Constraints and Opportunities of Co-Designing an Interdisciplinary MathCityMap Trail

Abstract

Interdisciplinary teaching enhances student learning but poses challenges in practical implementation. This study examines the co-design of an interdisciplinary MathCityMap trail by mathematics and physics educators, analyzing tensions and synergies through the ROGI framework (Resources, Orientations, Goals, Identity). Using qualitative methods, including reflective journals and co-design, we explore how disciplinary perspectives shape task design. A key constraint was the MathCityMap system’s requirement for verifiable outcomes, limiting open-ended inquiry, particularly in mathematics. While mathematics educators emphasized conceptual understanding, physics educators prioritized hands-on experimentation, leading to tensions in defining inquiry-based learning. Despite these challenges, the collaboration yielded innovative solutions, such as adapting tasks to balance system feasibility with inquiry-driven exploration. The study highlights the need for digital tools that accommodate both procedural and exploratory learning and emphasizes interdisciplinary collaboration as a valuable professional development approach. Future research should focus on enhancing digital platforms to support inquiry-based methodologies in STEM education.

1. Introduction

Interdisciplinary teaching, defined as the integration of two or more disciplines, has been a cornerstone of educational literature for decades and has resulted in the notion of STEM education. There have been several studies designing the interdisciplinary approach to teaching, aiming to empower students to develop new knowledge in both physics and mathematics. This approach fosters a deeper and more integrated understanding of these subjects compared to traditional teaching methods, offering students a richer learning experience that bridges conceptual gaps and emphasizes real-world application (Munier & Merle, 2009; Rabin et al., 2021).

Despite significant advancements in STEM education, a persistent challenge remains in bridging the gap between abstract classroom instruction and practical, real-world applications in that traditional teaching methods often struggle to engage students in ways that make STEM concepts meaningful and relevant to their everyday lives (Baran-Bulut & Yüksel, 2024). While technology has introduced new possibilities for interactive and inquiry-based learning, many existing approaches lack the ability to fully integrate real-world contexts with active problem-solving and critical thinking (Edelson et al., 1999). There is a need for innovative, technology-driven methodologies that connect abstract STEM concepts to authentic, experiential learning opportunities, providing students with a holistic understanding of how these disciplines function in real-world scenarios.

The mathematical trails offer an engaging and educational experience for groups of pupils, students, and the general public. While the concept of mathematical trails has existed for decades, the MathCityMap system introduced the innovative idea of integrating these trails with mobile technology. Participants on the trail walk between locations, solving mathematical tasks along the way. Each trail features at least four tasks, accessible via GPS coordinates. The app provides participants with a task title, an image of the associated object, task instructions, and hints to aid in problem-solving (Gurjanow & Ludwig, 2020; Laššová & Rumanová, 2023).

Teachers often face significant challenges in implementing pedagogical innovations in their practice. These challenges stem from both systemic and individual barriers. On a systemic level, the rigidity of curricula, time constraints, and traditional assessment systems limit the flexibility needed for inquiry-oriented approaches. Moreover, they often feel unsupported by their school environments, particularly in terms of collaboration with colleagues and access to professional development opportunities. These systemic barriers create a disconnect between the theoretical benefits of pedagogical innovations and the everyday teachers’ practice (Medova, 2020).

Teacher educators play a critical role in addressing these challenges by guiding teachers through the transition to innovative practices. However, the preparation and professional development of teacher educators themselves are often overlooked, despite their pivotal influence (Krainer & Spreitzer, 2020; Nachlieli, 2011). Effective teacher educators need deep knowledge not only of innovative pedagogies but also of how to mentor teachers in designing and implementing these strategies. Professional development for teacher educators should include opportunities for co-design, reflective practice, and engagement in inquiry communities. By equipping teacher educators with the tools and frameworks to support teachers effectively, professional development programs can create a domino effect, fostering innovative pedagogies, e.g., use of math trails, across classrooms (Karsenty et al., 2024; Krainer, 2014).

Training teacher educators requires a deep understanding of pedagogies. By incorporating activities such as MathCityMap trails, professional development can connect theoretical concepts to real-world contexts, enriching the learning experience. Each of these levels demands specific skills and knowledge, which together contribute to the overall improvement of educational quality.

