Spatial and Epistemic Agency When Engaging with Math Walks at a STEM Residential Camp

Abstract

Math walks, where participants explore physical spaces and make connections to mathematics, have been implemented in a wide variety of ways in different learning settings. Math walks connect place-based learning, the development of interest in STEM fields, and opportunities for students to exercise agency over their STEM learning. However, many enactments of math walks have been highly structured and directive in both the mathematics they promote and in the physical movements students are permitted to engage in as they walk. In the present exploratory, qualitative-dominant, mixed-methods bounded case study, we examine eight secondary students engaging in math walk activities as they participate in a STEM summer camp. We examine the nature and types of mathematical questions they ask during math walks, the ways in which they describe the spatial and epistemic agency they have during the experience, and their attitudes toward mathematics. We found that student-led math walks expose a consequential design problem: how to facilitate spatial and epistemic agency while retaining enough mathematical coherence to support depth and continued learning. The article enumerates implications for researchers, educators, and learning designers.

1. Introduction

Math walks, also called math trails, is a term that is used to refer to a broad range of pedagogical approaches where learners explore how mathematics relates to real-world spaces and locations (Lumb, 1980; Toliver, 1993). Math walks can be implemented in formal school settings (Barbosa et al., 2022; Buchholtz, 2021; Dickman & Feinberg, 2023), as well as informal learning settings, often focusing on STEM (Sager et al., 2024; C. Walkington et al., 2026a). Math walks can involve highly structured implementations where learners visit specific locations on a map and learn about particular concepts at each location, answering questions at different sites to practice specific mathematical skills (Barbosa et al., 2022; Jablonski et al., 2023). One advantage of such an approach is that learners and educators can engage in focused and coherent conversations around consistent mathematics concepts, with learners’ mathematical knowledge transforming over time. At the other end of the spectrum, math walks can involve learners choosing where to go and what places to visit, as well as what mathematics concepts to try to “pull out” of their environment based on their own interests and goals, such as when math teachers go on community walks in students’ neighborhoods (e.g., Aguirre et al., 2013; Turner et al., 2016). However, such research is relatively rare, making this an important gap in the research on K-12 students going on math walks. In practice, most math walk implementations lie somewhere between these two extremes.

The first approach, where the walk is very structured, may be effective when the outcome of interest is students’ learning of particular mathematics concepts. The second approach, where the walk is student-led, may be effective when the outcome of interest is enhancing students’ attitudes toward math and expanding their conceptualizations of mathematics as a discipline. Indeed, many students struggle with their motivation to learn mathematics (Gottfried et al., 2007), experience significant mathematics anxiety (Barroso et al., 2021), and often see mathematics they learn in school as disconnected from the world and their everyday experiences (Boaler, 1998; Gainsburg, 2008). In a formal classroom setting, most activities students engage in need to be tightly linked to specific conceptual learning outcomes. In informal learning settings like museums, clubs and afterschool programs, and nature centers, participation is often more voluntary and interest-driven, and learning is conceptualized in terms of a broader and more multifaceted set of outcomes than mastery of predetermined content (Rogoff et al., 2016); therefore, math attitudes and perceptions can be prioritized. These variables can have important downstream implications for students’ learning of mathematics (e.g., Bernacki & Walkington, 2018). However, there is still much that we need to learn about the ways in which mathematics experienced in informal environments, which students may not even recognize as mathematics (Pattison et al., 2017; Williams-Pierce et al., 2024), can change learners’ notions of mathematics as a discipline that spans informal and formal settings. This highlights a second important gap in the literature.

Place-based learning (Gruenewald, 2003; Sobel, 2004) is an instructional strategy that is especially congruent with the affordances of informal learning settings, as informal learning often involves students immersed in physical spaces that have significant affordances, infrastructure, or norms for open-ended STEM engagement. In place-based learning, the physical spaces that students inhabit or visit are brought into the learning process as resources for better understanding real-world applications of conceptual disciplinary ideas (Zimmerman & Land, 2014). While much attention has been paid in the educational literature to the idea of learners having epistemic agency to explore STEM concepts and engage in inquiry comparatively, less attention has been given to agency over physical spaces, where learners can move their bodies to different locations and immerse themselves perceptually in different environments to help them learn concepts. Although embodied cognition and immersive learning research demonstrate that physical movement and spatial interaction can enhance learning, these concepts have largely developed separately from discussions of learner agency in STEM education, which represents a third important gap in the literature.

