数学问题图式的等级性研究
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摘要
图式是学习新知识的前提,又是新知识学习的结果,在数学学习中,图式具有重要的作用,因此有关图式的研究成为数学教育心理学研究的热点。已有对数学应用题图式的研究表明,随着概括水平的不同,图式作为有组织的知识结构,呈现由低到高的等级序列;其中的一些研究从关系一表征复杂性和知识基础两方面解释了图式等级性的原因和本质,并由此确定问题的难度。
基于已有研究,本文采用理论思辨与定量验证的研究方法,将表征复杂性模型用到更复杂问题的图式等级性研究中。首先,研究了无背景问题图式的等级性:依据模型分成六个水平的模版(即问题类型图式),通过对问题进行关系复杂性和知识基础两方面的分析,将问题归入不同模版,同时对六个模版上的问题进行一定样本数据的定量设计与分析,采用两种图式测量技术以验证和支持有关问题图式的结论。其次,研究了有背景问题的难度:依据模型定性分析背景问题的难度,根据分析设计提示语,通过提示组和无提示组成绩的比较研究,检验背景问题难度的假设。此外,研究都区分了优、中、差三类学生,以获得各类学生的图式水平和受背景影响程度方面的信息。最后,依据图式等级理论,提出一些相关的数学教学策略。
研究结论:(1)个体的问题图式具有等级组织性,这种等级性基本可以用表征复杂性和知识基础两方面来解释,其中表征复杂性与个体的推理水平有关。(2)优、中、差三类学生的问题图式水平是有差异的,其中优生表征深度大,能运用知识基础表征隐含条件所反映的关系;中等生在解决关系较复杂、推理水平要求较高的问题上缺乏图式支持,不能有效运用具备的知识基础;差生的知识储备不足,在以“关系”形式包含至高一级关系的表征上有困难。(3)给问题加入背景后,会增加问题的难度。(4)给背景问题加入合适的提示后,会降低问题的难度,并且提示的作用在中等难度的背景问题中最为明显。(5)优秀生和普通生受背景影响程度不同,其中背景对优秀生的问题解决影响不大,而对普通生影响较大。(6)根据图式等级理论,提出四条数学教学策略:确定问题难度级差,教学做到循序渐进;分析学生问题图式水平,教学做到因材施教;认识背景问题难度成因,加强背景问题解决教学;有效运用提示语,提高学生解决背景问题的能力。
Schema is a key word in modern cognitive psychology. It was proposed as a form of mental representation for knowledge. From the point of view of mathematics learning, schema plays an important role. It can integrate existed knowledge and influence the acquisition of new knowledge. When refers to mathematical problem solving, it also has direct impact on representation and transfer. Therefore the quality of schema is crucial for mathematics learners.
The construction of schema of high quality should follow some cognitive principles. From schematic theory and some researches based on it, we know hierarchical ordering of schematic knowledge, which means forming schema of high quality should be based on the inferior one. Direct and indirect evidences were provided, and what's more, there is study explaining the essence of hierarchical ordering of templates based on the Relational - Representational Complexity Model (RRCM). According to the model, the order of a certain template can be interpreted by qualitative analysis without any experiment. Once validated, the model will offer an operable method to evaluate the problem complexity so that the learning materials can be organized in a scientific consequence helpful to teaching practice.
Previous study has indicated the model's theoretical value in distinguishing different ranks of templates for area-of-rectangle. In other word, we can analyze problem complexity according to the model. But the study refered to comparative simple problems fit for pupils solving. When it comes to more complex problems, is the model still effective? It is one of the main tasks of this research.
The research concludes two parts. One is about the hierarchical ordering of templates for Pythagorean theorem which are beyond the ability of pupils to solve. Guided by the relatitve research framework, test of internal and external performance validation of RRCM based on the data from 106 8th graders on two test formats to assess their schematic knowledge. The results showed that templates could be classified from the model only if reasoning level is recognized as a factor of representational depth, which reflects the complexity of representing relations of each rank especially on reasoning. It could also discriminate the level of representational complexity of excellent, normal and poor students.
