The change of title within the new EYFS for this area of learning and development is interesting. Calling it Mathematics is to be welcomed as maths is much more than just numeracy. Equally, problem-solving, like creativity, is to be found (and should be promoted) across the curriculum in all areas of provision. Although of particular importance in mathematics, it is not unique to this area.
There is a common assumption, perhaps reinforced by the two aspects of 'numbers' and 'shape, space and measures', that mathematics in the early years is about counting to 20 and simple computation, knowing the names of shapes and how many knives and forks are needed for lunch. But it is much more.
In the first place, views of what counts as mathematics are changing. Origami, for example, used to be thought of as a fun activity that could be found in children's annuals. Today it is a well-respected branch of mathematics which supports work on robotics and which was used, for example, in the construction of the Hubble telescope.
Second, while there can be no argument about the things that have been included in the early learning goals for Mathematics, there are important aspects of mathematics that have been given insufficient emphasis. An important one is pattern.
Although it is given a mention within 'Shape, space and measures', this in no way reflects the pivotal role that pattern plays in mathematical thinking and understanding - not only in relation to shape, space and measures but also in numbers. Keith Devlin, an American professor of mathematics, argues that mathematics is the science of patterns. This includes abstract patterns, number patterns, movement, shape, time and behaviour patterns and, as he says, even voting patterns.
Although problem-solving is now mentioned in both aspects of the early learning goals, no indication is given of the vital importance that both problem-solving and, equally importantly, problem-finding play in mathematics.
Maulfry Worthington and Elizabeth Carruthers argue that the way in which we approach mathematics should provide society not simply with 'adders and dividers' but with 'seekers and solvers of problems and makers of new mathematical meanings'. Boaler suggests that what is needed to address the gap between maths in real life and maths in schools is a problem-solving approach.
There are other components of mathematical understanding that are of fundamental importance in learning to think mathematically, but these are not made explicit within the early learning goals. These foundational elements include:
- abstract thought, which is developed through the use of imagination and symbolic representation
- the development of mathematical competences such as estimating, predicting or hypothesising - in other words, guessing
- developing the language of mathematics. It is widely agreed that becoming a mathematician is like learning another language.
Sadly, talking about aspects of mathematics is given less importance within the revised early learning goals for Mathematics than was the case in the previous version. And it is talking about maths which supports understanding.
IS MATHS EASY OR HARD?
Mathematics is one important way in which children (and adults) make sense of the world. Despite the fact that most of the population claim not to be good at maths, we are born mathematical, and it permeates every aspect of our lives. We are born problem-solvers and seekers of pattern - key elements of mathematical understanding.
It has been shown that in the first months of life babies can discriminate between groups of one, two or three objects, actions or sounds. They can, for example, make a connection between two sounds and a similar number of objects. In experiments involving a box with a screen, it has been shown that babies watching how many toys are added or removed can identify when the experimenter presents the wrong number of objects.
Children use mathematics in their block play and construction games, and in deciding whether things are fair. We all use it to cross the road, to get to work on time or prepare a meal. It is vital in skills such as roofing, tile-laying and dressmaking.
Why when we have these abilities do so many people claim to find maths so hard? Maths is hard because our brains find estimation easier than accuracy. It's hard because it is abstract. The Numeracy Strategy described mathematics as involving 'imagined worlds' - worlds involving numbers and quantities so huge as to be beyond imagination and so tiny as to be unthinkable. For educators at all stages, this poses particular challenges. For early years practitioners, many of whom never enjoyed maths at school and never felt competent at it, the difficulties may appear insurmountable.
Even if and when children can count reliably to 5, or 10, or even 20, doubling, halving and sharing require different kinds of thinking. Arguably the most difficult aspect of the early learning goal for numbers is that of requiring children to count back to find the answer to a problem. Practitioners often argue that there is no difficulty since children enjoy learning to count backwards. Yes - many do, but this is not the real issue. There are very few situations where we naturally count backwards in order to solve a problem. We frequently count on as, for example, when counting change.
Counting backwards to solve problems generally only occurs when we have a visual representation, such as number squares and number lines. But in order to use these, we are asking a great deal of young children.
In order to take three from ten, for example, we are asking a child to point to ten on a number line, jump to nine and say one, jump to eight and say two and jump to seven while saying three - and then to declare seven to be the answer. That's pretty demanding - not unlike pointing at red and saying yellow, or patting your head and rubbing your tummy.
Similarly, many of the concepts included in 'shape, space and measures' are complex. Size is the most common way in which children compare - but size goes way beyond big and small. Taller, heavier, wider, deeper are all relative terms which have to be compared to something else. In addition, some measures, such as weight and time, are very difficult concepts in their own right. Even some much older children have difficulty in understanding money: why isn't that huge pile of small shiny 5p and 10p coins more valuable than that dull brown £1 coin? And time - a concept adults continue to struggle with - long after they've learnt to tell the time.
