adults can support children’s considerable abilities by respecting and trying to identify their mathematical intentions. We do this very readily with spoken language – we are delighted when children use a single word and work hard to support them in making their meaning clear
BORN MATHEMATICAL? Most adults will readily agree that young children are competent learners but few realise just how competent. Current research techniques (Karmiloff and Karmiloff- Smith 2001) have given us new and exciting insights into very young children’s thinking. These techniques have enabled researchers and psychologists to determine babies’ interests or preferences, to understand what causes anxiety. Perhaps most startling of all, they have also revealed that babies have a range of surprising, inborn mathematical abilities (Devlin 2000). As Devlin reminds us, mathematics is not only about number but ‘about life’ (2000: 76) and involves thinking and learning processes such as the identification of pattern, handling information in abstract forms; and a wide range of problem-solving strategies. This chapter also considers the abilities of babies and toddlers to understand something of shape and space, measures and, most strikingly, of number and computation. Mathematical processes From birth, humans are programmed to seek out pattern. Our brains are good at recognising pattern, a strategy which we use to identify the thousands of faces with which we come into contact. We enjoy stories because we see in them patterns – ‘real or imagined, visual or mental, static or dynamic, qualitative or quantitative, utilitarian or recreational’ (Devlin 2000: 11). From their earliest days babies prefer ‘complex patterns of high contrast’ such as ‘checkerboards and bull’s-eyes’ (Gopnik et al. 1999: 64). Pattern is also an integral part of the musical interaction that permeates babies’ lives (Pound and Harrison 2003). In the first year of life babies show a remarkable ability not merely to identify but to anticipate patterns in songs and rhymes (Trevarthen 1998). The young child’s giggling sense of anticipation that precedes the final ‘tickling under there’ in Walking Round the Garden develops because he or she is aware that we are deviating from the rhythmic pattern of the rhyme. The ability to think about things that are not present or do not exist at all is ‘one characteristic feature of the human brain that no other species seems to possess (Devlin 2000: 117). This ability is at the heart of mathematics. The ability to categorise according to shape, colour or sound – one manifestation of abstract thought – is present at birth (Butterworth 1999). Language is itself a symbolic system and as language use is extended, young children’s ability to deal with abstract information develops. Use of language also accompanies young children’s increasing ability to categorise – which reflects their desire to deal with abstract information. At around 15 to 18 months of age they ask the names of objects obsessively. The words heard during this ‘naming explosion’ (Gopnik et al. 1999: 115) in response to the oft-repeated question ‘what’s that, what’s that?’ are learnt very rapidly because at this stage the baby is interested in categorising (Karmiloff-Smith 1994). New words can be linked together as a group or category and thus remembered or recalled more readily – a process known as ‘fast- mapping’ (Gopnik et al. 1999). Many young men have been embarrassed by a baby pointing persistently at them and shouting ‘dada’ – this familiar scene is simply evidence of fast-mapping. The unknown young man is being placed in the same category as the child’s familiar father. Humans also make use of a wide range of problem-solving strategies. Logic is commonly thought of as central to mathematical thinking. However, it is clear that logic is not the only means of reasoning at our disposal (Claxton 1997). Even the youngest baby hypothesises, reasons, predicts, experiments and guesses in just the way that scientists do (Gopnik et al. 1999; Devlin 2000), using these skills to make sense of the world. Gopnik et al. (1999) describe experiments where young babies learn to move a mobile when a ribbon is tied from it to their ankle. Several days later they will repeat the movement if the same mobile is used – but not if the mobile has been changed. This suggests that the baby’s hypothesis is that the movement has something to do with the mobile itself. Infants learn early that nappy changing and clattering crockery precede mealtimes. Learning to reach out, clutch an object, pull it towards the mouth and let go is a series of motor actions which require careful sequencing – pulling something towards you before you have grasped it is unrewarding. As children develop, their play includes sequences of action of increasing complexity. Despite their physical dependency, young babies are increasingly seen to be highly competent learners. One of the forces which appears to drive their actions is the desire to seek control of both their social world (Dunn 1988; Trevarthen 1998) and the physical world (Gopnik et al. 1999). They learn early that particular actions cause or produce specific effects and responses. Indeed babies’ survival depends on being able to capture and hold the attention of another person and make known to them their needs. This ability to act with intent is reflected in the development of language. Babies and toddlers demonstrate through their early use of there! or oh dear! that they have a plan or intention in mind. This occurs for example where they have succeeded or failed in building a tower of blocks. Shape, space and measures Just as these processes permeate our lives, so the more explicitly mathematical aspects of shape, space and measures are fundamental to everyday activity. Awareness of shape is linked to both our enjoyment of pattern and to our interest in abstract thought, as we exercise ‘the human brain’s ability to reason about the environment’ (Devlin 2000: 246). Spatial awareness is part of our evolutionary heritage. Survival in trees and life on the savannah required understanding of both two- and three-dimensional space (Devlin 2000). This is reflected in infancy as babies reach out towards a stimulating object, explore their own bodies, the bodies of others and the world around them. Exploratory play reflects children’s interest in shape and space as they push paper into cardboard tubes, squash tissues into boxes and use their fingers to probe all manner of objects, including the mouths and ears of others! On a larger scale, dropping objects from a height, throwing them high and steering wheeled toys through a tight space are all part of this learning process, which infants seem