The Volume of a Can of Worms

Following a recent thread of messages on Twitter about square centimetres vs. centimetres squared, triggered in part by this poll, I thought I would summarise my thoughts on the issue and its slightly wider context.

Healthy Debate

From the outset, I do want to emphasise that I see debates such as this as incredibly positive. Their participants are clearly eager to discuss practice, open up about potential misconceptions and, ultimately, ensure they are teaching their students correctly. All of this indicates a healthy, professional discussion.

However, points raised in this way run the risk of being misinterpreted (especially when half of the 140 characters go to usernames of the people involved!) or perhaps the discussion gets diverted down a series of blind allies and crossed wires. This is especially the case when there is not necessarily a single correct answer.

The question in the poll asked how one would read the units “cm²” and gave three options:

  • centimetres squared
  • square centimetres
  • it doesn’t matter

Right and Wrong in Mathematics

Many people love mathematics for being the subject where answers are either ‘right or wrong’: the formalism brings a sense of security. This certainty stems from the very precise definitions laid down for mathematical objects and these are often sufficient to settle arguments about misconceptions. Is 1 prime? No. Why not? Because the definition of prime has been very carefully crafted to deliberately exclude 1. (A much more interesting question which, sadly, people don’t often ask is why do we choose to exclude 1?  How much damage would it do? That particular path can take you a long way into abstract algebra. The paper What is the smallest prime? provides much interesting history.) Does the symbol √ represent both the positive and negative square roots of a number? No. Why not? Because it is defined to be a function (or operation if you prefer) from the set of non-negative real numbers to itself. (Again, try asking why we would define it that way. What would be the harm otherwise?) A couple of days ago I prepared some notes on this issue (pdf).

Beyond the Definitions

The difficulty in the current debate is that there are some issues that aren’t appropriately dealt with by a definition. Think about pronouncing the words ‘loci’ and ‘foci’. Are you in the low-kye or low-key or low-sye camp. Who’s right? Well, according to Oxford Dictionaries, we all are. One would hope that as an individual your own usage is consistent (mine notoriously isn’t, so I often opt for the exaggerated locussesesesss…) and perhaps it would even be worthwhile having consistency throughout a department, if only for the kids’ sanity. Although it is quite amusing at A level to hear students play teachers off against each other in the ‘shine’ or ‘sinch’ debate…

Square or Squared?

Returning to the debate that I set out to discuss, the question is whether ‘square centimetres’ or ‘centimetres squared’ is correct. And the hardest part of that question, unfortunately, is the word correct.

Mathematics won’t answer the question because it’s essentially one of pronunciation rather than one of satisfying conditions.

Oxford? Well, they offer up both alternatives essentially as synonyms:

Screen Shot 2016-02-18 at 22.32.21.png

Screen Shot 2016-02-18 at 22.32.04.png

To be honest, I think we are asking the wrong question. Neither can be called incorrect. So what actually matters here?

Better Questions

There are more valuable questions to debate and it is within these that the interesting discussions lie:

  • Should we be consistent in our own usage? (I.e. we choose one for ourselves and stick with it.)
  • Would it be advantageous to agree as a department on consistent usage?
  • Does either of the options risk causing ambiguity or creating misconceptions for students?
  • Does precluding one or the other usage cause potential conflict with future learning?
  • Do any formal assessments make use of one or other (or both or neither)?
  • How are units written and read more generally in science and engineering?

My Own Answers

Here is where I show my cards and give my own personal views on and partial responses to the above questions:

  • I think consistency is important to give students the security and confidence to express their ideas. Agreeing (one way or the other) across a department is eminently sensible. From a literacy point of view, however, I think it is no bad thing to explain to students that there are two common ways to read these units in day to day life.
  • One common argument against a phrase such as “5 centimetres squared” is that it could refer to square of side 5cm, with area 25cm². My own personal take on this is twofold: firstly, I feel it violates the order of operations (ok, I admit parentheses don’t fair well in spoken language): we happily read 5x² as 5-x-squared without concern for it being interpreted as (5x)²; secondly, there are much more careful phrasings for that situation: a square of side 5cm, a square measuring 5cm by 5cm etc. I would discourage anyone from using the phrasing “5 centimetres squared” to describe a square by its side length.
  • I have not looked too far into formal assessments but almost always units are given and expected in their symbolic form and not expressed ‘longhand’. However, consistent with day-to-day usage, I did find a Level 2 Functional Maths paper that refers to a “1 cubic metre bulk bag” of bark.
  • As students move to study higher levels of science, units play a more and more important role culminating, it could perhaps be said, in dimensional analysis. Teaching at A level, we usually write velocity and acceleration as ms-1 and ms-2. I often lazily read these as “m s to the minus one” and “m s to the minus two”, respectively. Or perhaps I say negative two. (That’s another can of worms for another day…). I have no issue with “metres per second squared” but “metres per square second” sounds awful. I have to own up and say I’m also not a big fan of “metres per second per second”.

So what’s my stance? I read them the same way I read algebraic variables: centimetres squared.

How did I vote? “It doesn’t matter.”


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