CURRICULUM LINKS
The wording of the early learning goals masks a great deal of complexity. However, the non-statutory guidance, Development Matters in the Early Years Foundation Stage, offers helpful advice about ways of supporting mathematical understanding.
The Characteristics of Effective Learning (included in the statutory framework of the EYFS, 1.10) are as true for mathematics as for any other area of learning and development. The way in which so many of us were taught mathematics may have led us to believe that these characteristics, or the ways in which we learn, are irrelevant. But the revised EYFS requires practitioners to pay attention not just to what children learn but to how they learn - in all areas of learning and development.
In order to support children's mathematical learning effectively, practitioners need to recognise the value of play and exploration. In play, children use all their senses -including balance and motion - and their actions underpin all conceptual development. They follow their interests and this signals to adults how children can best be engaged in learning. Above all, in play children are learning to make one thing stand for another - developing both the abstract thought essential to mathematical thinking; and an understanding of the way in which symbols come to represent ideas, which is also essential to mathematical understanding. Children's active learning encourages them to become persistent and involved - again important aspects of mathematics. Creating and thinking critically involves problem-solving and promotes the making of links or connections - such as pattern-spotting and making predictions.
LINKS WITH THE PRIME AREAS OF LEARNING AND DEVELOPMENT
Physical action is a vital part of mathematical development. The complex concepts involved mean that children need a great deal of experience in handling materials; in going through, under and over objects; in going around and between places. Communication of ideas is also a vital aspect of thinking, including mathematical thinking. This is much more than a question of vocabulary.
Three-year-old Hari was asked the shape of the sail on a small boat - he replied that it was square but later in the day playing with some plastic triangles he described them as being like the boat's sail. Without conversation, his thinking would have remained hidden. As it was, the adult was able to introduce the word 'triangle' informally - getting the word wrong was much less important in Hari's development than thinking about the concepts involved.
In popular thinking, Personal, Social and Emotional Development, the third of the Prime areas of learning and development, has little to do with mathematics. A moment's thought will correct that view. Almost all of those who regard themselves as being no good at maths will highlight emotional or interpersonal factors. They refer to hating the subject, to being anxious when it was time for a maths lesson. Motivation, enthusiasm, perseverance and other learning dispositions come from wanting to learn and, as Devlin says, 'the key to being able to do mathematics is wanting to'.
LINKS WITH SPECIFIC AREAS OF LEARNING AND DEVELOPMENT
Literacy
Reading and writing print share many features with learning to read and write numbers. Although they are separate systems, there is an overlap. Three may be written as a word, shown as a symbol (3) or represented as icons (O O O). Literacy is a concept that applies to both areas of learning and development.
An OECD report described mathematical literacy as involving sufficient mathematical understanding to ensure that in our private lives, our working lives, our social lives and our lives as a citizen we are able to play a full part. In our increasingly technological and scientific world, mathematical literacy, like being able to read and write, makes it possible to play that part. Although it's considered okay to say that we can't add up but not to say that we can't write, both undermine our ability to play a full part within our society. Some writers have even suggested that children have what they term 'a new civil right' to be taught mathematics in a way that ensures that they understand and can use its power in our everyday lives.
Understanding the World
Observations of the world around them enable children to see patterns and identify sequences. Witnessing changing seasons in the natural world and the recurrence of birthdays and festivals in the culture also contributes to children's developing sense of time. Interactions with technology support the growth of cause and effect relationships and ideas of sequence. These insights enable children to solve problems - a key aspect of mathematics as we have seen.
Expressive Arts and Design
Children need to use their imagination if they are to get to grips with the abstract ideas involved in mathematical thinking. Imagination involves thinking about things that are not present and so does abstract thought. Einstein famously developed 'thought experiments' in which he imagined himself, for example, accelerating through space. He wrote that 'imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world.'
Like literacy, the arts use one thing to represent another and thus support the growth of understanding about the symbols used in mathematics. The other major link with the expressive arts is to be found in music. In the 17th century, philosopher and mathematician Gottfried Leibniz argued that 'music is the pleasure the human mind experiences from counting without being aware that it is counting'.
Other writers link the development of human mathematical thinking to the use of fingers; to the development of dance or the role that rhythm played in bringing about an understanding of time. Whatever the truth there can be no doubt that music and mathematics are closely linked.
THEMES AND PRINCIPLES
Practitioners should not lose sight of the uniqueness of every child. Their learning journey will be different - their experiences of mathematics at home will be very different; their physical abilities may differ as will their interests. There has been a tendency in mathematics to teach as if it were simply something to be imparted by adults to passive child recipients. It is not - it is present in children's play, in their interactions with their physical and natural world.
Mathematical misconceptions are important in that they draw to all enabling adults' attention children's understanding - or lack of it - making the invisible visible. Enjoy the conversation, including the mistakes for what they are - a mirror on the child's thinking - and then reflect on how they can be helped in the development of their understanding.
