Conceptual and Procedural Knowledge in Mathematical Cognition

当面对数学问题时，孩子们是从概念上思考问题，还是通过一个程序来产生“正确答案”？本研究计划旨在以三种不同的方式更好地理解这个问题。首先，我的目标是进一步探讨是否有不同的学生群体谁依赖于不同的概念知识或程序知识。换句话说，我想看看是否有一些学生更依赖概念知识，一些学生更依赖程序知识，或者一些学生同样依赖两者。我实验室以前的研究已经证明了这些差异存在于分数和代数理解方面。本文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。有些人把它描述为算法的简单记忆，有些人强调它需要流畅地执行，还有一些人认为好的过程是包括概念理解的过程。这项研究将梳理出这些差异，看看哪些与数学成绩最相关。第三，我计划调查概念知识的具体方面，这些知识与我们表示数量的不同方式有关，以及其他数学能力（如数学事实的记忆）如何与此以及整体数学表现相关。 总之，本文的研究计划旨在扩展我们对数学认知领域中的概念知识、程序知识和数学事实自动性的理解。尽管我们对数学认知理解的任何进步都对改善我们的教育系统产生影响，但上面列出的三个因素已被确定为更好的数学理解的关键。这项研究计划将增强我们对数学认知这些关键方面的理解，这不仅会更好地理解数学和数字信息在人脑中的表现方式，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.