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Cardinal and ordinal numbers

Cardinal and ordinal numbers

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.

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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. 

A more subtle view of the topic

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).

A problem

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?

The standard solution

The standard counting technique is for a student to say, “He had ten apples. Nine, eight, seven. He has seven apples left.”

The slide show above demonstrates this.

An alternate solution

In one of the CGI videos, a student solves a similar problem in this way, “He had ten apples. Ten, nine, eight. Take away that; it’s seven.”

The slide show above illustrates this.

Relationship to cardinal and ordinal numbers

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".