The use of large language models (LLMs) for complex mathematical reasoning is an emergent area of research, with fast progress in methods, models, and benchmark datasets. However, most mathematical reasoning evaluations exhibit a significant linguistic bias, with the vast majority of benchmark datasets being exclusively in English or (at best) translated from English. We address this limitation by introducing {\sc Math-PT}, a novel dataset comprising 1,729 mathematical problems written in European and Brazilian Portuguese. {\sc Math-PT} is curated from a variety of high-quality native sources, including mathematical Olympiads, competitions, and exams from Portugal and Brazil. We present a comprehensive benchmark of current state-of-the-art LLMs on {\sc Math-PT}, revealing that frontier reasoning models achieve strong performance in multiple choice questions compared to open weight models, but that their performance decreases for questions with figures or open-ended questions. To facilitate future research, we release the benchmark dataset and model outputs.
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 analysis of mathematical reasoning in LLMs, enabling us to localize and compare model parameters that support mathematical reasoning across languages. We find that the extracted math-associated parameters exhibit partial cross-lingual overlap, with the strongest overlap concentrated in intermediate model layers. We further observe that English consistently produces the largest set of math-relevant parameters, whereas lower-resource languages reveal smaller sets of relevant parameters. These results suggest that math-related behavior in multilingual LLMs is neither fully language-invariant nor fully language-specific, but instead exhibits partial cross-lingual parameter overlap with systematic language-dependent differences.
Advances in large language models (LLMs) have spurred research into enhancing their reasoning capabilities, particularly in math-rich STEM (Science, Technology, Engineering, and Mathematics) documents. While LLMs can generate equations or solve math-related queries, their ability to fully understand and interpret abstract mathematical symbols in long, math-rich documents remains limited. In this paper, we introduce STEM-PoM, a comprehensive benchmark dataset designed to evaluate LLMs' reasoning abilities on math symbols within contextual scientific text. The dataset, sourced from real-world ArXiv documents, contains over 2K math symbols classified as main attributes of variables, constants, operators, and unit descriptors, with additional sub-attributes including scalar/vector/matrix for variables and local/global/discipline-specific labels for both constants and operators. Our extensive experiments demonstrate that state-of-the-art LLMs achieve an average accuracy of 20-60% under in-context learning and 50-60% with fine-tuning, highlighting a substantial gap in their ability to classify mathematical symbols. By improving LLMs' mathematical symbol classification, STEM-PoM further e
The evolution of mathematics is shaped importantly by interestingness: researchers choose which problems to pursue, and students choose which problems to engage with, based on expectations of interest and challenge. As AI systems, particularly large language models (LLMs) that operate flexibly over natural language and formal mathematics, are increasingly used in mathematics research and education, it becomes crucial to characterize how closely their judgments align with people from different mathematical backgrounds. We study whether LLMs align with human interestingness judgments by comparing LLM ratings with those of two populations, crowdsourced participants with college math experience and International Math Olympiad competitors. Although many LLMs broadly agree with human notions of interestingness, they largely fail to match the distribution of human judgments. They also weakly align with why humans find problems interesting, with low correlation to human-selected rationales. Finally, we evaluate LLMs' ability to generate interesting problems and find that, after filtering for validity, LLMs are able to generate engaging problems. We conclude with takeaways, including the ne
The demand for Large Language Models (LLMs) at multiple scales, capable of sophisticated and sound mathematical reasoning, continues to grow. However, the development of performant mathematical LLMs is often bottlenecked by the scarcity of useful training data containing problems with significant complexity. We introduce \textbf{SAND-Math} (\textbf{S}ynthetic \textbf{A}ugmented \textbf{N}ovel and \textbf{D}ifficult Mathematics problems and solutions), a pipeline that addresses this by first synthesizing high-quality problems from scratch and then systematically elevating their complexity via a our newly proposed \textbf{Difficulty Hiking} step. We demonstrate the effectiveness of our approach through two key findings: \textbf{(1)} Augmenting a strong post-training baseline with a small 500-sample SAND-Math dataset significantly boosts performance, outperforming the next-best synthetic dataset by $\uparrow$ 17.85 absolute points on AIME25 benchmark. \textbf{(2)} In a dedicated ablation study, we show the effectiveness of our Difficulty Hiking process in increasing average problem difficulty from 5.02 to 5.98. This step consequently lifts AIME25 results from 46.38\% to 49.23\%. The f
In this arxiv-post I present my solutions (published or not) to Problems that appeared in Amer. Math. Monthly, Math. Magazine, Elemente der Mathematik and CRUX, that were mostly done in collaboration with Rudolf Rupp. Some of them (including a few own proposals which were published) were also done in cooperation with Rainer Brück, Bikash Chakraborty, Pamela Gorkin, Gerd Herzog, Jérôme Noël, Peter Pflug and Amol Sasane.
