Proving is a fundamental skill that pre-service mathematics teachers are expected to develop and to support in their future students. This study aimed to examine the proving processes of pre-service middle school mathematics teachers. To this end, a total of 30 participants were selected through maximum variation sampling from among pre-service teachers who had just been admitted to a middle school mathematics teacher education program and pre-service mathematics teachers at all year levels of the program, with six participants from each level, and the data of this qualitative study were collected through clinical interviews. The data were collected in the context of the concepts of function and number. According to the results of the study, participants at the higher year levels generally showed greater success in constructing the formal-rhetorical part of a proof and in solving its problem-centered part, indicating a cross-year pattern rather than individual longitudinal change. It was also concluded that even when participants in the upper years possessed knowledge of proof methods, either their lack of sufficient conceptual knowledge to solve the problem-centered part or their deficiencies in the use of mathematical language hindered the proving process. Furthermore, it was observed that the participants were unable to use the full range of proof methods; whereas proof by counterexample showed the highest success in the proving process at every level, participants experienced difficulties in the direct proof and proof by contraposition processes. The findings suggest that the formal-rhetorical and problem-centered parts of proof should be addressed explicitly and systematically across all year levels of mathematics teacher education programs.
Angle is a fundamental concept that students need to learn correctly, as it is one of the building blocks of geometry and is embedded in many geometry topics. This study aimed to determine seventh-grade middle school students’ concept definitions and concept images of angle, as well as the ways in which they associated their concept definitions with their concept images. Adopting a qualitative research method, a form consisting of 23 open-ended questions was prepared by taking into account most of the angle-related learning outcomes up to the seventh grade. The form was administered to 168 seventh-grade students attending two state middle schools; after the written data were analysed, clinical interviews were conducted using the same form with 9 participants selected through criterion sampling. The data were analysed using qualitative data analysis and interpreted within Tall and Vinner’s concept image-concept definition framework. The findings showed that the participants held incorrect or incomplete images of angle and largely produced their personal concept definitions from their concept images. The participants could not define angle formally, confused angle with angle measure, and mostly emphasised the measure-related aspect of angle. They also had difficulty using the concept of angle in problem situations. The results point to the importance of learning environments in which the dynamic and static aspects of angle are associated and in which everyday-life contexts and dynamic geometry tools are used.
زاویه یکی از مفاهیم بنیادی است که دانش‌آموزان باید به‌درستی آن را بیاموزند، چرا که یکی از اجزای سازنده هندسه بوده و در بسیاری از مباحث هندسی جای گرفته است. هدف این پژوهش تعیین تعاریف مفهومی و تصاویر مفهومی دانش‌آموزان پایه هفتم متوسطه دربارهٔ زاویه و همچنین بررسی نحوه ارتباط تعاریف مفهومی آنان با تصاویر مفهومی‌شان بود. این پژوهش با اتخاذ روش کیفی انجام شد و فرمی شامل ۲۳ پرسش بازپاسخ با توجه به اکثر اهداف یادگیری مرتبط با زاویه تا پایه هفتم تهیه گردید. این فرم برای ۱۶۸ دانش‌آموز پایه هفتم که در دو مدرسه دولتی متوسطه تحصیل می‌کردند اجرا شد؛ پس از تحلیل داده‌های مکتوب، مصاحبه‌های بالینی با استفاده از همان فرم با ۹ نفر از شرکت‌کنندگان که به روش نمونه‌گیری مبتنی بر معیار انتخاب شده بودند، انجام گرفت. داده‌ها با استفاده از تحلیل داده‌های کیفی و در چارچوب تصویر مفهومی-تعریف مفهومی Tall و Vinner تفسیر شدند. یافته‌ها نشان داد که شرکت‌کنندگان تصاویر نادرست یا ناقصی از زاویه داشتند و عمدتاً تعاریف مفهومی شخصی خود را از تصاویر مفهومی‌شان استخراج می‌کردند. شرکت‌کنندگان قادر به تعریف رسمی زاویه نبودند، زاویه را با اندازه زاویه اشتباه می‌گرفتند و عمدتاً بر جنبه مرتبط با اندازه زاویه تأکید داشتند. همچنین در به‌کارگیری مفهوم زاویه در موقعیت‌های مسئله‌محور با دشواری مواجه بودند. نتایج بر اهمیت محیط‌های یادگیری که در آن‌ها جنبه‌های پویا و ایستای زاویه با یکدیگر مرتبط می‌شوند و از زمینه‌های زندگی روزمره و ابزارهای هندسه پویا استفاده می‌شود، تأکید دارد.
