A Study Of Algebra Instruction With Variation Based On Multi-Representations In Middle School

With the development of mathematics education in China, students have achieved good exam scores in mathematics that the world has admired, whereas they do not perform well in the practice and innovation of mathematics, which affects their quality-oriented development and even the future development of the country. Therefore, mathematics education is no easy task, and it is imperative to reform in mathematics education and such refonn should comply with the developing law of mathematics education:dialectical negation. The reform of teaching method in mathematics is the key to the success, thus, as a start in this study, the author inherits former successful experience and absorbs the advanced western theories and methods in mathematics education and teaching.Variant teaching features traditional mathematics education and guides daily mathematics teaching in China, however, Multi-representations, as the focus of western study in the psychology of mathematics education, is the training goal in "the Princi les and Standard of Mathematics Education in American School". What's more, Multi-representations is the content and also the tool of understanding mathematics. Currently, studying representations becomes the hot topic in cognitive science, education and other areas. In addition, Representations, Multi-Representations in particular, is the topic of International Group for the Psychology of Mathematics Education (PME), and a workshop is set up for studying Representation in mathematics in 1989. The theme of the study has gradually turned from the effect of diverse external representations in experimental conditions on learning into learning from and with multi-representations in daily teaching situations. It is worth exploring and studying "how is the learning multi-representations transplanted in traditional teaching theory and practice", since it has a good prospect. Variant teaching can promote learning multi-representations, and makes it easy to determine "potential distance" and "variable space" through the students'feedback on multi-representations, furthermore, teachers'variant and students'representation give vigor to classroom teaching in mathematics. Therefore, the theoretical foundation of this study is the proposition "Variant Teaching in Mathematics based on Multi-representations", since it is internally related between learning for multi-representations and teaching for variation.The idea of "Everyone learns algebra" is put forward in American mathematics education in order to train people's algebraic thinking, similarly, China should also attach importance to it. As a result, the objective of this study is how to train students' algebraic thinking efficiently in middle school and to develop their innovative awareness in new era, since junior year is the golden time of students' development.Considering the reasons mentioned above, the theme of this study is:"the Study of Algebra Instruction with Variation based on representations in middle school". It makes a tentative study of both localized and characterized development of mathematics education in China.Based on the idea of learning for multi-representations in mathematics, by the use of teaching variants in mathematics in China, this study is to build a theoretical model and to develop a practical strategy in teaching mathematics in middle school, with the algebra in middle school as the carrier, students'algebraic thinking and innovative awareness as the objective. The specific steps are as follows.First of all, there is the literature review of the multi-representations, variant teaching and some conclusions. The first part of literature review is based on understanding general meaning of representations, the author defines the connotation of multi-representations, analyzes the characteristics of multi-representations and its impact on cognitive development in mathematics, such as mathematics comprehension, problem-solving, meta-cognition and innovative way of thinking, and then explores teaching characteristics of multi-representations and its advantages and disadvantages. In order to deeply understand the connotation of multi-representations, the author elaborates the pluralism in learning multi-representations, and then gets the direction of development in learning and teaching multi-representations. The second part of the literature review is based on variant teaching. Variant teaching is put forward from new perspective after summarizing the research of variant teaching in China:variant teaching, a value-neutral teaching method, can be divided into'concept variant in process' and'question variant in process', which facilitates our explanation and application of variant teaching in practice. In order to grasp the essence and the developmental direction of variant teaching, the author conducts the philosophical essentiality analysis of variant teaching. On this basis, the author lays the theoretical foundation of this research:the integration of pluralism with essentiality. In other words, from this teaching-generated theory, a reflection of traditional Chinese philosophy in mathematical teaching environment, the author finds a way to realize mathematics teaching both in formalism and non-formalism. Under the philosophical guidance, the author finds the internal relationship between multi-representations and variant teaching, and then puts forward the theory:"multi-representation is the existing form of mathematical knowledge, and the variant is a variant of multi-representations." All these studies become the macroscopically theoretical system of "variant teaching based on multi-representations".Then the theoretical study of "variant teaching of algebra based on representations in middle school", or the application study of "variant teaching based on multi-representations" is conducted. The author first studies the characteristics of learning algebra in middle school:from its historical development, its form and structure to its feature of study, thus gets the conclusion that algebra is rich in language and structural in knowledge. Next, on the basis of analyzing the nature of learning algebra, the author defines the general features of algebra instruction: emphasizing "the four bases", strengthening multi-representations and structural model, guiding students to participate in activities, and so on. The focus of the study is the role of multi-representations in learning algebra when "process operation" and "object structure" in learning algebra are switching. In other words, in what way multi-representations develops students'algebraic thinking and innovation. Based on the analysis of learning and teaching multi-representations in mathematics and algebraic thinking, the author establishes the theory and the application plan of "Algebra instruction with Variation based on representations in middle school which is the theoretical foundation of "Its Teaching Model and Strategy Study"In addition, the author constructs "the teaching model for algebra with variation based on multi-representations in middle school", and "the teaching models for the concepts, skills and problem solving in algebra with variation based on multi-representations", and illustrates this systematical teaching process. Then the author analyzes whether this teaching model is rational from the perspective of systems theory, learning theory and epistemology. Then, the teaching objectives, process and method of this model are analyzed. Therefore, this part is an important supporting theory of "Algebra Instruction with Variation based on Representations in middle school".Based on this model, the author develops "the teaching strategy of algebra with variation based on multi-representations in middle school", and "the teaching strategy of concept, skill concept, and problem-solving in algebra with variation based on multi-representations", by using teaching theories of multi-representations, variant, and constructivism, as well as phenomenology theory and students'way of thinking in middle school. The teaching design and specific operations are further discussed, which can be used as a reference for teachers in their classroom teaching. This part is the application of the theory in a micro perspective.Then the author conducts a case study of "equation instruction with variation based on multi-representations for the first-year students in middle school", in order to testify whether the teaching model for algebra in middle school is rational and the teaching strategy of algebra in middle school based on multi-representations is valid. With the four continuous learning modules of "the introduction, understanding, problem-solving and application of equation" as the carrier, the author analyzes how the teaching facilitates students'multi-representations learning, and develops students'deep understanding and application of mathematical knowledge, so as to get the conclusion with practical characteristics, by adopting theory of mathematical cognitive psychology. It is an effective attempt in integrating teachers'variants with students'representations, and developing students'algebraic thinking and innovation.Finally, in the conclusion and suggestion part, the author analyzes whether this study is rational, feasible and scientific from the processes of topic, design, implementation and verification throughout the study, and then illustrates the depth and breadth of this study, so that the author puts forward constructive suggestions for algebra instruction, and guiding study advice for "the study of geometry, probability and statistics instruction with variation based on multi-representations in middle school".Based on the integration of multi-representations with variant teaching, the author theoretically discusses algebra instruction in middle school with developing students' algebraic thinking and innovative awareness as the objective, and innovates the theoretical system of "variant teaching based on multi-representations" so as to enrich teaching theory in algebra. Moreover, the author explores the specific operation of the theory in mathematical teaching in middle school in practice. On the whole, at the request of quality-oriented education and popular education, with mathematical teaching in middle school as the carrier, the author carries out a beneficial attempt of mathematical education and teaching from localization to internalization.This study is supported by the key project of "Research into teaching with variation based on multi-representations about junior algebra--a reform experiment on'teaching from the perspective of learning'(GOA107019)". The result of the case study as a monograph is published in "Advances in Mathematics Education" by well-known Springer Press.Of course, due to the limited time and energy, as well as the author's inexperience, the deficiency in this study is in existence. Any suggestions would be appreciated.