There’s a line from the original Star Trek where Khan says, “Improve a mechanical device and you may double productivity, but improve man and you gain a thousandfold.” Joan Horvath and Rich Cameron have the same idea about improving education, particularly autodidacticism or self-learning. They share what they’ve learned about acquiring an intuitive understanding of difficult math at the Hackaday Superconference and you can watch the newly published video below.
The start of this was the pair’s collaboration on a book about 3D printing science projects. Joan has a traditional education from MIT and Rich is a self-taught guy. This gave them a unique perspective from both sides of the street. They started looking at calculus — a subject that scares a lot of people but is really integral (no pun intended) to a lot of serious science and engineering.
You probably know that Newton and Leibniz struck on the fundamentals of calculus about the same time. The original papers, however, were decidedly different. Newton’s approach was more physical and less mathematical. Leibniz used formal logic and algebra. Although both share credit, the Leibniz notation won out and is what we use today.
Calculus Casualties
Unsurprisingly, a quarter of calculus students at Tier 1 universities get a D or lower. Quite a few of those who fail will leave science and engineering to either drop out or move to a less technical discipline. We always heard the joke that calculus was the biggest pre-business course on campus.
Joan Horvath and Rich Cameron entered the Hackaday prize with a plan to use 3D printed models to help teach concepts for calculus. The idea is to produce 3D printed objects that show the intuitive understanding of basics like the fundamental theorem of calculus. It’s like graphing out an algebra equation for better understanding, only this moves to three-dimensions and provides a tangible foothold for your brain to understand abstract concepts of traditionally difficult subjects.
Even if you don’t want to 3D print the models, they are in OpenSCAD, so you can experiment with them virtually in that environment. This approach has been so successful, that there’s an upcoming book that will expand on the topics at greater length.
Math Intuition
I’ve never been a big fan of learning math for math’s sake and a fair number of math classes I’ve had are organized that way. No one ever comes to your office and asks what’s X if 4X+13=2. They will come in and ask something like, how do I change this resistor to make the output of this power supply go from 5V to 5.2V? Algebra is part of the toolkit you can use to answer questions like that. Calculus is like the algebra of things that change and can answer a wider range of questions.
I’ve often thought that in modern times, teaching the intuition of math might be more important than the mechanics. You can always ask the computer to do the real work and get the right answer. The real skill is in formulating the correct question and frequently that requires a combination of algebra, calculus, and differential equations. Can you get by without it? Sometimes. Maybe even much of the time. But there’s always going to be certain cases where you really need advanced math tools. [Joan] and [Rich] can help.
Expand Your Horizons
Many things you just “know” by rote are actually applications of calculus. If you are taking the RMS value of a sine wave by dividing the peak value by the square root of two, you are using a result from calculus. If you have something other than a sine wave, you are going to need to do the math. If you know the time constant of an RC circuit, that’s a result of calculus, too. If you need to know the 63% point, you have your answer. But how about the time it takes the capacitor to charge to 10% or 80%? Calculus.
Hackaday has made attempts to demystify calculus in the past. I also love how modern computer graphics can make intuition easier to develop. Honestly, one of the best traditional calculus books I know of is Thompson’s Calculus Made Easy from 1914 (well, the second edition, anyway). His approach isn’t that different from [Joan] and [Rich] as he states:
Being myself a remarkably stupid fellow, I have had to unteach myself [overly difficult ways of doing calculus] and now beg to present to my fellow fools the parts that are not hard.
Although it is over a century old, calculus doesn’t really change much and you might enjoy reading it — along with the Hacker Calculus — if you plan to chase mathematical enlightenment.