Individual Differences in Mathematical Ability

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Individual Differences in Mathematical Ability

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Three Perspectives

• Psychometric
• Cognitive
• Behavioral

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Psychometric Perspective

• Method used Factor Analysis
• Results Ability Factors
• Criterion Factor stability or factor invariance
  -- the factor represents a basic domain of human
  ability, distinct from other cognitive abilities
  for educational, cultural, or biological reasons.
  
• Two general mathemttical domains numerical
  facility and mathematical reasoning

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Numerical Facility

• Stable across studies arithmetic represents a
  fundamental domain of human ability
• strongly defined by arithmetic computation tests
  tests that involve a conceptual understanding of
  number relationships and arithmetical concepts -
  basic arithmetical skills

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• Development Numerical Facility factor found in
• kindergarten children (Wechsler Intelligence
  Scale for Children WISC)
• 5 - 7 years of age also demonstrate (Meyers and
  Dingman 1960)
• samples of children, adolescents in elementary
  schools, junior high school, high school, and
  college

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• The factor becomes more exclusively arithmetical
  in nature with development.
• Arithmetic and working memory to exclusively
  arithmetical
• relationship between performance on numerical
  facility tests and tests of mathematical
  reasoning become weaker with age.

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Mathematical Reasoning

• Different labels arithmetical reasoning, general
  reasoning,
• ability to find and evaluate quantitative
  relationships and to draw conclusions on the
  basis of quantitative information
• Relationship between Numerical Facility and
  Mathematical Reasoning factors become weaker with
  age.

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• 8th grade the two factors clustered
• grades 9 to 11 and college students distinct
  Mathematical Reasoning
• college students the 2 unrelated.

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Other Mathematical Abilities

• Dot Counting Factor ability to quickly and
  accurately count arrays of dots.
• Digital Flexibility ability to manipulate, to
  arrange, and to compare (numbers), without
  performing arithmetic operations, e.g., making
  rapid greater than or less than comparisons of
  sets of numbers.
• Estimation Factor ability to make quantitative
  estimations ability to create and evaluate new
  hypothesis quickly in order to draw proper
  conclusions

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Cognitive Perspectives
Retrieving arithmetic facts from long-term memory
Numerical Facility
Memory span
Mathematical Ability
Mathematical Reasoning
Mental calculation ability
..
Psychometric
Cognitive
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Numerical Facility

• A battery of Paper-and-pencil Test
• numerical facility,
• perceptual speed (i.e., speed of encoding, or
  reading symbols such as numbers
• spatial ability

• Computer-administered arithmetic test
• number encoding (reading numbers off the computer
  screen
• retrieving arithmetic facts from long-term
  memory,
• carrying (or trading)
• Time required for each component process is
  recorded

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Relationship between Numerical Facility and
Cognitive Abilities

• The faster arithmetic facts could be retrieved
  from long-term memory and the faster the carry
  (or trade) operation could be executed, the
  better the performance on the paper-and -pencil
  numerical facility tests.
• Speed of executing these arithmetical processes
  was not directly related to performance on the
  perceptual speed and spatial tests.

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Implications

• Basic arithmetic skills underlying individual
  differences (agreed with psychometricians)
• numerical facility is arithmetical in nature.

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Other Studies (Geary Brown, 1991, )
Two variables
Arith
Spacial
Arith
Conceptual Knowledge
Working memory
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Developmental Issues

• Preschool and elementary school children
  conceptual knowledge and working memory appear to
  be a primary source of individual differences in
  arithmetic ability.
• As basic facts become committed to memory and
  procedures become automatic, speed with basic
  arithmetical processes such as carrying or
  trading become the major source.

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Mathematical Reasoning
For 112 Air Force recruits

• Battery of psychometric tests include
• Numerical facility,
• Perceptual Speed,
• Mathematical Reasoning (General Reasoning), 2
  sets of arithmetic-word-problem tests
• actually solve the problem
• determine the sequence of arithmetic operations
• Memory Span factors

• Cognitive measures
• speed of executing arithmetical operations such
  as carrying
• ability to perform arithmetical operations in
  working memory

80
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Individual Differences in Mathematical Reasoning

• Ability to mentally set up representations of
  arithmetic word problems
• the ease with problem-solving schemas are
  developed
• working memory, the ability to keep important
  information in mind while performing mathematical
  operations
• speed with which basic mathematical processes can
  be executed

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Behavioral Genetic Perspective

• heritability h2 estimate of the amount of
  variability in a particular trait for a given
  population that is due to individual differences
  in the constellation of genes that support the
  trait.
• 2 common methods
• comparing between sets of identical or
  monozygotic (MZ) twins and fraternal or dizygotic
  (DZ) twins
• familial resemblance studies the extent to which
  parents and children are similar on a given trait.

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Numerical Facility

• Median h2 across several studies .45 Vandenbery
  (1966)
• individual numerical facility tests such as
  addition .43 - .58 Vandenbery (1962)
• elementary grades in arithmetic .66 (Husen)
• arithmetical skills of parents and children ve
  correlated (DeFries)
• one half of the variability in arithmetical
  ability is due to genetic factors.

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Mathematical Reasoning

• h2.27 across several studies (Vandenberg)
• MZ and DZ h2 .36 to .66
• a small to moderate proportion of individual
  differences in mathematical reasoning ability are
  due to genetic factors.