Tung Duong Ta, Tim Oates, Thien Van Luong, Huan Vu, Tien Cuong Nguyen
Despite advances in mathematical reasoning capabilities, Large Language Models (LLMs) still struggle with calculation verification when using established prompting techniques. Chain-of-Thought prompting and its self-consistency variant lack evaluation of intermediate reasoning steps. Hierarchical techniques such as Tree-of-Thoughts (ToT) and Graph-of-Thoughts (GoT) evaluate abstract thoughts, which are difficult to score reliably. We present MDToC (Metacognitive Dynamic Tree of Concepts), a three-phase prompting technique built on the metacognitive stages of planning, monitoring, and reviewing. MDToC constructs a concept tree, develops accuracy-verified calculations for each concept, and employs majority voting to evaluate competing solutions. Evaluations across CHAMP, MATH, and Game-of-24 benchmarks demonstrate our MDToC's effectiveness, with GPT-4-Turbo achieving 58.1% on CHAMP, 86.6% on MATH, and 85% on Game-of-24, outperforming GoT by 5%, 5.4%, and 4% on all these tasks, respectively, without hand-engineered hints. MDToC consistently matches or surpasses existing prompting methods across all backbone models, yielding improvements of up to 19% over ToT and 13% over GoT. In addition, delegating auxiliary components to lighter models reduces cost while maintaining accuracy. These results establish metacognitive calculation verification as a promising direction for enhanced mathematical reasoning.