Understanding Numbers in Development

Susan Carey (2004)

- A competing theory that claims SOME innate knowledge but nevertheless argues that children acquire their understanding of number gradually and as a result of experience - Claims human infants are born with 'parallel individuation' system - this makes it possible for the to recognise and represent very small numbers exactly - She claims marked development from 1,2, and 3 objects in the first three years - The system allows infants to recognise '1' as having a distinct quantity - Although they do not first know that the words 'one' applies to this quantity - Later on, children are able to 'individuate' sets of 1 and 2 objects, and around aged 3/4 this extends to 1,2 or 3 - During this same period, children learn 'number words' - The association over time between the development of parallel individuation and learning the count eventually leads to what Carey calls 'Bootstrapping' - Bootstrapping = the children lifting themselves up by their own intellectual bootstrap

Different Counting Systems

- All counting systems must obey these principles, however they have found to vary across cultures - Saxe (1981) - The most obvious variation is in the words for numbers - e.g. in English the word 'eleven' is opaque. It is does not spell out 10 + 1 , however, 101 does - Whilst in French 'Quatre-vingt-dix' = 90 - Another variation is the base system - English are used to a decimal system - The system in Oksapmin i Papua New Guinea, use body parts for number words e.g. their counting starts with the word 'thumb' as 1, and continues up the arm and over the head - Having a base system in counting makes numerical calculations must easier for human beings

Wynn (1992, 1998, 2000) - Infants knowledge of addition and subtraction

- Also provides support for the idea of innate mathamatical structures - Looked at adding and subtracting in infants as young as 6 months - She enacted sums in front of the baby, sometimes leading to the correct answer and sometimes not - Argued that if baby can do the additions and subtractions, they should be surprised by incorrect outcomes and will look for longer

Counting

- An activity to establish the numerosity of a set, which in turn can be used as input for reasoning - Use of conventional number sequence and understanding counting are independent

a) Cardinality

- Any set of items with a particular number is equal in quantity to any other set with the same number of items in it - Cardinality is what makes the recognition of numbers more sophisticated that patterns - e.g. a group of four children in an orderly queue, stay the same numerically despite later running around in a playground

Critics of Wynns experiment:

- Bryant (1992) - claims she did not use subtraction in her second experiment and so one cannot be sure that infants of this age have a genuine understanding of subtraction - Wakley (2000) - says that infants could have attended more to the 1 + 1 = 3, than the 1 + 1 =2 because they were more interested in a larger number of toys

Cross-Cultural Diff in Shop Task

- Chinese, Japanese and Koreans do not have words life 'eleven' - rather they say the equivalent of ten-one, ten-two, ten-three etc. - As a result, these children are better at constructing numbers than European and American children - European and American children make more mistakes in counting tasks by producing the wrong word for numbers - Towse and Saxton (1997) - the most plausible explanation for this European/Asian difference is in linguistics, however must not rule out the possibility that there may be other cultural factors

However,

- Even if some implicit number sense is innate, social and linguistic experience as conveyed in number games with adults is needed to make these skills explicit - Current research is exploring how the different number systems in turn influence the development of children counting and later mathematical skills

The Decimal System

- Fuson (1988) - explains how the decimal system does not come easily to young children - Nunes and Bryant (1996) - Shop Task - Children given money to buy items - In some conditions, only one denomination needed (e.g. only 10p's) - In others, mix denominations needed (e.g. 24p) - These mix trials are a good test of understanding of the base 10 system - The mix trials were by far the hardest, as children usually go wrong by counting 10 pence as 1

PRINCIPLES BEFORE SKILL

- Gelman thinks that her five principles are innate - built in part of our cognitive system - Whilst she does acknowledge that children make mistakes when they count, she argues that this is a matter of procedural skills, which are never perfect

CULTURAL TOOLS

- How do children learn mathematics? - By looking at cultures, we can see that being taught some instructions is needed - Children cannot be left to reinvent these tools for themselves

Antell and Keating (1983)

- In a replication, found even neonates reacted in the same way

Feigenson, Dehaene, and Spelke

- In smalle number tasks (up to 3) infants react to perceptual cues that typically co-vary with number (e.g. amount or contour length of the objects), in large number tasks they seem to react to the numerical magnitude itself, but discrimination depends on having an appropriately large ratio between the numbers - This supports the idea of two number systems operative from infancy, a precise small number system that represents up to 3 individuals, and that may be a by-product of the object file system, and a large number system that represents approximate numerical magnitude

However, the interpretation of these results have been seriously questioned:

- Many ask, is this change really about the number? - Mix et al (1997) argues that infants made discrimination entirely based on continuous quantity in these experiments - They compared the effects of (a) varying the amount of material, while holding the number constant and (b) varying the number of items, while holding the total amount of material constant - In (a) the change did provoke instant interest in the babies - In (b) the babies showed no sign on revived interest - These results support the idea that the babies in Starkeys research might not have been attending to number - The issue of innate number discrimination remains

PIAGET IN NUMBER LOGIC

- No one disputes the fact that logical reasoning is needed in mathematics - The argument lies in whether children initially lack this logic and have to acquire it - Piaget (1952) claimed that children are at first held back by their lack of logic, and do have to acquire these abilities to understand mathematics - His view, was the direct opposite of the innate view - Piaget claims that young children may know number words quire well, yet they do not understand what they are doing when they count - The do not understand what number sequences mean as they have no idea or cardinality and ordinality

b) Ordinality

- Numbers come in an ordered scale of magnitude - 2 > 1, 3 > 2, and as a logical consequence, 3 > 1 - Being able to work this out is known as transitive inferences

