Developing an Interactive Computer Support System to Facilitate Teacher Learning of Proof-related Instruction in Mathematics

证明除了是数学中的一个核心概念外，还被认为是数学教育的一个重要组成部分，但在大多数课堂上仍然是一个陌生的概念。因此，教师在支持学生和解决他们的多重误解方面面临困难，但练习的机会有限。利用计算机科学的专业知识开发一种计算机系统，使教师能够在向虚拟学生传授证据之前，就像飞行模拟器如何帮助飞行员准备真实的飞行一样，在解决这个教育问题上取得进展。为了寻求一个既有教育价值又有技术可行性的计算机支持系统的设计框架，本研究使用了拉卡托斯的方法来证明，这种方法富有成效地使用例子和反例来做出和提炼猜想。虚拟学生的设计需要一个可靠的学生模型。利用数据收集和人工智能技术，将分别建立手动构建的学生模型和自动生成的学生模型，然后将其合并为一个综合模型。六名教师将向真实学生(控制组)或虚拟学生(试验组)讲授证明。将通过观察和访谈收集定性数据。除了有助于教师学习与证明相关的教学，从而间接地帮助学生学习证明之外，本研究还可能为学生建模和计算机辅助教师培训开辟新的途径。

Besides a central concept in Mathematics, proof is regarded as an essential component of Mathematics Education, but is still an alien concept in most classrooms. Thus, teachers face difficulties in supporting students and tackling their multiple misconceptions, but have limited chances to practise. Headway in addressing this educational problem can be made with the use of Computer Science know-how to develop a computer system that will enable teachers to teach proof to virtual students before real ones, similarly to how flight simulators help pilots prepare for real flights. Seeking a framework for the design of a both educationally valuable and technologically feasible computer support system, this study uses Lakatos' approach to proving, which productively uses examples and counterexamples to make and refine conjectures. The design of virtual students requires a reliable student model. Using data collection and artificial intelligence techniques, a manually constructed and an automatically generated student model will be built, respectively, and then combined into an integrated model. Six teachers will teach proof to either real (control group) or virtual students (experimental group). Qualitative data will be collected through observations and interviews. Besides contributing to teachers' learning of proof-related instruction and thus, indirectly, to students' learning of proof, this study may open new pathways to student modelling and computer-assisted teacher training.