Do LLMs Exhibit Coherent Knowledge Structures in Mathematical Reasoning? A Perspective from Knowledge Space Theory

Peng Cui; Heejin Do; Mrinmaya Sachan

cs.AISep 4, 2026

Do LLMs Exhibit Coherent Knowledge Structures in Mathematical Reasoning? A Perspective from Knowledge Space Theory

Authors: Peng Cui, Heejin Do, Mrinmaya Sachan

Organizations: ETH Zürich Department of Computer Science · ETH AI Center

Abstract

Human knowledge is inherently structured and interdependent: mastery of a concept requires prior mastery of its prerequisites, a principle formalized by Knowledge Space Theory (KST). While LLMs achieve strong performance on complex reasoning tasks, it remains unclear whether they exhibit coherent, human-like knowledge structure. We introduce a KST-grounded framework for evaluating LLM knowledge structure in mathematical reasoning, using it as a normative framework to analyze whether LLM behavior adheres to principled knowledge dependencies. Evaluating eight open- and closed-source LLMs against real human learners, we find that (1) LLMs do not adhere to human knowledge structure -- they frequently violate knowledge dependencies and fail to leverage related knowledge provided in context to improve performance on dependent questions; (2) LLMs do not share a consistent knowledge structure among themselves, as reflected by low overlap in their knowledge distributions. Furthermore, these structural deficiencies remain largely invisible to accuracy-based and LLM-as-judge evaluations. Together, our results provide behavioral evidence that current LLMs knowledge does not follow a human-like structure.

Figures & tables

Figure 1: Conventional evaluation measures average accuracy, treating knowledge as a flat unstructured collection. We propose a KST-based framework that models structured dependencies in knowledge, examining whether LLM exhibits coherent knowledge behavior.

Figure 2: (a) Concept Annotation : For each question, we prompt an advanced LLM to identify the relevant mathematical concepts from an expert-curated concept map. (b) Dependency Induction : We use the question-concept associations together with expert-defined concept dependencies to infer prerequisite relations between questions. (c) Behavioral Analysis of LLMs : We assess whether the knowledge structure and dynamics of LLMs are coherent with respect to the constructed knowledge space through a list of normative behaviors.

Interactions	# Students	$18,066$
Interactions	# Questions	$4,118$
Concepts	# Concepts	$480$
Concepts	# Concepts / question (avg.)	$1.07$
Dependency	# Prerequisites / concept (avg.)	$1.58$
Dependency	# Prerequisites / question (avg.)	$16.4$

Table 1: Statistics of the XES3G5M dataset and extracted concepts.

Model	Acc.	Avg. PSR		Question-level PSR distribution
Model	Acc.	Micro	Macro	$\mathbf{[0,0.2)}$	$\mathbf{[0.2,0.4)}$	$\mathbf{[0.4,0.6)}$	$\mathbf{[0.6,0.8)}$	$\mathbf{[0.8,1.0)}$	$\mathbf{=1.0}$
Mistral-7B-v0.3	0.217	0.299	0.353	30.25%	30.25%	22.22%	8.23%	0.82%	8.23%
LLAMA-3.1-8B-Instruct	0.559	0.638	0.618	7.21%	4.68%	29.09%	37.53%	8.83%	12.66%
LLAMA-3.1-70B-Instruct	0.716	0.790	0.748	5.23%	2.10%	9.20%	32.17%	30.49%	20.81%
Qwen2.5-7B-Instruct	0.771	0.828	0.820	2.12%	0.74%	5.33%	25.83%	38.66%	27.32%
Qwen2.5-32B-Instruct	0.849	0.862	0.817	3.69%	0.95%	5.86%	19.74%	44.33%	25.44%

Table 2: The accuracy and PSR results for all LLMs and human learners on all questions annotated by (XES full ). For question-level PSR (Eq. 2 ), we report the proportion falling within different intervals. For aggregate PSR, we report both micro (Eq. 4 ) and macro (Eq. 3 ) results. For Acc, the proportion of questions with

\text{PSR}=1.0

, and averaged PSR, we bold the best overall result and underline the best result among LLMs.

