5～6岁儿童数学问题解决的认知诊断评估工具
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摘要
基于认知诊断评估理论和技术,本研究采用文献分析、专家研讨和口语报告法构建5～6岁儿童"数与运算"领域数学问题解决的认知模型,对627名5～6岁儿童进行认知诊断评估,运用认知诊断模型—数据拟合对认知模型和诊断测验进行质量检验。结果显示,认知属性对题目难度的解释量为0.879,效应量为0.834,平均HCI为0.749;认知诊断测验题目的平均区分度为0.707(标准差为0.299);除两题的题目拟合度S-X~2指标的显著性水平<0.01,其他题目的拟合度都较好;测验的信度为0.95;平均掌握概率与测验总分呈单调递增曲线。以上结果说明,本研究构建的儿童数学问题解决认知模型良好,编制的认知诊断测验具有较理想的测量信度与效度,可以作为评估儿童数学问题解决的有效工具,并可以提供精细丰富的诊断信息,从而为相关教育教学及干预研究提供科学依据。
According to the theory and technology of cognitive diagnosis assessment,this research aimed to develop a cognitive diagnosis assessment of mathematics problem solving,which focuses on "number concept and operation",for children aged 5 to 6. Method:(1)Literature review,experts' judgment,children's oral were applied to specify the cognitive model(Cognitive attributes and their hierarchy).(2)A total of 627 Shanghai children aged 5～6 years old were assessed with cognitive diagnosis test and using the model-data fitting test to verify the cognitive model and cognitive diagnosis test. Result :(1)Cognitive attributes can explain 87.9% of the variation in the item difficulty,and the effect size was 0.834. The average HCI is 0.749;(2)Average discrimination of items was 0.707(SD = 0.299);besides two items with poor S-X~2 item fit statistic(<0.01),others were good item fit;the average classification consistency reliability of eleven attributes was 0.95;the relationship between average master probability and test score showed a monotone increasing curve. The above results suggested that the cognitive model was rationale and the test had high reliability and validity,which can be used as an effective measurement of mathematics problem solving cognitive diagnosis for children aged 5 to 6.
According to the theory and technology of cognitive diagnosis assessment,this research aimed to develop a cognitive diagnosis assessment of mathematics problem solving,which focuses on "number concept and operation",for children aged 5 to 6. Method:(1)Literature review,experts' judgment,children's oral were applied to specify the cognitive model(Cognitive attributes and their hierarchy).(2)A total of 627 Shanghai children aged 5～6 years old were assessed with cognitive diagnosis test and using the model-data fitting test to verify the cognitive model and cognitive diagnosis test. Result :(1)Cognitive attributes can explain 87.9% of the variation in the item difficulty,and the effect size was 0.834. The average HCI is 0.749;(2)Average discrimination of items was 0.707(SD = 0.299);besides two items with poor S-X~2 item fit statistic(<0.01),others were good item fit;the average classification consistency reliability of eleven attributes was 0.95;the relationship between average master probability and test score showed a monotone increasing curve. The above results suggested that the cognitive model was rationale and the test had high reliability and validity,which can be used as an effective measurement of mathematics problem solving cognitive diagnosis for children aged 5 to 6.
引文
[1]GEARY D C.Mathematics and learning disabilities[J].Journal of Learning Disabilities,2004,37(1):4-15.
[2]DUNCAN G J,DOWSETT C J,CLAESSENS A,et al.School readiness and later achievement[J].Developmental Psychology,2007,43(6):1428-1446.
[3]JORDAN N C,KAPLAN D,LOCUNIAK M N,et al.Predicting first-grade math achievement from developmental number sense trajectories[J].Learning Disabilities Research&Practice,2007,22(1):36-46.
[4]STOCK P,DESOETE A,ROEYERS H.Detecting children with arithmetic disabilities from kindergarten:evidence from a 3-year longitudinal study on the role of preparatory arithmetic abilities[J].Journal of Learning Disabilities,2010,43(3):250-268.
