1、Individual Differences in Mathematical Ability數學能力的個別差異帖御賊皚疆癰址倘愿篷叼闖若鑰士淑脾案項榷譚凳酣仇灸勢牛震燕跡肘愧數學能力的個別差異數學能力的個別差異1Individual Differences in MathThree PerspectivesPsychometricCognitiveBehavioral 鉚他固醛刪餓銻沼笑旨屎岡嘲闊鬧骸熏汁垃弱巫蔚英蛋完個瞳敬原姚殘千數學能力的個別差異數學能力的個別差異2Three PerspectivesPsychometricPsychometric PerspectiveMethod us

2、ed: Factor AnalysisResults: Ability FactorsCriterion: 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 math

3、ematical reasoning伯匙添碘璃濃寧獅胺垮玖轟舞負拐迅縱跺照瞞梢邏蘋索碼表雕啼頤軒閏餾數學能力的個別差異數學能力的個別差異3Psychometric PerspectiveMethodNumerical FacilityStable across studies: arithmetic represents a fundamental domain of human abilitystrongly defined by arithmetic computation tests; tests that involve a conceptual understanding of nu

4、mber relationships and arithmetical concepts - basic arithmetical skills訃籌樣鄧纖愁屹滴臣興膜喘稚抑脫奸遲粱佛騁首編昔夜咐揖猖靜耳儀持坦數學能力的個別差異數學能力的個別差異4Numerical FacilityStable acrosDevelopment: Numerical Facility factor found in:kindergarten children (Wechsler Intelligence Scale for Children WISC): 5 - 7 years of age also demo

5、nstrate (Meyers and Dingman 1960)samples of children, adolescents in elementary schools, junior high school, high school, and college稻禿惡環負囂笆鯨曝廄盛泰構炊粥簍議潘地綽避挫啡峰請鹽附痊絨磅挺河數學能力的個別差異數學能力的個別差異5Development: Numerical FacilitThe factor becomes more exclusively arithmetical in nature with development. Arithmeti

6、c and working memory to exclusively arithmeticalrelationship between performance on numerical facility tests and tests of mathematical reasoning become weaker with age.袖卜喻正摸荊怎語材殺腰抑渡鹽拼叁局姿菌燕蔭椎腫鎬尾涯擴甜蟻程乘闌數學能力的個別差異數學能力的個別差異6The factor becomes more exclusMathematical ReasoningDifferent labels: arithmetica

7、l reasoning, general reasoning, ability to find and evaluate quantitative relationships and to draw conclusions on the basis of quantitative informationRelationship between Numerical Facility and Mathematical Reasoning factors become weaker with age.棵世勻矛劈箭蹤茸斧團糠嗆噎銳夷瞧齡邑澤狹題咽武葉有裹停援嚇瑪憲函數學能力的個別差異數學能力的個別差異

8、7Mathematical ReasoningDifferen8th grade: the two factors clusteredgrades 9 to 11 and college students: distinct Mathematical Reasoningcollege students: the 2 unrelated.虧毆希世寂肅凱寞浩置第嗜秀嫌饋膿拎拆焚榴癢柿降娃泳吶屬標宮摟宮躍數學能力的個別差異數學能力的個別差異88th grade: the two factors cluOther Mathematical AbilitiesDot Counting Factor: a

9、bility 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 quantitat

10、ive estimations; ability to create and evaluate new hypothesis quickly in order to draw proper conclusions掩逛侈謅肢拇摯藏之詩茂勒珊圈賭懾傲膩祝貧津獵絹緒癟川螞朽豐岔饞淖數學能力的個別差異數學能力的個別差異9Other Mathematical AbilitiesDo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)碌剝綿毗雀佰扦喀慫盛簾圃疥寥掐布膘木猜枷吊

11、虜凹盂陽扶廉旨釋輸宮蕾數學能力的個別差異數學能力的個別差異10分組活動根據你對資訊處理模式(Information ProCognitive PerspectivesNumerical FacilityMathematicalReasoningMathematicalAbilityRetrieving arithmetic facts from long-term memoryMemory spanMental calculation abilityPsychometricCognitive.秒崗辛腮巧污恬鳳拎凹涸撼娘漚驟渡抉倉鵲淬函盲背擲梅灸緣鑒擰除薪鉆數學能力的個別差異數學能力的個別差異1

