Department of Psychology, Rutgers University-Newark, Newark, NJ, United States.
Department of Urban Education, Rutgers University-Newark, Newark, NJ, United States.
Cogn Psychol. 2023 Jun;143:101575. doi: 10.1016/j.cogpsych.2023.101575. Epub 2023 May 23.
Early emerging nonsymbolic proportional skills have been posited as a foundational ability for later fraction learning. A positive relation between nonsymbolic and symbolic proportional reasoning has been reported, as well as successful nonsymbolic training and intervention programs enhancing fraction magnitude skills. However, little is known about the mechanisms underlying this relationship. Of particular interest are nonsymbolic representations, which can be in continuous formats that may emphasize proportional relations and in discretized formats that may prompt erroneous whole-number strategies and hamper access to fraction magnitudes. We assessed the proportional comparison skills of 159 middle-school students (mean age = 12.54 years, 43% females, 55% males, 2% other or prefer not to say) across three types of representations: (a) continuous, unsegmented bars, (b) discretized, segmented bars that allowed counting strategies, and (c) symbolic fractions. Using both correlational and cluster approaches, we also examined their relations to symbolic fraction comparison ability. Within each stimulus type, we varied proportional distance, and in the discretized and symbolic stimuli, we also manipulated whole-number congruency. We found that fraction distance across all formats modulated middle-schoolers' performance; however, whole-number information affected discretized and symbolic comparison performance. Further, continuous and discretized nonsymbolic performance was related to fraction comparison ability; however, discretized skills explained variance above and beyond the contributions of continuous skills. Finally, our cluster analyses revealed three nonsymbolic comparison profiles: students who chose the bars with the largest number of segments (whole-number bias), chance-level performers, and high performers. Crucially, students with a whole-number bias profile showed this bias in their fraction skills and failed to show any symbolic distance modulation. Together, our results indicate that the relation between nonsymbolic and symbolic proportional skills may be determined by the (mis)conceptions based on discretized representations, rather than understandings of proportional magnitudes, suggesting that interventions focusing on competence with discretized representations may show dividends for fraction understanding.
早期出现的非符号比例技能被认为是后期分数学习的基础能力。已经报道了非符号和符号比例推理之间的正相关关系，以及成功的非符号训练和干预计划增强了分数大小技能。然而，对于这种关系的机制知之甚少。特别有趣的是非符号表示，它可以是连续格式，可能强调比例关系，也可以是离散格式，可能提示错误的整数策略，并阻碍对分数大小的访问。我们评估了 159 名中学生（平均年龄= 12.54 岁，43%为女性，55%为男性，2%为其他或不愿透露）的比例比较技能，跨越三种表示形式：（a）连续的，不分段的条形图，（b）允许计数策略的离散分段条形图，和（c）符号分数。我们使用相关和聚类方法，还检查了它们与符号分数比较能力的关系。在每种刺激类型中，我们都改变了比例距离，并且在离散和符号刺激中，我们还改变了整数一致性。我们发现，所有格式的分数距离都影响了中学生的表现；然而，整数信息影响了离散和符号比较表现。此外，连续和离散的非符号表现与分数比较能力有关；然而，离散技能解释的方差超过了连续技能的贡献。最后，我们的聚类分析揭示了三种非符号比较模式：选择具有最大分段数的条形图的学生（整数偏差）、机会水平表现者和高表现者。至关重要的是，具有整数偏差模式的学生在他们的分数技能中表现出这种偏差，并且没有表现出任何符号距离调制。总之，我们的结果表明，非符号和符号比例技能之间的关系可能取决于基于离散表示的（错误）概念，而不是对比例大小的理解，这表明专注于离散表示能力的干预措施可能会对分数理解产生好处。