"Is math useful?" might sound as a trick question. And it is. Of course math is useful, we live in a data-filled world and every aspect of life is totally entwined with math applications, both trivial and subtle applications, of both basic and advanced math. But we need to ask once again that question, in order to truly understand what is math useful for and what being useful means. Moreover, is it knowledge of math useful for a class of specialists, or for political leaders or for all people at large? Being more on a concrete level, why does math need to have a central role in education? Each section will be titled by a question. And each section will not give an answer, but -- at least I hope -- provide some food for tought to the reader, in order to try to come up with his or her own answers. I feel that these kind of questions are at home in a book devoted to the interplays between mathematics and culture: what is the space we should give to math in culture and what is math's role in becoming a complete citizen?
Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.
Math anxiety negatively relates to math performance. This negative relationship may be exacerbated in low-progress math learners. However, there are limited studies on math anxiety among low progress learners in a paradoxically high performing education system like Singapore. To fill this research gap, this research analysed the anxiety profiles of 151 students who were in the math learning support intervention program administered by the Ministry of Education, Singapore (MOE). We examined the complex relationship centred in math anxiety with relevant variables such as demographic characteristics, working memory and math performance. Limitations and future directions are discussed.
Solving mathematical problems requires advanced reasoning abilities and presents notable challenges for large language models. Previous works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results. However, our analysis of these datasets reveals severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries. Hypothesizing that difficult queries are crucial to learn complex reasoning, we propose Difficulty-Aware Rejection Tuning (DART), a method that allocates difficult queries more trials during the synthesis phase, enabling more extensive training on difficult samples. Utilizing DART, we have created new datasets for mathematical problem-solving that focus more on difficult queries and are substantially smaller than previous ones. Remarkably, our synthesis process solely relies on a 7B-sized open-weight model, without reliance on the commonly used proprietary GPT-4. We fine-tune various base models on our datasets ranging from 7B to 70B in size, resulting in a series of strong models called DART-MATH. In comprehensive in-domain and out-of-
Math anxiety is a highly prevalent problem in education that has consistently shown to lead to poor math performance. This study sought to investigate whether certain behaviours are predictive of math anxiety among students. This study involved elementary school students who were low-progressing in math, and is part of an educational intervention program. Ten classifications types of behavioural indicators were identified, such as counting out loud. A multiple linear regression was conducted, identifying three behavioural observations that were positively and significantly associated with their math anxiety. Implications and limitations are discussed.
Choreographer Reggie Wilson and mathematician Jesse Wolfson describe interactions of math and dance emerging from their 12+ year engagement with Black movement and music traditions as part of Wilson's research-to-performance/performance-to-research choreographic practice, with examples including fractals, braids and choreographic and mathematical notions of space, time and movement.
Many students struggle with math word problems (MWPs), often finding it difficult to identify key information and select the appropriate mathematical operations. Schema-based instruction (SBI) is an evidence-based strategy that helps students categorize problems based on their structure, improving problem-solving accuracy. Building on this, we propose a Schema-Based Instruction Retrieval-Augmented Generation (SBI-RAG) framework that incorporates a large language model (LLM). Our approach emphasizes step-by-step reasoning by leveraging schemas to guide solution generation. We evaluate its performance on the GSM8K dataset, comparing it with GPT-4 and GPT-3.5 Turbo, and introduce a "reasoning score" metric to assess solution quality. Our findings suggest that SBI-RAG enhances reasoning clarity and facilitates a more structured problem-solving process potentially providing educational benefits for students.
