Matthews Percival G, Ellis Amy B
Department of Educational Psychology, University of Wisconsin-Madison, Madison, WI, USA.
Department of Math and Science Education, University of Georgia, Athens, GA, USA.
J Numer Cogn. 2018;4(1):19-58. doi: 10.5964/jnc.v4i1.97. Epub 2018 Jun 7.
The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students' non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students' perceptually based intuitions about quantity and b) students' reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.
绝大多数培养早期数学思维的努力主要依赖于计数和相关的自然数概念。不幸的是，自然数和离散化思维与我们希望孩子们学习的大量数学概念不太契合。这种不匹配对教学造成了重要障碍。我们建议，规避这些陷阱的一种方法是利用学生的非数字体验，这些体验可以为基础数学概念提供直观的理解途径。具体而言，我们主张明确利用：a）学生基于感知的数量直觉，以及b）学生对变化和变异的推理，并探讨这种方法所带来的便利。我们认为，它可以支持一些思维方式，这些思维方式有时可能与待学习的数学思想更契合，因此与数字相比，对于特定的数学概念而言，它可能是一种有效的替代方法。我们以比例领域为例来说明这一观点，并且我们分别从作为认知心理学家和数学教育研究者所采用的不同学科视角来进行说明。最后，鉴于我们偏好的方法和一般认识论存在重大差异，我们讨论了进行富有成效的综合的可能性。
