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I never gave this a second thought until a friend who works in education brought it up the other day. Should we say that a fraction like $\frac{1}{2}$ "is" a number, or "represents" a number? In particular, should we say that $\frac{1}{2}$ and $\frac{2}{4}$ "are" the same number, or "represent" the same number? (Or something else entirely?)

My immediate reaction on hearing the question was that "obviously" they are the same number. I would explain the difference between $\frac{1}{2}$ and $\frac{2}{4}$ as being analogous to the fact that one person can have two names. If I go by "Mike" to my friends and "Dr. Shulman" to my students, and one of my friends is talking about me to one of my students, they might end up having to explain that "Mike and Dr. Shulman are the same person." They wouldn't say "Mike and Dr. Shulman represent the same person." They might say "'Mike' and 'Dr. Shulman' are names for the same person", which conveys about the same meaning as the latter, but in that case the two names would be quoted, because we're talking about them as bits of syntax rather than about the objects they denote. But when a name is not quoted, we're talking about its denotation rather than the name itself, e.g. when we say "$\frac{1}{2}$ is between 0 and 1" we don't mean that the symbol "$\frac{1}{2}$" is written in between the symbols "0" and "1" on a sheet of paper, we mean that the number denoted by "$\frac{1}{2}$" lies in between the numbers denoted by "0" and "1" with respect to the ordering of rational numbers. So we can say "$\frac{1}{2}$ and $\frac{2}{4}$ are the same number" but "'$\frac{1}{2}$' and '$\frac{2}{4}$' represent the same number".

However, this is just my intuitive reaction based on experience as a mathematician, and I've never actually tried to teach someone about fractions. I certainly wouldn't want to try to explain sense and reference to a third grader! So I'm curious about the consensus (or lack thereof) of people who do teach fractions and study people's understanding of them.

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Note that a string of decimal digits, e.g. "312", is itself just a representation of a number. Indeed you could ask the same question about Arabic and Roman numerals -- should we say that "312 and CCCXII are the same number", or that "312 and CCCXII represent the same number"? –  Jim Belk Apr 1 at 22:41
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I am not in elementary education, so this is just a comment, but I seriously doubt that getting into a discussion about Kantian philosophy is a good digression in an elementary school math class. –  Willie Wong Apr 2 at 8:08
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To those answering this question, please note that it is about the pedagogy of fractions, not about the status of fractions themselves. I don't think Mike is confused himself about what fractions are. (Or maybe ... has HoTT gotten as far as fractions yet, Mike?) –  Loop Space Apr 2 at 11:14
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@JoeHarper Thanks! Though as a general rule, answers should really be backed up by either personal experience (and I have none relevant to this question) or research, and should usually be longer than a sentence or two. A comment can be insightful and still be just a comment. –  Jim Belk Apr 2 at 13:54
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In the context of a classroom, making confusing philosophical distinctions would be defeating the role of the teacher. A teacher should teach, which necessarily includes engaging students in the subject in an inviting way. In elementary, you'll scare them away and make them hate Math and English. In college, some students may be attracted by the philosophical distinction and want to learn more. "Genius is the ability to reduce the complicated to the simple." –  Josiah42 Apr 2 at 16:41

12 Answers 12

Questions of this nature were addressed in some part by New Math in the late 50s and 60s.

They were also used as a point of criticism by its opponents. Most notable is Morris Kline's book Why Johnny Can't Add in which he writes relevantly in Chapter 1 :

Evidently the class is not doing too well and so the teacher tries a simpler question. "Is 7 a number?" The students, taken aback by the simplicity of the question, hardly deem it necessary to answer; but the sheer habit of obedience causes them to reply affirmatively. The teacher is aghast. "If I asked you who you are, what would you say?" The students are now wary of replying, but one more courageous youngster does do so: "I am Robert Smith."

The teacher looks incredulous and says chidingly, "You mean that you are the name Robert Smith? Of course not. You are a person and your name is Robert Smith. Now let us get back to my original question: Is 7 a number? Of course not! It is the name of a number. 5 + 2, 6 + 1, and 8 - 1 are names for the same number. The symbol 7 is a numeral for the number."

