The purpose of this study is to develop methods to facilitate studentʼs understanding of trigonometric function. To accomplish this purpose, this study develops a framework for characterizing a high school studentʼ s understanding of trigonometric function with covariational reasoning. Covariational reasoning is defined as the cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other. The framework was expressed in five levels of mental action to clarify studentʼs understanding and process of covariational reasoning: Coordination （MA-Ⅰ）, Direction （MA-Ⅱ）, Quantitative coordination （MA-Ⅲ）, Chunky continuous covariation （MA-Ⅳ）, Smooth continuous covariation（MA-Ⅴ）. The transition from MA-Ⅰto MA-Ⅴin the framework represented a deeper understanding of trigonometric function. The theoretical hypothesis was derived from how we can facilitate studentʼs understanding of trigonometric function, and this hypothesis was verified through a teaching experiment by a pairo f high school students. The conclusions of this study reveal that the following methods have certain effects to facilitate studentʼ s understanding of trigonometric function：⑴ Make the opportunity to perform covariational reasoning, ⑵ Facilitate of considering radians as quantities, ⑶ Facilitate of treating events as dynamic ones by manipulating objects and dynamic geometry software.

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The purpose of this study is to clarify the process of understanding the algebraic structure of learners and to characterize the transformation of learnerʼs understanding of the algebraic structure in school mathematics. One of the main claim in this study is to construct the theoretical framework for the empirical research. The method of this study is theoretical research based on interpretation of literature. First, the components of algebraic structure by Bourbaki (1951/1968a, 1964/1968b) are positioned as a prerequisite for constructing the theoretical framework. Then, based on the theoretical framework shown by Sasa (2003), a new theoretical framework is constructed by using Buxton (1978) and Sfard (1991). Finally, we construct “the theoretical framework for the process of understanding the algebraic structure in the set” , “the theoretical framework for the process of understanding the algebraic structure in the extension of sets”, and “the theoretical framework for the process of understanding the algebraic structure in the extension of number sets”. Furthermore, based on the new framework, we discuss the process of understanding the multiplication of negative numbers, the process of understanding irrational numbers and negative numbers, and the process of understanding an extension from the set of natural numbers to the set of integers. In the process of understanding the multiplication of negative numbers, we characterize the observational, the insightful, and the formal level in the process of transforming from low-level understanding to high-level understanding. In the process of understanding irrational numbers and negative numbers, we characterize the interiorization, the condensation, and the reification in the process of transforming from low-level understanding to high-level understanding. And, in the process of understanding an extension from the set of natural numbers to the set of integers, it is clarified that the condition that the set is closure under an operation affects the extension of the set of numbers. The result of this study is to construct the theoretical framework for describing the process of understanding the algebraic structure of learners and to characterize the process of understanding of the algebraic structure for each case mentioned above. The significance of this study is to construct a methodological foundation for empirical research and to analyze the understanding of the structure in other fields by using the theoretical framework.

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