Lecture 5: Number Concept + Reading Flashcards
What is the number concept?
(Definition)
- This refers to mathematical skills such as counting, comparing and judging quantities and ability to do arithmetic operations (like addition, subtraction, multipication)
There are number of rules
- We follow when it comes to numbers
5 Counting Principles
Gelman and Gallistel (1978)
Proposed 5 principles that govern counting:
What are those 5?
- One-to-one principle
- Stable-order principle
- Cardinal principle
- Order irrelevance principle
- Abstraction principle
5 Counting Principles
Gelman and Gallistel (1978)
What 3 counting principles define the counting procedure? (define the counting process/define how we count)
- One-to-one principle
- Stable-order principle
- Cardinal principle
Which principle do you really have to master later to show how counting works?
- Cardinal principle
Counting Principle
One-to-one principle
(2)
- According to this principle, when counting a set of objects, each object must be counted once.
- Each item in this set are assigned one unique label or “counting word”.
Counting Principle
One-to-One
Example + Violation Error
- For example, counting people in a front row of a lecturer theatre saying 1 ,2 , 3 ,4.
- Violation of one-to-one principle is if you assign more than one “counting word/label” to a person in a front row.
Counting Principle
Stable order principle
- “Counting words”/labels/tags must be used in a set order.
Example of Stable-order principle
+ Violation Error
- For example, when counting a set of 3 objects using 1,2,3 is an example of the stable-order principle.
- A violation of the stable-order principle is counting these 3 sets of objects as 1,3,2. This is because these “counting words” are not assigned to the objects in the correct order.
Counting Principle
Cardinal Principle
- The “counting word”/label for the final object in a set denotes the total number of items of the whole set.
Counting Principle
Cardinal Principle
Example
(2)
- For example when counting 3 cats in a garden, they would count each one with ‘one’ , ‘two , ’three’. ‘Three’ is the last cat in that set. Therefore there are 3 cats in that garden.
- For example ‘two’ refers to sets of two entities in that group/set.
Counting Principle
Order-irrelvance
- This refers to the understanding that objects in a set can be tagged/labelled in any sequence as long as the other counting principles such as one-to-one , stable-order and cardinal principle is not violated.
Counting Principle
Order irrelevance
Example
(2)
- For example there is a set of 4 sweets on the table.
- An individual may count these sweets from left to right or from right to left. Regardless of the direction of counting, the individual’s counting results in the total of 4 sweets.
Counting Principle
Abstraction principle
Definition.
This refers to that these counting principles can be applied to any collection of tangible and intangible objects.
Counting Principle
Abstraction Principle
Example
For example, When a child understand this principle, they are able to appreciate to count intangible objects such as counting the number of happy thoughts they have or tangible objects such as the number of cars they have in their toy box.
What is abstraction principle not?
- It is not labelling ‘cat’ to a cat.
- They are called tags and not labels in principles
Children’s knowledge of these counting principles
- It is implicit knowledge of these counting principles
- They are unable to articulate this knowledge of the principles but are able to follow these counting knowledge
Implicit vs Innate knowledge
Bess
(2)
- Innate knowledge is knowledge you are born with
- Implicit knowledge (could be innate) but not necessarily, knowledge you can demonstrate in various ways but can’t articulate it → important when discussing about young children.
From the literature it is demonstrated that these 5 counting principles are attainable by
the age of 5
Some are achievable earlier by the age of 3.
Overall, it shows these principles are achieved around preschool age in which big changes in how they are understanding the counting principles
How can we test for the children’s knowledge of these counting principles?
(3)
- Test this indirectly
- Control for performance demands (tweaks in tests)
Error detection task
Gelman and Meck (1983)
They tested the 3 main counting principles of one-to-one, stable -order and cardinal principles using an error detection task on a sample of…
3-5-year-olds
Gelman and Meck (1983)
In this error detection task:
puppet task paradigm
(3)
- In this task, the child is observing the counting performance of a puppet’s performance.
- The child’s ability to detect errors in the puppet’s application of one-to-one, stable order and cardinal counting principles in the puppet’s performance is measured.
- In this task, the children do not have to count themselves which elevates the possible restriction performance demands.
