  Large language models (LLMs) have demonstrated remarkable capabilities in
problem-solving. However, their proficiency in solving mathematical problems
remains inadequate. We propose MathScale, a simple and scalable method to
create high-quality mathematical reasoning data using frontier LLMs (e.g., {\tt
GPT-3.5}). Inspired by the cognitive mechanism in human mathematical learning,
it first extracts topics and knowledge points from seed math questions and then
build a concept graph, which is subsequently used to generate new math
questions. MathScale exhibits effective scalability along the size axis of the
math dataset that we generate. As a result, we create a mathematical reasoning
dataset (MathScaleQA) containing two million math question-answer pairs. To
evaluate mathematical reasoning abilities of LLMs comprehensively, we construct
{\sc MwpBench}, a benchmark of Math Word Problems, which is a collection of ten
datasets (including GSM8K and MATH) covering K-12, college, and competition
level math problems. We apply MathScaleQA to fine-tune open-source LLMs (e.g.,
LLaMA-2 and Mistral), resulting in significantly improved capabilities in
mathematical reasoning. Evaluated on {\sc MwpBench}, MathScale-7B achieves
state-of-the-art performance across all datasets, surpassing its best peers of
equivalent size by 42.9\% in micro average accuracy and 43.7\% in macro average
accuracy, respectively.
