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.
