The evolution of mathematical proof for students from grade eight to grade eleven. (c2005)

Abstract

This research study is an attempt to explore the evolution of the meaning and of the format of mathematical proof after one year of its introduction and after several years, for the purpose of addressing students' difficulties and finding proper strategies to introduce the students to the world of proof. Twelve 8 graders and twelve 11 graders, of a reputable school in Beirut, participated in this study. All the participants had the same mathematics teacher when they were in grade 7. Clinical interviews were performed with pairs of students. However, two 11 graders were interviewed individually, so that the influence ofthe social interaction could be assessed. In order to choose these students, a pretest was administered to 8 graders and to 11 graders. A comparison between their written solutions and their deductive thinking was conducted. A hierarchy was developed and the written so lutions of the pretest were analyzed and placed in the different categories. Accordingly, the sample was chosen representing all the categories of the hierarchy developed. For the purpose of comprehending the participants' understanding of the meaning of proof, as effected by the way they were introduced to proof, their grade 7 teacher was interviewed and her class sessions were observed. As a result, it was found that grade 8 students stress the format of the written solution much more than eleven graders. In addition, students from the 11th grade wrote proofs including more deductive thinking and more rigor. All 8th graders, except one, could not notice the incompleteness of their solutions, while all 8 graders wrote complete proofs. However, participants from both grade levels gave proof the same meanings of explanation, justification, communication, and conviction.