# Colin Foster's Mathematics Education Blog

## 19 January 2023

### Is zero really a number?

Zero is a strange number - learners sometimes even doubt if it is a number...

In my articles in Mathematics in School, I often address 'Questions pupils ask' (I even have a book of that name, Foster, 2017). In this blogpost, I'm going to address a question that came from a learner and that relates to zero and units:

“Is there any point writing ‘metres’ after zero, as it will just be equal to zero?”

In other words, is ‘zero metres’ just exactly the same as the number ‘zero’, without any units? (“Zero m is just zero!”)

Teachers often stress the idea that ‘$1$ metre’ is not at all the same as the number $1$. We know that $1$ is a dimensionless number, whereas $1$ m is a length. So, we would never dream of writing:

$1$ m $= 1$,

as this would be a dimensional catastrophe – as bad as saying something like ‘$1$ m $= 1$ kg’.

But is zero a different matter? If pupils think of ‘$1$ m’ as ‘$1$, multiplied by a metre’, as indeed it looks symbolically, then ‘$0$ m’ is ‘$0$, multiplied by a metre’, which is surely just zero, since zero multiplied by anything is just zero. Sometimes, when students are simplifying algebraic expressions (e.g., collecting like terms), they might simplify something like $$8a - 3m + a + m - 2a + 2m$$ by writing $7a + 0m$, but the teacher would probably say that there is no need to write the $+$ $0m$. Is it any more relevant to mention that "we haven't got any $m$'s" as it is to mention the absence of other quantities ($7a + 0m + 0r + 0mr + 0m^3$, etc.). In a particular context it might be worth being explicit about the zero $m$s (I sometimes find this useful when solving simultaneous equations by elimination, for instance, so as to keep everything nicely lined up in columns), but in general we wouldn't regard it as worth mentioning, so we would simplify to just $7a$. So, this feeling that $0m$ is just 'nothing' perhaps suggests that ‘zero metres’ should also be simplified to ‘zero’; i.e., nothing at all. Having no metres just means that you have nothing at all. Indeed, perhaps, on all the different dimensional scales of different quantities, the zeros coincide:

$0$ m $= 0$ kg $= 0$ °C $= 0$.

Clearly, it it not necessary to know any conversion factors to know that zero in any length unit, say, will be zero in any other length unit:

$0$ cm $= 0$ m $= 0$ inches $= 0$ furlongs $= ...$

But can it make any sense to write two zero measures on different dimensional scales as though they are equal, like $0$ m $= 0$ kg? This really looks wrong.

If it is right, does this mean that if a question says, “Give the units in your answer” that that is a subtle, unintended clue that the answer won’t be zero? Should a student not be penalised in an exam for omitting the units for a question where the answer is zero, such as this one?

The temperature on Tuesday is 2°C.
On Wednesday, it is 3°C warmer than it was on Tuesday.
On Thursday, it is 8°C colder than it was on Wednesday.
On Friday, it is 3°C warmer than it was on Thursday.
What temperature is it on Friday?
Give the units in your answer.

Surely not, as here the units are very necessary, since an answer of ‘$0$’ for temperature could be $0$ °F or $0$ K, since the zeroes of temperature certainly don’t coincide with one another, let alone with the zero for kilograms. Maybe ‘all zeroes are equal’ only applies to quantities with a ‘true zero’, as opposed to those with ‘arbitrary zeroes’, like temperature.

But can this be right? Zero oranges and zero apples do not necessarily represent the same state of affairs: just because I’ve run out of oranges, it doesn’t necessarily follow that I must have run out of apples too.

One possible response is to say that, with measurement, 'zero never really means zero'. So, a statement like '$0$ m' really means 'zero metres, to some degree of accuracy', so this represents not a point but an interval on the metres number line, such as $-0.5 \leq$ length $\lt 0.5$. (Although can length be negative? Possibly, say if it's a difference between two other lengths.) And now clearly $-0.5 \leq$ length $\lt 0.5$ is different from, say, $-0.5 \leq$ mass $\lt 0.5$, so the problem goes away. But in pure mathematics we can have an exact zero that is not rounded to any degree of accuracy.

Questions like this can make zero seem like ‘not a number’, or at least not like any other number, in ways that students may find disturbing. I remember a Year 10 (age 14-15) student remaining behind after a mathematics lesson to ask me a question she didn't want others to hear. I assumed it would be something personal, but it turned out she was embarrassed to ask ("This is probably a really silly question", etc.) the question: "I have always wondered, but is zero actually a number?"

It might be tempting to dismiss such questions. Of course zero is a number - it's on the number line. What else would we put half way between $1$ and $-1$? Would we want the number line to have a tiny, infinitesimally small gap 'at zero'? And if zero isn't a number, what else would it be? But there are instances where zero does indeed seem to be in a class of itself. One example is that it's neither positive nor negative. There are three kinds of real number: positive ones, negative ones, and then a class all of its own for the single number zero. I sometimes catch myself saying that the topic of 'directed numbers' refers to positive and negative numbers, or I might even call the topic 'positive and negative numbers', forgetting to say 'and zero', which is 'central' to the whole thing. It is interesting to contrast the question "Is zero a number?" with the (possibly related) question "Is infinity a number?", which I suspect different mathematics teachers would answer in different ways.

A question like the one I've discussed in this blogpost may feel very abstract, and why should we worry about such an unusual question? I am not claiming that lots of students are asking this particular question all the time. But uncertainty over things like this perhaps contributes to students' feelings that mathematics doesn't make any sense. Sometimes, I suspect, students who are confused or stuck in mathematics, and who we might regard as 'having difficulties', are in that position because they have thought further (rather than less) than their peers.

