Algorithmic foundations of mathematical knowledge

从计数到计算导数，学生掌握的许多数学技能都依赖于定义良好的程序步骤序列或算法。获得特定的算法技能，如计数，小数加法和分数推理，已经被深入研究，作为认知成就本身。然而，尽管算法在早期数学中发挥着关键作用，但人们对算法认知本身知之甚少。例如，我们不知道儿童一般如何发现、记忆和推理算法。该项目旨在促进我们对算法认知的理解，特别是在早期数学方面。也就是说，算法能力并不局限于数学;算法思维在音乐、语言、运动发育和推理等其他领域也无处不在。因此，记忆能力可能是人类最普遍的学习技能的一部分。在创建儿童算法能力的正式帐户时，该研究将促进对认知发展的普遍理解，同时更具体地阐明在STEM学习中发挥作用的认知机制。对儿童算法认知的正式理解将为针对这些系统的学习干预提供理论基础。它将直接揭示特定的表征和算法学习机制，作为干预早期数学发展的关键目标。实验数据和建模工具包将打包供公众使用，以鼓励额外的分析，并尽量减少该领域内的重复工作。该项目将把联合收割机计算方法和行为实验结合起来，让2至8岁的孩子了解算法思维的发展及其与数学思维的关系。它将特别关注长期存在的问题，即幼儿如何掌握计数和理解基数。这些早期的数字技能是后来在早期数学中取得成功的基础，但该领域还没有能够可靠地确定这些技能的核心认知前体。研究人员假设算法和计数技能之间存在双向影响;儿童算法技能的提高将立即先于学习计数，从而支持学习计数，但计数会导致这些算法的修改。这些实验将针对儿童如何对他们观察和推断的算法的行为和结果进行推理，以及他们如何在使用过程中自发地改进算法。算法与记忆系统、核心认知以及物理和概念对象的表征相结合;它们也由人类执行，人类的目标、动机状态和社会背景各不相同，因此必须加以考虑。因此，该项目将坚定地本着STEM学习的综合性解释的精神，强调导致成功获取的多种复杂途径。人工智能中计算建模技术的最新进展已经开发出了如何在特定领域获得算法的正式模型。这项工作展示了人类如何通过推断产生他们观察到的数据的计算过程来理解世界。在这些进展的基础上，该项目将实现早期算法认知的正式理论，这将为未来的实验工作提供可扩展的理论基础。这项工作将告知儿童用于学习数学的概念资源的理论，并对有针对性的干预措施的设计产生影响。该项目由EHR核心研究（ECR）计划资助，该计划支持STEM学习和学习环境的基础研究，扩大STEM的参与，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的评估来支持。影响审查标准。

From counting to computing derivatives, many mathematical skills that students master rely on well-defined sequences of procedural steps, or algorithms. The acquisition of specific algorithmic skills such as counting, small number addition, and fractional reasoning have been intensely studied as cognitive achievements in their own right. However, despite the key role that algorithms play in early mathematics, very little is known about algorithmic cognition itself. It is not known, for example, how children generally discover, remember, and reason about algorithms. This project aims to advance our understanding of algorithmic cognition, particularly with respect to early mathematics. That said, algorithmic abilities are not limited to mathematics; algorithmic thinking is ubiquitous in other domains like music, language, motor development, and reasoning. Algorithmic abilities are thus likely to be part of humans' most general learning repertoire. In creating a formal account of children's algorithmic capacities, the research will advance understanding of cognitive development generally, while illuminating cognitive mechanisms at play in much of STEM learning more specifically. A formalized understanding of children's algorithmic cognition will provide a theoretical foundation for learning interventions targeting these systems. It will directly reveal specific representations and algorithmic learning mechanisms as key targets for intervention in early mathematical development. Experimental data and modeling toolkits will be packaged for public use to encourage additional analysis and minimize duplicated effort within the field.The project will combine computational methods and behavioral experiments with two- to-eight-year-olds to understand the development of algorithmic thinking and its relation to mathematical thinking. It will specifically focus on the long-standing question about how young children come to master counting and understand cardinality. These early numerical skills are foundational for later success in early math, yet the field has not reliably been able to identify the core cognitive precursors of such skills. The researchers hypothesize bidirectional influences between algorithmic and counting skills; that improvement in children's algorithmic skills will immediately precede and thus support learning to count, but that counting leads to revisions of those algorithms. The experiments will target how children reason about the behavior and outcome of algorithms they observe and infer, and how they spontaneously improve algorithms during use. Algorithms interface with memory systems, core cognition, and representations of physical and conceptual objects; they are also executed by human beings whose goals, motivational states, and social contexts vary and must therefore be taken into account. The project will thus be firmly in the spirit of integrative accounts of STEM learning that emphasize the multiple, complex pathways leading to successful acquisition. Recent advances in computational modeling techniques in AI have developed formal models of how algorithms may be acquired in specific domains. This work shows how humans may understand the world by inferring the computational processes that generated the data they observe. Building on these advances, the project will implement a formal theory of early algorithmic cognition which will provide an extensible theoretical foundation for future experimental work. This work will inform theories of the conceptual resources children use to learn mathematics, with implications for the design of targeted interventions.This project is funded by the EHR Core Research (ECR) program, which supports fundamental research on STEM learning and learning environments, broadening participation in STEM, and STEM workforce development.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.