Author:
Christopher Danielson

To provide a more nuanced, conceptual explanation of the difference between cardinal and ordinal numbers.

Through examples, this packet presents an argument that there is more to ordinal numbers than the words first, second, etc.

Tutorial

The basic introduction to *ordinal* and *cardinal *numbers is this: Ordinal numbers refer to the *order* of things. If I was fifth in a race, *fifth *is the ordinal number. Cardinal numbers refer to *how many* things there are. If there are five fingers on one hand, *five* is the cardinal number.

At the introductory level, the distinction can be made linguistically: *first, second, third* are ordinal numbers while *one, two, three* are cardinal numbers.

This is an important distinction, but it's not really the whole story.

What do we mean when we say that “the Minnesota Twins are number 1″? We mean that they are the best, or that if we put all baseball teams in order from best to worst, they would be first.

The claim that “the Twins are number 1″ is a claim about *ordinal numbers*, but it doesn't use the word "first". That is, we don’t always make the linguistic distinction. Sometimes we say *one* when mean to refer to a cardinal number (the Twins are one team in the Major Leagues), sometimes we say *one* when we mean to refer to an ordinal number (the Twins are number one in the Major Leagues).

There is a lovely book titled *Children’s Mathematics *that reports results from the Cognitively Guided Instruction (CGI) research project at University of Wisconsin, Madison. The premise of the project is that young children come to school with powerful mathematical ideas, but that these ideas may differ from an adult’s way of thinking. The better teachers understand their students’ ideas, the better basis the teachers have for making instructional decisions. CGI set out to document the ways students think about addition and subtraction problems, and to associate these ways of thinking with strategies students use to solve problems.

The book comes packaged with a CD-ROM of second-grade children solving problems of different types, and demonstrating various strategies. One major strategy is counting up/back.

Consider this problem:

**Problem 1. **Griffin had 10 apples. He gave 3 apples to his sister Tabitha. How many apples does Griffin have left?

If a young child were to solve this problem by counting back, what would that look and sound like?

In the standard technique, we say “nine, eight, seven”. These are cardinal numbers. They represent *how many *apples are left after Griffin gave away each apple. He gave away one apple *NINE*, he gave away another *EIGHT*, he gave away another* SEVEN*. So after giving away three apples, there are seven left. Because we are counting back using cardinal numbers, the last number we say represents the number of apples remaining.

In the alternate solution, we say “ten, nine, eight”. These are ordinal numbers. They represent *which *apple Griffin is giving away at each step. He gave away apple number *TEN*, then he gave away apple number *NINE*, then he gave away apple number *EIGHT*. So after giving away three apples, he has given away apple number eight, there must be seven left. Because we are counting back using ordinal numbers, the last number we say is one bigger than the number of apples remaining.

Ordinal numbers are the names we give to the individual objects as we are counting. Cardinal numbers represent the total number of these objects in our set.

It is useful to think of ordinal numbers as *first*, *second, third, *etc. But we don't always call the first apple "first". Sometimes we call it "one".