Learning by Doing, Amplified: AI’s Role in Collaborative Math Education

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Author: Denis Avetisyan

A new approach to mathematics education leverages generative AI and action research to empower students and teachers through hands-on exploration and shared inquiry.

This review explores the integration of AI-powered tools within the action research tradition to foster student agency, teacher collaboration, and conceptual understanding in mathematics.

Traditional mathematics education often prioritizes procedural performance over conceptual understanding and student agency. This paper, ‘AI as a component in the action research tradition of learning-by-doing’, proposes a reimagining of learning through action research, integrating generative AI and computational tools to foster a ‘Language/Action’ approach. By framing mathematics education as a collaborative, inquiry-based process akin to professional mathematical modelling, we demonstrate how AI can augment human intellect and scaffold deeper conceptual understanding. Could this approach unlock more effective and engaging learning experiences, empowering both students and teachers in the evolving landscape of mathematics education?

The Illusion of Proficiency: Why Knowing How Isn’t Enough

Mathematics education frequently emphasizes the how of problem-solving, prioritizing memorization of procedures over a genuine understanding of why those procedures work. This approach fosters “fragile knowledge,” where students can successfully apply a formula in one context but struggle when faced with even slight variations or unfamiliar problems. Such learning relies heavily on surface-level pattern matching rather than internalizing underlying mathematical principles; a student might, for example, mechanically calculate the area of a rectangle without grasping the concept of area as a measure of two-dimensional space or the relationship between length, width, and total coverage. Consequently, this procedural fluency proves unreliable, diminishing as problems increase in complexity and hindering the development of adaptable, long-term mathematical competency.

A concerning indicator of this educational challenge lies in primary school performance, where approximately 26% of students do not meet expected standards. This isn’t simply a matter of individual struggles, but rather points to systemic issues within the curriculum and teaching methodologies. The statistic suggests a significant portion of young learners are leaving primary education without a solid foundation in mathematics, potentially impacting their future academic success and limiting their opportunities. This early shortfall creates a cumulative disadvantage, as gaps in understanding become increasingly difficult to bridge with more complex concepts later in their schooling, requiring targeted interventions and a fundamental re-evaluation of how mathematical concepts are initially introduced and reinforced.

The challenges faced by students in grasping core mathematical principles become increasingly pronounced as they advance through their education. Data reveals a significant proportion – 42% – fail to meet expected standards at the General Certificate of Secondary Education (GCSE) level, a critical juncture in their academic journey. This statistic isn’t simply a measure of current performance, but rather indicates a persistent difficulty with foundational concepts that accumulate over time. The failure to solidify understanding in earlier years translates into substantial hurdles when tackling more complex topics, suggesting that rote memorization and procedural learning, while potentially yielding short-term success, ultimately fail to provide the robust cognitive framework necessary for sustained mathematical competence. This pattern underscores the urgent need for pedagogical approaches that prioritize conceptual understanding over superficial skill acquisition.

The pursuit of mathematical proficiency extends far beyond memorizing procedures; a robust understanding hinges on grasping the underlying structures that govern numerical relationships. Rather than treating formulas as isolated steps, effective mathematics education emphasizes the ‘why’ behind each operation, revealing the interconnectedness of concepts like number sense, spatial reasoning, and algebraic thinking. This approach fosters not merely procedural fluency, but a deep conceptual understanding that allows individuals to adapt their knowledge to novel problems and confidently navigate increasingly complex mathematical landscapes. By prioritizing the foundational principles—the logical architecture of mathematics—students develop agency over their learning, transforming from passive recipients of information to active constructors of mathematical knowledge and empowered problem-solvers capable of applying these principles across disciplines.

Beyond Sets: Mapping Mathematical Structures

Conceptual Mathematics establishes a foundational theoretical basis for understanding mathematical structures by moving beyond set-theoretic approaches. This is achieved through the application of Category Theory, which focuses on relationships between mathematical objects rather than the objects themselves. Instead of defining objects by their internal composition, Category Theory defines them by morphisms – the mappings between those objects. This allows for a higher level of abstraction, enabling the identification of common patterns across diverse mathematical fields, such as algebra, topology, and computer science. A key concept is the idea of a category, consisting of objects and morphisms satisfying specific composition and identity laws, formalized by axioms defining functors and natural transformations. This framework facilitates a generalized and unified approach to mathematical reasoning, emphasizing structural similarities and enabling the transfer of knowledge between different mathematical domains.