driven to seek out in their search for control and problems to solve. Physical action is used to explore and think about shape and space. Spinning round, running up and down a slope, wriggling through a play-tunnel and hiding inside a blanket tent are a vital part of this process. Language is linked to the young child’s growing understanding. Early use of up or down to signify something to do with direction is commonplace. Either might serve to mean ‘pick me up’; ‘put me down’ or ‘I want the things on that high shelf’. This is illustrated in two-year-old Xav’s afternoon of activity. He placed some circular place mats on the floor – walking round them, repositioning them and walking on them like stepping stones, while maintaining the circular pattern. He ran out to the back garden and walked around and around the tree seat; then started running while balancing on the circle of bricks which edged the flower bed. He then ran in increasingly large circles around the tree. Later that afternoon he repeated similar movements in the front garden – taking a circular path around the flower bed, the tree and the whole garden. Throughout this frenetic activity, so characteristic of this age group, he recited to himself ‘round and round, round and round’. Language also reflects children’s understanding of measures. Big, little and more are similarly brought into action to denote height, weight or volume. Here too physical action remains important. In describing or drawing attention to something that is moving very fast, young children typically move their arms horizontally at speed, often accompanying these actions with running movement and appropriate sounds. Very large or very tall objects are depicted with sweeping gestures in the appropriate direction, arms and legs outstretched. Understanding number This is where our new found understanding of what appears to be happening inside babies’ heads becomes awe-inspiring. There is a wealth of studies (some of which are described in Dehaene 1997; Butterworth 1999; Devlin 2000) which make it clear that from the first day of life babies have some knowledge of number, or numerosity. (The ages shown below are indicative – the findings of studies should not be taken as representing the age at which some things can or cannot be done. Much depends on the context, the skill of the researcher and the personality of the baby involved. Some of the findings will relate simply to the age of the babies available when the researcher wished to carry out his or her experiments.) There are studies which indicate that babies as young as a day old can differentiate between cards with two or three dots (Antell and Keating, cited in Butterworth 1999); that babies of four days of age can differentiate between words with two or three syllables (Bijecjac-Babic et al., cited in Dehaene 1997). Studies looking at slightly older babies from five or six months generate responses which indicate that infants of this age can apply their awareness of number in a range of situations – recognising changes to a set of three or four moving objects on a computer screen; discriminating between the number of jumps made by a puppet; and, when shown sets of pictures and objects, preferring to link pictures and groups of objects which display the same number rather than depicting the same object (Butterworth 1999; Dehaene 1997). That is to say that, for example, babies of this age are likely to be more interested in the fact that a picture of buses is similar to a set of teddy bears because there are three of each in each case, than that there are three toy buses and a picture of three buses – numerosity is more important to them than other similarities (Starkey et al., cited in Butterworth 1999). Psychologists have been particularly interested in a link made by babies which appears to make their awareness of number even more surprising. The studies described above involve linking pictures and objects and demonstrate babies’ visual awareness. It is clear however from other studies that, in addition, babies have an innate oral awareness of number. When, for example, they areplayed a series of words with the same number of syllables, babies notice when the number of syllables changes. Perhaps even more interesting is the fact that they also make links between visual and oral information. Presented with sets of two or three objects, they will when played a number of drum beats choose to look at the set of objects which matches the number of drum beats heard (Starkey et al., cited in Dehaene 1997). So not only are babies interested in the numbers of things they can see but they are also able to make links between the number of things they can see and the number of sounds they can hear. Counting or guessing? Clearly, in none of these studies were the babies involved able to articulate the number of objects in any group or set – they were making connections or comparisons between the numbers within groups. The numbers involved in the studies outlined above were small – usually up to three or four. This ability is not the same process as counting – counting involves (Gelman and Gallistel 1978) being able to: •use one number name for one object (one-to-one principle) •remember to use the number names in the same order (stable-order principle) • understand that the last number name used gives the size of the group (cardinal principle) •recognise that anything can be counted – you can count a set of similar objects such as dolls, or a set of dissimilar objects which might include knives, forks, plates, cups and oranges (abstraction principle) • accept that no matter what order you count things in the answer will always be the same – so long as you apply the one-to-one and stable-order principles (order-irrelevance principle). What the babies in the studies cited by Butterworth and Dehaene were doing was not applying these principles and therefore counting but recognising a group of objects at a glance – a process known as subitising. Over time it appears that young children come to be able to recognise larger groups of objects – alongside, but different from, the development of the abilities that contribute to counting. Macnamara (1996) has studied this ability and has shown that children in nurseries are sometimes able to recognise groups of objects of five, six or seven in this way. Her studies also show that many children on entering formal schooling lose this ability and, rather than simply being confident that there are enough sweets for the group, begin to count everything. Macnamara (1996: 124) gives the example of a boy who had been a very confident ‘subitiser’ in the nursery. When asked to repeat the same test in the reception class he became