maths development 1 - cognition and skills Flashcards
define mathematical cognition
a field that seeks to understand the processes by which we come to understand mathematical ideas
components of mathematical cognition (3)
- 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
motivation for studying maths cognition (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)
order of developing 6 mathematical skills
- non-symbolic number (how many dots are there)
- learning the count list (count 1-10 - reciting these words doesn’t mean knowing their meaning)
- symbolic number (written number)
- arithmetic operations (+-/*)
- rational numbers (fractions)
- algebra
symbolic numbers
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
number word acquisition - rote counting
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
number word acquisition - do they actually understand what the numbers mean when they say it
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
5 counting principles
- one-to-one principle
- stable order principle
- abstraction principle
- order irrelevance principle
- cardinality principle
counting principle 1 - one-to-one principle
- 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
counting principle 2 - stable order principle
number words are recited in a fixed order
the order is meaningful
counting principle 3 - abstraction principle
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 …)
counting principle 4 - order irrelevance principle
order in which objects are counted does not matter
each order leads to the same results
counting principle 5 - cardinality principle
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
testing the cardinality principle of counting - 4 levels
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
arabic digit acquisition
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
