early maths development Flashcards
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
Q
indirect / informal examples on the questionnaire
A
- 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
Q
home numeracy environment and maths achievement
A
- 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
Q
home numeracy environment and parents factors
A
- 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 - 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 - 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 - 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
Q
the importance of early numeracy skills
A
- 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
Q
Duncan et al (2007) meta analysis
A
- 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
Q
numeracy skills beyond school years
A
- 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
Q
pathways to mathematics model (predictors of numeracy)
A
- 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
Q
quantitative skills
A
- 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
Q
subitising
A
- Quickly determining the number of items in a small set without counting
- Subitising in preschool/kindergarten predicted mathematics outcomes 2 years later
34
Q
non-symbolic arithmetic
A
- 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
Q
counting
A
- present kids with dots
- ask them to count them
- counting in kindergarten predicted arithmetic performance in Grade 1
36
Q
estimation
A
- ask kids to estimate number of dots without counting them
- estimation in kindergarten predicted arithmetic performance in Grade 1
37
Q
number comparison
A
- 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
Q
non-symbolic or symbolic use?
A
when considered together, symbolic number comparison demonstrates more predictive power than non-symbolic
39
Q
working memory
A
- 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
Q
visuospatial working memory
A
- 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
Q
verbal working memory
A
- 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
Q
evaluate visuospatial and verbal working memory
A
- 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
Q
linguistic skills
A
- 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
Q
receptive vocab and phonological awareness (linguistic skills)
A
- 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
Q
summarise pathways to mathematics
A
- 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
Q
early intervention
A
- 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
Q
criteria for evaluating interventions
A
- must use kids identified as being at risk
–> based on low numeracy performance (explicit and quantitative criterion to define low performance) - must include an at-risk comparison group
- group assignment (intervention/control) must be random
- must have a pre-test, immediate post-test and a delayed post-test
- numeracy outcome measures need to be reliable, valid and unbiased
- 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
