The Extension of Mathematical Knowledge: A Cognitive and Neuroscience Investigation

数学知识的扩展：认知和神经科学研究

• 批准号: 1007945
• 负责人: John Anderson
• 金额: $ 118.57万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007945&HistoricalAwards=false
• 关键词: Extension; Mathematical; Knowledge; Cognitive; Neuroscience; Extension; Mathematical; Knowledge; Cognitive; Neuroscience

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.

数学是人类发明的工具，旨在帮助处理商业，导航，农业和政府以及其他现实世界中的应用。他们经常要求我们将数学知识扩展到所教导的确切过程之外。 许多人已经掌握了他们在学校所教授的数学知识，但是当挑战将这些知识扩展到新情况并经常产生荒谬的答案时，他们仍然表现出严重的困难。这项研究将研究已经掌握了中学数学在内的青少年和大学生，包括分数和负数，并挑战他们扩展这种知识以应对一种新颖的数学概念。这项研究将集中在鲜为人知的数学问题上，称为pyramid问题或斜方形数字，具有自然的遗传学解释，以帮助他们指导他们的理解。 他们对公众的新颖性使它们成为学习的理想选择，并且不需要超越中学数学的计算。 将教会参与者解决涉及小整数的金字塔问题，然后将挑战他们扩展关系以处理大量，分数数量和负数。为了了解发展趋势，将将青少年与大学生进行比较。要收集的数据将涉及绩效指标，口头方案和fMRI脑成像模式的组合。单独的研究将研究成功知识扩展中个体差异的基础，元认知参与对成功的影响以及几何解释在指导有关数学关系的推论中的作用。这项研究将发生在ACT-R理论的理论框架内，ACT-R理论是数学问题解决的计算模型。该理论的关键检验将是它可以预测收集的数据丰富模式的能力。拥有这样的理论框架对于从金字塔问题概括结果来帮助学生更广泛地扩展其数学知识至关重要。这项研究的关键贡献是，它超出了如何教授特定数学能力的问题，并解决了如何使学生准备扩展其数学知识并发现新的数学关系的问题。在正式的认知计算模型的背景下，它将通过将理论移动到对定义明确的程序的学习到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。这项研究将在当前部署在许多美国教室中的认知导师的背景下进行，达到50万名学生。项目中开发的计算模型可以转变为这些导师，并能够在这些导师教授的各种能力方面做出重大增强。