Ratios summarize a scenario with a number, such as “income per day”. Unfortunately, this hides the explanation for how the result came about.
For example, look at two businesses:
- Annie’s Art Gallery sells a single, $1000 piece every day
- Frank’s Fish Emporium sells 250 trout at $4/each every day
By the numbers, they’re identical $1000/day operations, right?
Trig mnemonics like SOH-CAH-TOA focus on computations, not concepts:
TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery.
After months of work with the help of Neil, a great designer, and my Excel-blogging friend Andrew, I’m happy to launch a brand-new design.
Algebra is really about relationships. How are things connected? Do they move together, or apart, or maybe they’re completely independent?
Normal equations assume an “input to output” connection. That is, we take an input (x=3), plug it into the relationship (y=x2), and observe the result (y=9).
Hi all! I’m happy to announce a public availability of the BetterExplained Guide To Calculus. You can read it online:
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.
Summary: I’m building a calculus course from the ground-up focused on permanent intuition, not the cram-test-forget cycle we’ve come to expect.
Interested? Sign up for the mailing list to get news about the pilot, launching early August. The learning principles below will guide its development.
I’m thrilled to announce the print edition of Math, Better Explained is available on Amazon:
With the magic of print-on-demand, you can order the book with overnight shipping (Amazon Prime!), and be reading full-color insights tomorrow. Yowza.
Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh. Here’s how I learned to enjoy them:
- What is a limit? Our best prediction of a point we didn’t observe.
My first intuition about Bayes Theorem was “take evidence and account for false positives”. Does a lab result mean you’re sick? Well, how rare is the disease, and how often do healthy people test positive? Misleading signals must be considered.
This helped me muddle through practice problems, but I couldn’t think with Bayes.
The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations:
Yikes. Rather than deciphering it symbol-by-symbol, let's experience the idea. Here's a plain-English metaphor:
- What does the Fourier Transform do? Given a smoothie, it finds the recipe.
Despite two linear algebra classes, my knowledge consisted of “Matrices, determinants, eigen something something”.
Why? Well, let’s try this course format:
- Name the course “Linear Algebra” but focus on things called matrices and vectors
- Label items with similar-looking letters (i/j), and even better, similar-looking-and-sounding ones (m/n)
- Teach concepts like Row/Column order with mnemonics instead of explaining the reasoning
- Favor abstract examples (2d vectors!
It’s easy to forget math is a language for communicating ideas. As words, “two and three is equal to five” is cumbersome. Replacing numbers and operations with symbols helps: “2 + 3 is equal to 5″.
But we can do better.