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首页> 外文期刊>Journal of Mathematics Teacher Education >Preservice teachers’ knowledge of proof by mathematical induction
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Preservice teachers’ knowledge of proof by mathematical induction

机译：职前教师的数学归纳证明知识

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There is a growing effort to make proof central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ⇒ P(k + 1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.
机译：越来越多的努力使证明成为所有年级所有学生数学经验的核心。这个目标的成功在很大程度上取决于教师的证明知识，但是有限的研究已经检验了这一知识。本文通过研究职前中小学数学老师的数学归纳证明知识，为这一研究领域做出了贡献。该研究可以为数学教师教育者有效地向职前教师传授证明所需要的有关职前教师的知识提供信息。我们的分析基于95位参与者对特别制定的任务的书面答复以及对其中11位参与者的半结构化访谈。研究结果表明，两组的职前教师都存在以下困难：（1）归纳法基本步骤的实质； （2）在证明P（n）的范围内对任意k表示蕴涵P（k）⇒P（k + 1）的归纳步骤的含义； （3）通过数学归纳法证明的句子中的真理集包含其话语范围之外的值的可能性。与中学教师相比，学前基础课程的基础步骤和归纳步骤上的困难更为明显，但是对于归纳证明是否应尽可能涵盖所有方面的困难，对两组都同样重要。根据这些发现，讨论了对数学教师教育和未来研究的影响。

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