# 17 Introduction

When learning and teaching about arithmetic, it helps to have mental and physical *models* for what the operations mean. That way, when you are presented with an unfamiliar problem or a question about why something is true, you can often work it out using the model — this might mean drawing pictures, using physical materials (manipulatives), or just thinking about the model to help you reason out the answer.

### Think / Pair / Share

Write down your mental models for each of the four basic operations. What do they actually *mean*? How would you explain them to a second grader? What pictures could you draw for each operation? Think about each one separately, as well as how they relate to each other:

- addition
- subtraction
- multiplication, and
- division.

After writing down you own ideas, share them with a partner. Do you and your partner have the same models for each of the operations or do you think about them differently?

Teachers should have lots of mental models — lots of ways to explain the same concept. In this chapter, we’ll look at some different ways to understand the four basic arithmetic operations. First, let’s define some terms:

### Definition

**Counting numbers** are literally the numbers we use for counting: 1, 2, 3, 4, 5… These are sometimes called the *natural numbers* by mathematicians, and they are represented by the symbol .

**Whole numbers** are the counting numbers together with zero.

**Integers** include the positive and negative whole numbers, and mathematicians represent these with the symbol . (This comes from German, where the word for “number” is “zählen.”)

We already have a natural model for thinking about counting numbers: a number is a quantity of dots. Depending on which number system you use — Roman numerals, base ten, binary, etc. — you might write down the number in different ways. But the quantity of dots is a counting number, however you write it down.