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REVIEW 4 major objections 5 minor 15 references

CogMath: Assessing LLMs' Authentic Mathematical Ability from a Human Cognitive Perspective

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Standard math scores overstate AI competence; a nine-dimension cognitive probe cuts pass rates by 30–40%.

desk verdict Useful diagnostic framework, but the '30-40% overestimation' headline is an artifact of an unvalidated all-nine-dimension AND rule rather than a measured fact about LLM ability. read the letter →

arxiv 2506.04481 v1 pith:7AIMDSR3 submitted 2025-06-04 cs.AI

Jiayu Liu , Zhenya Huang , Wei Dai , Cheng Cheng , Jinze Wu , Jing Sha , Song Li , Qi Liu
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Shijin Wang Enhong Chen
This is my paper
classification cs.AI
keywords largelanguagemodelsmathematicalreasoningevaluationcognitivestagescounterfactualprobesmulti-agentknowledgeredefinitionbackwardbenchmarkoverestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

CogMath claims that the accuracy figures reported on standard math benchmarks overstate how well large language models actually understand mathematics. The paper formalizes human problem-solving into three stages—problem comprehension, problem solving, and solution summarization—and builds nine evaluation dimensions around them, from paraphrasing and scrambled-sentence detection to knowledge redefinition and backward reasoning. A model is counted as having mastered a problem only when it passes the original question plus all nine probes. Measured this way, seven mainstream LLMs lose 30–40 percentage points relative to vanilla accuracy on MATH, GSM8K, and a new K-12 exam set, which is the central evidence that current benchmark numbers are inflated. The stakes are practical: benchmark leaderboards are mis-estimating the capability gap that appears when a model has to actually use, adapt, and check its mathematical knowledge.

What carries the argument

The machinery is the Inquiry–Judge–Reference probe generator combined with an all-nine-pass gate. For each problem, an Inquiry agent writes a dimension-specific probe (paraphrase, disruption, missing condition, redundant condition, analogical variant, numerical variant, redefined knowledge, intermediate-step question, or backward masked-value question); a Judge agent checks and iteratively fixes the probe up to ten rounds; and a Reference agent produces the gold answer, except where the expected response is 'unsolvable'. The final pass rate is the fraction of problems on which the model answers the original correctly and passes every one of the nine probes, so the conjunction itself is what converts a coarse accuracy number into a cognitive-mastery claim.

What would settle it

Run the nine CogMath probes on mathematically trained humans who solve the original problems correctly. If most of them also fail at least one dimension—for example, by not answering 'unsolvable' to a disrupted sentence, or by resisting a redefined 'half' or area formula—then the all-nine criterion does not define authentic mastery and the 30–40% drop is an artifact of probe strictness. A quantitative version is to drop each dimension one at a time from the compound pass rate and measure how much the gap shrinks; if removing knowledge redefinition alone accounts for most of the drop, the overestimation claim is specific to that probe rather than to general mathematical ability.

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Extended reading notes

Core claim

CogMath's central discovery is that authentic mathematical ability—defined as passing cognitive probes at all three stages—is markedly lower than final-answer accuracy. On the paper's criteria, GPT-4 genuinely masters only 39.3% of MATH and 67.1% of GSM8K problems, against vanilla scores of 75.8% and 90.8%; the seven evaluated models show drops of roughly 30–40 percentage points. The paper further argues that this gap is not mostly memorization or benchmark contamination, because the same overestimation appears on MExam, a freshly collected exam set, and because stronger models show the largest absolute drops on harder problems. The diagnosis is stage-specific: smaller models stumble at problem comprehension, while capable models like GPT-4 and DeepSeek-V2.5 fail mainly at flexible knowledge application and backward verification.

Load-bearing premise

The load-bearing premise is that a model really masters a problem only if it passes all nine probes at once, including treating word-scrambled and missing-condition versions as unsolvable and re-solving under redefined formulas; if a genuinely capable solver could fail any single probe, the reported 30–40% overestimate partly measures the strictness of the compound test rather than a real deficit.

