NOTE: I get a lot of hits from Internet searches on “cardinal and ordinal numbers”. If that’s how you got here, consider going directly to an expanded and illustrated version of this post on Sophia. It is a better version of the same ideas that are here.

### overview

In my math content course for future elementary and special ed teachers, we are working on the difference between *cardinal *and *ordinal *numbers. I promised them a reading on the topic so they could work on it outside of class. But the first three pages of Google results (and really, who goes deeper than that?) all return the same basic idea, which is not as nuanced as the one I need to communicate to my students. So here is my take on things. If you find it useful, link to it please so that future students (mine and others’) will find it more easily.

### introduction to the topic

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 the distinction that is made on each and every one of the websites I looked at this afternoon. Here is a typical example.

### A subtler 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.* 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 league), sometimes we say *one* when we mean to refer to an ordinal number (the Twins are number one in the league).

### A problem

In class, we are reading 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?

**Problem 1 standard solution. **The standard counting technique is for a student to say, “He had ten apples. Nine, eight, seven. He has seven apples left.”

**Problem 1 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.”

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”. I argue that 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.

### A lovely question

A student asked a lovely question in class. Paraphrasing, she asked, “What if he were counting *up? *How would his strategy be different from the usual one?”

Surprisingly, his strategy wouldn’t be any different.

It would be highly unusual for a student to solve Problem 1 by counting up. So let’s consider another problem:

**Problem 2.** Tabitha had 3 apples. She picked some more. Now she has 10 apples. How many did she pick?

**Problem 2 standard solution.** The standard counting technique is for a student to say, “She had three apples. Four, five, six, seven, eight, nine, ten.” When the student says “four,” she puts up one finger then another for each successive number. After saying “ten,” she counts her fingers and says, “She picked seven apples.”

The numbers four, five, six, etc. are cardinal numbers. They each refer to how many apples Tabitha has after she picks another apple.

Or maybe they are ordinal numbers. Does “four” refer to Tabitha having four apples, or does it refer to the fact that the one she just picked is apple number four? Does *four* describe the* set of apples*, or does it describe the *fourth apple*? We can’t know from the information given. The ordinal and the cardinal counting sound identical when counting up.

And that is what makes the subtler distinction between ordinal and cardinal numbers so challenging. The easier distinction is based on the form of the words (*fourth-*ordinal; *four-*cardinal). But not all *fours* are *cardinal*.

When children are learning to count, they learn the sequence of words first: *one, two, three*. They do not fully understand that these words represent *how many* things there are. In that sense, it could be said that most children learn ordinal numbers first. The idea that the last number we say represents something about the whole group of things-that this is a cardinal number-comes later. It is not unusual to give a group of eight things to a 3-year old for her to count, to have her do so correctly, then to ask, “So how many are there?” and to have her say “twelve” or some other wrong answer. She has ordinal numbers-ascribing the correct number word to each object in the set; she does not have cardinal numbers-ascribing the correct number to the whole group.