Last time, we looked at some ideas about appropriate graph types, and the references I found put this in the context of identifying types of data. Here we’ll look at questions about two such classifications: nominal/ordinal/cardinal (with variants), and continuous/discrete. We’ll see that classifications can become distorted as they filter down from higher levels to elementary levels, or otherwise change their contexts.
Is time cardinal, ordinal, or nominal?
First, from a 5th grade teacher in 1999:
Numbers: Cardinal, Ordinal, Nominal? My math class is learning about cardinal, ordinal, and nominal numbers. The definition given in the book tells us that cardinal numbers tell how many (12 shirts per box), ordinal numbers tell position or order (1st place, 5th in line) and nominal numbers name things (number on a jersey, a telephone number). So far so good? That's what we thought until we were asked to determine which group of numbers "time" would fit into. We've had a lively discussion here at the International School in Brussels and are hoping that you can help clarify our dilemma.
I love the idea of a “lively discussion”! That’s exactly what should have happened here.
Doctor Jerry answered within minutes, from a mathematician’s perspective, saying that he had never heard of “nominal numbers”. Mr. Tomlinson replied:
Thank you for your quick reply to our last question (what kind of number is "time" - cardinal, ordinal or nominal) but of course, my class had additional thoughts on the matter. Yes we agree that time (30 seconds, 2 minutes, etc.) could be considered a cardinal number but what do you do with the "12th hour" or the "18th century?" Wouldn't this type of time be considered an ordinal number? And if they are ordinal numbers, then what about showing up for a meeting at "12:30" p.m.? Respectfully, Mr. Tomlinson P.S. The term "nominal" numbers is from the Harcourt Brace Math text for 5th grade. Are there nominal numbers or not? If there are, and they do in fact name things, then aren't you naming something (an hour, for example) when you say "12 o'clock"?
With the additional context, and several hours for thought and research, I replied:
Hi, Mr. Tomlinson. Like Dr. Jerry, I've never heard of "nominal numbers"; and for that matter, I haven't heard "cardinal" and "ordinal" numbers, in the elementary sense, used in higher math. In my mind they're really more a matter of English than of math - the words are worth knowing, to describe how we use numbers in our language, but we don't really do anything mathematical with ordinal numbers (in this sense).
There is a higher-level sense that definitely is not what they are discussing!
Now, before researching elementary-level meanings, I made a wild guess:
I suspect that some text writer fairly recently felt a need to respond to questions like yours from students, wondering whether, say, a phone number or a uniform number is cardinal or ordinal, and for that reason made up a new category, "nominal," where the number is purely arbitrary and has no implications of number or sequence. That makes some sense, though I'm not sure it really contributes anything to our understanding of numbers. The fact is that "cardinal" and "ordinal" aren't meant to cover every possible use of a number in the first place, so there's no real need to worry about it.
This guess is wrong, because the idea didn’t actually begin at this elementary level; but what I said is essentially valid.
Since time does involve sequence - you can compare or subtract two times, which is meaningless with telephone numbers - I would have to say, with Dr. Jerry, that time is not a mere "nominal" number, but fits whatever category you use for other measurements such as height. You're counting hours (or feet), so it fits the meaning of "cardinal" (except that "cardinal" usually only applies to whole numbers, which can be counted discretely, rather than to real numbers and continuous measurements of time or distance).
We’ll get to “continuous” and “discrete” later. But it appears that either this classification system has no place for fractions and decimals, or they have extended the meaning of “cardinal” to cover any quantity.
In addition, there is a difference between talking about an elapsed time of "1:25" (one hour and twenty-five minutes) and an actual time like "1:25" (twenty-five minutes after one o'clock). This is similar to the difference between a distance or interval (5 miles) and a location or coordinate (milepost 5, or the 50-yard line). The latter are used as names of a place; but rather than calling them "nominal" for that reason, I might call them a variety of ordinal, since they mean the same thing as "5th mile." I have to admit I can see elements of all three categories in a form such as "mile 5" or "5 o'clock"; and I'm reluctant to force it into one of two or three categories when none of them really fits. I think it's really a waste of time to try to classify every application of numbers this way.
I don’t think this system, taught at an elementary level, is meant to be analyzed too deeply!
On the other hand, "12th hour" and "18th century" are clearly ordinals; there you are very explicitly counting a position in a sequence. There's nothing wrong with the fact that we can use both cardinal and ordinal numbers in talking about time, any more than it's wrong to talk of both "5 students" and "the fifth student."
Significantly, it is not the thing itself (e.g. time), but the way we use a number to describe it, that can be cardinal, ordinal, or nominal. We can talk even about the same thing (time, or students) in different ways – not only to count something, but to locate or identify it.
