lecture 8 numeracy

Question 1

what are the main skills underlying mathematical abilities

Answer 1

understanding that symbols represent magnitude, amount, order
learning to count and learning of arithmetic

Question 2

how have we established that we have an innate sense of number (rats)

Answer 2

rats can distinguish between 2 and 4 light flashes

Question 3

how have we established that we have an innate sense of number (infants)

Answer 3

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)

Question 4

describe wynn 1992 (violation of expectation)

Answer 4

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

Question 5

problem with wynn violation of expectation study

Answer 5

simon et al
explain by knowledge of physical objects in world opposed to numerical understanding - object permanence - suprise because teddy has disappeared

Question 6

describe simon hespat and rochat improvement study on wynn

Answer 6

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

Question 7

results of simon hespat and rochat

Answer 7

look at possible

Question 8

what is subitising

Answer 8

infants have an ability to do simple additive reasoning

can understand number without need to count for small numbers ie up to 3

Question 9

describe xu spelke and goddard in discrimination of dots

Answer 9

infants discriminate between 8 and 16 dots but not between 1 and 2 dots
large no dealt with differently to small - analogue mag rep

Question 10

what is analogue magnitude representation

Answer 10

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

Question 11

what is webers law

Answer 11

perceptual discrimination depends on similarity of the stimuli intensity
underlying law of perception

Question 12

describe van oeffelen and vos in study of analogue mag rep

Answer 12

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

Question 13

does counting guarantee comprehension of number sense

Answer 13

no - count to 5 by 3 years but may not realize that it is a tool for comparing quantities

Question 14

what are counting principles

Answer 14

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)

Question 15

what does piaget argue about understanding of number

Answer 15

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

Question 16

define one to one correspondence

Answer 16

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

Question 17

problem in one to one correspondence

Answer 17

may just be focusing on the rhythm of counting (speed) and not the no items being counted

inaccurate finger pointing used

Question 18

define stable order/ordinality

Answer 18

child use same order in different situations - memorising long abstract lists (helped with rhyming/inonation)

Question 19

problem with stable order/ordinality

Answer 19

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

Question 20

define cardinality

Answer 20

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)

Question 21

problem with cardinality

Answer 21

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

Question 22

define abstraction

Answer 22

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

Question 23

define order irrelevance

Answer 23

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

Question 24

nativist explanation for number sense

Answer 24

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

Question 25

empiricist explanation of number sense

Answer 25

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

Question 26

problem with empiricist explanation of number sense

Answer 26

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

Question 27

define the interactionist theory of number sense

Answer 27

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)

Question 28

what is our number system

Answer 28

recursive (repetitve throughout)
base 10 (start again at every 10)

Question 29

what is the additive principle

Answer 29

up to 100, every number is a decade + no. ie 21 = 20 + 1

Question 30

what is the multiplicative principle

Answer 30

for every decade, its a decade x no ie 200 = 100 x 2

Question 31

what does grasping of the additive and multiplicative principle require

Answer 31

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

Question 32

how do number systems change across cultures

Answer 32

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

Question 33

what could explain exceeding maths skills in countries with fewer resources?

Answer 33

language may affect ability to count by providing a sense of ordinality

Question 34

describe miura, kim, chang and okamoto - language influence on ability to count

Answer 34

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

Question 35

possible soluitions to miura, kim, chang and okamoto - language influence on ability to count

Answer 35

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

Question 36

findings of miura, kim, chang and okamoto

Answer 36

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”

Question 37

describe vasilyeva et al 2015 language on number

Answer 37

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