# Math Operation Embeddings for Open-ended Solution Analysis and Feedback

arXiv: https://arxiv.org/abs/2104.12047 · HTML: https://mlsanity.com/paper/2104.12047/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 53 references that are in the map.
- First posted on arXiv 25 Apr 2021.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Convolutional Neural Networks for Sentence Classification](https://mlsanity.com/paper/1408.5882/) (2014)
- [Analysing Mathematical Reasoning Abilities of Neural Models](https://mlsanity.com/paper/1904.01557/) (2019)
- [Semantically-Aligned Universal Tree-Structured Solver for Math Word Problems](https://mlsanity.com/paper/2010.06823/) (2020)
- [Mathematical Language Processing: Automatic Grading and Feedback for Open Response Mathematical Questions](https://mlsanity.com/paper/1501.04346/) (2015)
- [Learning Program Embeddings to Propagate Feedback on Student Code](https://mlsanity.com/paper/1505.05969/) (2015)
- [Text2Math: End-to-end Parsing Text into Math Expressions](https://mlsanity.com/paper/1910.06571/) (2019)

## Built on by (most cited first, 8 of 8)

- [Automated Distractor and Feedback Generation for Math Multiple-choice Questions via In-context Learning](https://mlsanity.com/paper/2308.03234/) (2023)
- [Improving the Validity of Automatically Generated Feedback via Reinforcement Learning](https://mlsanity.com/paper/2403.01304/) (2024)
- [Automatic Short Math Answer Grading via In-context Meta-learning](https://mlsanity.com/paper/2205.15219/) (2022)
- [Exploring Automated Distractor Generation for Math Multiple-choice Questions via Large Language Models](https://mlsanity.com/paper/2404.02124/) (2024)
- [A Review of the Trends and Challenges in Adopting Natural Language Processing Methods for Education Feedback Analysis](https://mlsanity.com/paper/2301.08826/) (2023)
- [Improving Automated Distractor Generation for Math Multiple-choice Questions with Overgenerate-and-rank](https://mlsanity.com/paper/2405.05144/) (2024)
- [Algebra Error Classification with Large Language Models](https://mlsanity.com/paper/2305.06163/) (2023)
- [VEHME: A Vision-Language Model For Evaluating Handwritten Mathematics Expressions](https://mlsanity.com/paper/2510.22798/) (2025)

## Abstract (arXiv)

Feedback on student answers and even during intermediate steps in their solutions to open-ended questions is an important element in math education. Such feedback can help students correct their errors and ultimately lead to improved learning outcomes. Most existing approaches for automated student solution analysis and feedback require manually constructing cognitive models and anticipating student errors for each question. This process requires significant human effort and does not scale to most questions used in homework and practices that do not come with this information. In this paper, we analyze students' step-by-step solution processes to equation solving questions in an attempt to scale up error diagnostics and feedback mechanisms developed for a small number of questions to a much larger number of questions. Leveraging a recent math expression encoding method, we represent each math operation applied in solution steps as a transition in the math embedding vector space. We use a dataset that contains student solution steps in the Cognitive Tutor system to learn implicit and explicit representations of math operations. We explore whether these representations can i) identify math operations a student intends to perform in each solution step, regardless of whether they did it correctly or not, and ii) select the appropriate feedback type for incorrect steps. Experimental results show that our learned math operation representations generalize well across different data distributions.