Various studies focus on the preparation of teacher educators (Borko et al., 2021; Jacobs et al., 2017), their skills and knowledge (Lesseig et al., 2017; Peters-Dasdemir et al., 2023), as well as evolving practices (Kuzle & Biehler, 2015). Karsenty et al. (2023) developed a double-level conceptual framework for understanding facilitators’ practices and decisions. This is based on Schoenfeld’s (2010) original ROG (Resources, Orientations and Goals) model for teachers’ decision-making. Karsenty et al. (2023) extended this model with the addition of a fourth component—Identity—to create the ROGI framework. This model has been developed as a tool to better understand the decision-making processes of teacher educators.

2. Materials and Methods

This study examines how mathematics and physics teacher educators collaboratively designed an interdisciplinary MathCityMap trail, focusing specifically on the influence of their disciplinary identities, resources, orientations, and goals. Using qualitative methods, we applied the ROGI (Resources, Orientations, Goals, Identity) framework (Karsenty et al., 2023) to analyze key themes emerging from the collaborative process, the implementation of the trail with prospective teachers, and the subsequent reflective practices. The methodological approach of self-study (LaBoskey, 2004) enabled an in-depth exploration of the educators’ differing approaches to knowledge construction, capturing productive tensions as well as synergies characteristic of interdisciplinary collaboration.

Specifically, our research was guided by the following research question:

How do mathematics and physics teacher educators’ disciplinary identities, resources, orientations, and goals influence the interdisciplinary co-design of a MathCityMap trail, and what are the constraints and opportunities encountered in this process?

2.1. Participants

Three teacher educators (Janka, Ľubomíra, Veronika, co-authors of the paper) participated in a lesson study adaptation described in (Haringová & Medová, 2022). The implementation of the adaptation in this study was focused on the design of an interdisciplinary trail.

At the time of the study, Janka had 18 years of experience as a mathematics teacher educator. She has been teaching various mathematics courses within university-level mathematics teacher preparation programs, primarily focusing on linear algebra and combinatorics involving active-learning and inquiry-based pedagogies. She has participated as a researcher and educator in the Erasmus+ project MoMaTrE (Milicic et al., 2020), which resulted in the translation and adaptation of the MathCityMap system into Slovak, as well as in a national research project focused on the implementation of math trails in mathematics teacher professional development (Haringová & Medová, 2022).

Ľubomíra has 23 years of experience as a physics teacher educator and 10 years of practice as a physics teacher. She is the initiator of the “Physics Suitcase” project, which brings hands-on physics experiments into kindergartens. Her interactive physics shows, such as “Physics with all senses”, have reached hundreds of children and teachers. Her professional interests center on promoting inquiry-based approaches and hands-on experimentation in physics education at various educational levels. She has collaborated extensively with Janka in interdisciplinary mathematics and physics education, co-authoring several joint research articles exploring the integration and synergies between mathematics and physics teaching (Medová et al., 2022; Valovičová & Medová, 2019; Valovičová et al., 2020).

Veronika has a background as a mathematics and physics teacher. She first engaged with mathematical trails as a participant in an intensive study program focused on MathCityMap trails during her studies. Her professional interests include geometry education and outdoor mathematics activities (Bočková et al., 2020). At the time of the study, she was in her second year as a teacher educator in mathematics, with an ongoing research interest in utilizing mathematical trails to enhance student engagement and interest in geometry. Recently, she has begun collaborating with Janka and Ľubomíra, contributing her expertise in geometry to their interdisciplinary projects.

2.2. Data Collection

The process of adapted lesson study involved three key stages: (1) planning the interdisciplinary MathCityMap trail; (2) implementing the designed trail with future primary teachers; and (3) conducting a reflective meeting to evaluate the outcomes and suggest challenges. In addition, both educators maintained reflective diaries to document their observations and insights throughout the process. To bridge the gap between theory and practice, the educators designed tasks that integrated mathematical and physical principles while leveraging the affordances of the MathCityMap platform. The tasks aimed to balance conceptual exploration with procedural feasibility, ensuring that they were both inquiry-based and verifiable by the system. The educators also explored how MathCityMap trails could serve as innovative content for teacher professional development, linking real-world application with classroom teaching strategies.

