I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education.

Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). It all fits together.

Calculus is similarly enlightening. Don’t these formulas seem related in some way?

They are. But most of us learn these formulas independently. Calculus lets us start with “circumference = 2 * pi * r” and figure out the others — the Greeks would have appreciated this.

**Unfortunately, calculus can epitomize what’s wrong with math education**. Most lessons feature contrived examples, arcane proofs, and memorization that body slam our intuition & enthusiasm.

It really shouldn’t be this way.

Table of Contents

## Math, art, and ideas

I’ve learned something from school: **Math isn’t the hard part of math; motivation is.** Specifically, staying encouraged despite

- Teachers focused more on publishing/perishing than teaching
- Self-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject”
- Textbooks and curriculums more concerned with profits and test results than insight

‘A Mathematician’s Lament’ [pdf] is an excellent essay on this issue that 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 like this: **Kids, no fingerpainting in kindergarten.** Instead, let’s study paint chemistry, the physics of light, and the anatomy of the eye. After 12 years of this, if the kids (now teenagers) don’t hate art already, they may begin to start coloring on their own. After all, they have the “rigorous, testable” fundamentals to start appreciating art. Right?

Poetry is similar. Imagine studying this quote (formula):

“This above all else: to thine own self be true, and it must follow, as night follows day, thou canst not then be false to any man.” —William Shakespeare, Hamlet

It’s an elegant way of saying “be yourself” (and if that means writing irreverently about math, so be it). But if this were math class, we’d be counting the syllables, analyzing the iambic pentameter, and mapping out the subject, verb and object.

**Math and poetry are fingers pointing at the moon. Don’t confuse the finger for the moon.** Formulas are a *means to an end*, a way to express a mathematical truth.

We’ve forgotten that math is about ideas, not robotically manipulating the formulas that express them.

## Ok bub, what’s your great idea?

Feisty, are we? Well, here’s what I won’t do: recreate the existing textbooks. If you need answers *right away* for that big test, there’s plenty of websites, class videos and 20-minute sprints to help you out.

**Instead, let’s share the core insights of calculus**. Equations aren’t enough — I want the “aha!” moments that make everything click.

Formal mathematical language is one just one way to communicate. Diagrams, animations, and just plain talkin’ can often provide more insight than a page full of proofs.

## But calculus is hard!

I think anyone can appreciate the core ideas of calculus. We don’t need to be writers to enjoy Shakespeare.

It’s within your reach if you know algebra and have a general interest in math. Not long ago, reading and writing were the work of trained scribes. Yet today that can be handled by a 10-year old. Why?

Because we expect it. Expectations play a huge part in what’s possible. So *expect* that calculus is just another subject. Some people get into the nitty-gritty (the writers/mathematicians). But the rest of us can still admire what’s happening, and expand our brain along the way.

It’s about how far you want to go. I’d love for everyone to understand the core concepts of calculus and say “whoa”.

## So what’s calculus 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”. It’s correct, but not helpful for beginners.

Here’s my take: Calculus does to algebra what algebra did to arithmetic.

**Arithmetic**is about manipulating numbers (addition, multiplication, etc.).**Algebra finds patterns between numbers**: a^2 + b^2 = c^2 is a famous relationship, describing the sides of a right triangle. Algebra finds entire sets of numbers — if you know a and b, you can find c.**Calculus finds patterns between equations**: you can see how one equation (circumference = 2 * pi * r) relates to a similar one (area = pi * r^2).

Using calculus, we can ask all sorts of questions:

- How does an equation grow and shrink? Accumulate over time?
- When does it reach its highest/lowest point?
- How do we use variables that are constantly changing? (Heat, motion, populations, …).
- And much, much more!

Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them. **Like evolution, calculus expands your understanding of how Nature works.**

## An Example, Please

Let’s walk the walk. Suppose we know the equation for circumference (2 * pi * r) and want to find area. What to do?

**Realize that a filled-in disc is like a set of Russian dolls.**

Here are two ways to draw a disc:

- Make a circle and fill it in
- Draw a bunch of rings with a thick marker

The amount of “space” (area) should be the same in each case, right? And how much space does a ring use?

Well, the very largest ring has radius “r” and a circumference 2 * pi * r. As the rings get smaller their circumference shrinks, but it keeps the pattern of 2 * pi * current radius. The final ring is more like a pinpoint, with no circumference at all.

