Abstract We report a mixed-methods research study investigating the effect of quantitative reasoning on prospective mathematics teachers’ comprehension of a proof on real numbers. Nineteen prospective mathematics teachers engaged in quantitative reasoning while developing real numbers as rational number sequences in a series of instructional activities. All participants completed a proof comprehension assessment prior to and upon completion of the instruction. Six of the prospective mathematics teachers also participated in semi-structured interviews after the post-test. Results showed a significant difference in proof comprehension performance between the pre- and post-tests. Moreover, results from the interviews showed that prospective teachers reasoned quantitatively on the proof comprehension dimensions. Results suggest that engaging in quantitative reasoning during instruction may help to develop proof comprehension, particularly in situations involving the analysis of proofs entailing properties of the real number system. We recommend embedding quantitative reasoning in teacher education and professional development programs to facilitate mathematics teachers’ proof comprehension and proving activities.

中文翻译：

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定量推理对未来数学教师证明理解的影响：以实数为例

摘要 我们报告了一项混合方法研究，调查定量推理对未来数学教师对实数证明的理解的影响。19 位准数学教师在一系列教学活动中进行定量推理，同时将实数发展为有理数列。所有参与者在完成指令之前和之后都完成了证明理解评估。六名准数学教师也在后测后参加了半结构化面试。结果表明，前测和后测在证明理解能力方面存在显着差异。此外，访谈结果表明，未来教师在证明理解维度上进行了定量推理。结果表明，在教学过程中进行定量推理可能有助于培养对证明的理解，特别是在涉及对包含实数系统特性的证明进行分析的情况下。我们建议在教师教育和专业发展计划中嵌入定量推理，以促进数学教师的证明理解和证明活动。