Editorial extensions

If this is right

  • Benchmark accuracy should no longer be read as a ceiling for mathematical competence: users and providers should expect a 30–40 percentage-point drop when the same models are probed on comprehension, strategy transfer, knowledge flexibility, and backward checking.
  • The overestimation is not mainly data contamination: because the gap appears on a freshly collected K-12 exam set, simply adding new test questions will not fix the measurement problem.
  • Different models need different fixes: weak models first need problem-comprehension ability, while strong models like GPT-4 and DeepSeek-V2.5 need more flexible knowledge application and better backward self-verification.
  • Chain-of-thought and in-context learning are auxiliary, not foundational: they often fail to raise CogMath pass rates on MATH, can lower them, and ICL tends to hurt analogical reasoning, so they do not close the authenticity gap.
  • LLMs' weak backward reasoning means self-checking cannot be assumed reliable: even when forward answers are correct, inferring the masked condition from the conclusion remains a bottleneck, consistent with known self-correction failures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper does not pursue is to turn the nine probes into a training curriculum: fine-tune on knowledge redefinition and backward reasoning, then check whether CogMath pass rates improve more than vanilla accuracy; if they do, the framework is not just a metric but a diagnostic training target.
  • Because the probes, gold references, and quality checks are all produced by GPT-4, the measurement pipeline could have model-specific blind spots; swapping in independent human verification or a second probe generator would test how much of the 30–40% gap is generator-dependent.
  • The compound all-nine criterion mechanically guarantees a large drop whenever the nine probes are even partially correlated, so the more informative number for model builders is the per-dimension relative pass rate, which shows exactly which probe to target—for example, training on deliberately unsolvable problems to counter over-correction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes CogMath, a multi-agent evaluation framework that probes LLM performance on mathematics benchmarks along nine dimensions organized into three cognitive stages: problem comprehension, problem solving, and solution summarization. For each problem, an Inquiry agent generates a dimension-specific query, a Judge agent checks the query, and a Reference agent supplies an answer; a model is said to 'truly master' a problem only if it passes all nine inquiries. On GSM8K, MATH, and a newly collected MExam dataset, the authors report that seven mainstream LLMs' pass rates drop by roughly 30-40 percentage points relative to vanilla accuracy, with GPT-4 reaching 39.7% on MATH and 67.1% on GSM8K. The paper also analyzes stage- and dimension-level pass rates, reports an 'over-correction' tendency on unsolvable variants, and studies the effect of CoT and ICL prompting.

Significance. The framework is a useful complement to accuracy-only benchmarks if the mastery criterion is validated. The multi-agent pipeline is human-verified at roughly 95% for Judge and Reference agents on 500 sampled problems, the new MExam dataset provides a contamination-free testbed, and the per-dimension Relative Pass Rate offers interpretable diagnostics. However, the central quantitative claim of 30-40% overestimation is only as credible as the all-nine-dimensions conjunction rule, which is asserted rather than anchored to human judgments or external benchmarks. The paper's contribution is therefore valuable but currently conditional on a criterion whose necessity is not demonstrated.

major comments (4)
  1. [Section 3; Section 4.2] The headline 30-40% overestimation claim depends on the unvalidated conjunction criterion stated at the start of Section 3: 'only when an LLM passes all dimensions can we conclude that it has genuinely mastered the problem P'. The manuscript never independently establishes that every one of the nine dimensions is necessary for authentic mathematical competence. This matters because Dimension 7 (Knowledge Redefinition) is identified in Section 4.4 and Appendix D as a principal driver of the Stage-2 deficit (DeepSeek-R1 RPR = 0.617). A competent agent, human or machine, could reasonably resist recomputing under an arbitrary redefinition such as 'half means one-third' without lacking mastery of the original problem. Without a human baseline, a leave-one-dimension-out analysis, or an external criterion for 'genuine mastery', the reported drop is a property of the strict AND rule rather than an independently measured fact about the models.
  2. [Section 4.1; Table 6] The pass criterion for Dimensions 2 and 3 counts any response different from the original answer as a pass. The intended behavior in Table 6 is to identify the problem as 'unsolvable', but a different wrong answer, a refusal, or a nonsensical output also satisfies the implemented rule. Consequently, the reported pass rates on Dimensions 2 and 3, and the 'over-correction' interpretation in Section 4.4 ('current LLMs may inherently over-correct the problem into a solvable one'), are not supported by the metric. The scoring should require an explicit 'unsolvable'/cannot-be-determined response, or the authors should report sensitivity under a stricter matching rule.
  3. [Section 4.2; Appendix D; Table 7] The abstract and Section 4.2 state that LLM capabilities are 'overestimated by 30%-40%', but the manuscript's own numbers fall outside this range. In Table 1, GPT-4 drops by 28.3% and Llama3-8B by 28.0% on GSM8K; in Table 7 (Appendix D), DeepSeek-R1 drops by 53.4% on MATH. The range statement should be restricted to the seven models in Table 1 and the datasets reported there, or the range should be updated. As written, the headline is not consistent with the full evidence included in the paper.
  4. [Appendix C; Section 4.7] Appendix C reports that all Inquiry, Judge, and Reference agents are implemented with GPT-4. When the evaluated model is GPT-4 or a closely related system, the reference answers and judge decisions against which it is scored are produced by the same model family. The human verification in Section 4.7 (Tables 4 and 5) supports the quality of the generated queries and reference answers, but no inter-annotator agreement is reported and the check does not cover the counterfactual scoring rule for Dimensions 2 and 3. An independent judge/reference model or agreement statistics would reduce the risk that the framework's output reflects GPT-4's own biases.
minor comments (5)
  1. [Section 4.4] The RPR formula uses set intersection and set cardinality, but the sets Pass_i and Pass are not formally defined; please define them before the equation.
  2. [Section 4.5; Table 3] The conclusion that CoT and ICL 'fail to truly improve' mathematical ability is based on only three models; the text should state this limitation explicitly next to the claim.
  3. [Section 4.1; Appendix C] The MExam dataset is described only briefly; providing examples, basic statistics, and a public train/test split would make the contamination-free claim reproducible.
  4. [Section 4.2] No confidence intervals or significance tests are reported for any of the pass-rate differences; given the agent-based generation and subsampling, uncertainty estimates would strengthen the comparisons.
  5. [Appendix B.8.2] The Dimension 9 Judge prompt asks whether the question is solvable and whether the answer is the new answer, but the criterion for 'does not change the structure' is vague; a more operational definition would improve reproducibility.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline 30–40% overestimation result is largely built into the paper's definition of true mastery as passing all nine CogMath dimensions, making the central quantitative claim a consequence of the conjunction rule rather than an independently measured fact.

