AI To Transform Mathematics Education; Possibilities And Challenges Of Solving Mathematical Problems Using Large-Scale Language Models

3 main points
✔️ Large-scale language models play an important role in solving complex mathematical problems
✔️ Lack of a unified evaluation framework and adaptability to different problem types in current LLM math problem solving
✔️ Large-scale language models from an educational perspective

Large Language Models for Mathematical Reasoning: Progresses and Challenges
written by Janice Ahn, Rishu Verma, Renze Lou, Di Liu, Rui Zhang, Wenpeng Yin
(Submitted on 31 Jan 2024 (v1))
Comments: EACL 2024 Student Research Workshop.
Subjects: Computation and Language (cs.CL)

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The images used in this article are from the paper, the introductory slides, or were created based on them.

Summary

Mathematical reasoning is an essential component of human intelligence, and as the AI community continues to seek ways to address mathematical challenges, this pursuit will require further improvements in AI capabilities. A deep understanding of a variety of complex domains, from text comprehension to image interpretation, table analysis, and symbol manipulation, is required. as AI technologies evolve, a machine's comprehensive understanding of diverse aspects of mathematics is an important step beyond mere technical achievement to a more general-purpose, adaptive AI This is an important step toward more versatile and adaptive AI.

In particular, the emergence of large-scale language models has revolutionized the field of AI, establishing itself as a powerful tool in the automation of complex tasks. Large-scale language models have proven their ability as a valuable resource for uncovering subtle nuances in mathematical problem solving. These models provide new ways to explore the interplay between language and logic and are driving exploration in this area.

Despite the progress in this area, however, current large-scale languagemodel-based mathematical research remains challenging. The wide variety of problem types and the wide range of criteria, data sets, and techniques for evaluating them further complicate the situation. The lack of a unified framework makes it difficult to accurately assess progress and understand the ongoing challenges in this evolving field.

This paper focuses on the use of large-scale language models in mathematics and seeks to shed light on its complexities. It provides an in-depth exploration of types of mathematical problems and related data sets, an analysis of the evolution of problem-solving techniques with large-scale language models, the factors that influence problem solving, and insights into the persistent challenges in this rapidly growing field. It provides an overall understanding of how large-scale language models drive mathematical reasoning. It also provides new insights by exploring unexplored areas in the complex domain of language and logic.

Math Problems and Datasets

This section provides a brief introduction to the major types of math problems and their associated datasets: arithmetic, math writing problems, geometry, automatic theorem proving, and math in visual contexts.

The following is an arithmetic problem. It requires purely mathematical or numerical manipulation and does not require interpretation of text, images, or other contextual elements.

Question (Q): 21 + 97
Answer (A): 118

These problem formats reflect the fundamentals of arithmetic that are intuitive and easy to understand. The dataset "MATH-140" (Yuan et al., 2023) contains 401 arithmetic representations in 17 different groups, which help to deepen understanding of a wide variety of arithmetic problems.

Arithmetic forms the foundation of mathematics, and the clear problem sets in this category are very effective for learning the rudiments of mathematical thinking. Each problem is designed to promote an understanding of concrete numerical operations, providing a foundation for developing real-world computational skills.

Next is the Math Written Problem (MWP). These are problems presented through written or verbal explanations rather than in the form of direct equations. These problems require insight into the key mathematical concepts from the information provided and the ability to set up the appropriate equation and derive a solution.Math writingproblems mimic real-world situations and develop the ability to apply mathematical principles to problems faced in everyday life. These include problems such as.

The first is the question-answer format.

(e.g.) Lily received $20 from her mother. After spending $10 on books and $2.5 on candy, how much is left for her?(Answer) $7.5.

The second is a problem-equation-answer format.This provides an equation and a clearer mathematical solution.

(Example) Jack has 8 pens and Mary has 5. After Jack gives Mary 3 pens, how many pens does Jack have left?(equation) 8 - 3 (answer) 5

The third is a question-reasoning-answer format. Thisclarifies the reasoning process and provides explanations that guide complex problem solving.

(ex.) Beth bakes 4 dozen OR 2 dozen cookies per week. If this is divided among 16 people, how many cookies will each person get?(Rationale) Beth bakes a total of 4 x 2 = 8 dozen cookies and there are 12 cookies in each dozen, so that makes 96 cookies. Dividing that among 16 people, that is 6 cookies per person.(Answer) Six.

Thesemath writingproblems develop not only basic computational skills, but also critical thinking skills to interpret and apply information. Different types ofmath writingproblems exist, each containing different learning opportunities and challenges. This gives students the ability to understand and use mathematics in a broader context.

