Listiawati.
Kartika.
, & Arslan.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
https://doi.
org/10.
56855/jrsme.
Original scientific paper Received: 30 August 2025.
Revised: 23 November 2025.
Accepted: 16 December 2025.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Functions Enny Listiawati1* .
Hendra Kartika2 , yNidem Arslan3 1 Department of Mathematics Education.
STKIP PGRI Bangkalan.
Indonesia 2 Department of Mathematics Education.
Universitas Singaperbangsa Karawang.
Indonesia 3 Mathematics Education Department.
Bursa Uluda University.
Turkey Abstract Purpose: The rapid rise of generative artificial intelligence has reshaped discussions in mathematics education, particularly regarding the capacity of advanced systems such as ChatGPT and Gemini to support conceptual rigor.
This study aims to investigate how these generative AI tools define and explain vector functions, including the procedures for differentiating and integrating them, in order to evaluate their conceptual rigor of ai-generated mathematical explanations and pedagogical potential.
Methodology: Employing a qualitative case study design, the research analyzed responses generated by ChatGPT and Gemini to a structured mathematical prompt on vector functions.
The explanations were compared with authoritative calculus textbooks using qualitative content analysis and a standardized scoring rubric.
Findings: Findings reveal that both systems provide broadly accurate introductory descriptions of vector functions, highlighting their component-wise structure.
However, notable gaps emerge in mathematical precision, particularly in specifying domains, ranges, and the formal conditions underlying differentiability and integrability.
ChatGPT tends to include intuitive geometric interpretations, whereas Gemini provides concise procedural explanations, yet both models lack the rigorous logical framing found in standard mathematical texts.
Despite these limitations, the systems demonstrate consistent procedural accuracy in describing differentiation and integration of vector-valued functions.
Significance: The results underscore the educational potential of generative AI while highlighting the need for teachers to critically evaluate AI-generated mathematical content, particularly when these tools are used to support studentsAo conceptual learning in mathematics.
These findings also highlight important implications for AI literacy, instructional design, and future research in mathematics education.
Keywords: ChatGPT.
Conceptual rigor.
Generative artificial intelligence.
Gemini.
Mathematics education.
Qualitative analysis.
Vector functions.
A 2025 by the authors.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license .
ttps://creativecommons.
org/licenses/by/4.
0/).
* Corresponding author: Enny Listiawati, ennylistiwati@stkippgri-bkl.
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Kartika.
, & Arslan.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
Introduction The mathematical concept of a function is commonly defined as a relationship among two or more variables that can be represented graphically (Del Cerro Velyzquez & Myndez, 2.
As a foundational construct, the function serves as a unifying idea both within mathematics and in its connections to real-world phenomena.
Extensive scholarship over the past five decades has examined how functions are introduced and learned (Dubinsky & Wilson, 2.
, consistently emphasizing that a deep understanding of the function concept is essential for mastering calculus (Bardini et al.
, 2.
Calculus occupies a central position in the mathematical sciences due to its rich internal structure and its numerous interdisciplinary applications.
Its principles illustrate the versatility and explanatory power of mathematics in fields such as engineering, physics, and economics (Rodryguez-Nieto & Moll.
Yu & Cheng, 2.
Key conceptual connections within calculus include the inverse relationship between differentiation and integration, which is formalized in the Fundamental Theorem of Calculus (Garcya-Garcya & Dolores-Flores, 2.
Furthermore, vector calculus extends single-variable calculus to multivariable contexts, linking vector fields, line and surface integrals, curl, divergence, and curvature (Borji et al.
, 2.
These connections underscore the importance of strong conceptual foundations for advanced mathematical study.
Despite its centrality, the function concept continues to present challenges for learners at all levels.
Prior research has documented persistent difficulties and inconsistencies in studentsAo conceptions of functions, both within their own reasoning and in relation to formal definitions (Harel & Dubinsky, 1992.
Carlson, 1998.
Oehrtman et al.
, 2.
