Creating a Theory of Decimal Arithmetic Learning

熟练掌握有理数--分数、小数和百分比--是在代数等更高级的数学中取得成功的关键。它对职业成功也很重要；大多数白领和蓝领工人在工作中都使用有理数字。然而，许多孩子即使在接受了多年的教育后，仍在与有理数字作斗争。这个项目的目标是建立一个关于儿童在有理数的一个领域学习的理论：十进制算术。该项目将确定帮助儿童更容易学习十进制算术的知识类型，阐明这种便利发生的机制，并开发一个模拟学习十进制算术过程的计算模型。根据研究结果，本研究将提出改善儿童十进制算术学习的建议，其中包括：(1)将课堂和练习时间集中在成功学习者所使用的概念方法上；(2)特别强调对儿童构成困难的问题类型；(3)将课堂时间用于说明常见错误并解释其错误原因；以及(4)利用对常见错误的讨论作为说明一般概念的机会。这些建议预计将对改善数学教学产生总体影响。学习数学既包括学习概念，也包括学习过程。概念包括原则和关系；程序是解决问题的循序渐进的行动序列。对概念的理解被认为有助于儿童学习程序，但这是如何发生的尚不清楚。这个项目的目的是建立一种理论，说明概念理解--如果存在--如何促进对数学中一个特别困难和重要的领域--十进制算术--程序的学习。为了实现这一目标，该项目将采取三管齐下的方法，包括纵向、微基因和计算建模方法。纵向方法将确定预测成功学习十进制算术程序的特定类型的概念知识；微遗传学方法将为这些类型的概念知识促进学习的特定机制提供证据；计算模型将用于准确描述这些机制并模拟使用前两种方法观察到的经验现象。计算模型将建立和扩展以前在分数算术学习模型FARRA中采用的建模体系结构；它的成功将根据其生成准确性水平、错误模式以及类似于在儿童中观察到的概念知识和程序知识之间的相关性的能力来评估。这项拟议的研究将以三种方式促进科学知识的发展：通过将学习结果的个体差异与学习过程理论联系起来，通过增进对概念性知识和程序性知识之间关系的理解，以及通过将数字发展理论扩展到一个新的领域，十进制算术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Proficiency with rational numbers-fractions, decimals, and percentages-is essential for success in more advanced mathematics such as algebra. It is also important for occupational success; majorities of both white- and blue-collar workers report using rational numbers in their jobs. Yet, many children struggle with rational numbers even after years of instruction. The goal of this project is to create a theory of children's learning in one area of rational numbers: decimal arithmetic. The project will identify types of knowledge that help children to learn decimal arithmetic more easily, clarify the mechanisms by which this facilitation occurs, and develop a computational model that simulates the process of learning decimal arithmetic. Based on the results, recommendations will be generated for improving children's learning of decimal arithmetic including recommendations (1) to focus classroom and practice time on conceptual approaches that are used by successful learners, (2) to place special emphasis on types of problem that pose difficulty for children, (3) to devote classroom time to illustrating common errors and explaining why they are incorrect, and (4) to use discussion of common errors as an opportunity to illustrate general concepts. These recommendations are anticipated to have implications for improving mathematics instruction in general. Learning mathematics involves learning both concepts and procedures. Concepts include principles and relations; procedures are step-by-step action sequences for solving problems. Understanding of concepts is believed to help children learn procedures, but how this occurs is not known. This project aims to create a theory of how conceptual understanding - when present - facilitates learning of procedures within a particularly difficult and important area of math: decimal arithmetic. To accomplish this goal, the project will adopt a three-pronged approach including longitudinal, microgenetic, and computational modeling methods. Longitudinal methods will identify specific types of conceptual knowledge that predict success in learning decimal arithmetic procedures; microgenetic methods will provide evidence for specific mechanisms by which these types of conceptual knowledge facilitate learning; computational modeling will be used to describe these mechanisms precisely and to simulate the empirical phenomena observed using the previous two methods. The computational model will build on and extend a modeling architecture previously employed in a model of fraction arithmetic learning, FARRA; its success will be assessed based on its ability to generate levels of accuracy, patterns of errors, and correlations between conceptual and procedural knowledge similar to those observed among children. The proposed research will advance scientific knowledge in three ways: by connecting individual differences in learning outcomes with a theory of learning processes, by advancing understanding of the relations between conceptual and procedural knowledge, and by extending theories of numerical development into a new domain, decimal arithmetic.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.