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Do LLMs Exhibit Coherent Knowledge Structures in Mathematical Reasoning? A Perspective from Knowledge Space Theory
Peng Cui, Heejin Do, Mrinmaya Sachan
TL;DR
The paper addresses whether high-performing LLMs possess coherent, human-like knowledge structures rather than merely producing correct answers. It applies a KST-grounded framework to mathematical reasoning and compares eight LLMs with real human learners. The results show frequent prerequisite violations, weak use of prerequisite context, and inconsistent knowledge structures across models.
Problem
It remains unclear whether LLMs’ strong reasoning accuracy reflects coherent knowledge organized by human-like prerequisite dependencies.
Method
The study uses a KST-grounded normative framework, concept and dependency annotations, and comparisons of eight LLMs with real human learners.
Results
LLMs frequently violate prerequisite dependencies, fail to benefit reliably from prerequisite-grounded context, and show lower structural consistency than human learners.
Takeaways & Limitations
Current LLM knowledge appears fragmented rather than hierarchical, supporting structure-aware assessment as a complement to accuracy-based evaluation.
Takeaways & Limitations
The study focuses on mathematics, depends on expert-defined dependency graphs, and uses question-level correctness as an imperfect proxy for latent concept mastery.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper argues that accuracy-based and local reasoning evaluations miss whether LLM knowledge follows coherent prerequisite dependencies. It proposes a KST-grounded evaluation and finds systematic structural incoherence in mathematical knowledge.
- Motivation and framework: Local trajectory-based evaluations assess problems independently and cannot reveal whether success and failure patterns are globally consistent with knowledge structure.This motivates evaluating dependencies across related questions rather than only final answers or isolated reasoning paths.
- Motivation and framework: KST models knowledge as prerequisite-constrained concepts, enabling evaluation of whether LLM response patterns form coherent knowledge states rather than flat collections of answers.The framework treats mathematics as a domain with well-defined prerequisite relations.
- Research questions: The study asks whether LLMs follow human knowledge dependencies and whether they share a consistent knowledge structure among themselves.These questions define the paper’s two research questions.
- Findings: 72.7% of human learners’ correct answers satisfy prerequisite dependencies, whereas LLMs fail to leverage prerequisite knowledge to scaffold dependent questions.Human learners achieve moderate accuracy of 79.6%, while the LLM comparison indicates weaker human-like structural behavior.
- Findings: LLMs show significantly lower subsumption across performance levels than humans, suggesting that their knowledge acquisition is flatter and less consistently structured.Among humans, stronger learners’ knowledge states consistently subsume those of weaker learners.
- Contributions: The paper contributes a KST-grounded coherence framework, a concept-and-dependency-annotated dataset, and evidence of systematic incoherence in LLM mathematical knowledge.The analysis presents this incoherence as evidence that current models may not engage in genuine formal reasoning.
2 Related Work
The related work positions KST as a formal account of structured learning and contrasts it with evaluations that inspect reasoning locally. This paper extends that perspective to global dependency consistency in LLM behavior.
- Knowledge Space Theory: KST represents each learner by a mastered knowledge state, while the set of feasible states forms a knowledge space constrained by prerequisite relations.It also models learning as transitions between knowledge states, capturing hierarchical and cumulative learning.
- Knowledge Space Theory: Probabilistic KST extensions model response noise through factors such as lucky guesses and careless errors, supporting large-scale empirical assessment.These extensions adapt the deterministic framework to stochastic real-world responses.
- LLM reasoning evaluation: Recent LLM evaluations increasingly inspect reasoning trajectories because correct answers can arise from flawed or unfaithful reasoning chains.However, trajectory evaluations remain focused on isolated problem solving.
- LLM reasoning evaluation: The paper shifts evaluation toward structural dependencies, testing whether LLM behavior is consistent with the latent hierarchy in which complex concepts build on foundational prerequisites.This provides a more global measure of model reliability than superficial or isolated answer evaluation.
3 Framework
The framework uses Knowledge Space Theory to test whether LLM mathematical reasoning follows prerequisite structure and exhibits coherent knowledge dynamics. It constructs question-level dependencies from expert-curated concepts, then evaluates normative behaviors including prerequisite satisfaction, scaffolding, and cross-model knowledge overlap.
