- We have rings spanning from radius 0 up to “r”. For each possible radius (from 0 to r), we position the unrolled ring at that corresponding location.
- The total area of this “ring triangle” = $frac{1}{2} text{ base} cdot text{height} = frac{1}{2} (r) (2 pi r) = pi r^2$, which is precisely the formula for the area of a circle!
Incredible! The combined area of all the rings equals the area of the triangle, which in turn equals the area of the circle!
This was a brief illustration, but did you grasp the core principle? We took a disc, decomposed it into smaller parts, and rearranged these segments in a different configuration. Calculus revealed that a disc and a ring are fundamentally connected: a disc is essentially a collection of rings.
This is a recurring theme throughout calculus: Large entities are constructed from smaller entities. And often, these smaller components are simpler to analyze and manipulate.
A Word on Practical Examples in Calculus
Many calculus examples are rooted in physics. While valuable, they can sometimes be abstract. Realistically, how often do you know the equation for velocity of an object in everyday situations? Probably infrequently.
I prefer starting with concrete, visual examples because they align with how our minds naturally process information. The ring/circle demonstration we just explored? You could physically construct it using pipe cleaners, separate them, and straighten them to form a rough triangle to verify the math. This level of tangible interaction is less feasible with abstract velocity equations.
A Note on Rigor for Math Enthusiasts
I anticipate some mathematically inclined readers might be concerned about rigor. A brief note on this:
Did you know that we don’t learn calculus in the exact way Newton and Leibniz originally conceived it? They utilized intuitive notions of “fluxions” and “infinitesimals,” which were later replaced by limits because of the question: “It may work practically, but does it hold up theoretically?”.
We’ve developed intricate theoretical frameworks to “rigorously” prove calculus, but in doing so, we’ve often sacrificed intuition.
It’s akin to analyzing the sweetness of sugar at a neurochemical level, instead of recognizing it as nature’s signal: “This is energy-rich. Consume it.”
My aim is not to teach advanced mathematical analysis or train researchers. Would it be detrimental if everyone understood calculus at the “non-rigorous” level that Newton initially did? A level where it transforms their perspective of the world, as it did for him?
Premature emphasis on rigor can discourage students and make math seem unnecessarily complex. For instance, ‘e’ is technically defined by a limit, but the intuitive understanding of growth is how it was initially discovered. The natural logarithm can be defined as an integral, or as the time required for growth. Which explanations are more beneficial for beginners?
Let’s engage in some “finger painting” initially and incorporate the “chemistry” gradually. Happy learning!
(PS: A kind reader has created an animated PowerPoint slideshow that visually enhances these concepts (best viewed in PowerPoint for animations). Thank you!)
Note: I have developed a complete intuition-first calculus series in the style of this article:
https://betterexplained.com/calculus/lesson-1
