*Teachers of mathematics seem not to be particularly fond of crocodiles/alligators when they are used to give meaning to the inequality signs $<$ and $>$. What are the problems with doing this, and should we resist all anthropomorphising or zoomorphising of mathematical symbols?*

It's long been noticed that a crocodile's mouth looks a little bit like an inequality symbol, $<$. Furthermore, crocodiles are (apparently?) greedy and, when given the choice, always eat the larger object (Figure 1). And so we can use this as a rationale for writing $3<4$ and $4>3$.

*Figure 1. Will a hungry crocodile eat a mouse or an elephant?*

Now, framed in this way, clearly this is a bit silly. Even very young children are likely to wonder:

What if the crocodile isn't hungry?

What if the smaller animal is tastier, more nutritious, easier to catch, or less likely to attack than the largeranimal?

Is this 'just a bit of fun', not to be taken too seriously? Are we showing a 'sense of humour failure' by making a fuss? Or is this perhaps where, even at a really young age, children begin to subtly experience being asked to suspend all common sense - and, indeed, their age-appropriate knowledge of biology - when doing mathematics? Perhaps this is an example of when children begin to get used to the idea that success in mathematics means accepting nonsensical statements and claims? Before long, learners will talk as though some things are "true in maths but not in real life". Is this *zoomorphism* of an inequality symbol the thin end of a $<$-shaped wedge that we would just be much better off without?

I tend to think that the crocodiles are really unnecessary here. Certainly, the $<$ symbol has a 'small end' and a 'large end'. The greater quantity goes at the larger end, and that's all you need to know. There is no need to bring animals with long jaws into it at all. But are my observations on the shape of the $<$ symbol also playing with symbols in an unhelpful way and part of the same kind of problem - trying to make the 'abstract' symbol more 'iconic' than it really is?

What does the research say? Wege, Batchelor, Inglis, Mistry and Schlimm (2020) found that mathematical signs that visually resembled the concepts they represent were easier to use than those that didn't, and they advised, for instance, "choosing symmetric symbols for commutative operations and asymmetric symbols for non-commutative operations" (p. 388). It seems that even experienced mathematicians find it easier to work with symbols whose properties mirror those of the concept being expressed. Imagine being forced to use $<$ to represent 'greater than' and $>$ to represent 'less than'. There is more than unfamiliarity to overcome there, but an obstinate reversal of the 'natural' way round that fits the meaning of these symbols. Although symbols like this are clearly arbitrary, in that they could have been otherwise, that doesn't mean that they were created totally at random, without rhyme or reason.

The equals sign $=$, for example, originates from the idea of expressing ‘equality’ of the left-hand side (LHS) and the right-hand side (RHS) by using two, parallel, ‘equal’ lines:

LHS $=$ RHS

Understood this way, one of those lines can be thought of as representing the value of the LHS of the equation and the other one the value of the RHS of the equation. But it is unclear which is which. This doesn’t really matter, I suppose, but perhaps this symbol would be more transparent if if it were rotated through $90^\circ$:

LHS $\lvert \rvert$ RHS

Now, the left-hand vertical line represents the value of the LHS and the right-hand vertical line represents the value of the RHS, and these being the same length represents equality of the two sides.

I suppose there could be the danger with this of thinking that $\lvert \rvert$ was the number eleven, but we get round that kind of ambiguity all the time with mathematical symbols. For instance, we use modulus symbols to write things like

$\lvert -11 \rvert=11,$

carefully making the modulus lines a little longer that the $1$s, and we seem to get away with having at least five different meanings for a short line segment in this equation.

But the really nice thing about using $\lvert \rvert$ for equality would be that we could use very natural symbols for greater than and less than:

$3 < 4$ becomes $3 $ |$\rvert \; 4 $

$4 > 3$ becomes $4 \; \lvert$|$ \; 3$

The ‘rule’ (if you even need to call it that) is that the shorter line refers to the smaller side and the longer line refers to the greater side. But it hardly even needs saying. You could just start using symbols with this level of transparency, and learners would quickly infer what was going on.