[5]CHETTY R,FRIEDMAN J N,HILGER N,et al.$320,000 kindergarten teachers[J].Phi Delta Kappan,2010(92):22-25.
[6]MORGAN P L,FARKAS G,WU Q.Kindergarten children’s growth trajectories in reading and mathematics:who falls increasingly behind[J].Journal of Learning Disabilities,2011(5):472-488.
[7][37]许晓晖.儿童早期日常数学问题解决能力发展及其影响因素研究[D].北京:北京师范大学,2005:4-5.
[8][18]LEIGHTON J P,GIERL M J.Cognitive diagnostic assessment for education[M].Cambridge:Cambridge University Press,2007:3,88-99.
[9]戴海琦,谢美华,丁树良.我国大陆认知诊断研究的文献计量分析[J].南京师大学报(社会科学版),2013(6):88-97.
[10]李峰,余娜,辛涛.小学四、五年级数学诊断性测验的编制:基于规则空间模型的方法[J].心理发展与教育,2009(3):113-118.
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[13]康春花,吴会云,陈婧,等.小学数学“图形与几何”认知诊断测验的编制[J].教育测量与评价:理论版,2015(10):4-8,28.
[14]范晓玲,王珺,周路平,等.基于规则空间模型的3~4年级阅读障碍儿童语素意识的认知诊断[J].教育测量与评价,2017(5):12-16.
[15]SARAMA J,CLEMENTS D H.Early childhood mathematics education research:learning trajectories for young children[M].New York and London:Routledge,2009:27.
[16]STOCK P,DESOETE A,ROEYERS H.Predicting arithmetic abilities:the role of preparatory arithmetic markers and intelligence[J].Journal of Psychoeducational Assessment,2009,27(3):237-251.
[17]JORDAN N C,KAPLAN D,RAMINENI C,et al.Early math matters:kindergarten number competence and later mathematics outcomes[J].Developmental Psychology,2009,45(3):850-867.
[19]GIERL M J,ROBERTS M,ALVES C,et al.Using judgments from content specialists to develop cognitive models for diagnostic assessments[A].The annual meeting of the National Council on Measurement in Education.San Diego,CA.USA,2009.
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[21]MAYER R E.Thinking,problem solving,cognition(2nd ed.)[M].New York:W.H.Freeman and Company,1992.
[22]喻平.数学问题解决认知模式及教学理论研究[D].南京:南京师范大学,2002:21-23.
[23]LUCANGELI D,TRESSOLDI P E,CENDRON M.Cognitive and metacognitive abilities involved in the solution of mathematical word problems:validation of a comprehensive model[J].Contemporary Educational Psychology,1998,23(3):257-275.
[24]DEEK F P,TUROFF M,MCHUGH J A.A common model for problem solving and program development[J].IEEE Transactions on Education,1999,42(4):331-336.
[25]梁好翠.论数学应用问题解决的认知过程模式[J].钦州学院学报,2008(6):107-111.
[26]魏雪峰,崔光佐.小学数学问题解决认知模型研究[J].电化教育研究,2012(11):79-85.
[27]CARPENTER T P,MOSER J M.The development of addition and subtraction problem-solving skill[C]//CARPENTER T P,MOSER J M,ROMBERG T A.Addition and subtraction:a cognitive perspective.Hillsdale,NJ:Lawrence Erlbaum Associates Inc,1982:9-24.
[28]HELLER J I,GREENO J G.Semantic processing of arithmetic word problem solving[A].Presented at the Meeting of the Midwestern Psychological Association,Chicago,1978.
[29]LEWIS A B,MAYER R E.Student’s miscomprehension of relational statements in arithmetic word problems[J].Journal of Educational Psychology,1987,79(4):363-371.