The primary data for this study were collected from audio recordings of the planning session, recordings of the reflective meeting, and entries from the reflective diaries of two of the educators. The reflective meeting was conducted with aim to overcome the constraints observed while designing the tasks to the interdisciplinary MathCityMap trail. All data were transcribed and analyzed to identify themes and patterns in the co-design process and its alignment with interdisciplinary teaching practices.

2.3. Data Analysis

The transcribed data were analyzed using the ROGI framework (Resources, Orientation, Goals, Identity) proposed by Karsenty et al. (2023). This framework provided a structured lens to examine how the educators’ disciplinary perspectives and teaching identities influenced the design and implementation of the trail. The analysis focused on the resources utilized during planning and implementation, the orientations toward inquiry-based versus result-driven learning, the goals set for the trail’s educational impact, and the professional identities and disciplinary philosophies of the teacher educators. Each of the four components plays a key role in shaping the decisions and professional growth of teachers and their educators. The individual components can be characterized as follows:

Resources include the intellectual, material, contextual, and social tools that educators draw upon in their teaching and design practices. In the context of mathematics and physics education, these resources encompass mathematical concepts, physical phenomena, experimental setups, pedagogical knowledge, teaching materials, and digital educational platforms such as MathCityMap.

Orientations encompass educators’ beliefs, values, attitudes, and preferences related to student learning and their approach to teaching. For mathematics and physics educators, this includes their beliefs about inquiry-based learning, conceptual understanding versus procedural fluency, and their attitudes toward integrating real-world contexts into their teaching. Facilitators’ orientations strongly influence their decision-making during interdisciplinary task design.

Goals represent the explicit and implicit educational objectives educators set for their learners. Within interdisciplinary mathematics and physics education, these include content goals (e.g., understanding specific concepts from both disciplines), process goals (e.g., fostering inquiry-based learning, problem-solving skills, or experimentation), and broader educational aims, such as helping students recognize the relevance of mathematics and physics in everyday life.

Identity refers to how educators position themselves professionally in relation to their disciplines and within collaborative settings. Mathematics and physics educators often negotiate multiple identities—for instance, as teachers, researchers, facilitators, or subject-matter specialists. Their disciplinary identities shape how they perceive their roles, interact with colleagues, and engage in interdisciplinary teaching practices. Understanding these identities provides insights into their collaborative dynamics and pedagogical choices.

3. Findings

The MathCityMap (MCM) system, which was used as a core resource in the co-design of the interdisciplinary trail, played a crucial role but also imposed significant limitations. MathCityMap offers a platform for designing outdoor trails, where students solve tasks in real-world contexts and receive immediate feedback. However, one of its major constraints is that tasks must be designed with checkable outcomes that the system can verify. This requirement shaped the design of the tasks in the trail. This constraint can be exemplified as follows:

Veronika:

Why can’t I just assign that a person walks at an average speed of 4 km/h?

Ľubomíra:

Because that’s not physics, you want them to measure something.

Veronika:

But they are measuring something.

Ľubomíra:

But the point is, I would like them to calculate the speed themselves… the distance and time… they won’t measure anything if it’s already given.

Janka:

They’ll measure the distance.

Ľubomíra:

the speed…

Janka:

…will be different for each person, it’ll take them different amounts of time.

Veronika suggests giving students a predefined resource—the average walking speed of 4 km/h—as a way to help them complete the task efficiently. This would allow students to focus on other aspects of the problem, like measuring distance and time, without having to derive the speed themselves. Veronika sees providing these data as a way to simplify the problem and keep it manageable. Ľubomíra argues that simply providing the walking speed is not in line with inquiry-based learning, particularly in physics, where the process of measuring and calculating is crucial to understanding. For her, the value lies in students performing their own measurements to calculate the speed based on the distance and time they have measured. By giving the speed, the task loses an essential inquiry component. Janka emphasizes that students will still be measuring distance and hints that this aspect of the trail retains some inquiry-based learning. However, Janka recognizes the limitations of trails, particularly when scaffolding is absent. She may be more willing to compromise by allowing certain parameters to be predefined (like speed) if it means making the task more feasible or checkable by the system.

