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.
