Dr Alison J Price of Oxford Brookes University explains why understanding the relationship between numbers, and the connections between calculations, is an important part of developing mathematical awareness, and how this can influence delivery of the curriculum.
What is mathematics?
Four year old Mark when hearing about ‘maths’ asked, ‘is it a kind of game for grown ups?’ (Owen and Rousham 1996). I recently asked a group of students studying for an Early Years Foundation degree, all of whom work with children in foundation stage settings, ‘What is mathematics?’ The answers that I got were predictable – It’s about sums, numbers, shapes, calculations, numerals, etc, but one student exclaimed ‘I have never thought about what it is – it’s just what they taught me at school – but what is it?’
The Chambers Twentieth Century Dictionary (1977) defines it as ‘the science of magnitude and number, and of all their relations’, but in fact we have extended this to include aspects of shape and space and algebra. It is interesting in passing that we then invented the word numeracy to clarify that we meant the number aspect of mathematics, but have now redefined numeracy (as in The national numeracy strategy [NNS] 1999) to include everything again.
My answer to the students was that mathematics is one particular way of making sense of the world.
Let’s consider a tree
We could look at it from the point of view of language and describe it or write a poem about it; from art and paint it; from history and explore how and when it came from Australasia to Europe; from geography and chart its travels or examine its habitat; from science and discuss growth and respiration; or from mathematics and measure it, count its branches, calculate its loss of leaves over time, etc.
The foundation stage curriculum gives mathematical development as a separate topic, but in fact it is only another way of ‘Making Sense of the World.’ ‘So’, says, one of my students, ‘if it is a way of making sense of the world, then it ought to make sense!’ Why did it not make sense for this student, and many of her colleagues?
Making sense of mathematics
Let me give you a task first I want you to remember the following letters of the alphabet in this order as quickly as you can
B A T F L E U I U
In order to do this you probably first looked at the letter to see whether they made any sort of sense. BATFLEUIU? If they don’t, the task becomes more difficult.
What if we break it up in the following way B A T F L
E U I U
B E A U T I F U L
Would this help you to remember the letters in the right order? Perhaps you could generalise a rule that says something like ‘out of the word beautiful, take first the odd letters (1st, 3rd, 5th…) and then the even letters in order’. Could you remember it then? Once you have got the bigger picture, the detail slots into place.
One of the reasons why many people find that mathematics does not make sense is that they have never been able to see the bigger picture. So, Julia Anghileri says that ‘what characterises children with “number sense” is their ability to make generalisations about the patterns and processes that they have met, and to link new information to their existing knowledge’ (Anghileri 2000). And this sense making starts in the foundation stage. So, how can we help children to make sense of mathematics in this way?
When I studied the relationship between teaching and learning early number in reception and year one classes I found two different approaches to the curriculum mathematics. In one, which I described as Atomistic, the topics were broken down into small steps (like breaking a material down into its constituent atoms) and taught with little reference to other topics. In the other, which I call Holistic, the teacher started with an overview and was able to relate the parts of the whole as the work progressed.
The children who were taught with a holistic approach to the curriculum were more successful at mathematics than those taught in an atomistic way. Because they could see the links, they could understand and remember more – just as you could learn the letters in the sequence more easily once you could see how they were related.
I want to look at some aspects of these relationships which are crucial to arithmetic at an early years level.
Starting from where the children are
Traditionally the early years curriculum was heavily based on psychology research, especially that of Piaget and his ideas of conservation of number. So the curriculum comprised lots of ordering, sorting, matching and one- to- one correspondence which were seen as ‘pre-number’ skills which would lead into counting and calculation (Thompson 1997).
However, we now know much more about young children’s understanding of number. For example, we know that even small babies can recognise the difference between small sets of objects and will get concerned if something unexpected seems to have happened. Wynn (1998) has been able to show that babies as young as five months are able to distinguish between sets of two and three objects and by 13 months can distinguish between four and five. By the age of two or three years children can tell the difference between three sweets and five sweets shown to them but are unable to say whether five is more than three if asked as an abstract question. They rely more on perception than language for understanding.
From these, we conclude that Piaget was incorrect in thinking that children could not think mathematically before the age of seven or so, but their thinking may be more embedded in what can be seen and is relevant to them.
So, the early years curriculum now has a much higher emphasis on number and counting, than it had previously. Children do need experience of sets of objects to count and compare but the emphasis is on the numerical – this set has more objects that that one because this has five and that has four – rather than pre-counting skills. And, while one-to-one correspondence is an important part of counting, to match one counting word to one object, Ian Thompson (1997) reports on one piece of research that indicates that teaching ordering and classifying ‘pre-number skills’ does not develop counting skills, while teaching counting also develops the ‘pre-number’ skills.
