How Proportional Reasoning Relates to Whole Number Operations and Numerical Estimation in Elementary School Children

数字信息通常使用比例表示。然而，研究表明，小学生很难进行比例推理，并且难以处理分数和小数，而分数和小数是在科学、技术、工程和数学(STEM)学科中取得成功的关键概念。这种困难的来源很大程度上是因为比例由相对量组成，这些相对量与更常见的绝对量不遵循相同的原则(例如，3/4大于6/10，尽管3或4都不大于6或10)。事实上，最近的研究发现，孩子们在比例方面的一些困难是由于过度使用经常使用的整数解决数学问题造成的。该项目的重点将是检查个体差异和在整个儿童发展过程中对一系列量化任务的潜在背景影响。在第一项研究中，小学生将完成一系列计算机化任务，这些任务将衡量比例等价性、整数运算和数字估计能力。研究人员预测，在较小的儿童中，那些在整数运算问题上表现良好的人在比例等价性问题上的表现将较差，但在较大的儿童中，问题类型之间将存在正相关。在第二项研究中，孩子们将完成一系列的相对或绝对量问题，然后是比例推理任务。研究人员预测，那些首先解决相对量问题的人在比例推理任务上的表现会比那些最初解决绝对量问题的人更好。这个项目将为许多数学思维的基础发展进程提供信息。研究结果将有助于理解如何最好地对儿童进行数学概念教育，并对STEM教育实践产生直接影响。如果早期对比例推理概念的理解与整数运算之间存在相反的关系，如果比例推理能力如预测的那样随着直接解决问题的背景而变化，那么这表明在向儿童介绍这些概念时必须格外小心。这项工作还将提出可能采用的技术，以最有效地介绍这些概念。

Number information is frequently presented using proportions. Research has shown, however, that elementary school-aged children have great difficulty reasoning proportionally and struggle with fractions and decimals, which are crucial concepts for success in the Science, Technology, Engineering, and Mathematics (STEM) disciplines. The source of much of this difficulty is because proportions are composed of relative quantities that do not follow the same principles as more familiar absolute quantities (e.g., 3/4 is greater than 6/10, though neither 3 nor 4 is greater than either 6 or 10). Indeed, recent research has found that some of children's difficulties with proportions are attributable to an over-application of well-practiced mathematical problem solving using whole number. The focus of this project will be to examine individual differences and potential context effects on a range of quantitative tasks throughout children's development. In a first study, elementary school students will complete a battery of computerized tasks that will measure proportional equivalence, whole number operations, and numerical estimation abilities. The investigators predict that in younger children, those who perform well on whole number operations problems will perform poorly on the proportional equivalence problems, but in older children there will be a positive correlation between problem types. In a second study, children will complete a series of relative or absolute quantity problems followed by the proportional reasoning task. The investigators predict that those who solve relative quantity problems first will perform better on the proportional reasoning task than those who initially solve absolute quantity problems. This project will inform the developmental progression that underlies much of mathematical thinking. The results will contribute to an understanding of how best to instruct children in mathematical concepts, with direct implications for STEM educational practices. If there is an inverse relationship between early understanding of proportional reasoning concepts and whole number operations, and if proportional reasoning abilities vary with the immediate problem-solving context as predicted, then this would suggest extra care must be used in introducing these concepts to children. This work will also suggest techniques that might be adopted to introduce these concepts most effectively.