Now here’s where things get funky. **Let’s unroll those rings and line them up.** What happens?

- We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles).
- One side has the smallest ring (0) and the other side has the largest ring (2 * pi * r)
- We have rings going from radius 0 to up to “r”. For each possible radius (0 to r), we just place the unrolled ring at that location.
- The total area of the “ring triangle” = 1/2 base * height = 1/2 * r * (2 * pi * r) = pi * r^2, which is the formula for area!

Yowza! The combined area of the rings = the area of the triangle = area of circle!

This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings.

This is a recurring theme in calculus: **Big things are made from little things.** And sometimes the little things are easier to work with.

## A note on examples

Many calculus examples are based on physics. That’s great, but it can be hard to relate: honestly, how often do you know *the equation for velocity* for an object? Less than once a week, if that.

I prefer starting with physical, visual examples because it’s how our minds work. That ring/circle thing we made? You could build it out of several pipe cleaners, separate them, and straighten them into a crude triangle to see if the math really works. That’s just not happening with your velocity equation.

## A note on rigor (for the math geeks)

I can feel the math pedants firing up their keyboards. Just a few words on “rigor”.

Did you know we don’t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because **“Sure, it works in practice. But does it work in theory?”**.

We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process.

We’re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature’s way of saying “This has lots of energy. Eat it.”

I don’t want to (and can’t) teach an analysis course or train researchers. Would it be so bad if everyone understood calculus to the “non-rigorous” level that Newton did? That it changed how they saw the world, as it did for him?

A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but the intuition of growth is how it was discovered. The natural log can be seen as an integral, or the time needed to grow. Which explanations help beginners more?

Let’s fingerpaint a bit, and get into the chemistry along the way. Happy math.

(PS: A kind reader has created an animated powerpoint slideshow that helps present this idea more visually (best viewed in PowerPoint, due to the animations). Thanks!)

**Note: I’ve made an entire intuition-first calculus series in the style of this article:**

https://betterexplained.com/calculus/lesson-1

## Other Posts In This Series

- A Gentle Introduction To Learning Calculus
- Understanding Calculus With A Bank Account Metaphor
- Prehistoric Calculus: Discovering Pi
- A Calculus Analogy: Integrals as Multiplication
- Calculus: Building Intuition for the Derivative
- How To Understand Derivatives: The Product, Power & Chain Rules
- How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
- An Intuitive Introduction To Limits
- Why Do We Need Limits and Infinitesimals?
- Learning Calculus: Overcoming Our Artificial Need for Precision
- A Friendly Chat About Whether 0.999... = 1
- Analogy: The Calculus Camera
- Abstraction Practice: Calculus Graphs

## Leave a Reply

327 Comments on "A Gentle Introduction To Learning Calculus"

Sign me up! I did all that crazy “area under the curve” stuff at school, but never understood how it REALLY worked. y=2x^2 => dy/dx = 4x…sure, but what the heck is going on. They lost me when the sines and cosines joined the calculus party.

I’ve nevertheless remained interested in maths over the 20 years since, and here’s the crack: your article is brilliant. I can absolutely get what you’re talking about. Your circle example is dynamite, and I also found the idea that calculus “finds patterns between equations” very intuitive.

Now let me remember, my little equation is differentiation. That’s like taking pi.r^2 back to 2pi.r So what you showed was integration; which makes sense to me as you did take the area under a curve. So, to differentiate pi.r^2 I don’t ask for dy/dx, but rather something/dr I don’t see any other letter, an pi is just a number, so maybe the use of y is just convention? So…dy/dr?

Keep up the good work.

Beautiful commentary. I’m currently taking Calculus III, and have already finished Differential Equations. For my degree, these would be the final mathematics courses I would need. However, recently I’ve felt that it’s all starting to make sense and come together, and I’ve found that mathematics is quite elegant. After a certain point, I don’t feel awed by its complexity, but rather it’s simplicity. How an incredibly seemingly complex relationship can be broken down into a symbolically simple idea is truly beautiful. I’ve decided to continue taking some mathematics courses in my next semester, and see how far I want to go in that direction.

It really is a shame that the way mathematics is presented creates a negative impression from grade school on. Conceptually, it is beautiful and elegant and explanatory and all-encompassing. If I had been introduced to mathematics in that form when I was younger, I would have probably been hooked then.