Undergraduate students encounter functions repeatedly across coursesAifrom calculus and linear algebra to differential equations and abstract algebraAiyet such encounters do not always lead to deeper conceptual understanding (Zandieh et al.
, 2.
Studies conducted at the upper secondary, tertiary, and teacher-education levels have revealed a range of misconceptions related to functions (Viirman, 2.
Related work also highlights challenges in learning vector-related concepts, including limited conceptual understanding and difficulties with vector and scalar product procedures (Tairab et al.
, 2.
Nevertheless, research examining conceptions of vector functions generated by generative AI remains extremely limited, even as the use of such tools by students and teachers continues to grow.
The past year has witnessed remarkable advancements in artificial intelligence (AI), accompanied by its unprecedented influence on human creativity, productivity, and knowledge generation (Ali et al.
Badshah et al.
, 2.
A pivotal milestone in this trajectory was the release of ChatGPT, a Generative Pre-Trained Transformer (GPT) developed by OpenAI, in November 2022, which significantly reshaped global discourse on generative AI .
enAI) capabilities and limitations (Tlili et al.
, 2.
Soon after.
Google DeepMind introduced Gemini on 06 December 2023, a state-of-the-art multimodal AI model built on Visual Language Model (VLM) technology and positioned as a direct competitor to ChatGPT.
GPT-4, and GPT-4V (Coles, 2023.
Perera & Lankathilaka, 2.
Gemini has demonstrated considerable utility in domains such as reinforcement learning, deep learning, and digital education, further expanding the scope of genAI applications (Imran & Almusharraf, 2.
The integration of AI technologies into education has accelerated substantially over the past two decades, with adaptive learning platforms, automated assessment tools, predictive analytics, and virtual tutoring systems emerging as transformative innovations (Wong, 2.
These AI-driven systems promise to enhance personalization, improve access to high-quality learning experiences, and support greater global collaboration (Luckin & Holmes, 2.
However, their rapid adoption also raises critical queshttps://journals.
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, & Arslan.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
tions regarding their pedagogical, ethical, and epistemological implications, underscoring the need for rigorous inquiry into both their potential benefits and inherent challenges (Lebovits, 2018.
Varsik & Vosberg, 2.
As genAI tools continue to gain widespread traction, closer examination of their educational affordancesAiboth within general education and within specific disciplinary contextsAihas become increasingly essential (Yoon et al.
, 2.
Recent studies indicate that genAI tools can support various educational purposes, including teacher-focused, student-focused, and text-focused tasks (Daher & Anabousy, 2.
The present study focuses specifically on the capacity of genAI systems to assist teachers in the design and construction of lessons, with particular attention to their ability to articulate conceptions of vector functions.
To this end, we examine two leading genAI platformsAiChatGPT and Gemini.
Although these tools have recently attracted considerable scholarly attention in educational research (Ali et al.
, 2023.
Pan et al.
Ram et al.
, 2.
, limited work has investigated their conceptual competence, particularly in explaining mathematical definitions and conceptual structures.
Addressing this gap, the present study adopts an exploratory approach to analyze how ChatGPT and Gemini define, explain, and engage with the concept of vector functions, and to identify the factors shaping the nature and quality of their conceptual output.
Building on the theoretical and empirical gaps highlighted in the Introduction, this study is guided by the following overarching research question: How do generative AI systemsAispecifically ChatGPT and GeminiAiconceptually define vector functions and explain the procedures for differentiating and integrating them, when evaluated against authoritative mathematical references? This inquiry is further elaborated into two sub-questions: .
What conceptual characteristics appear in AI-generated explanations of vector functions? and .
How do these systems describe the differentiation and integration of vector-valued functions, and to what extent do these descriptions demonstrate mathematical rigor? These questions provide a coherent analytical frame for examining the depth, accuracy, and pedagogical value of AI-generated mathematical explanations.
Methods This study employed a qualitative case study design.
A qualitative case study offers a robust methodological framework that facilitates an in-depth, contextually grounded investigation of contemporary phenomena using multiple sources of evidence, thereby enabling analysis from diverse theoretical and analytical perspectives (Baxter & Jack, 2008.
