Hi all! I’m happy to announce a public availability of the BetterExplained Guide To Calculus. You can read it online:

http://betterexplained.com/calculus/

And here’s a peek at the first lesson:

(Like the new look? I’ve been working with a great designer and will be refreshing the main site too.)

The goal is an intuition-first look at a notoriously gnarly subject. This isn’t a replacement for a stodgy textbook — it’s the friendly introduction I wish I’d had. A few hours of reading that would have saved me *years* of frustration.

The course text is free online, with a complete edition available, which includes:

- Course Text
- Video walkthroughs for each lesson
- Per-lesson class discussions
- PowerPoint files for all diagrams
- Quizzes to check understanding (In development)
- Print-friendly PDF ebook (In development)
- Invitations to class webinars (In development)

The price of the course will increase when the final version is released, so hop onto the beta to snag the lower price.

## Building A Course: Lessons Learned

A few insights jumped out while making the course. This may be helpful if you’re considering teaching a course one day (I hope you are).

**Incentives Matter**

I struggled with what to make free vs. paid. I love sharing insights with people… and I also love knowing I can do so until I’m an old man, complaining that newfangled brain-chip implants aren’t “real learning”.

Incentives always exist. I want to make education projects sustainable, designed to satisfy readers, not a 3rd party.

Similar to the fantastic Rails Tutorial Book, the course text is free, with extra resources available. Having the core material free with paid variations & guidance helps align my need to create, share, and be sustainable.

**Being Focused Matters**

Historically, I’m lucky to write an article a month. But this summer, I wrote 16 lessons in 6 weeks. What was the difference?

Well, pressure from friends, for one: I’d promised to do a calculus course this summer. But mostly, it was the focus of having a single topic, brainstorming on numerous analogies/examples, and carving a rough path through on a schedule (2-3 articles/week).

I hope this doesn’t sound disciplined, because I’m not. A combination of fear (*I told people I’d do this*) and frustration (*Argh, I remember being a student and not having things click*) pushed me. When I finished, I took a break from writing and vegged out for a few weeks. But I think it was a worthwhile trade — in my mind, a year’s worth of material was ready.

**Fundamentals Matter**

There’s many options for making a course. Modules. Quizzes. Interactive displays. Tribal dance routines. Hundreds of tools to convey your message.

And… what *is* that message, anyway? Are we transmitting facts, or building insight?

Until the fundamentals are working, the fancy dance routine seems useless. I’d rather read genuine insights from a pizza box than have an interactive hologram that that recites a boring lecture.

When lessons are lightweight and easy to update, you’re excited about feedback (*Oh yeah! A chance to make it better!*).

The more static the medium, the more you fear feedback (*Oh no, I have to redo it?*). A fixed medium has its place, ideally after a solid foundation has been mapped out.

I’ll be polishing the course in the coming weeks, feedback is welcome!

-Kalid

I’m very excited for this course after reading your book. I went through (and enjoyed) a lot of higher math for my degree. Even with that background though, getting back to the basics the way you present it is awesome and fun. Makes me wish I’d had this stuff available when I was slogging through the hard parts the first time around.

Do you have a favorite go-to source for the historical stories you mix into the lessons (like Archimedes unrolling the rings of a circle to find the area)? I’ve been trying to get a good collection of those kinds of stories for my kids to use for intuition building as they learn math and science, but most of what I’ve found so far hasn’t been that great.

Thanks Erik — really glad the approach is clicking for you (and think it’s awesome you’re looking for similar ways to explain it to your kids).

You know, I’ve started to take a more biographical approach to math. So when we have some result (Limits are the theory that XYZ) I’ll usually look on Wikipedia for a quick overview of the history. Not the math, but just the people involved, what they were thinking, earlier versions, etc.

Eventually you wind up on ideas like the Archimedean Property (http://en.wikipedia.org/wiki/Archimedean_property) which is an ancient principle, but comes down to whether you “allow” there to be infinitely large (or small) elements in your numbers (that’s a decision?). Despite years of math classes, I’d never heard of that choice being spelled out, let along that it’s an ancient idea! It makes a lot of things snap into place for me. Some number systems allow them, some don’t. If you do, you can have infinitesimals (what Newton/Leibniz used), if you don’t, you get the theory of limits (1800s math formalizations).

I’d like to put in more footnotes to Wikipedia, many of the stories are things that seemed interesting as I poked around biographies.

For example, unrolling the triangle was in Archimedes treatise called “Measurement of a Circle”:

http://en.wikipedia.org/wiki/Measurement_of_a_Circle

Why do we so often show what he found (pi*r^2) but not how he found it? Madness :).

Am reviewing algebra (“Forgotten Algebra”) and will then tackle calculus for non-math majors (“Forgotten Calculus”). My goal is to learn higher math (not for college–I’m 64), just for me. What would you suggest after I go through your book? Any texts you are ardently fond of?

Hi Len! I haven’t looked closely at enough books, but I like the general approach of the “Cartoon guides to XYZ” or “The Manga guide to ABC”. My general philosophy is to get a blurry overview of a subject, building an intuition, then refine the details with the more formal / technical descriptions. Having a medium like a cartoon forces the author to use diagrams and visualizations :).

K–

In learning the Maclaurin series, I was struck by how the description of a whole curve–or at least the seeds of that description–could be contained in something as nebulous as the nth derivative. It reminded me of fractal geometry, where the concept of self-similarity can be found on every scale, from the smallest to the largest.

Hey Tim, thanks for the comment. Great point. There’s something neat that a curve holds so much information in the derivative — it’s almost like a hologram (or a cell in a body) where any tiny portion contains information about the greater whole. The equation for each derivative can be used to create the parent curve.

hi khalid….,,, i have thought of a small intuitive idea on “laplace transform” need your suggestions on it….

Laplace transform are basically used to solve differential equation. we know that derivative of e^x =e^x the same function…. so if we can express a function f(t) in terms of e^x all the D operators will vanish… and we will get a simple picture … something like watching a function wearing exponential glasses .. and how do we do that… by division…like how many 2 are there in 10 ??? =10/2=5 …. likewise how much (scaling factor) of e^xt in f(t)..??= f(t)/e^xt…. but since the function f(t) is extending in time ,so we have to integrate to get real expression of f(t) in e^xt. thus we get L{f(t)} =integration f(t)*e^-xt dt..

i also think the term “s” used to denote variable in laplace transform is misleading… it should be “sigma” (i don’t know how to write its symbol :D)…

my textbook uses s>o ,s<k like terms where s is complex variable…and i have learned that complex variables do not have ordering property (i.e they can't be compared by equality or inequality ) and that confused me …. i even had a argument with the teacher….

need some intuitive lessons on complex integration and analytic functions….help to rescue