Gelman & Meck (1983)
One-to-One principle
Using puppet paradigm
- 3 types of trial
- The one-to-one principle study involved the puppet counting a row of red and blue objects.
Gelman & Meck (1983)
One-to-One principle
Using puppet paradigm
Correct trial
The correct trial is in which the puppet is counting the objects in the row correctly by counting each object in the set once.
Gelman & Meck (1983)
One-to-One principle
Using puppet paradigm
In-error trial
In the in-error trial, the puppet either skipped or double-counted objects in that row.
Gelman & Meck (1983)
One-to-One principle
Using puppet paradigm
Pseudo-error trial + example (2)
- In the pseudo error trial, the puppet when they are counting the items in a set in an unfamiliar way but not necessarily violating the counting rules
- For example, the puppet may begin to count in the middle of the set of objects and continue until the end of the row before returning to the beginning of the row for the remaining items that need to be counted.
Gelam & Meck (1983)
Stable-order principle
- The puppets counted small trinkets that varied in size and colour in this stable-order principle study.
- They had 2 types of trials in the stable-order principle condition
Gelam & Meck (1983)
Stable-order principle
In-error trial + correct
Correct trial
In-error
The convential order of two items in a set was reversed e.g 1,3,2,4;
Using a randomly ordered list 3,1,4,2
A list where one or more tags are skipped 1,3,4
Gelman & Meck (1983)
Cardinality
Instructions + Set size
(3)
- Same procedure as the stable-order experiment however its instructions was somewhat different as children were asked to indicate whether the puppet gave the right answer after its finished counting and the children were encouraged to tell the puppet what the right answer was.
- 3 and 4 year olds.
Gelman & Meck (1983)
Cardinality: 2 types of trial
Correct
In-error trial
- Correct (explain the principle)
- In-error
- (nth value +1 like 1,2,3,4 I have 5 items)
- less than N like 1,2,3,4, I have 3 items
- or attributes the final tag of the sequence to an irrelevant feature of an object such as 1,2,3,4 I have blue items)
Gelam & Meck (1983)
Results
Specific percentages
(Remember bold)
- Showed child’s performance was highly accurate on correct trials
- High accuracy on incorrect trials
Gelam & Meck (1983)
Results
Pseudo error and set size
- Pseudo-errors detected as peculiar but not incorrect
- Even able to articulate to experimenter why in some cases
- Show understanding of order-irrelevance (implicit, not articulate)
Gelam & Meck (1983)
Results
Older children + set size.
- Older children performed better, but.
- Success rates not affected by set size (how far they count) - even for young children.
Gelam & Meck (1983)
Conclusions
- Children as young as 3 can understand and demonstrate this knowledge of the 3 main principles, even though they can not articulate them
- Understanding demonstrated even in set sizes that are too big for children to count in.
- Children therefore show implicit knowledge of these principles.
Baroody (1984)
Not sure about Gelman’s and Meck (1983)’s conclusions
What did Baroody want to investigate/test
- Order-irrelevance and cardinality in 5-7-year-olds specifically.
Baroody (1984)
Not sure about/convinced Gelman’s and Meck (1983)’s conclusions
Wanted to contest Gelman and Meck’s conclusion as Baroody argued that…
(Bess) - 4.
- They may know that the puppet can count 1,2,3,4 or count backwards as 1,2,3,4, they may understand that this is okay.
- However, this does not necessarily imply that they have understanding of both order-irrelvance and cardinality.
- Does not tell that the ‘4’ for example represents there are 4 items in the set.
- Do they understand how the cardinal principle works alongside with order-irrelvance principle?
Baroody (1984)
Not sure about Gelman’s and Meck (1983)’s conclusions
Wanted to contest Gelman and Meck’s conclusion as Baroody argued that…
(Bess)
- The ability to understand that the tags can be assigned arbitrarily in a series of items in a set, understanding of the order-irrelevance principle, does not necessarily imply that they have the understanding that differently ordered counts will produce the same cardinal desgination/same number of items.
Baroody (1984)
Not sure about Gelman’s and Meck (1983)’s conclusions
Method
- The children count themselves (not error detection task) since they are older like 5-7 than 3-5.
