Numerical Cognition

1. Cross-Cultural Variability

1.1. Principle of Linear Order: Ordering elements linearly according to some dimension, such as magnitude.

1.2. Left-to-Right Mapping Bias: In Western societies, there is a tendency to associate small numbers with the left side of space and large numbers with the right side of space. Pitt et al. (2021) contends that this bias is influenced by culture, rather than being innate

1.3. Culture Dependent Experiences: Unique experiences that individuals within a given culture are exposed to. Such experiences can influence mental mapping. In societies with left-to-right mapping, there happen to be many left-to-right oriented experiences, such as reading left to right.

1.4. Whorf's Hypothesis: Suggests that the language one speaks could influence how numerosity is represented.

2. Adolscent Numerical Development

2.1. Whole Number Bias: Tendency to focus on the whole number parts of fractions (numerator and denominator) instead of viewing the fraction as a single number. Can interfere with how kids understand a fraction's magnitude

2.2. Braithwaite & Siegler (2016): Explores how affects one's understanding of fraction magnitude. Results found that whole number bias decreases as age and fraction experience increase. This reflects increased attention to the magnitudes of fractions.

2.3. Fraction Representations: Componential fraction representations reflect the magnitude of a fraction's whole number components. Hybrid fraction representations reflect the influence of component magnitude, as well as intergrated fraction sizes.

2.4. Algebraic Thinking: Encompasses analytical, structural, and functional thinking.

3. Numerical Skills in Education

3.1. Grounded Cognition: Notion that one's thinking is linked to one's physical experiences, as well as one's interactions with their environment.

3.2. Embodied Cognition: Notion that learning occurs through physical actions in one's environment (ex. gestures), in addition to one's perceptions/physical senses.

3.3. Principles of embodied math experiences: 1. Learners' movement should have mathematical meaning. 2. Learners should take advantages of their body's ability to move dynamically. 3. To establish connections, learners should move from concrete to abstract concepts. 4. Encourage peer collaboration among learners.

3.4. PUMP Curriculum: Connects math to the real world through the use of math problems involving every day life.

4. Child Numerical Development

4.1. Cardinal Principle: The number applied to the last item in a set is representative of the set's total items. Sarnecka & Carey (2008) describe it as a principle which states that a numeral's ordinal position in a list defines its cardinal meaning.

4.2. Subset Knowers: A one knower is a child who understands the quantity, 'one.' They assocociate the term 'one' with one object. However, they don't know to associate larger numbers such as two with the corresponding number of objects. The same applied up to 'four.' such children are described as subset knowers. Sarnecka & Carey (2008) describes children who understand numbers greater than four as cardinal-rule-knowers.

4.3. The Integrated Theory of Numerical Development: 1. Rational numbers have their magnitudes represented on a MNL. 2. Magnitude representations go from a logarithmic to linear distribution. 3 All numbers have magnitudes. 4 Association and analogy are important to numerical magnitude knowledge. 5. Understanding whole and rational numbers correlated to understanding in other areas of math. 6. Interventions which strengthen understanding of whole and rational magnitudes have positive effects on many other math outcomes.

4.4. Gunderson et al. (2012): Explored how spatial skills influence numerical development in children. Results indicate that spatial skills support numerical development in children by strengthening the linearity of how they represent the number line.

5. Math Affect and Anxiety

5.1. Math Anxiety (MA): Feelings of anxiety of and apprehension that occur when faced with math, thus impairing mathematical ability. As discussed in Ashcraft (2002), it can result in personal, educational, and cognitive consequences.

5.2. Working Memory (WM): Cognitive system that can temporarily store info at a limited capacity. As discussed in Ashcraft (2002), it helps explain why MA has more impact on two digit problems than one digit problems. Two digit problems likely require more WM, and MA interferes with WM.

5.3. Math Attitudes: How one feels about math, whether they think math is useful, and whether they choose to engage with math or avoid it. Math anxiety can have a negative impact on math attitudes.

5.4. Math Identity (MI): One's identity as it pertains to math. Includes how good one feels they are at math and whether one feels they belong in math related spaces. As explained by Miller-Cotto & Lewis (2020), MI's are dynamically constructed, changing based on environment and context

6. Mathematical Learning and Disabilities

6.1. Dyscalculia: Impaired understanding of quantity and difficulty connecting number systems with the magnitudes they're represenentative of. Symptoms can include difficulty with approximate estimation, impaired representation of numerosity, and using fingers to count at older ages.

6.2. Cognitive Structures: The intraparietal sulcus is responsible for learning new arithmitic facts while the angular gyrus helps retrieve learned facts. As discussed in Butterworth et al. (2011), the organization of basic numerical activity shifts to the parietal and occipital-temporal regions as one ages. Those with dyscalculia may have altered brain processes.

6.3. Dyscalculia Interventions: The Number Race is a computer game designed to make the inherited approximate numerosity system more precise in those with Dyscalculia. Grapho-Game-Maths is a computer game designed to help those with dyscalculia represent and manipulate quantity sets.

6.4. Wilson et al. (2006): Examined whether 'The Number Race' game helped strengthen math ability in subjects with dyscalculia. They found that subjects improved in subtraction, subitizing, and number comparisons

7. Infant Numerical Development

7.1. Approximate Number System (ANS): Mental system that lets humans and animals estimate quantities without the need for exact counting. The ANS is both approximate and ratio dependent.

7.2. Subitizing: The ability to instantly recognize small quantities without needing to count. Perceptual subitizing involves quickly recognzing 3-4 items. Conceptual subitizing involves recognizing large quantities by organizing small sets.

7.3. Weber Fraction: Corresponds to the smallest, and thus most difficult, ratio that can be reliably differentiated. A smaller Weber fraction means a more precise ANS.

7.4. Izard et al. (2009): Explores whether infants are capable of representing abstract amodal numbers. Results showed that are born with abstract numerical representations, thus supporting the notion that number approximation is an innate ability.

7.5. Berger et al. 2006): Presented infants with correct and incorrect math equations (by showing them dolls) and measured their looking time and levels of brain activation. Elevated brain activity showed that infants could detect mathematical errors, thus indicating that the cognitive mechanisms needed for error detection form within the first year of life.

8. Animal Cognition

8.1. Mental Number Line (MNL): Association between number and space in which magnitude is mentally represented across a horizontal line. Smaller numbers are associated with one side (usually the left) and larger numbers are associated with the other (usually the right). Apparent in humana and some animals.

8.2. Ditz & Nieder (2016): Examined the numerical discrimination ability of crows. Found that crows can understand large quantities, have an ANS, and have a logarithmic MNL.

8.3. Cantlon & Brannon (2007): Compared humans' and monkeys' nonverbal arithmitic abilities. Results indicate that the ability to accurately combine numerical values in monkeys is ratio dependent. The authors thus concluded that humans and nonhuman primates seem to possess similar systesm for basic nonverbal aritmitic.

8.4. Ferrigno et al. (2016): Examined quantitative reasoning in infant monkeys. As the ratio between experimental quantities increased, trial accuracy decreased. This indicates that when solving tasks, infant monkeys used analog quantitative reasoning. Thus, nonverbal quantitative appears during the first year of life in monkeys.