[30]GREENO J G.Some example of cognitive task analysis with instructional implication[C]//SNOWR E,EDERICO P F,MONTAGU W E.Aptitude,learning,and instruction hillsdale.NJ Lawrence Erlbaum Associates Inc,1980:1-21.
[31]REED S K.Word problems:research and curriculum reform[M].New York:Routledge,1999:15-20.
[32]RILEY M S,GREENO J G.Developmental analysis of understanding language about quantities and of solving problems[J].Cognition and Instruction,1988(5):49-101.
[33][64]CUMMINS D D,KINTSCH W,REUSSER K,et al.The role of understanding in solving word problems[J].Cognitive Psychology,1988,20(4):405-438.
[36]付建慧,韦小满.小学分数问题解决的认知模型研究[J].考试研究,2016(1):82-89.
[38]杨海清,庞象辰,周碧光,等.幼儿数概念和运算能力的发展[J].心理科学杂志,1985(5):57-59.
[39]仇佩英.幼儿数概念形成及加减运算掌握情况调查[J].上海教育科研,1986(4):25-26.
[40]周晶.5~7岁儿童数学过程性能力构成要素探索与模型建构[J].学前教育研究,2018,278(2):12-24.
[41]NUNES T,BRYANT P,EVANS D,et al.Assessing quantitative reasoning in young children[J].Mathematical Thinking and Learning,2015,17(2-3):178-196.
[42]LEIGHTON J P,GIERL M J,HUNKA S M.The attribute hierarchy method for cognitive assessment:a variation on tatsuoka’s rule-space approach[J].Journal of Educational Measurement,2004,41(3):205-237.
[43][57]GIERL M J,WANG C,ZHOU J.Using the attribute hierarchy method to make diagnostic inferences about examinees’cognitive skills in algebra on the SAT[J].Journal of Technology,Learning,and Assessment,2008,6(6):1-53.
[44]丁树良,杨淑群,汪文义.可达矩阵在认知诊断测验编制中的重要作用[J].江西师范大学学报(自然科学版),2010(5):490-494.
[45]丁树良,汪文义,罗芬.认知诊断中Q矩阵和Q矩阵理论[J].江西师范大学学报(自然科学版)2012,36(5):441-445.
[47]JUNKER B W,SIJTSMA K.Cognitive assessment models with few assumptions,and connections with nonparametric item response theory[J].Applied Psychological Measurement,2001,25:258-272.
[48]HARTZ S M.A bayesian framework for the unified model for assessing cognitive abilities:blending theory with practicality(unpublished doctoral dissertation)[D].IL:University of Illinois at UrbanaChampaign,Champaign,2002.
[49]TEMPLIN J L,HENSON R A.Measurement of psychological disorders using cognitive diagnosis models[J].Psychological Methods,2006,11:287-305.
[50]MARIS E.Estimating multiple classification latent class models[J].Psychometrika,1999,64(2):187-212.
[51]DA LA TORRE J.The generalized DINA model framework[J].Psychometrika,2011,76(2):179-199.
[52]DA LA TORRE J.An empirically based method of Q-matrix validation for the DINA model:development and applications[J].Journal of educational measurement,2008,45(4):343-362.
[53]ORLANDO M,THISSEN D.Likelihood-based item-fit indices for dichotomous item response theory models[J].Applied Psychological Measurement,2000,24:50-64.
[54]ORLANDO M,THISSEN D.Further investigation of the performance of S-X2:an item fit index for use with dichotomous item response theory models[J].Applied Psychological Measurement,2003,27:289-298.
[55]TEMPLIN J,BRADSHAW L.Measuring the feliability of diagnostic classification model examinee estimates[J].Journal of Classification,2013,30(2):251-275.
[56]ROBITZSCH A,KIEFER T,GEORGE C A,et al.CDM:cognitive diagnosis modeling(R package version 4.6-0,2015)[R/OL].(2016-03-12)[2018-12-12].http://CRAN.R-project.org/package=CDM.