- The children have displayed 8 items in front of them and they count them from left to right and then indicate what the cardinal value is in that set.
- Then the experimenter asked the children “Can you make this number 1?” while pointing to the item on the right-hand side of the set. (had to count backwards).
- The experimenter said “We got N counting this way, how many would you think we would get counting the other way?”
- While the experimenter asked this, they blocked the linear array so the child had to predict the cardinal value in the other direction, this is called a How-Many task.
Baroody (1984)
Not sure about Gelman’s and Meck (1983)’s conclusions
Results
- Only 1 child could recount in the opposite direction (without the linear array being blocked → control procedure)
- However…
- 45% of 5-year-olds and 87% of 7-year-olds were successful in the prediction task.
- This contrast the high accuracy in Gelman’s in 5 year olds as compared to 3 year olds.
- 45% of 5-year-olds and 87% of 7-year-olds were successful in the prediction task.
Baroody (1984)
Not sure about Gelman’s and Meck (1983)’s conclusions
Conclusion
- This understanding of order-irrelevance develops gradually with age and that..
- Young children’s understanding of the principles in earlier studies such as Gelman and Meck are overestimated.
Gelman, Meck & Merkin (1986)
Gelman and colleagues respond to Baroody’s study and their conclusion by saying:
- The task affects the children’s performance.
- The failure in these tasks is the result of the children misinterpreting the instructions rather than showing evidence of the lack of understanding of these counting principles.
- Sees the question/instructions the experimenters asked as a challenge.
- Such as asking the child ‘you got 8 counting this way, what are you going to get now?’ The child may be thinking 8 isn’t right answer and maybe have a similar answer around 8.
- Sees the question/instructions the experimenters asked as a challenge.
- The failure in these tasks is the result of the children misinterpreting the instructions rather than showing evidence of the lack of understanding of these counting principles.
Gelman, Meck, Merkin (1986)
The failure in these tasks is the result of the children misinterpreting the instructions rather than showing evidence of the lack of understanding of these counting principles.
How Many Task with college students
- Similar findings were found in Gelman’s (1993) ‘How-Many’ task in which college students were asked how many questions regarding 18 blocks, all of the college students had counted them but only one of them repeated the last count word they said. Repeats of this ‘how many’ questions elicited confusion and recoutning.
Gelman, Meck & Merkin (1986)
Method
- Replicated Baroody’s (1984) study
- 3 groups: Baroody group, count 3 times group and altered question group.
- However, they gave the children 3 opportunities to count the linear array first.
- Then they altered the question instead of saying ‘we got N counting this way, how many will you get the other way?’, they just said ‘can you start counting with N?’ ‘How many will there be/What will you get’?
Gelman, Meck & Merkin (1986)
Results
Graph
- Same exact results in Baroody replication group
- Count 3 times group (Baroody) → performance is higher as they have increased confidence in their answer with having opportunities to count the linear array 3 times (7/12)
- Altered question group ( + Baroody) , their performance was significantly higher than the count 3 times group (9/12) as perhaps they seen the instruction and question the experimenter asked in Baroody as challenging.
Conclusion of all 3 studies
Baroody, Gelman and Meck, Gelman, Meck and Merkin
+ N task
- Young children demonstrate some understanding of counting principles
- As they have an implicit knowledge of this (can’t articulate this)
- Young children might be limited by larger sets in some cases or not.
- The task matters
- Counting verus error-detection
- Subtle changes in types of questions used
- Give N is most typically used to date.
All of these studies of Gelman and Meck, Baroody and Gelman and colleagues have led to
Give ‘N’ task and knower levels.
Task
- Give-N is an alternative task to the ‘How many task’ to measure cardinal-principle knowledge.
- The child is asked to give ‘N’ a number of items.
- The experimenter initially starts by asking the child to ‘give one lemon’ to the puppet.
- When the child responded correctly to the experimenter’s request for N, request for N+1 was added. If they respond incorrectly, the next request for N-1, this request was continued until eh child succeeded two of them.
- N continued until highest numerals of 5 and 6.
All of these studies of Gelman and Meck, Baroody and Gelman and colleagues have led to
Give ‘N’ task and knower levels.