Structured Drawing and the Cuisenaire-Gattegno Approach offer tangible methods for engaging with abstract mathematical concepts. Structured Drawing utilizes geometric constructions and visual representations to map relationships between elements, allowing learners to externalize and analyze complex structures. The Cuisenaire-Gattegno Approach employs colored rods of varying lengths, representing numerical values, to facilitate the exploration of arithmetic, algebra, and number theory through physical manipulation. Both methods prioritize experiential learning, enabling learners to move beyond symbolic representation and develop a deeper, intuitive understanding of mathematical principles by actively building and deconstructing concepts with concrete materials.

Haskell is a statically typed, purely functional programming language particularly well-suited for representing and manipulating mathematical structures due to its strong support for abstract data types and function composition. A key feature is its advanced type inference system, which automatically deduces the types of expressions, reducing boilerplate code and allowing developers to focus on the logic of their implementations. This is valuable when modeling mathematical concepts, as types can directly correspond to mathematical categories and their associated morphisms. For example, a type representing a group can enforce the group axioms at compile time. Haskell’s lazy evaluation further enhances its utility by allowing the definition of infinite data structures and computations, useful in exploring concepts like limits and series. The language also supports parametric polymorphism, enabling the creation of generic functions and data structures applicable across multiple mathematical domains.

Interactive Development Environments (IDEs) facilitate a cyclical process of hypothesis and validation crucial for conceptual learning. These environments allow learners to immediately translate abstract mathematical ideas – such as function composition or category-theoretic transformations – into executable code. The resulting output provides direct feedback, enabling rapid testing of understanding and identification of errors in implementation or conceptualization. Features like debuggers and step-by-step execution further enhance this process by allowing learners to trace the flow of computation and observe the effects of different inputs. This dynamic interplay between coding, execution, and observation is significantly more effective than static examples or passive reading, allowing for active construction of knowledge and deeper conceptual understanding.

Testing the Waters: A Collaborative Approach to Validation

Action research functions as the primary methodological framework by integrating the practical expertise of educators with the systematic inquiry of researchers. This collaborative process moves beyond traditional research models, which often separate theory and practice, by emphasizing cyclical phases of planning, action, observation, and reflection. The iterative nature of action research allows for continuous refinement of the educational approach based on real-time data collected from the learning environment. This ensures that interventions are not only theoretically sound but also demonstrably effective in specific contexts, and that adjustments can be made promptly to optimize outcomes. The collaborative dynamic also fosters a sense of ownership and shared responsibility among all stakeholders involved in the research and implementation process.

The educational approach prioritizes experiential learning, actively involving both teachers and students in the learning process. This is achieved through practical application and direct engagement with the material, moving beyond passive reception of information. Teachers participate in implementing and reflecting on the approach in real-world classroom settings, while students learn through hands-on activities and problem-solving. This emphasis on “Learning by Doing” aims to foster deeper understanding, skill development, and the ability to transfer knowledge to new situations for both groups, creating a cycle of continuous improvement based on practical experience.

Dialogue and negotiation, as theorized by Winograd and Flores, are central to the iterative refinement of the educational approach because they move beyond simple information exchange to address issues of mutual understanding and commitment. Their framework posits that communication isn’t merely about transmitting data, but about coordinating action and resolving breakdowns in understanding. Specifically, it distinguishes between ‘requests’ and ‘assertions’—requests require a commitment from the receiver, while assertions offer information. Effective negotiation, therefore, involves clarifying whether a statement is an assertion or a request, and addressing any implicit requests that may hinder shared understanding. This focus on clarifying commitments and resolving breakdowns allows practitioners to collaboratively adapt the educational approach to the unique demands of their specific contexts, ensuring alignment between intentions and observed outcomes.

The action research framework intentionally cultivates self-awareness in learners by integrating reflective practices into the learning cycle. This is achieved through systematic opportunities for students to examine their own comprehension, identify knowledge gaps, and articulate their learning strategies. Learners are encouraged to analyze the effectiveness of different approaches, assess their progress toward defined learning objectives, and adjust their methods accordingly. This process of metacognitive monitoring – thinking about one’s thinking – enables students to become more autonomous learners capable of self-directed improvement and a deeper understanding of their individual learning processes.

Navigating the System: Agency and the Future of Education

The dynamic between educational agents – encompassing teachers, technologies, and learning materials – and individual agency, or a learner’s capacity to act purposefully and make choices, is foundational to effective pedagogy. Recognizing this interplay moves beyond simply delivering content; it emphasizes a co-creation of the learning experience. A student’s agency isn’t merely about selecting from pre-defined options, but actively shaping their learning path, posing questions, and pursuing areas of interest. Simultaneously, the role of the teacher shifts from sole knowledge provider to facilitator and guide, curating resources and fostering environments that nurture student initiative. This reciprocal relationship, where both agent and agency are valued and actively engaged, is crucial for cultivating not just knowledge acquisition, but critical thinking, problem-solving skills, and a lifelong love of learning.

