Roxanne-WANG/Integration_Problem_Set

Dataset Project: Integration Problem Set

Author: Ruoxin Wang

Date: Nov.26.2024

GitHub Repository

The code is available at https://github.com/WendyWAAAAANG/Integration_Problem_Set

Executive Summary

This project aims to create a dataset specifically designed for the automatic generation of integration problems. It will serve as the basis for training LLMs to generate integration problems. The problems will be labeled by difficulty and category, making it easy for users to select problems.

Motivation

Generating diverse and enough integration problems for practice or exams can be time-consuming. We provide this dataset for fine-tuning LLMs and ask LLMs to generate integration problems. Compared to letting LLMs generate problems directly, a dataset built with symbolic computation and verified mathematical problems ensures higher accuracy and strict adherence to mathematical rules than GPT-generated questions. Secondly, this dataset offers diverse problem types and controlled difficulty levels, something GPT might struggle with in maintaining consistency. Automating the generation of integration problems can provide students with endless opportunities for practice and help them get a deeper conceptual understanding. And teachers can benefit from automated systems that reduce the workload of creating custom problem sets. This system could also be valuable in building AI educational technology platforms.

Potential applications

• Training LLMs for mathematics education.
• Building educational apps that recommend problems tailored to user needs.
• Developing interactive tutoring systems with step-by-step explanations.

Uniqueness About This Dataset

Nowadays, there is no large-scale dataset available in the public domain explicitly focusing on integration problems, although the collection of integration problems in a systematic manner can be an effective way to meet the requirement of problem generation. Current datasets are insufficient for teaching advanced calculus problem generation and do not contain detailed step-by-step problems which is an important component of learning.

1. Comprehensive Coverage of Integration Techniques: The dataset will incorporate all forms of integration issues with basic, intermediate, and advanced integration issues covered. It includes basic integration, substitution, integration by parts, etc Integration by parts computes the integral with the help of integration of other functions.
2. Diversity of Functions: The dataset will include integrals of various classes of functions which in fact will include polynomial, trigonometric, logarithmic, exponential, and hyperbolic functions.

Description of Data

The dataset consists of integration problems in JSON format, including:

• difficulty: The difficulty level of the question, the labels include: "Basic", "Moderate", and "Advanced".
• category: The category of the question which involves "Trigonometric Substitution", "Constant Rule", "Multiple Integrals", "Integration by Parts", "Power Rule", and "Partial Fractions".
• question: The content of the integration question.

Example Entry:

  [
      {
          "difficulty": "Advanced",
          "category": "Integration by Parts",
          "question": "\u222b\nxtan(x)\ncos2(x)\n dx"
      },
      {
          "difficulty": "Basic",
          "category": "Partial Fractions",
          "question": "\u222bcos(3t)sin(8t)dt"
      },
  ]

Power Analysis

Assume that:

• Alpha: 0.05 (Significance Level)
• Effect Size: 0.5 (Moderate Effect)
• Power: 0.8 (80% Probability of Correctly Rejecting the Null Hypothesis)

Conclusion: According to the power curve, the required sample size is 64. This demonstrates that moderate effects are easier to detect with fewer samples, making this scenario practical for most dataset collection projects.

Data Collection Protocol

1. Textbooks: Stewart’s Calculus, Apostol’s Calculus
2. Web Scraping from Open-Source Online Platform:Symbolab, Paul’s Online Math

Explory Data Analysis

1. Category Distribution

• The most frequent category is "Trigonometric Substitution", followed by "Constant Rule", and "Multiple Integrals".
• Less frequent categories include "Integration by Parts", "Partial Fractions", and "Power Rule".
• Trigonometric Substitution appears to dominate, likely indicating its prevalence in integration problems or its importance in the dataset focus.
• Graph Summary: A horizontal bar chart highlights the frequency of each category, emphasizing the dominance of certain categories over others.

2. Question Length Distribution

• Most questions have a length between 8 and 14 characters, indicating a tendency for a concise representation of problems.
• There is a right-skewed distribution, with fewer questions having lengths exceeding 25 characters.
• Longer questions might represent more complex integrals with multiple terms or variables.
• Graph Summary: A histogram with a KDE overlay shows the distribution of question lengths, with a peak of around 10 characters.

3. Difficulty Distribution

• Questions are evenly distributed across the three difficulty levels: Advanced, Moderate, and Basic.
• Advanced problems slightly outnumber the others, suggesting a focus on challenging questions.
• The uniform distribution ensures representation across skill levels, making the dataset comprehensive for different audiences.
• Graph Summary: A vertical bar chart illustrates the similar proportions of problems in each difficulty tier.

4. Word Cloud Analysis

• Common terms in questions include symbols like x, dx, sin, cos, and numerical constants such as 2, 3, and 4.
• These terms focus heavily on single-variable integrals and common trigonometric or polynomial components.
• Patterns like tan, ln, and higher powers suggest questions often involve transcendental functions or logarithmic integrations.
• Graph Summary: A word cloud visualizes the most frequently occurring terms, emphasizing the focus on fundamental and advanced mathematical symbols.

Ethical Statement

The Integral Problem Set project supports educational objectives and facilitates advancements in AI-driven learning tools. This ethical statement outlines the principles and considerations guiding the development and use of this dataset.

Purpose

This dataset aims to provide high-quality, diverse, and accurate integration problems for:

• Educational platforms and applications.
• Training AI models, including Large Language Models (LLMs), for mathematics education.
• Assisting students, educators, and researchers in advancing mathematical problem-solving tools.

Data Sourcing

1. Open and Publicly Available Sources:

   • The problems are collected exclusively from publicly available resources, such as free online educational platforms, open-access textbooks, and reliable mathematics tools like Symbolab, Paul's Online Math Notes, and Wolfram Alpha.

2. No Proprietary Content:

   • The dataset does not include proprietary, copyrighted, or restricted materials. All content is sourced in compliance with the terms of service of the platforms used for collection.

3. Transparent Collection:

   • The methods and tools for data collection (e.g., web scraping scripts) are openly documented and included in the project repository, ensuring reproducibility and transparency.

Use of the Dataset

1. Encouraging Ethical Applications:

   • The dataset is intended for educational, research, and academic purposes. Potential applications include training AI models for educational platforms, automating problem generation, and building interactive math tutoring systems.

2. Preventing Misuse:

   • The dataset should not be used for unethical purposes, such as:
     • Academic dishonesty (e.g., cheating in exams or assignments).
     • Developing systems that infringe on intellectual property rights.

3. Accessibility:

   • The dataset is open-source under the MIT License, ensuring accessibility to educators, students, and researchers while requiring proper attribution.

Commitment to Responsible AI

This project aligns with the principles of responsible AI development:

• Transparency: All data collection methods, tools, and validation techniques are openly shared.
• Fairness: The dataset is diverse and balanced, catering to learners of varying skill levels.
• Accountability: The project contributors are committed to addressing ethical concerns and improving the dataset through community feedback.

License

This dataset is licensed under the MIT License.