Cognitive and developmental aspects of mathematical skills

Latest papers

• Order-Invariant Answers, Order-Sensitive Representations in Mathematical Reasoning

  Zhixu Silvia Tao · 24 September 2026

  Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings …

• Toward a Unified Mathematics of Concepts

  Chen Shani · 22 September 2026

  Concepts are commonly defined as abstract, compact representations of knowledge and treated as basic units of intelligent behavior. Yet, cognition, psychology, and AI lack a shared mathematical language for them. Modern systems represent concepts as vectors, distributions, symbols, graphs, and other…

• AdaR: A Framework for Equipping LLMs with Adaptive Reasoning

  Zhejian Lai, Xiang Geng, Zhijun Wang, Yang Bai, Jiahuan Li, Rongxiang Weng, Jingang Wang, Xuezhi Cao, Xunliang Cai, Shujian Huang · 25 August 2026

  Mathematical reasoning is a primary indicator of large language models (LLMs) intelligence. However, existing LLMs exhibit failures in robustness and generalization. This paper attributes these deficiencies to spurious reasoning, wherein generated reasoning traces are driven by superficial correlati…

• Probing the Origins of Reasoning Performance: Representational Quality for Mathematical Problem-Solving in RL vs. SFT Fine-Tuned Models

  Antyabha Rahman, Akshaj Gurugubelli, Omar Ankit, Kevin Zhu, Aishwarya Balwani · 30 July 2026

  Large reasoning models trained via reinforcement learning (RL) have been increasingly shown to outperform their supervised fine-tuned (SFT) counterparts on mathematical reasoning tasks; Yet the mechanistic basis for this advantage remains unclear. We therefore ask, what internal representational dif…

• Hard or Just Unreached? Diagnosing the Sampling Blind Spot in Math-Reasoning Difficulty Estimation

  Luca Zhou, Sajel Shah, Emanuele Rodol\`a, Roberto Dess\`i · 19 June 2026

  Math and science reasoning benchmarks rely on pass@k, the fraction of sampled chains that reach gold, as the canonical per-example difficulty signal. The same signal drives RL with verifiable rewards, math data curation, synthetic curricula, and verifier training. We show this proxy has a persistent…

• LLM Parameters for Math Across Languages: Shared or Separate?

  Behzad Shomali, Luisa Victor, Tim Selbach, Ali Hamza Bashir, David Berghaus, Joachim Koehler, Mehdi Ali, Markus Frey · 18 June 2026

  Large language models (LLMs) exhibit substantial cross-lingual variation in mathematical reasoning performance, but it remains unclear whether these differences reflect language-specific parameters or a shared mechanism that manifests differently by language. We present a cross-lingual mechanistic a…

• Who Brought Easter Eggs to Eid? Auditing Cultural Translation of Math Word Problems Across Diverse Languages and Regions

  Parisa Suchdev, Juniper Lovato · 10 June 2026

  Large language models are increasingly used to adapt math word problems for personalized learning at scale, but it remains an open question whether those adaptations are consistent across models, preserve cultural diversity at scale, and reveal which cultural entities models treat as most salient. W…

• Benchmarks in Leipzig

  Andrei Balakin, Mikl\'os B\'ona, Marie-Charlotte Brandenburg, Clara Briand, Veronica Calvo Cortes, Shelby Cox, Jesus A. De Loera, Danai Deligeorgaki, Hannah Friedman, Tim Gehrunger, Chiara Giardino, Stephen Griffeth, Baran Hashemi, Elena Hoster, Alexander Ivanov, Nupur Jain, Aryaman Jal, Leonie Kayser, Joris Koefler, Kevin K\"uhn, Mario Kummer, Felix Lotter, Ren\'e Marczinzik, Victor S. Miller, Alejandro Morales, Greta Panova, Gianni Petrella, Nathan Pflueger, Lakshmi Ramesh, Nikolas Rieke, Carlos Rodriguez, Andrea Rosana, Flavio Salizzoni, Otto T. P. Schmidt, Sven Ulf Schmitz, Lina Maria Simbaqueba Marin, Luca Sodomaco, Christian Stump, Bernd Sturmfels, Alexander Taveira Blomenhofer, Simon Telen, Philipp Tuchel, Emil Verkama, Carl Felix Waller, Julian Weigert, Annette Werner, Nathan Williams, Claudius Zibrowius · 5 June 2026

  Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences i…

• Temporal Stability and Few-Shot Prompting in Math Task Assessment

  Danielle S. Fox, Brenda L. Robles, Elizabeth DiPietro Brovey, Christian D. Schunn · 29 May 2026

  As AI tools become increasingly integrated into educational contexts, questions arise about both their stability over time and their responsiveness to prompt engineering techniques. This longitudinal study focused on different AI tools' ability to use the Task Analysis Guide (TAG; Stein \& Smith, 19…

• Transformers Linearly Represent Highly Structured World Models

  Roman Kniazev, Nathana\"el Fijalkow · 20 May 2026

  Do transformers, when trained on sequential reasoning traces, build internal models of the underlying task? And if so, does the structure of those internal representations mirror the structure of the domain? We train an 8-layer transformer on Sudoku solving traces and perform a mechanistic analysis …

• What Really Improves Mathematical Reasoning: Structured Reasoning Signals Beyond Pure Code

  Yuze Zhao, Junpeng Fang, Lu Yu, Zhenya Huang, Kai Zhang, Qing Cui, Qi Liu, Jun Zhou, Enhong Chen · 20 May 2026

  Code has become a standard component of modern foundation language model (LM) training, yet its role beyond programming remains unclear. We revisit the claim that code improves reasoning through controlled pretraining experiments on a 10T-token corpus with fine-grained domain separation. Our finding…

• Leveraging Speech to Identify Signatures of Insight and Transfer in Problem Solving

  Linas Nasvytis, Judith E. Fan · 14 May 2026

  Many problems seem to require a flash of insight to solve. What form do these sudden insights take, and what impact do they have on how people approach similar problems in the future? In this work, we prompted participants (N = 189) to think aloud as they attempted to solve a sequence of five "match…

• Discovering Learning-Friendly Generation Orders for Sequential Computation

  Yuta Sato, Kazuhiko Kawamoto, Hiroshi Kera · 11 May 2026

  Sequential computation via autoregressive generation can make difficult tasks learnable, but the generation order of intermediate states strongly affects whether training succeeds. We address the problem of discovering a learning-friendly target order automatically, rather than relying on task-speci…

• Math Takes Two: A test for emergent mathematical reasoning in communication

  Michael Cooper, Samuel Cooper · 27 April 2026

  Although language models demonstrate remarkable proficiency on mathematical benchmarks, it remains unclear whether this reflects true mathematical reasoning or statistical pattern matching over learning formal syntax. Most existing evaluations rely on symbolic problems grounded in established mathem…

• Schoenfeld's Anatomy of Mathematical Reasoning by Language Models

  Ming Li, Chenrui Fan, Yize Cheng, Soheil Feizi, Tianyi Zhou · 24 April 2026

  Large language models increasingly expose reasoning traces, yet their underlying cognitive structure and steps remain difficult to identify and analyze beyond surface-level statistics. We adopt Schoenfeld's Episode Theory as an inductive, intermediate-scale lens and introduce ThinkARM (Anatomy of Re…

• Language Models Learn Universal Representations of Numbers and Here's Why You Should Care

  Michal \v{S}tef\'anik, Timothee Mickus, Marek Kadl\v{c}\'ik, Bertram H{\o}jer, Michal Spiegel, Ra\'ul V\'azquez, Aman Sinha, Josef Kucha\v{r}, Philipp Mondorf, Pontus Stenetorp · 24 April 2026

  Prior work has shown that large language models (LLMs) often converge to accurate input embedding for numbers, based on sinusoidal representations. In this work, we quantify that these representations are in fact strikingly systematic, to the point of being almost perfectly universal: different LLM …

