U-MATH & μ-MATH: Assessing LLMs on university-level math
Why U-MATH?
U-MATH is the largest university-level math benchmark, designed to address the gap in existing math benchmarks, which are limited in scope and size.
— 1,100 problems from math courses across 6 key subject areas
— 20% of tasks include images to challenge LLMs
— Practical applications for industry and education
What is µ-MATH?
The Meta-Evaluation Benchmark is a set of 1084 meta-evaluation solutions designed to rigorously assess the quality of LLM judges, based on U-MATH problems.
Together, these benchmarks support comprehensive evaluation of LLM proficiency on university-level math.
LLM leaderboard on U-Math
Both U-Math and μ-MATH datasets were collected with the help of Gradarius - a learning platform that helps students master calculus through a step-by-step approach, providing immediate feedback to guide them through problem-solving.
Insights into LLM performance on U-Math
Reasoning models boast breakthrough performance, but university-level problems are still a challenge.
Gemini emerges as the overall winner across the board — from smaller models to reasoner systems.
Open-weight models are rapidly closing the gap on text-only tasks, but continue to lag on multi-modal problems.
Specialization trumps size: domain-specific models such as Qwen Math beat models an order of magnitude larger.
Integrating visual reasoning proves tough: U-MATHv scores lag significantly behind text-only performance, and adding visual capabilities to a model typically leads to degradation.
Gemini models are consistently most adept at visual reasoning, exhibiting a large U-MATHv margin within all the model groups.
Comparison of models’ accuracy on our U-MATH benchmark and its subjects. Scores for various mathematical categories, including text and visual analysis, are displayed. For each subject 2 numbers are provided - text-only (T) and visual (V) problems. Asterisk denotes a small number of samples (<15). Free-form solutions judged by gpt-4o-2024-08-06. Images are not included in the prompt for text-only models, only the problem statement. Bold indicates the best result in each group.
What makes
U-MATH stand out
Challenging problems to test deep understanding
and advanced reasoning skills
Covers 6 subjects: Algebra, Precalculus, Differential Calculus, Integral Calculus, Multivariable Calculus, Sequences & Series
Problems and solutions are sourced from real coursework and checked by experts to meet academic standards
Existing auto-evaluation math benchmarks with corresponding test samples published, visual samples percent,and percent of free-form answers. Levels include university-level benchmarks as well as lower educational levels. Some of the problems in U-MATH and mu-MATH are sourced from OpenStax under CC BY 4.0.
U-Math Dataset Collection Process
To identify the most challenging problems from tens of thousands of examples available for our benchmark we used a multi-stage selection process:
Filter out
unsuitable problems
We exclude easy problems, those requiring extensive calculations, and multiple-choice problems.
Test LLMs
Solve the selected problems using popular small LLMs
Analyze results
Choose the most challenging problems in each subject area
Expert validation
The final set of problems
is checked by experts from Stevens Institute of Technology
U-Math Data Samples
μ-MATH Meta-Evaluation Benchmark insights
Problem-solving performance ≠ judgment performance. A tradeoff between these skills emerges in non-reasoners.
The tradeoff yields distinctive judgment styles: proprietary models are more conservative, minimizing false positives, while Qwens tend to be more lenient.
Reasoning models push through the Pareto frontier, typically inducing substantial imbalances. o1 pushes even further and exhibits a far more weighted performance.
Judging is a non-trivial skill. The maximum attainable performance is imperfect, even with reasoners, so we need to account for errors when using auto-evaluation.
A balanced mix of training data leads to well-rounded performance, as evidenced by Qwen2.5 and Qwen2.5-Math.
Reducing the model size, on the contrary, exacerbates bias. Balanced performance requires the model to generalize over the training mixture.
Which LLMs excel at judging math solutions?
U-MATH shows accuracy in solving math problems. μ-MATH reflects accuracy in judging solutions
μ-MATH Dataset Collection Process
Robust test for LLM judges
Judgment errors and biases in evaluations are often overlooked, creating uncertainty and unreliability. Meta-evaluations are essential to identify, quantify, and address these issues, yet they remain scarce, especially in math.
Dataset construction
— Hand-picked ~25% of U-MATH problems (271 in total), selected for their judgement complexity and representing university-level math
— Generated 4 solutions for each: using Qwen2.5-72B, Llama3.1-70B, GPT4-o and Gemini1.5-Pro, resulting in 1084 problem-solution pairs
— Supplied each pair with a correct judgment verdict via a combination of Toloka's math experts labeling and Gradarius formal autoverification API
— Treat the judgment problem as a binary classification task, compute standard binary metrics, with macro F1 as the main one (so that both positive and negative labels contribute equally)
Testing and metrics
During testing, a model is provided with a problem statement, a reference answer, and a solution to evaluate. We treat this as a binary classification task.
Primary metric:
Macro-averaged F1 score to minimize the effect of class imbalance
Fine-grained metrics:
Positive Predictive Value (PPV, or Precision) and True Positive Rate (TPR, or Recall) for the positive class
Negative Predictive Value (NPV) and True Negative Rate (TNR) for the negative class
Data Sample
An example problem from the µ-MATH meta-evaluation benchmark, illustrating the comparison between the golden (reference) answer and the answer generated by an LLM.
Frequently Asked Questions
Toloka’s U-MATH dataset is tailored for working with advanced mathematical problems. It provides structured data to simplify problem-solving processes and analyze complex mathematical ideas, supporting a wide range of applications in academic research, education, and AI training with an emphasis on measuring mathematical problem-solving skills for both models and students.
Here are answers to some common questions about the math dataset concept and Toloka's U-math dataset.