### Questions to reflect on

1. Do you think that $0$ cm is 'the same thing' as $0$ kg? Would you dare to put an equals sign between them? Why/ why not?

2. Would you mark a zero answer wrong for not having the units?

3. What other issues do you see students having with zero?

### References

Foster, C. (2017). Questions Pupils Ask. Mathematical Association.

## 05 January 2023

### Proportionality

If 'proportional' or 'multiplicative' thinking/reasoning is the central idea in age 11-14 mathematics, then how might we do a better job of making approaches to this consistent across the curriculum, so that students develop a deeper understanding?

At Loughborough University, with colleagues Tom Francome and Chris Shore, we are currently working on putting together a complete set of free, editable teaching resources for mathematics at Key Stage 3 (ages 11-14), and we are particularly trying to do so in a 'coherent' fashion, meaning that everything is connected together into a story that makes sense (Note 1; Foster, Francome, Hewitt, & Shore, 2021). We hope it will be ready to release later in 2023.

In this blogpost, I'm going to reflect on some of the thinking that has gone in the 'Multiplicative Relationships' Unit in Year 7 (ages 11-12). This Unit isn't yet finished, but we have a structure and a bit of the detail, which I wanted to share here.

It seems to me that 'proportionality' is the central idea in 11-14 mathematics that so much else is related to, and so we make a big deal within the 'story of the curriculum' of $y=mx$. Moving too quickly on to $y=mx+c$ just muddies the main point, so we save the '$+c$' for later on (Foster, 2022) (Note 2). Initially, we want students to become very comfortable working with $y=mx$. Think of all the things that come under the umbrella of $y=mx$ and can be viewed as instances of this:

• straight-line graphs through the origin
• ratio and proportion
• similar triangles
• gradient
• multiplication and division as inverses
• speed, density and other 'rates'
• rearranging formulae like $V=IR$ and $F=ma$
• the basis for trigonometry (see Foster, 2021)

Often each of these content areas comes with a different notation, and this is something that we want to address in the name of 'coherence'. For example, we might write $y=mx$ for line graphs but $y=kx$ for 'proportionality', and students may miss that the 'constant of proportionality' $k$ is precisely the gradient of the corresponding $x$-$y$ graph. There is much to gain by seeing all multipliers and rates as gradients, so we try to use consistent notation to highlight this.

In addition to the different notations across topics, multiple different representations are commonly used, even just within the topic of ratio and proportion, such as ratio tables, double number lines, etc. Perhaps because everyone agrees that proportional reasoning is hard, there is a temptation to throw everything at it, piling lots of different ideas on students, hoping that something will make sense and stick. Instead, in our curriculum design work, we have tried to avoid this, and instead choose one powerful approach and then use it consistently and probe into it deeply.

Because $x$-$y$ graphs are not just a representation but part of the content of the curriculum (unlike more 'optional' representations, like ratio tables, that some teachers use and others might not), we focus throughout the LUMEN Curriculum on number lines and Cartesian graphs (which we see as two number lines coming together at right angles, intersecting at the origin). So, before we tackle 'proportionality' as such, we spend a lot of time making sense of multiplication through the family of graphs of $y=mx$.

We begin by taking two identical number lines and stretching one of them and (eventually) rotating it by 90°.

This develops into the idea of a rule, linking two number lines via points in the Cartesian plane.

We go on to stress multipliers, like the gradient, $m$, as the key number that takes you, by multiplication, from any (non-zero) number to any other number:

We use lots of examples like this to practise finding multipliers and missing numbers:

We then tackle a mixture of proportion problems with 'nice' numbers as well as ones with 'hard' numbers. Below, we begin with a situation that uses 'nice' numbers. First we raise and discuss the 'additive' error:

Then, we work multiplicatively, first 'between variables' (i.e., from one variable to the other), using the same arrow notation for multipliers as we used earlier:

Then we find 'within variables' multipliers:

We contrast multipliers 'between variables' (which we call rates) with multipliers 'within variables', which we call scale factors. Scale factors are always dimensionless, whereas rates sometimes have units (e.g., here Rosie's multiplier was £/km).

Whether a rate or a scale factor is more convenient depends on the numbers; hence, tasks like this:

This eventually builds up to being able to use any of the elements from this kind of diagram:

(The diagram looks overwhelming with everything included. But in any situation you would only use 2 of these arrows at once, always the same colour as each other.)

This is all very much work in progress, and we'd be very glad of any thoughts or criticisms of what we're doing!

### Questions to reflect on

1. Do you agree about the centrality of proportionality in the lower secondary mathematics curriculum?

2. What do you like and dislike about the approach outlined here?

3. In what ways is it similar to or different from what you typically do?

### Notes

1. To find out more about the LUMEN Curriculum, go to https://www.lboro.ac.uk/services/lumen/curriculum/

2. And, when it comes, I think that $y=mx+c$ might be better encountered as $y-c=mx$. This way, rather than seeing $y=mx+c$ as a non-example of proportionality, we see it as another example of proportionality, but we just 'have the wrong origin'. So, the shift to $y-c$ as our variable, rather than $y$, makes it understandable as another instance of a proportional relationship.

### References

Foster, C. (2021). On hating formula triangles. Mathematics in School, 50(1), 31–32. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20On%20hating%20formula%20triangles.pdf

Foster, C., Francome, T., Hewitt, D., & Shore, C. (2021). Principles for the design of a fully-resourced, coherent, research-informed school mathematics curriculum. Journal of Curriculum Studies, 53(5), 621–641. https://doi.org/10.1080/00220272.2021.1902569

Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. For the Learning of Mathematics, 42(3), 21–27. https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf

### Teaching specific tactics for problem solving

This is my final blogpost, as my year as President of the Mathematical Association draws to a close, so I've allowed myself to go on at ...