Task Results
Pre-numerical, one knower, two knower, three and four knower,
- Give-N task classifies the children into ‘knower’ levels.
- This pre-numerical knowledge showed that they may give one object to a puppet or a handful of objects where its quantity is unrelated to the number the experimenter requested.
- The ‘one’ knower is in which the child understands the concept of one and gives exactly one lemon to the puppet when asked for one lemon, then when asked a numeral other than one they give two or more objects to the puppet.
- The ‘two’ knower was able to give exactly one and two lemons when the experimenter request for one and subsequently two lemons. However, they are unable to distinguish among numerals, three, four and five.
- ‘Three’ and ‘four’ knowers in which at this elvelt hey are known as subset knowers.
- After some time the child has been a subset-knower, there are able to dramatically produce right cardinality for numerals five and higher. With children progressing gradually through the subset-knower levels, they seem to acquire higher meanings of numeral all once, they are known as cardinal-principle knowers → CP knowers.
CP knowers perform qualitatively different than sub-set knowers and succeed across tasks.
CP knowers
Sarnecka & Carey (2008)
(3) + Method
Unit Task
- The statistical tests found that CP knowers significantly performed better than lower knower level group (pre-number, one-knowers , two knowers, three knowers) and four knowers.
- Therefore shows that..
- Only CP knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.
CP knowers perform qualitatively different than sub-set knowers and succeed across tasks.
CP knowers
Sarnecka & Carey (2008)
(2) Method
Unit Task
- In a unit task, the experimenter begins each trial by telling the child that they are putting four frogs in a box, then asked the child after the box was closed how many frogs are there? After the child answered this correctly, the experimenter will then add one or two more items in the box then asking the child ‘Now is it five or six?’
- The test trials were 4+1, 4 +2 , 5+1 , 5 +2
After conclusion of 3 studies?
- How much of numerical knowledge is innate?
- How much born, readily able to do.
Where does our numerical knowledge come from?
Give two main ideas
Empiricsm
- Empiricism: Numerical knowledge comes from experience and develops gradually.
Where does our numerical knowledge come from?
Give two main ideas
Nativism
Nativism: They have a core knowledge in which they have an innate understanding of some elements of number concept. The core knowledge is these perceptual building blocks that children are born with makes it easier to make quantity judgements etc…
Where does our numerical knowledge come from?
Give two main ideas
- Do very young infants show understanding of numbers?
- What about non-human animals (core knowledge possibly having evolutionary roots?; no language, no exposure, to number system?)
Habituation studies
Can use with very young infants to gauge innate knowledge
Methodology - dots
- The procedural example used with quantity judgements (to test those)
- Habituation to 4 dots for a long period of time, then lose interest (habituates to that stimuli).
- Followed with the exposure to 2 dots,
- If tell difference between two quantities, dishabituate then show renewed interest to 2 dots.
- If they still show no interest, can’t tell difference
Habituation studies
Can use with very young infants to gauge innate knowledge
Dots - Results and Conclusion
Results:
Looked longer at 2 dots
Conclusion:
Understand numerosity
Some argued as basic pattern discrimination unable to judge quantities.
Example for:
Xu and Spekle (2000) controlling for pattern discrimination:
Methodology and control for pattern discrimination
- They controlled for pattern discrimination by using large numbers and controlling for other properties of arrays. For example, the infant in the habituation phase was displayed 8 vs 16 dots that were presented to them 6 times. These stimuli also varied in each presentation during the habituation phase in terms of the size and location of the dots.
- After the habituation phase was over, the infant was tested with the dot stimuli (either 8 or 16 dots) that was not displayed in the habituation phase.
- Example for:
- Xu and Spekle (2000) controlling for pattern discrimination:
- Results and conclusion
- Results showed that 6-months old looking time was significantly longer for the novel dot stimuli regardless if they were habituated to 8 or 16 dors.
- Therefore, demonstrating that 6-month-old infants are able to discriminate between 8 and 16 dots (Feigenson et al, 2004)
Example for:
Xu and Spekle (2000) controlling for pattern discrimination:
Further experiments of Xu and Spekle showed that…
- 6-months olds were also able to discriminate 16 vs 32 dots, however failed to discriminate between 8 vs 12 and 16 vs 24 dots.
