Numeracy development

Question 1

What did Piaget (1952) argue re: numeracy? (3 points)

Answer 1

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

Question 2

Some argue number knowledge may be innate & so evidence prior to counting. Yet nobody argues that infants can: (3 points)

Answer 2

1) recognise numerals or Maths symbols
2) add or subtract actively
3) understand advanced number properties

Question 3

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?

Answer 3

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

Question 4

A key limitation on 2 vs. 3 circle habituation findings:

Answer 4

Infants may discriminate on the basis of continuous perceptual differences between 2 and 3 circles rather than discrete numerical differences = perceptual vs. conceptual understanding

Question 5

How did Wynn (1992) use the VOE paradigm with Mickey Mouse dolls to test 5m ability to add and subtract?

Answer 5

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

Question 6

Wynn (1992) criticised her first experiment for___. She ruled out this possibility by changing the quantity of dolls even on incorrect trials and found___

Answer 6

Not demonstrating that 5m were making precise calculations, just that they expected some sort of change in quantity

5m looked longer at incorrect displays

Question 7

Are Wynn’s (1992) Mickey Mouse addition/ subtraction findings on 5m replicable?

Answer 7

No

Question 8

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?

Answer 8

Due to problems in remembering or applying principles e.g. integrating them with motor responses = performance limitations

Question 9

What are the 5 counting principles

Answer 9

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

Question 10

Name 2 methods used to test counting principle understanding and evaluate the first one

Answer 10

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?

Question 11

In opposition to principles-first theory, is…

Answer 11

Procedures-first theory: children follow rote procedures. They only extract principles after experiencing counting in different contexts

Question 12

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

Answer 12

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

Question 13

G & G’s (1978) criteria for understanding the cardinal word principle may be too lenient. Why?

Answer 13

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

Question 14

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?

Answer 14

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

Question 15

From Wynn (1990), what can we conclude about the age at which children understand the cardinal word principle? How does this relate to predictions?

Answer 15

At 3.5y = earlier than Piaget suggested but later than G & G proposed

Question 16

Do children perform better on a ‘give me x’ or ‘how many’ task?

Answer 16

On a ‘how many’ task…but does this require understanding of the principle?

Question 17

How did Baroody (1984) test understanding of the order irrelevance principle? At what age did children show understanding?

Answer 17

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

Question 18

There were 2 flaws in Baroody’s (1984) study which Cowan (1996) corrected in her test of order irrelevance understanding. What were they?

Answer 18

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

Question 19

Re: order irrelevance, the ‘how many’ vs. ‘will the answer be the same’ distinction may be an example of…

Answer 19

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

Question 20

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)

Answer 20

Cardinal word. Exceptions in both directions. Predicting the result of adding or subtracting items without recounting. 4

Question 21

Cowan (1996) found a strong r/ship between counting proficiency and understanding the___principle but exceptions also existed

Answer 21

Order irrelevance

Question 22

Findings that procedure proficiency does not correlate with principle understanding in every child suggests that neither___. Instead, these support___theory which argues___

Answer 22

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

Question 23

Sarnecka & Carey (2009) tested whether cardinal word principle understanding was a prerequisite to adding and subtraction. Is it? What was their method?

Answer 23

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

Question 24

In Piaget’s conservation task, Greco (1962) asked children to count the rows, which they did successfully, before comparing them. Under 6s…

Answer 24

Still said the longer row contained more items!

Question 25

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?

Answer 25

3-5y usually incorrectly judged counting both rows together as the correct strategy in the comparison task = they understand absolute but not relative quantities

Question 26

Understanding order irrelevance & invariance in conservation tasks is mastered in the__th year

Answer 26

5

Question 27

Is arithmetic ability made up of one or several components? E.g.___. Do children always master component A before B?

Answer 27

Several components e.g. you may have strong precise calculation skills but weak estimation skills. No

Question 28

Preschoolers can’t do 2 + 1 in the abstract sense & don’t understand plus or minus signs but can…

Answer 28

Add 1 dog to to 2 dogs to get 3 dogs I.e. simple word problems

Question 29

Arithmetic strategies become more sophisticated. 4 commonly used strategies in order of acquisition are…

Answer 29

1) using concrete objects, 2) using fingers, 3) calculating mentally, 4) retrieving facts from memory

Question 30

How is memory important for arithmetic?

Answer 30

LTM for facts e.g. times tables & procedures

WM for monitoring a procedure, performing it in the right order & not becoming distracted

Question 31

What is the counting span procedure?

Answer 31

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

Question 32

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

Answer 32

Age, arithmetic performance, cause & effect, counting

Question 33

Visuospatial WM is more important for___arithmetic performance, whereas phonological WM is more important for___arithmetic performance

Answer 33

6ys’, older children’s

Question 34

Arithmetic principles are used to___. What is the commutativity principle?

Answer 34

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

Question 35

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?

Answer 35

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

Question 36

What is the most difficult principle and why?

Answer 36

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

Question 37

Bryant et al. (1999) demonstrated that understanding the addition-subtraction inverse principle is not all or nothing. How?

Answer 37

With concrete materials, 5-6y applied the ‘addition cancels out subtraction’ principle = a degree of understanding

Question 38

How is commutativity linked to counting? What does this show?

Answer 38

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

Question 39

What advantage do Asian languages have for arithmetic? = the 200-year-old H1

Answer 39

More regular counting systems e.g. the chinese word for eleven is ten-one, ten-two for twelve etc

Question 40

At what number does the counting development of 4 & 5y diverge in America vs. China? The chinese are___ahead after this

Answer 40

12. 1y

Question 41

What difference to representing numbers in tens & units does a regular vs. irregular counting system make?

Answer 41

6y users of regular counting systems were more likely to use tens & units blocks rather than just units blocks for e.g. 42

Question 42

What advantage does studying Welsh-medium school attending children have over Japanese children? Which 2 advantages do they show in arithmetic development?

Answer 42

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

Question 43

What 4 number tasks did Butterworth (2008) give speakers of languages with no numbers above 3?

Answer 43

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

Question 44

What did Butterworth (2008) find? Has this been replicated?

Answer 44

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

Question 45

What did Carraher (1985) find re: the transfer of arithmetic knowledge from the street to school?

Answer 45

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

Question 46

What is dyscalculia? How common is it? What brain region might be involved & how is this controversial?

Answer 46

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

Question 47

Dyscalculia may reflect a deficit in a ___ & ___ number module critical for recognising ___ quantities. Or separate ___ & ___ modules may be selectively impaired

Answer 47

Universal, innate, small, approximate, exact arithmetic

Question 48

Iuculano (2008) found one dyslexic who had difficulty in ___, whilst the other had difficulty in ___. This shows…

Answer 48

Recognising small quantities, approximate arithmetic, dyslexia is an umbrella term for different deficits