Changes in Problem Representation as a Mechanism of Knowledge Change

理解发展的关键之一是理解儿童的知识是如何随时间变化的。因此，发展研究的一个重要挑战是明确知识变化的机制。本研究的重点是数学知识发展变化的一个潜在机制：即儿童如何表达问题的变化。我将问题表征定义为对问题的特定特征在内部的心理复制。我认为表征是儿童数学概念知识与解决数学问题策略知识之间的重要联系。这项研究在学习数学等价概念的儿童中测试了这一说法，数学等价概念是指方程的两边代表相同的量。孩子们对这个概念的认识是用3+4+5=_+5的公式来测试的。本研究主要探讨了两个关于问题表征变化作为知识变化机制的问题。首先，问题表征的变化如何影响儿童的知识？本研究验证了儿童问题表征的改变会导致他们对问题解决策略的理解发生变化，并导致他们对相关概念的认识发生变化的假设。其次，是什么使表征发生变化？提出的研究探讨了可能导致表征变化的三种因素：(a)概念知识的获得，(b)学习新的解决问题的策略，以及(c)在问题领域获得经验。本研究将揭示表征在数学学习中的重要性，以及导致和影响表征变化的因素。我的首要目标是构建一个明确的理论模型，说明问题表征的变化如何作为认知变化的机制发挥作用。

One of the keys to understanding development is understanding how children's knowledge changes over time. Thus, an important challenge for developmental research is to specify the mechanisms of knowledge change. This research focuses on one potential mechanism of change in the development of mathematical knowledge: namely, changes in how children represent problems. I define problem representation as the creation of an internal, mental copy of particular features of a problem. I argue that representation is an important link between children's knowledge about mathematical concepts and their knowledge about strategies for solving mathematical problems. The proposed research tests this claim among children learning the concept of mathematical equivalence, which is the notion that the two sides of an equation represent the same quantity. Children's knowledge of this concept is tested using equations of the form 3+4+5=_+5. The proposed studies address two main questions about changes in problem representation as a mechanism of knowledge change. First, how do changes in problem representation influence children's knowledge? The proposed studies test the hypothesis that changes in children's problem representations can lead to changes in their understanding of problem-solving strategies, and to changes in their knowledge of related concepts. Second, what makes representations change? The proposed studies explore three types of factors that may contribute to changes in representation: (a) gains in conceptual knowledge, (b) learning new problem-solving strategies, and (c) gaining experience within a problem domain. This research will shed light on the importance of representation in mathematical learning, and on factors that lead to and influence representation change. My overarching goal is to construct a well-specified theoretical model of how changes in problem representation function as a mechanism of cognitive change.