Mathematical thinking Flashcards
(20 cards)
What is transitivity in mathematics?
If A and B have a relation to each other and B and C have the same relation, this allows us to use information to make logical conclusions about relationships between elements
Transitivity is a fundamental property in mathematics that enables reasoning about relationships.
What does cardinality represent?
Being represented by a cardinal number allows us to work out the exact number in a set, even if it is too large to estimate
Cardinality is a key concept in set theory and helps in understanding the size of sets.
Define ordinality.
Numbers to indicate their order in relation to one another
Ordinality is essential for understanding rankings and sequences within sets.
What are additive relations?
Quantities are the same if nothing is added or subtracted
Additive relations are crucial for understanding basic arithmetic operations.
At what developmental stage did Piaget argue children must understand relational information?
Concrete operational stage at 7-11 years
This stage is characterized by the development of logical thinking and understanding of relationships.
What are the two systems of number cognition proposed?
System 1 = approximate magnitude; System 2 = precise representations of distinct individuals
Understanding these systems helps in recognizing how different cognitive processes relate to mathematical understanding.
What happens if we exceed the limits of the cognitive systems in mathematics?
Maths becomes hard
Each system has signature limits that affect mathematical problem-solving abilities.
List the tasks used to test the limits of each cognitive system.
- Habituation trials
- Violation of expectation tasks
- Cracker choice experiments
- Manual search experiments
These tasks are designed to explore numerical understanding in infants and adults.
At what age can infants estimate approximate magnitudes?
As young as 6 months
This indicates the early development of numerical cognition.
Who proposed three ways of learning in number cognition?
Carey
Carey’s theory provides insight into how children develop numerical understanding.
What is the analogue system in number cognition?
A system allowing for approximate numerical understanding, part of Carey’s model
This system is essential for grasping basic numerical concepts without precise counting.
What does the parallel individuation system allow?
Allows children to learn how to connect number with the counting system, only for up to 3 items
This system is crucial for early number learning and understanding small quantities.
What is bootstrapping in the context of number cognition?
Learning ordinal properties of number 1-3, leading to ‘cardinal principle knowers’
Bootstrapping illustrates how early numerical concepts can support later learning.
What is set-based quantification?
Understanding singular/plural distinction, dependent on language
This concept highlights the relationship between language and numerical understanding.
What numerical understanding did the Toupinambos exhibit?
Didn’t have words for numbers above 5 but showed numerical understanding above 5
This indicates that numerical cognition can exist independently of numerical language.
How did the Piraha and Munduruku differ in their use of numerical language?
- Piraha - uses words 1 and 2 but inconsistently
- Munduruku - 123 consistently, 4 and 5 inconsistently
These examples illustrate variability in numerical language and cognition across cultures.
True or False: Language is important for exact calculations but not approximate ones.
True
There is some disagreement about the nature of the role language plays in numerical understanding.
What difficulties do subset knowers face?
Difficulty with word concept mapping
This highlights challenges in learning numerical concepts among bilingual children.
What challenges do cardinality principle knowers experience?
Difficulty with concept
This reflects the complexities of understanding numerical principles in different contexts.
