Explaining and Tuning Transformer-based LLMs in Arithmetic Tasks with Human Strategies
Authors: Luyu Qiu, Jianing Li, Hwanhee Kim, Xiaoyong Wei, Yueyuan Zheng, Janet Hsiao, Lei Chen
Organizations: 1The Hong Kong University of Science and Technology · 2The Hong Kong Polytechnic University · University of California, Berkeley
Abstract
Transformer-based large language models (LLMs) continue to achieve state-of-the-art performance across various natural language processing tasks. However, their subpar performance on seemingly elementary problems, such as basic arithmetic, raises concerns about model reliability, safety, and ethical deployment. In this study, we demonstrate that the performance of a vanilla Transformer model trained on integer arithmetic tasks can be improved using methods effective for human learners. We begin by decomposing the arithmetic task into well-defined subtasks and conducting loss convergence order analysis together with ablation studies for each subtask. Our findings reveal that LLMs exhibit learning patterns similar to those of human learners, with a faster learning speed for simpler subtasks compared to more complex ones. In addition, we successfully improved the accuracy of LLMs by applying problem-solving strategies and cognitive empowerment methods shown to enhance the performance of human learners. This suggests that transformer-based LLMs may share cognitive processes with human learners in arithmetic. Lastly, we provide a comprehensive demonstration of our method's effectiveness, including significant accuracy improvement experiments, visualization verification, and explanation-based analysis to illuminate the intricacies of LLMs in arithmetic learning. In general, this work explores the potential similarities between transformer-based LLMs and human learners, supported by explainable AI (XAI) verifications, ultimately fostering trust in LLMs for critical and high-stakes applications.
Large language models (LLMs) have demonstrated impressive capabilities, yet their internal mechanisms for handling reasoning-intensive tasks remain underexplored. To advance the understanding of model-internal processing mechanisms, we present an investigation of how LLMs perform arithmetic operations by examining internal mechanisms during task execution. Using early decoding, we trace how next-token predictions are constructed across layers. Our experiments reveal that while the models recognize arithmetic tasks early, correct result generation occurs only in the final layers. Notably, models proficient in arithmetic exhibit a clear division of labor between attention and MLP modules, where attention propagates input information and MLP modules aggregate it. This division is absent in less proficient models. Furthermore, successful models appear to process more challenging arithmetic tasks functionally, suggesting reasoning capabilities beyond factual recall.
Tanja Baeumel, Josef van Genabith, Simon Ostermann
We investigate whether methods of human mathematics pedagogy can guide the training of language models toward arithmetic reasoning. Building on the GASING method -- an Indonesian pedagogy that solves basic arithmetic through a left-to-right procedure aligned with the causal order of token generation -- we operationalize each operation as a computational procedure whose execution trace is serialized into natural-language Chain-of-Thought (CoT) supervision. A small GPT-2 decoder (86M parameters) with a syllabic-agglutinative TOBA tokenizer for Indonesian is trained from scratch on this data using only a next-token prediction objective, without reinforcement learning or reward-based optimization. Monitoring training reveals three distinct learning phases, and mechanistic analyses -- attention-masking interventions on the CoT information graph, residual-stream probing, and logit-lens inspection -- show that the model first internalizes a procedural pathway and subsequently develops an associative, ``mental-arithmetic'' capacity that retrieves intermediate results without explicit step-by-step computation. The trained model reaches over 80% accuracy on held-out problems and attains competitive performance against substantially larger language models, indicating that targeted, pedagogically grounded training can yield strong and economical arithmetic capability at small scale.
Large language models achieve strong performance on arithmetic reasoning benchmarks, and one common response to arithmetic brittleness is to delegate computation to code. Yet models are still often used in settings where they must reason directly from natural language, and trustworthy models should solve small-number arithmetic word problems without external tools. Prior work shows that LLMs are sensitive to numerical variation: a model may solve an original problem but fail on structurally similar variants requiring the same reasoning procedure with different numbers. We ask whether this fragility persists under a stricter setting involving small, schema-preserving numeric changes that retain the original reasoning program and avoid large-number stress tests. We introduce an automatic algorithm for generating numeric-remapping attacks on arithmetic word problems. Unlike template-based perturbation methods requiring manual schemas or constraints, our approach derives problem-specific symbolic representations, generates constrained numeric remappings, recomputes gold answers, and realizes transformed questions through deterministic edits guided by LLM-generated edit plans. Stage-wise validation and a high-confidence audit retain reliable attacks, making the pipeline scalable with limited human intervention. We evaluate DeepSeek-R1 (70B), Gemma4 (31B), and GPT-OSS (120B) on GSM8K, MAWPS, and MultiArith. On GSM8K, completed runs show conditional accuracy drops of 12.16 to 25.82 percentage points. MAWPS and MultiArith are far more stable, with most attacked accuracies near or above 98%. These results show that numeric-remapping robustness depends strongly on dataset structure: GSM8K remains sensitive even when reasoning programs are preserved and answers are recomputed, while shorter, more regular datasets are more robust.