Abstract The teaching and learning of translation and scaling related to graphs of functions or equations often emphasize the algebraic aspect the representation. The teaching experiments described in the paper used the manipulable, dynamic, linked, multiple representation computer environments to help students understand the relationship of the movement or variation of the equation graph and its algebraic representation. In these environments, students could manipulate (translating or scaling) the graphic objects and the coordinate system. Contrasting three kinds of translation and scaling was emphasized-transforming the graph only, transforming the coordinate system only, and trans-forming both the graph and the coordinate system simultaneously. Three students with different levels of mathematical learning experience participated in this study. They were in algebra I , algebra. II and algebra III courses respectively. Students were found to think in three planes-representational, physical, and operational planes. Although all of the students often used the function provided in the environments to manipulate the graphic objects (thinking in the physical plane), there were differences in thinking operationally. The degree of operational thinking seems to correspond in reverse to their mathematical experiences. The student with the most mathematical experience appeared to think less creatively and less operationally than the other two students. The student's algebraic sophistication may have inhibited her operational thinking, while the explicit discussion of the inverse action of translation seems to have been helpful in developing this operational thinking. Two students were found to use Piaget's INRC group relationship. In the realm of translation and scaling, these relationships helped them solve problems and organize their knowledge. But the basic scheme all the students used was the backward-substitution scheme. The object movement or variation approach seemed aslo to provoke some visual and intuitive errors. The reversal errors were found in both translation and scaling of graph and axes. Students often confused the point-to-point transformation with the function-to-function transformation when they relied purely on their intuition. Students were also found to rely on external stimuli to provide cognitive conflicts and did not appear to monitor their thinking although they were all viewed as "good" in mathematics. Helping students to develop an automatic feedback system to monitor the thinking process and hence confirm the results automatically by themselves seems to be a very important aim to mathematics education.
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