My favorite moments in horrible math classes in high school and junior high would be when the teacher would digress and just talk about the nature of zero or infinity or other interesting concepts. Of course, the teacher would usually end with something like, “Well, anyway, to get back on topic…” and resume with some cumbersome proof.

I’m not saying that a conceptual presentation of mathematics should precede basic grade school necessities like arithmetic, but it should definitely have its place. By misrepresenting the elegant nature of mathematics, we are restricting students who would otherwise begin to take interest.

Again, great article!

I like these sorts of examples for people who have never seen calculus before because, honestly, the subject is not that hard. Give me an above-average student and I can teach them the basics of calculus in less than a week.

But it’s rarely the basics that get people. These methods, after all, were how calculus developed up into the mid-19th century — nary a delta or an epsilon in sight.

Euler was the master of these types of proofs. It wasn’t until mathematicians like Weierstraß started getting counter-intuitive results with these so-called “intuitive” methods that they decided an absolutely rigorous foundation for calculus (and all of mathematics) was necessary.

So, the only caveat is that while these methods might be intuitive and help people just learning calculus, there are limits at which this type of reasoning breaks down and we simply can’t reconcile what is true with what our intuition says is true.

Dude, you rock!

Being an Engineer, I understand the pain a naive student goes through when he is burdened with truck load of Calculus books having tons of theorems, proofs and unimaginable number of weird questions that have absolutely no relevance to the real world!

I scored well in my engineering mathematic subjects but I never really understood the point of learning that stuff. Heck, I don’t even remember half of it now.

I wish we had someone like you who could paint such a wonderful picture and make the subject more relevant to students.

I look forward to whatever article you come up with next in the series.

God Bless You!

(BTW, where are you from? I wud love to meet a genius like you sometime!)

I just wanted to say I’ve been reading your blog for some time now, but I just had to let you know every article is great and very informative, I just wish you wrote more often =) (j/k I know it must be a lot of time to put together these articles, but thanks again!)

This was just great. Now can someone out there with the requisite skills (I don’t have them) *please* make the circle into triangle thing into a video and post a link to youtube?

You said: “Instead, let’s share the core insights of calculus. Equations aren’t enough — I want the “aha!” moments that make everything click.” Amen! Those “aha!” moments make live worth living (or math worth learning ;) )

“[…] they decided an absolutely rigorous foundation for calculus (and all of mathematics) was necessary”

Well “they” may have decided that, but they failed. No mathematical system is absolutely perfect. There are always holes to poke. This is the essence of Gödel’s work. Your system will never be rigorous enough to always be right, but it might be rigorous enough to work for the problems you care about.

Godel’s work doesn’t imply it’s incorrect, just incomplete, but that’s fairly far away as a concern for most mathematics. The mid 19th-century foundation allows it to solve problems where it failed previously.

Perhaps relegating this approach to mathematical analysis courses, or maybe some honors Calculus courses is useful pedagogically, however.

Wow, thanks for the comments guys!

@Paul: You got it — we were essentially integrating the equation for circumference. But if you call it that from the outset, and define it rigorously, people’s eyes will glaze over :).

And as you said, the use of x (input) and y (output) are conventions. So the regular way would be to say the equation is really 2 * pi * x, where x is the radius (never mind that we always learned it as 2 * pi * r). dy/dr is a perfectly fine way of saying it too.

One interesting thing about integration is seeing how something that doesn’t “look” like a curve (a bunch of rings) can be twisted into a format that does.

@Mike: Thanks for the awesome comment! You really nailed it, there are such beautiful ideas buried in math, which could really encourage people, but don’t have a chance because we jump into the details.

Conceptual discussions & drills have their place. It may be like listening to fun music (rock, rap, etc.) and being inspired to play. Then you start learning an instrument and memorize scales (doing drills). Drills are much more manageable when you have an appreciation for why you’re doing them.

Those side discussions you mention can be awesome — it highlights the discovery side of math. For every equation, there was someone seeing it for the first time and saying “whoa”.

@Jesse: That’s a very good point. I see it similar to teaching Physics: we start with Newtonian mechanics, which are “intuitive” to a degree. Then, as people advance, we teach them about the exceptions: strange things happen at the speed of light (relativity) and when you get really small (quantum mechanics).

But if we started off with relativity and quantum we’d lose everyone along the way.

