1
Q
what is mathematical cognition?
A
A field that seeks to understand the processes by which we come to understand mathematical ideas
2
Q
What are mathematical cognition researchers interested in?
A
- how maths understanding and performance develops across the lifespan
- factors that explain individual differences in maths achievement
- understanding why some people find maths so difficult
3
Q
who struggles with maths?
A
- approximately 24% of adults in the UK have numeracy below that needed to function in everyday life
–> e.g., understand food prices, pay household bills - globally, 1/5 of adults unable to accurately deal with two-step calculations or understand irrational numbers
–> decimals, percentages, fractions
4
Q
the 6 stages of developing mathematical skills (in order)
A
- non-symbolic numbers
- learning the count list
- symbolic numbers
- arithmetic operations
- rational numbers
- algebra
5
Q
what are symbolic numbers?
A
- abstract and exact representations of numerosity
- human invention to describe numerosities
–> typically 2 forms:
–> number words and arabic digits - words for small numbers (1, 2, 3) are among the first words learnt
- arabic digits are learnt slightly later
6
Q
number word acquisition
A
- children learn the count sequence by rote before understanding the numerical meaning of number words and Arabic numerals
–> rote counting = reciting the number words in sequence - children acquire the meaning of ‘one’ at a young age but they do not automatically grasp the meaning of “two”
–> English-speaking children: 24-36 months
–> culture-dependent (e.g., plural markers of nouns)
–> morphological bootstrapping hypothesis
7
Q
Morphological Bootstrapping Hypothesis
A
- languages with plural and singular words allow kids to learn the meaning of one quicker
- if there is an ‘s’ there is more than one, if not there is one
- languages that don’t have the clear difference between singular and plural learn the meaning of one later on
8
Q
the 5 counting principles
A
- The one-to-one principle
- The stable order principle
- The abstraction principle
- The order irrelevance principle
- The cardinality principle
9
Q
the one-to-one principle
A
- Each object can only be counted once
- Each number word has to be paired with one and only one object
- Each object can only be paired with one number word
- All objects are paired with a number word
10
Q
the stable order principle
A
The number words are recited in a fixed order
11
Q
the abstraction principle
A
- any array or collection of sets can be counted
–> dogs, cats, people present, people absent, thoughts, actions… - we count the collection of sets the same way regardless of their characteristics
–> regardless of colour, shape or size
12
Q
the order irrelevance principle
A
- The order in which objects are counted does not matter
- Each order leads to the same result
13
Q
the cardinality principle
A
- The last number in the count sequence also describes how many objects there are in the total set
- Not only describes the order of the object but also the quantity of the whole set
- stops kids from just saying “1, 2, 3, 4, 5”
- they actually know they’re are 5 cars
–> not just counting up like habit
14
Q
Give-N-Task (testing the cardinality principle)
A
- task aim = ask kid to give a number of items and see how many they give
- Grabbers
–> take a random amount without thinking - Pre-number-knowers
–> know a specific amount is wanted but don’t know how many to grab
–> either give the same regardless of number or give a random amount - Subset-knowers (one-knower, two-knower, three-knower, four-knower)
–> can grab the right number but only if they know it
–> if they are a two knower, can grab one and two but not three - Cardinal principle (CP)-knower
–> know all the numbers and their meanings and so can be successful in this task - children typically become CP-knowers around 3-4 years of age, but there is large inter-individual variation
15
Q
Arabic digit acquisition
A
- Arabic digits also represent exact numerosities
- children acquire the meaning of Arabic digits slightly later than the meaning of number words
- correlated with the onset of schooling
–> children learn to write the numbers and connect the number names with written symbols
16
Q
connecting spoken number words, arabic digits and quantity (Lira, et al., 2017)
A
- 2 to 4 year old children
- kids tested on their ability to match:
–> quantity-to-number word
–> number word-to-quantity
–> number word-to-digit
–> digit-to-number word
–> quantity-to-digit
–> digit-to-quantity
17
Q
Connecting Spoken Number Words, Arabic Digits, & Quantity (Hurst et al., 2017)
A
- 3 to 4 year old children
- six mapping tasks
- found a mediation model
–> relationship between quantity-word dyad and quantity-numeral dyad is moderated by their knowledge of word-numeral dyad
18
Q
ordinality
A
- The relation between items in sequence
- Ordinality emerges later than cardinality
- In kindergarten and Grade 1, children have a strict definition of order that is tied to knowledge of count sequence (i.e., can order adjacent but not non-adjacent sequences)
- By 7-12 years of age, children quite accurate for both adjacent and non-adjacent sequences
19
Q
ways to assess ordinality
A
- number ordering task
–> put the numbers in the correct order
–> quickly and accurately - order judgement task
–> decide if the sequence of numbers on the screen is in the right order
–> do better when the sequence IS in order
–> also do better when the numbers are adjacent (whether it is correct or not)
20
Q
evaluate ordinality tasks
A
- can assess ordinality with either a number ordering task or an order judgment task
- performance on the two tasks are highly correlated and they tap into the same cognitive skill
- number ordering task is more appropriate for young children
21
Q
what is the home numeracy environment?
