The purpose of this study is to investigate how first and second graders in middle school take in integrated understanding about the concept of function. The integrated understanding on function means that the students can notice the relationship between varying quantities and then, define function and understand the meaning of symbols and the values of function. As a result, they can interpret various tasks based on the concept of function.
The data for this study was collected through the questionnaire conducted by the first and second-year students at A, B middle school in Cheongju. The questionnaire consisted of 14 questions related to the extent of understanding a concept of function, the ability to express function and to translate function. To improve the validity of the questions, the questionnaire has been developed from several reference materials such as the revised textbooks under the 2009 Korean National Curriculum, other related studies. The results are summarized as follows.
First, the percentage of correct answer made a difference according to the types of representation. That is, questions leading students to translate a task into a table or an equation such as 1-1 and 1-2, 2-1 and 2-2 showed quite high correct response rates. However, questions like 1-3, 2-3 asking students to translate a task into graphs showed high incorrect responses. This shows that the ability to translate some tasks into a variety of functional forms is essential to comprehend the concept of function, but it is not possible to grasp function only with the ability. Shortly, it will not mean that students completely understand the essence of function though students are able to represent a task into function. Linking among the various ways to represent function can help students to understand the concept of function.
Second, the result shows that students have the different viewpoints depending on their grades when they have to determine whether the suggested situation belongs to function. The first-year students tended to consider function as the concept of ''definition''(38.2%) and ''regularity''(21.5%). On the other hand, the second-year students emphasized ''regularity''(30%) and ''equation''(28%) of function. This gap enables us to confirm the first-year students perceive the concept of function better than the second-year students. The reason could be that the first ones learn the definition of function through their regular textbooks but the second ones usually deal with a linear function in class.
Finally, only a few students can distinguish the various situations and representations into the definition of function. There were only 12 first-graders(14.6%) who notice the definition of function from the suggested tasks but, there was no second-grader. In detail, the number of the students who determine more than six from eight tasks as to the definition of function was 24 in the first graders(15%), while only 2 second graders(2.6%) was. Given that most of students couldn''t determine the tasks of the questionnaire using the definition of function, this result shows that students didn''t get the integrated understanding of the concept of function.