# Students’ Comparison of Their Trigonometric Answers with the Answers of a Computer Algebra System in Terms of Equivalence and Correctness

## Abstract

The answer offered by computer algebra systems (CAS) can sometimes differ from those expected by the students or teachers. Such answers could serve as a catalyst for rich math-ematical discussion (see (Pierce and Stacey, 2010)). An important topic is the equiva-lence/non-equivalence of the students’ answers with the CAS answers. It is also relevant in case of equations and could be particularly helpful when working with trigonometric equa-tions, which can often have quite sophisticated answers. Are students able to ascertain equivalence/non-equivalence? How do they understand correctness of the answers? Are there any differences in this regard between different types of equations and answers?The paper is based on the lessons of a course in elementary mathematics to first-year stu-dents who had very limited experience with CAS. The students worked in pairs and their dis-cussions were audio-recorded. They had worksheets with trigonometric equations and ques-tions. The equations to be solved were in a prescribed order. For each equation there was a particular CAS that gave a different answer from the expected answer of the students.

Initially, the students solved an equation (correctly or not) without a CAS. Then they solved the same equation with a particular CAS. They were given questions, guiding them to ana-lyze the differences, equivalence and correctness of their own answers and CAS answers.

Ca 200 instances of equation-solving by ca 50 pairs of students were analyzed. The data consists of the students’ worksheets and audio records of their discussions. The worksheets provide information on equivalence/non-equivalence of the students’ and CAS answers. It is determinable through mathematical reasoning by researchers. The second dimension is the students’ opinion about the equivalence/non-equivalence. That is based on an analysis of the worksheets and audio data. Different types of equations can be associated with different ob-stacles to clarification of equivalence/nonequivalence. For example, there seems to be quite a high degree of confusion about the meaning of n in the answers to trigonometric equations, even if it is used correctly in solutions.

It seems that comparing their answers with CAS answers was an interesting task for the stu-dents. In most cases this led to active discussions on trigonometry. The productivity of the discussions is a subject matter for another study but some points are also briefly mentioned in this paper.