Roussel Rahman

Roussel Rahman

An essential element of human mathematical reasoning is our number sense -- an abstract understanding of numbers and their relationships -- which allows us to solve problems involving vast number spaces using limited computational resources. Mathematical reasoning of Large Language Models (LLMs) is often tested on high-level problems (such as Olympiad challenges, geometry, word problems, and puzzles), but their low-level number sense remains less explored. We introduce "Numberland," a 100-problem test to evaluate the numerical reasoning abilities of LLM-based agents. The tasks -- basic operations, advanced calculations (e.g., exponentiation, complex numbers), prime number checks, and the 24 game -- aim to test elementary skills and their integration in solving complex and uncertain problems. We evaluated five LLM-based agents: OpenAI's o1 and o1-mini, Google Gemini, Microsoft Copilot, and Anthropic Claude. They scored 74-95% on the first three tasks that allow deterministic steps to solutions. In the 24 game, which needs trial-and-error search, performance dropped to 10-73%. We tested the top 24 solver (o1 with 73% accuracy) on 25 harder problems, and its score fell to 27%, confirming search as a bottleneck. These results, along with the types of mistakes, suggest a fragile number of LLMs, which is a bit surprising given their prowess in challenging benchmarks. The limits of LLM numerical reasoning highlight the scope of simple, targeted tests to evaluate and explain LLM math skills to ensure safe use.

Roussel Rahman, Aashwin Ananda Mishra

Large Language Models (LLMs) have demonstrated remarkable emergent capabilities, yet the robustness of their numerical reasoning remains an open question. While standard benchmarks evaluate LLM reasoning on complex problem sets using aggregated metrics, they often obscure foundational weaknesses. In this work, we probe LLM mathematical numeracy by evaluating performance on problems of escalating complexity, from constituent operations to combinatorial puzzles. We test several state-of-the-art LLM-based agents on a 100-problem challenge comprising four categories: (1) basic arithmetic, (2) advanced operations, (3) primality checking, and (4) the Game of 24 number puzzle. Our results show that while the agents achieved high accuracy on the first three categories, which require deterministic algorithmic execution, they consistently failed at the number puzzle, underlining its demand for a heuristic search over a large combinatorial space to be a significant bottleneck. These findings reveal that the agents' proficiency is largely confined to recalling and executing known algorithms, rather than performing generative problem-solving. This suggests their apparent numerical reasoning is more akin to sophisticated pattern-matching than flexible, analytical thought, limiting their potential for tasks that require novel or creative numerical insights.

Roussel Rahman, Jeff Shrager

Strategy Choice Theory (SCT)\footnote{``Strategy Choice Theory'', ``Distributions of Associations'', and ``Overlapping Wave Theory'' have been used to refer to this line of work, emphasizing different aspects.}\citep[e.g.,][]{siegler1984strategychoices, siegler2000rebirth} explains important aspects of children's arithmetic learning based upon principles including learning from developmentally naturalistic data, probabilistic representation, confidence-based retrieval, and the phase-like importance of scaffolding strategies, such as finger-counting. Here we recast SCT as a ``Small Math Model'' (SMM), employing a neural-network-based architecture analogous to LLMs. The SMM extends SCT to include counting practice\footnote{The original SCT model was pre-biased in accordance with the supposed experience of counting.}, symbol (number) embedding, and gated attention. Similar to earlier work, the SMM demonstrates constructive and destructive interference between counting and addition, and the ``wave-like'' use of finger-counting as sum recall improves. We plan to extend the SMM to later aspects of the decades-long SCT program, including adaptive strategy choice and eventually strategy discovery, providing a unified platform to investigate the understanding of numerical characteristics and relationships essential for mathematical reasoning -- as it can emerge in LLM-based agents.

Roussel Rahman, Aashwin Ananda Mishra, Wan-Lin Hu

Both humans and machine learning models learn from experience, particularly in safety- and reliability-critical domains. While psychology seeks to understand human cognition, the field of Explainable AI (XAI) develops methods to interpret machine learning models. This study bridges these domains by applying computational tools from XAI to analyze human learning. We modeled human behavior during a complex real-world task -- tuning a particle accelerator -- by constructing graphs of operator subtasks. Applying techniques such as community detection and hierarchical clustering to archival operator data, we reveal how operators decompose the problem into simpler components and how these problem-solving structures evolve with expertise. Our findings illuminate how humans develop efficient strategies in the absence of globally optimal solutions, and demonstrate the utility of XAI-based methods for quantitatively studying human cognition.