Automated Grading of Handwritten Mathematics Using Vision-Capable LLMs

Open-ended mathematics exams are valuable because they assess reasoning, proof construction, algorithmic thinking, and communication of intermediate steps. They are also difficult to grade at scale because instructors must apply partial-credit rubrics consistently while giving feedback that helps students repair misconceptions. This paper evaluates six contemporary large language model (LLM) configurations, Gemini 3.1 Pro Extended, Gemini 3.5 Flash, ChatGPT 5.5 Pro Extended, ChatGPT 5.5 Thinking, Claude Pro Opus 4.7, and Claude Sonnet 4.6, as grading assistants for an undergraduate discrete mathematics examination. The study compares two grading policies. The BASELINE policy uses a stricter rubric-following prompt that emphasizes explicit evidence and complete justification. The LIBERAL policy was added after preliminary grading showed that the baseline condition sometimes applied harsh point deductions and failed to recognize valid partial reasoning. Agreement with human grading is measured at both the question and exam-total levels using mean absolute error, root mean squared error, normalized root mean squared error, Pearson correlation, and exact agreement. The results show that liberal partial-credit prompting reduces average question-level error for every evaluated model family. ChatGPT 5.5 Thinking (LIBERAL) has the lowest average question-level MAE (1.87) and RMSE (2.53), while Gemini 3.1 Pro Extended (LIBERAL) has the lowest total-score MAE (8.00) and RMSE (10.66). However, the strongest total-score Pearson correlation occurs under Gemini 3.1 Pro Extended (BASELINE) at 0.58, showing that point calibration and rank preservation remain distinct goals. We also report practical usability observations.

Criterion-level grading connects examination performance to learning outcomes, but manual marking introduces workload and variation between markers. This study evaluates vision-language models (VLMs) for handwritten outcome-based assessment across five dimensions: accuracy, human agreement, repeated-run reliability, error concentration, and explanation quality. Using 1,982 criterion-level records from 485 undergraduate examination answers, we compare 20 configurations spanning Qwen2.5-VL, InternVL3, Pixtral, a Donut baseline, and a cascade ensemble. Evaluation setups include zero-shot prompting, few-shot prompting, partial fine-tuning, and Low-Rank Adaptation (LoRA). Two independent faculty markers regraded all 291 test criteria, providing a human agreement baseline on the same assessment materials. Qwen2.5-VL with LoRA achieved Quadratic Weighted Kappa (QWK) of 0.727 and mean absolute error of 0.435 marks against the examiner, compared with mean human-pair QWK of 0.551. This comparison reflects calibration to the examiner's training marks. LoRA outperformed partial fine-tuning for all three instruction-tuned VLMs, while few-shot prompting reduced QWK in every configuration with valid prompted scores. Aggregate reliability and exact repeatability diverged: intraclass correlations ranged from 0.790 to 0.874, yet 50.2-63.6% of criteria changed marks across five sampled runs. Attention-guided deletion showed no statistically significant advantage over random masking, and four faculty reviewers reached no consensus on explanation usefulness. These findings highlight the need for rubric-specific calibration, repeatable scoring, review of consequential errors, and separate validation of explanations. The released evaluation protocol supports criterion-level assessment research and grading tools with teacher oversight.

The integration of large language models (LLMs) into educational assessment represents a transformative shift in classroom grading practices. While automated scoring systems and machine learning techniques have existed for decades, generative AI (GenAI) now enables educators to implement standards-based grading (SBG) with unprecedented efficiency and scale. This paper examines the theoretical foundations and evaluates an LLM grader that uses commercially available foundation models with context and prompt engineering to score student work against a rubric. Drawing on an empirical interrater agreement study using Massachusetts Comprehensive Assessment System (MCAS) data, we observed the Quadratic Weighted Kappa (QWK) and Proportional Reduction in Mean-Squared Error (PRMSE) across mathematics, science, and ELA, using Claude Sonnet 4, Haiku 4.5, GPT-5, and GPT-5 Mini. The results demonstrate that LLM graders, especially when based on foundational models with more parameters, achieve substantial agreement with human raters in mathematics and science assessments, while the performances vary in ELA, suggesting generic foundation models can be effective at scoring in given contexts. Additional analysis of teacher and student feedback reveals strong acceptance of AI-generated narrative feedback but skepticism toward numerical scores, suggesting that LLMs function most effectively as formative tools rather than summative evaluators. Our findings indicate that thoughtfully designed hybrid models that combine AI efficiency with teacher judgment can reduce workload, enhance feedback quality, and support equitable assessment practices without displacing professional expertise.