Understanding and Cultivating the Connections Between Students' Natural Ways of Reasoning and Mathematical Ways of Reasoning

许多人认为数学推理是一种基本的数学技能，与了解和使用数学密不可分。然而，尽管它的重要性，数学教育研究继续描绘了一幅学生数学推理能力的暗淡图景。与此相反，认知科学研究揭示了儿童在非数学领域推理能力的惊人优势，表明儿童能够发展复杂和抽象的因果理论，并使用强大的归纳推理策略。因此，这就产生了一个悖论：为什么儿童在非数学领域的推理能力如此之强，而在数学领域的推理能力却如此之差？本研究的目的就是探讨这一看似矛盾的现象。特别是，我们的目标是将认知科学研究扩展到数学教育领域，更具体地说，扩展到中学数学领域。我们试图了解学生的推理和数学的优点和缺点，了解学生在不同领域的推理之间的联系，并最终提高学生的数学推理能力。

Many consider mathematical reasoning to be a basic mathematical skill and inseparable from knowing and using mathematics. Yet despite its importance, mathematics education research continues to paint a bleak picture of students' abilities to reason mathematically. In contrast, cognitive science research has revealed surprising strengths in children's abilities to reason in non-mathematical domains, suggesting that children are capable of developing complex and abstract causal theories, and of using powerful strategies of inductive inference. Thus, this raises something of a paradox: Why are children so good at reasoning in non-mathematical domains, yet so poor at reasoning in mathematical domains? The purpose of this study is to explore this seeming paradox. In particular, our goal is to extend the cognitive science research into the domain of mathematics education and, more specifically, into the domain of middle school mathematics. We seek to understand the strengths and weaknesses of students' reasoning in and out of mathematics, to understand the connections between students' reasoning in different domains, and, ultimately, to improve students' abilities to reason mathematically.