In the present study, we examine youth engaging in place-based math walks as they explore a university campus during a summer camp. We show how an explicitly unstructured approach to math walks can promote both forms of learner agency. We examine the nature of the mathematics concepts themselves that tend to be surfaced by students and how students perceive these concepts through mathematical question asking. Finally, we explore students’ attitudes toward and beliefs about mathematics over the course of the experiences. With this work, we seek to contribute to scalable approaches for STEM place-based learning in informal settings, where student agency is a central point of leverage.

1.1. Literature Review

Math walks serve a variety of purposes; they support participants in understanding the relevance of math ideas (Barbosa et al., 2022) and provide new opportunities to engage in mathematical thinking and pose problems about the world (Dalton & Yantz, 2025). Research on math walks has documented several key benefits, including increases in student engagement and interest (Buchholtz, 2021; Milton et al., in press; Wang et al., 2021), students recognizing mathematics within their community (Dalton & Yantz, 2025), and enhancement of algebraic thinking (Barbosa et al., 2022).

MathCityMap (Cahyono, 2018) is a GPS-enabled mobile app where teachers and learners can explore objects or situations at particular places in the world, which are shown on a digital map where students are directed to explore particular places one after another. Mathematics problems are created by researchers or teachers that correspond to these locations, with these tasks including (1) a question, (2) information about the object and tools needed to solve the problem, and (3) hints and/or feedback. Research on MathCityMap indicates that students using the app show indicators of intrinsic motivation and identified regulation (Cahyono & Ludwig, 2017) and that using the app leads to statistically significant gains in mathematics performance compared to learning mathematics in the regular setting (Cahyono & Ludwig, 2019). A version of the app with augmented reality elements that display geometric shapes that students can solve problems and make calculations about was associated with stronger modelling performance over the implementation (Cahyono et al., 2020). The app’s hints and feedback allow students to solve more tasks and fail less and, in some cases, rethink their modelling approach (Jablonski et al., 2023). In addition, students use the digital feedback to verify their result and may be less likely to engage in mathematical elaboration when given the app’s support (Jablonski, 2024).

The Mathfinder app (v1.0) is a mobile augmented reality iPad app that also allows students to go on and create their own walk stops (Figure 1). However, rather than focus on answering questions during math walks like MathCityMap, this app focuses on asking questions during math walks. Students are immersed in a storyline in the app where they are an explorer from space (Figure 1, top left); they create a human avatar (Figure 1, top right) and then can explore an interactive map of their area with professional walk stops populated on it (Figure 1, middle left). Clicking on these professional walk stops brings up pre-produced videos where informal learning leaders in the community discuss how mathematics appears at a specific location at a specific site (e.g., the giraffe habitat at a zoo; Figure 1, middle right). Students can also take GPS-tagged pictures in the app (Figure 1, bottom left), access their “Field Journal” to annotate those pictures, and ask math questions related to those pictures (Figure 1, bottom right) and answer the questions they ask. They can then submit these photos with questions and answers as “Crew Discovery Reports.” Once approved, these reports can appear on the interactive map in the app for everyone to see. Finally, the app incorporates a variety of augmented reality tools that operate through the iPad’s camera feed (Figure 2). These include tools to measure lengths, areas, and angles; tools to count; and tools to overlay different geometric shapes onto the environment. Students are identified in the app by a random space-themed game name, which we use to identify students in the present article.

Research on math walks in the context of the Mathfinder app has also been conducted. Studies suggest that the app improves learners’ attitudes toward mathematics (Milton et al., in press; C. Walkington et al., 2026b) and that learners can make sophisticated mathematical connections between real-world settings and mathematics principles (Sager et al., 2023; Sherard & Walkington, in press). We also see youth taking part in self-directed mathematical investigations and posing questions about how math relates to spaces in their communities (Milton et al., in press; Sager et al., 2023). Students pose mathematical questions for their walk stops at various levels of complexity—ranging from identifying math concepts in their environment to asking questions about measurements and strategies for measurements and to posing problems about more complex cause-and-effect relationships between variables (Sherard & Walkington, in press). Practices of question-asking are directly related to increases in students’ math interest (C. Walkington et al., 2026b).