The part two discussed how to explain the complexity of problems with background. In former part it was found once with the background, the problem was more difficult than before. So what influence the complexity become the question tried to be answered in this part. It offered a hypothesis the complexity of problems with background could be divided into two parts. For each part it could also be explained by RRCM The hypothesis was confirmed by the data from two opposite groups one of which accepted clue out of the hypothesis about the explanation of problem- complexity during problem solving test. It was also found that faced problems with background normal students were more vulnerable than excellent peers.
At last, based on the researches carried out, a few didactical suggestions were proposed.
In conclusion, this thesis further clarified the definition of representational complexity especially the depth of representation so that it can be used to ascertain the complexity of more difficult problems. Furthermore, RRCM is applied to explain the complexity of problems with background which developed the effect of the model. It enriched the study of hierarchical ordering of schematic knowledge at advanced mathematics.
基于已有研究,本文采用理论思辨与定量验证的研究方法,将表征复杂性模型用到更复杂问题的图式等级性研究中。首先,研究了无背景问题图式的等级性:依据模型分成六个水平的模版(即问题类型图式),通过对问题进行关系复杂性和知识基础两方面的分析,将问题归入不同模版,同时对六个模版上的问题进行一定样本数据的定量设计与分析,采用两种图式测量技术以验证和支持有关问题图式的结论。其次,研究了有背景问题的难度:依据模型定性分析背景问题的难度,根据分析设计提示语,通过提示组和无提示组成绩的比较研究,检验背景问题难度的假设。此外,研究都区分了优、中、差三类学生,以获得各类学生的图式水平和受背景影响程度方面的信息。最后,依据图式等级理论,提出一些相关的数学教学策略。
研究结论:(1)个体的问题图式具有等级组织性,这种等级性基本可以用表征复杂性和知识基础两方面来解释,其中表征复杂性与个体的推理水平有关。(2)优、中、差三类学生的问题图式水平是有差异的,其中优生表征深度大,能运用知识基础表征隐含条件所反映的关系;中等生在解决关系较复杂、推理水平要求较高的问题上缺乏图式支持,不能有效运用具备的知识基础;差生的知识储备不足,在以“关系”形式包含至高一级关系的表征上有困难。(3)给问题加入背景后,会增加问题的难度。(4)给背景问题加入合适的提示后,会降低问题的难度,并且提示的作用在中等难度的背景问题中最为明显。(5)优秀生和普通生受背景影响程度不同,其中背景对优秀生的问题解决影响不大,而对普通生影响较大。(6)根据图式等级理论,提出四条数学教学策略:确定问题难度级差,教学做到循序渐进;分析学生问题图式水平,教学做到因材施教;认识背景问题难度成因,加强背景问题解决教学;有效运用提示语,提高学生解决背景问题的能力。
Schema is a key word in modern cognitive psychology. It was proposed as a form of mental representation for knowledge. From the point of view of mathematics learning, schema plays an important role. It can integrate existed knowledge and influence the acquisition of new knowledge. When refers to mathematical problem solving, it also has direct impact on representation and transfer. Therefore the quality of schema is crucial for mathematics learners.
The construction of schema of high quality should follow some cognitive principles. From schematic theory and some researches based on it, we know hierarchical ordering of schematic knowledge, which means forming schema of high quality should be based on the inferior one. Direct and indirect evidences were provided, and what's more, there is study explaining the essence of hierarchical ordering of templates based on the Relational - Representational Complexity Model (RRCM). According to the model, the order of a certain template can be interpreted by qualitative analysis without any experiment. Once validated, the model will offer an operable method to evaluate the problem complexity so that the learning materials can be organized in a scientific consequence helpful to teaching practice.
Previous study has indicated the model's theoretical value in distinguishing different ranks of templates for area-of-rectangle. In other word, we can analyze problem complexity according to the model. But the study refered to comparative simple problems fit for pupils solving. When it comes to more complex problems, is the model still effective? It is one of the main tasks of this research.
The research concludes two parts. One is about the hierarchical ordering of templates for Pythagorean theorem which are beyond the ability of pupils to solve. Guided by the relatitve research framework, test of internal and external performance validation of RRCM based on the data from 106 8th graders on two test formats to assess their schematic knowledge. The results showed that templates could be classified from the model only if reasoning level is recognized as a factor of representational depth, which reflects the complexity of representing relations of each rank especially on reasoning. It could also discriminate the level of representational complexity of excellent, normal and poor students.