Mtisi, 2.
As emphasized by Yin .
, case studies are particularly suited to addressing how and why questions, allowing the researcher to generate explanatory and evaluative insights.
Thus, this design is well aligned with the aim of the present study, namely to evaluate ChatGPTAos and GeminiAos conceptualizations of vector functions.
The subjects of this study were ChatGPT and Gemini, selected for their prominence and rapidly expanding influence across educational and professional domains.
ChatGPT, in particular, has become a focal point in contemporary educational discourse (Delima et al.
, 2.
Since its public release on 30 November 2022, interest in leveraging ChatGPT within educational contexts has increased substantially, driven by its demonstrated pedagogical potential (Daher & Gierdien, 2.
Both ChatGPT and Gemini represent state-of-the-art generative AI systems distinguished by extensive user bases and wide-ranging applications across multiple disciplinary fields (Colaco & Antao, 2.
Data were collected by inputting a prompt into both chatbots.
The prompt was adapted from Stewart .
and consisted of questions concerning the definition of a vector function and the procehttps://journals.
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, & Arslan.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
dures for determining its derivative and integral.
This prompt was selected because it offers a discipline-validated and conceptually grounded instrument for eliciting participantsAo understanding of vector The two questions are aligned with the AuUnderstandAy level of the Revised BloomAos Taxonomy (Anderson & Krathwohl, 2.
, as they require participants to explain, interpret, and summarize the definition of a vector function and the procedures for determining its derivative and integral.
These items therefore capture essential conceptual and procedural indicators while maintaining methodological focus.
The brevity and open-ended design of the prompt promote the articulation of participantsAo reasoning without constraints, and its grounding in an authoritative calculus source strengthens methodological rigor, transparency, and replicability.
The prompt used in this study is presented in Table 1.
The responses generated by each chatbot were captured using screenshots and stored as image files.
Table 1.
Question prompt Indicator Question Understanding of the definition of a What is a vector function? vector function Understanding of the methods for find- How do you find its derivaing the derivative and integral of a vec- tive and its integral? tor function A qualitative content analysis was conducted to examine the conceptual understanding and completeness of the responses.
The outputs were compared against two authoritative calculus textbooks:
Stewart .
and Rogawski et al.
Conceptual alignment was additionally evaluated using established learning theories and models.
To assess response quality and reliability, the study adopted a standardized scoring rubric proposed by Ozdemir and Ozdemir .
This rubric evaluates responses on a four-point scale: 1 = entirely incorrect or irrelevant.
2 = partially correct with misleading or inaccurate elements.
3 = generally accurate but insufficient or incomplete.
and 4 = fully accurate, comprehensive, and conceptually sound.
To evaluate reproducibility, each prompt was submitted twice from two different computers at different times.
Responses identical across devices were classified as Auconsistent,Ay whereas differing responses were categorized as Auinconsistent.
Ay In cases of inconsistency, only the first response was Reproducibility rates were analyzed by question category and compared across chatbots.
Ethical approval was not required as the study did not involve human participants.
Although broader ethical issues associated with AIAisuch as privacy, fairness, non-discrimination, and transparencyAiare frequently highlighted in the literature (Cotton et al.
, 2023.
Mhlanga, 2.
, the procedures employed in this study adhered to established ethical standards and did not infringe upon any ethical guidelines (Baytak, 2.
Results and Discussions In this study, the concept of a function is defined as follows.
Let D and S be two nonempty subsets of the real numbers.
A function from D to S is a rule that assigns exactly one element of S to each element of D.
This requirement of assigning a unique output to every input is commonly referred to as https://journals.
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Kartika.
, & Arslan.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
the one-valuedness property of a function.
The sets D and S are termed the domain and codomain, respectively (Borke, 2.
This section addresses the first research question: What is a vector function? An initial analysis of responses generated by ChatGPT and Gemini indicates that both systems provide conceptually aligned explanations, particularly in their characterization of vector functions as mappings whose outputs are vector-valued quantities.
