On December 11th, 2017, Eno Tõnisson defended his PhD thesis on the topic "Differences between Expected Answers and the Answers Offered by Computer Algebra Systems to School Mathematics Equations" ("Oodatavate vastuste ja arvutialgebra süsteemide vastuste erinevused koolimatemaatika võrrandite puhul")
Dots. Rein Prank (Institute of Computer Science, UT)
Prof. PhD. Zsolt Lavicza (Johannes Kepler University, Linz, Austria)
Prof Cand. Sc. Peeter Normak (Tallinn University, Tallinn, Estonia)
It is possible to solve most mathematical problems, including equations of school mathematics, with the help of Computer Algebra Systems (CAS). The answers offered by CAS (CAS answers) often coincide with the answers that are expected in the school context (school answers), but sometimes not. Such unexpected CAS answers are often correct, but based on different standards, for example in complex domain. A systematic review of the differences between CAS answers and school answers is useful for development of CAS and organizing the teaching process. A review of the differences between CAS answers and school answers and their reasons in case of school mathematics equations is provided in this dissertation. The spectrum of differences is explained by using two possible classifications. A key criterion of the first classification is comparing whether the CAS answer includes a larger or a smaller number of solutions than the expected answer. The other classification is more content-oriented, highlighting the issues of the form, completeness, dependence on the number domain, branching and automatic simplification of answers. The differences caused by number domain and branching are discussed separately in greater depth in separate chapters. The differences between school answers and CAS answers can be used in teaching and learning. This dissertation proposes a pedagogical approach that is based on comparative discussions on students' answers and CAS answers in pairs. In addition to teaching and learning, the format is also suitable for collecting data on students' understandings and misunderstandings. The proposed approach was used in lessons on trigonometric equations. The focus was on analyzing whether students can adequately identify the equivalence/non-equivalence and correctness of their answer and CAS answer. It is found that even if a student's solution looks to be correct, students can have misunderstandings and knowledge gaps.