# Discreating

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- Discreating

## Discreating

We’ve all done it – found ourselves with leftover food that we decide to keep, so we select a container – maybe a bowl, a mug or a plastic box – and start to fill it… only to realise that the container isn’t big enough. I often reflect on this as I continually repeat the process of convincing myself that a packet of pasta will fit inside a particular jar, and only when I start pouring it in do I realise that, once again, I have completely misjudged the capacity of my container. Again and again I do this – my cupboard is full of jars filled to the brim with dried pasta, beans, lentils, etc., and next to them a large plastic box that contains all the left-over bits that didn’t actually fit in the jar I had allocated. Sometimes I get bold and decide that to tidy things up I will transfer one type of pasta from the large jar it currently occupies into a smaller one. Inevitably this doesn’t work either, as I still fail to estimate correctly the volume of the pasta to be transferred…

It fascinates me that I seem to find this process of comparing capacities of different containers so difficult. I consider myself to have reasonably good number sense and spatial awareness – but I almost always get it wrong!

In the classroom there is often an unspoken assumption that children should recognise when answers to calculations are unreasonable and much of this is to do with them relating the numbers they have obtained to their own ‘real’ experience of a particular scenario. Volume and/or capacity is a prime example of this as children are often dealing with numbers without any sense of their relative size. I have been reflecting on this of late. How many times have I asked children to calculate the volume of cuboids but never actually asked them to consider how big that would be in reality?

The issue of capacity vs volume is also an interesting distinction in real-life situations. Dried pasta is always tricky to transfer to a container because it’s bulky and the pieces don’t ‘fit’ together in a particularly efficient manner. Sugar, on the other hand, is of course much easier, due to its shape and size. So in real terms the capacity of a container is dependent upon the thing that I want to put inside it.

Of course, in the image above, the notion of discrete and continuous is problematic as it’s all a matter of proportion and scale. What I mean by this is that if we were trying to put potatoes in the rice jar, there might be obvious problems. If, however, humans were ‘scaled up’ to a height of, for instance 20m, then potatoes might appear to them as rice appears to us… food for thought?

My other problem of same item – different container is related to variation and degrees of freedom. Placing a selection of mathematically similar containers in order from least to most capacity is considered a relatively elementary task. However, in real life, the containers we are dealing with may well vary in too many ways to be able to compare them well without numbers.

It is often forgotten – or perhaps not considered at all – that giving children opportunities to estimate measures, not just the answers to numerical calculations, can improve their general sense of number and encourage them to make connections among the mathematics concepts they are learning and the skills they are developing (e.g. Sriraman & Knott, 2009). This, in turn, can contribute to better reasoning with numbers, flexible thinking and understanding of relationships between numbers.

Maybe we should be spending more time asking children (and all of us for that matter) to pour quantities from one container into another, using the language of volume and capacity and comparing common as well as unusually-shaped containers or objects in terms of what they can hold. This is something that my colleague Rachael Horsman feels very strongly about and it has influenced her work on the Cambridge Mathematics Framework.

If you want to challenge children’s intuitions about volume and capacity, you could look at something like this, and expose them to those things that are usually avoided in the mathematics classroom because they don’t belong to the set of things composed of cuboids and other ‘standard’ objects.

And, the next time we feel the need to have our students calculate the volumes of cuboids, perhaps instead we could consider doing a ‘hands-on’ activity like this. Maybe we could even go as far as to use something like pasta or rice to compare those volumes and talk informally about other mathematical ideas like multiplicative relationships (this box holds nearly twice as much as this one, this box holds about a quarter of this one).

There is also the lovely idea that we can keep adding different-but-smaller items to a container, filling in the gaps, which renders the idea of ‘full’, well, full of philosophical and mathematical issues to discuss. Does this work ad infinitum? Is there a connection to a sort of inverse Sorites paradox here?

How would you define something as ‘full’, and why?

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