Mathematical competence, teaching, and learning. Reflections on 'Challenges in Mathematical Cognition' by Alcock et al. (2016)

In this commentary, I focus on the notion of competence and issues related to the distinction between knowing how mathematical problems are solved versus knowing how to teach mathematics. Although definitions of competence may necessarily be affected by value judgements and thus less amenable to factual answers, providing a defensible definition is important because it affects eligibility for intervention and treatment. One way to tackle this issue is to focus on the identification of prerequisite skills and concepts needed for particular domains of mathematics. Recent work on fraction and algebra has shown that long held assumptions may need to be re-examined. On knowledge versus application, some cautionary notes are made on the importance of not losing sight of translating our knowledge of processes involved in mathematical problem solving into better pedagogical practices. [Commentary on: Alcock, L., Ansari, D., Batchelor, S., Bisson, M.-J., De Smedt, B., Gilmore, C., . . . Weber, K. (2016). Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41. doi:10.5964/jnc.v2i1.10] notReviewed publishedVersion

本评论聚焦于能力（competence）概念，以及“知晓数学题的求解方法”与“知晓如何教授数学”二者间的区分相关议题。尽管能力的定义必然会受到价值判断的影响，因此难以通过事实性解答予以界定，但提供一个站得住脚的定义仍至关重要——因其直接影响干预与治疗的资格准入。破解这一议题的路径之一，是聚焦于特定数学分支所需的前置技能与概念的识别工作。近期针对分数与代数的研究表明，长期以来秉持的诸多假设或需重新审视。就知识与应用之辨而言，本文提出若干警示，强调切勿忽视将数学问题求解过程的相关知识转化为更优质的教学实践的重要性。【评论对象：Alcock, L.、Ansari, D.、Batchelor, S.、Bisson, M.-J.、De Smedt, B.、Gilmore, C. 等……Weber, K.（2016）。《数学认知领域的挑战：一项协作衍生的研究议程》，《数值认知期刊》，2卷，第20-41页。DOI:10.5964/jnc.v2i1.10】未评审 正式出版版本