maths development 1 - cognition and skills

Question 1

define mathematical cognition

Answer 1

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

Question 2

components of mathematical cognition (3)

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

motivation for studying maths cognition (3)

Answer 3

• use maths in our daily lives
  –> make financial and health decisions, interpret statistics reported by the media, calculate discounts, follow a recipe
• scientific and technological innovation at the forefront of economic climate
  –> avoiding careers in maths may limit future employment and economic opportunities for students
• a lot of people struggle with maths
  –> 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, one-fifth of adults unable to accurately deal with two-step calculations or understand rational numbers (decimals, percentages, fractions)

Question 4

order of developing 6 mathematical skills

Answer 4

1. non-symbolic number (how many dots are there)
2. learning the count list (count 1-10 - reciting these words doesn’t mean knowing their meaning)
3. symbolic number (written number)
4. arithmetic operations (+-/*)
5. rational numbers (fractions)
6. algebra

Question 5

symbolic numbers

Answer 5

abstract and exact representations of numerosity

human invention

2 forms:

• number words (three)
• arabic digits (3)

words for small numbers are among the first words learnt whereas Arabic digits are learnt slightly later

number skills before formal schooling is a good predictor of later maths skills

Question 6

number word acquisition - rote counting

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

just because they can say the numbers doesn’t mean they understand the meaning of it

Question 7

number word acquisition - do they actually understand what the numbers mean when they say it

Answer 7

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

other cultures learn it later as they may not have plural markers like in english “-s” - helps them understand it sooner

Question 8

5 counting principles

Answer 8

• one-to-one principle
• stable order principle
• abstraction principle
• order irrelevance principle
• cardinality principle

Question 9

counting principle 1 - 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

counting principle 2 - stable order principle

Answer 10

number words are recited in a fixed order

the order is meaningful

Question 11

counting principle 3 - abstraction principle

Answer 11

any array of sets can be counted

we count the collection of sets the same way regardless of their characteristics

can count regardless of colour, size, shape, or whether it is an abstract concept (thoughts, actions, people present, people absent …)

Question 12

counting principle 4 - order irrelevance principle

Answer 12

order in which objects are counted does not matter

each order leads to the same results

Question 13

counting principle 5 - cardinality principle

Answer 13

last number in the count sequence = how many objects there are in the total set

describes the order of the object and also the quantity of the whole set

Question 14

testing the cardinality principle of counting - 4 levels

Answer 14

Give-N-Task (Wynn, 1990)

ask a child to “give me N number of those”

results:
children fall into different categories with numerical development:

• grabbers
  –> just take a handful without thinking, or always the same number of objects
• pre-number-knowers
  –> know you want a specific quantity but can’t work out how many that means
• subset-knowers (one-knower, two-knower, three-knower, four-knower)
  –> depends what number meanings they know as to whether they are successful
  –> e.g. if they know up to 3 but not 4 they will fail if asked to give them 4 but succeed with lower numbers
• cardinal principle (CP)-knower
  –> know all number word meanings so are successful in the task, know each number is one more than the last

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 represent exact numerosity’s

meanings of arabic digits are acquired later than the meaning of number words

correlated with onset of schooling –> learn to write numbers and connect number names with written symbols

Question 16

study connecting spoken number words, arabic digits, and quantity - 2-4 year olds

Answer 16

Jiménez Lira et al. (2017)
2- to 4-year-old children

ask to match between two types of stimuli:

• quantity - to - number word + vv
• number word - to - digit + vv
• quantity - to - digit + vv

Question 17

study connecting spoken number words, arabic digits, and quantity - 3-4 year olds

Answer 17

Hurst et al (2017)
3-4 year olds

6 mapping tasks

found quantity – numeral connection was mediated by quantity – word and word – numeral

check slide!!

Question 18

define ordinality

Answer 18

relation between items in a sequence

Question 19

ordinality development

Answer 19

emerges later than cardinality

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)

7-12 years of age = quite accurate for both adjacent and non-adjacent sequences

Question 20

ordinality tasks

Answer 20

assess ordinality with either a number ordering task or an order judgment task:

• ordering = have to quickly and accurately tap numbers in the correct order on a screen
• order judgement = determine whether sequences are in ascending order or not

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 - other could be confusing instructions (don’t know word “ascending”)