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arxiv: 2505.16646 · v5 · submitted 2025-05-22 · 💻 cs.AI

SMART: Self-Generating and Self-Validating Multi-Dimensional Assessment for LLMs' Mathematical Problem Solving

Yujie Hou , Mei Wang , Yaoyao Zhong , Ting Zhang , Xuetao Ma , Hua Huang This is my paper

Pith reviewed 2026-05-22 13:42 UTC · model grok-4.3

classification 💻 cs.AI
keywords LLM evaluationmathematical reasoningbenchmarkcognitive dimensionsproblem solvingAll-Pass ScorePolya's theory
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The pith

A new benchmark splits mathematical problem-solving into four cognitive dimensions to show that LLMs have uneven capabilities hidden by standard tests.

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

The paper contends that existing evaluations of LLMs on math tasks reduce reasoning to simple input-output checks and fail to capture its multi-stage nature. It proposes SMART, which decomposes problem-solving into four dimensions—Semantic Understanding, Mathematical Reasoning, Arithmetic Computation, and Reflection & Refinement—each assessed through dedicated tasks drawn from Polya's theory. Application to 22 leading open- and closed-source models reveals large performance differences across these dimensions. The work introduces the All-Pass Score, which counts only cases where a model succeeds in every dimension, as a stricter gauge of genuine problem-solving skill. A sympathetic reader would care because this approach identifies concrete weaknesses rather than accepting overall accuracy scores at face value.

Core claim

Mathematical problem-solving consists of four distinct cognitive dimensions that current benchmarks do not measure separately. The SMART benchmark introduces dimension-specific tasks to isolate and evaluate Semantic Understanding, Mathematical Reasoning, Arithmetic Computation, and Reflection & Refinement. When run on 22 state-of-the-art LLMs, the results expose substantial discrepancies in model performance across these areas. These gaps indicate real limitations in current systems and support the All-Pass Score as a metric that requires success on all dimensions to reflect true problem-solving capability more faithfully than final-answer accuracy alone.

What carries the argument

The SMART benchmark and its dimension-specific tasks that separately measure the four cognitive processes of Semantic Understanding, Mathematical Reasoning, Arithmetic Computation, and Reflection & Refinement.

If this is right

  • LLMs display inconsistent strengths and weaknesses across the four dimensions instead of uniform capability.
  • Many models succeed on final answers yet fail on reflection and refinement tasks.
  • The All-Pass Score offers a stricter metric that better distinguishes genuine problem-solving from partial success.
  • Model development should target improvements in each dimension separately rather than overall accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Developers could design training methods that strengthen the weakest dimension for a given model.
  • The same multi-dimensional breakdown might help evaluate LLM performance in non-mathematical domains such as scientific reasoning.
  • Single-metric benchmarks likely overestimate real reasoning ability by overlooking dimension-specific failures.

Load-bearing premise

The four cognitive dimensions are distinct, non-overlapping processes that can be isolated and measured independently by the introduced tasks.

What would settle it

Re-evaluating the 22 models and finding that scores on the four dimension-specific tasks are highly correlated with each other and with standard final-answer accuracy, with no meaningful gaps or added value from the All-Pass Score.

Figures

Figures reproduced from arXiv: 2505.16646 by Hua Huang, Mei Wang, Ting Zhang, Xuetao Ma, Yaoyao Zhong, Yujie Hou.

Figure 1
Figure 1. Figure 1: Comparison of evaluation paradigms for LLM [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of SMART benchmark construction. First, we collect seed questions from datasets of varying [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The performance across the varying difficulty settings for each SMART dimension. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The reasoning step statistics of the seed ques [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A data sample in the SMART benchmark [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An overview of the SMART framework for evaluating the mathematical problem-solving process. The [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The prompt for LLMs to extract context from a seed question. [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The prompt for LLMs to convert the seed question to a symbolic expression. [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The prompt for LLMs to convert the SMT-LIB expression to an arithmetic notation problem. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example of CoT with different errors. System Prompt: You are an expert evaluator tasked with assessing the semantic similarity between a reference answer and a model-generated answer. User Prompt: Your goal is to determine how closely the model-generated answer aligns with the reference answer in terms of meaning, content. Instructions: Read both answers carefully: Understand the core information, details… view at source ↗
Figure 11
Figure 11. Figure 11: The prompt for LLM-as-a-Judge for evaluating the Understanding task. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The prompt for LLMs to solve the arithmetic notation problem. [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The prompt for LLMs to detect mistakes in the CoT. [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The prompt for LLMs to detect more than one mistake in the CoT. [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The prompt for LLMs to correct the mistakes in the CoT. [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The prompt for LLMs to self-refine the Reasoning dimension. [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The prompt for LLMs to self-refine the Arithmetic dimension. [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Example of questions with a different number of noise sentences. [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Example of questions with different reasoning steps. [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Example of arithmetic questions with numbers in different digits. [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The confusion matrix of the final answer and other dimensions. P means Positive, and N means Negative. [PITH_FULL_IMAGE:figures/full_fig_p027_21.png] view at source ↗
read the original abstract