As you can see, your concern extends beyond rational numbers (and is at least a half century old).

Today teachers are unlikely to draw such a fine distinction between whether e.g., 1/2 is a number or represents a number or is the name of a number. Instead, they are likely to focus more on whether the students are speaking about the mathematics in a way that makes sense. (This relates to sense making; cf. the Common Core State Standards for Mathematics or some of my previous posts, e.g., here.)

Probably the majority of early mathematics teachers will tell you 1/2 is a number: it's a fraction. 2/4 is also a number: it's a fraction. Furthermore, 1/2 and 2/4 are the same number even though they are written differently. Nothing like equivalence relations would be explicitly covered in these classes (I'd write "needless to say," but people try some wacky stuff).

Implicitly, it helps to talk about a fraction in its simplified form (which is effectively asking for a representative from the infinite equivalence class to which it belongs). This ought to be covered early on.


With regard to your title question: I would say, e.g., 1/2 is a number. (Talking specifically about the context you have put forward, i.e., teacher discourse in elementary school education.)

If a teacher were to write 1/3 and 8/12 on the board and say, "Okay, add up these numbers!" then I would be concerned by students who get stuck upon (or object to) hearing 1/3 and 8/12 referred to as such. Much better, I think, would be to note the latter is equal to 2/3 and say, "1."

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That sort of philosophical approach for a class of students who aren't doing well seems like it could easily backfire. It could work for an algebra class, where the teacher is trying to show that the name of a variable has no relation to the value of the variable (leaving aside systems of equations, where the point is to determine a consistent value for a set of named variables) or some sort of math class involving working with multiple radixes, but if you declare that a number's symbolic representation isn't the number then a class of low-performing students is more likely to get confused. –  JAB Apr 2 at 13:04
    
(Or is that the point? I don't really know the context of the quote.) –  JAB Apr 2 at 13:06
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Seems incorrect, anyway. 7 is a number. "7" is the name of a number. That's not a deep fact about mathematics or about names, it is the correct way in English to indicate the difference between a statement about the thing referred to by a word (or in this case symbol) vs. a statement about the word. I suspect Kline's point here is that the teacher is pointlessly confusing the students with this guff ;-) –  Steve Jessop Apr 2 at 20:15
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@JAB With regard to context: The quotation comes from a well-known opponent of New Math (you can find more about New Math curricular materials here: matheducators.stackexchange.com/questions/369/…) who was speaking somewhat hyperbolically. A lot of people agreed with Kline, which led to the abolition of New Math, and what was called the Back to Basics movement that followed it. –  Benjamin Dickman Apr 2 at 20:19
    
Teaching one's children and teaching classrooms is quite different, but I <b>did</b> teach my son about equivalence relations when he was about 5 years old. I think it helps a lot, because if $\frac{1}{2}$ and $\frac{2}{4}$ are the same, then what are we doing taking common denominators? His intuition also tells him they are different- two quarters of cake is different from one half of cake, even if they have the same weight and size. Conversely, 8-2 cookies and 7-1 cookies are the same. There's no need to be super-rigourous about it, but I think I'd teach equivalence relations for fractions. –  Daniel Moskovich Apr 3 at 2:18

The other answers are good, and/but some of those points are reinforced by a diagnostic that has ever-more been important to me: is intention being communicated? And a different, more linguistic point mentioned in other answers: toleration of ambiguity, allowing context (where intent is adequately clear) to disambiguate.

Thus, yes, it is perverse to entrap/prank little kids by exaggerating the distinction between a thing and its name, a referent and the reference. Sure, it can matter, but it's not usually the main issue. Such a distinction can be presented as a funny thing, but not to prank kids. It's not constructive, although it has interest.