As a result, the tasks tend to lean toward procedural problem-solving rather than open-ended inquiry, which limits the depth of conceptual exploration. Ľubomíra was not satisfied.

Ľubomíra:

What challenge did we encounter when we wanted to create this trail? Our main problem was that you wanted a result.

Janka:

Something that is verifiable by the system.

While MathCityMap enabled students to independently complete tasks and receive feedback, its reliance on system-verifiable results constrained the opportunity for deeper, process-oriented learning, particularly in mathematics. This contrasts with the inquiry-based approach in physics, where Ľubomíra did not perceive similar constraints and focused more on the process of exploration rather than a definitive, checkable result. Thus, while MathCityMap was a valuable resource for structuring the trail, its limitations influenced the design decisions and shaped the overall learning experience. It was expressed by Janka in her reflection diary:

“I would also like to use the trails in the way Ľubomíra envisions, but the absence of the teacher during task-solving is a significant obstacle. Yes, for me as well, the process is more important than the result, but based on our experience, I know that without help, children often can’t move forward. Assigning problem-solving tasks during the trails can be very demotivating for students and can affect their productive disposition”.

3.1. Mathematics: Procedural vs. Conceptual Resources

The co-design process revealed a marked difference in the way each educator viewed resources within their respective disciplines.

Ľubomíra viewed resources in mathematics procedurally, focusing on formulas and predefined methods for solving problems. In her words, “Mathematics is about getting to a number, a result that I have given by a formula, and I know how to get to it”. This approach reflected her belief that mathematical resources are tangible and immediately applicable, aligning more with the tools she uses in physics education, where resources like experimental setups are key to exploration.

For Janka (mathematics), on the other hand, the key resources were conceptual tools that help students build deep understanding. She emphasized that students should not merely apply formulas but should understand the reasoning behind them. As she stated, “Mathematics is WHY!… While you get to $V=abc$, there’s the mathematics”. Here, Janka saw mathematical concepts as the main resource, supporting the development of critical thinking and problem-solving skills.

3.2. Physics: Experimental Resources and Inquiry Tools

In physics, Ľubomíra’s approach to resources was more aligned with hands-on, experimental tools. She viewed the MathCityMap trail as an opportunity for students to engage directly with physical phenomena, using real-world resources to explore concepts such as speed, distance, and time. She insisted that students should calculate variables like speed themselves, based on their own measurements, rather than relying on predefined values such as average walking speed. This approach reflects the importance of inquiry-based resources in physics, where the process of measurement and experimentation is central to learning.

By contrast, Janka had to adapt the tasks in mathematics to be more procedural, designing activities that students could complete without immediate teacher support:

“My problem with the trail is that when students are solving tasks, the teacher isn’t there. As a teacher, you don’t have the opportunity to provide scaffolding on the spot—even physics can’t be done solely through MathCityMap”.

(Janka, reflection)

Janka’s orientation is more focused on the teacher’s role in facilitating understanding. She sees the absence of immediate teacher intervention as a hindrance to effective learning, as students may struggle without the necessary support, potentially misinterpreting key concepts. In her opinion this fact limits the richness of the conceptual resources available to students during the trail but was necessary to ensure that tasks were “checkable by the system”.

3.3. Solution of the Constraints Caused by the Discrepancy in Goals

After the exchange analyzed above, Janka checked the task already prepared by Authors 2, 3, and 4. She found out that there was an object in the surroundings moving at a constant speed: the self-closing door.

Janka:

You calculate how quickly the door closes.

Ľubomíra:

Not here, that’s a task for the primary level. They just measure the time.

Veronika:

Can’t it be for the lower-secondary one, too?

Ľubomíra:

It can, but we need to fine-tune it.

Janka:

Then let’s make students estimate its speed, they have a constant time and a constant distance there. They may calculate a quarter of a circle.

In this situation, the team overcame a limitation in the task design due to the system’s need for checkable outcomes. The team realized that with proper adjustment, the task to estimate the speed at which the door is closing could also challenge lower-secondary students. Janka suggested focusing on the constant speed of the door, as it allowed students to measure both time and distance—factors that could easily be adapted to system-based verification. By incorporating the concept of a quarter-circle calculation, they introduced a more advanced mathematical element that would be both feasible and checkable by the MathCityMap system. This adjustment allowed the task to maintain its inquiry-based nature while also satisfying the system’s need for measurable and verifiable outcomes.