• Language Models Learn Universal Representations of Numbers and Here's Why You Should Care

  Michal \v{S}tef\'anik, Timothee Mickus, Marek Kadl\v{c}\'ik, Bertram H{\o}jer, Michal Spiegel, Ra\'ul V\'azquez, Aman Sinha, Josef Kucha\v{r}, Philipp Mondorf, Pontus Stenetorp · 23 April 2026

  Prior work has shown that large language models (LLMs) often converge to accurate input embedding for numbers, based on sinusoidal representations. In this work, we quantify that these representations are in fact strikingly systematic, to the point of being almost perfectly universal: different LLM …

• Disentangling Mathematical Reasoning in LLMs: A Methodological Investigation of Internal Mechanisms

  Tanja Baeumel, Josef van Genabith, Simon Ostermann · 20 April 2026

  Large language models (LLMs) have demonstrated impressive capabilities, yet their internal mechanisms for handling reasoning-intensive tasks remain underexplored. To advance the understanding of model-internal processing mechanisms, we present an investigation of how LLMs perform arithmetic operatio…

• Large Language Models for Math Education in Low-Resource Languages: A Study in Sinhala and Tamil

  Sukumar Kishanthan, Kumar Thushalika, Buddhi Jayasekara, Asela Hevapathige · 20 April 2026

  Large language models (LLMs) have achieved strong results in mathematical reasoning, and are increasingly deployed as tutoring and learning support tools in educational settings. However, their reliability for students working in non-English languages, especially low-resource languages, remains poor…

• Minimal Embodiment Enables Efficient Learning of Number Concepts in Robot

  Zhegong Shangguan, Alessandro Di Nuovo, Angelo Cangelosi · 14 April 2026

  Robots are increasingly entering human-interactive scenarios that require understanding of quantity. How intelligent systems acquire abstract numerical concepts from sensorimotor experience remains a fundamental challenge in cognitive science and artificial intelligence. Here we investigate embodied…

• Exploring Natural Language-Based Strategies for Efficient Number Learning in Children through Reinforcement Learning

  Tirthankar Mittra · 9 April 2026

  In this paper, we build a reinforcement learning framework to study how children compose numbers using base-ten blocks. Studying numerical cognition in toddlers offers a powerful window into the learning process itself, because numbers sit at the intersection of language, logic, perception, and cult…

• 4OPS: Structural Difficulty Modeling in Integer Arithmetic Puzzles

  Yunus E. Zeytuncu · 27 March 2026

  Arithmetic puzzle games provide a controlled setting for studying difficulty in mathematical reasoning tasks, a core challenge in adaptive learning systems. We investigate the structural determinants of difficulty in a class of integer arithmetic puzzles inspired by number games. We formalize the pr…

• Weber's Law in Transformer Magnitude Representations: Efficient Coding, Representational Geometry, and Psychophysical Laws in Language Models

  Jon-Paul Cacioli · 24 March 2026

  How do transformer language models represent magnitude? Recent work disagrees: some find logarithmic spacing, others linear encoding, others per-digit circular representations. We apply the formal tools of psychophysics to resolve this. Using four converging paradigms (representational similarity an…

• Integrating Arithmetic Learning Improves Mathematical Reasoning in Smaller Models

  Neeraj Gangwar, Suma P Bhat, Nickvash Kani · 19 March 2026

  While large models pre-trained on high-quality data exhibit excellent performance on mathematical reasoning (e.g., GSM8k, MultiArith), it remains challenging to specialize smaller models for these tasks. Common approaches to address this challenge include knowledge distillation from large teacher mo…

• On the mechanical creation of mathematical concepts

  Asvin G · 11 March 2026

  Any act of problem-solving combines prior knowledge, local search, and a third element that is less often discussed: the extraction of information from search to update understanding. I propose a model of mathematical problem-solving as a belief-update loop in which the mathematician generates auxil…