September 16, 2025
Theory-Driven Digital Learning Environments (DLEs) for Mathematics
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Many digital learning environments (DLEs) for mathematics are available to educators and students. For middle and high-school mathematics, graphing calculators like Desmos and Geogebra are powerful and popular tools. They provide opportunities for different representations of mathematical objects, visualizations, and real-time manipulations[1]. In higher education and for more advanced concepts, computer programming-based tools like Wolfram Mathematica and MatLab are widely used. Research on the effectiveness of such environments on building mathematical knowledge of students has shown promising results[2]. However, research has also shown that their effectiveness depends on a range of factors like curricular alignment, teacher readiness, student motivation, and prior knowledge[3],[4]. These DLEs offer tools, interactions, and affordances for students to experiment and test mathematical conjectures and hypotheses. Yet, from an instructional perspective, they are often criticized for the lack of pedagogical foundations in their design, guided instruction, and structured problem solving[5]. They are designed to be used independently by the learners with little scope for scaffolding, feedback, addressing misconceptions, and adaptive learning. Finally, their integration for learning requires careful decision making, intense lesson planning, content creation, and curation by the teachers[6]. To address these concerns researchers have argued that DLEs be grounded in strong theoretical foundations that address mathematical thinking and pedagogy[7]. For concepts from middle and high school math curricula, such theoretical framing is needed to address their complex and hierarchical nature, promote mathematical ways of thinking, and identify gaps in student knowledge for remedial instruction.
DLEs for Mathematics
Existing theory-driven learning environments for math problem solving have shown promising results. For example, the Graspable Math System relies on a framework of gestures and dynamic manipulation of algebraic notations to assess and build students’ understanding of middle school algebra[8]. Similarly, MathPad also relies on a framework of gestures and ensuring consistency in algebraic problem solving to connect different representations of mathematical expressions[9]. While studies have built evidence for the effectiveness of these systems, the capabilities of these DLEs are limited to only a few math topics (e.g., algebra). More broadly, Vretta’s MathemaTIC[10] platform uses Polya’s four stage problem solving framework[11] to guide students as they solve math problems from a wide range of topics from primary to early secondary grade mathematics. It encourages students to first understand the problem and identify the key pieces of information, then devise a plan of how they might solve it, apply their mathematical knowledge to carry out the plan, and finally reflect and review their solution. This is done by providing students with specific tools needed at each step. For example, students can highlight relevant pieces of information in the problem, create a table of values, and use a digital logbook to take notes. Moreover, MathemaTIC utilizes the power of narrative and storytelling for math instruction by introducing students to the stages of Polya’s framework as four characters with unique abilities as shown below.
Studies have found such context and narrative-based techniques to enhance students’ engagement and motivation, alleviate boredom, and alleviate negative math attitudes[12],[13],[14]. Further, MathemaTIC uses a system of log-file data collection to record students’ actions and tool use (e.g., timestamps, digits entered). This can be used for learning analytics and gaining insights into students’ problem solving strategies. Such analysis can identify students’ misconceptions, recommend remedial instruction, and feedback. A recent study by Schipper and colleagues (2025) identified five commonly adopted strategies by students (N=802) when solving a pre-algebra problem and made connections between strategy use and math ability[15]. Results show that high scoring students adopted a diverse range of strategies indicating multiple facets of mathematical ability.
Overall, theory-driven DLEs for math have the potential to not only provide interactive user interfaces and tools to support math learning but also provide opportunities for assessment, analysis, and feedback.
The Way Forward - Features of the Theoretical Framework
No single DLE can address all aspects of an ideal problem solving environment for middle and high school math. While graphing calculators do not have embedded layers of assessment and feedback, DLEs like Graspable Math are limited by the type of problems and math topics they address, and MathemaTIC’s underlying problem solving framework (Polya’s four stages of problem solving) is not math specific. As a discipline-agnostic framework, it does not consider the complex and hierarchical nature of math concepts.
In the endeavour to create a powerful DLE that provides students with the tools to solve complex problems at a higher level, it is essential to clearly identify and define its theoretical foundations. And before that, it is important to list some key features of such a theoretical framework:
It should be discipline specific: It is critical that the theoretical framework is math specific. Discipline-agnostic theories do not consider factors like students’ cognitive models of math objects, stages towards mastery of math concepts, and mathematical ways of thinking (e.g., constructivism, cognitive load theory)[16].
It should not be sub-domain specific: The theoretical framework should not be specific to sub-domains in math. For example, the Van Hiele Model of Geometric Thinking[17] can only be applied to geometry topics and is not applicable to other sub-domains like algebra and calculus. Therefore, the theory should be specific to math yet generalizable enough to address the wide range of math concepts.
It should be age-appropriate: The theoretical framework should account for the complexity of concepts in middle and high school math curricula. As concepts get interrelated and hierarchical in senior grades, the theory should account for such complex nature of math concepts.