The finalized version of the task was as follows: Observe the self-closing door at the entrance to Block B. Open the door approximately to a right angle (90°), so that when released, it closes automatically along a quarter-circle path. Calculate the average speed of the key in the keyhole. Provide the results in meters per second (m/s) rounded to two decimal places.

3.4. Inquiry-Based vs. Result-Oriented Learning

The orientations of the two educators were another source of tension in the co-design process, particularly regarding their views on inquiry-based learning. Both participants valued inquiry but framed it differently within their disciplines.

Janka’s orientation toward mathematics was rooted in constructivist principles, where the process of inquiry was central to understanding. For her, the goal was for students to develop their own conceptual knowledge through exploration and problem-solving. She frequently emphasized that mathematics is about understanding the “why” behind formulas and procedures. However, the practical limitations of the trail, such as the absence of teacher support, pushed her to design tasks that were more result-oriented than she would have liked. This shift was driven by the need for tasks that students could complete independently, with results that could be checked automatically.

In contrast, Ľubomíra’s orientation in physics remained strongly aligned with inquiry-based learning throughout the co-design process. She emphasized the importance of experimentation and the process of discovery, stating, “For me, the process of inquiry—how students approach the task—is more important than the result”. Her focus was not on whether students arrived at the correct answer but on the steps they took to explore and interpret physical phenomena. This orientation allowed her to design tasks that encouraged open-ended investigation, even if they did not always lead to a single, correct solution needed for MathCityMap trail.

3.5. Goals in Mathematics: Balancing Inquiry and Feasibility

The goals of the participants in designing the MathCityMap trail reflected the ongoing tension between fostering deep inquiry and meeting practical constraints. Janka’s ultimate goal was to foster deep, conceptual understanding through inquiry. She aimed to create tasks that helped students see the real-world relevance of mathematical concepts and understand the logic behind them. However, she had to compromise these goals in order to design tasks that students could complete independently during the trail. Her focus shifted toward ensuring that tasks were “checkable by the system”, which provided students with a sense of progress and achievement, even if it limited the depth of conceptual inquiry.

Janka’s goals also included making mathematics visible in everyday life. As she stated, “The trail is mainly focused on students seeing math in real life”. This reflects her desire to connect abstract mathematical concepts to the world around students, using the trail as a tool to bridge the gap between theory and practice.

3.6. Goals in Physics: Process-Driven Learning

Ľubomíra’s goals in physics were more firmly centered on the process of inquiry and experimentation. She was less concerned with students arriving at the correct result and more focused on how they approached the task; “The steps students take to approach the problem are more important to me than the result”, she explained. Her tasks were designed to encourage students to engage in scientific inquiry by making their own measurements and forming hypotheses. This process-driven goal reflects the nature of physics as an experimental science, where understanding often comes from engaging directly with physical systems.

The contrast between these goals—result-driven learning in mathematics and process-driven learning in physics—created both opportunities and challenges in the co-design process. While Ľubomíra’s inquiry-based goals aligned more naturally with the open-ended nature of the trail, Janka’s goals were constrained by the need for tasks to be feasible and verifiable in the absence of teacher support.

3.7. Goals in Incorporating Trails in Education

Ľubomíra’s questions, “What is the benefit of creating a math trail for math teachers?” and “What is the goal of a math teacher when they take a math trail and implement it with students?” reflect her curiosity (and perhaps skepticism) about the educational purpose and goals of math trails in the context of mathematics education. Given her focus on inquiry-based learning and exploration in her own field of science, Ľubomíra seems to be probing what real value a math trail can offer for mathematics, particularly in terms of deeper conceptual understanding.

