the training set. Specifically, we propose to bootstrap the questions in both forward and backward
reasoning directions. For the forward direction, we have the original and LLM-rephrased questions.
For the backward direction, we have the self-verification question [68] and FOBAR question [28].
To construct backward reasoning questions, we mask a token in a question using an identifier “x”
and ask the model to predict the masked token if the answer is provided. Different from [28, 68]
that apply backward reasoning for inference verification, we use it as a form of question for lan-
guage model fine-tuning. For answers, we adopt an answer augmentation method based on rejection
sampling [69], where diverse reasoning paths are generated and only those with correct answers
are used. After combining both forward and backward mathematical questions with augmented
answers, we construct a new dataset for fine-tuning, called MetaMathQA. By fine-tuning LLaMA-2
METAMATH