SHAPE of Thinking: Mathematical Heuristics for Analyzing and Improving LLM Reasoning

Jonghyun Song; Sangjun Song; Minjae Oh; Haesung Pyun; Sungsik Lee; Yohan Jo

cs.AIJun 28, 2026

SHAPE of Thinking: Mathematical Heuristics for Analyzing and Improving LLM Reasoning

Authors: Jonghyun Song, Sangjun Song, Minjae Oh, Haesung Pyun, Sungsik Lee, Yohan Jo

Organizations: Graduate School of Data Science, Seoul National University, Seoul, Korea · Department of Mathematics Education, Seoul National University, Seoul, Korea

Abstract

Large language models (LLMs) achieve strong performance on mathematical reasoning benchmarks, yet final-answer accuracy reveals little about the strategies they use. We introduce SHAPE (Structured Heuristic Analysis for Problem-solving Evolution), a framework grounded in mathematics-education research that characterizes local mathematical actions in reasoning trajectories using a problem-independent heuristic taxonomy. Using SHAPE, we examine two previously hypothesized phenomena in LLM reasoning. First, we revisit the hypothesis that LLMs reuse learned solution techniques under problem perturbations. Across six models on MATH-Perturb, perturbations requiring a different solution approach yield lower accuracy but greater heuristic shifts than those preserving it. Second, we characterize mode-seeking in RLVR through mathematical heuristic profiles. Across four base--RL pairs, successful RL solutions overlap substantially with base-model profiles. We further introduce Heuristic-Augmented Plan-GRPO (HA-Plan-GRPO), which guides RLVR planning through heuristic selection and applicability reasoning. Across three models and five evaluation sets, it improves mean Avg@16 by up to 7.0% absolute and mean Pass@16 by up to 2.4% absolute over planning-only RLVR. Our code is available at https://github.com/holi-lab/shape-of-CoT.

Figures & tables

Figure 1: Illustrative SHAPE annotations on a synthetic CoT trajectory. Each row pairs a content unit with its heuristic or non-heuristic labels. Unit 2 illustrates multi-label annotation with H9a (exploring particular cases) and H11f (reflection on rigor), while unit 4 receives the non-heuristic final-answer label N4.

Code	Category
H1	Changing the register of semiotic representation
H2	Cognitive reinterpretation
H3	Introduce symbolic representation, formalization, and structural augmentation
H4	Problem classification / rephrase the problem and goal
H5	Wishful thinking (simplify / reduce the problem and conditions)
H6	Explicit case analysis, decompose into subproblems

Table 1: Heuristic and non-heuristic categories used in SHAPE .

Figure 2: Overview of the automated SHAPE annotation pipeline. Given a question and a chain-of-thought (CoT) trace, the pipeline segments the trace into content units $1,\ldots,k$ and assigns one or more heuristic or non-heuristic labels to each unit. Both stages use Grok or Qwen as annotators.

Annotator	F1 (%)	Gwet’s AC1
Grok-4.1-Fast	77.16	0.909
GPT-5	71.56	0.859
GPT-5-mini	66.13	0.850
Gemini-3-Flash	65.11	0.860
Gemini-3.1-Flash-Lite	71.22	0.890
Qwen3.5-27B	70.44	0.894

Table 2: Annotator model agreement with the human gold standard.

Model	Pass@1 (%)			$D_{\mathrm{JS}}$ (Full)		$D_{\mathrm{JS}}$ ( $\leq$ 5)
	Original	Simple	Hard	O–S	O–H	O–S	O–H
Qwen3-8B	96.0	94.0	84.0	.043	$.066^{*}$	.160	$.190^{*}$
Qwen3-32B	92.0	90.0	76.0	.047	$.070^{*}$	.150	$.200^{*}$
Nemotron-Cascade-8B	92.0	95.0	71.0	.049	$.072^{*}$	.170	$.210^{*}$
Olmo-3-7B-Think-RLVR	97.0	94.0	76.0	.041	$.065^{*}$	.165	$.205^{*}$
Qwen2.5-7B (SimpleRL-Zoo)	61.7	58.3	31.3	.152	$.253^{*}$	.158	$.254^{*}$

Table 3: Pass@1 and heuristic-frequency divergence on MATH-Perturb. O, S, and H denote original, simple, and hard problems.

