Mathematical Thinking Flashcards
(11 cards)
Importance of maths: 1. Transitivity
Important as it allows us to use info we have about relationships between elements tto mae logical conclusions about relations between other elements
If A and B have relation to each other….
And B and C have same relation…
Then A and C also have same relation
Importance of maths: 2. Cardinality
Being represented by a cardinal number
Enables us to work out exact number in a set, even when number is too large to estimate
If one set of elements can be paired with another set of element, with none left over, then they have same cardinality
This means we can work out approximate equivalence of any number of terms, even if they’re too big to count
Importance of maths: 3. Ordinality
Use of numbers to indicate their order in relation to one another
- Eg knowing 6 is greater than 4, so set of 6 will always be bigger than set of 4
Importance of maths: 4. Additive relations
Quantities stay same if nothing is added/ subtracted .. or if same number is added then subtracted
Children can count before they can add/ subtract
Intermediate phase:
- significant sep towards understanding units that make up quantity
- at this stage they struggle with not being able to see units, so they try represent them in perceptible format (eg fingers)
According to Piaget:
- children must understand relational info
- only occurs at Concrete Operational age (7-11 yrs)
- conversation tasks - juice in 2 cups, child says equal- puts one cup of juice into tall slim glass- younger child says they have diferent amounts, older can understand its same amount of juice
Number cognition: two system- 1. Approximate magnitude (Spelke 2004)
Each system has signature limits: if we go beyond limits, maths becomes hard
Approximate number system = how humans count without counting
Several types of paradigm that can be used with adults + young infants
- habituation experiment
- violation of expectation experiment: infants look longer at events that violate their expectation (surprising= more looking)
- manual search experiments
- Infants as young as 6 months can estimate approximate magnitudes
- 6-month-olds can do 1:2 rations
- but not 2:3
- improves with age (10-month-olds can do 2:3, adults can do 7:8)
- infants as young as 10 months can keep track of distinct individuals
Number cognition: two system- 2. Precise representations of distinct individuals
Adults maintain birth systems across lifespan
Use one or other depending on circumstance
Infants as young as 10 months can keep track of distinct individuals
Upper limit of 3
Can also track continuous variables
Conversation errors- not until 7 yrs
- different methodology (explicit, verbal tasks)
- even if we are capable of something, it doesn’t mean we always do it
3 ways of learning number (Carey, 2004)
- Analogue system (the 2 systems)
- Parallel individuating system
- Set-based quantisation
3 systems: 2. Parallel individuation system
- allows kids to learn how to connect number with counting system
- Learn association between quantity and counting
- recognise + represent small numbers exactly
- only for up to 3 items
Children learn 1st 3 numbers gradually, over several months across first 18-24 months
Bootstrapping
1. Learn ordinal properties of numbers 1-3
2. Learn list of numbers from 3+
3. Use this knowledge to infer that the rest of the number list works in same way
3 systems: Set-based quantification
Understanding singular/ plural distinction
Understanding quantifiers
Eg crackers task
- fail to understand number
- also, fail to understand plurality
- dependent on language
Sue Carey
- language is mechanism for re-structuring non-verbal notions of numbers
- bootstrapping: learnig list of numbers is placeholder for learning concepts
Rochelle Gelman
- language and number cognition are separate
- Amazonian cultures- no words for numbers above 5- still show numerical understanding
-
Semantic dementia
- very poor language production and comprehension
- scored at chilling on numerical calculation tasks
Language and maths
Language seems to be important for exact calculations…. But not approximate ones
Disagreement about nature of this role
Why does learning of number words relate to number understanding so gradually ?
1. Difficulty understanding concepts
2. Difficulty mapping words onto concepts
Language and maths
Wagner et al. (2015) — bilingual kids
Bilingual children
If slowness is due to conceptual difficulties, bilingual children learn them in both languages at the same time
If due to word-concept mapping, should learn them in each language separately
Subset-knowers: neither counting ability nor subset level in primary language predicted secondary language ——> difficulty with word-concept mapping
Cardinality-principal-knowers: CP-knowledge in primary language predicted secondary language —> difficult with concept
