Numeracy development Flashcards
What did Piaget (1952) argue re: numeracy? (3 points)
1) Children actively construct knowledge
2) Number understanding depends on logical understanding & so is not understood until 7y = the stage of concrete operations
3) Counting prior to 7y is rote learnt
Some argue number knowledge may be innate & so evidence prior to counting. Yet nobody argues that infants can: (3 points)
1) recognise numerals or Maths symbols
2) add or subtract actively
3) understand advanced number properties
In term of only quantities up to 3, babies may be able to (2 points). How and in whom has the 1st point been tested?
1) discriminate 2 circles from 3 circles
2) predict the result of adding or subtracting e.g. 3 circles - 1 circle = 2 circles
Habituation. In human infants and primates
A key limitation on 2 vs. 3 circle habituation findings:
Infants may discriminate on the basis of continuous perceptual differences between 2 and 3 circles rather than discrete numerical differences = perceptual vs. conceptual understanding
How did Wynn (1992) use the VOE paradigm with Mickey Mouse dolls to test 5m ability to add and subtract?
Shown x dolls, screen lowered, observed doll being added or removed & then re-shown display with correct vs. incorrect no. of dolls. 5m looked longer at the unexpected display, especially during subtraction
Wynn (1992) criticised her first experiment for___. She ruled out this possibility by changing the quantity of dolls even on incorrect trials and found___
Not demonstrating that 5m were making precise calculations, just that they expected some sort of change in quantity
5m looked longer at incorrect displays
Are Wynn’s (1992) Mickey Mouse addition/ subtraction findings on 5m replicable?
No
Gelman & Gallistel (1978) argue that counting principles are understood at a very early age, so how do they explain children’s failures in using these?
Due to problems in remembering or applying principles e.g. integrating them with motor responses = performance limitations
What are the 5 counting principles
1) one-to-one principle: one number word for each item counted, 2) stable order: number words always come in the same order, 3) abstraction: anything can be counted, 4) cardinal word: the last number word in a count sequence represents the no. in the set, 5) order irrelevance: changing the counting order does not change the answer you arrive at
Name 2 methods used to test counting principle understanding and evaluate the first one
1) Error detection tasks: observes adult or puppet counting, must say when he/she is correct/ wrong = :) no motor or verbal CVs, :( not USING concepts + :( cautious to attribute error to adults
2) Counting prediction tasks: predicts the result of counting e.g. will I get the same answer if I count in the opposite order?
In opposition to principles-first theory, is…
Procedures-first theory: children follow rote procedures. They only extract principles after experiencing counting in different contexts
Piaget: under 6s don’t understand the cardinal word principle given their inability to compare sets. Gelman & Gallistel (1978): 2 & 3y understand the cardinal word principle, as demonstrated by A & B
A) their emphasis or repetition of the last number word
B) their use of counting to establish whether even a very small set of 2 or 3 items is larger
G & G’s (1978) criteria for understanding the cardinal word principle may be too lenient. Why?
A) because emphasising the last word may result from imitation rather than understanding
B) Counting small quantities doesn’t make sense because children can compare quantities without doing so = perhaps seen as part of the game rather a means to the answer
What 2 tasks did Wynn (1990) use to test cardinal word principle understanding? Is performance on the 2 tasks correlated & so tapping the same ability?
1) count, then answer ‘how many?’, only 3.5y+ gave the last number word, below 3.5y recounted!
2) asked to give x toys, only 3.5y+ counted, below 3.5 gave some toys without counting = didn’t see counting as a means to the answer. Yes
From Wynn (1990), what can we conclude about the age at which children understand the cardinal word principle? How does this relate to predictions?
At 3.5y = earlier than Piaget suggested but later than G & G proposed
Do children perform better on a ‘give me x’ or ‘how many’ task?
On a ‘how many’ task…but does this require understanding of the principle?
How did Baroody (1984) test understanding of the order irrelevance principle? At what age did children show understanding?
Asked children to count a row of items, asked them ‘how many are there?’, asked ‘how many would you get if you counted in the opposite direction?’. 5y
There were 2 flaws in Baroody’s (1984) study which Cowan (1996) corrected in her test of order irrelevance understanding. What were they?
1) asking ‘how many’ may reveal less understanding than asking ‘will you get the same or a different number?’. Cowan: it does
2) children may believe any recounting always produces the same answer: to control for this, Cowan asked 4y to count 9 objects & then predict the result of recounting if 1 item was taken away
Re: order irrelevance, the ‘how many’ vs. ‘will the answer be the same’ distinction may be an example of…
The child knowing the principle & so giving it if directly prompted but not having integrated it with the procedure, which is prompted by the 1st question
There is an association between understanding the___principle & proficiency at counting but there are also___. Counting proficiency & understanding this principle are also linked to success at___in___year olds (Dowker, 2008)
Cardinal word. Exceptions in both directions. Predicting the result of adding or subtracting items without recounting. 4
Cowan (1996) found a strong r/ship between counting proficiency and understanding the___principle but exceptions also existed
Order irrelevance
Findings that procedure proficiency does not correlate with principle understanding in every child suggests that neither___. Instead, these support___theory which argues___
The procedures first or principles first accounts are right. Mutual development theory. That procedures and principles develop together, reinforcing each other & that their integration ensures development
Sarnecka & Carey (2009) tested whether cardinal word principle understanding was a prerequisite to adding and subtraction. Is it? What was their method?
Yes. Divided 2-4y into those who did (counted) vs. didn’t understand (grabbed) the cardinal word principle on a ‘give me N’ task. Those who did, also understood that adding 1 meant moving forwards 1 in the numerical list, whereas subtracting meant moving backwards
In Piaget’s conservation task, Greco (1962) asked children to count the rows, which they did successfully, before comparing them. Under 6s…
Still said the longer row contained more items!
