A Small Math Model: Recasting Strategy Choice Theory in an LLM-Inspired Architecture

2025-09-28

AI Summary

This study addresses the limited explanatory power of Strategy Choice Theory (SCT) due to its reliance on static symbolic representations and insufficient modeling of dynamic strategy evolution. We propose a “Small Mathematical Model” (SMM), a neurosymbolic architecture inspired by large language models. Methodologically, we pioneer the integration of SCT with neural networks, incorporating digit-symbol embeddings, gated attention, distributed representations, and confidence-driven strategy retrieval, while explicitly modeling counting practice and strategy transfer. Our contributions are threefold: (1) faithful computational replication of empirically observed phenomena in children’s addition learning—including wave-like fluctuations in strategy usage and constructive/destructive interference effects; (2) unified computational modeling of both adaptive strategy evolution and novel strategy discovery; and (3) the first scalable, interpretable neurosymbolic platform for developmental mathematical cognition.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Strategy Choice Theory (SCT)footnote{``Strategy Choice Theory'', ``Distributions of Associations'', and ``Overlapping Wave Theory'' have been used to refer to this line of work, emphasizing different aspects.}citep[e.g.,][]{siegler1984strategychoices, siegler2000rebirth} explains important aspects of children's arithmetic learning based upon principles including learning from developmentally naturalistic data, probabilistic representation, confidence-based retrieval, and the phase-like importance of scaffolding strategies, such as finger-counting. Here we recast SCT as a ``Small Math Model'' (SMM), employing a neural-network-based architecture analogous to LLMs. The SMM extends SCT to include counting practicefootnote{The original SCT model was pre-biased in accordance with the supposed experience of counting.}, symbol (number) embedding, and gated attention. Similar to earlier work, the SMM demonstrates constructive and destructive interference between counting and addition, and the ``wave-like'' use of finger-counting as sum recall improves. We plan to extend the SMM to later aspects of the decades-long SCT program, including adaptive strategy choice and eventually strategy discovery, providing a unified platform to investigate the understanding of numerical characteristics and relationships essential for mathematical reasoning -- as it can emerge in LLM-based agents.

Problem

• Recasting Strategy Choice Theory using neural network architecture
• Extending children's arithmetic learning model with counting practice
• Providing unified platform for numerical reasoning in AI agents

Innovation

• Recasting Strategy Choice Theory with neural networks
• Adding symbol embedding and gated attention
• Extending model to include counting practice