  1. self definitional [Abstract and Section 3 (CogMath framework, page 4)]
    "An LLM is considered to truly master a problem only when excelling in all inquiries from the 9 dimensions. ... For humans, truly mastering a mathematical problem requires a solid performance at each dimension. Hence, in CogMath, only when an LLM passes all dimensions can we conclude that it has genuinely mastered the problem P."

    The paper's central claim is that LLMs' mathematical abilities are overestimated by 30%-40%, computed as Vanilla accuracy minus CogMath pass rate. But CogMath pass rate is, by the paper's own definition, the intersection of passing the original problem and passing all eight additional inquiries. This intersection is mathematically guaranteed to be no larger than the vanilla rate. The existence and direction of a 30-40% drop is therefore not an empirical discovery about the models; it is a formal consequence of defining 'true mastery' as a conjunction of nine stricter conditions. The only empirical content is which dimensions fail, not that a gap exists.

  2. self definitional [Section 3.2, Dimension 7 (Knowledge Redefinition); Section 4.4; Appendix D]
    "This understanding is flexible—if the problem redefines the formula of “triangle area”, a human who truly grasps the concept will adapt her reasoning to fit the new definition. ... Dimension 7 accounts for the low pass rate discussed in Section 4.3."

    Dimension 7 defines mastery of mathematical knowledge as the ability to solve problems under arbitrary redefinitions such as 'half means one-third' or a modified area formula. The paper then reports that Dimension 7 is a major driver of the CogMath drop, and Appendix D says DeepSeek-R1's RPR on Dimension 7 is only 0.617. This supports the headline overestimation conclusion only if one already accepts the paper's stipulation that flexible knowledge necessarily includes complying with arbitrary redefinitions. A mathematically competent model could reasonably fail such a probe without lacking mathematical ability.

full rationale

The CogMath numbers themselves are honestly reported and the per-dimension robustness results have independent descriptive value: they show which perturbation types cause models to fail, and the MExam results argue against a pure data-contamination explanation. The multi-agent pipeline is also partially anchored by human verification of 500 inquiries and reference answers, so the agent self-reference (all agents are GPT-4) is not by itself the main circular step. However, the paper's headline quantitative claim—that authentic mathematical ability is overestimated by 30%-40%—reduces, by the authors' own definitions, to the difference between vanilla accuracy and the conjunction of nine probe pass conditions. Since 'true mastery' is defined as passing all nine dimensions, the drop from vanilla to CogMath is guaranteed to be positive for any model that fails at least one probe. The magnitude is empirical, but the existence and interpretation of the gap as 'overestimation' are built into the metric. Dimension 7 is the clearest case: the paper first stipulates that genuine knowledge mastery includes adapting to arbitrary redefinitions, then uses failure on that probe as evidence that models lack genuine mathematical ability. No human baseline or external criterion validates that necessity. These are self-definitional rather than fitted-input or self-citation circularities; hence the score is 6 rather than higher. There is no load-bearing self-citation chain and no fitted parameter renamed as a prediction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The framework's load-bearing elements are its definitions: the nine-dimension decomposition, the all-pass mastery threshold, the D2/D3 pass rule, and reliance on GPT-4 as generator and judge. These are hand-set rather than derived, and each affects the headline numbers.

