# An Intuitive Introduction To Limits

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.

# Understanding Bayes Theorem With Ratios

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.... Read article

# An Interactive Guide To The Fourier Transform

The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations:

$\displaystyle{X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i 2 \pi k n / N}}$

$\displaystyle{x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{i 2 \pi k n / N}}$

Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor:

• What does the Fourier Transform do?

# An Intuitive Guide to Linear Algebra

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!

# Math As Language: Understanding the Equals Sign

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.... Read article

# Why Do We Learn Math?

I cringe when hearing "Math teaches you to think".

It's a well-meaning but ineffective appeal that only satisfies existing fans (see: "Reading takes you anywhere!"). What activity, from crossword puzzles to memorizing song lyrics, doesn't help you think?

Math seems different, and here's why: it's a specific, powerful vocabulary for ideas.... Read article

# A Brief Introduction to Probability & Statistics

I’ve studied probability and statistics without experiencing them. What’s the difference? What are they trying to do?

This analogy helped:

• Probability is starting with an animal, and figuring out what footprints it will make.
• Statistics is seeing a footprint, and guessing the animal.

# Understanding Algebra: Why do we factor equations?

What’s algebra about? When learning about variables (x, y, z), they seem to “hide” a number:

$\displaystyle{x + 3 = 5}$

What number could be hiding inside of x? 2, in this case.

It seems that arithmetic still works, even when we don’t have the exact numbers up front.... Read article

# Finding Unity in the Math Wars

I usually avoid current events, but recent skirmishes in the math world prompted me to chime in. To recap, there’ve been heated discussions about math education and the role of online resources like Khan Academy.

As fun as a good math showdown may appear, there’s a bigger threat: Apathy.... Read article

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# How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms

Last time we tackled derivatives with a “machine” metaphor. Functions are a machine with an input (x) and output (y) lever. The derivative, dy/dx, is how much “output wiggle” we get when we wiggle the input:

Now, we can make a bigger machine from smaller ones (h = f + g, h = f * g, etc.).... Read article

# How To Understand Derivatives: The Product, Power & Chain Rules

The jumble of rules for taking derivatives never truly clicked for me. The addition rule, product rule, quotient rule — how do they fit together? What are we even trying to do?

Here’s my take on derivatives:

• We have a system to analyze, our function f
• The derivative f’ (aka df/dx) is the moment-by-moment behavior
• It turns out f is part of a bigger system (h = f + g)
• Using the behavior of the parts, can we figure out the behavior of the whole?