A Study on High School Students' Use of Theorems and Symbols in Problem Solving in the Integration by Substitution Section of the Textbook

The structure and narrative style of textbooks are factors that directly affect students' mathematics learning and teachers' lesson plans. This study investigated how high school students use theorems in problem-solving and how they understand and use symbols in the integration by substitution section of textbooks to find out the relationship between the structure and narrative style of textbooks and students' mathematical problem-solving and use of symbols. First we examined students' ways of thinking through open-ended questionnaire responses, and then conducted interviews with some students. As a result of the study, first, students had various views on the symbol of ∫f(x)dx and the role of dx in it. Second, in solving problems in the integration by substitution section of the textbook, students did not use the <theorem> presented just before the problem. They responded that they did not use <theorem>, and they reported that it would be confusing if <theorem> was used. Third in the integration by substitution, students sometimes used the symbol  dt/dx, as if dt and dx were separate quantities, similar to a division, and even though the students themselves used such calculation methods, they questioned the validity of the process. The results of this study will help teachers design lessons, reconstruct the curriculum, and write textbooks by showing students' perceptions of the structure and use of symbols in the integration by substitution section of the textbook.