The rise of Multi-Academy Trusts (MATs) introduces a significant tension within the educational landscape. While intended to improve efficiency and standards, these large-scale organizations can inadvertently centralize control, potentially diminishing the responsiveness of schools to their specific communities and student needs. This centralization poses a challenge to the principle of localized education, where curricula and teaching methods are tailored to reflect the unique characteristics of each school’s context. Successfully integrating innovative approaches, such as personalized learning facilitated by AI, within a MAT requires deliberate navigation; careful consideration must be given to balancing the benefits of scale with the necessity of maintaining autonomy and fostering a learning environment that genuinely reflects the needs of each individual school and its students.

The history of Mathematics Circles offers valuable insight into fostering genuinely collaborative and exploratory learning environments, extending beyond traditional classroom structures. Originally conceived as informal gatherings for students and mathematicians to engage with challenging problems and novel concepts, these circles prioritize the process of mathematical discovery over rote memorization or standardized testing. This model, emphasizing open-ended investigation and peer-to-peer teaching, demonstrates that rich mathematical understanding can flourish when students are empowered to construct knowledge collectively. Increasingly, educators are recognizing the potential of adapting these principles – the emphasis on problem-solving, justification of reasoning, and shared inquiry – to broaden access to engaging mathematics education, even within the constraints of formal schooling and diverse educational landscapes. Successful implementation requires a shift in pedagogical approach, but the precedent established by Mathematics Circles suggests that scaling such models is not only feasible, but offers a promising pathway toward cultivating deeper mathematical fluency and fostering a more positive attitude toward the subject.

The integration of Generative AI Agents into conceptual mathematics education holds considerable promise for transforming learning experiences and alleviating teacher workload. These agents aren’t intended to replace educators, but rather to function as powerful assistants, capable of dynamically generating supplementary materials tailored to individual student needs and learning paces. Research indicates teachers currently dedicate between one and three hours weekly to resource curation, a time commitment that often detracts from lesson planning and direct student interaction. By automating the creation of practice problems, explanatory content, and varied examples – all aligned with established mathematical principles – AI agents can recapture this valuable time. This allows educators to focus on fostering deeper conceptual understanding, providing personalized support, and cultivating critical thinking skills, ultimately leading to more effective and engaging mathematics instruction. The potential for scalable, personalized learning, combined with a reduction in administrative burden, positions these AI agents as a significant advancement in educational technology.

The pursuit of elegantly automated learning, as proposed within this reimagining of mathematics education through action research, feels… familiar. The article’s emphasis on fostering student agency via computational tools and generative AI is admirable, yet inevitably invites a certain skepticism. It’s a beautifully constructed system, designed to unlock deeper conceptual understanding – until production, in the form of unpredictable student interactions and unforeseen edge cases, begins to exert its influence. As Linus Torvalds once stated, “Most developers think lots of planning is good. I think that’s a sign of a poorly understood problem.” The very nature of learning-by-doing, especially when mediated by complex technologies, guarantees that the carefully crafted frameworks will encounter realities not foreseen in the initial design. The promise of AI-assisted mathematics circles is compelling, but the inevitable entropy of real-world application remains a constant.

What Comes Next?

The reimagining of mathematics education, as explored within this work, predictably introduces as many practical challenges as theoretical opportunities. The integration of generative AI, while promising increased student agency, simply relocates the source of error. The system moves from fallible human calculation to fallible algorithmic interpretation—a substitution, not an elimination. Type inference, for all its elegance, will inevitably encounter the ambiguous intent inherent in natural language, and the resulting debugging will feel…familiar.

The emphasis on teacher agency, while laudable, risks becoming another layer of expectation atop an already burdened profession. The question isn’t whether collaborative frameworks can improve instruction, but whether they can survive the realities of standardized testing and resource constraints. Legacy systems – existing curricula, assessment methods – aren’t replaced; they’re awkwardly accommodated, forming a palimpsest of innovation and inertia.

The true test won’t be the elegance of the computational tools, but their resilience in the face of sustained, unglamorous use. A ‘proof of life’ isn’t a successful demonstration; it’s a confirmed bug report. The field doesn’t move forward by solving problems, it learns to live with them—prolonging the suffering of the system, one iteration at a time.

Original article: https://arxiv.org/pdf/2511.11445.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-17 13:58