The part two discussed how to explain the complexity of problems with background. In former part it was found once with the background, the problem was more difficult than before. So what influence the complexity become the question tried to be answered in this part. It offered a hypothesis the complexity of problems with background could be divided into two parts. For each part it could also be explained by RRCM The hypothesis was confirmed by the data from two opposite groups one of which accepted clue out of the hypothesis about the explanation of problem- complexity during problem solving test. It was also found that faced problems with background normal students were more vulnerable than excellent peers.
At last, based on the researches carried out, a few didactical suggestions were proposed.
In conclusion, this thesis further clarified the definition of representational complexity especially the depth of representation so that it can be used to ascertain the complexity of more difficult problems. Furthermore, RRCM is applied to explain the complexity of problems with background which developed the effect of the model. It enriched the study of hierarchical ordering of schematic knowledge at advanced mathematics.
引文
[1]F.C.Bartlett.黎炜译.记忆:一个实验的与社会的心理学研究[M].杭州:浙江教育出版社,1998.
[2]皮亚杰.王宪钿等译.发生认识论原理[M].北京:商务印书馆,1995.
[3]Rumelhart,D.E.(1980).'Schernata:the building blocks of cognition'.In R.J.Spiro,B.C.Bruce & W.F.Brewer(Eds.),Theoretical issues in Reading Comprehension.Hillsdale,New Jersey;Lawrence Erlbaum Associates.
[4]Rumelhart,D.E.and Ortony,A.(1977).'The representation of knowledge in memory'.In R.C.Anderson,R.J.Spiro & W.E.Montage(Eds.),Schooling and the Acquisition of knowledge.Hillsdale,New Jersey;Lawrence Erlbaum Associates.
[1]Thorndyke,P.W.(1984).'Application of schema theory in cognitive research'.In J.R.Anderson & S.M.Kosslyn(Eds.),Tutorials in Learning and Memory.San Francisco:Freeman.
[2]MeClelland,J.L.& Rumelhart,D.E.,Eds.(1986).Parallel Distributed Processing:Explorations in the Microstrueture of Cognition Vol.2:Psychological and Biological Models.MIT Press
[3]Alan,P.(1996)."Schema theory and the use of hypertext-style computer programs'.British Journal of Educational Technology,27.
[1]徐敏毅.儿童解决算术应用题认知加工过程的实验研究[J].心理发展与教育,1994,(2):33-39.
徐敏毅.4-8岁儿童解决算术应用题认知加工过程的实验研究(Ⅱ)[J].心理发展与教育,1995,(4):16-21.
[2]刘广珠.儿童解决算术应用题认知加工过程及比较图式形成的实验研究[J].心理发展与教育,1996.(2):1-5.
[3]辛自强.问题解决中图式的建构:一项应用题分类研究[J].心理发展与教育,2005,(1):69-73.
[1]Low,R.& Over,R.(1992)'Hierarchical Ordering of Schematic Knowledge Relating to Area-of-rectangle problems'[J].Journal of Educational Psychology,1992,84(1):62-69.
[2]辛自强.关系-表征复杂性模型的检验[J].心理学报,2003.35(4):504-513.
[1]Mayer,R.E.(1981).'Frequency norms and structural analysis of algebra story problems into families,categories and templates'.Instructional Science,10,135-175.
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[2]Kintsch,W.& Greeno,J.G.(1985)'Understanding and solving word arithmetic problems'.Psychological Review,92,109-129.
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[1]根据Strnberg(2000,pp.15-17)的观点,模型有内部和外部效度,分别对应有内部和外部两种检验形式。内部检验指判断一个理论在多大程度上解释了该理论涉及的任务领域中的数据,外部检验的一种方式是考察它能否说明理论所涉及的任务领域之外的数据。对于关系-表征复杂性模型的内部检验主要是验证它能否解释或预测问题难度的序列;外部检验比如不同学业水平的学生所能达到的表征复杂性是否被区分开。
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[1]辛自强.关系表征复杂性模型[J]心理发展与教育,2007(3):122-125.
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[1]沿用研究中表示问题在表征关系复杂性和知识基础两方面指标的表示方式,其中横坐标表示问题所涉及的知识基础的个数,纵坐标表示问题所需的表征关系最高层级数。
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(36)Mayer, R. E. (1982a). 'Memory for algebra story problems'. Journal for Educational Psychology, 74.
