Cognitive analysis of students' errors and misconceptions in variables, equations, and functions

The fundamental goal of this study is to explore why so many students have difficulty learning mathematics. To achieve this goal, this study focuses on why so many students keep making the same errors over a long period of time. To explore such issues, three basic algebra concepts - variable, equation, and function ? are used to analyze students? errors, possible buggy algorithms, and the conceptual basis of these errors: misconceptions. Through the research on these three basic concepts, it is expected that a more general principle of understanding and the corresponding learning difficulties can be illustrated by the case studies. Although students? errors varied to a great extent, certain types of errors related to students? misconceptions occurred frequently and repeatedly after one year of additional instruction. Thus, it is possible to identify students? misconceptions through working on students? systematic errors. The causes of students? robust misconceptions were explored by comparing high-achieving and low-achieving students? understanding of these three concepts at the object (structural) or process (operational) levels. In addition, high achieving students were found to prefer using object (structural) thinking to solve problems even if the problems could be solved through both algebra and arithmetic approaches. It was also found that the relationship between students? misconception and object-process thinking explained why some misconceptions were particularly difficult to change. Students? understanding of concepts at either of two stages (process and object) interacted with either of two aspects (correct conception and misconception). When students had understood a concept as a process with misconception, such misconception was particularly hard to change. In addition, other concerns, such as rethinking the misconception of the ?equal sign,? the influence of prior experience on students? learning, misconceptions and recycling curriculum, and developing teachers? initial subject knowledge were also discussed. The findings of this study demonstrated that the fundamental reason of misconception of ?equal sign? was the misunderstanding of either side of equation as a process rather than as an object. Due to the existence of robust misconceptions as stated in this study, the use of recycling curriculum may have negative effect on students? understanding of mathematics.