用类大模型架构重做儿童数学策略理论，模拟算术学习过程。
A Small Math Model: Recasting Strategy Choice Theory in an LLM-Inspired Architecture
- 基于类LLM的神经网络重构儿童算术策略理论
- 再现计数与加法间的建设性/破坏性干扰现象
- 适合研究数学推理如何在智能体中自然涌现
策略选择理论（SCT）解释了儿童算术学习的关键机制，包括从发展自然数据中学习、概率表征、基于信心的检索，以及支架策略（如数手指）的阶段性重要性。本文将SCT重构为一个‘小数学模型’（SMM），采用类似大语言模型的神经网络架构，引入计数练习、数字嵌入和门控注意力机制。与早期工作一致，SMM再现了计数与加法之间的建设性与破坏性干扰，并表现出随着求和回忆能力提升，数手指行为呈波浪式使用模式。未来计划将SMM扩展至更长期的SCT研究方向，包括自适应策略选择乃至策略发现，为探究数学推理能力在基于大模型的智能体中如何自然形成提供统一平台。
原文摘要 · Abstract (English)
Strategy Choice Theory (SCT; Siegler and Shrager, 1984; Siegler, 2000) 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 practice, 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.
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