Introduction
My job has always been teaching. Even when I wasn’t officially working as a teacher, I was always
explaining something to someone. Helping people understand new things was always what I
ended up doing, whether it was running lunch hour calculus lessons for my senior classmates,
explaining to my daughter how to solve systems of equations with matrices as we drove along a
dark country road, or emailing explanations of linear programming or third grade multiplication
to friends and family across the country. So it’s not really a surprise that I’m writing this for you.
I don’t know if I’m a “typical” teacher, but there are two ideas that have always guided my
teaching. The first is that successful teaching and successful learning require that the teacher
understand what the student doesn’t understand. That doesn’t just mean that the teacher is better
educated. It means that the person doing the teaching actually sees why the other person finds an
idea difficult or confusing. People tend to become teachers because they’re good at a subject, but
people who are good at a subject sometimes find it hard to see what’s difficult and why. I’ve spent
almost 40 years trying to understand, and I’m grateful to the hundreds of students who have
taught me. I’ve tried to bring that understanding to this book.
The other guiding principle is that the teacher’s job is to find another way to explain. And
another, and another, and another, until one works. In my classrooms, that has led to silly stories
about sheep, rules and formulas set to music, and quizzes that students giggle their way through.
Whatever works, works, and language isn’t just for language classes. How you tell the story can
make all the difference for understanding it. I’ve tried to give you the benefit of what my stu-
dents have taught me about the ways to explain math that work for them.
Part of the successful storytelling and the successful learning is creating a world your readers
can imagine, visualize, and understand. This book is my attempt to take you into the world of
numbers for a work-study tour. I hope you’ll enjoy the trip.How This Book Is Organized
This book is presented in five sections.
In Part 1, The World of Numbers, you’ll journey from the counting numbers, through the
integers, and on to the rational numbers and the irrational numbers. You’ll take a tour of the uni-
verse that mathematicians call the real numbers. This is no sightseeing tour. You’ll work your way
through the natural numbers, the integers, and the rational numbers, presented as both fractions
and decimals. You’ll practice all the arithmetic you need to know and explore different ways of
writing numbers and the relationships among them.