It should have a layer of assessment: The theoretical framework should factor in the analysis of learners’ actions and problem solving strategies. This is needed to identify their current knowledge level and misconceptions. This information can then be used to further identify remedial instruction and interventions.
It should represent learners’ mental models: The theoretical framework should be able to consider the diversity in problem solving strategies and the different approaches adopted by learners. For example, when asked to solve the equation 2(3x-4)=6, students can use different approaches. They can apply the distributive property of multiplication on the left hand side as the first step and then solve for x, or, divide by 2 on both sides as the first step, then apply the distributive property to solve for x. In this case, the type and the order of actions taken by the learners represent different levels of understanding of linear equations.
Establishing these criteria is the first step towards identifying the right theoretical approach for building interactive DLEs for math. While a theory-driven design is necessary and critical for its effectiveness in improving learning outcomes, it poses some interesting challenges.
Limitations and Challenges
All approaches to teaching and learning, instructional design, and creation of DLEs have certain limitations. A theory-based approach aligned with all the features presented in the previous section also poses some limitations and challenges.
Choosing the appropriate theoretical framework: Many theoretical frameworks of math instruction and learning have been proposed by researchers in the past[18]. It is highly likely that a universal framework that addresses all math concepts, considers all aspects of learning and assessment (e.g., conceptual and procedural knowledge, misconceptions, thinking processes) does not exist. All frameworks have certain underlying assumptions that also limit their applicability to the wide range of math topics. For example, the generalizability feature of a theoretical framework to the entire discipline of math also leads to its limited deeper application to sub-domains like geometry, algebra, etc.
Application of theories to design DLEs is resource-intensive: Applying theoretical frameworks to guide the design of a DLE is a resource intensive exercise requiring time and expertise in mathematics, science of learning, and technology. The research and development cycle is iterative and time consuming involving several steps. These are: hypothesising based on the theoretical framework, applying theory to math concepts, mapping learner interactions in the DLE to the theoretical model, building and testing prototypes, collecting experimental data with students, data analysis, updating the model based on findings, and making technological changes to the DLE.
Individual biases: The application of a theoretical framework is influenced and limited by the knowledge and biases of the team members. Decisions about how learner actions and tasks are defined and operationalized within the DLE, mapped to the affordances and interactions offered by the DLE, mapped to the concepts, and the underlying assumptions are susceptible to personnel bias. Including a diverse range of experts from instructional design and learning sciences is recommended to address individual biases.
As future DLEs adopt an evidence-based, theory-driven design, it is critical that developers are aware of such limitations to avoid pitfalls that could lead to loss of learning, reinforcing misconceptions, and over generalization.
Conclusion
It is recommended that future DLEs are grounded in discipline-specific, age appropriate, and relevant theoretical models of student learning. DLEs have the potential to capture large amounts of student data as they engage in the problem solving process. Such data can then be analyzed to provide deeper insights into students’ thinking processes. Hence, as DLEs for mathematics and other disciplines utilize the potential of advanced technologies like AI and learning analytics, a guiding theoretical framework is important to address risks associated with algorithm and interpretation bias. Further, it is crucial to understand the limitations and challenges of the theoretical framework being applied. All theories have certain underlying assumptions and represent hypothetical models of students’ understanding. They need to be tested empirically. Developers should be cognizant and try to address such limitations of the theoretical framework.
About the Author
Robin Sharma is a creator of science-based, digital learning environments (DLEs). He has over seven years of international experience in conceptualizing, implementing, and evaluating educational programs. As Vretta’s Learning Scientist, he is working with the Innovation Team to create an AI-driven problem solving canvas for high school mathematics grounded in the science of mathematical thinking. Robin has expertise in mathematics education, digital-game based learning, and curriculum development. He also works at the Technology, Learning, and Cognition Lab at McGill University where he studies factors influencing teachers’ adoption of digital games into classrooms and theory-based design of math games.
Previously, he managed the “Games for Learning” program at UNESCO MGIEP. He developed curriculum guides, educator toolkits, and several MOOCs on educational games. He led and co-authored the UNESCO guidelines on digital learning, a set of principles for ethical and responsible development of DLEs aligned with UNESCO’s framework on Education for Sustainable Development (ESD) and social-emotional learning (SEL). Robin has also worked in the videogame industry, developing the world’s first, interactive, gaming curriculum guides for teachers. These guides aim to support educators’ adoption of the immersive Assassin’s Creed Discovery Tour video games by Ubisoft. He has won several awards, scholarships, and grants including the International Doctoral Research Award from IDRC Canada and the FRQSC Doctoral Fellowship from the Government of Quebec.
Robin is an active member of the EdTech Impact Network managed by the International Center for EdTech Impact, Mathematics Teachers Association, and Game Research and Design Community (GRADE).