The Extension of Mathematical Knowledge: A Cognitive and Neuroscience Investigation

数学是人类发明的一种工具，用来帮助处理商业、导航、农业、政府和其他现实世界的应用。它们通常要求我们扩展我们的数学知识，超越我们所学到的确切程序。 许多人已经掌握了学校里教给他们的数学知识，但当他们面临挑战时，他们仍然表现出严重的困难，将知识扩展到新的情况，并经常产生荒谬的答案。本研究以掌握了分数和负数等初中数学的青少年和大学生为对象，挑战他们将这些知识扩展到处理新的数学概念。本研究将关注一类鲜为人知的数学问题，称为金字塔问题或梯形数，这些问题具有自然的几何解释，有助于指导他们的理解。 它们对公众的新奇使它们成为学习的理想选择，它们不需要超出中学数学的计算。 参与者将被教导解决涉及小正整数的金字塔问题，然后他们将面临挑战，以扩展关系来处理大数字，分数和负数。为了了解发展趋势，青少年将与大学生进行比较。收集的数据将涉及性能测量，口头协议和功能磁共振成像脑成像模式的组合。单独的研究将调查成功的知识扩展的个体差异的基础，元认知参与对成功的影响，以及几何解释在指导数学关系的推理中的作用。本研究将在ACT-R理论的理论框架内进行，ACT-R理论是数学问题解决的计算模型。对这一理论的关键检验将是它预测所收集数据的丰富模式的能力。有这样一个理论框架将是重要的推广结果，从金字塔问题，以帮助学生扩大他们的数学知识更普遍。这项研究的关键贡献是，它超越了如何教一个特定的数学能力的问题，并解决了如何准备学生扩展他们的数学知识和发现新的数学关系的问题。如果将其置于认知的正式计算模型的背景下，它将使理论超越定义明确的程序学习，转向负责产生新知识的机制，从而为认知科学和神经科学做出重大贡献。这项研究将在认知导师的背景下进行，这些导师目前部署在许多美国教室中，覆盖超过50万名学生。该项目中开发的计算模型可以过渡到这些导师，并将使这些导师教授的能力得到重大提高。

Mathematics is a tool humans invented to help deal with commerce, navigation, agriculture, and government, and other real-world applications. They often require that we extend our mathematical knowledge beyond the exact procedures that we have been taught. Many people have mastered the mathematical knowledge they have been taught in school, but they still display serious difficulties when challenged to extend that knowledge to new situations and often produce nonsensical answers. This research will study adolescents and college students who have mastered middle-school mathematics including fractions and negative numbers, and challenge them to extend this knowledge to deal with a novel mathematical concept.This research will focus on a little-known class of mathematical problems, called pyramid problems or trapezoidal numbers, that have a natural geometrical interpretation to help guide their understanding. Their novelty to the general public makes them ideal for study and they require no calculations that go beyond middle school mathematics. Participants will be taught to solve pyramid problems that involve small positive integers and then they will be challenged to extend the relationships to deal with large numbers, fractional numbers, and negative numbers. To understand developmental trends, adolescents will be compared with college students. The data to be collected will involve a combination of performance measures, verbal protocols, and fMRI brain imaging patterns. Separate studies will investigate the basis of individual differences in successful knowledge extension, the effect of metacognitive engagement on success, and the role of the geometric interpretation in guiding inferences about the mathematical relationships. This research will take place within the theoretical framework of the ACT-R theory, a computational model of mathematical problem solving. The critical test of this theory will be its ability to predict the rich pattern of data collected. Having such a theoretical framework will be important for generalizing the results from pyramid problems to helping students extend their mathematical knowledge more generally. The critical contribution of this research is that it goes beyond the question of how to teach a specific mathematical competence and addresses the question of how to prepare students to extend their mathematical knowledge and discover new mathematical relationships. Placed in the context of a formal computational model of cognition, it would make a major contribution to cognitive science and neuroscience by moving theory beyond the learning of well-defined procedures to the mechanisms responsible for the generation of new knowledge. This research will take place within the context of the Cognitive Tutors, which are currently deployed in many American classrooms, reaching over 500,000 students. The computational model developed in the project can be transitioned to these tutors and would enable a major enhancement in the kinds of competences that these tutors teach.