# Colin Foster's Mathematics Education Blog

## 16 February 2023

### Don't forget the units?

Sometimes, the units (e.g., cm) that come with a quantity can really help to make sense of what's going on. But do we always need units?

Recently, I was with some teachers who were arguing about a question like this:

What is the area of this rectangle?

Some of the teachers were complaining that the question was ‘wrong’, because the question writer had apparently ‘forgotten the units’! This was seen as ironic, because we are always telling students, “Don’t forget to include the units”, and yet here was a situation where this error had apparently been made in the writing of the question.

“You can’t have an area of 8,” someone said – “it has to be 8 somethings, like 8 centimetres squared.” The whole question was completely unspecified – what on earth is a “4 by 2” rectangle – imagine going into a shop to buy a carpet that is “4 by 2” – without some units it is completely useless!

I didn’t agree. I am quite happy to have a line segment of length 4 or a rectangle of area of 8. In pure mathematics, these things are dimensionless numbers. When we calculate the area enclosed between the curve $y=x^2$ and the curve $y=x(2-x)$, the answer is $\frac{1}{3}$. It doesn’t have any units, even though it really is an area. It isn’t $\frac{1}{3}$ of anything in particular, although I suppose you could say that it is $\frac{1}{3}$ of a unit square, if you wanted to. It would certainly be absurd to write the answer as $\frac{1}{3}$ cm$^2$ unless you were in some applied context in which you’d stated that the scales along the $x$ and $y$ axes were marked off in centimetres. The same applies if we use a formula like $$\int_a^b \sqrt{1+\left( \frac{dy}{dx} \right)^2} dx$$ to calculate the length along a curve: the answer has no units.

Confusion about units with things like this leads to students thinking that they need to write the words ‘square units’ after a definite integral. Similar confusions sometimes lead students to want to write the word ‘radians’ after $$\int_0^1 \frac{1}{1+x^2} dx=\tan^{-1} ⁡1=\frac{\pi}{4},$$ or possibly even writing $$\int_0^1 \frac{1}{1+x^2} dx=45°,$$ which makes no sense at all! (How would you respond to the question: "When you do a definite trigonometric integral, should you give the answer in radians or degrees?")

The debate around units seems to be one where both sides think that the other side is demonstrating some kind of dangerous misconception. Contexts are very important, as are the applications of mathematics, but I am not convinced that everything is always made clearer by setting it in context. The abstract concept of area can be used to solve real-world problems, like painting walls and laying carpets, but there is also just the abstract notion of area, which is measured in dimensionless numbers. If you are happy that you can have a number like 8, all on its own, which isn't a measure of anything in any particular unit, then it ought to be OK to have a length of 8 or an area of 8 too.

A similar issue arises when people object to tasks like:

Write down 5 positive integers with a mean of 7.

Write down 5 positive integers with a mean of 7 and a median of 4.

etc.

Sometimes, teachers object that when you are calculating a mean of a real-life dataset the data points are very unlikely to be nice, neat positive integers. And would it really make sense to calculate summary statistics from data sets containing so few numbers? How meaningful is a median when there are just 5 data values altogether? People who object in this kind of way would be much happier if all of the data points had a couple of decimal places, and ideally would like us to have 500 data points, rather than 5, and handle them in a modern fashion using technology.

This all seems very valuable to me, and I am all for students grappling with realistic, messy datasets, with all the opportunities they present for data cleaning, examining outliers and using descriptive statistics to get a handle on what’s going on. In such a task, there is a purpose - something you want to find out from the data - and the focus becomes less on the nitty gritty of adding up and dividing and more on asking meaningful questions and using the mathematics to figure out meaningful answers (i.e., mathematical modelling).

But I don’t see that kind of work as an alternative to tasks like the ones above. The arithmetic and geometric means (as well as the harmonic mean and other kinds of mean) are all essentially (and certainly were originally) pure mathematics concepts. Certainly, they have important applications to statistics, and elsewhere, but if you are dealing with the AM-GM inequality, for instance, there is no reason to think that the quantities being averaged must constitute some kind of 'realistic data set'.

There are times when context really helps students to get a sense of the underlying mathematics, but there are also times when context can get in the way. It seems likely to me that learners might get a better understanding of what the mean ‘means’ by using - at least at first - simple, easy to apprehend numbers. Tasks, for instance, in which you add an additional small integer value to a small, simple data set, and notice if the mean increases or decreases, or combine two small, simple data sets of different sizes, and explore what happens to the mean, seem very valuable to me. It can be through this kind of work that learners build a sense that the mean is the 'equal shares' value that 'balances' all the values in the set. In such tasks, it would be impossible to notice anything amid the noise of vast quantities of awkward numbers. Later on, of course, when applying the concept of the mean to the real world, we can bring those insights to bear on larger, more realistic data sets, but having messiness from Day 1, as the default, seems undesirable to me. I think there is no reason to feel guilty about asking learners to find the mean of a few small positive integers.

This is not to say that it's always wrong to 'begin with complexity'. Often that can be motivating and lead to powerful mathematics. But paring back the complexity at times, so as to see the mathematical structure, can also be really insightful and can support powerful generalisations. When we say that a 2 by 4 rectangle has area 8, we are effectively making a general statement that subsumes all of these:

A 2 cm by 4 cm rectangle has an area of 8 cm$^2$.

A 2 km by 4 km rectangle has an area of 8 km$^2$.

A 2 mile by 4 mile rectangle has an area of 8 mile$^2$.

etc.

For me, to say "a 2 by 4 rectangle has area 8" isn't wrong, even though it doesn't apply directly to any one specific real-world area.

### Questions to reflect on

1. Do you feel that 'an area of 8' is wrong? Why / why not?

2. When do you think that contexts are helpful and when do they get in the way?

1. I agree totally with all of this. I would add that when I taught standard deviation I would always get the class to do one example (no more) - usually {4, 6, 7, 8, 10}, or {4, 6, 7, 7, 8, 10} if you insist on using (n - 1) - wholly without a calculator. I believe that "getting their hands dirty" (just the once) gave them an understanding of the formula and its meaning that would be lacking if all their calculations were simply "substituting into a formula" or using calculator software.
What I would really like to know is how we can set public exams that test things like mean and SD realistically, i.e. requiring candidates to use computer software to analyse large and messy data. Surely we are at the stage when taking examinations online need not be restricted to candidates with learning difficulties?

2. This is my recent twitter thread. An engineer says "the speed of a particle that travels d meters in t seconds is d/t meters per seconds". d and t are numbers, so you can put them into a calculator to work out d/t, and you don't have to tell the calculator what the units or dimensions are. In this usage, "d m" is a length, whilst "d" is a number.
OTOH, a mathematician, at least at University level, will talk about a distance d and a time t and a speed d/t. So here "d" is a distance, it has a dimension but no units. A distance can be divided by a time, but not on a calculator. The relationship can be illustrated by saying "this distance" (indicating two points) and "this time" (snapping fingers twice), and "this speed" (sweeping a finger from one point to the other in the indicated time). Understanding this equation does not require a system of units. So "d" can either mean a number (of meters) or a physical length. I detect some lack of clarity in school mathematics about which approach is being used, and I don't think that the distinction is taught either at school or university. The engineer will say the mathematician has forgotten to put in the units, but not so. The engineer's "d" is a completely different class of entity than the mathematician's.
Sio normally I'd say a numeric length needs a unit. But then we talk about unitless lengths and areas in the Cartesian (or Argand) plane, so here we are talking about the geometry of our abstract number system, and perhaps the words "length" and "area" are being misapplied in this context

### 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 ... 