- However, it is shown that this numerical discrimination increases precision over development.
- This is shown as 6 months old were able to discrminate 1:2 ratios (imprecise ratios) but not 2:3 ratios whereas 10 month old infants can succeed in both types of ratios. Also adults can detect even more precise ratios as small as 7:8 (Barth et al., 2003; Feigenson et al., 2004).
Addition and Subtraction Study
Wynn (1992)
General Information
- The study consisted of 32 5 months old infants.
- They used a looking time procedure which displayed to these infants different mathematical outcomes (possible and impossible outcomes)
- They wanted to investigate whether infants are able to compute arithmetical computations showing that they demonstrate the numerical concept.
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
the 32 5 month infants were divided into 2 groups
1+1
- ‘1+1’ group is in which the infants were shown a single item that was placed in an empty display area. A small screen is lifted up so that the infant isn’t able to see this object anymore.
- Then the experimenter would bring a second item and place in the display area so it was in clear view of the infant and then the item was placed behind the screen so this item was no longer visible to the infant.
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
the 32 5 month infants were divided into 2 groups
2+1 group
- In the ‘2-1’ group, a similar sequence of events was performed however, when the two items were visible to the infant and when the small screen lifted up, the experimenter removed one of the items (this was clearly visible to the infant).
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
End of ‘2-1’ and ‘1+1’ group:
What was measured?
- For both of ‘2-1’ and ‘1+1’ group after these sequencesof events were performed, the screen lifted downwards to either reveal 1 or 2 items in the display area/platform.
- 1 being the impossible event for ‘1+1’ group, 2 being possible.
- ‘2-1’ , 2 being impossible and 1 being possible.
- During this, the infant’s looking time at the display was recorded.
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment 1
Preference pre-test trials
- In the pre-test trials, the infants were introduced with a display that had 1 item and a display of 2 items and their looking time was measured for both to record the baseline looking preferences.
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
Results
Pre-test trials
- Pre-test trials showed that there was no difference in infant’s looking times to 1 or 2 objects.
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
Results
Test trials
- In both groups, the infants looking time was significantly longer at the impossible outcome as compared to the possible outcome.
- In the ‘1+1’ group, the infants looked longer at 1 item than 2 items.
- In the ‘2+1’ group, the infants looked longer at the 2 items than the 1 item.
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
Conclusion from experiment 1 and experiment 2.
- Therefore, showing that infants are able to compute addition and subtraction operations and that infants expect that the result of this operations leads to a numerial change (change in the number of items)
Addition and Subtraction Study
Wynn (1992)
Looking time procedure methodology
Experiment1:
The conclusion from experiment 1 and experiment 2
Reason for experiment 3
- However, they may simply expect that adding an item will result in a number other than 2 and that substracting an item will result in a number other than 2.
- Therefore, experiment 3 was conducted to test these following 2 hypotheses:
- Infants have the ability to compute precise results of simple addition and subtraction operations.
- Secondly, whether infants have an expectation that an arithmetic operation will result with a numerical change , with no expectation of the size or direction of this change.
Experiment 3
Wynn (1992)
Precise addition counterpart
Same correct direction
(3)
- A similar methodological procedure used in experiments 1 and 2 was used added a more precise addition counterpart of 1+1 = 2 OR 3.
- In this case, the possible outcome (2) and the impossible outcome (3) was in the same correct direction (Wakeley)
Experiment 3
Wynn (1992)
Precise addition counterpart
Same ordinal direction
Results
- The results showed that the infants looking time was significantly longer/preferred the impossible outcome, 3, than the possible outcome 2.
- In the pre-test trials, the infants did not prefer either 2 or 3.
Wynn (1992) Conclusions
- 5 months old can compute the precise results of simple arithmetical (addition and subtraction) operations.
- Infants have true numerical concepts, therefore, suggesting humans innately and readily have the ability to perform these simple arithmetic operations.
McCrink & Wynn 2005
Wynn 1992 Conclusions
- Replicated using larger sets such as 10 vs 5 items.
- Infants still able to do this.
BUT: Wakeley et al. (2000)
Aim.