free parameters (3)
  • Maximum inquiry generation iterations (delta) = 10
    Used in the Inquiry-Judge loop; problems that fail after 10 attempts are omitted from that dimension. Chosen by the authors, not derived; affects which problems are scored.
  • Dimension pass criterion for D2 and D3 = response differs from original answer
    Section 4.1 defines pass for counterfactual dimensions as any answer not equal to the original answer, rather than requiring the correct 'unsolvable' response. This hand-set threshold can inflate pass rates.
  • All-dimension mastery conjunction = must pass all 9 dimensions
    Section 3 states mastery only if all inquiries pass; this equal-weight conjunction is a modeling choice, not fitted, but it determines the headline drop.
assumptions (4)
  • domain assumption The three cognitive stages and nine dimensions are a necessary decomposition of authentic mathematical ability.
    The paper cites psychological theories (Schoenfeld, Lesh and Doerr, Dehaene) but provides no empirical validation that these dimensions are necessary for LLM proficiency. Section 3 and Table 6.
  • ad hoc to paper Passing all nine dimensions is necessary and sufficient for genuine mastery.
    Section 3: 'only when an LLM passes all dimensions can we conclude that it has genuinely mastered the problem P.' This criterion is stipulated, not independently validated.
  • domain assumption GPT-4-generated reference answers and judge decisions are accurate enough to serve as ground truth.
    Section 4.7 reports human verification at 95-98.6% consistency on 500 samples, but remaining errors in reference answers (e.g., D7 at 95.2%) propagate into pass-rate measurements.
  • ad hoc to paper For dimensions 2 and 3, any response different from the original answer indicates the model recognized the problem as unsolvable.
    Section 4.1 defines the pass rule this way; a model giving a different but incorrect number also passes, so the rule does not actually test recognition of unsolvability.

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Cite this review

Pith. "Pith review of CogMath: Assessing LLMs' Authentic Mathematical Ability from a Human Cognitive Perspective." pith.science (2026). https://pith.science/paper/7AIMDSR3

@misc{pith2026250604481,
  author       = {Pith},
  title        = {Pith review of: CogMath: Assessing LLMs' Authentic Mathematical Ability from a Human Cognitive Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7AIMDSR3}},
  note         = {Machine review of arXiv:2506.04481}
}
read the original abstract

Although large language models (LLMs) show promise in solving complex mathematical tasks, existing evaluation paradigms rely solely on a coarse measure of overall answer accuracy, which are insufficient for assessing their authentic capabilities. In this paper, we propose \textbf{CogMath}, which comprehensively assesses LLMs' mathematical abilities through the lens of human cognition. Specifically, inspired by psychological theories, CogMath formalizes human reasoning process into 3 stages: \emph{problem comprehension}, \emph{problem solving}, and \emph{solution summarization}. Within these stages, we investigate perspectives such as numerical calculation, knowledge, and counterfactuals, and design a total of 9 fine-grained evaluation dimensions. In each dimension, we develop an ``\emph{Inquiry}-\emph{Judge}-\emph{Reference}'' multi-agent system to generate inquiries that assess LLMs' mastery from this dimension. An LLM is considered to truly master a problem only when excelling in all inquiries from the 9 dimensions. By applying CogMath on three benchmarks, we reveal that the mathematical capabilities of 7 mainstream LLMs are overestimated by 30\%-40\%. Moreover, we locate their strengths and weaknesses across specific stages/dimensions, offering in-depth insights to further enhance their reasoning abilities.

Figures

Figures reproduced from arXiv: 2506.04481 by the authors.

Figure 1
Figure 1. Illustration of our CogMath framework. 3. CogMath To achieve a comprehensive evaluation, we draw inspiration from how humans solve mathematical problems. Specifi￾cally, psychological theories indicate that human reasoning process consists of three stages: problem comprehension, problem solving, and solution summarization (Schoenfeld, 2014; Lesh & Doerr, 2003; Dehaene et al., 1999). They build upon each other, with e… view at source ↗
Figure 2
Figure 2. Relative Pass Rate (RPR) of different LLMs in each dimension. Overall, from [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Relationship between LLM performance with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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