| Development of students’ mathematical thinking ability,especially mathematical higher-order thinking(M-HOT for short),is not only the basic task and core target of mathematical teaching,but also the classical and hot subject of mathematics education research.Under the background of developing students’ key competence in China s basic education,there is the new connotation and significance of the times for exploring the teaching approach for developing students’ M-HOT in this paper.A large number of teaching researches have shown how to develop students’mathematical thinking ability at home and abroad from different perspectives.Such as a lot of teaching experimental researches have been done by Lin Chongde and his team since the 1980s,which took cultivating thinking quality as a breakthrough,and teaching strategies as methods;Gu Lingyuan has explored the Chinese variant teaching in mathematical teaching;the american QUASAR program,which aims at improving American mathematics teaching,emphasizes the teaching practice withhigh-level cognitive mathematical tasks;and any more.Based on these studies,this paper puts forward this study s basic research approach which analyzes the characteristics of teaching tasks,teaching strategies and task design strategies in teaching practice oriented to developing students’ mathematical thinking flexibility for developing students’ M-HOT.In particular,this study has done teaching case studies with choosing chinese daily mathematics lessons in junior high school mathematics,which focus three teaching elements about the characteristics of the mathematical task of developing students’mathematical thinking flexibility,and its teaching strategies and tasks design strategy based on the design,implementation and feedback of the mathematics classroom teaching.This study was done arounding three research questions:(a)What features do mathematical tasks have for developing students’ mathematical thinking flexibility(MTF for short),which bases on the analysis of teaching activities?(b)What teachingstrategies are there for developing students’ MTF,which bases on the features of mathematical tasks?(c)What design strategies of mathematical tasks are there for developing students’ MTF,which bases on the features of mathematical tasks?On the basis of former researches,the characteristic of thinking flexibility is conceptualized into three basic elements:change,transformation and association.Further,this study constructs the system of representation of MTF by the three types of mathematics teaching topics:concept understanding,skill training and problem solving(see 3.2.2).It constructs the mathematical task analysis framework(see 3.2.3)based on the characteristics of mathematics from three aspects:subject properties,the orientation of thinking and teaching strategies.At last,it forms the research plan to carry out lesson-case studies on above two aspects.From the end of 2013 to the autumn of 2016,the researchers carried out a three-year teaching practice study with three teaching teams(a free team and two top-teacher workgroups).But it is a gradual development process for understanding the research framework,research approaches and teaching topics and still is not a mature system.At last,this study has explored the teaching elements to develop students’mathematical thinking flexibility by only selecting three typical mathematics cases(linear function and proportional function,review of factorization,Pythagorean theorem)in junior high schools.It has formed a "toolbox" of teaching elements(see table below).See 8.1,8.2 and 8.3 for details.List of teaching elements oriented by developing students MTF(?)Each of the teaching elements in this "toolbox" can be traced back to the analysis of the teaching tasks of the three lesson-cases in this study.It is only orientation of mathematical tasks,and therefore it means that these elements simply give some"inspiring" elements as a design reference for subjects to develop students’ MTF. |
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