Averages: Thoughts on Language and Sequencing (Part 1)

Hi there!

For those who don’t know me, I’m Josh. I’m a maths teacher based in Kent. I’ve worked in a number of secondary schools and FE colleges, all of which are based on the south-east Kent coast. That translates to fairly high PP, SEND and EAL catchments, even more so post-Covid. This all goes to say that I have spent lots of time with my respective departments trying to find ways to close attainment gaps. Two of the most powerful methods of achieving this have been expert modelling, and precise language. The latter is very much the focus of this topic.

Mathematics and linguistics have always had a mutual relationship to me. I find beauty in the mathematics of language, and find power in the language of mathematics. The algorithmic nature of both has always been fascinating to me, and many mathematical terms, particularly of Greek origin, really help to unlock key concepts.

One of the most demonstrative topics for this, in my opinion, is averages and spread.

Many of the definitions, formulae and general shortcuts when teaching averages can be riddled with ambiguity. This often leads to gross misconceptions, or outrageously caveated and long-winded explanations to compensate. It boggles the mind how we expect pupils to memorise these, never mind all their exceptions and fringe cases.

I presented the following ideas and discussions at Complete Mathematics’ Mathsconf34 in Bristol, to an audience which included Peter Mattock, Sam Blatherwick and Jo Morgan (the latter of whom arrived unannounced when referenced in my slides, utterly terrifying for a first conference!). Included are my own musings, and some of the discussions that they spawned.

I began the session with a Think, Pair, Share activity among delegates. If you choose to present these findings to your department, I would highly recommend this as a starter to get the ball rolling and see the (in)consistencies between your colleagues. A range of answers were given, but among which were found the most popular (albeit not most accurate in my view) definitions: ‘divide by how many there are’ [mean], ‘middle number of an ordered set’ [median], ‘most common/occurring’ [mode] and so on.

Without any further fanfare, here are my go-to definitions/formulae:

Mean = Sum/Frequency

Median = Medium

Mode = Most frequent

Range = Distance between

When it comes to mean, there is sadly no mnemonic which will do the job. However, 1 out of 4 not utilising the word itself is fairly good going, I think. The precision in the words that we do use, though, will pay off. Sum is succinct, allowing the focus to be elsewhere. It also does not place too much emphasis on addition. Yes, in raw form, we will be adding up numbers. But to even get there in a table, we must first multiply. Quite rightly, we will explain to pupils that this is, in fact, repeated addition. However, we must recognise that in the environment where they most need it, without a safety net, many will not necessarily appreciate the nuance. If this means a compromise of them remembering a weird looking method, but associating it with the word Sum, then so be it.

Peter Mattock raised a poignant question, about whether confusion would arise if pupils are given a question when the sum has already been found (e.g. mean weight of a sack of apples which has already been measured). My counter to this is that we still want pupils to recognise that a sum has been found; we do not necessarily have to be the ones who have calculated it.

Median seemed to be the idea which gained the most traction post-conference, particularly when put into the ether by Jo Morgan in her takeaway tweets. Middle has, ironically, always been the safe mid-ground for teachers when it comes to remembering median. And yet, it is rife with ambiguity, leading to one of the most common misconceptions whereby pupils find the geographic middle, as opposed to ordering them first.

I cannot think of any good reason against making the switch to Medium as your descriptor ahead of middle. For one thing, it sounds more like the original word than middle does (at times I found people even forgetting which one sounded most like middle, very useful!). More importantly though, the word medium already implies size order. This is shown very easily visually. A simple sketch of some item of clothing (I often use a sideways-facing rack of trousers) will show pupils that finding the medium does not always mean to look geographically centrally.

I often do this on the whiteboard, but this photo from IKEA will do the job. If I ask pupils to find the medium pair, they are far more likely to indicate the right-hand pair. This may even lead to comments like ‘Oh, so we don’t always look in the middle for the medium sized option?’. Follow this up immediately with a small, odd set of numbers:

4, 7, 5

Which is the medium number? Then with a set of 5, and one larger set (say 11) to demonstrate the point. This can then hook them into the methodology, rather than just taking our word for it. All of a sudden, this has become a subliminal transition from I Do to We Do, when previously the pupil may be yet to practice or comprehend these ideas.

The altering of mode’s definition may seem more pedantic in nature. However, you will not think this when you see pupils grappling with a grouped frequency table, all the while cursing their chosen deities that they are not using a ‘commonality table’ or ‘how many there are table’. Our pupils are more than capable of understanding what frequency refers to. If you ask them what it means to be a frequent bus service, or to have a high frequency of guests, they will usually know with very little provocation that it means ‘often’. Why, then, do so many teachers avoid exploiting and exploring this?

I often think that the trend of ‘dumbing down’ language, particularly for lower attainment learners, is both insulting and harmful. I would much rather have the learner remember a few more words for specificity, than choose the ‘easy’ way out, and have them not actually comprehend the concept the word is conveying. This is without even comprehending maths’ everyday role in supporting literacy standards. Alas, this Part 1 blog is already nearing the length of an article in The Athletic, and that idea deserves its own time.

To round off the less utilised terms is the application of distance to range. Firstly, it again means that the word itself lends clues to the pupil. This can be hugely significant in exam environments, where the wording is your only potential ally. Over the last few years I have some go-to analogies to fit this definition, including driving range and a range of goods. Even the store The Range encompassing a large distance due to its range of goods. One of my latest favourites was actually suggested by a student this year: Range Rover.

Secondly, not only does this word give its own clue to identity, but also implicitly conveys its own method. I currently teach FE, to Level 2 Functional Skills pupils. Many begin disillusioned with methods they’ve already seen, and tricks to memorise which have never worked. However, whenever I ask them to find the ‘distance between’ this range of numbers, they immediately look to subtraction. I almost never have to hear ‘biggest take smallest’, let alone vocalise it myself.

I really hope, even if you feel you have superior ideas to this, that it can generate discussion amongst your department or trust. Take time to think about why you use the wording that you do. And if there is not sufficient reason why you’ve chosen it, ask yourself if there should be one.

In the next part, I will talk about how I would sequence the topic to also allow these ideas to germinate to their fullest. If there are any replies to this, I may also dedicate a later post to answering any queries or suggestions made. Until then, I hope that you enjoyed this first deep-dive into averages, and look forward to exploring more.