A
- Parents’ involvement with mathematics, including mathematical experiences, artifacts, and parent-child talk
- Researchers use a range of semi-distinct, but overlapping, terminology or use alternative conceptualisations for categorizing activities
–> can be difficult when determining what the home numeracy environment is - no consensus on the specific components that should be included to capture this parents’ involvement in home numeracy
22
Q
ways to measure the home numeracy environment - observational studies
A
- parent number talk
–> parent utterances of number words (e.g., one, two, three) and words related to magnitude comparison (e.g., more, less) during children’s infancy and early toddlerhood - quantify mathematical language use by counting the number of times parents make mathematics related utterances based on observations of parents and children either in the home or a more controlled laboratory setting
23
Q
ways to measure the home numeracy environment - questionnaires
A
- The home numeracy environment refers to the mathematics-related activities that parents share in the home
- Direct (formal) activities
–> explicit instructional activities directly targeting numeracy/mathematics - Indirect (informal) activities
–> everyday activities that incidentally involve numeracy/mathematics - can ask parents about these
24
Q
direct / formal examples on the questionnaire
A
- I teach my child to count objects to indicate quantity
- I teach my child to recognize Arabic numerals
- I teach my child to sort and classify objects by colour, shape and size
- I teach my child to write numbers
- I teach my child simple addition and subtraction
- I help my child with mathematics problems
25
indirect / informal examples on the questionnaire
- When we shop together, I discuss the price with my child
- We sing counting songs
- We play games that involve counting, adding, or subtracting
- We play board games or cards
- When I cook, I ask my child to count the quantity of ingredients
- When we watch television together, we discuss questions involving numbers
26
home numeracy environment and maths achievement
- small, positive relation between home numeracy environment and children’s maths achievement
--> associations vary widely between studies
--> varies based on age
- activity type matters
--> advanced, but not basic home numeracy activities associated with children’s maths skills
27
home numeracy environment and parents factors
1. mothers vs fathers
- most home numeracy studies only have data from mothers
- when both parents participate, only mothers’ reports of formal activities linked to children’s maths skills
2. parent education level
- mothers’ education levels predict maths activities beyond the home maths environment
- mothers with higher education levels provide advanced numeracy activities to their children more frequently
3. parent attitudes and expectations
- parents’ beliefs and expectations regarding children’s maths abilities and the importance of maths influence their children’s maths beliefs and performance
4. parent maths anxiety
- parents who are maths anxious may engage in fewer numeracy and mathematics activities at home
- maths anxiety moderates relation between home numeracy environment and children’s numeracy skills
28
the importance of early numeracy skills
- large individual differences, even as early as kindergarten
- on average, 7-year span in ability within a simple primary classroom
- numerous studies have shown that children who enter kindergarten with poor numeracy skills do not catch up
29
Duncan et al (2007) meta analysis
- Early maths skills strong predictor of later maths skills
- Early maths skills predict children’s later reading skills
- Early maths strongest predictor of later academic performance
30
numeracy skills beyond school years
- Numeracy skills are important for life outcomes:
--> employment opportunities
--> obtaining and retaining employment
--> promotion opportunities
--> owning a home, income
--> quality of healthcare
--> mental health
31
pathways to mathematics model (predictors of numeracy)
- Summarises early cognitive precursors to later numeracy skills
- Three pathways:
1. Quantitative
2. Working Memory
3. Linguistic Skills
- all concurrent and lead to later maths achievement
32
quantitative skills
- Early numeracy skills of quantifying, labeling, comparing, and manipulating sets
- How have quantitative skills been measured:
--> Subitising
--> Non-symbolic arithmetic
--> Counting
--> Estimation
--> Number comparison
33
subitising
- Quickly determining the number of items in a small set without counting
- Subitising in preschool/kindergarten predicted mathematics outcomes 2 years later
34
non-symbolic arithmetic
- Adding/subtracting with manipulatives
--> how many animals are in the barn now?