@Prateek: Thanks for the kind words! Just a curious learner here. I know what you mean — I’ve taken many math classes, but the formulas just seemed to stay there, and didn’t really change how I viewed the world.

I’m usually in the Boston or Seattle area, and if you’re around feel free to drop me an email (kalid@instacalc.com).

@Justin: Thank you for the kind words, that really means a lot. Yeah, I wish I posted more frequently too :).

The articles can be time consuming (10-15 hours) but I think my brain is the bottleneck — procrastination, perfectionism, and sometimes it’s a struggle to have a “good enough” insight (I don’t want to rewrite what’s already on wikipedia). Maybe I can find a way to trick myself into writing more :).

@James: That would be awesome. Unfortunately I don’t have any animation skills either.

@Rodrigo: I agree — math would be a boring place if it was only about pushing numbers around :).

@x: You hit the nail on the head. Math, at its core, depends on unprovable axioms and assumptions — at some point you have to say “this seems to work, it’s good enough, let’s run with it”.

Unfortunately the quest to make calculus rigorous turned it into something which isn’t as easily understood for beginners.

This is something I’ve learned from my quite limited independent study of calculus, which is my personal way of looking at it: calculus is all about how things change. The derivative is one tiny change, and the integral is the sum of many tiny changes. That explanation works quite well, to me, for setting up equations that use calculus. It also makes the fundamental theorem of calculus very simple to understand.

I have to agree about math education; I’m reminded every day that there are people intelligent enough to understand math who don’t get it because it’s not explained in a way that makes sense intuitively. It wasn’t even until about a year or two ago that I started to really understand math and not just use the equations I was given.

I’m sick of the way the education system teaches math, so much that I’ve considered writing a textbook in the style I think math should be taught. To me, it’s simple: learn the way that it was originally discovered. It was discovered through intuition, and that’s the best way to learn it.

I’ll cut short my rambling here. I’ve given you too much to read as it is.

Hi Zac, thanks for the comment. Yep, seeing the derivative and integral that way (in terms of changes) can really give an intuitive feel — and the fundamental theorem becomes that much clearer.

I agree with you about math education — I think many people are capable of learning the subject, but it’s not presented in the best way. We tend to show the final result without all the steps along the way — and those steps are what build intuition. It surprises me that people don’t often write about their own insights (vs. formulas), so just trying to take a stab at it.

Always appreciate an interesting discussion!

Another good explanation. Thanks Kalid.

You’re welcome Viru, glad you enjoyed it.

[…] A Gentle Introduction To Learning Calculus | BetterExplained I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education. (tags: math education learning calculus toread kids) […]

Hi,

Wow! You have communicated a beautiful simplicity. I have several books on calculus (Calculus for Dummys, Math for the Millions, etc. etc.–never was able to read them) but your explanation is what I have needed all these years. Congratulations, and thanks.

Doug Hogg

Former Prinicpal of Pinewood Academy

P.S. Since it only communicates to people who know calculus, I think you could leave this line out:

“I’d feel I cheated if I called calculus “the study of limits, derivatives, integrals, and infinite series”.

“You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity).”

Sounds like just as strong an argument for Creation if you ask me! Sugar and fat are provided to aid survival, and our bodies are designed to make use of them in an optimum way. Sweet fruits encourage consumption and hence spreading of seeds for survival; sounds like a good “plan” to me.

I enjoy your articles, but weakly weaving religion into an article on math is unnecessary and, frankly, I didn’t think it was your style.

I have always, ALWAYS hated math. I’m actually pretty decent at it when I understand it, but it is such a painful process to get to where I understand it that by the time I do, I’m sick of it and don’t want to do it anymore. I would be so much better at it if I bothered to practice it, but I hate it so much that I don’t WANT to practice it. I’m in my first year of college, and the placement test put me in trigonometry (I don’t know how, because I only made it through a year and a half of high school algebra before I gave up), but I only have to take college algebra to transfer, so that’s what I’m going to do next fall because it stands a chance of not making me crazy.

But reading this post…well, it kind of made me want to learn how to like math. It made me CURIOUS about numbers, which has honestly never happened before. The rings-into-triangle thing was the biggest “AHA!” moment I’ve ever had regarding math. It made sense, so I liked it. (I like things when I understand them, see. Like, solving gigantic equations is ridiculously fun, because I know how to do it.)