Although minor differences appear in phrasing and presentation, both responses consistently communicate that a vector function assigns a vector to each value in its domain.
The detailed outputs from ChatGPT and Gemini, together with definitions drawn from authoritative mathematics textbooks, are summarized in Table 2.
Table 2 Definitions of Vector Functions from Multiple Sources Source Explanation ChatGPT Gemini Stewart .
AuA vector function is a function whose domain is a set of real numbers and whose range is a set of vectors.
Ay Rogawski et al.
AuA vector-valued function is any function r.
of the form in .
whose domain O is a set of real numbers and whose range is a set of position vectors.
The variable t is called a parameter, and the functions x.
, y.
, z.
are called the components or coordinate We usually take as domain the set of all values of t for which r.
is defined-that is, all values of t that belong to the domains of all three coordinate func- https://journals.
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Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
tions x.
, y.
, z.
Ay Table 2 presents a comparative overview of definitions of vector functions derived from generative AI models (ChatGPT and Gemin.
alongside established mathematical references, namely Stewart .
and Rogawski et al.
A critical examination reveals a substantial disparity in the level of precision, completeness, and formal rigor across these sources.
While the textbook definitions offer comprehensive and mathematically robust descriptions, the AI-generated explanations remain noticeably limited in scope.
The definitions provided by Stewart .
and Rogawski et al.
clearly articulate the essential properties of vector-valued functions, emphasizing that such a function maps a set of real numbers .
ts domai.
to a corresponding set of vectors .
ts rang.
Rogawski et al.
further extend this formalism by specifying the structure of a vector function in terms of its coordinate functions , the role of the parameter , and the requirement that all component functions share a com- mon domain.
These features reflect the mathematical rigor expected in standard calculus and analysis In contrast, the definitions generated by ChatGPT and Gemini are generally accurate at a conceptual level but remain insufficient and incomplete for scholarly use.
As previously noted, neither model explicitly states the fundamental mapping relationship from real numbers to vectors, nor do they identify the domain or range with the precision required in mathematical discourse.
The omission of these foundational elements results in definitions that lack the structural clarity and formal exactness present in authoritative sources.
This indicates that, although AI-driven explanations may serve as accessible introductory descriptions for non-expert audiences, they currently fall short of the definitional standards demanded in academic mathematics.
Table 3 presents the explanations provided by ChatGPT.
Gemini, and a standard textbook regarding the derivative and integral of vector functions.
This table addresses the second research questionAiHow do you find its derivative and its integral?Aiby comparing how each source describes the component-wise procedures involved in differentiating and integrating vector-valued functions.
Table 3 Conceptions of Derivative and Integral Vector Functions from Multiple Sources Source Explanation ChatGPT AuTo find the derivative, differentiate each component:
This derivative gives the tangent vector to the curve.
Ay AuIntegrate each component:
Gemini C is a constant vector.
Ay AuThe derivative of a vector function is found by simply taking the derivative of each of its component functions.
The derivative, denoted , is itself a vector Ay https://journals.
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, & Arslan.
Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
Journal of Research in Science and Mathematics Education (J-RSME), 4.
, 122-132.
AuThe indefinite or definite integral of a continuous vector function is found by simply taking the integral of each of its component functions.
Ay Stewart .
AuTo find the derivative or integral, we can differentiate or integrate each component of the vector function.
Ay Rogawski et al.
Aur.
is differentiable if and only if the components are differentiable.
In this case, .
is equal to the vector of derivatives Ay AuThe integral exists if each of the components is integrable.
Ay The findings presented in Table 3 provide important insights into how derivative and integral vector functions are conceptualized across different knowledge sourcesAispecifically, generative AI systems (ChatGPT and Gemin.
and established calculus textbooks (Stewart, 2012.
Rogawski et al.
, 2.
The central pattern emerging from this comparison is the overall convergence toward a component-wise interpretation of both differentiation and integration.
All four sources adhere to the principle that a vector function can be differentiated or integrated by operating independently on each of its scalar components.