[58]CHEN J,TORRE J,ZHANG Z.Relative and absolute fit evaluation in cognitive diagnosis modeling[J].Journal of Educational Measurement.2013,50(2):123-140.
[59]陈兴国.数学课堂中问题解决教学的理论与实践研究[D].昆明:云南师范大学,2006:8-10.
[60]MEVARECH Z,KRAMARSHI B.Critical math for innovative societies and the role of matecognitive pedagogies[M].Paris:OECD Publishing,2014:2-15.
[61]CARPENTER T P,ANSELL E,FRANKE M L,et al.Models of problem solving[J].Journal for Research in Mathematics Education.1993,24(5):428-441.
[62][63]KAMII C.Young children reinvent arithmetic(2nd ed.)[M].New York and London:Teachers College Press,2000.
[65]CARPENTER T P,HIEBERT J,Moser J.The effect of problem stricture on first graders’initial solution processes for simple addition and subtraction problems[A].The annual meeting of The American Educational Research Association.Sanfrancisco,California,1979:1-45.
[66]HEGARTY M,MAYER R E,Monk C A.Comprehension of arithmetic word problems:a comparison of successful and unsuccessful problem solvers[J].Journal of Education Psychology,1995,87(1):18-32.
[67]陈英和,仲宁宁,田国胜,等.小学2~4年级儿童数学应用题表征策略差异的研究[J].心理发展与教育,2004(4):19-24.
[68]韩瑽瑽,曹仕莹,陈英和.4~6岁儿童加减法反演律概念的发展与影响因素[J].心理发展与教育,2012(2):121-13.
[69]任培晓.4~6岁幼儿对数学加减逆反原则理解和运用的发展研究[D].上海:华东师范大学,2008:21.
[2]DUNCAN G J,DOWSETT C J,CLAESSENS A,et al.School readiness and later achievement[J].Developmental Psychology,2007,43(6):1428-1446.
[3]JORDAN N C,KAPLAN D,LOCUNIAK M N,et al.Predicting first-grade math achievement from developmental number sense trajectories[J].Learning Disabilities Research&Practice,2007,22(1):36-46.
[4]STOCK P,DESOETE A,ROEYERS H.Detecting children with arithmetic disabilities from kindergarten:evidence from a 3-year longitudinal study on the role of preparatory arithmetic abilities[J].Journal of Learning Disabilities,2010,43(3):250-268.
[5]CHETTY R,FRIEDMAN J N,HILGER N,et al.$320,000 kindergarten teachers[J].Phi Delta Kappan,2010(92):22-25.
[6]MORGAN P L,FARKAS G,WU Q.Kindergarten children’s growth trajectories in reading and mathematics:who falls increasingly behind[J].Journal of Learning Disabilities,2011(5):472-488.
[7][37]许晓晖.儿童早期日常数学问题解决能力发展及其影响因素研究[D].北京:北京师范大学,2005:4-5.
[8][18]LEIGHTON J P,GIERL M J.Cognitive diagnostic assessment for education[M].Cambridge:Cambridge University Press,2007:3,88-99.
[9]戴海琦,谢美华,丁树良.我国大陆认知诊断研究的文献计量分析[J].南京师大学报(社会科学版),2013(6):88-97.
[10]李峰,余娜,辛涛.小学四、五年级数学诊断性测验的编制:基于规则空间模型的方法[J].心理发展与教育,2009(3):113-118.
[11][34]涂冬波,戴海崎,蔡艳,等.小学儿童数学问题解决认知诊断[J].心理科学,2010,33(6):1461-1466.
[12][35]康春花,辛涛,田伟.小学数学应用题认知诊断测验编制及效度验证[J].考试研究,2013(6):24-43.
[13]康春花,吴会云,陈婧,等.小学数学“图形与几何”认知诊断测验的编制[J].教育测量与评价:理论版,2015(10):4-8,28.
