This study is designed to determine errors in student\'s learning quadratic function, to classify the task of quadratic-function graphs into interpretation, prediction, scaling and translation tasks, to confirm errors in middle school seniors learning the task of quadratic-function graphs and to find out if there is a change in error correction after learning based on explanation-based study and task-represeantation study and if there is a change made through task expression study based on the movement of expressions, diagrams and graphs. The survey reveals that the effects of error correction in explanatory class and task-expression study differed in accordance with the categories of errors. The researcher discovered that some of the subjects\' errors changed in the desirable direction of understanding the different represeantation ways of the same functions in the situation of task expression study. This study was carried out to inquire into the categories of errors by focusing on the errors of represeantation. And the researcher also made a suggestion of changes in the teaching environment of task represeantation study as a vehicle for correcting mathematics errors. But this seems to be limited as a means of correcting mathematical errors related to quadratic graphs. Therefore it is necessary to grope for specific teaching methods with a view to treating mathematical errors. And the researcher tried to effect changes in teaching and learning environment with the help of interviews as to already formed mathematical errors, or feedback-utilized task-represeantation study.