| It is more frequently for the pedagogy of variation to be used in mathematics teaching than in other subjects. So far, the theoretical and experimental studies of the pedagogy of variation , however, are still scanty . Gu Lingyuan thinks that the variation consists of conceptual variation and procedural variation. And the principle of variation for learning and the idea, "teaching as the construction of the space of variation " put forward by some abroad scholars, such as, F. Marton, have enriched the theories on variation, and opened a new perspective for the research of the pedagogy of variation.The author , as both a researcher and a participator, has done the cooperative action research with correlative mathematics teachers for one year or so. The date in this study is obtained in the following ways: a questionnaire for 102 mathematics teachers from more than ten junior high schools in Shanghai; a test for 399 students in Grade 8 in junior high schools in Shanghai; and the typical lessons study, based on the method of instrumental case study.Part 1. The research results on teachers' understanding and action to pedagogy of variation show:Few teachers simultaneously regard variation as a teaching means, a kind of idea of teaching or a model of teaching. To the most of teachers, the practice with variation is the main form of pedagogy of variation. So teachers pay more attention to the variation of the ways of problem solving, and pursuit the diversity of the ways.Most of the teachers think that it is the need for students' cognition to use variation, and also for problem solving. They are more regardful of the "attainments" of students but not for impartment, while they use variation .Most of the teachers think that repeated practice is helpful to memory, as well as understanding. Usually , teachers have a design for variation using in the class.According to the result of the comparative study between the teachers with no more than 7 teaching years and the ones with more than 7 teaching years, this study suggests that the novice teachers should pay attention to the role of pedagogy of variation, use variation with design, take the variation of methods of teaching and learning into account and probe the suitable dimensions of variation in the process of teaching.Part 2. The teaching forms and the meanings of the pedagogy of variation are probed in terms of five cases and one test.1. On the variation of basic figures. The basic figure, as a starting point, decidesthe orientation of teaching. The procedural variation based on movement and construction .reveals the formative process of the knowledge and the relationship among the knowledge.The students get the variety figures by themselves, then can really understand the essential consistency in the variety figures. Meanwhile, variation must not be used on the surface.2. On the variation of context. Contexts should be designed by different levels. In this study, context is divided into three levels: para-reality context, para-mathematical context and mathematical context. The context in different levels guides students' different activities by helping students recognize the intention of the context. As for the hierarchy of context, it is very fundamental to control the appropriate potential gap between the different contexts. Meanwhile, the sequence and diversity of context should be considered.3. On the variation of examples. A space of learning is constructed with "one dimension and three values ". Because of the only one dimension of variation, three values' relation is just linear. The procedure of implement of the lesson corresponds to the transform among the three values. So it is easier for the knowledge to analogy and transfer.The examples chosen by teachers should cover all special values in every dimension. Meanwhile, the generality and individuality of examples should be dealt with reasonably.4. On variation of mathematical activities. There are three dimensions in this space of mathematical activities. They are respectively,...
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