lecture 8 numeracy Flashcards
what are the main skills underlying mathematical abilities
understanding that symbols represent magnitude, amount, order
learning to count and learning of arithmetic
how have we established that we have an innate sense of number (rats)
rats can distinguish between 2 and 4 light flashes
how have we established that we have an innate sense of number (infants)
infants have preverbal knowledge - 10 months detect equality, 14months detect ‘less than’ in habituation (measure looking time after habituated to different quantities)
can distinquish quantity across modality
violation of expectation (wynn)
describe wynn 1992 (violation of expectation)
5months - teddy placed in box with door screen up 2nd teddy put into box remove screen - only 1 teddy - violates infants expectation so look sig longer
problem with wynn violation of expectation study
simon et al
explain by knowledge of physical objects in world opposed to numerical understanding - object permanence - suprise because teddy has disappeared
describe simon hespat and rochat improvement study on wynn
same paradigm but when reveal
- 2 teddy (possible)
- 1 teddy (arithmetically impossible)
- 1 teddy 1 clown (identity impossible)
- 1 clown (identity and arithmatically impossible)q
results of simon hespat and rochat
look at possible
what is subitising
infants have an ability to do simple additive reasoning
can understand number without need to count for small numbers ie up to 3
describe xu spelke and goddard in discrimination of dots
infants discriminate between 8 and 16 dots but not between 1 and 2 dots
large no dealt with differently to small - analogue mag rep
what is analogue magnitude representation
internal mental representation of continuous quantities
similar to webers law
based on distance between quantities - internal mental representation of numbers on a continuum
therefore number judgement should be ratio sensitive
what is webers law
perceptual discrimination depends on similarity of the stimuli intensity
underlying law of perception
describe van oeffelen and vos in study of analogue mag rep
adults who must decide if there are 12 dots on a display are less precise when there are 10/11 compared to 4 or 20
- symbolic distance effect which marks analogue coding
ROUGH AND READY CODING OF QUANTITY
does counting guarantee comprehension of number sense
no - count to 5 by 3 years but may not realize that it is a tool for comparing quantities
what are counting principles
gelman and gallistel one to one corresponsence (how to count) stable order/ordinality (how to count) cardinality (how to count) abstraction (applying counting) order irrelevance (applying counting)
what does piaget argue about understanding of number
cannot fully comprehend number until have equivalence of sets - recognise tat quantity of things remain the same across modalities ie 5 children is the same as 5 sweets
define one to one correspondence
understand that each item has only one tag - physically/mentally tracking items counted and to be counted, one at a time and recognising they are unrelated to the item themselves
problem in one to one correspondence
may just be focusing on the rhythm of counting (speed) and not the no items being counted
inaccurate finger pointing used
define stable order/ordinality
child use same order in different situations - memorising long abstract lists (helped with rhyming/inonation)
problem with stable order/ordinality
cant do if have not yet commited order to memory
ordinality just order of number names - magnitude requires more than this - not just that 3 comes after to but that 3 represents something greater than 2
lacks deeper understanding of magnitude in number
define cardinality
understanding that the final number represents the size of the set as a whole
requires understanding last no is final number, relates to quantity and is progresive (from start to finish)
problem with cardinality
may know the answer but not relate to quantity
cardinality too simplistic- also about the relation between sets of numbers ie that set of 8 > set of 4
define abstraction
understand that both real and imagines things can be counted
ie events or ideas as well as objects
ie count sheep to go to sleep
define order irrelevance
doesn’t matter in what order the items are counted
requires understanding of one to one, stable order, cardinality and abstraction
recognize items are things not simple tags of numbers and that tags are temporary
recog order irrelevance doesn’t affect cardinality
nativist explanation for number sense
2 innate abilities that give rise to number sense:
approx non symbolic no system from birth that allows to make approx judegements about quantity
have inante knowledge of counting principles
gradually more precise but not explain how the approx system develops to do so
empiricist explanation of number sense
3 ways that infants learn to represent number:
- analogue approx system
- parallel individuation system - children learn by bootstrapping and recognition that number system is related to quantity ie those who know what ‘1’ means have diff knowledge of no compared to thsoe who know what ‘2’ means
- set based quantification
language help in learning number
problem with empiricist explanation of number sense
emphasises induction and language - children with poor language still show an understandig of larger quantities
bootstrapping as a circular idea - knowledge of number requires an understanding of number - does not identify the mechanism underlying bootstrapping
define the interactionist theory of number sense
piaget
actions initially reflexes
understanding of relation between quantity based on development of action schemas
representation of an action applied to object
scientist child predicts an outocme based on experienc
2 core insights: equivalence of sets (5dogs=5cats), order and class inclusion (hierachy)
what is our number system
recursive (repetitve throughout) base 10 (start again at every 10)
what is the additive principle
up to 100, every number is a decade + no. ie 21 = 20 + 1
what is the multiplicative principle
for every decade, its a decade x no ie 200 = 100 x 2
what does grasping of the additive and multiplicative principle require
additive composition of number - any number = total of two other numbers
multiplicative - units can have different values ie 1, 10, 100- each digits value depends on its location
how do number systems change across cultures
most european irregular up to 100 ie english base 10, french 60+12 = 72
aisan consistent with base 10
other cultures inconsistent names for same numbers but can still differentiate quantities
what could explain exceeding maths skills in countries with fewer resources?
language may affect ability to count by providing a sense of ordinality
describe miura, kim, chang and okamoto - language influence on ability to count
compareed 6-7 years in america, china, japan and korea
gave 10 base block and single unit blocks
ask to represent numbers using blocks ie 34
THEN ask for alternative way of representing the same number
possible soluitions to miura, kim, chang and okamoto - language influence on ability to count
one to one collection - 34 as 34 single blocks
canonical base - 3x10 blocks and 4 single
non canonical base - ie 2x10 blocks and 14 single base
findings of miura, kim, chang and okamoto
Asian children prefer use of construction of tens and ones to represent two-digit numbers.
American children preferred to use a collection of units.
more Asian children than American children able to
construct numbers in two different ways, which suggested “greater flexibility of mental number manipulation”
describe vasilyeva et al 2015 language on number
compare korean, taiwan, american and russian children 5-6 year olds on tiowse paradigm
no diff between lang groups on use of 10 base strategies but difference on instrictuional condition - all found large numbers more difficult