But wait … there’s more!
It’s a little different in statistics
Because I'm curious about this terminology, I searched the web and ran across a couple of references to it. It turns out that the terms "ordinal" and "nominal" are used in statistics, an area in which I have little experience.
I gave a link to a site that is no longer there. I’ll be providing current links to the same concepts.
The meaning of these categories is a little different from what you are discussing; it claims only to categorize statistical variables, not all uses of numbers, and "nominal" variables don't even have to be numbers. There are four categories, "nominal," "ordinal," "interval," and "ratio," with increasing mathematical content in terms of the operations that can be applied (=, >, -, /), and roughly corresponding to "set," "ordered set," "group," and "field."
This concept is called “Level of measurement“, and is used to determine what sorts of calculation make sense.
- Nominal data merely gives a name to a category; in statistics, this is usually not a number, but something like male/female, colors, or states. The important thing is that, even if it’s a number, no arithmetic performed on it would make sense. All you can do is ask if they are the same (=).
- Ordinal data gives some sort of order or rank to an item; this might be a number (such as the number of stars in a movie review, or pain ranked on a scale from 1 to 10), or it might just be words (low/medium/high, or very unsatisfied/unsatisfied/neutral/satisfied/very satisfied) that might be encoded as a number. Again, no arithmetic calculations would make sense; but we can compare them (<, >).
- Interval data is numerical data in which not only the mere order, but the amount of difference between numbers, is meaningful. The classic example is temperature scales such as Celsius, where a different of 10 degrees, say, has a specific meaning regardless of the temperature, but 0 has no real meaning, so you can’t talk about something being twice as hot as another (though a temperature difference may be twice as much as another). Another example might be shoe sizes, where size 8 is not (in America, at least) twice as long as size 4, but the difference of 4 represents an actual difference in length. This means that addition (as in finding an average) and subtraction (of two values) make sense.
- Ratio data goes beyond interval data in having a meaningful zero. Most measurements we make, like height, length, and weight, are this type, since zero means there is no height, length, or weight. Here, multiplication (by a scaling factor) and division (by one another, to find a scaling factor) make sense.
Back to the elementary level
I also found the following lesson plan on the subject at your grade level, which likewise has four categories, the last two being called "natural" and "cardinal," which correspond quite closely to the statistical categories. He calls street addresses "ordinal," but unfortunately never mentions time. (I think he would call 1 o'clock a "natural" number, along with Celsius temperature.) Intermediate (4-6) Math Lesson Plans What Are Numbers? (5 or 6), Fred Jacquot http://www.edu-orchard.net/PROFESS/LESSON/MATH/MATH46/ma46fjbl.html
(This link, too, is dead, but I’ve linked to it at archive.org.)
In using the terms Nominal, Ordinal, Natural (Interval), and Cardinal (Ratio), he is taking ideas from statistics, but using names that will make more sense to elementary students. And he presents the topic as a first-day-of-class discussion to get students thinking about numbers, not as something important in the curriculum.
In this presentation the categories make some sense, though I've never heard their names used in quite this way; but I think the point of it is not to introduce important terms that the students will ever see again, but to get them thinking about how numbers are used. If that's the purpose of your text's discussion too, maybe you can get them thinking even more by trying to decide together whether they have been given too few categories to choose from, and letting them come up with their own category for coordinates (times and mileposts) if they think it's needed. After all, math is not always a matter of following known rules; sometimes we have to think for ourselves and invent new categories or concepts by looking for patterns or parallels in need of a name. A discussion like this can give them a more realistic picture of what mathematicians (or, in this case, perhaps linguists or philosophers) do.
This is why I loved Mr. Tomlinson’s “lively discussion”. The fact that the given list of categories doesn’t quite match reality is a great opportunity to think like a mathematician, which students do all too rarely.
I'd be interested to hear how your text defines the terms. If they are anything like those in the last reference, I would call both "30 minutes" and "1:25" cardinal numbers (or Jacquot's "natural"); I'd still call "20th century" ordinal, though a case could be made against it under these rules.
Today, I would answer a little differently. In terms of statistics, which I have done much more with since 1999, I would say that 30 minutes, as a measurement of elapsed time, is Ratio (Jacquot’s Cardinal), and 1:25, as a moment of time, is Interval (Jacquot’s Natural), because the zero time is arbitrary; we can subtract two times, but not divide them and say that 2:50 is twice as late as 1:25! And “20th” in itself is an ordinal number in the classic sense, but “20th century” is much like a time, with a logically arbitrary zero point, making it Interval.
Here is a more typical presentation at the elementary level:
Cardinal Numbers
A Cardinal Number says how many of something, such as one, two, three, four, five, etc.