[14]范晓玲,王珺,周路平,等.基于规则空间模型的3~4年级阅读障碍儿童语素意识的认知诊断[J].教育测量与评价,2017(5):12-16.
[15]SARAMA J,CLEMENTS D H.Early childhood mathematics education research:learning trajectories for young children[M].New York and London:Routledge,2009:27.
[16]STOCK P,DESOETE A,ROEYERS H.Predicting arithmetic abilities:the role of preparatory arithmetic markers and intelligence[J].Journal of Psychoeducational Assessment,2009,27(3):237-251.
[17]JORDAN N C,KAPLAN D,RAMINENI C,et al.Early math matters:kindergarten number competence and later mathematics outcomes[J].Developmental Psychology,2009,45(3):850-867.
[19]GIERL M J,ROBERTS M,ALVES C,et al.Using judgments from content specialists to develop cognitive models for diagnostic assessments[A].The annual meeting of the National Council on Measurement in Education.San Diego,CA.USA,2009.
[20][46]涂冬波,漆书青,戴海琦,等.教育考试中的认知诊断评估[J].考试研究,2008(4):4-15.
[21]MAYER R E.Thinking,problem solving,cognition(2nd ed.)[M].New York:W.H.Freeman and Company,1992.
[22]喻平.数学问题解决认知模式及教学理论研究[D].南京:南京师范大学,2002:21-23.
[23]LUCANGELI D,TRESSOLDI P E,CENDRON M.Cognitive and metacognitive abilities involved in the solution of mathematical word problems:validation of a comprehensive model[J].Contemporary Educational Psychology,1998,23(3):257-275.
[24]DEEK F P,TUROFF M,MCHUGH J A.A common model for problem solving and program development[J].IEEE Transactions on Education,1999,42(4):331-336.
[25]梁好翠.论数学应用问题解决的认知过程模式[J].钦州学院学报,2008(6):107-111.
[26]魏雪峰,崔光佐.小学数学问题解决认知模型研究[J].电化教育研究,2012(11):79-85.
[27]CARPENTER T P,MOSER J M.The development of addition and subtraction problem-solving skill[C]//CARPENTER T P,MOSER J M,ROMBERG T A.Addition and subtraction:a cognitive perspective.Hillsdale,NJ:Lawrence Erlbaum Associates Inc,1982:9-24.
[28]HELLER J I,GREENO J G.Semantic processing of arithmetic word problem solving[A].Presented at the Meeting of the Midwestern Psychological Association,Chicago,1978.
[29]LEWIS A B,MAYER R E.Student’s miscomprehension of relational statements in arithmetic word problems[J].Journal of Educational Psychology,1987,79(4):363-371.
[30]GREENO J G.Some example of cognitive task analysis with instructional implication[C]//SNOWR E,EDERICO P F,MONTAGU W E.Aptitude,learning,and instruction hillsdale.NJ Lawrence Erlbaum Associates Inc,1980:1-21.
[31]REED S K.Word problems:research and curriculum reform[M].New York:Routledge,1999:15-20.
[32]RILEY M S,GREENO J G.Developmental analysis of understanding language about quantities and of solving problems[J].Cognition and Instruction,1988(5):49-101.
[33][64]CUMMINS D D,KINTSCH W,REUSSER K,et al.The role of understanding in solving word problems[J].Cognitive Psychology,1988,20(4):405-438.
[36]付建慧,韦小满.小学分数问题解决的认知模型研究[J].考试研究,2016(1):82-89.
[38]杨海清,庞象辰,周碧光,等.幼儿数概念和运算能力的发展[J].心理科学杂志,1985(5):57-59.
[39]仇佩英.幼儿数概念形成及加减运算掌握情况调查[J].上海教育科研,1986(4):25-26.
[40]周晶.5~7岁儿童数学过程性能力构成要素探索与模型建构[J].学前教育研究,2018,278(2):12-24.
[41]NUNES T,BRYANT P,EVANS D,et al.Assessing quantitative reasoning in young children[J].Mathematical Thinking and Learning,2015,17(2-3):178-196.