This points to the durability and widespread acceptance of the component-wise approach in vector calculus across diverse knowledge systems.
However, the deeper analysis reveals substantial differences in epistemic framing, linguistic precision, and theoretical rigor among the sources.
These variations provide critical insight into the capabilities and limitations of generative AI in mathematical explanation.
ChatGPTAos descriptions demonstrate a pedagogically oriented tendency, mixing formal procedure with intuitive geometric interpretation.
By framing the derivative as providing Authe tangent vector to the curve,Ay ChatGPT adds a layer of conceptual visualization not present in the textbook sources.
While such contextualization may aid studentsAo understanding, it also signals a deviation from the formal definitional style of mathematical exposition.
Similarly, the mention of a Auconstant vectorAy in integrationAithough correctAilacks formal justification and is not situated within a theorem-driven framework.
Gemini, meanwhile, offers concise and technically correct descriptions that closely resemble standard textbook phrasing.
Yet.
Gemini also refrains from articulating the logical and conditional structure that determines the existence of a derivative or integral.
Its explanations are operationally valid but lack the rigorous conditional statements that characterize formal mathematical definitions.
This suggests that Gemini, like ChatGPT, predominantly internalizes procedural patterns rather than the deeper logical architecture of calculus.
In contrast, the textbook sources provide definitions grounded in explicit logical structure.
Stewart .
maintains a pedagogical yet precise emphasis on component-wise computation, reflecting the didactic design of introductory calculus resources.
Rogawski et al.
, however, articulate a more theoretically mature perspective, framing differentiability and integrability in terms of necessary and sufficient conditions.
Their use of biconditional language (Auif and only ifA.
underscores the logical dependency relationships foundational to rigorous mathematical reasoning.
These contrasts reveal a crucial insight into the role of AI in mathematical understanding.
While generative AI systems can reproduce correct procedural knowledge, they often lack the capacity to articulate the formal conditions, logical dependencies, and structural coherence that underpin higher-level mathematical concepts.
This discrepancy highlights an epistemic boundary: AI-generated mathematical explanations are useful as complementary learning aids but should not be treated as authoritative sub- https://journals.
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Conceptual Rigor of AI-Generated Mathematical Explanations: The Case of Vector Function.
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, 122-132.
stitutes for human-authored, peer-reviewed mathematical texts.
Qadir .
cautions that the use of ChatGPT and other generative AI systems may inadvertently perpetuate biases or disseminate misinformation inherent in their training data.
Consequently, human users must exercise critical judgment and systematically verify the accuracy of AI-generated responses (Daher & Gierdien, 2.
Furthermore.
Cong-Lem et al.
highlight ChatGPTAos capacity to generate personalized practice tasks tailored to studentsAo proficiency levels, thereby providing pedagogical support for both teachers and learners, particularly when students encounter difficulties in understanding specific mathematical concepts.
Conclusions The comparison further suggests important implications for educational practice, particularly for teachers who increasingly encounter AI-generated explanations in mathematics instruction.
As AI-based tools become more prevalent in learning environments, educators must remain attentive to the risk that students may uncritically accept AI-generated content that appears procedurally correct yet lacks conceptual rigor.
Such over-reliance may obscure underlying theoretical inaccuracies and foster superficial rather than structural understanding of mathematical ideas.
Consequently, teachers need to develop the capacity to critically evaluate AI-produced explanations, identify conceptual omissions or inconsistencies, and guide students toward more coherent and mathematically sound reasoning.
The findings also underscore broader implications for future research in AI literacy and mathematics education.
Further investigation is needed to examine how students interpret and evaluate AI-generated mathematical content, how reliably AI systems can produce higher-level mathematical arguments, and what forms of pedagogical scaffolding best support critical engagement with AI outputs.
As generative AI capabilities continue to evolve, a nuanced understanding of their epistemic strengths and limitations will be essential for ensuring ethical, effective, and pedagogically robust integration into mathematics teaching, learning, and scholarly practice.
Declaration of Conflicting Interests The author.
declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
References