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In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion.

I am a mathematics teacher in the UK and there are questions in national GCSE exams phrased like this:

Put these numbers in order from smallest to biggest: 3, -1, 7, -5, 13, 0.75

These questions are in exams designed for low ability students and testing their knowledge of place value and ordering numbers and the correct solution in the exam would be: -5, -1, 0.75, 3, 7, 13.

I think if the question says "smallest to biggest" the correct solution should be 0.75, -1, 3, -5, 7, 13. Even though it doesn't seem to bother most people, I think the precise mathematical language is important and "smallest to biggest" should be avoided but if it is used it should refer to the absolute value of the numbers.

So my question is: Which is bigger, -5 or -1?

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I don't know if this qualifies as a real question... Nevertheless I think the exam problem is phrased unambiguously as it doesn't mention absolute values in any way. – Sebastian Jun 1 '11 at 13:11
I'd at least add a "soft question" tag to this. Mathematicians will use all sorts of language, informally. When written formally, the distinction between $|x|<|y|$ and $x<y$ is a bit clearer. If we were thinking in terms of vectors, we will often say one vector is "bigger" than another if its norm is bigger - a vector is a length and a direction, but you can't really compare directions, just lengths. In the case of the real line as a trivial vector space, this becomes the notion of comparing absolute values. – Thomas Andrews Jun 1 '11 at 15:23
I get reminded of this… – user17762 Jun 1 '11 at 16:13
Let's say I'm tens of thousands of dollars in debt because of student loans and you have exactly one thousand dollars; which of us has "more" money? (This is just one way of looking at it.) – anon Sep 27 '11 at 21:28
@anon: If you are 100 dollars in debt, while I am 1000 dollars in debt, who of us has "more debt"? The meaning of less and more depends on the direction you declare as ascending. - If you have a primitive economy, you may either have no gold nuggets at all, or a positive number of them, so "more" and "less" are meaningful concepts for abstract numbers. But if you introduce financial economy and invent debts, negative numbers become useful, but I think the meaning of numbers changes. – shuhalo Sep 28 '11 at 12:17

Like all too many test questions, the quoted question is a question not about things but about words.

Roughly speaking the same question will have appeared on these exams since before the students were born. And in their homework and quizzes, students will have seen the question repeatedly.

Let's assume that the student has a moderately comfortable knowledge of the relative sizes of positive integers. It is likely that the student has in effect been trained to use the following algorithm to deal with questions like the one quoted.

  1. Arrange the numbers without a $-$ (the "real" numbers, negatives are not really real) in the right order.

  2. Put all the things with a $-$ to the left of them, in the wrong order. Why? Because your answer is then said to be right.

  3. goto next question

Even if there has been a serious attempt by the teacher to discuss the "whys," at the test taking level, the whys play essentially no role.

The OP's suggestion that "size" might be more intuitively viewed as distance from $0$ is a very reasonable one. That is part of what gives the ordering question some bite. Students who follow their intuition can be punished for not following the rules.

Sadly, in our multiple choice world, questions are often designed to exploit vulnerabilities and ambiguities.

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The problem is that we have two notions of "bigger", coming from the two operations, addition and multiplication (or alternately, one comes from the fact that $\mathbb{R}$ is ordered, one comes from the fact that $\mathbb{R}$ is a vector space), and they coincide for positive numbers. In almost every situation except the negative numbers, either only one of the notions make sense or they both agree. Consequently, people don't feel the need to be careful in distinguishing the two notions.

Personally, when you say "smallest to biggest" I think the way you do, that we are looking at size in terms of absolute value (the multiplicative notion of bigger), but if you say "least to most" you would give the answer the test is looking for. In my mind, the latter is referring to quantity, while the former is referring to magnitude. As Geryy said, there is a context to where I would use these terms, and the context determines what I would consider the natural thing to be looking at. However, I don't think that everybody uses language the same way I do, and on a national exam where you can't ask for clarification, making assumptions like that is an easy way to get things wrong.

The one thing that would sway me towards their interpretation of the wording is this: have the students discussed absolute value in any great length, viewing it as the size of the number? Also, is there any chance that this question is a multiple choice question, and that only one of the "right" answers is an option?

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If this is a test for "low ability students", it is likely a bad idea to come across with mathematics as something abstract and an end in itself. The prerequisite for higher mathematics is to be fully comfortable with basics (say, arithmetics up to integration).

Hence the usual approaches, and in particular rigour axiomatics, don't lead to the desired educational results. It is better to introduce mathematics as tool, even if it hurts ones understanding of "mathematical purity" and so on. For these people, mathematics is a tool at best, and should be related to applications.

