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BORN MATHEMATICAL?

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BORN MATHEMATICAL? Powered By Docstoc
					                                 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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Description: 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