Previous neural solvers of math word problems (MWPs) are learned with full supervision and fail to generate diverse solutions. In this paper, we address this issue by introducing a weakly-supervised paradigm for learning MWPs. Our method only requires the annotations of the final answers and can generate various solutions for a single problem. To boost weakly-supervised learning, we propose a novel learning-by-fixing (LBF) framework, which corrects the misperceptions of the neural network via symbolic reasoning. Specifically, for an incorrect solution tree generated by the neural network, the fixing mechanism propagates the error from the root node to the leaf nodes and infers the most probable fix that can be executed to get the desired answer. To generate more diverse solutions, tree regularization is applied to guide the efficient shrinkage and exploration of the solution space, and a memory buffer is designed to track and save the discovered various fixes for each problem. Experimental results on the Math23K dataset show the proposed LBF framework significantly outperforms reinforcement learning baselines in weakly-supervised learning. Furthermore, it achieves comparable top-1 and much better top-3/5 answer accuracies than fully-supervised methods, demonstrating its strength in producing diverse solutions.

Solving algebra story problems remains a challenging task in artificial intelligence, which requires a detailed understanding of real-world situations and a strong mathematical reasoning capability. Previous neural solvers of math word problems directly translate problem texts into equations, lacking an explicit interpretation of the situations, and often fail to handle more sophisticated situations. To address such limits of neural solvers, we introduce the concept of a situation model, which originates from psychology studies to represent the mental states of humans in problem-solving, and propose SMART, which adopts attributed grammar as the representation of situation models for algebra story problems. Specifically, we first train an information extraction module to extract nodes, attributes and relations from problem texts and then generate a parse graph based on a pre-defined attributed grammar. An iterative learning strategy is also proposed to further improve the performance of SMART. To study this task more rigorously, we carefully curate a new dataset named ASP6.6k. Experimental results on ASP6.6k show that the proposed model outperforms all previous neural solvers by a large margin, while preserving much better interpretability. To test these models' generalization capability, we also design an out-of-distribution (OOD) evaluation, in which problems are more complex than those in the training set. Our model exceeds state-of-the-art models by 17% in the OOD evaluation, demonstrating its superior generalization ability.