Janka’s practical view of math trails was visible in her response; “To solve application tasks. The trail is mainly focused on students seeing math in real life. I am not building a concept, the system is not for that, the trail is not for that”. This revealed her focus on the application of mathematics in real-life contexts, rather than the development of abstract concepts. She acknowledges that the primary goal of math trails, as they are currently designed, is to help students see the practical relevance of mathematics in their everyday surroundings. The technical limitations she faces, including the absence of teacher support and the need for tasks to be self-checkable, prevent the math trails from being a space where students can build robust conceptual knowledge in the way she ideally envisions. Janka’s statement reveals a significant distinction between her ideal educational aims (building concepts) and the practical goals (solving application tasks) that she can realistically achieve within the context of a math trail. The trail’s focus on real-world applications reflects a more surface-level engagement with mathematics compared to the deeper conceptual understanding that Janka would like to foster. However, the trail’s structure and the lack of teacher–student interaction push her to prioritize tasks that can show the relevance of mathematics to daily life without requiring heavy conceptual scaffolding. Janka’s understanding of the goal for MathCityMap trail is also confirmed in another turn:

Ľubomíra:

Approach to physics: by doing experiments, they should be able to see physics around them… an ideal physics teacher teaches it that way.

Janka:

An ideal math teacher builds the concept and then goes outside to show it to them; we agree on that.

This highlights a crucial difference: Janka sees the trail as a tool for demonstrating concepts that have already been built in the classroom, rather than for constructing new conceptual knowledge on its own. This ties back to her concern that the technical limitations of math trails prevent them from being fully effective as tools for deeper inquiry. However, Janka is not satisfied by this distinction and continues the communication.

Janka:

Experimenting and building concepts is important for us too, which is why we have a second lesson after the trail. You have the trail at a different phase of the cognitive process.

Ľubomíra:

That’s it! That’s the fundamental difference we’ve been running into.

This exchange between Janka and Ľubomíra further underscores the fundamental differences in their approaches to using trails in teaching, particularly in relation to when they integrate trails within the broader learning process. The conversation points to a key distinction in how each educator views the trail’s role in students’ cognitive development and their approach to inquiry.

3.8. Teacher Identity and Disciplinary Philosophies

The final component of the ROGI framework—identity—proved critical in shaping how Janka and Ľubomíra approached the MathCityMap trail. Their professional identities, deeply rooted in their respective fields, influenced their expectations for the trail and their interactions during the co-design process.

Janka’s identity as a mathematics educator was closely tied to her belief in fostering deep conceptual understanding through inquiry. She saw herself as a facilitator of student-driven exploration, where the goal was to help students develop their own mathematical reasoning rather than simply apply formulas. This identity influenced her desire to use the MathCityMap trail as a tool for conceptual exploration, even though practical constraints forced her to shift her focus toward more procedural tasks. Janka’s frustration with these limitations is rooted in her strong identity as a proponent of inquiry-based learning.

Ľubomíra’s identity, as a physics educator, was shaped by her background in empirical, experimental science. For her, the MathCityMap trail represented an opportunity for students to engage in hands-on experimentation and inquiry, where they could explore the world around them and construct their understanding through direct engagement with physical phenomena. Her focus on process over results reflects her identity as an educator who values exploration and discovery. Unlike Janka, Ľubomíra did not face the same constraints in designing her tasks, as the nature of physics allowed for open-ended experimentation that aligned more closely with her teaching philosophy.

3.9. The Two Identities of Ľubomíra

Ľubomíra’s approach to mathematics reflects her understanding of it as a practical tool, primarily aimed at solving problems through the application of predefined formulas. For her, mathematics is procedural, where the goal is to arrive at a correct result by following established steps—she often sees it as a method for producing specific outcomes efficiently, stating, “Mathematics is about getting to a number, a result that I have given by a formula, and I know how to get to it”.

In contrast, her view on physics education is deeply rooted in inquiry-based learning, where the emphasis is on experimentation and the process of discovery. In physics, Ľubomíra encourages students to explore, hypothesize, and experiment to understand the “why” behind physical phenomena: “The approach to physics is such that I see the physics around me, but I need to adapt it to the fact that I am conducting an experiment. For me, the steps of inquiry are important—how students approach the task—and the result itself is not important, as they cannot interpret it. We are interested in why it turned out that way”.

The contrast is striking: while mathematics is approached as a toolbox for obtaining results, physics is seen as a field where the process of inquiry, not the outcome, is the focal point of learning. This dual perspective reflects how her teaching orientations shift depending on the discipline, with physics fostering deeper exploration and critical thinking, while mathematics focuses on procedural accuracy.

