Calculation–Conceptualization Gap

• The paper synthesizes empirical evidence and theoretical models to reveal a systematic gap between accurate calculations and coherent conceptual reasoning.
• Research shows that targeted instructional frameworks, like dual pathway assessments, enhance the integration of formal methods with conceptual understanding.
• Benchmark studies in AI and education reveal that while precise calculations can be performed, linking them to meaningful concepts remains a persistent challenge.

The calculation–conceptualization discrepancy denotes a persistent gap—empirically and theoretically observed across domains—between the ability to execute formal mathematical procedures and the ability to construct, interpret, or sense-make using mathematical concepts. It manifests where agents, whether students, domain experts, or machine learning systems, perform correct calculations but fail to link those operations coherently to the underlying conceptual structure of a task. The phenomenon is systematic, affecting instructional design, assessment, decision support, benchmarking in AI, statistical inference, and even foundational problems in high-energy physics. This article synthesizes empirical evidence, formal definitions, methodological frameworks, and approaches for resolving the discrepancy.

1. Empirical Manifestations in Human Reasoning

Quantitative educational studies show that university students in calculus-based physics can often carry out algebraic or arithmetic procedures, yet their performance drops sharply when required to use mathematical concepts for sense making or when numerical complexity and abstraction increase (Brahmia et al., 2016). Success rates in structured assessments hover near 50%, with systematic error patterns:

• Context dependence: Students show marked decline when ratio reasoning moves from familiar contexts (“olive oil” pricing) to abstract or physics-specific ones (“traxolene” mass).
• Numerical-complexity effects: Replacing integers with decimals or variables reduces correct responses by 15–20 percentage points.
• Recurring error patterns: Common mistakes include using differences instead of ratios, syntactic translation errors, neglect of units, and destabilization by symbolic quantities.

The “idiosyncratic cognitive blend” that characterizes expert physicists’ mathematization—where conceptual sense-making, units, and algebra fuse—contrasts sharply with students’ often procedural, unit-blind manipulation.

2. Instructional and Assessment Frameworks

Research in physics education recommends explicitly targeting coherence between calculations and concepts. The calculation–concept crossover assessment framework tracks whether students flexibly alternate between mathematical and conceptual approaches. Assessment items are designed with dual pathways—one that can be attacked mathematically and another conceptually—testing for genuine mathematical sensemaking (Kuo et al., 2019).

Quantitative findings indicate:

• Increased use of crossover approaches predicts higher correctness, especially on qualitative tasks where conceptual-only approaches are highly error-prone.
• Curricula that foreground coherence (via three-stage prompts, clicker questions, and symbolic-forms instruction) foster deeper sensemaking, as evidenced in randomized controlled studies.

Instructional recommendations emphasize making mathematization processes explicit and rewarding coherence-seeking (not just final correctness). This includes mapping units, verifying dimensional consistency, and contextualizing formulas within physical meaning.

3. Epistemological Dynamics and Conceptual Resistance

Case studies reveal that calculation–conceptualization discrepancy may stem from epistemological stances rather than deficits in formal skills (Gupta et al., 2010). For instance, engineering students may over-trust formal equations as “fact-bearing laws,” regarding reconciliation with everyday or intuitive reasoning as unnecessary or implausible. However, stances shift dynamically: targeted prompts (e.g., sign checks) can trigger productive sense-making, realigning mathematical results with physical intuition. Instructional models must therefore engage not just skill gaps but students’ epistemological expectations.

4. Manifestations in AI: Benchmarking, Symbolic Integration, and Embedding Interpretation

LLMs and related AI systems exhibit analogous calculation–conceptualization discrepancies. Empirical benchmarking on real-world quantitative tasks finds LLMs articulate correct conceptual chains of reasoning but frequently err in exact computation, with accuracy ranging from 45–63% on calculator-verified tasks (Herambourg et al., 4 Nov 2025). Error decomposition shows that mechanical calculation and rounding issues account for the majority of failures:

• Correct concepts, numeric slips: E.g., accurate application of BMI formula but incorrect rounding.
• Conceptual missteps, accurate arithmetic: E.g., valid calculations under a wrong domain assumption.
• Refusals or deflection: Declining to apply known formulas due to illusory uncertainty.

Augmentation strategies—such as program-aided models, symbolic execution, and ensemble methods—can partially mitigate this gap by decoupling conceptual planning from numeric precision (Kadlčík et al., 2023). Chain-of-thought frameworks, with external calculator calls, double arithmetic accuracy on benchmark datasets.

In representation learning, embedding vectors produced by neural networks are opaque to human conceptualization. Post-hoc projection into conceptual spaces (e.g., via cosine similarity with human-annotated concept phrases) enables interpretable mapping, closing the discrepancy for model explainability, debugging, and semantic tracing (Simhi et al., 2022).

5. Formal Definition and Theoretical Analysis in Decision Support and Inference

Formal discrepancy metrics are crucial in contexts where calculation (algorithmic synthesis) and conceptual judgment must align. In pairwise comparison ranking, the discrepancy factor $\Delta(A, w)$ directly quantifies the maximal deviation between input expert judgments and the computed eigenvector-based ranking. Sufficient conditions for order preservation guarantee conceptual intent only if raw judgments exceed calculational error bounds (Kułakowski, 2013).

In computational social science, conceptualization errors—i.e., misaligned or underspecified codebooks in text classification—lead to irreducible bias not removable by improved calculation (model accuracy) or post-hoc correction. Simulations establish that only complete, community-vetted codebooks secure unbiased downstream inference, highlighting conceptualization as a first-order methodological concern (Halterman et al., 3 Oct 2025).

6. Technical Resolutions and Geometric Insights

Bridging calculation and conceptualization involves:

• Explicitly marking and validating calculation steps in neural reasoning: CALC-X markup and external-symbolic execution.
• Mapping vector calculations into conceptual basis: CES algorithm for embedding interpretation.
• Formulating geometric explanations for complex calculations: Viewing kernel distances as ensembles of circular discrepancies in two-sample testing yields both efficient computation and conceptual clarity (Zhao et al., 2014).
• Revisiting foundational mathematical procedures: In high-energy theory, mismatches in entanglement entropy calculations are resolved by tracking total derivative terms, reconciling calculational prescriptions with underlying conceptual invariants (Astaneh et al., 2014).

7. Synthesis and Implications

The calculation–conceptualization discrepancy is domain-general, appearing wherever formal procedures and conceptual interpretation may decouple. Its resolution demands integrated assessment, instruction, benchmarking design, and post-hoc interpretability methods. Across human education, machine learning, statistical inference, and mathematical physics, effective practice requires the ongoing reconciliation of the procedural with the conceptual, ensuring neither calculation nor conceptualization occurs in isolation.

References:

• “Obstacles to Mathematization in Introductory Physics” (Brahmia et al., 2016)
• “Mathematical Sensemaking as Seeking Coherence…” (Kuo et al., 2019)
• “Beyond deficit-based models of learners' cognition…” (Gupta et al., 2010)
• “The ORCA Benchmark: Evaluating Real-World…” (Herambourg et al., 4 Nov 2025)
• “Calc-X and Calcformers…” (Kadlčík et al., 2023)
• “Interpreting Embedding Spaces by Conceptualization” (Simhi et al., 2022)
• “Notes on discrepancy in the pairwise comparisons method” (Kułakowski, 2013)
• “What is a protest anyway? Codebook conceptualization…” (Halterman et al., 3 Oct 2025)
• “FastMMD: Ensemble of Circular Discrepancy…” (Zhao et al., 2014)
• “Entropy discrepancy and total derivatives in trace anomaly” (Astaneh et al., 2014)