Mathematical Problem Schema Level,

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.