1.2. Theoretical Framework

1.2.1. Place-Based Education in Mathematics and the Development of Interest

The concept of place-based education (PBE) has evolved from an implicit aspect of experiential learning research into an explicit pedagogical framework that intentionally synthesizes learner-centered, context-relevant learning experiences intended to connect learners, communities, and physical environments. The concept of PBE was popularized by Gruenwald and Smith (2014) as educational experiences designed to connect learners with content knowledge in culturally, geographically, and/or ecologically relevant contexts, with consideration of the intersection between learner identities and learning contexts to empower and engage learners. PBE provides an umbrella under which to describe, analyze, and compare initiatives that emphasize the relationship between the learner and the place where learning occurs (Yemini et al., 2025). While original frameworks of PBE referenced outdoor education a great deal (Gruenwald & Smith, 2014), the term has evolved to often include indoor and even online experiences (Yemini et al., 2025). Place-based education has been associated with increased student engagement and academic learning, while also providing opportunities for civic and community engagement (Powers, 2004; Yemini et al., 2025).

We define mathematics broadly as “study of patterns and relationships among quantities, numbers, and space” (National Academy of Engineering & National Research Council, 2014, p. 14) and mathematical reasoning as the process of drawing on social practices from different communities, building on research on participation in socially and cultural organized practices (Lave & Wenger, 1991), to ask questions about or inquire into patterns and relationships. Research suggests that math outcomes are not always emphasized in integrated STEM activities (National Academy of Engineering & National Research Council, 2014) and that although important math concepts are in many STEM activities, unless math connections are brought to learners’ attention in an explicit way, learners struggle to “see” the math (Nathan et al., 2017; C. A. Walkington et al., 2014). Similarly, students use complex and interesting forms of mathematics in their everyday activities (Pattison et al., 2017; C. A. Walkington et al., 2014), but everyday mathematics is often very different from and difficult to reconcile with school mathematics (C. Walkington & Hayata, 2017). For this reason, it can be difficult for students to see the mathematical activities they do in informal contexts as being mathematics, limiting the extent to which efforts to use such student-driven contexts to improve students’ attitudes toward mathematics.

Indeed, one proposed benefit of both math walks and place-based learning is that they have the potential to enhance students’ mathematics-related attitudes and beliefs—including enhancing mathematics interest, mathematics ability beliefs, and mathematics self-efficacy and reducing mathematics anxiety. We define mathematics interest according to Hidi and Renninger (2006) as the process of engaging and the predisposition to re-engage with particular activities or ideas. Interest is a dynamic variable that can be triggered, maintained, and transformed from fleeting situational interest in elements of a learning task to more enduring and stable individual interests in specific domains or topics (Hidi & Renninger, 2006). We define mathematics ability beliefs as whether and to what level students perceive themselves to be mathematically able (Perez-Felkner et al., 2017). We define mathematics self-efficacy as learners’ beliefs about their capability to successfully perform future mathematics tasks, such as solving mathematical problems (Bandura, 1997; Pajares & Miller, 1994). Mathematics interest, mathematics ability beliefs, and mathematics self-efficacy show important relations to students’ mathematics performance and achievement (e.g., Kriegbaum et al., 2015; Parker et al., 2014; Schöber et al., 2018). We next turn to research on math walks, describing their characteristics that have the potential to facilitate changes in students’ attitudes toward mathematics.

1.2.2. Spatial and Epistemic Agency in Informal Learning Contexts

Our analysis revealed the ways in which math walks incorporate learner agency. Although agency was not an a priori element of our theoretical framework, during the initial thematic analysis, this construct emerged from the data. We subsequently drew on scholarship concerning spatial and epistemic agency to interpret these patterns. We briefly review the theory behind this construct in order to lay the groundwork for interpreting the results where agency arose as a central theme of math walks in our context.

