Errors in proof schemes with complex numbers

Abstract

The written productions of students can be subjected to analysis for research and teaching purposes. In the present work, one of the few on errors and complex numbers, the productions presented by Higher Education students for a proof task of a particular case of the parallelogram identity are analysed. In a work that may give clues for the teaching practice in High School and Higher Education, we attempt to classify the proof schemes of those productions, to identify the errors and to connect these two strands of analysis. The nature of the research is qualitative, making use of content analysis with, on the one hand, known criteria of categorization of proof schemes and, on the other hand, new criteria of categorization of errors. The students produced, mainly, external conviction proof schemes, included in a new subcategory called not valid in the universe, and deductive proof schemes, included in a new level that is characterized by the inchoate discernment of the comprehension of the task, of the mathematical context, of the hypotheses and thesis, and the previous knowledge that was needed. The dominant errors in the productions, for which possible explanations are presented, are related to the understanding of the concepts of linearity of a function and of modulus of a complex number. Although a significant relation between the proof schemes produced by the students and the errors is not identified, material for the construction and the discussion of concept questions in the context of complex numbers is obtained.

Mathematics education; Proof scheme; Error; Parallelogram identity; Complex number