Conceptual and Procedural Knowledge in Mathematical Cognition

当面对一个数学问题时，孩子们是概念性地思考这个问题，还是通过一个程序来生成“正确答案”？这个研究项目旨在从三种不同的角度更好地理解这个问题。首先，我的目的是进一步探索是否存在不同的学生群体，他们对概念性知识或程序性知识的依赖不同。换句话说，我想看看是否有一些学生更依赖概念性知识，一些依赖程序性知识，或者一些同时依赖两者。我实验室之前的研究已经证明，在分数和代数理解方面存在这些差异。这里的目的是扩展这些发现，并测试这些个体差异是学习轨迹中的暂时停止，还是反映了更根深蒂固的学习偏见。其次，我将探讨程序性知识的确切含义。一些人将其描述为算法的简单记忆，另一些人强调需要流畅地执行，还有一些人坚持认为好的程序包括概念理解。这项研究将梳理出这些差异，看看哪些与数学表现最相关。第三，我计划调查与我们表示数量级的不同方式相关的概念知识的具体方面，以及其他数学能力，如数学事实的记忆，是如何与此以及整体数学表现相关的。总之，这里提出的研究计划旨在扩展我们对数学认知领域的概念知识、程序知识和数学事实自动性的理解。尽管我们对数学认知的理解的任何进步都对改善我们的教育系统有影响，但上面列出的三个因素已被确定为更好地理解数学的关键。这个研究项目将增强我们对数学认知的这些关键方面的理解，这不仅将使我们更好地理解数学和数字信息是如何在人脑中表示的，而且还可能对如何提高人们的数学理解产生影响。它还将培养具有统计、编程和研究专业知识的高素质人才，为数学认知领域做出贡献。

When faced with a mathematical problem, do children think about the problem conceptually, or do they instead work through a procedure to generate the "right answer"? This program of research is aimed at better understanding this question in three different ways. First, I aim to further explore whether there are distinct groups of students who rely differently on conceptual knowledge or procedural knowledge. In other words, I want to see if there are some students who rely more on conceptual knowledge, some who rely on procedural knowledge, or some who rely equally on both. Previous research in my lab has demonstrated these differences exist in regard to fraction and algebra understanding. The aim here is to extend these findings and test whether these individual differences are temporary stops in the learning trajectory or reflect more entrenched learning biases. Second, I will explore what exactly is meant by procedural knowledge. Some characterize it as the simple memorization of algorithms, others emphasize that is needs to be executed fluently, and still others maintain that good procedures are ones that include conceptual understanding. This research will tease out these differences and see which are most related to math performance. Third, I plan to investigate the specific aspects of conceptual knowledge related to the different ways in which we represent magnitude, and how other math abilities, like the memorization of math facts, are related both to this and to overall math performance. In sum, the research program proposed here seeks to extend our understanding of conceptual knowledge, procedural knowledge, and math fact automaticity in the field of mathematical cognition. Although any advancement in our understanding of mathematical cognition has implications for improving our education system, the three factors listed above have been identified as being key to better mathematical understanding. This program of research will enhance our understanding of these key aspects of mathematical cognition, which will not only lead to a better understanding of how mathematical and numerical information is represented in the human brain, but could also have implications about how to improve people's mathematical understanding. It will also train Highly Qualified Personnel with the statistical, programing and research expertise to contribute to the field of mathematical cognition.