Here, we conceptualize learner agency as occurring within sociocultural contexts as learners engage in discursive practices with others (Arnold & Clarke, 2014) and are positioned with, perceive, and act upon opportunities to shape consequential aspects of an activity. In this way, agency is something that learners can achieve as they engage with an instructional ecology over time, rather than a fixed ability that they possess or do not possess (Biesta & Tedder, 2007). Agency in informal learning spaces is often conceptualized in terms of interconnected elements of expressions of agency related to the physical environment, having choice and autonomy to shape their experiences as they interact with others and having choice to pursue options of personal relevance that can allow for empowerment. Informal learning environments can have characteristics that position students for agency, setting the stage for learners’ actions and interactions. Students then perceive these structures and decide what to explore and how, with support from teachers or facilitators (Letourneau et al., 2025). In these ways, students can be set up to be positioned for agency, they can perceive the level of agency they believe they have based on the opportunities they are given, and they can choose to act with agency (Miller et al., 2018). Agency is not the same thing as open-endedness, in that students need a system of support and guidance to engage effectively in their own ways (Letourneau et al., 2025).

Related to Letourneau et al. (2025) conceptualization of agency related to the physical environment, we use the term spatial agency to refer to learners’ interactions as they make consequential choices about how and where they move when learning, with these choices having implications for which objects, perspectives, and features in the environment they focus on. Research on outdoor informal science learning shows that movement through personally relevant places can support learners in noticing disciplinary features, adopting new perspectives, and producing representations and knowledge artifacts grounded in the physical environment (Zimmerman & Land, 2014). This relates to the construct of students’ independent mobility, which involves the extent to which students can access and move through spaces. Different environments have different actualized affordances that are perceived by students as containing possibilities for future movement (Kyttä, 2004). Our perspective recognizes that learners’ movements in traditional classrooms may be viewed as being characterized by immobility and that the presence of mobility can provide learning opportunities through embodied engagement (Leander et al., 2010). Learners construct trajectories through places, with their movements reorganizing relations among people, materials, and knowledge. In this way, students’ mobilities come to shape the learning that happens in an environment (Leander et al., 2010). Adult authority can further shape and control the spaces that youth inhabit, limiting youth’s opportunities to experience time and space independently (Kyttä, 2004). In addition, much of youth’s new mobilities have become virtual with the rise of new technologies and digital media (Franco & Birenboim, 2024).

We also highlight the possibility of math walks for enhancing epistemic agency, which refers to the learner’s actions that show control over decisions about what knowledge is worth pursuing and how forms of mathematical knowledge can be developed and shared. This relates to Letourneau et al.’s (2025) dimension of choice and autonomy as vehicles for agency in informal learning spaces, as well as visitors being able to pursue personal relevance. Epistemic agency involves youth being given opportunities to build knowledge and create knowledge products for others, as well as foregrounding students’ knowledge and lived experiences in ways that students themselves control (Miller et al., 2018). Giving students epistemic agency has been identified as a key practice when implementing ambitious instruction in science classrooms (Stroupe, 2014). Students take and use cognitive authority, hold each other accountable, and use materials and intellectual resources, allowing them to engage in legitimate participation in knowledge building. Curricula can redistribute epistemic agency by opening particular decisions to students, including which phenomena to investigate, which questions to pursue, what methods to use, and what explanations or models to construct (Ko & Krist, 2019). This follows the work of Williams-Pierce et al. (2024) in mathematics education, where participants’ interests and goals are centered, and learners are not given goals explicitly or directed to learn a particular structure and engage in implicit mathematization of the world. They highlight that rich mathematical activity may not look very mathematical at first glance, particularly when students engage in math walks using little formal mathematical notation, which is the case with math walks.

Spatial and epistemic agency can co-occur during math walks in important ways to shape the nature of the mathematics that learners experience. Each of these two kinds of agency can be supported in the task and environment, perceived by the students, and then acted upon in practice.

Table 1. Dimensions of spatial and epistemic agency.

	Lower Epistemic Agency	Higher Epistemic Agency
Lower spatial agency	Students’ experiences are constrained to externally assigned places/routes, and the nature of their mathematical questions and methods for addressing those questions are pre-specified.	Students are assigned specific places, routes, and objects, but they then decide what mathematics to pursue and how to pursue it with respect to the places.
Higher spatial agency	Learners choose routes, places, and aspects of places to focus on, but complete predetermined mathematical tasks.	Students choose which places, objects, and routes to focus on and the mathematical inquiries that emerge for these foci, with support.

shows how each kind of agency may occur in activity structures. Most prior work on math walks has occurred in the upper two quadrants of the table, with very little work in the lower right quadrant specifically.

