Consider the seemingly simple question What is ten?
Quantity. This refers to how many things there are. If ten is a quantity, then it refers to this many things: ***** *****
Numeration. This refers to how we write how many things there are. If ten is a set of symbols, then it refers to this: 10.
Number language. This refers to how we say how many things there are. If ten is a word, then it refers to this word: ten.
To illustrate the difference, ask a French person to read this number: 10. Then ask a fifth grader what this Roman numeral stands for: X. Finally ask a computer programmer what number this refers to in binary: 10.
In order, the French person’s dix illustrates that we can use different number language for the same numeration and quantity. The fifth grader’s ten illustrates that we can use different numeration for the same number language and quantity. And the programmer’s two illustrates that we can use the same symbols to represent different quantities.
If I weren’t so lazy, I’d link to a Karen Fuson reference for the research details. Maybe I’ll get to that sometime. But she’s the go-to person on this.
Ten can also indicate quantity in a different way – how much, instead of how many. The ceiling is 10 feet high.
It’s also the name of one denomination of US currency: “I’d like change for this bill: please give me a ten and two fives.”
Can anyone say Bo Derek?