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程序。他们经常要求我们扩展我们的数学知识，而不是我们被教授的确切程序。许多人已经掌握了他们在学校里学到的数学知识，但当他们被要求将这些知识推广到新的情况下时，他们仍然表现出严重的困难，而且经常会得出荒谬的答案。这项研究将研究已经掌握了包括分数和负数在内的中学数学的青少年和大学生，并挑战他们扩展这一知识来处理一个新的数学概念。这项研究将专注于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，以帮助指导他们的理解。对于普通大众来说，它们的新颖性使它们成为学习的理想工具，而且它们不需要超越中学数学的计算。学员将被教授解决涉及小正整数的金字塔问题，然后他们将被挑战将关系扩展到处理大数、分数和负数。为了了解发展趋势，我们将青少年与大学生进行比较。要收集的数据将涉及性能测量、口头协议和功能磁共振脑成像模式的组合。单独的研究将调查个体差异在成功扩展知识方面的基础，元认知投入对成功的影响，以及几何解释在指导关于数学关系的推理中的作用。这项研究将在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。对这一理论的关键考验将是它预测收集到的丰富数据模式的能力。拥有这样的理论框架对于推广金字塔问题的结果，帮助学生更广泛地扩展他们的数学知识将是重要的。这项研究的关键贡献在于，它超越了如何教授特定数学能力的问题，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。放在正式的认知计算模型的背景下，它将把理论从明确定义的程序的学习转移到负责产生新知识的机制上，从而对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，目前许多美国教室都部署了认知导师，覆盖了50多万名学生。项目中开发的计算模型可以过渡到这些导师身上，并将使这些导师教授的能力得到极大的提高。

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.