Use of Computer Algebra Systems (CAS) and written solutions in a CAS allowed Year 12 mathematics subject: teachers’ beliefs and students’ practices

© 2014 Dr. Lynda Maree BallA computer algebra system (CAS) automates symbolic work in mathematics, perturbing traditional pen-and-paper practices. This study investigated teacher beliefs, from three experienced Year 12 mathematics teachers, about relative roles of pen-and-paper and CAS in Year 12 mathematics. It also investigated students’ use of CAS in Year 12 mathematics examinations and features of students’ examination solutions. The context for the study was the first implementation of Year 12 Mathematical Methods (CAS) in Victoria, Australia; the first Year 12 subject in Australia where students were expected to have CAS in high-stakes external examinations. The study students were the only students enrolled in this pilot study. Teachers and students were pioneers in using CAS in Year 12 mathematics. The three teachers and the students needed to come to personal decisions about use of CAS and pen-and-paper for teaching, learning and doing mathematics. Teachers needed to make decisions about how to help students learn important mathematical concepts and skills, as well as prepare students for their examinations. In examinations students could choose to use pen-and-paper or CAS for any problem. In the context of high-stakes examinations, that contributed to a score used for university entrance, students needed to make good decisions about when to use CAS or pen-and-paper techniques to solve part or all of each problem, as well as communicate solutions to access marks in the situation where some (or all) intermediate working may be carried out within CAS. For teachers new to teaching Year 12 mathematics with CAS some tension existed about the balance between pen-and-paper and CAS in mathematics. The study found that teachers believed that pen-and-paper made significant contributions to mathematical understanding and had a role in understanding CAS, an important consideration when working with CAS and when students wrote solutions in examinations. Teachers believed that different students had different choices to make in terms of CAS or pen-and-paper. Teachers wanted students to become good users of CAS and understand that there was a choice to be made about pen-and-paper or CAS, rather than defer to one approach immediately. Choice of CAS or pen-and-paper for solving examination questions was an important issue for teachers and they believed that choice depended on speed, number of marks for a problem and difficulty of pen-and-paper techniques. For the four examination problems analysed in this study the students used CAS well. Their examination solutions contained mostly standard mathematics, with words used to describe CAS use. Some new features of written solutions employed by the CAS students were identified through comparison with a student group studying the parallel subject, Mathematical Methods (a graphics calculator-expected subject). This included greater use of words and inclusion of particular words, such as ‘solve’, which could be directly attributed to CAS use. The study provides direction for future advice for CAS-allowed subjects. Future classrooms should institutionalise practices with CAS to help students develop exemplary practices when working with CAS, in developing appreciation of the need for CAS and pen-and-paper, and for promoting exemplary communication of solutions in a CAS environment