Physical action increases mathematical understanding not just in its spatial aspects. This means that children with limited movement need additional help if they are to have similar benefits. New insights from neuroscience indicate that mirror neurons ensure that watching others doing the things they are not able to do will impact on their brains and thus support their learning in similar ways.
Equally, it is important to provide a wide range of resources that ensure that children with other disabilities can access resources. For children with impaired vision, for example, this may include providing tactile number cards, and number lines with large print, as well as resources with distinctive shapes or textures.
One of the most important tasks for practitioners wanting to improve the effectiveness of teaching and learning in relation to mathematics is to develop positive relationships with parents. The majority of parents, like the rest of the population, are likely to have negative views of mathematics. They may have been unknowingly transmitted to children or it may mean that they approach mathematics from an entirely different standpoint than that of the practitioners.
Most parents are unlikely to see the importance of story, music or play in their children's mathematical development, so it is important to explain it. Helping them to understand why you do things the way you do is vital. It takes a lot of trust to have confidence in the idea that time spent on the climbing frame is at least as valuable as time spent reciting numbers. This does not mean, of course, that we will never recite numbers, but the activity needs a context, children, buttons, stairs and so on.
DAY-TO-DAY MATHEMATICAL PRACTICE
The phase in which we work is the foundation stage and this means that we are laying the foundation for later learning. This is often interpreted as the basics - but basics are not simply the simplest aspects of a subject. The job of the early years practitioner is to open children's minds to possibilities; to help them to lay down pathways in the brain with which later learning can connect. The real basics are the tools that children will need to help them to think mathematically. This means that in day-to-day practice babies and children of all ages need:
- an environment alive with mathematical possibilities, indoors and out - exciting things to compare, to puzzle, to explore, to count, to describe. Don't forget some older children's enthusiasm for big numbers - having a large number of similar objects invites counting while small groups can easily be compared visually
- an ethos that encourages conjectural thinking, and promotes creative approaches to problem-solving by nurturing flexibility, perseverance and a 'what if?' approach to thinking and learning. Errors are accepted and children encouraged to 'have a go'
- a wealth of songs, stories, books and rhymes that reflect mathematical ideas, represent concepts and nurture imagination. For some ideas, see Tom Thumb's Musical Maths or Judith Stevens' Maths in Stories
- an environment in which pattern is evident and discussed - in resources, in routines, in music and in the way in which festivals and changing seasons are celebrated
- interactions with adults who accept children's misconceptions, value their guesses and competence
- practitioners who recognise and seize opportunities for mathematical discussion and development in every aspect of their work with children and who work closely with parents to encourage them to do the same.
WHAT HAS CHANGED
New early learning goals for mathematics
Numbers: children count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer. They solve problems, including doubling, halving and sharing.
Shape, space and measures: children use everyday language to talk about size, weight, capacity, position, distance, time and money to compare quantities and objects and to solve problems. They recognise, create and describe patterns. They explore characteristics of everyday objects and shapes and use mathematical language to describe them.
Changes from Problem-solving, Reasoning and Numeracy
This replaces Numbers as labels and for counting and calculating.
- The most obvious change is in counting from 1-20 rather than 1-10.
- There is less clarity about what is actually involved in counting.
- There is a change in emphasis with less mention of talk.
- There is less emphasis on the processes of mathematical thinking and problem-solving and more explicit reference to specific kinds of numerical problems such as halving, doubling, etc.
Distance and time have been added to the listed areas but the list is not exhaustive.
- The phrase 'use developing mathematical ideas and methods to solve practical problems' has been downplayed.
- 'Talk about patterns' has been replaced with 'describe patterns', which underplays the richer conversations which support children's understanding.
REFERENCES AND FURTHER READING
- The Elephant in the Classroom: Helping children learn and love mathematics, J Boaler, (2009), London: Souvenir Press
- The Maths Gene: Why everyone has it, but most people don't use it, K Devlin, (2000), London: Weidenfeld and Nicolson
- The Language of Mathematics: Making the invisible visible, K Devlin, (2002), New York: Henry Holt and Co
- Early Education (2012) Development Matters in the Early Years Foundation Stage (non-statutory guidance). London: BAECE (www.early-education.org.uk)
- Tom Thumb's Musical Maths, H MacGregor, (1998), London: A&C Black
- Measuring student knowledge and skills: a new framework for assessment, OECD (Organisation for Economic Co-operation and Development) (1998), Paris: OECD
- Thinking and Learning about Mathematics in the Early Years, L Pound, (2008), Abingdon: Nursery World/Routledge
- Maths in Stories, J Stevens, (2008). London: BEAM
- Children's Mathematics: Making marks, making meaning, M Worthington and E Carruthers, (2006), London: Sage.
Photographs at Sheringham Nursery School and children's centre, London, By Justin Thomas.
