The purpose of this study is to design and apply tasks that satisfy various levels of learning goals in consideration of field suitability. To accomplish this, we developed and applied a rectangular puzzle task reflecting the design guidelines of a good task. The teaching experiment was conducted on 10 students in the second grade of middle school, and their learning process and application results were analyzed. As a result, students showed various levels of learning achievement, improved mathematical ability by instructional treatment, and positive affective characteristics. These findings provide pedagogical implications for the design of tasks for enhancing mathematical competence.

Trigonometric ratios are difficult concepts to teach and learn in middle school. One of the reasons is that the mathematical terms (sine, cosine, tangent) don't convey the idea literally. This paper deals with the understanding of a concept from the learner's standpoint, and searches the orientation of teaching that make students to have full understanding of trigonometric ratios. Such full understanding contains at least five constructs as follows: skill-algorithm, property-proof, use-application, representation-metaphor, history-culture understanding [Usiskin, Z. (2012). What does it mean to understand some mathematics? In: Proceedings of ICME12, COEX, Seoul Korea; July 8-15,2012 (pp. 502-521). Seoul, Korea: ICME-12]. Despite multi-aspects of understanding, especially, the history-culture aspect is not yet a part of the mathematics class on the trigonometric ratios. In this respect this study investigated the effect of history approach on students' understanding when the history approach focused on the mathematical terms is used to teach the concept of trigonometric ratios in Grade 9 mathematics class. As results, the experimental group obtained help in more full understanding on the trigonometric ratios through such teaching than the control group. This implies that the historical derivation of mathematical terms as well as the context of mathematical concepts should be dealt in the math class for the more full understanding of some mathematical concepts.

The purpose of this study was to explore the possibility of interconnection between non-fractional representation and non-circular decimal representation in irrational teaching and learning of school mathematics and to examine the possibility of field application. First, through the analysis of previous studies, it was found that the core of understanding irrational concepts is the intuitive perception of convergence and the inference of acyclicity. And through mathematical analysis, we confirmed that finite decimal sequences and approximate fractional sequences should be learning contents. In this regard, we applied engineering tools in teaching and learning activity of irrational numbers and analyzed the process by which engineering tools become learners’ mathematical tools and then learning occurs, that is, instrument genesis. As a result, in the activity of finding finite fractions of square roots, students gained a meaningful understanding by focusing on the meaning of square root symbols and square roots as objects from inputting functions and operations and using drag. And in the activity to solve the task ‘Can a graph cross a grid point?’, Students could infer the acyclicity of the decimal representation of irrational numbers by observing the approximate fraction as the slope. Thus, this study is expected to be a basic reference of teaching and learning of irrational numbers to construct a mental image of real numbers completeness axiom.

This study analyzes the historical development process of continuum and discusses the pedagogical implications of continuity of real numbers based on this. As a result of historical analysis, the following was confirmed. First, the process of constructing real numbers was a very difficult process that took a long time surrounding understanding and justification of infinity. Second, reductio ad absurdum was a premise condition to revealing that completeness is equivalent to the various ways of constructing real numbers. Teaching implications for this are as follows. First, the method of constructing real numbers from rational numbers can be divided into various levels from intuitive understanding to mathematical justification, and pedagogical treatment appropriate to students' understanding level should be provided. Second, it is not enough to distinguish the properties of rational numbers from the continuity of real numbers in the number line model, so the alternatives need to be studied. These findings provide significant implications for teaching and learning about the continuity of real numbers.

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