relational mapping
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How Children Discover Mathematical Structure through Relational Mapping
A core question in human development is how we bring meaning to conventional symbols. This question is deeply connected to understanding how children learn mathematics—a symbol system with unique vocabularies, syntaxes, and written forms. In this talk, I will present findings from a program of research focused on children’s acquisition of place value symbols (i.e., multidigit number meanings). The base-10 symbol system presents a variety of obstacles to children, particularly in English. Children who cannot overcome these obstacles face years of struggle as they progress through the mathematics curriculum of the upper elementary and middle school grades. Through a combination of longitudinal, cross-sectional, and pretest-training-posttest approaches, I aim to illuminate relational learning mechanisms by which children sometimes succeed in mastering the place value system, as well as instructional techniques we might use to help those who do not.
Abstract Semantic Relations in Mind, Brain, and Machines
Abstract semantic relations (e.g., category membership, part-whole, antonymy, cause-effect) are central to human intelligence, underlying the distinctively human ability to reason by analogy. I will describe a computational project (Bayesian Analogy with Relational Transformations) that aims to extract explicit representations of abstract semantic relations from non-relational inputs automatically generated by machine learning. BART’s representations predict patterns of typicality and similarity for semantic relations, as well as similarity of neural signals triggered by semantic relations during analogical reasoning. In this approach, analogy emerges from the ability to learn and compare relations; mapping emerges later from the ability to compare patterns of relations.
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