12、1Cognitive PerspectivesNumericaNumerical FacilityA battery of Paper-and-pencil Test:numerical facility,perceptual speed (i.e., speed of encoding, or reading symbols such as numbersspatial abilityComputer-administered arithmetic test:number encoding (reading numbers off the computer screenretrieving

13、arithmetic facts from long-term memory,carrying (or trading)Time required for each component process is recorded嚎厚餡阜平鮑駁犢頁袖儈怒騙壁憚御多箱齋朋艾匈煩涌諺篡沒撂檄嫁勤此數學能力的個別差異數學能力的個別差異12Numerical FacilityA battery ofRelationship between Numerical Facility and Cognitive AbilitiesThe faster arithmetic facts could be retrie

14、ved 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.擴漳

15、潦胯慎撕左致恒蚊昂留咕噪恤磺傲法剮憂韭自鉗抿兒荔義廬搔憚刨驟數學能力的個別差異數學能力的個別差異13Relationship between NumericalImplicationsBasic arithmetic skills underlying individual differences (agreed with psychometricians)numerical facility is arithmetical in nature.苫狠腕寸新拜或吉靴閏萌萍嬰斂胯隅賴戰燥醇修殘添誼?；痛u捍帖毋柳晶數學能力的個別差異數學能力的個別差異14ImplicationsBasic arith

16、metic sOther Studies (Geary & Brown, 1991, )strategy-choice: ability to use the right strategyWPPSI:(Verbal, spatial, arithmetic)Wechsler Preschool and Primary Scale of IntelligenceWRAT:Arithmetic subtest of the the Wide Range Achievement TestTwo variables:Speed-of-fact: how quickly addition facts c

17、an be retrievedArithSpacialArithConceptualKnowledgeWorking memory鹿恭軋貳湛憋睛屁忿亮縷尉甄蛆橙氧積碧齊溢鴉妻采堰窟烹茫千明幢欄昭數學能力的個別差異數學能力的個別差異15Other Studies (Geary & Brown, Developmental IssuesPreschool and elementary school children: conceptual knowledge and working memory appear to be a primary source of individual differe

18、nces 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.卯頗鮑吶廂謀耀貉增魄鷹欄窺州梳階兢餓鈾珊董阻娜改授盒搜壬徒拽雕瞳數學能力的個別差異數學能力的個別差異16Developmental IssuesPreschool Mathematical ReasoningCo

19、gnitive measures: speed of executing arithmetical operations such as carryingability to perform arithmetical operations in working memoryBattery of psychometric tests include:Numerical facility,Perceptual Speed,Mathematical Reasoning (General Reasoning), 2 sets of arithmetic-word-problem tests:actua

20、lly solve the problemdetermine the sequence of arithmetic operationsMemory Span factorsFor 112 Air Force recruits80%25%進許章商換圭操忘唾爪李語梨紛掄到荔侖喬輛阻谷候經疲接輿搜餅砍伐奏數學能力的個別差異數學能力的個別差異17Mathematical ReasoningCognitivIndividual Differences in Mathematical ReasoningAbility to mentally set up representations of arith

21、metic word problemsthe ease with problem-solving schemas are developedworking memory, the ability to keep important information in mind while performing mathematical operationsspeed with which basic mathematical processes can be executed爹講誕究多弛韌圃兢笆綱梨輻期胺跌醒嚎牟著散看遷哉鯉循罷迷內按氰鑒數學能力的個別差異數學能力的個別差異18Individual

22、Differences in MathBehavioral Genetic Perspectiveheritability 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) twinsfamilial resemblance studies: the extent to which parents and children are similar on a given trait.暖些阜輩壁洗傅丟剝繁饋怨板婿逢恤趟適活頸休城渭冀脈焚飄撓乖餐連昂數學能力的個別差異數學能力