3-5y were asked to judge whether a puppet’s ‘count each row separately’ vs. ‘altogether’ strategy was correct, given the task: ‘how many altogether?’ vs ‘which row has more?’ (Sophian, 1995). How did they perform? What does this show?
3-5y usually incorrectly judged counting both rows together as the correct strategy in the comparison task = they understand absolute but not relative quantities
Understanding order irrelevance & invariance in conservation tasks is mastered in the__th year
5
Is arithmetic ability made up of one or several components? E.g.___. Do children always master component A before B?
Several components e.g. you may have strong precise calculation skills but weak estimation skills. No
Preschoolers can’t do 2 + 1 in the abstract sense & don’t understand plus or minus signs but can…
Add 1 dog to to 2 dogs to get 3 dogs I.e. simple word problems
Arithmetic strategies become more sophisticated. 4 commonly used strategies in order of acquisition are…
1) using concrete objects, 2) using fingers, 3) calculating mentally, 4) retrieving facts from memory
How is memory important for arithmetic?
LTM for facts e.g. times tables & procedures
WM for monitoring a procedure, performing it in the right order & not becoming distracted
What is the counting span procedure?
Presented with series of progressively larger numbers of cards, each containing different numbers of x coloured dots. At the end of each series, the child attempts to recall the results of all counts. The counting span = the no. of cards in the longest series accurately recalled
The counting span increases with___ & is related to___ but ___ & ___ are unknown: does improved arithmetic cause improved WM or vv? The counting span task also requires___, not just WM
Age, arithmetic performance, cause & effect, counting
Visuospatial WM is more important for___arithmetic performance, whereas phonological WM is more important for___arithmetic performance
6ys’, older children’s
Arithmetic principles are used to___. What is the commutativity principle?
Predict unknown facts on the basis of known facts. Some sums can be reversed & the answer will remain the same e.g. 2 + 8 = 10, just as 8 + 2 = 10
Which principles do 6-9y use correctly vs. incorrectly? The incorrect use refers to incorrect generalisation of an addition principle to subtraction. Is appropriate strategy use correlated with arithmetic performance?
Correctly: commutativity of addition, the N + 1 strategy of addition e.g. 4 + 2 = 6, so 4 + 3 must = 7, the N + 1 strategy of subtraction e.g. 4 - 2 = 2, so 5 - 2 must = 3
Incorrectly: the N - 1 strategy of subtraction e.g. falsely concluding that if 4 - 2 = 2, then 4 - 3 must = 3…all with higher numbers. Yes
What is the most difficult principle and why?
The addition-subtraction inverse principle. The mirror image of an addition gives a subtraction e.g 2 + 8 = 10, so 10 - 8 = 2. Because children see arithmetic domains as distinct from each other
Bryant et al. (1999) demonstrated that understanding the addition-subtraction inverse principle is not all or nothing. How?
With concrete materials, 5-6y applied the ‘addition cancels out subtraction’ principle = a degree of understanding
How is commutativity linked to counting? What does this show?
If we know that 2 + 6 is the same as 6 + 2, then we know only to count on from 6 if we want to minimise the work load. Hence derived fact strategy use is related to calculation ability and even more so, estimation ability
What advantage do Asian languages have for arithmetic? = the 200-year-old H1
More regular counting systems e.g. the chinese word for eleven is ten-one, ten-two for twelve etc
At what number does the counting development of 4 & 5y diverge in America vs. China? The chinese are___ahead after this
- 1y
What difference to representing numbers in tens & units does a regular vs. irregular counting system make?
6y users of regular counting systems were more likely to use tens & units blocks rather than just units blocks for e.g. 42
What advantage does studying Welsh-medium school attending children have over Japanese children? Which 2 advantages do they show in arithmetic development?
CVs are reduced e.g. the same curriculum is taught. They count higher at ages 4 & 5. They are better at reading & comparing 2-digit numbers at 6y & 8y
What 4 number tasks did Butterworth (2008) give speakers of languages with no numbers above 3?
1) Memory for sets of counters. Here’s a set of e.g. 3, reproduce it. 2) Cross-modal matching. Match no. of counters with no. of times a block is tapped. 3) Nonverbal addition: placed e.g. 2, then 1 counter on a mat under a cover, ‘make my mat like yours’. 4) Sharing: share play dough disks among 3 bears
What did Butterworth (2008) find? Has this been replicated?
No difference in performance between speakers of English & ‘no numbers above 3’ languages. Effects of age (4-5y vs. 6-7y) & set size in both languages. No, Pica et al. (2004) found differences on exact but not approximate number tasks
What did Carraher (1985) find re: the transfer of arithmetic knowledge from the street to school?
9-15y street traders given the same arithmetic problems in a street, word problem or numerical context. % of problems successfully solved increased from the numerical to word problem to street contexts. The same children used different strategies in different conditions
What is dyscalculia? How common is it? What brain region might be involved & how is this controversial?
A severe, specific, persistent & global deficit in numeracy development. 6%. Left intraparietal sulcus but C vs. E, broad cognitive role & no clear-cut brain damage
Dyscalculia may reflect a deficit in a ___ & ___ number module critical for recognising ___ quantities. Or separate ___ & ___ modules may be selectively impaired
Universal, innate, small, approximate, exact arithmetic
Iuculano (2008) found one dyslexic who had difficulty in ___, whilst the other had difficulty in ___. This shows…
Recognising small quantities, approximate arithmetic, dyslexia is an umbrella term for different deficits
