数学能力的个别差异
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17
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
第一季
第二季
第三季
第四季
2
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
• 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 9 conclusions
14
Other Studies (Geary & Brown, 1991, …)
WPPSI:(Verbal, spatial, arithmetic) Wechsler Preschool and Primary Scale of Intelligence WRAT: Arithmetic subtest of the the Wide Range Achievement Test Arith Two variables: Speed-of-fact: how quickly addition facts can be retrieved strategy-choice: ability to use the right strategy
分組活動
• 根據你對資訊處理模式(Information Processing Model) 的認識, 試找出5種可能 發生差異之處, 說明所在部份, 所相差的 能力 (例如: 速度, 容量…)
10
Cognitive Perspectives
Retrieving arithmetic facts from long-term memory
16
Mathematical Reasoning
For 112 Air Force recruits Cognitive measures: • speed of executing 80% arithmetical operations such as carrying • ability to perform 25% arithmetical operations in working memory 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
4
• 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
12
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. 13
Implications
• Basic arithmetic skills underlying individual differences (agreed with psychometricians) • numerical facility is arithmetical in nature.soning
• 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.
Conceptual Knowledge
Working memory 15
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.
7
• 8th grade: the two factors clustered • grades 9 to 11 and college students: distinct Mathematical Reasoning • college students: the 2 unrelated.
8
Other Mathematical Abilities
Individual Differences in Mathematical Ability
數學能力的個別差異
1
Three Perspectives
• Psychometric • Cognitive • Behavioral
90 80 70 60 50 40 30 20 10 0 東部 中部 北部
Mathematical Ability
Numerical Facility Memory span
Mathematical Reasoning
Mental calculation ability ……..
Psychometric
Cognitive
11
Numerical Facility
A battery of Paper-andpencil 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
5
• 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.
3
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
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
第一季
第二季
第三季
第四季
2
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
• 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 9 conclusions
14
Other Studies (Geary & Brown, 1991, …)
WPPSI:(Verbal, spatial, arithmetic) Wechsler Preschool and Primary Scale of Intelligence WRAT: Arithmetic subtest of the the Wide Range Achievement Test Arith Two variables: Speed-of-fact: how quickly addition facts can be retrieved strategy-choice: ability to use the right strategy
分組活動
• 根據你對資訊處理模式(Information Processing Model) 的認識, 試找出5種可能 發生差異之處, 說明所在部份, 所相差的 能力 (例如: 速度, 容量…)
10
Cognitive Perspectives
Retrieving arithmetic facts from long-term memory
16
Mathematical Reasoning
For 112 Air Force recruits Cognitive measures: • speed of executing 80% arithmetical operations such as carrying • ability to perform 25% arithmetical operations in working memory 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
4
• 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
12
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. 13
Implications
• Basic arithmetic skills underlying individual differences (agreed with psychometricians) • numerical facility is arithmetical in nature.soning
• 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.
Conceptual Knowledge
Working memory 15
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.
7
• 8th grade: the two factors clustered • grades 9 to 11 and college students: distinct Mathematical Reasoning • college students: the 2 unrelated.
8
Other Mathematical Abilities
Individual Differences in Mathematical Ability
數學能力的個別差異
1
Three Perspectives
• Psychometric • Cognitive • Behavioral
90 80 70 60 50 40 30 20 10 0 東部 中部 北部
Mathematical Ability
Numerical Facility Memory span
Mathematical Reasoning
Mental calculation ability ……..
Psychometric
Cognitive
11
Numerical Facility
A battery of Paper-andpencil 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
5
• 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.
3
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