A Gentle Introduction To Learning Calculus

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

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/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 :wink: )

“[…] 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.

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.

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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.


  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?