- 3 Framework: The framework evaluates whether LLM knowledge structures are coherent through normative behaviors addressing human dependencies and consistency across models.It defines knowledge states from correctly answered questions and measures multiple behavioral properties.
- 3.1 Background of KST: Knowledge Space Theory models knowledge as concepts linked by prerequisite dependencies, with valid knowledge states closed under those prerequisites.A learner cannot master a concept without mastering its prerequisites.
- 3.2 Knowledge Space Construction: The study applies KST to mathematics using New York State standards and their expert-defined concept dependencies.Mathematics provides a structured domain with extensively documented prerequisites.
- 3.2 Knowledge Space Construction: Questions are annotated with relevant concepts, then concept dependencies are used to infer conservative prerequisite relations between questions.The annotation procedure selects concepts from a predefined standards list, using a two-stage cluster-then-concept process.
- 3.3.1 Do LLMs Adhere to Human Knowledge Dependencies? (RQ1): NB1 requires mastery of a question to entail mastery of its prerequisites, measured by Prerequisite Satisfaction Ratio.PSR is computed only for correctly answered questions and captures the proportion of prerequisites also answered correctly.
- 3.3.1 Do LLMs Adhere to Human Knowledge Dependencies? (RQ1): NB2 predicts that prerequisite or same-concept knowledge should improve performance on dependent questions, tested by providing relevant examples through in-context learning.Scaffolding Gain measures the accuracy change when relevant knowledge is supplied in context.
- 3.3.2 Do LLMs Share a Coherent Knowledge Structure? (RQ2): Cross-model coherence is measured with normalized Knowledge Overlap Coefficient, which adjusts overlap for chance agreement on a shared question set.The normalized value ranges from chance-consistent overlap at 0 to perfect subsumption at 1.
4 Experimental Setup
The experiments use real student problem-solving records and concept annotations to compare eight LLMs with human learners under KST-based structural evaluation. Models are tested across prerequisite satisfaction, in-context scaffolding, and knowledge-state overlap.
- The study uses XES3G5M, a mathematics knowledge-tracing dataset derived from real student problem-solving logs with correctness labels and full question content.
- The evaluation spans strong closed-source and open-source LLMs across multiple model families and scales.
- Concept annotation uses a predefined NYS standards concept list, with 400-question manual verification producing 79.3% annotation accuracy.
- The study compares human and LLM conformity to three KST-derived normative behaviors using corresponding structural metrics.
- Scaffolding experiments compare no context, random, same-skill, similarity-based, and prerequisite exemplars, using three in-context examples per condition.
5 Results
Across prerequisite satisfaction, scaffolding, and knowledge-state overlap, LLMs often achieve strong accuracy without matching human-like dependency structure. Human knowledge states are more coherent and ordered, whereas LLM scaffolding benefits and cross-model overlap are inconsistent.
- 5.1 Prerequisite Satisfaction (NB1): Human learners achieve PSR values of 0.936/0.942 at 0.796 accuracy, while LLMs generally show less prerequisite consistency even when matching or exceeding human accuracy.QWEN3-80B-INSTRUCT slightly surpasses humans in micro-PSR but remains lower in macro-PSR.
- 5.1 Prerequisite Satisfaction (NB1): 72.7% of human questions achieve PSR = 1.0, compared with 81.6% for humans versus 55.7% for the best LLM on the verified subset.The verified subset has fewer prerequisites per target question, making perfect prerequisite satisfaction easier to achieve.
- 5.2 Scaffolding Effect (NB2): Prerequisite contexts do not consistently outperform alternatives; QWEN2.5-7B-INSTRUCT records SG=−0.56, while QWEN2.5-32B-INSTRUCT and QWEN3-80B-INSTRUCT obtain only +0.70 and +0.50.Same-skill and semantically similar exemplars usually provide the strongest gains, and random examples often match prerequisite retrieval.
- 5.3 Knowledge-State Overlap (NB3): Human learner groups show high knowledge overlap consistent with stronger states subsuming weaker ones, whereas LLM–human overlap is substantially lower.
- 5.3 Knowledge-State Overlap (NB3): Human learners exhibit near-perfect knowledge subsumption across ability levels, while stronger LLMs can diverge further from human knowledge progression patterns.
6 Conclusion
The paper introduces a KST-grounded framework for evaluating structural coherence in LLM mathematical knowledge. Across eight LLMs and more than 18,000 human learners, high accuracy coexists with fragmented prerequisite structure and no clear prerequisite-scaffolding advantage.