Model	Acc.	PSR
Model	Acc.	Micro	Macro	=1.0
Mistral-7B-v0.3	0.217	0.325	0.387	0%
LLAMA-3.1-8B-Inst.	0.560	0.616	0.614	5.63%
LLAMA-3.1-70B-Inst.	0.717	0.826	0.789	32.1%
Qwen2.5-7B-Inst.	0.771	0.822	0.834	21.3%
Qwen2.5-32B-Inst.	0.849	0.848	0.802	22.1%

Table 3: Accuracy and PSR results on the verified subset XES verified . We bold the best overall result and underline the best result among LLMs.

Figure 3: Knowledge scaffolding across various context types on the common subset.

Figure 4: Normalized knowledge overlap coefficients (Eq. 7 ) between the knowledge states of different LLMs and of different human learner groups. Models are ordered by performance from weaker to stronger within both the LLM and human groups.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

ID	Question	LLM-annotated concepts
1	Distance between A and B is 350 km. A car leaves from A at 8:00 with speed 40 km/h toward B. After 2 hours, another car leaves from B toward A at 50 km/h. At what time do the two cars meet on the road? ( )	Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
2	A rectangle with perimeter $20$ meters and length $7$ meters; what is its width ( ) meters?	Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
3	From the numbers $5$ , $6$ , and $7$ , choose two numbers to form the largest two-digit number ( ).	Understand that the two digits of a two-digit number represent amounts of tens and ones.
4	There are $8$ boxes of apples, with $40$ apples per box. If evenly distributed among $4$ classes, how many per class?	Use multiplication and division within 100 to solve word problems involving equal groups, arrays, and measurement quantities, for example by using drawings and equations with a symbol representing the unknown number.
5	A fraction with denominator 6 and numerator 5 is ____.	Understand a fraction 1/b as the quantity formed by one part when a whole is divided into b equal parts, and understand a fraction a/b as the quantity formed by a parts of size 1/b.
6	On one side of a road that is $24$ meters long, plant a tree every $4$ meters, with neither end planted. How many trees can be planted along this road?	Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end. Understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to cases where the object is measured by a whole number of length units with no gaps or overlaps.

Appendix

Table 4: Examples of questions and their LLM-annotated concepts on the XES dataset.

Figure 5: Raw knowledge overlap coefficients (Eq. 6 ) between the knowledge states of different LLMs and human learner groups.

Figure 6: Comparison of reasoning scores (a) and prerequisite satisfaction (b) across five open-source models.

Reasoning Consistency in Language Models

Reasoning consistency in language models measures whether conclusions and reasoning remain reliable across equivalent prompts, samples, steps, and contexts.

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When Does Structured Knowledge Help Neural Theorem Proving?

Sareh Nabi, Roland Vogl, Marzieh Nabi

Stanford University

Automated Theorem Proving Knowledge Augmentation for Language Models LLM Reasoning with Graphs Lean Theorem Proving Large Language Model-Guided Theorem Proving

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Does structured mathematical knowledge help LLMs prove theorems in Lean 4? If so, for which models, and does the answer vary by problem? Formal libraries such as Mathlib encode 285,000+ verified theorems with syntactic dependencies, but the semantic layer mathematicians rely on for discovery (analogies, generalizations, cross-domain bridges) remains implicit. We introduce MathAgent, which builds this layer as a knowledge graph, MathKG, and uses it to augment LLM theorem provers. MathKG connects 364 Mathlib theorems and definitions by 9,434 typed semantic edges inferred via LLM-based relation extraction anchored to verified Mathlib declarations. We run a controlled ablation across four augmentation modes (no context, knowledge-graph context, Mathlib retrieval, both) and five models: Qwen3-8B/32B, their Lean-specialized derivatives Goedel-Prover-V2-8B/32B, and Claude Sonnet 4.6, on miniF2F, plus PutnamBench and MathOlympiadBench for Sonnet. Three findings emerge. (i) Specialization dominates augmentation: Lean fine-tuning adds 33-38 percentage points of solve rate in every mode, and a specialized 8B model beats a

4\times

larger general one by 29-35 points, while no augmentation mode improves solve rate by more than 3 points. (ii) Augmentation is capability-conditioned: knowledge-graph context helps small models but hurts large ones, with the specialized model gaining more relative to its general base at every scale. (iii) Yet the augmentation modes solve different problems: an oracle selecting the best mode per problem solves 6% to 58% more than the unaugmented prover, a complementarity effect that strengthens on harder problems (32% more on PutnamBench). These results motivate adaptive strategies that select augmentation by model capability and problem. Code, data, and artifacts are available at https://github.com/sarehnabi/mathagent

Abstract

Reasoning Consistency in Language Models

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