Mathematics does lend itself to contriving formally-correct but pointless, non-constructive entrapment situations, "gotchas". Pranking people, whether young or old, by perpetrating such stunts is close to immoral, and certainly mildly anti-social. Certainly misrepresents what I consider to be the (positive) true nature of mathematics (which, yes, I admit is not much reflected in k-12 or undergrad math, but...)

That is, operationally, what do we do with fractions "in real life"? Not what we might do formally, formal-linguistically, formal-mathematically, formal-logically. Don't create fake issues, fake distinctions.

And, then, there is the sad side-effect of belief in the essential formality of mathematics that we are supposedly "not allowed" to use context-dependent or ambiguous language. Of course we are "allowed".

... but while small kids probably do have experience with context-disambiguation of usage of ordinary language, they will not have been taught (give the ambient tradition) that they should exercise common sense for mathematical issues. Pity.

Perhaps just as "naive set theory" is much more useful than "formal set theory" to the rest of mathematics, and similarly "naive category theory" (my own term) rather than "formal...", perhaps "naive mathematics" is more useful than "formal mathematics". For me this is a guide to how to explain things... Honesty, rather than "formal correctness".

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This is strictly context dependent. Consider the following sentence:

Is banana a fruit?

Well, is it? The answer, of course, depends on what exactly you're talking about. It can be interpreted in two ways:

1. Is the physical object known as banana a fruit?

In this case, yes, it obviously is. But consider this interpretation:

2. Is the string of characters "banana" a fruit?

In this case, again, obviously, answer is no. The string "banana" merely represents a fruit called banana.

So, in conclusion, I'd say that both "X is a number", and "X represents a number", are corect, in their suitable contexts. Which one you should use is depended on what you are trying to comunicate.

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Would you really ever interpret the question Is banana a fruit?, without quotation marks around "banana", in the second way? –  Mike Shulman Apr 2 at 20:58
    
The OPs question is about verbal comunication. You don't see the quotes. –  Davor Apr 3 at 6:51
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I am the OP. A lack of quotes in verbal communication doesn't mean that a spoken sentence can be interpreted to have any meaning obtained by inserting quotes in arbitrary places. –  Mike Shulman Apr 3 at 21:53
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Botanically speaking, bananas are berries! –  mrf Apr 3 at 22:07
    
@MikeShulman That doesn't really change anything. Your examples are exactly about this distinction that I pointed out. –  Davor Apr 4 at 6:14

I would say that $\frac12$ is a number and that "$\frac12$" represents (or names) that number. I should therefore not say things like "the denominator of $\frac12$ is 2", because the denominator depends not only on the number but on its name. I confess, however, to being rather careless about such things most of the time.

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You would object to an elementary school teacher saying: "The denominator of $\frac{1}{2}$ is 2"? –  Benjamin Dickman Apr 2 at 20:14
    
@BenjaminDickman No. Since I, knowing that I ought to be careful about such things, confess to being careless most of the time, I can hardly object when an elementary-school teacher does the same. –  Andreas Blass Apr 2 at 20:42
    
Interesting; I am reading your response comment as: "No, I make that same mistake, so it would be hypocritical/unfair to criticize others for doing the same." (Please correct me if my interpretation is wrong!) To this end, how would you complete the following sentence so that it is, in your view, correct? "The denominator of $\frac{1}{2}$ is..." –  Benjamin Dickman Apr 3 at 0:40
    
Amusingly, this very question just came up on math.stackexchange. Perhaps one of you would like to chime in? math.stackexchange.com/questions/737535/… –  Steven Gubkin Apr 3 at 2:22
    
@BenjaminDickman Whether something is a mistake depends on the context. In the present context of a question about such a technicality, a statement like "the denominator of $\frac12$ is 2" is a mistake because it ignores the technicality, the main point of the question. In elementary school (and even in most of my conversations), the same statement is not a mistake but innocent carelessness, because nobody cares about this technicality. [continued in next comment] –  Andreas Blass Apr 3 at 14:07

This makes me think of the intensional/extensional distinction. Here's a sentence you probably see all the time: "Consider the function $f(n) = n^{2}$." Of course, on the natural numbers, this function yields the same set of ordered pairs as the rule $n \mapsto \sum_{k=1}^{n} 2k-1$ that says "given $n$, return the sum of the first $n$ odd numbers".

