Research On Mathematical Fundamental Activity Experience

“Mathematical fundamental activity experience” as part of the course objectives of theexperimental version of the Chinese compulsory education mathematics curriculum standardswas first proposed in2001. In2011, it was proposed as one of “the four basic requirements ofthe course objectives” in the current version of the compulsory education mathematicscurriculum standards. Research was undertaken to investigate the nature of “mathematicalfundamental activity experience” and why it should be included as a key component of thecourse objectives. These are important questions in the context of curricular reform in Chinaand investigation of the classroom implementation of “mathematical fundamental activityexperience” provides significant insight into the reform process and the challengesconfronting mathematics teachers in China and elsewhere.The research reported in this paper defined mathematical fundamental activity experienceand constructed a theoretical framework, including its dimensions and associated hierarchicalstructure, with which to investigate its implementation in the classroom. On the practical side,the paper identifies the conditions governing junior middle school students’ experience of thefundamental mathematical activity through questionnaires and the investigation of the dailyclass teaching of “linear function” for eighth-grade students. The research methods includedboth qualitative and quantitative approaches, such as hierarchical analysis of the relativelyobjective scores of the test questions, and cluster analysis of measures of the level of students’fundamental mathematical activity experience.The main conclusions are the following:First, the fundamental mathematical activity experience is based on the cultivation ofinnovative talents, which are the requirement of the times and of developments in thediscipline of mathematics. Mathematical activity experience is different from mathematicalknowledge and also different from mathematical ability. Knowledge can be told; ability canbe divided into parts, while mathematical activity experience must be a combination ofpersonal practice and comprehension, and takes a long time to develop.Second, mathematical fundamental activity experience provides the basis for theformation of the individual’s mode of mathematical thinking, which should be theconsequence of personal experience of the process of mathematical inductive and deductivereasoning over an extended period of time. The outcome of this process is the formation ofmathematical intuition. This is visible in the mathematical performances of primary andsecondary school students as the mathematical thinking mode of beginning with the simplestquestions and exploring the laws of mathematics (and its nature) step by step. Among these,“exploring the law” is the thinking process of analysing special cases, and of discovering common and idiosyncratic characteristics of mathematical objects and procedures. The threehierarchical divisions of the process of “exploring the law” are imitation, nature, and essence.Third, the dimensions of mathematical fundamental activity experience are observationand association, inductive guess, expression, and verification or proof. As with “exploring thelaw,” the hierarchical levels of students’ fundamental mathematical activity experience areimitation, nature, and essence.Fourth, the results from researcher-developed student questionnaires indicate that theoverall situation of students’fundamental mathematical activity experience is not satisfactory;the results indicate a normal distribution in the inductive guess dimension, but skeweddistribution in the other dimensions. The results of the cluster analysis indicate that manystudents are at the “imitation level” and “nature level”, a few students are the “nature level”,while very few students can reach the level of essence.Finally, through classroom observation, we discover that teachers are not able to helpstudents to consciously develop the mathematical thinking mode of “starting with the simplestsituation and exploring the law step by step.” Students have their own original intuitions, andeven if the students can grasp the essence of the questions, it is not at all certain that this willlead to the development of a rich imagination–a key goal of the new curriculum.In brief, the main idea of the paper is to enable students to develop mathematical thinking.We should help the students to accumulate mathematical fundamental activity experiences of“begin with the observation, reveal by special case, and come to guess by inductivereasoning”, on the basis of which students should be capable of forming rich mathematicalconnections, leading to sophisticated mathematical understanding, intuition and imagination.