The table below lists most data sets in three categories: question-answer, question-equation-answer, and question-evidence-answer. Note that the letters in the table below are:E = Elementary, M = Middle School, H = High School, C = College, H = Hybrid.

Next up is TABLE FORMAT MATH PROBLEMS (TABMWP).Tabular MathProblems is the first dataset of open-domain, tabular, contextual math problems. This dataset is large in size and each problem is represented in the form of either an image, semi-structured text, or structured table.

(e.g.)Henrik bought 2.5 kilograms of oval beads. How much did he pay? (Unit:$) (Answer:5)

This section also discusses the generation ofmath writingproblems.Techniques have been developed in this area to generate new problems rather than simply answer math problems, and evolved models such as GPT-2 and GPT-3 have been trained to generate math writing problems from specific equations to test the effectiveness of problem generation. Studies have shown that GPT-4 tends to revise human-written problems to increase readability and lexical diversity, while using more minor words.

These advances provide a more dynamic and hands-on approach to mathematics education and AI learning. This will not only improve real-world problem-solving skills, but also dramatically expand the scope and efficiency of AI applications.

Next up is geometry.Geometry problems are different in difficulty from math writing problems.Whereasmath writingproblems revolve around logical reasoning and arithmetic operations, geometry requires a spatial understanding of shapes, magnitudes, and their interrelationships. The solution of geometry problems requires the application of geometric principles, theorems, and formulas, which analyze and derive the properties of shapes.

Symbolic methods and pre-defined search heuristics are predominantly used in modern geometry. This indicates the specialized strategies required by geometry and the expertise required in this area. These differences in problem-solving approaches illustrate the diversity of mathematical challenges and the breadth of skill sets required in different mathematical domains.

(Example) a=7 inches; b=24 inches; c=25 inches; h=5.4 inches; What is the area of this figure in square inches?(Answer) 24.03 square inches

The table below, which includes key data sets, is also referenced to provide resources for solving geometry problems. This allows the reader to gain a better understanding of complex geometry problems and apply them to real-world calculations and designs.

Automatic Theorem Proving (ATP) is a specialized field in mathematics that aims to automatically construct proofs for specific conjectures. The field presents unique challenges, including the need for logical analysis, a deep understanding of formal languages, and an extensive knowledge base; ATP plays a particularly important role in the verification and development of software and hardware systems.

Key datasetsinclude theMINIF2F dataset (Zheng et al., 2022), theHOList benchmark (Bansal et al., 2019), andthe COQGYM dataset (Yang and Deng, 2019).These datasets illustrate the diversity of methodologies and skill sets in automatic theorem proving and reflect the multifaceted nature of mathematical problem solving; the evolution of ATP is opening up new possibilities not only in mathematics, but also in many practical technical domains.

Finally, there is the mathematical problem in visual language contexts.Research and datasets in this area demonstrate the complexity and diversity of mathematical reasoning.

Key datasetsincludeCHARTQA (Masry et al., 2022) andMATHVISTA (Lu et al., 2023a).These datasets demonstrate how to linguistically analyze visual information and utilize multiple inference methods to solve mathematical problems. Mathematics in visual language contexts is becoming an emerging trend in education and research, especially in the current era in which data visualization plays an important role.

Analysis: Robustness of Large-Scale Language Models in Mathematics

Prior to the introduction of large-scale language models, tools for solving mathematical writing problems relied primarily on encoder-decoder models with LSTMs. These models used superficial heuristics to achieve high performance on simple benchmark datasets. In a subsequent study, a more challenging data set, SVAMP, was introduced, which was created by selecting samples from an earlier data set and making careful modifications.

Subsequently, the 2023 study added distractors to the original problem in the CMATH dataset andevaluated the robustness ofseverallarge-scale languagemodels. As a result, GPT-4 has been able to remain robust while other models have failed.Inaddition,a new dataset, ROBUSTMATH, has been proposed to evaluate the robustness oflarge-scale languagemodels' ability to solve mathematics. Their extensive experiments show that adversarial samples from high-accuracy large-scale language modelsare also effective at attackinglow-accuracylarge-scale languagemodels, that complexmath writingproblems are particularly vulnerable to attack, and that prompting a small number of shots with adversarial samples can improvethe robustness ofmath writingproblems shown to improve the robustness of math writing problems.