Are these the "same function"? Sometimes the right answer is yes, and sometimes it's no---it depends on the context. I don't think such subtleties should be entirely glossed over in a math class, but I do think it's important to cultivate in one's students an ability to suffer abuses of terminology in stride, and moreover to default to the intended level of abstraction in standard contexts. This is the same ability an author relies upon any time they write (or fail to write) "by an abuse of notation".

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I go out of my way to avoid the phrase "consider the function $f(n)=n^2$" and its comrades. It doesn't take much extra effort to say "consider the function $f$ given by $f(n)=n^2$." This type of caution seems warranted in situations where confusions about identity arise. Function-talk is one such situation. I've not encountered this type of confusion regarding fraction-talk, but that may be idiosyncratic. –  Shay Apr 3 at 17:10

In elementary education, "is" would be the right term since students hardly know what "represent" means. However in teacher education when discussing the rationals, it's different. Once you understood equivalence relations, you should notice that they appear many times. Teachers should be aware of the fact that operations might not be well-defined in these contexts. This can prevent mistakes and even help students to see that some operations (like cancelling digits in a fraction) cannot be correct.

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When I was in elementary school, we distinguished between numbers and numerals. A number is a number, and a numeral is the symbol we use to represent it. So we can represent the number five with the numeral '5' in the Arabic system, or the numeral 'V' in the Roman system, or in any other number of ways. This took about thirty seconds to explain and I don't think anyone in my class found it confusing.

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I'd personally say the hint is in the name "fraction."

A fraction of a cake is still cake. So, a fraction of a number is still a number.

But also, seriously, a fraction is essentially a sum, a half is (1/2), id est... a number. But this produces the argument, does a sum represent the resultant number, or is it in itself THE resultant number?

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"A fraction of a cake is still cake." What if the fraction is $\frac{0}{1}$? Then there is no cake. What if the fraction is negative? –  Joel Reyes Noche Apr 2 at 13:28
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Please don't take my comment above too seriously. One of my favorite jokes is to say that "a fraction of the profits will go to charity" and then to say that the fraction is $\frac{0}{1}$. –  Joel Reyes Noche Apr 2 at 13:29

I would like to say that 1/2 and 2/4 and 3/6 and $\pi /2\pi$ are different fractions however they are all the same number. There are many different ways to express the number, but they are all the same number. This is what we mean by number. Implicit within the concept of number is the assumption of equivalence of fraction. Therefore, when I write $1/2 = 2/4$ I mean to say they are the same number. Or, when I write $0.999\dots =1$ again, I mean to say they are the same number. Are they different expressions? Sure. Do they indicate differing mathematical structures. Absolutely.

I simply object to making students doubt themselves on questions like "is $1/2$ really $2/4$ ?" There is a time and a place to examine such questions, in my view, that time and place is when you have the tools to properly do such. That is in the context of a course where you are building numbers from the base up. Perhaps a second course in algebra, but certainly not in elementary school. That is a time to build up confidence about how to do things with numbers, not to breed uncertainty about questions which they cannot hope to properly dissect.

There is always something foundational to learn. However, I don't think that everything I know now is worthless just because there are gaps in my foundations. I hope that I am aware of the gaps, especially when I task myself on a problem where the gaps have bearing. But, more to the point, it's quite important to be able to do math at different levels of abstraction. Often, it is more important to understand the structure of an object than to be able to give an explicit construction thereof. For example, what is $a+ib$ where $a,b \in \mathbb{R}$ and $i^2=-1$? Is it an extension field of $\mathbb{R}$? Is it the set of $2 \times 2$ matrices $\left[ \begin{array}{cc} a & -b \\ b & a\end{array} \right]$? Is it a pair $(a,b)$ with the multiplication defined in the manner of Gauss? I would offer, for most purposes, the answer that $a+ib$ is the quantity that when multiplied with $c+id$ gives $(a+ib)(c+id) = ac-bd+i(bc+ad)$. Of course, this can be derived from any of the models of complex numbers I mention. Moreover, apart from dispelling a certain air of mystery as to what a complex number is I don't think it a good use of time in an elementary complex course to belabor the construction. Much better to use them and get to applications and theorems which show the utility and beauty. It is the same with real numbers. Just use them, if you want to discuss their construction, save it for a time when you have tools.

