The purpose of this study is to investigate if top-ranked high school students do integrated understanding about the concept of a differential coefficient. For here, the meaning of integrated understanding about the concept of a differential coefficient is whether students understand tangent and velocity problems, which are occurrence contexts of a differential coefficient, by connecting with the concept of a differential coefficient and organically understand the concept, algebraic and geometrical expression of a differential coefficient and applied situations about a differential coefficient.
For this, 38 top-ranked high school students, who are attending S high school, located in Cheongju, were selected as subjects of this analysis. The test was developed with high-school mathⅡ textbooks and various other books and revised and supplemented by practising teachers and experts. It is composed of 11 questions. Question 1 and 2-(1) are about the connection between the concept of a differential coefficient and algebraic and geometrical expression, question 2-(2) and 4 are about the connection between occurrence context of the concept and the concept itself, question 3 and 10 are about the connection between the expression with algebra and geometry. Question 5 to 9 are about applied situations. Question 6 is about the connection between the concept and application of a differential coefficient, question 8 is about the connection between application of a differential coefficient and expression with algebra, question 5 and 7 are about the connection between application of a differential coefficient, used besides math, and expression with geometry and question 9 is about the connection between application of a differential coefficient, used within math, and expression with geometry.
The research shows the high rate of students, who organizationally understand the concept of a differential coefficient and algebraic and geometrical expression. However, for other connections, the rates of students are nearly half of it or lower than half. Even though, top-ranked high school students were investigated, it is found that not many students do integrated understanding about the concept of a differential coefficient. The result means that classes are failing to establish a connection between algebraic and geometrical expression and only focus on calculating a differential coefficient without trying to understand the meaning, when dealing with applied situations.
To improve students'' understanding on the concept of a differential coefficient, realistic problems, including problems about tangent and velocity, which are historical occurrence context, should be suggested. Furthermore, the activities, through which students can understand importance of the concept and deduce the meaning and apply those to algebraic and geometrical expression to understand a differential coefficient, are needed.
The study has a limit in that it only analyzed the understanding of students based on the integrated understanding. Thus, further studies should invent realizable teaching methods, helping students do the integrated understanding on the concept of a differential coefficient, and analyze changes in understanding about the concept through experiments.