Integral of Sin(x): Geometric Intuition

Pythagorean Theorem As Sweeping Area

Intuition for the Quadratic Formula

Intuition For Polynomials

Analogy: Math and Cooking

Intuition for Taylor Series (DNA Analogy)

What does an exponent really mean?

Intuition For Graphed Functions

Imaginary Multiplication vs. Imaginary Exponents

Q: Why is e special? (2.718…, not 2, 3.7 or another number?)

Intuition for Slope-Intercept Form

Slope-intercept form is written in a very mathy way: y = mx + b. Let's turn this into plain English.

Learning Math: Find The Implied Subject

In English, we often drop the subject of a phrase, such as in these signs: Who are these signs written for? It's really *You, stop* or *You, yield* or *You, be alert for bears* (I'm not doing it). After internalizing a language, we can take hints without explicit instructions. But, to put it politely, math isn't usually well-internalized. Let's get clear about who the "math signs" are referring to.

Math and Analogies

Here's a talk (video and transcript) on "Math And Analogies" I did for the Center on Contemporary Art in Seattle. The gist: good analogies make math a joy.

Quick Insight: Intuitive Meaning of Division

While working on some math colorizations I ran across some interpretations of division.

Multiplication can be repeated addition, scaling, rotating (via imaginary numbers), and more.

What about division? Let's take a look.

Common Definitions of e (Colorized)

Learning Tip: Seek Math Wisdom, Not Facts

Learning Math With Psychological Safety

Why do we multiply combinations?

Quick Insight: Subtracting Negative Numbers

Math Learning Tip: Experience Ideas Directly

Intuition For The Golden Ratio

Understanding Discrete vs. Continuous Growth

How to Add 1 through 100 using Calculus

Learning Math (Mega Man vs. Tetris)

Math Proofs vs. Explanations (aka Nutrition vs. Taste)

Math Learning Tip: Find An Idea’s True Name

A Programmer’s Intuition for Matrix Multiplication

The Math Inside the US Highway System

Quick Insight: Easier Arithmetic With Calculus

Empathy-Driven Mathematics

Abstraction Practice: Calculus Graphs

Learning to Learn: Math Abstraction

Analogy: The Calculus Camera

Different Interpretations for the Number Zero

Easy Trig Identities With Euler’s Formula

Trig identities are notoriously difficult to memorize: here’s how to learn them without losing your mind.

Vector Calculus: Understanding the Cross Product

Intuition For The Law Of Cosines

Intuition For The Law Of Sines

Learning math? Think like a cartoonist.

What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma? I’d say it’s an eye for simplification, capturing the essence of an idea.

How To Think With Exponents And Logarithms

Understand Ratios with “Oomph” and “Often”

How To Learn Trigonometry Intuitively

Go beyond SOH-CAH-TOA and visualize trig.

A Quick Intuition For Parametric Equations

BetterExplained Calculus Course Now Available (Public Beta)

It’s Time For An Intuition-First Calculus Course

**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.

An Intuitive Introduction To Limits

Understanding Bayes Theorem With Ratios

An Interactive Guide To The Fourier Transform

An Intuitive Guide to Linear Algebra

Math As Language: Understanding the Equals Sign

Why Do We Learn Math?

A Brief Introduction to Probability & Statistics

Understanding Algebra: Why do we factor equations?

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

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

Calculus: Building Intuition for the Derivative

Vector Calculus: Understanding the Dot Product

Using Logarithms in the Real World

Understanding Pythagorean Distance and the Gradient

Understanding Why Complex Multiplication Works

Intuitive Understanding of Sine Waves

Understanding Calculus With A Bank Account Metaphor

Understanding Why Similarity Works

Surprising Patterns in the Square Numbers (1, 4, 9, 16…)

Intuitive Understanding Of Euler’s Formula

Fun With Modular Arithmetic

A Friendly Chat About Whether 0.999… = 1

Why Do We Need Limits and Infinitesimals?

A BetterExplained Guide To Calculus

Navigate a Grid Using Combinations And Permutations

Understanding the Monty Hall Problem

A Calculus Analogy: Integrals as Multiplication

Learning How to Count (Avoiding The Fencepost Problem)

Understanding Exponents (Why does 0^0 = 1?)

Developing Your Intuition For Math

Learning Calculus: Overcoming Our Artificial Need for Precision

Intuitive Guide to Angles, Degrees and Radians

Prehistoric Calculus: Discovering Pi

A Gentle Introduction To Learning Calculus

How To Analyze Data Using the Average

How To Understand Combinations Using Multiplication

Rescaling the Pythagorean Theorem

How to Develop a Sense of Scale

A Visual Guide to Simple, Compound and Continuous Interest Rates

Intuitive Arithmetic With Complex Numbers

A Visual, Intuitive Guide to Imaginary Numbers

Rethinking Arithmetic: A Visual Guide

A Quirky Introduction To Number Systems

How to Develop a Mindset for Math

How To Measure Any Distance With The Pythagorean Theorem

Surprising Uses of the Pythagorean Theorem

Another Look at Prime Numbers

Demystifying the Natural Logarithm (ln)

An Intuitive (and Short) Explanation of Bayes’ Theorem

An Intuitive Guide To Exponential Functions & e

Understanding the Birthday Paradox

Vector Calculus: Understanding Circulation and Curl

Vector Calculus: Understanding the Gradient

Techniques for Adding the Numbers 1 to 100

Easy Permutations and Combinations

The Rule of 72

Mental Math Shortcuts

Types of Graphs

Vector Calculus: Understanding Divergence

Vector Calculus: Understanding Flux