School Students Understanding Of The Function Transform

The function concept is one of the central topics in school mathematics and it appears and reappears like a thread throughout high school mathematics. The transformation of function plays an important role in high school mathematics textbooks. One of the most important problems on transformations in which students always have difficulties is: What is the relationship between the two graphs when the one analytically represented function is transformed into another one?This thesis focuses on high school students and mathematics teachers' conceptions of function translation and dilation. Findings of the questionnaire survey and interviewing reveal that most of the students use the memorized rules to explain the horizontal translation of function ("do the opposite: plus-left and minus-right") but don't understand why these rules work. Similarly, they rely on the memorized rules to do the dilation problems without any desire to understand the underlying reasons. Other explanations provided by the students include considering the function point-wise, operation of the analytic expressions, graph plotting, attending to the axes of symmetry, zero or vertex of the function, finding the period, substitution, etc.Teachers provided a variety of explanations of these two types of transformations, including citing the rules, considering the function point-wise and attending to the zero of the function, etc. However, none of the strategies used by the students or teachers are helpful for the students to understand transformations of function, and they are often misleading. Students' errors in function translation and dilation include confusions of horizontal and vertical shifts, right and left shifts, up and down shifts, stretching and shortening, and the distance of the horizontal or vertical shift.The students' obstacles in their understanding of the function transformations are analyzed and some teaching implications suggestions are given at the end of the paper.