Transitioning from “It Looks Like” to “It Has To Be” in Geometrical Workspaces: affect and near-to-me attention

Abstract Within a practitioner researcher framework, this paper draws on a particular mathematics education theory and aspects of neuroscience to show that, from a learner’s perspective, moving to a deductive reasoning style appropriate to basic Euclidean geometry, can be facilitated, or impeded, by emotion and/or directed attention. This shows that the issue of a person’s deductive reasoning is not a merely cognitive one, but can involve affective aspects related to perception – particularly perception of nearby sense data – and emotion. The mathematics education theory that has been used is that of the Espace de Travail Mathématique, the English translation of which is known as Mathematical Working Spaces (MWS). The aspects of neuroscience that have been used pertain to the distinct processing streams known as top-down and bottom-up attention. The practitioner research perspective is aligned with Mason’s teaching-practice-based ‘noticing’; qualitative data analysed in this report include individual interviews with school teachers on in-service courses and reflective notes from teaching. Basic Euclidean geometry is used as the medium for investigating transition from ‘it looks like’ to a reasoned ‘it has to be’.

摘要 在从业者-研究者框架下，本文依托特定的数学教育理论与神经科学相关内容，从学习者视角论证：转向适配基础欧几里得几何的演绎推理思维模式的过程，既可能受情绪与/或定向注意的推动，也可能受到其阻碍。这表明，个体演绎推理能力的相关问题并非仅涉及认知层面，还可能关联与感知（尤其是对邻近感官数据的感知）及情绪相关的情感维度。本文所采用的数学教育理论为法语命名的*Espace de Travail Mathématique*，其英文译名为数学工作空间（Mathematical Working Spaces, MWS）。本文所引用的神经科学相关内容，涉及被称为自上而下注意通路与自下而上注意通路的两类独特加工通路。本研究采用的从业者研究视角，与梅森基于教学实践提出的“觉察（noticing）”理念相一致；本报告所分析的质性数据，包含针对在职培训课程中中小学教师的个别访谈记录，以及教学反思笔记。本研究以基础欧几里得几何为载体，探究学习者从“看起来如此”到经论证的“必然如此”的思维转变过程。