An investigation of pre-service teachers\u27 and professional mathematicians\u27 perceptions of mathematical proof at the secondary school level

The National Council of Teachers of Mathematics [NCTM] states that by the time students graduate high school, they should learn to present arguments consisting of logically rigorous deductions of conclusions from hypotheses (NCTM, 2000, p. 56) in written forms that would be acceptable to professional mathematicians (NCTM, 2000, p. 58). Research studies indicate, however, that students and teachers have difficulty with many aspects of mathematical proof, including its nature and meaning. In addition, there appears to be a disconnect between school teachers\u27 and university mathematicians\u27 expectations for their respective students regarding mathematical proof. This study examined the perceptions of thirteen pre-service teachers and eight professional mathematicians with regard to mathematical proof in both the discipline of mathematics and proof in high school mathematics. Participants were asked to complete a questionnaire composed of open-ended questions related to their perceptions of mathematical proof on these two dimensions, for example, their perceptions about the purpose and importance of mathematical proof, their perceptions about what is acceptable and valid as mathematical proof, and their expectations for students. The participants\u27 responses to the questions on the questionnaire served as the primary data source. Analysis of the data occurred in three phases: (1) a coarse reading through all the data to get a feel for the responses; (2) a line-by-line microanalysis and coding of the data; (3) a global analysis whereby responses were collected and chunked into episodes pertaining to a single concept or topic. As the data were being analyzed, hypotheses were formed. The hypotheses were then compared with the data to help bolster the plausibility of the hypotheses or to provide direction or modification of the hypotheses. Results indicate that there are important differences in the perceptions between pre-service teachers and professional mathematicians regarding the nature and meaning of mathematical proof and its place in the high school curriculum. Some of the observed differences include: (1) professional mathematicians value the content of an argument over its form, while pre-service teachers place more importance on the details of the form of an argument, in some cases, to the exclusion of its content; (2) professional mathematicians\u27 view of what constitutes proof is flexible and context dependent, while pre-service teachers\u27 perceptions about what is acceptable is much less context dependent, and in some cases, rigid and unyielding; (3) pre-service teachers expect high school students to know specific derivations of formulas and particular formats for arguments, while professional mathematicians expressed the desire that high school students know about the nature of proof in mathematics