number sense and learning to count

Question 1

what is mathematical cognition?

Answer 1

seeks to understand how individuals understand math ideas

• neuro and behaviour are influenced by environment and influence each other and cognition
• cognitive system processes numerical info
• advantage in society to be good with math
• 24% adults have below needed level
• 20% adults can’t do simple calculations

Question 2

what are the 2 systems in numerosity perception?

Answer 2

small exact and large approximate

Question 3

what is subitizing?

Answer 3

the fast and accurate enumeration of small numerical quantities (3-4)

Question 4

small exact system (OTS, PI)

Answer 4

1-3 items between 2-5 years

3-4 items from child-adulthood

Question 5

where does limited capacity for subitizing come from?

Answer 5

visuospatial WM

• children with math disability have steeper subitizing curve
• reduced subitizing in children with math disability

Question 6

approximate number system (ANS)

Answer 6

• estimation of large numerosities, no time for counting
• ratio effect = ability to discriminate between large numerical quantities varies as a function of the ratio of numerical sets
• as ratio gets further from 1, easier to discriminate
• 6 month olds habituate to 8 dots then look at 16 bc notice difference. notice ratio of 2 (8 vs 16) but not 1.5 (8 vs 12)
• 10 months can do 1.5 (8 vs 12) but not 1.2 (10 vs 12)
• predicts later math achievement
• acuity reduced in children with dyscalculia

Question 7

ANS and weber fraction

Answer 7

the width of curves=a measure of overlap between numbers

the lower the width, the better

Question 8

why is the ability to discriminate between numerical quantities found in different species?

Answer 8

• groups of predators
• food
• reproduction

Question 9

difficulties when studying ANS

Answer 9

• reliability low
• perceptual confounds: density, area, perimeter. congruent vs incongruent trials
• executive skills: inhibition may explain relation between ANS and math achievement
• sensory-integration theory: numerosity estimation is a product of weighted combo of non-numerical sensory cues related to non-symbolic stimuli

Question 10

learning to count

Answer 10

• number words (e.g. one) are learn’t before arabic numbers (e.g. 1)
• learn the meaning of the number words at 24-36 months

Question 11

what are the counting principles?

Answer 11

• stable order: recited in fixed order
• one-to-one correspondence: each object only counted once, all objects paired with one number word
• abstraction: ant set of objects can be counted
• order irrelevance: order in which items are counted is irrelevant
• cardinality: understanding that the last pronounced number represents the numerosity of the set

Question 12

stages of acquisition of cardinality principle

Answer 12

• pre-number knower: numerosity given is unrelated to requested number
• one-knower: child gives one when asked for one but gives more than one for higher
• two-knower
• three-knower
• four-knower
• cardinal-principle knower: knows exact meaning of all number words as high as they can count (no longer a subset knower)

Question 13

how long does it take to become a cardinal-principle knower?

Answer 13

takes 1.5 years from ages 2-4

• children from less privileged background at 48 months
• home numeracy predicts cardinality knowledge

Question 14

ways that magnitude representations support children acquiring the meaning of number words

Answer 14

• mapping of number words to ANS could give approx numerical meaning - scaffolding
• first learn meaning of small number words by linking to object representations in OTS. for large they conceptually shift by discovering similarity in structure between lots of number words and objects–> one more object in set corresponds to next number

Question 15

transparency

Answer 15

• transparent languages reflect place-value explicitly in number words e.g. in mandarin 11 = ten-one, 23 = two-ten-three
• fewer distinct number words to learn
• asian children superior performance

Question 16

inversion

Answer 16

order of number words for multi-digits is reversed e.g. in german 23 = three-and-twenty

Question 17

mirroring of words and digits

Answer 17

tends to disappear at 7 but might be present in children with math and reading difficulties

Question 18

single digit processing: digit comparison task

Answer 18

• distance effect: the closer the digits in numerical value, the longer it takes to decide which is larger
• size effect: the larger the two digits, the longer is takes to decide which is larger number
• distance + size effect: ratio effect - longer responses for ratios close to 1
• size congruity effect: longer reaction times for incongruent trials (smaller number shown visually larger than bigger number)
• evidence that you are accessing non-symbolic representations

Question 19

when does the size congruity effect emerge?

Answer 19

7-8 years

Question 20

place-value understanding

Answer 20

• the position of an arabic digit in a multi-digit number represents its meaning (different in different places)
• young children struggle until end of primary school

Question 21

transcoding

Answer 21

=translating between number words and numerals

• verbal number system: lexicon composed by different entries (units, decades)
• arabic: place-value (10 elements + 1 principle)
• additive rule
• multiplicative rule
• zero: they are spoken absent in the middle of multidigit
• errors in transcoding at 7-9 years relate to math performance 5 years later