Large Language Models (LLMs) have achieved remarkable performance across a wide range of mathematical benchmarks. However, concerns remain as to whether these successes reflect genuine reasoning or superficial pattern recognition. Existing evaluation methods, which typically focus either on the final answer or on the intermediate reasoning steps, reduce mathematical reasoning to a shallow input-output mapping, overlooking its inherently multi-stage and multi-dimensional cognitive nature. Inspired by Polya's problem-solving theory, we propose SMART, a benchmark that decomposes mathematical problem-solving into four cognitive dimensions: Semantic Understanding, Mathematical Reasoning, Arithmetic Computation, and Reflection & Refinement, and introduces dimension-specific tasks to measure the corresponding cognitive processes of LLMs. We apply SMART to 22 state-of-the-art open- and closed-source LLMs and uncover substantial discrepancies in their capabilities across dimensions. Our findings reveal genuine weaknesses in current models and motivate a new metric, the All-Pass Score, designed to better capture true problem-solving capability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces SMART, a benchmark that decomposes mathematical problem-solving into four cognitive dimensions inspired by Polya's theory—Semantic Understanding, Mathematical Reasoning, Arithmetic Computation, and Reflection & Refinement—and uses self-generated, self-validated dimension-specific tasks to evaluate LLMs. It reports results on 22 open- and closed-source models, documents substantial cross-dimension discrepancies, and proposes the All-Pass Score as a stricter metric for genuine problem-solving capability.

Significance. If the dimension-isolation claim holds, the work would provide a more granular diagnostic tool than existing single-score math benchmarks, directly addressing concerns about superficial pattern matching versus multi-stage reasoning. The self-generation/self-validation pipeline and the All-Pass Score are concrete, falsifiable contributions that could be adopted or extended by the community.

major comments (2)
  1. [§3.2 and §4.1] §3.2 (Dimension-Specific Task Construction) and §4.1 (Validation Procedure): The central claim that the four dimensions are distinct, non-overlapping processes rests on the assertion that the introduced tasks isolate each cognitive component. No ablation, correlation matrix, or difficulty-matched control is reported showing that, for example, Semantic Understanding items are solved independently of Mathematical Reasoning ability. Without such evidence the reported discrepancies and the All-Pass Score cannot be interpreted as dimension-specific profiles rather than correlated general capabilities.
  2. [§4.2 and Table 2] §4.2 (Experimental Results) and Table 2: The All-Pass Score is defined as requiring success on all four dimension-specific tasks for a given problem. Because the tasks may share latent factors (e.g., overall model scale or instruction-following), the metric risks simply re-ranking models by a stricter but still unidimensional threshold; a sensitivity analysis or comparison against a single combined task is needed to establish added value.
minor comments (2)
  1. [Abstract] The abstract and introduction repeatedly use “self-validating” without a concise operational definition; a one-sentence gloss in the abstract would improve readability.
  2. [Figure 3] Figure 3 (dimension radar plots) would benefit from error bars or per-model variance across the self-generated task sets to indicate stability of the reported discrepancies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and will incorporate revisions to strengthen the empirical support for our claims.

read point-by-point responses

  1. Referee: [§3.2 and §4.1] §3.2 (Dimension-Specific Task Construction) and §4.1 (Validation Procedure): The central claim that the four dimensions are distinct, non-overlapping processes rests on the assertion that the introduced tasks isolate each cognitive component. No ablation, correlation matrix, or difficulty-matched control is reported showing that, for example, Semantic Understanding items are solved independently of Mathematical Reasoning ability. Without such evidence the reported discrepancies and the All-Pass Score cannot be interpreted as dimension-specific profiles rather than correlated general capabilities.

    Authors: We acknowledge that the manuscript does not report ablations, inter-dimension correlations, or explicit difficulty-matched controls to empirically demonstrate the independence of the four dimensions. The task construction in §3.2 is intentionally designed to isolate processes (e.g., Semantic Understanding tasks require only problem parsing without solution steps, while Mathematical Reasoning tasks assume semantic input and focus on strategy), and §4.1 validates task quality via self-validation. However, we agree that additional evidence is needed to rule out shared latent factors. In the revised manuscript we will add a correlation matrix of model performances across dimensions, report any available controls from the generation pipeline, and discuss the implications for interpreting the discrepancies as dimension-specific. revision: yes

  2. Referee: [§4.2 and Table 2] §4.2 (Experimental Results) and Table 2: The All-Pass Score is defined as requiring success on all four dimension-specific tasks for a given problem. Because the tasks may share latent factors (e.g., overall model scale or instruction-following), the metric risks simply re-ranking models by a stricter but still unidimensional threshold; a sensitivity analysis or comparison against a single combined task is needed to establish added value.

