이차함수 단원에서 학생오류에 대한 교사의 인식 연구

다국어 초록 (Multilingual Abstract)

ABSTRACT

A Study of the Teacher's Recognition about the Students' Errors of Quadric Functions in the Middle School

Bark, Hyang Gi

Department of Mathematics Education
Graduate School of Education, Kongju National University

Supervised by Prof. Ahn, Jea Man.


This study aims to comprehend errors of third grade middle school students about quadric functions and the degree of teacher's perception of the errors of the students in Korea. To achieve this goal, research questions as follows were set up.

1. What are students' errors shown in <9-A> ‘Quadric Functions?’
2. How teachers perceive the students' errors shown in <9-A> ‘Quadric Functions?’

In order to solve these research questions, an examination study and literature research through the test and questionnaires were conducted. 116 third grade middle school students from four classes and 17 teachers were selected. The tests for the students and the questionnaires to the teachers used in the research were prepared directly by the researcher referring to previous researches and textbooks for third grade middle school students and instruction guides for teachers, reviewed by experts in math education and math teachers. Summing up the result of the tests for the students and the questionnaires for the teachers prepared accordingly, the conclusion of this study regarding the research questions were summarized as follows:

First, for questions about coefficient of quadratic term, the ‘error which they responded that the smaller the modulus, the greater the width gets’ was most(28.2%). Although this only cannot be a perfect reason, considering that they answered correctly, it was judged that these were errors that they omitted the word, coefficient of quadratic term or ones caused by carelessness. Also, the students who wrote the answers only without explanation were 21.2% among those who answered wrong. The ‘error that they expressed coefficient of quadratic term as ‘degrees’’ was 20.0%, the third place among the order of errors. This seems to be because they applied over-generalizing the concept used in primary functions to quadric functions. There were differences between the order of errors expected by the teachers and that shown by the actual students. The teachers expected that the error which they responded that ‘the greater the modulus of coefficient of quadratic term, the wider the width gets.’ However, actually this error was at the fifth place in the order of errors shown by the students. The teachers expected that the students might have errors od the concept of coefficient of quadratic term; however, actually they knew the concept better than the teachers thought. In addition, the teachers predicted that the ‘error that they expressed coefficient of quadratic term as ‘degrees’’ would be at the fifth place in the order of errors. However, actually in the order of errors shown by the students, this error took the third place in the order. This shows that much more students than the teachers expect express the width of quadric function graph as ‘degrees.’
Second, in a graph with a form of , the percentage of correct answers was relatively as low as 27.6% to the questions regarding the coordinate of the apex and quadric function formula. The students who made an error in which the apex found in the suggested graph and that applied to the formula were different were 17.3%, and those who made an error that they suggested only the apex or the quadric function formula were the most, 17.3%. This means that the students don't connect the graph to the formula successfully. Next, the second to the most errors, 15.4% was one that they answered quadric function formula as ‘ type or type’ which means that they think the coefficient of in the quadric function is always 1 in their perception structure or the coordinate of the apex determines the coefficient of quadratic term, which is wrong. Also, 13.6% of the students committed an error that they confuse between the coordinate of the apex and the coordinate. There were a great difference between the order of errors expected by the teachers and that shown by the students for these questions. In particular, the teachers predicted that the most students would show an error that they confuse between the coordinate and the coordinate of the apex. However, actually, in the order of errors shown by the students, this error took the fifth place. Also, the teachers expected that the ‘error that they substitute the apex found with type’ would be the second place in the order of errors. However, actually it took the sixth place in the students' order of errors.
Third, for the properties of quadric function, in the questions regarding the relationship between the axis and the apex, the students who answered wrong because not explaining after finding the equation of the axis with the coordinate of the apex were 28.9% among the entire students who answered wrong, which was most. Next, 21.1% of the students committed an ‘error that although they knew the concept of the apex, they couldn't connect that to the equation of the axis.’ Also, the students who made an ‘error that they didn't know the concept of the apex and substitute and develop the coordinate (2, 0)’ were 18.4% while those who made an ‘error of the expression of the equation of the axis’ were 18.4%. Most of the errors the teachers showed were committed due to their not being aware of the concept of the equation of the apex and the axis accurately. There were difference between the order of errors expected by the teachers on these questions and the actual order of errors shown by the students. The teachers predicted that the students would make the most error of the expression of the equation of the axis; however, in the actual order of errors the students showed, this error was at the third place. And, the order of errors of the students who answered wrong as writing the answers without explanation was the first place; however, the teachers expected that this would be the fifth place.
Fourth, to the questions regarding the translation and graph drawing of quadric function, the ‘error that they knew the movement but couldn't draw the graphs’ were 41.3%, which was most. This seems to be caused by their acquiring that as procedural knowledge through methodological understanding of the concept of translation. In other words, the students who committed this error didn't know the meaning of the translation of the graph. Next, 39.1% of those who made an error committed an ‘error that they neither knew the movement nor drew the graph.’ The order of errors expected by the teachers on these questions and that shown by the students actually were consistent exactly.
Fifth, to the questions regarding the absolute maximum and minimum values of the quadric function graphs, the percentage of correct answers was 13.8%, which was the lowest percentage of correct answers among all the questions. Among the wrong answers, the ‘error that they answered only absolute maximum’ was the most (39.5%), most while 15.8% of the students committed an ‘error that although they knew the absolute maximum, answered minimum value wrongly.’ It seems that many students knew the concept of the absolute maximum of the graph bulging up, but didn't know the concept of minimum value. Also, the students who committed an error of perfect square expression in the course of changing the quadric function formula from general type to standard type, were 18.4%, through which it can be found that the students have great difficulties in the course of changing the general type to the standard type of the quadric functions. There were a slight difference in the order of errors between the expectation of the teachers and the actual order of errors of the students regarding this question. The teachers predicted that ‘perfect square expression error’ would be most, but in the actual order of errors of the students, this error was at the second place. Also, the teachers expected that the students would commit the ‘error that they answered absolute maximum only’ as the second to the most error; however, this was in fact the most error of the students. In other words, the teachers predicted that the error of the perfect square expression committed in the course of the calculation correcting the general type to the standard type of the form of the quadric function formula to find the absolute maximum; however, actually the students showed the most error committed because they didn't know the concept of the minimum value.
Sixth, for the questions regarding the ranges of the increase and decrease of the quadric function graphs, the ‘error that they responded that they were ’ was most(30.7%). This seems to be an error coming from the lack of their basic knowledge of the increase and decrease. Students who committed other errors were 23.1% while those who responded ‘it was quadrant’ 15.4%, who made the error because they didn't know the basic concept of the process of change centering around the axis of the quadric function graph. There was a difference between the order of errors expected by the teachers regarding the questions and that actually shown by the students. The teachers expected that most students would make the ‘error of expression’ however, actually in the order of errors shown by the students, this error was the least-committed one, the third place. In addition, the teachers expected that the students would commit the ‘error that they responded that it was ’ as the second most one; however, in fact, the students committed this error the most. In other words, the teachers expected that the error committed because they didn't express the concepts of the increase and the decrease of the quadric functions though they knew them; however, actually the students made the most errors because they didn't know the basic concept of the increase and decrease of the graphs and one with which they can find the ranges of the increase and the decrease centering around the axis of the quadric function graphs.