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This veers towards http://en.wikipedia.org/wiki/Haddocks%27_Eyes.

It seems to me that distinctions have to serve differences. I very much doubt that you will help an elementary-level student by calling '1/2' a process or not a number, or getting involved in whether '1/2' and '2/4' are two numbers with the same value or one number with two representations. Learning math is all about harnessing naive intuition when possible, and correcting it when it hurts.

My view is that you should not need to be concerned with the distinction between numbers and their representations until you get involved in multiple bases or the reals. When discussing the reals, it's quite important to recognize that .999999... is the same as 1.0, which very strongly leads in the direction of distinguishing representation from the thing represented.

Until you get there, you are perilously close to teaching Platonism to Penguins.

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They are not the same. For example, $$1\over1+x$$ is clearly a fraction, but it is not in general a number but rather a variable expression. We usually represent rational numbers as a fraction of integers, and if we want to have a unique representation, in lowest terms and with a positive denominator.

But that does not mean that rational numbers are the same as fractions.

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And yet it evaluates to a number for any number x. (except x = -1) –  daviewales Apr 5 at 15:28

I think it's helpful to separate representation from semantics.

Maybe if we rephrase the question this can help accomplish that goal.

  1. Is 1/2 a number (an abstraction of quantity) or

  2. Is 1/2 a representation (a process or symbol) of a number.

What "is" a number, as others have explained it is an abstract concept of quantity. Those abstractions can be assigned symbols for us to understand and discuss, for example quantity 2, can written in symbolic form as "2", "II", or "..", etc... Different symbols, which indicate the same abstract quantity.

There are two meanings of representation that we have to distill.

Definition of representation (as you mean it):

  1. The symbol that identifies something ("2", "0.5", "pi").
  2. The process that identifies something.

Ask yourself is "1/2" actually a symbol like "2"? Do you have a "1/2" key on your keyboard? No.

Therefore, "1/2" actually represents a process to determine a number. Think about it, "1/2" is actually a combination of 3 distinct symbols that when combined have inherent meaning as to a process. That process is mathematical division. It is literally "1 divided by 2." From this thought process I would conclude that "1/2" is NOT the number, but a representation (definition #2) of it. This is distinctly different than looking at "0.5", this is the symbol that identifies that value (there is no procedure to this representation). Now some numbers (irrational ones, for example) cannot be shown symbolically in decimal form without losing accuracy, and in that case the fractional representation may be the most accurate one, but it still fundamentally represents a process, not the number itself. So I think this question hinges on your quest for semantics, which is interesting to think about. :)

Disclaimer: I'm not a mathematician, so take my answer worth a grain of salt. :)

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Greetings and welcome to the site! I'd say that it is totally fine to post here without being a mathematician. However, I'm not sure that your answer fits the original question. You have attempted to answer "Is it true that fractions 'are' numbers or 'represent' numbers," but the question was "[When teaching fractions], should we say that fractions 'are' or 'represent' numbers?" –  Chris Cunningham Apr 3 at 16:50
    
@ChrisCunningham, thanks for the feedback. I'm positive there are more "academic" answers to this questions, e.g., by those with credentials greater than mine in this specific field. My answer is an attempt to get at the meaning of the question which I found interesting. I apologize if it was too tangential. But if I can clarify I attempted to answer the original question and I'll summarize: the symbol (1, 2, 3, etc...) is the number, the combination of symbols "1/2" is not the number, but represents how to find the number. Again, this is a non-academic answer, so it's worth a grain of salt. –  TechMedicNYC Apr 3 at 21:00

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