So. Number Bases. Very few primary teachers even know what these are, and yet they are required to teach at least 2 of them to children before the age of 7.
Consider these questions: “What is a carrot?” “What is a labrador?”
These are now the two questions with which I often start PD sessions on Place Value and Number Sense. They generate some raised eyebrows, yet these questions turn out to have an important part to play in children’s understanding of place value.
Anyone who has studied the new primary curriculum will be acutely aware that without a deep understanding of place value, children will struggle to progress successfully from Key Stage 1 to Key stage 2.
In late 2013 and early 2014 I had the opportunity to attend a training event aimed at people involved in delivering professional development in schools. One of the resources used there was straws, which we traditionally put in bundles to demonstrate the ‘ten-ness of ten’. But that was where it stopped, and I have always felt that powerful though that image is, there is potential for more.
Serendipitously at the time, I was reading a book by Zoltan Dienes, Hungarian Mathematician and inventor of Dienes blocks, who coincidentally had died only that week. Dienes puts a huge emphasis on children playing with numbers, and bases, and this combined with the nagging feeling that we were missing a trick with the straws set off a train of thought in my mind which has led to this article.
The pattern that we call ‘place value’ is of course the number system known as ‘base 10’. Simply put, we arrange numbers in columns, in such a way that each column represents ten times as much as the column to its right. But this is a complex idea, and base 10 is far from the only base in town; we are free to choose to count things in ANY number base, such as binary (base 2), hexadecimal (base 16), time (base 60 or base 24) and so on. We use base 10 for convenience but it is simply a subset of a whole range of counting systems.
So – how did you answer the carrot and labrador questions? In my experience, over 90% of people will use the word ‘ vegetable’ or the word ‘dog’ in their explanations.
Why is this so? Undoubtedly because we aim to define things by seeing them as a type of something, or part of something we already understand; we then look for similarities and differences to help us define things. So it is with base 10; it is a part of something highly structured, yet my concern is that 75% of primary teachers with whom I come into contact have NEVER studied any number bases other than base 10.
How then can we expect teachers and children to have a truly deep understanding of place value if they have only ever seen a single example, limited to a single base? It felt to me that the straws offered an opportunity to work with teachers and children in a multi-sensory way, learning about a range of number bases. I set myself the goal of making it so simple to do that Reception and Year 1 children would cope with the activity, and hence build a solid understanding of place value.
And so ‘Six-land and Beyond’ was born. It is based on a conversation I had some years ago with Dr Tony Wing, my inspirational university maths lecturer, and I think may be reminiscent of an activity he used to do with children, so is by no means original. So far I have trialled it in three schools, and many other teachers who have been on my training courses have tried it with their own classes, with many reporting encouraging results.
Equipment: have ready drinking straws with flexible tops, rubber bands or equivalent, and sand trays or jars, and a flipchart or large sheet of paper and pen.
Introduction: Start with a group of five or six Reception or Y1 children, Explain to the children that today we are going to visit ‘Six-land’. Check that the children can count up to six objects reliably, as this is a pre-requisite skill. Explain that they each will have some straws, and they must count them. But they have to watch out – in Six-land, nobody can count higher than six!
Ensure that some children have fewer than six straws, some have exactly six, while others have between seven and fifteen in order to set up the ‘problem’ of how we count MORE than six things.
Allow the children with fewer than six straws to count and say how many they have. Explain that there are some trees in Six-land, and you need exactly six straws to make a tree. Ask who has enough straws to make a tree, and allow the children with exactly six straws to bundle them to make a tree, and those with more to bundle as many sixes as they can (those children with more than eleven straws will be able to make two trees. Children may need help putting rubber bands around groups of six straws. The ends of the straws can be bent down to make the branches, and the tree can be put in the sand tray or a jar.
We can now talk with the children about what we have (how many trees, how many straws left over.) When appropriate, we can make a note of what we have done. We do this by drawing a straw (actually a ‘1’ but the bendy top is folded down!) and the letter ’t’ as the first letter of the word ‘tree’ and its initial sound.
Encourage the children to count the trees, and either you or they write down how many they have, under the ’t’. They should then count the loose straws that remain, and write that down under the ‘1’. This diagram shows how this would look for children who have fifteen straws. It is preferable if the children themselves are able to record what they have, but initially it might be wise to give them a clear model.
Special cases: zero plays an important role in this activity – by always starting with trees, those children who have six or twelve straws should be encouraged to write ’10’ and ’20’ rather than merely ‘1’ and ‘2’; this emphasises the concept of using zero as a placeholder from a very early stage.
Important note: I strongly believe that wherever possible, writing down maths should only be about recording something that we have actually DONE.
This can be done with small groups of children, who thus begin to learn to count larger amounts of things by grouping them.
The intention is to work with the group daily, visiting Sixland and finding out how many trees they can possibly make and then how many straws they have left. It also provides an opportunity for children to practise writing their numerals.
Over the week, gradually increase the number of straws that children have to count. If they are able to make 6 trees, explain that 6 trees make a wood, and they can group 6 trees together (see picture) to make a wood. They can if they wish put a larger band around the 6 trees and ‘plant it’ in a separate sand tray (see picture). This can be added to the recording as a ‘1’ under a ‘W’ for ‘wood’.
After children are comfortable with this activity, perhaps after four or five days, they should then be invited to visit ‘Seven-land’. Of course, in Seven-land, we can count all the way up to 7, and (amazingly) it takes exactly 7 straws to make a tree. And, even more amazingly, it takes 7 trees to make a wood. But you probably knew that.
After a week or so here, children can visit Eight-land for a few days, then Nine-land, and even, you guess it, Ten-land. Please do not rush to get to Ten-land, or children will miss the connection. Also vary the land that you visit – perhaps have some 6,7,8 and 9 numbers in a bag and allow children to pick one out or simply choose which of the ‘number-lands’ they will visit that day. This means that they might be in EIghtland one day but Seven-land the next, for example.
Crucially, during this process, no mention whatsoever is made of columns, tens, bases, place value or anything else that might get in the way of building these powerful concept images for children.
Over time children will start to build a rich and connected sense of how these numbers fit together and the underlying system behind place value, rather than simply the rules by which base 10 operates.
In my next article I will explain how this develops, and how it can potentially lead to very strong mental and written calculations in Reception and KS1.