Algorithmic foundations of mathematical knowledge

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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.