(3)
- Aim is to investigate the robustness of infant’s arithmetic ability.
- They conducted 3 experiments using Wynns procedure.
- They replicated the exact conditions in Wynn’s experiments 1 and 2 in which the infants showed addition (1+1 = 1 or 2) and subtraction (2-1 = 1 or 2).
Waekely et al. (2000)
Substraction counterpart to Wynn (1992)
(2)
Then in Wynn’s experiment 3, they added a precise subtraction counterpart which was 3-1 = 1 or 2 (both correct and incorrect result are in the expected ordinal direction)
They controlled for the possibility that the preferred answer was always the greater number of items - the incorrect result is the smaller number of items.
Wakeley
Results
(!)
- They found that there was no systematic preference for the incorrect vs correct across experiment 1, 2 and 3. This means the infants looked equally long at either outcome (impossible vs possible)
Wakeley (2000)
Results
- They found that there was no systematic preference for the incorrect vs correct.
Wakeley
Conclusions
Review of literature = inconsistent results
Moore
- These earlier findings of numerical competence is not replicated as there is variation across studies regarding infant’s reaction to the impossible vs possible outcome of arithmetic operations.
- For example, Moore (1997) found using Wynn’s (1992) 1+1 = 1 OR 2 and 2-1 = 1 or 2, showing that female infants looked at the incorrect result more as compared to males.
Wakeley
Conclusions
Review of literature = inconsistent results
Infant’s reactions are variable..
- Infant’s reactions are variable and their numerical competences are not robust in the literature.
Nativist view
Carey 2009
- Argue that we are born with some innate ability that expands with age and experience (Carey, 2009) - Ratio.
- That this inborn ability shared with other animals?
Wakeley
Conclusions
Argue that arithmetic competencies develop gradually.
- Number of studies investigated the development of addition and subtraction in toddlers and preschoolers using measures such as reachig, object manipulation and verbal response that has demonstrated that young children’s performance on these simple addition and subtraction tasks with small sets of objects indiciates by their second year that they know addition yields more and subtractio yields less (Sophian & Adams, 1987).
- The ability to calculate precisely results of simple subtraction ad addition operations does develop gradually during their early childhood.
Wynn’s Response
- There is procedural differences affected the attentiveness of infants.
- Therefore, use of computer program versus experimenter to determine the start of experiment.
- If infant is not looking then did not ensure infants saw complete trial because of this.
- Exclusion of “fussy” infants higher in Wynn’s (and other) studies. If fussy babies included then skew sample.
Additional evidence?
Are core numerical abilities shared with other animals?
- Animals can discriminate amounts (sounds, arrays, food)
- Counting
- Arithmetic operations?
- Core knowledge hypothesis.
Experience is still important though..
Empiricism view
Gobel et al. 2011
- Cross-cultural evidence demonstrates that language, counting practices impact representation and processing of number.
Experience is still important though..
Empiricism view
Levine et al 2010
- Evidence showed that the amount of number talk the parents engage in with their children is robustly linked with mathematical development (cardinal number knowledge)
Experience is still important though..
Empiricism view
Gunderson & Levine 2011
- Levine & Gunderson (2011) characterises the different types of number talk that parents produce to investigate which type is most predictive of children’s later cardinal number knowledge.
- The findings showed that parents’ number talk that includes counting sets of visible objects is strongly associated with children’s later cardinal knowledge than other number talks that involve objects that are not visible.
Experience is still important though..
Empiricism view
Gunderson & Levine 2011
Conclusion
Therefore, shows that experiece plays a role in their development of the number concept.
Experience is still important though..
Empiricism view
Gunderson & Levine 2011
Bess
- Within cultures: Number-talk from parents predict CP knowledge, related to later performance in school
Evidence of innate abilities
- Numerosity (habituation studies)
- Arithmetical operations
How robust are the findings?
–Also evidence for gradual accumulation of this knowledge
–Born with limited ability, which then expands with age/experience? (Carey, 2009)
Conslusions
- •Children as young as 3 seem to have implicit knowledge of counting principles
- •Evidence of innate abilities
- •How robust are the above?
- •Task and procedure have large impact on results and age at which we see these abilities