This is not to say that ordering, sorting and matching are not needed in the foundation stage since they contribute to an understanding of handling data which is later seen in diagrams and graphs, but they are no longer seen as essential pre-number skills.
So, we can see how the curriculum for mathematics in the early years has drawn on an understanding of what young children can already do, and focuses on making links between these emergent number skills and the foundation stage curriculum. So, how can we help children to make sense of counting?
Using patterns to make sense
The ability to see a pattern allows us to remember and understand something more easily.
Let’s try another task. How long will it take you to learn the following sequence of numbers?
1234, 1243, 1252, 1261, 1270, 1279, …
Your first reaction was probably – ‘Oh dear, biggish numbers, is there a pattern?’ You might have noticed that the tens are going up by one and the units down by one. Until you get to the last one!
If you are still in the dark, try adding 9 to the first number – you should get the next and so on. So, you could learn this sequence by saying to yourself, ‘start with 1234 (not too difficult to remember), then add 9 each time’.
One of the key patterns in early number is that of the shapes that the sets make when arranged in an ordered manner.
Such patterns are used in dice and dominoes. Tacon and Wing have used such patterns to develop a structured apparatus called Numicon (Tacon, Atkinson & Wing, undated) which is based on learning numbers through such patterns and using these to work on addition and subtraction.
This structure allows the children to see how the particular number is related to the other numbers around it, numbers on either side, odd and even numbers, etc. You might want to consider what you have in your setting which shows the patterns in number and how you could use these to develop the children’s understanding.
Making sense of the number system
Before the NNS many of the mathematics schemes for reception-aged children emphasised the cardinal aspect of number (how many). Children would be asked to focus on one number and see how it could be represented in different ways.
The following week they would ‘do’ the number five, etc. But there was little emphasis on the relationship between these numbers.
This relationship is most easily seen in terms of ordinal number – which relates to 1st, 2nd, 3rd etc, but more importantly relates to the order that the numbers come in; 1, 2, 3, 4, 5… One of the most important things about four is that it is one more than three and one fewer than five. If the numbers are taken individually this relationship is not clear and it is this relationship which is at the heart of mathematics. For, if you do not know what one more than four is, how can you solve 4 + 1 and then go on to solve eg 4 + 3? (Anghileri 2000, p17 explains this further).
So, making explicit the ordinal relationships between numbers, using a number line or track, will encourage an holistic learning of the number system rather than learning each number in isolation.
Making sense of calculation
In schools, prior to the NNS, lessons would often focus on a single number operation; addition, subtraction, multiplication or division, with little discussion of the relationships between them.
When calculating mentally people often use the inverse relationship:
- between addition and subtraction (for example: to subtract one number from another they add on from the smaller one to the larger),
- or between multiplication and division by solving 56 ÷8 because they know 7 x 8 is 56.
They use repeated addition when they do not know the multiplication tables for calculating eg 7 fives, by counting in 5s: 5, 10, 15, 20, 25, 30, 35.
Similarly, it is possible to use repeated subtraction to solve a division problem, but more often people will use two relationships here. For example, when solving 42 divided by 7, they use repeated addition of sevens to get to 42 and then use the fact that multiplication and division are inverse to solve the problem.
So the four number operations can be seen as clearly related to one another.
While this is mostly beyond the mathematics you may be teaching in foundation stage, it is important that you bear it in mind as you begin to teach addition and subtraction concepts. Many people in the past created an understanding of these relationships for themselves without being taught them, they were the ones who had made sense of mathematics, but many others were not able to do this for themselves. Teachers are now encouraged to be explicit about them so that all children have the opportunity of making mathematics meaningful.
To sum up
The Curriculum Guidance for the Foundation Stage guides teachers into what to teach and how. In this article I have set out to clarify what the connections are that you will need to be making with your pupils.
So, by keeping in mind the relationships and patterns between numbers and between number operations, perhaps you will be able to help the next generation of school leavers so that they will not be surprised to hear that mathematics is about making sense.
Dr Alison J Price is a senior lecturer in primary mathematics education, Oxford Brookes University
- Anghileri J, (2000), Teaching Number Sense, London: Continuum.
- Owen A and Rousham L, (1996) ‘Maths – Is that a New Kind of Game for Grown-ups?’ in Whitebread D, Teaching and Learning in the Early Years, London: Routledge.
- Tacon, R, Atkinson, R, and Wing, T, (Undated), Learning about Numbers with Patterns: Using Structured Visual Imagery (Numicon) to Teach Arithmetic, London: BEAM.
- Thompson, I, (1997), Teaching and Learning Early Number, Buckingham: Open University Press.
- Wynne K, (1998), Numerical Competence in Infants, in Donlan, C, (ed) The Development of Mathematical Thinking, Hove: Psychology Press.