[42]LEIGHTON J P,GIERL M J,HUNKA S M.The attribute hierarchy method for cognitive assessment:a variation on tatsuoka’s rule-space approach[J].Journal of Educational Measurement,2004,41(3):205-237.
[43][57]GIERL M J,WANG C,ZHOU J.Using the attribute hierarchy method to make diagnostic inferences about examinees’cognitive skills in algebra on the SAT[J].Journal of Technology,Learning,and Assessment,2008,6(6):1-53.
[44]丁树良,杨淑群,汪文义.可达矩阵在认知诊断测验编制中的重要作用[J].江西师范大学学报(自然科学版),2010(5):490-494.
[45]丁树良,汪文义,罗芬.认知诊断中Q矩阵和Q矩阵理论[J].江西师范大学学报(自然科学版)2012,36(5):441-445.
[47]JUNKER B W,SIJTSMA K.Cognitive assessment models with few assumptions,and connections with nonparametric item response theory[J].Applied Psychological Measurement,2001,25:258-272.
[48]HARTZ S M.A bayesian framework for the unified model for assessing cognitive abilities:blending theory with practicality(unpublished doctoral dissertation)[D].IL:University of Illinois at UrbanaChampaign,Champaign,2002.
[49]TEMPLIN J L,HENSON R A.Measurement of psychological disorders using cognitive diagnosis models[J].Psychological Methods,2006,11:287-305.
[50]MARIS E.Estimating multiple classification latent class models[J].Psychometrika,1999,64(2):187-212.
[51]DA LA TORRE J.The generalized DINA model framework[J].Psychometrika,2011,76(2):179-199.
[52]DA LA TORRE J.An empirically based method of Q-matrix validation for the DINA model:development and applications[J].Journal of educational measurement,2008,45(4):343-362.
[53]ORLANDO M,THISSEN D.Likelihood-based item-fit indices for dichotomous item response theory models[J].Applied Psychological Measurement,2000,24:50-64.
[54]ORLANDO M,THISSEN D.Further investigation of the performance of S-X2:an item fit index for use with dichotomous item response theory models[J].Applied Psychological Measurement,2003,27:289-298.
[55]TEMPLIN J,BRADSHAW L.Measuring the feliability of diagnostic classification model examinee estimates[J].Journal of Classification,2013,30(2):251-275.
[56]ROBITZSCH A,KIEFER T,GEORGE C A,et al.CDM:cognitive diagnosis modeling(R package version 4.6-0,2015)[R/OL].(2016-03-12)[2018-12-12].http://CRAN.R-project.org/package=CDM.
[58]CHEN J,TORRE J,ZHANG Z.Relative and absolute fit evaluation in cognitive diagnosis modeling[J].Journal of Educational Measurement.2013,50(2):123-140.
[59]陈兴国.数学课堂中问题解决教学的理论与实践研究[D].昆明:云南师范大学,2006:8-10.
[60]MEVARECH Z,KRAMARSHI B.Critical math for innovative societies and the role of matecognitive pedagogies[M].Paris:OECD Publishing,2014:2-15.
[61]CARPENTER T P,ANSELL E,FRANKE M L,et al.Models of problem solving[J].Journal for Research in Mathematics Education.1993,24(5):428-441.
[62][63]KAMII C.Young children reinvent arithmetic(2nd ed.)[M].New York and London:Teachers College Press,2000.
[65]CARPENTER T P,HIEBERT J,Moser J.The effect of problem stricture on first graders’initial solution processes for simple addition and subtraction problems[A].The annual meeting of The American Educational Research Association.Sanfrancisco,California,1979:1-45.
[66]HEGARTY M,MAYER R E,Monk C A.Comprehension of arithmetic word problems:a comparison of successful and unsuccessful problem solvers[J].Journal of Education Psychology,1995,87(1):18-32.
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