# 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

# 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?

# Learning To Learn: Embrace Analogies

Why do analogies work so well? They’re building blocks for our thoughts, written in the associative language of our brains.

At first, I thought analogies had to be perfect models of the idea they explained. Nope.

“All models are wrong, but some are useful” – George Box

Analogies are handles to grasp a larger, more slippery idea.... Read article

# Site Update: Ahas and FAQs for articles

I’ve just added a new feature to the site: an Aha / FAQ section for each article.

You can add an aha! moment or question, and vote / discuss them individually. This extends aha.betterexplained.com, making mini-posts for key ideas in an article.... Read article

# Calculus: Building Intuition for the Derivative

How do you wish the derivative was explained to you? Here's my take.

Psst! The derivative is the heart of calculus, buried inside this definition:

$\displaystyle{ f'(x) =\lim_{dx\to 0} \frac{f(x+dx)-f(x)}{dx}}$

But what does it mean?

Let's say I gave you a magic newspaper that listed the daily stock market changes for the next few years (+1% Monday, -2% Tuesday...).... Read article