I am not saying this would be worth doing. This post is not really making any practical suggestions; rather, it is a thought experiment. When might there be a benefit in replacing something arbitrary with something a little bit less arbitrary (though still arbitrary!)? Or in involving learners in discussing what kinds of symbols they think might appropriately represent various operations or relations? Should we always go straight to the conventional, correct symbol that 'mathematicians' use? Or are there times when it might be worth using more transparent but informal alternatives, while learners are in the process of getting to grips with the concepts, and then later transition to the more formal symbols, perhaps after appreciating some of the inconveniences with the less formal versions?

After all, writing $3 $ |$\rvert \; 4 $ is all very well, but how would you express $3 \le 4$, and how would you handle double (or more) inequalities, like $3 < 4 < 5$? I think it would get quite messy and awkward. In our usual notation, I do like the way in which we show 'approximately' by making the straight lines wiggly in $3 \approx 4$ and I like how we can show 'much less than' as $3 \ll 4000$. I like how in $\LaTeX$ we can even extend this by using the code '\lll' and write $3 \lll 10^{100}$, with a symbol composed of *three* less-than symbols.

I am left wondering whether it is any worse to call the $<$ symbol 'a crocodile' than it is to refer to $\bar{x}$ as '$x$ bar' or $\hat{x}$ as '$x$ hat'? Can these really be 'informal' names if they are what 'everyone' calls them? I notice as I type these in $\LaTeX$ that 'bar' and 'hat' are precisely the words I need to type to produce them, so knowing these names is actually quite useful.

### Questions to reflect on

1. When is anthropomorphising or zoomorphising mathematical symbols OK and when is it not?

2. When are 'informal' names for symbols OK and when should they be avoided?

2. Is it ever worth introducing made-up, informal versions of symbols (or names for them), with learners? Can they be a useful stepping stone towards formal symbols, or are they just extra things that learners will have to 'unlearn' later?

### Reference

Wege, T. E., Batchelor, S., Inglis, M., Mistry, H., & Schlimm, D. (2020). Iconicity in mathematical notation: Commutativity and symmetry. *Journal of Numerical Cognition, 6*(3), 378-392. https://doi.org/10.5964/jnc.v6i3.314

Thanks as ever for an interesting and thoughtful blog. I would say it is okay when it makes things more fun (with obvious caveats to avoid racism or engendering unhelpful beliefs for life). I think its main use is when introducing ideas - I remember learning the game of chess from a book where pieces talked to one another, which I liked.

ReplyDeleteInformal terms: to be controversial (ignorant?) I actually think we should use the terms "top of a fraction" and "bottom of a fraction" - I think they are just as precise, and IMHO clearer. I think informal terms can be helpful to aid understanding, but there is a big potential danger of then not recognising standard terms (with "top" and "bottom" above I these should be the standard terms) or being confused by having two terms for the same thing (since a very reasonable assumption is that if they were the same, there wouldn't be two terms!)

Using own terms I think is helpful for a sense of ownership, and particularly good in say exploring a problem where the student spots some property, and it's really helpful to give it a name. In the debrief, I think it is helpful for the teacher to map these to any standard terms/concepts e.g. "parity", "odd function".

Really interesting thoughts, Colin. I wonder to what degree relating inequality signs to magnitudes hinders when we get to negative numbers, though. Is thinking of 3<4 in terms of 'smaller' and 'bigger' problematic when we get to -4<-3, when -4 is a 'bigger' negative than -3?

ReplyDeleteReally interesting. With your alternate vertical lines, and visually linking equality with inequality, I think there is also the potential to link the horizontal lines - an equal sign becomes an inequality by the two lines getting closer together at one end and therefore further apart at the other - no need for the crocodile, but still exploring the visual

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