It does not have fractions or decimals, it is only used for counting.
How to remember: “Cardinal is Counting”
Ordinal Numbers
An Ordinal Number tells us the position of something in a list: 1st, 2nd, 3rd, 4th, 5th and so on.
How to remember: “Ordinal says what Order things are in”.
Nominal Numbers
A Nominal Number is a number used only as a name, or to identify something (not as an actual value or position).
Years later, I taught a course on Mathematics for Elementary Teachers, which (briefly) mentions these three terms in an introductory discussion, similar to Jacquot’s. The text talks about “the three types of numbers”, nominal, ordinal, and cardinal, which it describes (more accurately, I think) as three uses of number, describing a cardinal number as the number of objects in a set. There is clearly no intention to really cover all numbers (e.g. decimals); it just lists uses of whole numbers.
The book has one exercise on this topic. The solution says that in “June 13”, 13 is an ordinal number (which, indeed, in this restricted system, is the best you can do); and it calls a grade of 93 a cardinal number (representing the number of points out of 100). They don’t overemphasize these ideas.
Discrete and continuous functions (and data)
We got a question about a different classification (which came up last time) in 2016:
Discrete Functions, Broken Down Which situation represents a discrete function? A. the distance a runner ran during training, and the time spent running B. the cost of a bag of jelly beans, and the number of pounds bought C. the number of chairs needed for an assembly, and the number of people attending the assembly D. the height in inches of the juice in a bottle, and the amount of juice that you drink I am stuck between three answer choices. When I first went through them, I thought it was B; but as I read more, it seemed as though A and C were also correct. My weakness is word problems, so when it said "verbal," I freaked out. I have now read these at least five times, and all this reading is messing me up.
I answered:
Hi, Zhaniah. I can empathize: often, the hardest parts of math, ironically, are those dealing with ... words! Words are hard to be precise about; it's their imprecision that makes it hard to fit them into math.
In “word problems”, the hard part is translating a story into symbols; the math is generally easy. Well, easier.
I have to say I don't like the way this is worded, myself. A question about functions should explicitly state what is the input and what is the output. I'll interpret "x and y" as "x is a function of y." Here is a rewritten version:
Which situation represents a discrete function?
A. the distance a runner ran during training as a function of
the time spent running
B. the cost of a bag of jelly beans as a function of
the number of pounds bought
C. the number of chairs needed for an assembly as a function of
the number of people attending the assembly
D. the height in inches of the juice in a bottle as a function of
the amount of juice that you drink
Now, what does “discrete function” mean? We talked about continuous functions last time.
The key word you are being asked about is "discrete." A function is discrete when the inputs can't vary continuously, but can only take separate values, such as whole numbers. For example, foot lengths are continuous, but shoe sizes are discrete; so a function of shoe size would be discrete. So you can really just focus on whether the *input* values are discrete. Here's a new version of the question that just asks that: Which quantities are discrete? A. the time spent running B. the number of pounds bought C. the number of people attending the assembly D. the amount of juice that you drink
We saw this last time, too. (Technically, a function whose output is discrete would also not be continuous; but I’m pretty sure this is what was intended.)
Now, some quantities can be interpreted differently in different situations, which complicates things a little. Time, for example, is really a continuous quantity: you can spend an hour, or any part of an hour, running; it doesn't come in separate chunks. But if I say "the number of hours," that starts to sound as if I meant whole hours only. (I don't.) And what about money? You can spend 1.15 dollars, not just whole numbers of dollars. On the other hand, you can't spend 1.15 cents. So it would seem like there's a point at which money becomes discrete. ... or is there? Now that much of our spending is kept track of by bank computers rather than pieces of metal in our pockets, there's no reason not to say something costs 1.15 cents. The discrete nature of money is just a superficial, practical matter, not a real aspect of money itself, and I'd call it continuous.
Subtle? Yes.
All of which is to say that you can get confused if you overthink this. Just ask yourself for each of these situations, does the quantity come in whole chunks that can't be broken down, or can it have any number in between as well? Can I run for 1.15 hours? Can I buy 1.15 pounds of jelly beans? Can 1.15 people attend? Can I drink 1.15 something-or-others of juice?
I hope the answer is clear by now: It’s people that are discrete, so the answer is (c).
If you still aren't quite sure, write back using the link for more help, tell me what your reasoning is on each side of any part you wonder about, and we can look into that in more detail. But, frankly, I think the authors of such questions hope students won't think too hard! They're meant to be easy, not to provoke deep thoughts -- and the authors probably didn't put as much thought into it as we are.
Zhaniah replied:
The way you explained it lit a light bulb in my mind, so thank you so, so much.