# Learning by Applying: A General Framework for Mathematical Reasoning via Enhancing Explicit Knowledge Learning

arXiv: https://arxiv.org/abs/2302.05717 · HTML: https://mlsanity.com/paper/2302.05717/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 22 of its 64 references that are in the map.
- First posted on arXiv 11 Feb 2023.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Language Models as Knowledge Bases?](https://mlsanity.com/paper/1909.01066/) (2019)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [ConceptNet 5.5: An Open Multilingual Graph of General Knowledge](https://mlsanity.com/paper/1612.03975/) (2016)
- [Effective Approaches to Attention-based Neural Machine Translation](https://mlsanity.com/paper/1508.04025/) (2015)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [EKT: Exercise-aware Knowledge Tracing for Student Performance Prediction](https://mlsanity.com/paper/1906.05658/) (2019)
- [Generate & Rank: A Multi-task Framework for Math Word Problems](https://mlsanity.com/paper/2109.03034/) (2021)
- [Probabilistic Logic Neural Networks for Reasoning](https://mlsanity.com/paper/1906.08495/) (2019)
- [Topology-Aware Correlations Between Relations for Inductive Link Prediction in Knowledge Graphs](https://mlsanity.com/paper/2103.03642/) (2021)
- [Learning to Reason Deductively: Math Word Problem Solving as Complex Relation Extraction](https://mlsanity.com/paper/2203.10316/) (2022)
- [MWPToolkit: An Open-Source Framework for Deep Learning-Based Math Word Problem Solvers](https://mlsanity.com/paper/2109.00799/) (2021)
- [Knowledge-Enhanced Hierarchical Graph Transformer Network for Multi-Behavior Recommendation](https://mlsanity.com/paper/2110.04000/) (2021)
- [Abductive Knowledge Induction From Raw Data](https://mlsanity.com/paper/2010.03514/) (2020)
- [The Gap of Semantic Parsing: A Survey on Automatic Math Word Problem Solvers](https://mlsanity.com/paper/1808.07290/) (2018)
- [Unit Dependency Graph and its Application to Arithmetic Word Problem Solving](https://mlsanity.com/paper/1612.00969/) (2016)
- [Curvature Regularization to Prevent Distortion in Graph Embedding](https://mlsanity.com/paper/2011.14211/) (2020)
- [KLearn: Background Knowledge Inference from Summarization Data](https://mlsanity.com/paper/2010.06213/) (2020)

## Built on by (most cited first, 5 of 5)

- [Consistent Autoformalization for Constructing Mathematical Libraries](https://mlsanity.com/paper/2410.04194/) (2024)
- [RePair: Automated Program Repair with Process-based Feedback](https://mlsanity.com/paper/2408.11296/) (2024)
- [EduNLP: Towards a Unified and Modularized Library for Educational Resources](https://mlsanity.com/paper/2406.01276/) (2024)
- [Foundation of Intelligence: Review of Math Word Problems from Human Cognition Perspective](https://mlsanity.com/paper/2510.21999/) (2025)
- [A Knowledge-Injected Curriculum Pretraining Framework for Question Answering](https://mlsanity.com/paper/2403.09712/) (2024)

## Abstract (arXiv)

Mathematical reasoning is one of the crucial abilities of general artificial intelligence, which requires machines to master mathematical logic and knowledge from solving problems. However, existing approaches are not transparent (thus not interpretable) in terms of what knowledge has been learned and applied in the reasoning process. In this paper, we propose a general Learning by Applying (LeAp) framework to enhance existing models (backbones) in a principled way by explicit knowledge learning. In LeAp, we perform knowledge learning in a novel problem-knowledge-expression paradigm, with a Knowledge Encoder to acquire knowledge from problem data and a Knowledge Decoder to apply knowledge for expression reasoning. The learned mathematical knowledge, including word-word relations and word-operator relations, forms an explicit knowledge graph, which bridges the knowledge "learning" and "applying" organically. Moreover, for problem solving, we design a semantics-enhanced module and a reasoning-enhanced module that apply knowledge to improve the problem comprehension and symbol reasoning abilities of any backbone, respectively. We theoretically prove the superiority of LeAp's autonomous learning mechanism. Experiments on three real-world datasets show that LeAp improves all backbones' performances, learns accurate knowledge, and achieves a more interpretable reasoning process.