Anyway. I am rambling. But thank you, thank you! I feel like there’s a glimmer of hope that I might be able to get a handle on math if I just look at it differently. I never thought of it being ideas; it was just brain-numbing formula memorisation until now. And I hate it when I’m unable to do something, so I really would love to be able to do math and not excuse myself by saying it isn’t my subject. Your definition of calculus made so much more sense than the ones I’ve heard.

@Mark

2nd Paragraph:

“Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs create stronger germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). It all fits together.”

I guess I don’t see where Kalid is “weakly weaving religion into an article on math.” Or was your comment meant to be taken sarcastically?

[…] Como no soy una persona de lo más iluminada ni inteligente, reiteradas veces necesito que me expliquen las cosas como si tuviese 2 años, entonces me puse a pensar ¿ qué no todo se puede explicar de esta manera ? ¿ no debería ser acaso la manera normal de explicar las cosas ? entonces me puse del otro lado y entendí que para poder explicar algo de esta manera hay que contar con un completo dominio de los conocimientos cosa que la mayoría de los docentes no tienen. Sobre todo para temas “difíciles” como matemática, álgebra, cálculo para las que el sistema de educación no puede ofrecer formas de entusiasmar a los alumnos, sino una bola de fórmulas las cuales nosotros tratamos de combinar para llegar a un resultado, perdiendo la dimensión del problema y la capacidad de resolverlo usando nuestra mejor herramienta, la imaginación (Einstein se revuelca en su tumba). Esta nota me levantó el ánimo viendo que no soy solo yo el único que piensa así, realmente creo que vale la pena leerlo (está en inglés) […]

@Doug: Thanks for the note, I’m glad you found it helpful! Good point on the note — I changed the wording a bit. It makes me chuckle when I see complex subjects (calculus) explained in terms of other complex subjects (limits, integrals, etc.), without at least _some_ plain-english explanation. How is a beginner looking up what calculus means supposed to have an idea of what it does?

@Mark: I’m not sure I understand the connection to creation — the goal was to use evolution as an example of a simple, unifying theory that can explain a lot of natural behavior.

Animals that hated sugar, fat and other high-calorie foods probably starved when times were tough. But their siblings with a sweet tooth probably survived, which selected for that trait. Evolutionary pressure gives an explanation of why sugar would seem sweet to us today (I’m not a biologist, there may be other reasons too).

Anyway, the point is that calculus finds similar connections/underlying themes between math — there are nice (simple) reasons why the formulas are linked.

Without calculus, the similarity in the equations just looks like a happy coincidence, much like “sugar is sweet and spoiled food tastes bad” might seem like a lucky coincidence without the theory of evolution. Hope this helps clarify what I meant.

@Kat: That’s awesome! I love getting those “aha” moments and I’m happy you were able to get excited about calculus ideas (it’s a rare thing in this day and age).

You definitely can get a handle on math — I really believe it’s a skill like writing. Once upon a time, everyone thought reading & writing were “hard” and only for scribes; today everyone does it.

The hardest part about math can be staying interested and keeping your motivation, so hang in there! Seeing it as just another way to talk about an idea can help get the big picture. And you’re right, when you get it, even solving gigantic equations can be fun :).

[…] Anyway, this ramble came about as a result of reading A Gentle Introduction to Learning Calculus. […]

@Kalid:

Your implication appears to be that evolution is THE theory that provides the “aha” level of understanding the natural world. Yet the example you provided is just as easily explained by creation. It came off a bit preachy to me and detracted from an otherwise well-written article.

I always wanted to learn this calculus stuff. Tho I seemed to have survived the last 40 years of electronics and computer theory without it, I’ve always had a curiosity about just what all those squiggly lines were on the old chalk boards. I think you have succeeded in clearing up some of the fog. (so far so good anyway) Please keep up the good work you have been doing on this web site. I really have enjoyed all of your articles.

@Mark: Point taken, and happy for the discussion. I think the key point behind it all is that the sweetness of sugar serves a purpose (to help us survive) — but if we don’t notice this underlying theme then we miss many of connections that exist in the real world.

@Paul: Thanks for dropping in, and for your comment! Glad to make things clearer as I can — the funny thing is that despite using the squiggly lines many times, they tended to stay in the realm of abstract symbols without much inherent meaning. So I’m trying to go back and relearn the stuff with the viewpoint of “it has to mean something!”. I’ll keep writing as best I can :).