#
Mathematical

Salad

## Seeing Spots

29 June 2018

How many spots did you see?

How did you know that?

Try again. Did you count them? Or did you ‘just know’?

The tendency of the human brain to spot patterns in the world around us is well known: it’s how we learn to recognise faces and familiar shapes and develop language. From a mathematical point of view, pattern spotting can contribute to the development of counting and number sense. Iconic patterns such as the arrangement of dots on dice are some of the patterns encountered in childhood and can be significant building blocks for the development of ideas such as estimation and additive reasoning.

Until very recently, I didn’t really ever think about how I knew that, for example, this was six and that was five – I just did. I hadn’t thought about the fact that this ability to ‘subitise’ numbers (Piaget: *an ability to instantaneously recognise the number of objects in a small group*) was something that developed when I was growing up – mostly at home I suspect – playing games that happened to familiarise me with these images. It’s something that I unconsciously assumed my students also possessed, but I never once thought to dig deeper into how and why I might need explicitly to encourage it in support of their mathematical thinking and reasoning.

So, aside from being able to count small groups of objects quickly, what mathematical benefit does this have? It’s more about our ability to look for and identify an underlying structure; in this case with numbers and our number system. For example, if presented with a larger number of dots, we tend to subitise into smaller groups in order to determine how many there are. See what you do with this example:

Do any of the images below match your own visualisations of the dots?

Consider the rich dialogue that we could encourage and facilitate with a group of children based on this example:

- • Recognising that the same quantity can be partitioned in different ways
- • Noticing the additive structure in the ‘part-whole’ relationship and talking about this using lots of examples
- • Getting a sense of more general additive properties such as commutativity and associativity
- • Thinking about notions of equivalence
- • Talking about the relationships between numbers of dots, introducing symbolic notation if we wish, as a helpful common language and short-hand for the number sentences that we identify
- • Finding that some multiplicative relationships emerge. In particular, there are also some interesting connections to be made to the rectangular array-structure, something that is likely to be built on as children develop additive into multiplicative reasoning and an understanding of area and area models of multiplication.

Subitising is clearly only one instance of pattern-spotting, but as a ‘sneaky’ context for thinking more generally about mathematical structure it is powerful.

With this in mind, we come to a game, Mexican Train, that recently caught my interest. It is a simple game based on dominoes, in which the object is for a player to play all the dominoes from their hand onto one or more chains, or ‘trains’, emanating from a central hub or ‘station’.

The most appealing feature of this game for me however, is the pieces themselves – the game is played with a double-twelve set of dominoes! I found this intriguing and the pieces particularly lovely to handle and work with. I don’t know about you, but I certainly found myself having to think much harder about the tiles beyond the typical six-dot arrangement and it forced me to think differently about my counting strategies. I started to ask myself questions about the arrangements and what made them easy or hard to identify. As an example, take a step back and consider the four arrangements of dots pictured below. Try not to actually count the dots. Think about these questions:

- • Are there an even number of dots? How do you know?
- • What’s the same and what’s different between these images?
- • Can you see image 1 in any of the other images?
- • Can you see image 2 in any of the other images?
- • Are images 3 and 4 the same?
- • How many dots are there in each image?

Many suggest (for example, Clements, 1999) that subitising is an important mathematical skill that may be fostered from an early age, and which can enhance overall number sense. Perhaps, in this technological world, children are playing fewer ‘hands-on’ games involving dice, cards and dominoes and so simple ways of exposing them to such resources can surely only serve to enhance their mathematical repertoire.

Reference:

Clements, D. H., 1999. Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5(7), p. 400.