(37)Mayer, R. E. (1982b). 'Different problem solving strategies for algebra word and equation problems'. Journal for Experimental Psychology: Learning, memory and cognition, 8.
(38)Mayer, R. E. (1986). 'Mathematics'. In R. F. Dillon & R. J. Sternberg (Eds.), Cognition and Instruction. Orlando: Academic Press.
(39)McClelland, J.L. & Rumelhart, D.E., Eds. (1986). Parallel Distributed Processing: Explorations in the Microstructure of Cognition Vol. 2: Psychological and Biological Models. MIT Press.
(40)Reed S K. A Structure-mapping Model for Word Problems [J]. Journal of Experimental Psychology: Learning, Memory, Cognition, 1987,113(1).
(41)Rumelhart, D. E. and Ortony, A. (1977). 'The representation of knowledge in memory'. In R. C. Anderson, R. J. Spiro & W. E. Montage (Eds.), Schooling and the Acquisition of knowledge. Hillsdale, New Jersey; Lawrence Erlbaum Associates.
(42)Rumelhart, D. E. (1980). 'Schemata: the building blocks of cognition'. In R. J. Spiro, B. C. Bruce & W. F. Brewer (Eds.), Theoretical issues in Reading Comprehension. Hillsdale, New Jersey; Lawrence Erlbaum Associates.
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(44)Thorndyke, P. W. (1984). 'Application of schema theory in cognitive research'. In J. R. Anderson & S. M. Kosslyn (Eds.), Tutorials in Learning and Memory. San Francisco: Freeman.
[2]皮亚杰.王宪钿等译.发生认识论原理[M].北京:商务印书馆,1995.
[3]Rumelhart,D.E.(1980).'Schernata:the building blocks of cognition'.In R.J.Spiro,B.C.Bruce & W.F.Brewer(Eds.),Theoretical issues in Reading Comprehension.Hillsdale,New Jersey;Lawrence Erlbaum Associates.
[4]Rumelhart,D.E.and Ortony,A.(1977).'The representation of knowledge in memory'.In R.C.Anderson,R.J.Spiro & W.E.Montage(Eds.),Schooling and the Acquisition of knowledge.Hillsdale,New Jersey;Lawrence Erlbaum Associates.
[1]Thorndyke,P.W.(1984).'Application of schema theory in cognitive research'.In J.R.Anderson & S.M.Kosslyn(Eds.),Tutorials in Learning and Memory.San Francisco:Freeman.
[2]MeClelland,J.L.& Rumelhart,D.E.,Eds.(1986).Parallel Distributed Processing:Explorations in the Microstrueture of Cognition Vol.2:Psychological and Biological Models.MIT Press
[3]Alan,P.(1996)."Schema theory and the use of hypertext-style computer programs'.British Journal of Educational Technology,27.
[1]徐敏毅.儿童解决算术应用题认知加工过程的实验研究[J].心理发展与教育,1994,(2):33-39.
徐敏毅.4-8岁儿童解决算术应用题认知加工过程的实验研究(Ⅱ)[J].心理发展与教育,1995,(4):16-21.
[2]刘广珠.儿童解决算术应用题认知加工过程及比较图式形成的实验研究[J].心理发展与教育,1996.(2):1-5.
[3]辛自强.问题解决中图式的建构:一项应用题分类研究[J].心理发展与教育,2005,(1):69-73.
[1]Low,R.& Over,R.(1992)'Hierarchical Ordering of Schematic Knowledge Relating to Area-of-rectangle problems'[J].Journal of Educational Psychology,1992,84(1):62-69.
[2]辛自强.关系-表征复杂性模型的检验[J].心理学报,2003.35(4):504-513.
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[1]根据Strnberg(2000,pp.15-17)的观点,模型有内部和外部效度,分别对应有内部和外部两种检验形式。内部检验指判断一个理论在多大程度上解释了该理论涉及的任务领域中的数据,外部检验的一种方式是考察它能否说明理论所涉及的任务领域之外的数据。对于关系-表征复杂性模型的内部检验主要是验证它能否解释或预测问题难度的序列;外部检验比如不同学业水平的学生所能达到的表征复杂性是否被区分开。
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