--> originally 5 and 2 are removed
- Non-symbolic arithmetic in preschool/kindergarten predicted mathematics outcomes 2 years later
35
counting
- present kids with dots
- ask them to count them
- counting in kindergarten predicted arithmetic performance in Grade 1
36
estimation
- ask kids to estimate number of dots without counting them
- estimation in kindergarten predicted arithmetic performance in Grade 1
37
number comparison
- Common measure of quantitative skill
- Numerical comparison proposed as key foundational capacity for numeracy
- Two types of tasks:
1. Non-symbolic
--> Compare one set of dots to another
--> Particularly useful when working with younger children
--> Non-symbolic number comparison in kindergarten predicted arithmetic performance in Grade 1 and mathematical fluency in Grade 2
2. symbolic
--> Compare two Arabic digits (is 5 or 9 bigger?)
--> Symbolic number comparison in kindergarten predicted many maths outcomes in Grade 1 and Grade 2
38
non-symbolic or symbolic use?
when considered together, symbolic number comparison demonstrates more predictive power than non-symbolic
39
working memory
- Cognitive system responsible for the active maintenance and temporary storage of task-relevant information
- Often measured using span tasks to determine how many items can be held in working memory
- In mathematics, working memory supports:
--> Performance of multiple steps (counting, arithmetic, problem solving)
--> Ability to keep track of intermediate results
--> Ability to visualise problems and solutions
- Two common subtypes measured:
1. Visuospatial working memory
2. Verbal working memory
40
visuospatial working memory
- Responsible for the maintenance and storage of visual and/or spatial information
--> e.g. copy the frog's path
--> have to watch and replicate the frog's route across lily pads
- Visuospatial working memory in preschool and kindergarten predicted mathematics outcomes 2 years later (i.e., calculation, numeracy, geometry, measurement)
- Visuospatial working memory in kindergarten predicted arithmetic and word problem performance in Grade 1
41
verbal working memory
- Responsible for the maintenance and storage of verbal information
- Kindergarteners’ working memory (composite of visuospatial and verbal) predicted performance and growth in maths from Grades 1-9
42
evaluate visuospatial and verbal working memory
- Repeat these numbers in reverse order
- Visuospatial and verbal in kindergarten predicted word problems and applied problems, respectively, in Grade 1
- Visuospatial and verbal working memory contribute equally to skill in mathematics
- younger kids tend to be better at visuospatial
43
linguistic skills
- Early linguistic skills include phonological awareness
--> i.e., knowledge of the sound structure of language
- Include receptive vocabulary
--> i.e., words the child understands
- Support the learning of mathematics vocabulary
--> e.g., number names and numerals, more than/less than/equal to - Supports rules of the number system
44
receptive vocab and phonological awareness (linguistic skills)
- Linguistic skills in preschool and kindergarten predicted mathematics outcomes 2 years later
- Kindergarten phonological awareness predicted Grade 1 word-problem performance
- Phonological awareness at school entry found to be strongest predictor of both mathematics grades and national mathematics test scores 2 years later
45
summarise pathways to mathematics
- Strong support for quantitative skills, working memory, and linguistic skills as kindergarten predictors of later numeracy
- Relations between working memory and linguistic skills in kindergarten and later mathematics skill are mediated by quantitative skills
- Children vary considerably in their numeracy skills
- In combination, the three pathways account for a lot of the variability in arithmetic (44-79%), word problems (53-61%), number system knowledge (48-64%), and geometry (26-84%) 1+ years later
- Performance on these measures can be used to identify which children are likely to struggle to gain numeracy skills
46
early intervention
- identify and then intervene
--> Change developmental trajectory
--> Improve numeracy outcomes
- without intervention, over 60% of children identified as having maths difficulties in kindergarten continued to have difficulties in Grade 5
- With respect to the three pathways, interventions focus on building quantitative skills:
--> only interventions to improve numeracy outcomes involved training on quantitative skills
--> domain-general interventions for working memory and linguistic skills have not convincingly been shown to improve numeracy
--> children’s working memory and linguistic skills can help guide selection of suitable interventions
47
criteria for evaluating interventions
1. must use kids identified as being at risk
--> based on low numeracy performance (explicit and quantitative criterion to define low performance)
2. must include an at-risk comparison group
3. group assignment (intervention/control) must be random
4. must have a pre-test, immediate post-test and a delayed post-test
5. numeracy outcome measures need to be reliable, valid and unbiased
6. Interventions must demonstrate numeracy gains, compared to an appropriate control group, that are both statistically significant and meet the Institute of Educational Sciences criterion for meaningful intervention effects (g ≥ 0.25)
--> Hedge’s g statistic is used to measure the effect size for the difference between means
L2 Developmental S2
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