Tracing proof schemes: some patterns and new perspectives

The aim of this paper is to review some studies conducted with different learning areas in which the schemes of different participants emerge. Also it is about to show how mathematical proofs are handled in these studies by considering Harel and Sowder's classification of proof schemes with specific examples. As a result, it was seen that the examined studies were addressed in the learning areas of Analysis, Geometry, Algebra, Linear Algebra, Elementary Number Theory, Probability, Combinatorics, and MIX. Students in early grades tend more towards external and empirical proof schemes. On the other hand, the characteristics of the proof schemes become more sophisticated as the participants' profiles change to pre-service teachers or as they become more specializing in mathematics. Some results are as follows: the academic achievement levels, genders, and grade levels of the participants in the studies examined in this paper have indicated that they have similar traces with the schemes they use. In addition, it has been determined that new perspectives such as examining Harel and Sowder's classification with new lenses, revealing the overlooked roles of some dynamics in proof, or improving the framework provide an important research area in terms of revealing students' potentials.