D_{\mathrm{JS}}

is Jensen–Shannon divergence between frequency distributions over eleven heuristic families, using base-2 logarithms. Full uses the entire trajectory;

\leq

5 uses the first five content units, or the entire trajectory when fewer units are available. Means are computed over matched problems with valid O/S/H distributions, separately for each condition. ∗ denotes O–H

>

O–S in a one-sided paired Wilcoxon signed-rank test (uncorrected

p<.05

). For most SimpleRL-Zoo trajectories,

\leq

5 includes the entire CoT.

Reference	Target	$N_{\text{ref}}$	$N_{\text{target}}$	Density	Coverage
Post-trained
Qwen3-1.7B-Base	Qwen3-1.7B-GRPO	834	886	1.219	0.871
Olmo-3-7B	Olmo-3-7B-Think-RL-Zero	1229	1307	1.250	0.707
Qwen2.5-7B	Qwen2.5-7B-SimpleRL-Zoo	1280	1294	1.511	0.888
DeepSeek-Math-7B-Base	DeepSeek-Math-7B-SimpleRL-Zoo	463	500	1.525	0.950
Cross-model baselines

Table 4: Density and Coverage for base–post-trained pairs and cross-model baselines in the original annotation-frequency space (

k=3

, cosine distance, successful trajectories only, aggregated across all perturbation conditions). Density measures target overlap with base neighborhoods; Coverage is the fraction of base neighborhoods containing at least one target trajectory. The left model defines the reference neighborhoods;

N

counts successful trajectories.

Figure 3: Pooled successful heuristic-frequency profiles of Olmo-3-7B and Olmo-3-7B-Think-RL-Zero projected onto the first principal component (PC1). The RL distribution has a sharper peak near the base-model mode and a reduced left tail.

Method

Original

Simple

Hard

MATH500

Minerva

Mean

Avg.

Pass

Avg.

Pass

Avg.

Pass

Avg.

Pass

Avg.

Pass

Avg.

Pass

DeepSeek-Distill-Llama-8B

Base

58.5

80.9

56.7

81.7

41.3

67.0

62.3

81.8

15.6

34.2

46.9

69.1

Plan-GRPO

54.4{\scriptstyle\pm 6.2}

81.7{\scriptstyle\pm 1.7}

52.1{\scriptstyle\pm 7.0}

83.2{\scriptstyle\pm 2.7}

35.5{\scriptstyle\pm 4.7}

68.1{\scriptstyle\pm 4.3}

58.8{\scriptstyle\pm 5.4}

82.9{\scriptstyle\pm 1.8}

13.0{\scriptstyle\pm 1.6}

32.6{\scriptstyle\pm 2.9}

42.8

69.7

HA-Plan-GRPO

\mathbf{63.6}{\scriptstyle\pm 1.8}

\mathbf{84.6}{\scriptstyle\pm 2.7}

\mathbf{61.6}{\scriptstyle\pm 0.9}

\mathbf{84.9}{\scriptstyle\pm 2.2}

\mathbf{42.4}{\scriptstyle\pm 1.9}

\mathbf{68.4}{\scriptstyle\pm 1.3}

\mathbf{66.1}{\scriptstyle\pm 1.6}

\mathbf{86.6}{\scriptstyle\pm 2.4}

15.2{\scriptstyle\pm 1.6}

\mathbf{35.8}{\scriptstyle\pm 2.8}

49.8

72.1

Qwen3-1.7B-Base

Table 5: Avg@16 and Pass@16 (%) across five evaluation sets. Trained variants report mean

\pm

standard deviation across three training seeds; Base results are point estimates. Mean columns average the five evaluation sets with equal weights. Light gray rows denote HA-Plan-GRPO (ours). Bold indicates the highest mean within each model, evaluation set, and metric.