Piaget's work focused mainly on cardinality in infants:

- Piagets word on conservation was directly concerned with children's use and understanding of number words - Greco (1962) gave 4 - 8 y/o children three versions of the conservation task: 1) Children saw two identical sets and judged correctly that the two were equal in number. After seeing the appearance of one changing, they recognised the quantity had changed 2) The same, accept after transformation the children were asked to count one set, and infer the amount of the other 3) Children required to count both sets and asked whether they were equal - Most children under 6 failed task 3 - Despite counting both and knowing one had more, they still said that the one which is spread out had more than the other - They judged that one set with 'eight' objects, was more than the other set with 'eight' object - Piaget however states this is because they don't know what the word 'eight' means

PreSchoolers

- Preschoolers cannot only count, they can also add/subtract in some concrete contexts - In fact, even preverbal infants can discriminate numbers and seem to understand the effect of adding or subtracting one object from a display

In her first study, she enacted 1 + 1 by:

- Putting mickey mouse on a platform in front of the child, then raising a screen in front of this toy, so it was not seen - Placed another MM behind the screen (the child saw this) - Lowered the screen so the child could now see two toys - On half of the trials, 2 toys were there, but on the other half only 1 was (thus an incorrect outcome) - Subtraction (2-1) was carried out in the same way

Gelman and Gallistel (1978)

- Recorded how well children, aged 2-5, count sets of objects - Children given sets that varied in number from 2-19, and they had to count each set -They assessed whether children always produced number words in the same order, always counted each object once and whether they seemed to recognise that the last number counted signified the number of the set - Results: found that set size had an effect - The children followed counting principles more for the small number sets - They call this the 'Principles before skills' hypothesis - Children grasp the principles, but mistakes in larger sets are due to difficulties in applying the right procedures in difficult circumstance - Claim children lose their way in one-to-one counting with large sets because of forgetting which items they have already counted

The claim for innate number understanding is unimpressive

- Research on number discrimination in babies may not be about number and research on principle-before-skills leave out some essential principles - We need to examine the alternative idea, which is that children acquire an understanding of number gradually and as a result of much experience

WORKING WITH INFANTS TO STUDY NUMBERS

- The idea that humans are born with the ability to reason mathematically is taken very seriously

Results:

- The infants did look longer at the wrong outcomes - Wynn recognised that these results could be explained in other ways - Carried out the exact same study, however this time 3 were revealed rather than 2 - Thus, both the correct and incorrect outcomes were different to the starting point - Wynn found the infants still showed more interest in the incorrect outcome - Concluding that babies but have been able to work out the 1 +1 addition

Summary

- The main question posed regard how much of childrens learning must depend on instruction and how much do they learn for themselves without the help of others - The answers are roughly the same as children learning to read and do mathematics - In both cases, either innately or through their own experience, children do acquire some crucial knowledge, long before school, which prepared them for the formal learning they eventually have to do - This takes the form of an early awareness and use of one-to-one correspondence in the case of mathematics - It still takes quite a while to learn, with the help of formal instructions, - When concluding on how children make this connection, it is important to recognise that both their own informal knowledge and of the intellectual demands of the cultural tool which they learn from others

Universal counting principles

- There are essential characteristics of numbers - These principles are a set of principles that must be obeyed in order for our number system to work - Unless we do sol, we are not really counting

Same number = same quantity

- This is a challenge for many young children

LOGIC PRINCIPLES

- Unless a system conforms to logic principles it cannot be called a number system a) Cardinality b) Ordinality

Learning Numbers

- When children learn numbers they have to come to terms with a mixture of universal, logical principles and human interventions (cultural tools) that very quite considerably from place to place

Number and Counting Systems require a formidable list of logic!

-There are clearly many difficulties on the way to learning about numbers - Some psychologists have suggested that children have the help of a remarkable resource during this learning = an innate understanding of numbers - The evidence for this claim comes mostly from work with infants

R.Gelman and Gallistel (1978) - 5 principles

1-3 = 'How to count principles' (1) One-to-one principle = when counting a set of objects, each object must be counted once and once only (2) Stable-order principle = Number words must be produced in a set order, and in the same set order each time (e.g. counting 1-2-3 on one occasion, you must not count 3-2-1 on another) (3) Last number counted = Represents the value of that set (4) Abstraction Principle = The number in a set is independent of any qualities of the members in that set (rules for counting a heterogeneous set is the same for a homogeneous one) (5) Order Irrelevance Principle = The order in which members of a set are counted does not affect the number of items in the set (left to right will get the same answer as right to left)

Gellman and Meck (1983)

supported this hypothesis from research on babies that was done by herself and her colleagues who used the technique of habituation - Together with Starkey and Cooper (1980) they studies whether 4 month old babies could discriminate numbers - Showed babies, over a series of trials, a certain number of dots - In each trial the same number of dots were shown, but the arrangement varied - Some were shown few, whilst others were shown many - Babies attention to the number displays declined - In the second face, number of dots were changed and babies were assessed to see if their interest returned - Found: When number of dots were small, an increase led the baby to look at the displays for a relatively long time - Thus they recognised a change in their numbers