    Authors: The All-Pass Score is proposed to capture the requirement of succeeding at every stage of problem-solving rather than permitting compensation across dimensions, which distinguishes it from standard aggregate accuracy. We recognize that without further analysis it could be viewed as a stricter unidimensional threshold. To address this, the revised manuscript will include a sensitivity analysis that compares the All-Pass Score against performance on a single combined task, examines its correlation with model scale and instruction-following ability, and quantifies the additional diagnostic value it provides over existing metrics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; benchmark and metric are independently constructed

full rationale

The paper defines four dimensions inspired by Polya's problem-solving theory, introduces dimension-specific tasks to measure them, applies the resulting SMART benchmark to 22 LLMs to obtain empirical performance data, and defines the All-Pass Score as a new aggregate metric based on those measurements. No load-bearing step reduces by construction to a fitted parameter, self-referential definition, or self-citation chain. The discrepancies and metric are direct outputs of the evaluation rather than tautological restatements of the input design. The derivation chain is self-contained against external benchmarks and does not rely on any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a new benchmark and metric; it relies on the domain assumption that Polya's theory supplies a useful four-way decomposition but introduces no free parameters or new entities.

axioms (1)
  • domain assumption Polya's problem-solving theory supplies a valid decomposition of mathematical reasoning into four distinct cognitive dimensions that can be separately assessed.
    The benchmark is explicitly inspired by Polya's theory for the dimension breakdown.

pith-pipeline@v0.9.0 · 5710 in / 1222 out tokens · 55908 ms · 2026-05-22T13:42:05.477860+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 2 internal anchors

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    Check: Ensure all the variables in SMT-LIB formula are declared

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    Three-shot Examples [Examples] The Given Question: [Seed question] Now, analyze the next math problem

    Do not write comments. Three-shot Examples [Examples] The Given Question: [Seed question] Now, analyze the next math problem. Generate the symbolic expression of the math word problem. Strictly following the steps and formatting provided. Be precise, logical, and concise in your responses. The Answer of Task: [SMT-LIB] Figure 8: The prompt for LLMs to con...

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  14. [14]

    Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180

    Arithmetic Number Error Definition: Randomly selects a number in the CoT and replaces it with a different, randomly generated number. Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180. In the first two months, the total number of downloads of the program was 180+60 = <<180+60=280>>280. In the third mon...

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  15. [15]

    Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180

    Skipped Step Error Definition: Randomly removes one sentence (a segment delimited by a period) from the CoT. Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180. In the first two months, the total number of downloads of the program was 180+60 = <<180+60=240>>240. In the third month, the number of downloa...

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  16. [16]

    Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180

    Hallucinatory Insertion Error Definition: Randomly selects a sentence from another CoT and inserts it into a random position in the primary CoT. Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180. In the first two months, the total number of downloads of the program was 180+60 = <<180+60=240>>240. In th...

    work page

  17. [17]

    Example: In the first two months, the total number of downloads of the program was 180+60 = <<180+60=240>>240

    Logical Order Error Definition: Randomly selects two sentences in the CoT and swaps their positions. Example: In the first two months, the total number of downloads of the program was 180+60 = <<180+60=240>>240. The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180. In the third month, the number of downloads of th...

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  18. [18]

    Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180

    Redundant Output Error Definition: Randomly selects a sentence in the CoT, duplicates it, and inserts the copy into a random position. Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180. In the first two months, the total number of downloads of the program was 180+60 = <<180+60=240>>240. In the third mo...

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  19. [19]

    Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180

    Incorrect Operator Error Definition: Replaces a mathematical operator (+, -, ×, ÷) in the CoT with a different, randomly selected operator. Example: The number of downloads of the program in the second month increased to 3*60 = <<3*60=180>>180. In the first two months, the total number of downloads of the program was 180+60 = <<180+60=240>>240. In the thi...

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  20. [25]

    Please carefully review each step in the CoT and determine whether any of the above types of errors are present

    Modify Operator: Replaces a mathematical operator (+, -, ×, ÷) in the CoT with a different, randomly selected operator. Please carefully review each step in the CoT and determine whether any of the above types of errors are present. If you find errors, output the corresponding error type name and numbers. Three-shot Examples [Examples] The Given Question:...

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  21. [26]

    Change Number: Randomly selects a number in the CoT and replaces it with a different, randomly generated number

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  22. [27]

    Delete Segment: Randomly removes one sentence (a segment delimited by a period) from the CoT

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  23. [28]

    Insert Segment: Randomly selects a sentence from another CoT (cot2) and inserts it into a random position in the primary CoT

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  24. [29]

    Swap Segments: Randomly selects two sentences in the CoT and swaps their positions

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    Duplicate Segment: Randomly selects a sentence in the CoT, duplicates it, and inserts the copy into a random position

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    Refinement complete – reasoning strengthened

    Modify Operator: Replaces a mathematical operator (+, -, ×, ÷) in the CoT with a different, randomly selected operator. Please carefully review each step in the CoT and determine whether any of the above types of errors are present. More than one errors are in the CoT. Output all the corresponding error type namse and numbers. Three-shot Examples [Example...

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