Abstract
Proving is a process that has important roles in terms of learning and teaching in almost all the areas of mathematics. Because the process of proof constructions an extensive process that includes skills as mathematical thinking, reasoning and making connections. Reasoning is one of the most important components of this process. However, most students have difficulty in making a good reasoning and they make various reasoning errors in the process. The purpose of the study is to investigate the reasoning errors that pre-service mathematics teachers exhibit during proof construction. This study was carried out with 80 university students from second, third, fourth and fifth grade levels. An open-ended exam based on abstract mathematics and algebra was used. To deeply examine reasoning errors in the proving process, clinical interviews were conducted with pre-service teachers. A scale was developed by considering the literature review and the expert opinions; this was used to analyse the data about the reasoning errors. The results illustrate that the reasoning errors mostly do not show differences for all grade levels. However, the percentages of reasoning errors according to the grade levels and to the upper classes these errors show resistance to decrease the deficiencies. It is important to design a learning environment enabling students to experience proof construction in order to reduce or eliminate the reasoning errors.
Highlights
Proof has an important place in the historical of humanity
The reasoning errors of the different grade levels of pre‐service mathematics teachers in the proof construction process were classified; similar reasoning errors, deficiencies and gaps were categorized across all the grade levels
; in this study, reasoning deficiencies and gaps are divided into sub‐components; they are classified into different categories as the reasoning gaps and deficiencies have different meaning from reasoning errors
Summary
Proof has an important place in the historical of humanity. Despite the fact that the term of “Proof” started to play an active role systematically in the area of mathematics since the Ancient Greece, “Proof” starts together with the history of the human being. Because of man’s nature, he feels the need of seeing the validity of a statement or something like that or the need of clearing the suspicions in his mind. This need leads to seeking convincing answers to the questions of “Why?”, “What for?”. In this sense, proof is one of the concepts that constitute the fundamentals of the mathematics, and mathematical proof is, undoubtedly, one of the most important elements that characterize mathematics and that distinguish it from the other disciplines. What makes “Proof” gain this qualification is its forming a basis that shows whether a mathematical statement or judgement is valid or invalid (Tall & Mejia‐Ramos, 2006)
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