두 대응관계를 연결한 과제에 대하여 역함수 존재 여부에 대한 학생의 질문에 관한 소고

A study on the student's question about the existence of the inverse function for the task that connects the two correspondence relations

Abstract

본 연구에서는 특정한 학생의 질문에서 시작된 고민을 다루고 있으며, 교과서 분석을 통하여 해당 질문이 충분히 제기 가능한 질문임을 확인하였다. 또한 면담을 거쳐 두 대응관계가 연결된 새로운 대응관계에서 학생들이 합성함수 여부 판정과 역함수 존재 여부 판정에서 정의역에 대한 고민이 학생들에게서 어떻게 의미 있는 수학 지식으로 재구성 되어가는 지에 대한 사례를 관찰하였다. 본 연구에서 제시된 사례들은 제한된 상황에서의 특정한 사례라는 한계가 있기 때문에 곧바로 교수학습 상황에 적용될 수는 없으나 다른 연구자들에게 함수 학습 관련 연구에 대하여 통찰의 기회를 제공할 수 있을 것으로 기대된다.

This study deals with the anxieties that originated from specific student questions. Through the analysis of the textbooks, we confirmed that the question was a sufficiently plausible question. Third interviews were also held with three high school students. Through the interviews, we analyzed students' expressions about the new correspondence relationship that the two correspondence relations are linked. In the determination of the composite function and the determination of the inverse function existence, We have observed a case of how the worries about domain are being reconstructed from students into meaningful mathematical knowledge. Through this, we confirmed that the question will be confusing to students in the field. In this study, we observed the transfer of domain in relation to student domain in composite function. In particular, a present study revealed that the students involved in the interview were influenced by this domain transfer phenomenon in determining whether the task given in the interview was a function. This was the same in determining the existence of a inverse function. The examples presented in this study are limited to specific cases in limited circumstances. Therefore, it can not be applied directly to teaching and learning situations. However, it is expected that this study will provide other researchers with insight into function learning related research.

[Fig. 3] Figure about spot of interview

[Fig. 4] Tasks of interview

[Fig. 5] Mathematics II in the 2009 revised mathematics curriculum, table of contents of function in textbooks

[Fig. 7] Mathematics II in the 2009 revised mathematics curriculum, composite function configuration method

[Fig. 8] When introducing a composite function, the difference between the range of the function and the domain of the function

[Fig. 9] Mathematics II in the 2009 revised mathematics curriculum, inverse function configuration method

[Fig. 10] Procedure to obtain inverse function in textbook

[Fig. 11] Example of a textbook showing a composite function including a function rather than a one-to-one function⁷⁾

[Fig 12] Example of a textbook showing a composite function including a function rather than a one-to-one function

[Fig 13] A picture of the correspondence relationship, student B presents to student A

[Fig. 1] Correspondence relationship that a student’s question about a function

[Fig. 2] Modified Figure ‘Correspondence relationship that a student’s question about a function’

[Fig. 6] Mathematics in the 2015 revised mathematics curriculum, table of contents of function in textbooks

[Table 1] Sequence of interview

[Table 2] An analysis table of the textbook introduction materials

[Table 3] A table showing students' expressions about ‘function’, ‘composite function’, ‘inverse function’