So where negative numbers actually come from? It is probably not by counting things. You do not pile up coins and obtain -5 dollars, as much as you do not measure length and obtain -23.5 meters.

A feeling for negative numbers is better conceived if you introduce a sense of direction and a pivot point.

For example, take a look at the thermometer. You have a pivot point "degree 0" ( in Celsius, I am continental European), and temperature compares with this point. The sign of the temperature tells you whether below or above the pivot, and the magnitude how much. Equally important accountancy. Having a surplus of money or being in debt can be conceived as a sense of balance, around the pivot "0 dollars".

Second, A sense of direction can be perceived if we measure relative positions and movement. Say, on a street you step backward and forward. This can be perceived as moves in positive or negative directions, and in this case, you do not even have a fixed pivot.

Third, I would like to give a perspective partly inspired by geometry. If you have positive and negative numbers, you have an orientation. The sign of the thermometer or your account status are arbitrary. You could measure coldness and poverty instead of warmth and prosperity.

Maybe these three points are helpful for mathematical didactics. Actually I know think that bigger or smaller is indeed the wrong naming. You can have more debts, you can go farther backwards, temperature can be more freezing, comparativly. In so far, "bigger" and "smaller" is a convention that does not contain the full meaning or applicability of negative numbers.

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I'd say $-1$ is bigger because the difference $-1-(-5)$ is positive. But I admit I don't consistently take this view. When I'm talking about $x\to-\infty$ in a class that hasn't studied limits, I'll often talk about "very large negative $x$." As long as ones meaning is clear from context, no harm is done. The problem with an exam question is that there isn't any context.

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Your first sentence is equivalent to saying that you use x < y rather than |x| < |y|... – The Chaz 2.0 Jun 1 '11 at 16:17
@The Chaz - yes, is that a bad thing? – Gerry Myerson Jun 2 '11 at 0:52
Not at all! It was meant as a point of clarification for others (esp. in light of Thomas' comment above). I debated whether to use "Your..." or "The..." - internet etiquette is confusing :) – The Chaz 2.0 Jun 2 '11 at 3:05
@The Chaz, OK, thanks. – Gerry Myerson Jun 2 '11 at 3:25
@GerryMyerson: In other words, you're saying that you use "bigger" to use "more", and (presumably) "smaller" to mean "less". (See OP's first sentence.) I'm not saying this is a bad thing (or incorrect); just pointing out that this is exactly the distinction the OP is talking about. – ShreevatsaR Sep 28 '11 at 10:04

This is a well known problem for math teaching. First, in the context of $\mathbb{R}$, "$x$ is bigger than $y$" means $x > y$, nothing less or more.

When teaching calculus, we often have to remind students that if a function changes value from $-4$ to $-1$ then the function has, in fact, increased, so we expect the derivative to be positive somewhere. Quite a few students will think that a change from $-4$ to $-1$ means the function has decreased instead. But in that case we would have to say that the function $y = x$ is decreasing on the interval $[-4, -1]$, which would be very strange.

The difficulty is that it is hard at first to learn to use definitions. It takes a particular skill to separate the formal meaning of a term from "false cognate" associations, and to be able to fall back on the literal wording of the definition when needed. This skill can take time to develop, and questions like "is -5 bigger than -1?" help students to develop it.

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So "x is bigger than y" is the same as "x is more than y" (and "x is greater than y")? I have not actually seen "bigger" used in this way, to be honest. Only "more" and "greater" – ShreevatsaR Sep 28 '11 at 17:34
@ShreevatsaR: I wouldn't use "bigger" in a research paper, but if that is the term that was used in this class then it makes sense to me as "increased = got bigger". Of course, if they never talked about this, and it just showed up on an exam, then the answer by André Nicolas matches my sentiments. – Carl Mummert Sep 29 '11 at 11:59
Carl: "increased = got bigger" is true only for positive numbers. :-) (At least it's self-evidently true only for positive numbers. If you're defining "bigger" that way for negative numbers you must justify why, and this is precisely the OP's question.) – ShreevatsaR Sep 29 '11 at 17:22
Otherwise one has the unfortunate outcome that a number can increase and simultaneously became smaller. My point is that this arises frequently when students learn to interpret derivatives in calculus, and it takes my students some time to get used to what it means for a negative number to increase. – Carl Mummert Sep 29 '11 at 19:01
Yes, one could say that when -2 increases to -1, it simultaneously becomes smaller. What's wrong? I do agree that students get confused between things increasing and their magnitude increasing, but that is IMHO the same confusion being exhibited here. (I agree that it's ultimately it's just a matter of defining "bigger" and "smaller", but I still don't see your justification for defining it your way.) – ShreevatsaR Sep 30 '11 at 3:20

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