Benson Ian, Marriott Nigel, McCandliss Bruce D
School of Education, University of Roehampton, London, United Kingdom.
Ian Benson and Partners Ltd., London, United Kingdom.
Front Psychol. 2023 Aug 28;14:1116555. doi: 10.3389/fpsyg.2023.1116555. eCollection 2023.
The ability to reason about equations in a robust and fluent way requires both instrumental knowledge of symbolic forms, syntax, and operations, as well as relational knowledge of how such formalisms map to meaningful relationships captured within mental models. A recent systematic review of studies contrasting the Cuisenaire-Gattegno (Cui) curriculum approach vs. traditional rote schooling on equational reasoning has demonstrated the positive efficacy of pedagogies that focus on integrating these two forms of knowledge.
Here we seek to replicate and extend the most efficacious of these studies (Brownell) by implementing the curriculum to a high degree of fidelity, as well as capturing longitudinal changes within learners via a novel tablet-based assessment of accuracy and fluency with equational reasoning. We examined arithmetic fluency as a function of relational reasoning to equate initial performance across diverse groups and to track changes over four growth assessment points.
Results showed that the intervention condition that stressed relational reasoning leads to advances in fluency for addition and subtraction with small numbers. We also showed that this intervention leads to changes in problem solving dispositions toward complex challenges, wherein students in the CUI intervention were more inclined to solve challenging problems relative to those in the control who gave up significantly earlier on multi-step problems. This shift in disposition was associated with higher accuracy on complex equational reasoning problems. A treatment by aptitude interaction emerged for both arithmetic equation reasoning and complex multi-step equational reasoning problems, both of which showed that the intervention had greatest impact for children with lower initial mathematical aptitude. Two years of intervention contrast revealed a large effect (d = 1) for improvements in equational reasoning for the experimental (CUI) group relative to control.
The strong replication and extension findings substantiate the importance of embedding these teaching aides within the theory grounded curricula that gave rise to them.
以稳健且流畅的方式对等式进行推理的能力，既需要对符号形式、句法和运算的工具性知识，也需要了解这些形式体系如何映射到心理模型中所捕捉到的有意义关系的关系性知识。最近一项关于对比库埃纳 - 加泰尼奥（Cui）课程方法与传统死记硬背式教学在等式推理方面的研究的系统综述表明，专注于整合这两种知识形式的教学法具有积极效果。
在此，我们试图通过高度保真地实施该课程，并通过一种新颖的基于平板电脑的等式推理准确性和流畅性评估来捕捉学习者的纵向变化，从而复制并扩展这些研究中最有效的一项（布朗内尔研究）。我们将算术流畅性作为关系推理的函数进行考察，以使不同组的初始表现相等，并在四个成长评估点跟踪变化。
结果表明，强调关系推理的干预条件会使小数加减法的流畅性得到提高。我们还表明，这种干预会导致解决复杂挑战的问题解决倾向发生变化，其中，与在多步骤问题上更早放弃的对照组学生相比，接受CUI干预的学生更倾向于解决具有挑战性的问题。这种倾向的转变与复杂等式推理问题的更高准确性相关。对于算术等式推理和复杂多步骤等式推理问题，均出现了处理与能力的交互作用，这两者都表明该干预对初始数学能力较低的儿童影响最大。两年的干预对比显示，相对于对照组，实验组（CUI）在等式推理方面的改善有很大效果（d = 1）。
强有力的复制和扩展结果证实了将这些教学辅助工具嵌入产生它们的基于理论的课程中的重要性。
