Schoenfeld's Anatomy of Mathematical Reasoning by Language Models

Schoenfeld’s Anatomy of Mathematical Reasoning by Language Models Ming Li*, Chenrui Fan*, Yize Cheng*, Soheil Feizi, Tianyi Zhou University of Maryland, College Park {minglii,cfan42,yzcheng,sfeizi}@umd.edu, tianyi.david.zhou@gmail.com ‡ Project: https://github.com/MingLiiii/ThinkARM Abstract Large language models increasingly expose reasoning traces, yet their underlying cogni- tive structure and steps remain difficult to iden- tify and analyze beyond surface-level statis- tics. We adopt Schoenfeld’s Episode Theory as an inductive, intermediate-scale lens and intro- duce ThinkARM (Anatomy of Reasoning in Models), a scalable framework that explicitly abstracts reasoning traces into functional reasoning steps such as Analysis, Explore, Im- plement, Verify, etc. When applied to math- ematical problem solving by diverse models, this abstraction reveals reproducible thinking dynamics and structural differences between reasoning and non-reasoning models, which are not apparent from token-level views. We further present two diagnostic case studies showing that exploration functions as a criti- cal branching step associated with correctness, and that efficiency-oriented methods selectively suppress evaluative feedback steps rather than uniformly shortening responses. Together, our results demonstrate that episode-level represen- tations make reasoning steps explicit, enabling systematic analysis of how reasoning is struc- tured, stabilized, and altered in modern lan- guage models. 1 Introduction Large language models (LLMs) have demonstrated remarkable performance on complex reasoning tasks (OpenAI, 2024c; Marjanovi´c et al., 2025; Comanici et al., 2025; Qwen Team, 2025a), partic- ularly when generating explicit chains of thought (CoT) (Wei et al., 2023). Consequently, evaluation of reasoning models has predominantly focused on outcome-oriented metrics such as accuracy, solu- tion length, or aggregate token counts (Lightman et al., 2023; Jiang et al., 2025a). While these mea- sures are effective for comparing final performance, they provide limited insight into how these models *Co-First Authors. organize their reasoning traces and how different reasoning behaviors emerge across models. It is still unclear which parts of a generated chain- of-thought correspond to problem understanding, exploration, execution, or verification, making it difficult to interpret model behavior beyond sur- face statistics. This opacity is particularly salient when studying “overthinking” (Chen et al., 2025b; Fan et al., 2025; Kumar et al., 2025), where longer or more elaborate reasoning does not necessarily translate into improved correctness (Feng et al., 2025b), yet the underlying thinking dynamics and structure are hard to characterize systematically. Various reasoning behaviors that are often dis- cussed conceptually, e.g., abstract planning versus concrete execution, open-ended exploration versus evaluative checking, lack rigorous formulation and quantitative comparison across models. To better interpret such reasoning traces, a nat- ural question is whether they contain meaning- ful structure at an intermediate level of abstrac- tion beyond individual tokens. Motivated by prior work (Li et al., 2025c) that introduced Schoenfeld’s Episode Theory (Schoenfeld, 1985) as a framework for characterizing problem-solving behaviors, we adopt episode-level representations as an inductive lens for analyzing LLM reasoning traces. Episode Theory conceptualizes problem solving in terms of functional episodes, thereby providing an in- terpretable intermediate-scale representation that bridges low-level token statistics and high-level reasoning intents. A condensed example is shown in Figure 1. While earlier work (Li et al., 2025c) first lever- aged this theory to reasoning models, their scope is limited to one model and one dataset. Thus, it remains unclear whether such an episode-level ab- straction can reveal systematic, reproducible struc- ture in LLM reasoning at scale. In this work, we build upon this foundation and extend episode-level annotation to a broader, comparative setting, us- 1 arXiv:2512.19995v1 [cs.CL] 23 Dec 2025 Let 𝑓𝑟= ∑ ! "! #$$% "&# = ! #! + ! '! + ⋯+ ! #$$%! . Find ∑ 𝑓(𝑘) ( )&# 1. Read 2. Analyze 3. Plan 4. Implement 5. Explore 6. Verify 7. Answer 8. Monitor Question: Categories: ❶ ❶ ❶ ❽ ❸ ❸ ❷ ❷ ❷ ❷ ❷ ❺ ❻ ❻ ❻ ❼ ❼ ❹ ❽ ❸ Figure 1: A condensed example of a reasoning trace that is annotated in our framework. Each sentence in response is tagged with one of the eight episode categories. ing it to examine what principal patterns emerge when diverse reasoning traces are viewed through a theory-grounded abstraction. Our study is orga- nized around three research questions. RQ1: Do reasoning traces exhibit consistent linguistic and dynamic structure when viewed through episode- level representations? RQ2: How do reasoning dynamics differ across models and reasoning styles, as indicated by episode sequencing and transition patterns

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