In her reflection, Janka expressed frustration and surprise at the fundamental differences in how she and Ľubomíra approach the teaching of mathematics, particularly noting the lack of alignment in their educational philosophies despite years of collaboration:

“I had no idea that Ľubomíra approaches math and its teaching in such a transmissive manner, and that over the years, she hasn’t been influenced by my approach to math, even when we collaborate on so many things”.

3.10. The Intersection of Identities in the Co-Design Process

The intersection of these two identities—one rooted in conceptual exploration (mathematics) and the other in empirical experimentation (physics)—created both challenges and opportunities during the co-design process. While their differing perspectives led to productive tensions, they also fostered a deeper understanding of how interdisciplinary collaboration can enrich both fields.

For example, Janka’s emphasis on making mathematics visible in real-world contexts aligned with Ľubomíra’s belief that students should see physics around them. Both educators recognized the importance of connecting abstract concepts to everyday experiences, even if their approaches to achieving this differed. This shared understanding of the importance of real-world application provided a strong foundation for their collaboration, even as they navigated the tensions between their respective disciplinary identities.

Janka’s reflective diary entry reveals a deeper level of frustration and a sense of personal and professional conflict, which adds another layer to the previous analysis of their contrasting teaching philosophies. Her statement offers insight into the tension between her and Ľubomíra’s differing views on how students perceive and engage with mathematics and physics in everyday life:

“I was offended that Ľubomíra thinks the same students see physics around them but not mathematics. I know that they often don’t see math (and that’s precisely why I put so much energy into math trails), but I don’t think it’s the opposite with physics. I don’t know where Ľubomíra gets this certainty. Maybe it’s because of the children she works with, but they aren’t a typical population”.

4. Discussion

This study explored the interdisciplinary co-design of a MathCityMap trail by mathematics and physics educators, highlighting both challenges and opportunities in integrating these disciplines into an outdoor learning environment. The MathCityMap system was used to structure the tasks, but its requirement for checkable results imposed significant constraints on the design process.

One of the central findings of this study is the contrasting perspectives on teaching mathematics and physics between the two educators, Janka and Ľubomíra. While both agreed on the importance of inquiry-based learning, their approaches to each subject differed significantly. Janka viewed mathematics as a process of conceptual exploration, emphasizing the importance of building a deep understanding of underlying principles. In contrast, Ľubomíra approached mathematics in a more procedural manner, seeing it as a tool for solving problems and arriving at specific results, much like how she views the application of formulas in physics.

Ľubomíra demonstrates dual orientations toward mathematics and physics, reflecting her distinct approaches to each subject. In mathematics, she adopts a more procedural orientation, viewing it as a tool for solving problems and arriving at concrete results using predefined formulas. This result-driven approach contrasts with her inquiry-based orientation toward physics, where she emphasizes experimentation, exploration, and the process of discovery. In physics, Ľubomíra encourages students to engage with real-world phenomena through hands-on experiments, prioritizing the investigative process over the final outcome. These contrasting orientations highlight her ability to navigate between a tool-based view of mathematics and a more process-oriented, exploratory approach in physics education.

The contrasting views on mathematics between Ľubomíra and Janka became particularly evident during the design of the trail tasks. Janka, despite her emphasis on inquiry, had to adapt her approach due to the practical limitations of the MathCityMap system. Tasks needed to have measurable, system-verifiable outcomes, which pushed her to focus on more procedural tasks, such as calculating distances and speeds, rather than fostering deeper conceptual engagement.

The tension between Janka’s constructivist approach to mathematics and Ľubomíra’s more result-driven approach highlighted a broader challenge in interdisciplinary teaching. While Janka aimed to build mathematical concepts before applying them in the real world, Ľubomíra preferred a more practical approach where the result was key. This tension was exacerbated by the limitations of the MathCityMap system, which required tasks to be checkable by the system, reducing the opportunity for open-ended inquiry, especially in mathematics.

The case of the self-closing door is illustrative. Initially, Ľubomíra suggested using the door’s constant speed as base for a task about measuring the time, but Janka saw this as an opportunity to design a task about speed verifiable by a system. It fulfilled Ľubomíra’s preference that students should measure the time and distance themselves to calculate speed, maintaining an inquiry-based approach. These tasks allowed both aspects to be incorporated—students would calculate the speed but in a structured way that could be verified by the system, allowing the task to remain within the system’s constraints while still promoting inquiry.