distressed because his teacher had told him he must count everything. There are some important lessons to be learnt from Macnamara’s work. Any failure to support and build on young children’s abilities may cause them to lose confidence. A very important aspect of mathematics is the ability to estimate. Indeed, much of the mathematics that we use in our day-to-day lives requires approximate answers – in instances like buying paint, deciding how much money to take on holiday or planning a journey we are not trying to come up with accurate answers – merely good enough estimates. We do this alongside the important job of learning to recite number names and count groups of objects using Gelman and Gallistel’s five principles. Macnamara gives further examples of children in the reception class who remained good at recognising groups – to the point where some could identify groups of nine or ten objects at a glance. When questioned it appeared that these children achieved success by a combination of subitising and counting on. One child commented ‘I remember some and then I count the rest’, by which he meant that while looking at the group of dots or objects he was able to identify (by subitising) a group of six or seven objects and then to count on mentally the three or four additional objects which had not been part of the first group. Given that much time is often taken up early in formal schooling helping children to learn to count on – being able to hold one number in mind while adding a second set or number to the first – it may be that our failure to recognise children’s ability to make use of both sets of skills is contributing to their later difficulties. Both guessing (or subitising) and counting have a part to play in children’s mathematical development and practitioners need to support both. Adding and subtracting As if all of this were not surprising enough, it seems that in the first half of the first year of life babies seem able to predict what ought to happen when objects are added to or taken away from an existing set of objects. A mother’s naturalistic observation of her five-month-old daughter (cited by Karmiloff-Smith 1994: 173) perfectly reflects the phenomenon unveiled in complex experiments: Sometimes we play games after she’s finished eating in the high chair. She loves one where I take a few of her toys, hide them under the table top shouting ‘all gone’ and then making them pop up again just after. She squeals with delight. I once dropped one of the three toys we were playing with by mistake, and I could swear she looked a bit puzzled when I put only two toys back on her table. These studies of addition and subtraction underline what was found in the studies of recognition of groups. The baby is more interested in number than in the objects themselves. The toys used could be exchanged – cuddly toys being replaced by coloured balls – but so long as the number of objects was correct the baby was not concerned (Butterworth 1999). Counting on our fingers Human understanding of number appears to be linked to our use of fingers for counting and computation. This is because the two functions are governed by closely related areas of the brain (Ramachandran and Blakeslee 1999). Understanding of number is from the earliest stages of learning linked to the use of the hands, the control of which takes up a larger proportion of the brain than any other part of the body (Greenfield 1997). The relationship between this part of the body and numbers can be seen in babies and toddlers, as they take up one, then two objects in their hands. They may signal a number of objects by using their fingers, holding up all ten fingers fully spread to signify a large number – sometimes accompanied by language – ‘I want this many’. At this stage the ability to recognise a group of objects is not linked to an ability to count. A group of two objects is recognised as a group of two, without any link to language or any facility to compare one group with another. Learning to recite number names in order – although a vital skill at a later stage – is a separate part of a highly complex learning process. Here too the human mind has, in addition, two important advantages. One is the use of fingers and the other is the role of music in infant development (Pound and Harrison 2003). Cultures around the world use songs and action rhymes, which combine the use of fingers, the role of rhythm and the pleasure of physical and playful contact to make number memorable. So what should practitioners do? Many of these exciting findings related to the mathematical abilities of young children demand only what practitioners working with young children have thought of as common sense. Providing a rich range of experiences and materials for problem-solving and exploration enables children to develop connections in the brain which will support future thinking, including mathematical thinking. Counting songs – especially those which include the use of fingers and movement – also reinforce learning, enabling children to memorise, represent and recall numbers. The findings should also encourage practitioners to look for evidence of these apparently innate abilities. Being able to communicate mathematical ideas enables children to think mathematically since communication and thought are mutually supportive (Goldschmied and Selleck 1996; Siegel 1999). In order to help young children to communicate their ideas we should learn to enjoy their problem-solving abilities; notice when they are identifying or creating patterns – whether musical, based in their movements, talk or in other aspects of their play. We should also encourage imagination, as it contributes to abstract thought – imagining things which are not actually present. This occurs through play and stories, and by talking about things that we are going to do and have already done. Where this can be supported by photographs and relevant objects this will support children in visualising abstract ideas. Above all, adults can support children’s considerable abilities by respecting and trying to identify their mathematical intentions. We do this very readily with spoken language – we are delighted when children use a single word and work hard to support them in making their meaning clear. We accept that milk might mean water or orange – what they want is a drink. We also accept their imaginative ideas – taking delight in their ability to pretend that a block is a banana or a mobile phone. Let’s work equally hard to ensure that we don’t get too caught up in feeling that their mathematical ideas must always be wholly accurate. Let’s encourage mathematical play and guessing!
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