A BetterExplained Guide To Calculus
I’ve struggled with how to write about calculus. The standard techniques seem to be:
- The “bag of formulas”: memorize ‘em and move on
- The anal-retentive, rigorous treatment: written by math robots, for math robots!
- The happy smiles tour: oversimplifications without examples (Calculus helps scientists solve problems!)
No, nyet, nein! I know what I need: intuition (What does it really mean?) followed by examples to back it up. I want a calculus series that lets calculus be calculus — wild, interesting, and fun.
The Explanatory Approach
I started writing in a vacuum, but realized I don’t remember calculus. I need a refresher — in fact, I need the insights I want to share! These articles are for us both (it’s what I’d want to relearn the subject), and here’s my approach:
- I’m reading Elementary Calculus: An Infinitesimal Approach [free pdf]. It teaches calculus using its original approach (infinitesimals), not the modern limit-based curriculum. My goal is intuition, so this works well.
- As I study the chapters, I’ll share the insights I find and the concepts I struggled with.
- I’ll sprinkle examples along the way. They’re a gut check, not the focus (if you want practice problems, the book has plenty).
It’s a lack of insights, not information, that makes calculus hard. We don’t need another course repeating the definitions that confused us the first time (Here’s the definition of a limit, again!).
We shouldn’t be struggling with the true meaning of a subject centuries after its invention. This is my intuition-laced hat in the ring.
The Calculus Articles
The goal is to be concise, informal, and fun. Dabble, skim and ignore the examples if needed — focus on the insights. The elegance of calculus can be appreciated progressively: we don’t need astrophysics to enjoy a starry night.
Learning Math
Calculus Overview
Small numbers: Limits and Infinitesimals
- Learning Calculus: Overcoming Our Artificial Need for Precision
- Understanding the need for small numbers (in progress)
Measuring Changes: Derivatives
Accumulating Changes: Integrals
This post is the table of contents for the series. Happy math.
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Have a question? Know an explanation that caused your own a-ha moment? Write about it here.
When I was taught the math robot guide to calculus, it felt BetterExplained to me. All of a sudden there were REASONS for things!
Anonymous — November 9, 2009 @ 1:21 am
Hi Kalid, you might as well like the book by Michael Comenetz: Calculus: The Elements. Read the readers’ reviews on Amazon. I’m relatively at the beginning of the book, but so far I completely agree with 5-stars reviews (i.e. 100% of them in the time of writing this comment).
Thanks for the link to the free book, did not know it.
Martin — November 9, 2009 @ 5:15 am
Struggling with this myself, so good luck! Understanding is definitely not linear – a nice graph to illustrate:
http://abstrusegoose.com/191
Dan — November 9, 2009 @ 5:16 am
Have you read Iverson’s “Calculus”? It’s online (free) at http://www.jsoftware.com/jwiki/Books#Calculus, and its unique point — aside from the fact that its notation is actually a programming language — is that it emphasizes polynomial approximations, so that there’s more time to cover really advanced stuff (it’s the only calc book I’ve seen that teaches fractional calculus, i.e. taking fractional powers of the differential operator).
Wm Tanksley — November 9, 2009 @ 8:22 am
@Anonymous: I’ll admit that I enjoy the math-robot style, but after I’ve found the intuition
.
@Martin: Thanks for the pointer! I’m always interested in seeing how other people tackle this problem of how to present the material.
@Dan: That’s a great pic, I agree that understanding comes in waves.
@Wm: That’s really interesting, thanks for the pointer! As a cs guy I think it might make things clearer (one “problem” with math is that assignment and equality both use =).
Kalid — November 9, 2009 @ 11:25 am
I am currently reviewing calculus too. My book of choice has always been Thomas’s Calculus which was the only book I was able to digest. Stewart’s calculus for me has always been an example of a book how calculus should never be taught (horrible-horrible book!!!). After reading your article and some other reviews online i decided to give Keisler’s book a try as it seems to be one of a kind. Thanks as always Kalid!
RF_Guy — November 11, 2009 @ 7:46 am
I was first introduced to calculus by the infinitesimal approach in Silvanus P. Thompson’s 1910 book Calculus Made Easy (1998 edn. revised by M. Gardner) http://bit.ly/2Jqjcv. Then moved on to the limits approach without difficulty. Currently revising my knowledge of the subject. Thanks for the tip, Kalid. From what I’ve read so far Keisler’s book looks really interesting.
Elliot — November 11, 2009 @ 6:32 pm
@RF_Guy: You’re more than welcome! I’d love to check out Thomas’s book, I’m always on the lookout for new teaching approaches.
@Elliot: Thanks for the pointer. I’m starting to understand the role of limits myself, after revisiting them a few times and comparing them with infinitesimals. Look for a post soon
.
Kalid — November 13, 2009 @ 5:28 am
Amazing articles. I now have your blog feed on my RSS reader. Could you tell me other similar blogs that teach math.
Arvind — November 19, 2009 @ 5:03 am