Cardinal and ordinal numbers

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.

10 responses to “Cardinal and ordinal numbers”

1. Neil Seeley

I was just searching this out on my quest for different views of quantity. I am still digging things up but heres a nice link that ties ordinal and cardinal pretty well together:
http://mathworld.wolfram.com/CardinalNumber.html

• Christopher

I dig the Wolfram article, Neil. As you can imagine, it’s a bit weighty for my future elementary teachers. But it’s also missing the human connection that I wanted to illustrate for my students. I wanted to be sure that they could see that the distinction isn’t just (a) linguistic, as is suggested by most sites for educators, nor (b) purely formal, as the Wolfram article suggests. I wanted to provide support for thinking about this subtle idea that would also motivate the topic for people who care first about kids.

2. Neil Seeley

I was hoping coincidence would carry the weight of your needs 😉

I did follow a reference to a mathematician who does 100 square fun stuff and he emphasizes know the difference between ordinal and cardinal in a short reaction paper. I can send it in email if you like.
1-100 Rules Okay? by Ian Thompson
ERIC has the full text

This insight has been really helpful for me as well, so thank you for the inspiration!

3. Christy

Griffin’s apples!
This takes me back to our conversation in Kodiak about Math Recovery and early numeration in children. In Math Recovery language, the child that says “10,9,8 the number of apples Griffin has left is 7” is a discrete counter. He is saying that we have taken away the discrete items “Mr. 10, Mr. 9, Mr. 8” and we have “Mr. 7” and everyone else left behind. The child who says “9,8,7 the answer is 7” is an interval counter, as they’re counting the jump back, or the interval from 10 back to 9 etc. The important thing is not to decide which strategy is ‘best’, as some children are intuitively interval counters and some are discrete. The challenge is to think about how we teach them, and if they keep getting the wrong answer (one off), rather than trying to make them into the other sort of counter, to put them in a situation which helps them see how their strategy can be used successfully.

• Christopher

Thanks for reading, Christy. I agree wholeheartedly that “best” is not the issue here. And what is interesting to me about the video in question is that the discrete counter is correct and sure he is correct. It is such a useful tool to use with future teachers because of this. It helps them to understand that people really do think in different, correct mathematical ways. Our ways of understanding as teachers are not the only ones that work.
You might also be interested in my more polished version of this post on Sophia: http://sophia.org/packets/cardinal-and-ordinal-numbers

4. stephanie

Your conversation about ordinal vs cardinal is very interesting. I think that when you are talking about a 3 year old and their understanding of numbers you are correct in thinking that they may be rote counting- ie following a pattern, counting out loud 1,2,3,4 and yes they may not yet be able to show a group of 8. This requires a greater understanding of numbers and the ability to count with one to one correspondence. Recognizing that 5 means 5 objects. This is not yet developed in young children. When I think of ordinal numbers, I only think of them as a position a way to tell me who was 1st in a race or who is last in a line. If I have a line of 10 children, my kindergarten children can tell who is first, and who is last in the line. In fact it is often a big deal in terms of who is the line leader.
I can’t help but think that we should be using a 10 frame and plastic manipulatives to help children see the above problem. I think fingers and number lines work to, but at least with the children I teach which are a fair bit younger ( 4,5, year olds) it is really important that they understand and see physically 10 plastic objects- placed inside the frame- they then physically take 3 objects away. I totally agree in the ideas that children come to school and see things differently than adults might. Maybe we should be giving children a “problem” and letting them work in small groups to come up with the solution. Then bringing them back as a group and writing down some of the different ways that they solved it- more of a problem solving based model.

5. Thanks for the nice, little read. I stumbled upon this while researching some stuff for a totally unrelated paper I am writing on number theory, and couldn’t help but read it.
I would only like mention that ordinal numbers don’t always line up with cardinal numbers when counting up. The best example I can think of is when we say “the 17th century” to talk about the 1600’s. But we do it at other times too. My son is in his first year of life, but is not yet one year old.
So I am not sure this is really an issue with cardinal vs. ordinal numbers, and think it might just be a question of knowing just where the counter started counting. For instance, using only cardinal numbers, a student could say “he had ten, gave away one, had nine, gave away another, had eight, and gave away a third; now he must have seven,” and with respect to being ‘one off’ is the same as saying “he gave away the tenth, ninth, and eighth, so now he must have seven,” even though the student is thinking of cardinal numbers in the first sentence, and ordinals in the second.
I am not qualified to give advice in education, so I should probably just shut up. But I think this is a wonderful thing you are pointing out, and that it can really help children who do math a little differently. The main thing is that you make them aware of this thing that they are doing, and teach them how to keep it all straight in their own way. In my experience, the main thing is picturing what is happening as I go along. I have worked as a carpenter for a long time, and it is amazing how many mistakes I have seen because of this exact thing you are pointing out. Just last week we were a fence post short because of it…
So, as long as I am giving out useless advice, I would say that this would be a great time to start working on some little problems like figuring out how many cuts are required to make 8 one foot boards out of one 8 foot board, or conversely, how many posts are needed to hold up 100′ of fence with a post every ten feet (we verified experimentally that 11 are required).
My feeling is that the cardinal / ordinal thing is mostly bunk. The real distinction, to my way of thinking, is that ordinal numbers are attached to each specific thing, whereas cardinal numbers refer to the group as a whole. For instance, to get back to the ten apples, with cardinal numbers, whatever apple you give away first was number ten, and now that it’s gone there is no number ten apple, only one through nine remain. But with ordinal numbers based on order of acquisition, you could give away the second apple you acquired, and you would be left with apples 1,3,4,5,6,7,8,9,10. That would be a group of nine apples, but when you say it is a group of nine apples, you aren’t using ordinal numbers any more, you are using nine as a cardinal number. You could also use ordinal numbers to order your apples by size, and if you give away your third largest apple, you would then be left with apples 1,2,4,5,6,7,8,9,10. But, now that the third largest apple is gone you could, in this case, re-order them based on size of apples currently in your collection, and have 1,2,3,4,5,6,7,8,9, and still be talking about ordinal numbers. It’s confusing, and largely useless.
At some point someone must have thought that if they could figure out the exact rules for different kinds of numbers, they would be able to go straight to heaven in a fiery chariot like the prophet Elija. But it turns out that it is usually just a really obtuse way to get to something pretty obvious.

6. Christopher

Thanks for adding your thoughts and real-world experience, Mike. If you find this sort of thing interesting, you should come join the fun over on Talking Math with Your Kids, where math and kids’ thinking come into conflict and harmony on a regular basis.

Indeed, the century thing has come up there in the context of playlists I maintain for my little ones.