Figure 4: Paraphrased evaluation-time solutions from Qwen3.5-4B checkpoints trained with Plan-GRPO and HA-Plan-GRPO. Both models generate the response to the same zero-shot CoT prompt without heuristic guidance at inference.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Strategy Name	Description & Sources
H1. Changing the register of semiotic representation	This strategy involves translating the problem’s representation from one semiotic register to another. It includes converting between natural language, algebraic, geometric, and visual representations to facilitate understanding or solving. Sources: Creating a model (K); Change the semiotic representation register (Changer de registre de représentation sémiotique) (F)
H2. Cognitive Reinterpretation	This involves changing the way an object or property in the problem is interpreted. It redefines the identity or attributes of an element in a way different from the initial presentation, without necessarily changing the register. Sources: Consider another way of interpreting the problem’s objects (Envisager une autre façon d’interpréter les objets du problème) (F); Adopting a different point of view (P&K)
H3. Introduce Symbolic Representation, Formalization, and Structural Augmentation, including…
H3a. Introduce Symbolic Representation and Formalization	The act of introducing new variables, labeling unknowns, or performing substitutions to make ambiguous targets operationally manageable. Sources: Notation; Setting up equations (P); Creating a model (K); Introduce names or notations (Introduire des noms ou des notations) (F)
H3b. Structural Augmentation	Constructing auxiliary objects, lemmas, or entirely new mathematical frameworks that are not present in the original problem. This is a creative addition to the problem space, such as drawing auxiliary lines, defining new functions, or shifting the problem into a new structural representation Sources: Auxiliary elements, Lemma (P); Introducing an auxiliary element (K); Introduce auxiliary elements (Introduire des éléments auxiliaires) (F)
H4. Problem Classification / Rephrase the Problem and Goal, including…

Appendix

Table 10

Strategy Name	Description & Sources
H6. Explicit Case Analysis, Decompose into Subproblems	Logically decomposing the problem into distinct cases, non-overlapping subsets, or sub-problems that, when combined, yield the full solution. The cases should ideally be exhaustive and mutually exclusive. Sources: Decomposing and recombining (P); Décomposer le domaine du problème et travailler cas par cas (Decompose the problem domain and work case by case) (F)
H7. Arguing by contradiction	A proof strategy where the negation of the proposition is assumed to derive a contradiction, thereby proving the original statement. Sources: Reductio ad absurdum and indirect proof (P); Arguing by contradiction (K)
H8. Analogy and Presenting Related Theorems, including…
H8a. Analogy	Recalling previously solved problems, known methods, or applying a recently established logical procedure to a new target within the same problem. This involves recognizing structural similarities and transferring a strategy from one context (or one part of the equation) to another. Sources: Have you seen it before?; Do you know a related problem? (P); Activating a previous experience (K); Make a connection with a previously encountered problem (Faire le lien avec un problème déjà rencontré) (F); Solving a simpler analogous problem (P&K)
H8b. Presenting Related Theorems, Tools, or Properties	Introducing specific mathematical theorems, formulas, identities, or properties that are not provided in the problem statement but are necessary to advance the solution. Sources: Connect with a mathematical tool (theorem, property) (Faire le lien avec un outil mathématique (théorème, propriété)) (F)
H9. Experimental and Pattern Exploration, including…

Appendix

Table 6: Taxonomy of Problem-Solving Strategies. This table summarizes the code, strategy name, detailed description, and theoretical sources associated with each heuristic. Here, P = Pólya (1945) , K = Koichu et al. (2007) , F = Favier (2022) , and P&K = Posamentier and Krulik (2008) .

Name	Description
N1. Literal Repetition	Merely repeating the problem statement or giving a generic opening/closing without mathematical substance or added insight.
N2. Technical Performance	Routine calculations or simple algebraic manipulations that do not involve strategic planning or new insight; “doing the math” after the plan is set.
N3. Alien Statements	Unrelated thoughts or off-topic remarks.
N4. Answer	Stating the final answer.

