Mathematical thinking in differential equations through a computer algebra system

Abstract

This study is an effort to promote the mathematical thinking of students in differential equations through a computer algebra system. Mathematical thinking enhances the complexity of the mathematical ideas as an important goal of mathematics education which has not been widely achieved yet in mathematics instruction. This study was conducted in two parts comprising the teaching experiment in the main study and task based interviews in the follow up study. The experiment was conducted with an undergraduate class of differential equations with thirty-seven chemical engineering students in a public university in Malaysia. Maxima, an open source software, was the computer algebra system chosen to be used as a cognitive tool in the learning activities. The instruments included the worksheets designed by the researcher based on instrumental genesis, Three Worlds of Mathematics, and prompts and questions. Seventeen observation sessions and twelve semi-structured task based in-depth interviews with six students were conducted in the main study. In addition, eighteen interviews were carried out in the follow up study with the same six students. Qualitative analysis was used to classify the type of mathematical thinking powers as well as the mathematical structures. The findings showed that mathematical thinking powers to make sense of mathematical structures were interwoven and students used them in a non-sequential manner. The students applied specializing powers, imagining and expressing, changing, varying, comparing, sorting, and organizing, and checking the calculation in general to make sense of mathematical structures such as facts, techniques, and representations. In addition, the relationships among the main contributing factors that support this innovative approach were determined which include the type of tasks, the role of the teacher, class discourse, and the capabilities of technology. The approach can be incorporated not only in the mathematics curriculum at the tertiary level but could also be extended to schools.