Using critiques as a strategy for improving the proofs of the equivalence of ring theoretic concepts

The aim of the study was to investigate if critiques of proofs presentations by the class and the instructor had an impact on writing proofs for the equivalence of definitions of ring theoretic concepts. For a ring A and an ideal J of A, ring- theoretic concepts are defined describing a certain property of J. In the cases that we consider, the property of J has an equivalent definition in terms of the elements in the ring A/J. Eight problems involving equivalence type relations were selected for research (see Appendix). Students worked in pairs on the problem allocated to them to prove the equivalence of two definitions for a particular concept. Each pair of students presented their proof during the weekly tutorial class, one proving the sufficient ("if") part and the other the necessary ("only if") part. The instructor and the students critiqued these proofs, after which the students were instructed to improve their proofs based on the critiques. Both the original and the improved version of the proofs were submitted to the instructor for grading. All the students were then given an unannounced test in which they were instructed to do any two of the eight problems they had worked on. Of the 16 students, 14 attempted their own problem and one new problem in the test. Three weeks later one of the eight problems was selected by the instructor as a final examination question for the course. The Seldon and Seldon (2009) theoretical framework was used to look at the logical construction path, the hierarchical, the rhetorical and the problem solving aspects of the proof. An 8-point marking rubric based on the framework was drawn up and used to grade the proofs. The main problems the students experienced were (a) knowing how to start the proof; (b) deciding on the type of proof (direct, contradiction, cases); (c) poor knowledge of definitions (zero divisor, nilpotent element, etc.), assumptions and technical terms; (d) gaps in their logical reasoning; (e) not giving reasons for statements; and (f) not applying the rules for equality of cosets, addition and multiplication of cosets correctly. The results indicate that students were able to carry-over some knowledge and experience of proof development from the critiqued class presentations to the test when the question was familiar (first test question), but they were not so successful when they had to develop a proof for a less familiar problem (second test question and the exam question). The research is significant in that it (a) highlights the difficulties students have in writing proofs of the equivalence of definitions of ring theoretic concepts; (b) presents a teaching strategy that should lead to improvements in proof writing, student participation and communication. Further research is necessary to find strategies that can maintain the initial gains.