Appendix

Table 7: Non-heuristic categories capturing reasoning steps that do not contribute strategic insight, including repetition, routine computation, irrelevant statements, and final answer reporting.

Solver	$n$	Qwen	Grok
Nemotron-Cascade 8B	115	$0.019$ $[0.008,0.029]$	$0.023$ $[0.014,0.032]$
Olmo-3-7B Think-RLVR	115	$0.022$ $[0.012,0.033]$	$0.023$ $[0.015,0.032]$
Qwen3-8B	107	$0.024$ $[0.014,0.035]$	$0.022$ $[0.014,0.032]$
Qwen3-32B	113	$0.021$ $[0.009,0.032]$	$0.022$ $[0.014,0.031]$

Appendix

Table 8: Mean full-trajectory hard-minus-simple JSD contrasts under Qwen3.5-27B and Grok-4.1-Fast. Brackets contain problem-bootstrap 95% confidence intervals;

n

is the number of seed problems with complete matched O/S/H triples.

Reference	Target	$N_{\text{ref}}$	$N_{\text{target}}$	$k=3$		$k=5$		$k=10$
				Density	Coverage	Density	Coverage	Density	Coverage
Post-trained
Qwen3-1.7B-Base	Qwen3-1.7B-GRPO	834	886	1.219	0.871	1.175	0.952	1.157	0.999
Olmo-3-7B	Olmo-3-7B-Think-RL-Zero	1229	1307	1.250	0.707	1.272	0.806	1.272	0.897
Qwen2.5-7B	Qwen2.5-7B-SimpleRL-Zoo	1280	1294	1.511	0.888	1.339	0.964	1.215	0.989
DeepSeek-Math-7B-Base	DeepSeek-Math-7B-SimpleRL-Zoo	463	500	1.525	0.950	1.338	0.989	1.248	1.000

Appendix

Table 9: Neighborhood-size sensitivity using the same successful trajectory pools and closed cosine neighborhoods as Table 4 .

N_{\text{ref}}

and

N_{\text{target}}

are fixed across

k

; Density and Coverage are reported separately. The final row adds the Olmo Think-RLVR checkpoint.

Parameter	Value
Training Batch Size	32
Rollouts per Prompt	4
Number of GPUs	2 $\times$ B200
Optimizer	AdamW
Temperature	1.0
Top- $p$	1.0

Appendix

Table 10: Hyperparameters used for GRPO training.

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Knowing What to Solve Before How: Preplan Empowered LLM Mathematical Reasoning

Shaojie Wang, Liang Zhang

Hong Kong University of Science and Technology (Guangzhou)

LLM Planning Structured Reasoning LLM Mathematical Reasoning

Open paper AskPDF arXiv

Current plan-based reasoning methods improve large language models (LLMs) by inserting a planning stage before execution, giving rise to the question

\rightarrow

plan

\rightarrow

cot paradigm. While effective, a closer examination reveals an inherent paradigm-level gap: both the planning and its execution stages decide how to solve a problem, while the prior question of what to solve; recognizing the problem type, the applicable tools, and the foreseeable pitfalls; remains entirely implicit. To bridge this gap, we propose PPC (Preplan-Plan-CoT), a framework that introduces an explicit problem-understanding stage, the preplan, yielding a new question

\rightarrow

preplan

\rightarrow

plan

\rightarrow

cot paradigm. Realizing this paradigm requires safeguarding the conceptual integrity of preplan at both ends. Specifically, we design a three-stage synthesis pipeline with a spoiler-score detector that filters out leakage and spoiler failures to build clean preplan supervision, and a composite GRPO reward enforces that the generated plan genuinely follows from the preplan. Experiments across four backbones and five mathematical reasoning benchmarks show that PPC achieves the best results on 39 of 40 metrics, improving maj@16 and pass@16 by +2.23 and +3.06 over the strongest baseline without introducing additional inference token overhead.

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