Is -5 bigger than -1?

Question

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?

Answer 1

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.

Answer 2

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?

Answer 3

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.

Answer 4

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.

Answer 5

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.