  Large-scale pre-trained language models (PLMs) bring new opportunities to
challenging problems, especially those that need high-level intelligence, such
as the math word problem (MWPs). However, directly applying existing PLMs to
MWPs can fail as the generation process lacks sufficient supervision and thus
lacks fast adaptivity as humans. We notice that human reasoning has a dual
reasoning framework that consists of an immediate reaction system (system 1)
and a delicate reasoning system (system 2), where the entire reasoning is
determined by their interaction. This inspires us to develop a cooperative
reasoning-induced PLM for solving MWPs, called Cooperative Reasoning (CoRe),
resulting in a human-like reasoning architecture with system 1 as the generator
and system 2 as the verifier. In our approach, the generator is responsible for
generating reasoning paths, and the verifiers are used to supervise the
evaluation in order to obtain reliable feedback for the generator. We evaluate
our CoRe framework on several mathematical reasoning datasets and achieve
decent improvement over state-of-the-art methods, up to 9.6% increase over best
baselines. Our codes are available at https://github.com/TianHongZXY/CoRe
