early maths development

Question 1

what is mathematical cognition?

Answer 1

A field that seeks to understand the processes by which we come to understand mathematical ideas

Question 2

What are mathematical cognition researchers interested in?

Answer 2

• 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

Question 3

who struggles with maths?

Answer 3

• 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

Question 4

the 6 stages of developing mathematical skills (in order)

Answer 4

1. non-symbolic numbers
2. learning the count list
3. symbolic numbers
4. arithmetic operations
5. rational numbers
6. algebra

Question 5

what are symbolic numbers?

Answer 5

• 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

Question 6

number word acquisition

Answer 6

• 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

Question 7

Morphological Bootstrapping Hypothesis

Answer 7

• 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

Question 8

the 5 counting principles

Answer 8

1. The one-to-one principle
2. The stable order principle
3. The abstraction principle
4. The order irrelevance principle
5. The cardinality principle

Question 9

the one-to-one principle

Answer 9

• 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

Question 10

the stable order principle

Answer 10

The number words are recited in a fixed order

Question 11

the abstraction principle

Answer 11

• 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

Question 12

the order irrelevance principle

Answer 12

• The order in which objects are counted does not matter
• Each order leads to the same result

Question 13

the cardinality principle

Answer 13

• 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

Question 14

Give-N-Task (testing the cardinality principle)

Answer 14

• 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

Question 15

Arabic digit acquisition

Answer 15

• 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

Question 16

connecting spoken number words, arabic digits and quantity (Lira, et al., 2017)

Answer 16

• 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

Question 17

Connecting Spoken Number Words, Arabic Digits, & Quantity (Hurst et al., 2017)

Answer 17

• 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

Question 18

ordinality

Answer 18

• 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

Question 19

ways to assess ordinality

Answer 19

1. number ordering task
   –> put the numbers in the correct order
   –> quickly and accurately
2. 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)

Question 20

evaluate ordinality tasks

Answer 20

• 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

Question 21

what is the home numeracy environment?

Answer 21

• 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

Question 22

ways to measure the home numeracy environment - observational studies

Answer 22

• 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

Question 23

ways to measure the home numeracy environment - questionnaires

Answer 23

• 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

Question 24

direct / formal examples on the questionnaire

Answer 24

• 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

Question 25

indirect / informal examples on the questionnaire

Answer 25

• 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

Question 26

home numeracy environment and maths achievement

Answer 26

• 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

Question 27

home numeracy environment and parents factors

Answer 27

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

Question 28

the importance of early numeracy skills

Answer 28

• 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

Question 29

Duncan et al (2007) meta analysis

Answer 29

• 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

Question 30

numeracy skills beyond school years

Answer 30

• Numeracy skills are important for life outcomes:
  –> employment opportunities
  –> obtaining and retaining employment
  –> promotion opportunities
  –> owning a home, income
  –> quality of healthcare
  –> mental health

Question 31

pathways to mathematics model (predictors of numeracy)

Answer 31

• 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

Question 32

quantitative skills

Answer 32

• Early numeracy skills of quantifying, labeling, comparing, and manipulating sets
• How have quantitative skills been measured:
  –> Subitising
  –> Non-symbolic arithmetic
  –> Counting
  –> Estimation
  –> Number comparison

Question 33

subitising

Answer 33

• Quickly determining the number of items in a small set without counting
• Subitising in preschool/kindergarten predicted mathematics outcomes 2 years later

Question 34

non-symbolic arithmetic

Answer 34

• 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

Question 35

counting

Answer 35

• present kids with dots
• ask them to count them
• counting in kindergarten predicted arithmetic performance in Grade 1

Question 36

estimation

Answer 36

• ask kids to estimate number of dots without counting them
• estimation in kindergarten predicted arithmetic performance in Grade 1

Question 37

number comparison

Answer 37

• 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

Question 38

non-symbolic or symbolic use?

Answer 38

when considered together, symbolic number comparison demonstrates more predictive power than non-symbolic

Question 39

working memory

Answer 39

• 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

Question 40

visuospatial working memory

Answer 40

• 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

Question 41

verbal working memory

Answer 41

• 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

Question 42

evaluate visuospatial and verbal working memory

Answer 42

• 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

Question 43

linguistic skills

Answer 43

• 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

Question 44

receptive vocab and phonological awareness (linguistic skills)

Answer 44

• 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

Question 45

summarise pathways to mathematics

Answer 45

• 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

Question 46

early intervention

Answer 46

• 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

Question 47

criteria for evaluating interventions

Answer 47

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