In this way, the educators maintained the integrity of both disciplines while adapting to the constraints of the system. This process of adaptation demonstrates the flexibility required when designing interdisciplinary tasks, particularly in the context of digital tools like MathCityMap. While the system’s limitations initially appeared to hinder the depth of inquiry, the collaborative approach of the educators allowed for creative solutions that ensured both mathematics and physics were effectively integrated into the trail.

However, this study also highlights the need for more flexible tools that support open-ended inquiry without compromising system checkability. While MathCityMap provided a valuable structure for the trail, its limitations in fostering deeper inquiry in mathematics suggest that future iterations of such systems should offer more room for conceptual exploration, particularly in disciplines where the process is as important as the result.

The use of self-study as a research approach provided deeper insights into the tensions and opportunities inherent in interdisciplinary co-design from an insider perspective. Although self-study as a research approach is sometimes questioned due to concerns of subjectivity, it is particularly valuable for examining the nuanced decision-making processes, reflections, and adjustments that educators undergo in authentic educational settings. Furthermore, according to Karsenty et al. (2023), self-study enables researchers “to pursue the unpacking and the operationalisation of core constructs in the field, which is … necessary for its advancement. The tendency to engage in self-study is therefore typical of the early stage in which the MTE (mathematics teacher educators) literature currently stands” (p. 149). Thus, adopting self-study allowed us to critically explore and operationalize our disciplinary differences and educational philosophies explicitly, contributing to both our professional growth and the ongoing discourse in interdisciplinary mathematics and physics education. Through reflective diaries and critical dialogue, this approach facilitated a detailed exploration of how the educators’ disciplinary identities, orientations, and goals influenced their pedagogical choices, thus providing rich, contextualized understanding that may not have emerged through external observation alone.

5. Conclusions

The interdisciplinary co-design of the MCM trail demonstrated both the potential and the challenges of integrating mathematics and physics in an outdoor learning environment. The limitations imposed by the MathCityMap system pushed the educators to adapt their tasks in creative ways, ensuring that both subjects retained their educational value. By balancing the need for system-verifiable outcomes with the principles of inquiry-based learning, the educators were able to create a trail that engaged students in both conceptual exploration and practical problem-solving. This study highlights the importance of interdisciplinary collaboration and suggests pathways for improving digital tools to better support inquiry-based learning in future educational projects.

By applying the ROGI framework to the co-design process, it becomes clear that the constraints and opportunities in interdisciplinary collaboration are deeply influenced by the resources, orientations, goals, and identities of the participants. While Janka and Ľubomíra faced significant challenges in reconciling their differing approaches to inquiry and task design, the MathCityMap trail provided a space for them to integrate their expertise in a way that enriched both subjects.

The differences in their resources and orientations—conceptual vs. procedural, inquiry-based vs. result-driven—highlighted the tensions that arise when experts from different disciplines collaborate. However, these tensions also created opportunities for interdisciplinary synergy, particularly in the integration of real-world contexts and inquiry-based learning across disciplines.

Ultimately, the co-design process underscored the importance of balancing practical constraints with pedagogical goals, as well as the value of interdisciplinary collaboration in enriching student learning. By navigating the tensions between their identities as mathematics and physics educators, Janka and Ľubomíra were able to create a MathCityMap trail that fostered both conceptual understanding and hands-on experimentation, offering students a richer and more holistic learning experience.

Future research should focus on examining how digital tools, such as MathCityMap, could better support the diverse disciplinary needs of mathematics and physics education. Specifically, there is a need to explore ways to extend digital platforms to accommodate both procedural and conceptual approaches to learning. Developing advanced feedback mechanisms or scaffolding features could enable students to engage in deeper inquiry, while preserving the system’s ability to verify outcomes automatically. Further studies could investigate the impact of these enhanced features on students’ conceptual understanding, motivation, and engagement across different STEM disciplines. Additionally, future research could explore how interdisciplinary co-design processes influence teacher educators’ professional identities and pedagogical practices over the long term.