- The framework evaluates LLM knowledge coherence through normative behaviors grounded in Knowledge Space Theory.
- Across eight LLMs and more than 18,000 human learners, the strongest model fully satisfies prerequisites for 48.16% of correct answers, versus 72.7% for humans.
- Prerequisite-grounded context provides no clear advantage over surface-similar baselines, indicating that LLM knowledge is not reliably used as human-like scaffolding.
- The findings characterize current LLM knowledge as fragmented rather than hierarchical and motivate structure-aware assessment alongside accuracy evaluation.
7 Limitations
The framework has three main limitations: it depends on expert-defined concept dependencies, focuses on mathematics, and uses question-level correctness as a proxy for latent concept mastery.
- The framework assumes an expert-defined concept dependency graph, which may be difficult to construct reliably in domains with less explicit knowledge dependencies.
- The study focuses exclusively on mathematics, so whether its observations extend to science, programming, or general factual knowledge remains open.
- Question-level correctness enables large-scale evaluation but is only an imperfect proxy for underlying knowledge states.Future work could estimate concept-level mastery to align the evaluation more closely with KST.
8 Ethical Statement
The study uses publicly available mathematics datasets and benchmark questions for research purposes, without personal or sensitive information.
- The data consist of publicly available mathematics datasets and benchmark questions used only for research purposes.
- The datasets do not contain personal or sensitive information, and AI assistance was limited to language editing and proofreading.
A Prompts
The appendix specifies prompts for concept extraction, LLM evaluation, and scaffolding in-context evaluation, with constrained outputs and step-by-step reasoning instructions.
- LLM Evaluation Prompt: The LLM evaluation prompt supplies reference skills, a question, and its solution for evaluation.
- Concept Extraction Prompt: The concept extraction prompt asks an expert model to select required skills only from a provided reference skill list.
- Concept Extraction Prompt: The extraction prompt requires a JSON array containing selected skill IDs, or an empty list when no listed skill applies.
- Scaffolding In-Context Prompt: The scaffolding prompt provides prerequisite or related problems with solutions before asking the model to solve a target problem.
- Scaffolding In-Context Prompt: The prompts instruct the model to reason step by step and place its final answer within boxed notation, with answer choices supplied for multiple-choice targets.
B Evaluation of LLM-based Concept Annotation on NYS Example Questions
The appendix evaluates LLM-based concept annotation on NYS mathematics examples and illustrates annotation quality on XES questions. The evaluation finds imperfect recall and alignment, while excluding unidentified concepts from later computations.
- Among 480 NYS mathematics concept descriptions, 325 have associated example problems used as ground-truth annotation pairs.
- 192 of 325 instances are correctly annotated, 74 are incorrect, and 59 are unannotated, indicating limitations in annotation recall.
- Questions without identified concepts are discarded, which may reduce the prerequisite relations discovered while prioritizing precision over noisy relations.
- Excluding cases without identified concepts, LLM annotation accuracy reaches 72%, leaving substantial room for improvement.
- XES examples include accurate item–concept alignments as well as annotations that are overly broad or exceed the question’s numerical scope.
C Reasoning Scores and PSR Provide Complementary Views of Capability
Reasoning scores and prerequisite satisfaction (PSR) assess different aspects of model capability. Conventional scores evaluate individual traces, whereas PSR tests consistency with prerequisite relations across questions.
- Reasoning Scores: LLM-as-judge scores each reasoning trace for Relevance, Coherence, and Accuracy.GPT-4.1-MINI is used as the judge.
- Comparison: Among the three strongest models, reasoning scores differ modestly, while the PSR= 1.0 rate remains more variable and non-monotonic.For example, Relevance rises from 4.76 for LLAMA-3.1-70B-INSTRUCT to 4.86 for QWEN2.5-7B-INSTRUCT and 4.93 for QWEN2.5-32B-INSTRUCT.
- Comparison: Figure 6 compares reasoning scores with prerequisite satisfaction across five open-source models.
- Complementary Metrics: Reasoning scores measure local relevance, coherence, and correctness, whereas PSR measures cross-question consistency with prerequisite relations.
- Complementary Metrics: Models with similar conventional reasoning scores may still differ in strict prerequisite satisfaction.PSR therefore serves as a complementary diagnostic of whether response patterns respect prerequisite structure.