Research suggests that tutors should adopt a strategic approach when addressing math errors made by low-efficacy students. Rather than drawing direct attention to the error, tutors should guide the students to identify and correct their mistakes on their own. While tutor lessons have introduced this pedagogical skill, human evaluation of tutors applying this strategy is arduous and time-consuming. Large language models (LLMs) show promise in providing real-time assessment to tutors during their actual tutoring sessions, yet little is known regarding their accuracy in this context. In this study, we investigate the capacity of generative AI to evaluate real-life tutors' performance in responding to students making math errors. By analyzing 50 real-life tutoring dialogues, we find both GPT-3.5-Turbo and GPT-4 demonstrate proficiency in assessing the criteria related to reacting to students making errors. However, both models exhibit limitations in recognizing instances where the student made an error. Notably, GPT-4 tends to overidentify instances of students making errors, often attributing student uncertainty or inferring potential errors where human evaluators did not. Future work
This is a corrigendum to Acta Math. 196 (2006) as well as to the follow-up publications JFA 259 (2010) and to JFA 260 (2011).
Recent progress in large language models (LLMs) like GPT-4 and PaLM-2 has brought significant advancements in addressing math reasoning problems. In particular, OpenAI's latest version of GPT-4, known as GPT-4 Code Interpreter, shows remarkable performance on challenging math datasets. In this paper, we explore the effect of code on enhancing LLMs' reasoning capability by introducing different constraints on the \textit{Code Usage Frequency} of GPT-4 Code Interpreter. We found that its success can be largely attributed to its powerful skills in generating and executing code, evaluating the output of code execution, and rectifying its solution when receiving unreasonable outputs. Based on this insight, we propose a novel and effective prompting method, explicit \uline{c}ode-based \uline{s}elf-\uline{v}erification~(CSV), to further boost the mathematical reasoning potential of GPT-4 Code Interpreter. This method employs a zero-shot prompt on GPT-4 Code Interpreter to encourage it to use code to self-verify its answers. In instances where the verification state registers as ``False'', the model shall automatically amend its solution, analogous to our approach of rectifying errors duri
This is a concise version of the original article in [arXiv:2203.17115] that will be published in the String Math 2022 Proceedings by the American Mathematical Society.
We correct the proof of Theorem 4.1 from [C. R. Math. Acad. Sci. Soc. R. Can. \textbf{44} (2022), no. 4, 88--112].
Math word problem (MWP) solving aims to understand the descriptive math problem and calculate the result, for which previous efforts are mostly devoted to upgrade different technical modules. This paper brings a different perspective of \textit{reexamination process} during training by introducing a pseudo-dual task to enhance the MWP solving. We propose a pseudo-dual (PseDual) learning scheme to model such process, which is model-agnostic thus can be adapted to any existing MWP solvers. The pseudo-dual task is specifically defined as filling the numbers in the expression back into the original word problem with numbers masked. To facilitate the effective joint learning of the two tasks, we further design a scheduled fusion strategy for the number infilling task, which smoothly switches the input from the ground-truth math expressions to the predicted ones. Our pseudo-dual learning scheme has been tested and proven effective when being equipped in several representative MWP solvers through empirical studies. \textit{The codes and trained models are available at:} \url{https://github.com/steven640pixel/PsedualMWP}. \end{abstract}
In the winter semester of 1890--1891 Adolf Hurwitz delivered a lecture course at the Albertina University in Königsberg entitled -Theorie der algebraischen Gleichungen-. These lectures contain a particularly clear presentation of the ideas of Evariste Galois and, in particular, a proof of the fundamental theorem of Galois theory formulated in the language of substitutions. The present paper analyzes Hurwitz's treatment of this result on the basis of his lecture notes preserved in the ETH Library in Zurich (Hs 582:66), together with material from his Mathematisches Tagebuch 23 (Hs 582:23). After placing the Königsberg lectures in their historical context, we give an overview of their mathematical content and reconstruct in detail Hurwitz's argument leading to the fundamental theorem.