Analysis: Factors Affecting Large-Scale Language Models in Mathematics

The comprehensive evaluation from the 2023 study covers OpenAI's GPT series (GPT-4, ChatGPT2, and GPT-3.5) and various open source large-scale language models. The analysis systematically examines factors that affect the arithmetic skills of large-scale language models, such as tokenization, pre-training, prompting techniques, interpolation and extrapolation, scaling laws, chains of thoughts (COT), and in-context learning (ICL).

A comprehensive evaluation by the 2023 study highlights the important role tokenization plays in the arithmetic performance of large-scale language models. In particular, models such as T5 that do not have dedicated tokenization for arithmetic are less effective than models that use advanced methods such as Galactica and LLaMA. This indicates that the frequency of tokens in prior learning and the method of tokenization is critical for arithmetic performance.

In addition, the advanced arithmetic skills of large language models are correlated with the codes and LATEX in the pre-training data. For example, Galactica, which uses a large amount of LATEX, shows superior performance on the arithmetic task, while models such as Code-DaVinci-002, which excels in theoretical reasoning, lag in arithmetic, highlighting the distinction between arithmetic and reasoning skills.

The nature of input prompts has a significant impact on the arithmetic performance of large language models. Lack of prompts reduces performance, and models such as ChatGPT, which responds to educational system-level messages, illustrate the importance of the type of prompt. Instructional tuning in prior learning is also an important factor.

Furthermore, with respect to model size,there is a clear correlation between the number of parameters and the arithmetic performance of large language models. While larger models generally perform better, performance plateaus are also observed for the 30B and 120B parameters, as shown by Galactica. However, this does not always mean superior performance, and smaller models such as ChatGPT can outperform larger ones.

Analysis: Pedagogical Perspectives in Mathematics

In machine learning,large-scale languagemodels emphasize mathematical problem-solving skills, but in actual educational settings, their primary role is to support student learning. Therefore, an important consideration is how to understand students' needs, abilities, and ways of learning, rather than simply improving their mathematical performance.Some of the benefits oflarge-scale languagemodelsin mathematics educationinclude.

• Promotes critical thinking and problem-solving skills: Large-scale language models provide comprehensive answers and foster students' critical thinking and problem-solving skills through rigorous error analysis.
• Detailed and ordered hints: Educators and students report a preference for detailed hints with clear and consistent narratives generated by large-scale language models.
• Introducing a conversational style: Large-scale language models are an important asset in mathematics education by introducing a conversational style to the problem-solving process.
• Providing Deep Insight and Understanding: The use of large-scale language models goes beyond computational support to provide deep insight and understanding in areas such as algebra, calculus, and statistics.

On the other hand,the followingshortcomings ofnarratives inmathematics educationhave also been identified

• Potential for misunderstanding: large language models can be confusing when students misunderstand questions or make explanatory errors. This can reinforce misunderstandings and compromise the quality of education.
• Limitations in responding to individual learning styles: Large-scale language models rely on algorithms and may have difficulty fully capturing the unique needs of each student. In particular, they may not provide sufficient support for learners who benefit from hands-on activities or visual aids.
• Privacy and Data Security Challenges: Lack of appropriate security measures when collecting and analyzing large amounts of student data creates the risk of privacy violations through unauthorized access and misuse of data.

Summary

While current research trends focus on curating broad data sets, the lack of robust generalization to different data sets, grade levels, and types of math problems remains a challenge. To address this, it may be necessary to move from examining how humans acquire math solving skills to employing continuous learning to help machines improve their abilities.

Large-scale language models also expose several vulnerabilities in mathematical reasoning. These include inconsistent performance on questions expressed in different text formats, reaching different conclusions on multiple attempts at the same question, and vulnerability to adversarial input.

Current large-scale language model-based mathematical reasoning does not adequately take into account the needs and comprehension abilities of real users. In particular, the GPT-3.5 and GPT-4 are problematic in that they misinterpret questions from young students and provide overly complex hints. This calls for a more active inclusion of human factors in AI research.

The paper delves into various aspects of large-scale language models in mathematical reasoning, their capabilities and limitations, and discusses the persistent challenges for different mathematical problems and data sets. It also highlights advances in large-scale language models and their applications in educational settings, as well as the need for a human-centered approach to mathematics education. It is hoped that this paper will provide suggestions for future research in the large-scale language modeling community and promote further advances and practical applications in a variety of mathematical contexts.

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Categories related to this article

• Large Language Models
• Machine Learning
• Deep Learning
• Language Processing
• Transformer
• Language Generation
• Generative Model
• ChatGPT
• GPT