To solve the problems drawn out from this study, in the areas where the degree of the teacher's recognition of the errors of the students is low, the teachers should make efforts to grasp the errors of the students first, and in the areas where their level of recognition of the students' errors is high, investigation on the teachers' instruction methods and the development of new instruction methods are necessary. Thus, based on the results of this study, the following suggestions would be desired.
First, this study suggested the errors of the third grade students in quadric function and the level of the teachers' recognition of the errors of the students depending on the tests for the students and the responses to the questionnaires to the teachers only, but a qualitative research on the process of thinking of recognition of the errors and a field survey on how the teachers actually perceive the errors of the students in actual classes and how they instruct their students are necessary.
Second, not only studies on the level of the teachers' recognition of the errors of the students in the Unit, ‘Quadric Functions’ but also on the actual methods of instructions helpful for the improvement of the errors by studying and applying the instruction methods with regard to the curricula are necessary.

목차 (Table of Contents)

• I. 서론 1
• A. 연구의 목적 1
• B. 연구문제 2
• C. 연구의 제한점 3
• II. 이론적 배경 4
• A. ‘이차함수' 단원의 교육과정 4
• B. 이차함수에 관한 선행연구 8
• C. 함수학습의 인식론적 장애 10
• D. 수학교육에서의 오류 14
• III. 연구 방법 및 절차 18
• A. 연구대상 18
• B. 검사도구 18
• C. 연구 방법 및 절차 20
• D. 자료의 처리 및 분석 21
• Ⅳ. 결과 및 논의 23
• A. 중학교 3학년 학생들의 이차함수 단원에서 보이는 오류 23
• B. <9-가> ‘이차함수’ 단원에서 보이는 학생 오류유형에 대한 교사들 의 인식 33
• V. 결론 및 제언 44
• 참고문헌 48
• Abstract 51
• 부 록 56