1.3. Research Purpose and Questions

Previous research on math walks in mathematics education has largely focused on students answering closed-ended mathematics questions at particular physical locations (Cahyono, 2018). Research on the Mathfinder app has focused more on students asking questions rather than answering them (e.g., C. Walkington et al., 2026a), but the app has mainly been used in formalized ways where students have an explicit and highly structured sequence of activities in the app, relating to particular mathematical topics embedded in professional videos. Despite the fact that many studies on the Mathfinder app occur in informal learning environments, little research has been conducted where the student agency and choice that is possible in these environments is fully leveraged and capitalized upon. This was one of the primary purposes of the present investigation. This approach to math walks brings up further unique considerations relating to facilitating student interests and allows students to experience new kinds of epistemic and spatial agency in place-based settings. In the present study, we used a self-directed model of student engagement within the Mathfinder app, where students had substantial choice over the routes and locations they explored, the environmental features they examined, and the mathematical questions they posed. We let the secondary students who participated in the study choose places they wanted to visit, as well as what math concepts or physical and spatial properties of different locations they found compelling or interesting. Our research questions are as follows:

(1)

What mathematical connections and types of questions emerged as students engaged with professionally authored and student-authored math walk stops?

(2)

How did students describe the affordances of the math walk experience?

(3)

What exploratory patterns of change were observed in students’ mathematics-related attitudes and beliefs over the camp experience, and how did these patterns converge with, complement, or diverge from students’ qualitative accounts of the math walk experience?

2. Materials and Methods

The present study was conducted as an exploratory, qualitative-dominant, mixed-methods bounded case study. Mixed-methods research intentionally combines and integrates qualitative and quantitative forms of evidence within a single inquiry to develop an understanding of a phenomenon that neither form of evidence could provide as fully on its own (Creswell & Clark, 2018). The study was qualitative-dominant because students’ math walk submissions, responses during the focus group, and open-ended survey responses provided the primary basis for interpreting their mathematical activity and experiences of agency, whereas quantitative survey measures provided a supplementary source of evidence about possible patterns of affective change (Morse & Niehaus, 2009). Case study research involves an in-depth examination of a particular case within its real-world context, with the case understood as an integrated system whose boundaries can be identified (Yin, 2018). In the present study, the case was bounded temporally by the one-week residential camp, spatially by the university campus, socially by one cohort of eight campers, and substantively by the Mathfinder activities in which those students participated. Qualitative and quantitative findings were integrated during interpretation to develop a case-level account of how spatial and epistemic agency related to students’ mathematical noticing, question posing, and mathematics-related attitudes.

The positionalities of the authors were as follows. The first and second authors were clinical faculty members with a background in video game design and instructional technology who jointly led the summer camp. Their positions at the university were largely focused on program coordination and/or teaching courses rather than research, and they worked closely with and formed relationships with the students over the course of the camp. The third author was in a research-focused position at the university, with a focus on mathematics education and educational technology. While she did not attend the summer camp, she was the director of the project that created the Mathfinder app and regularly engaged in Mathfinder camps with youth.

2.1. Participants and Context

Youth participants included eight students (seven female, one non-binary) who were enrolled in a residential summer camp at a private university in the South, run by its video game design program. The students ranged in age from 11 to 16 years old (mean = 14) and ranged in grade level from seventh to eleventh grade, with two identifying as White, five as Black, and one identifying as Hispanic. These students were recruited through counselors and teachers at local STEM-based schools and programs, who were sent emails with information for the camp. There was no cost for participating in the camp, and every interested student was offered a spot. The students all obtained parental consent and gave their assent to participate in the research. The study was approved by an IRB at Southern Methodist University (21-073). The students received $25 for participating in the focus group portion of the research study. The camp was facilitated by two research assistants and two instructors. The research assistants were both undergraduate students and had attended previous summer camps run by the same personnel, which provided an opportunity for young girls to be introduced to STEM careers. The previous campers were invited to attend this university camp as mentors. One instructor was the director of those previous summer camps and a faculty member in the university’s video game design program. The other instructor was one of the directors of the video game design program at the university itself and had previously directed video game design summer camps.

2.2. Procedures

This camp was a 1-week residential camp in which women in STEAM fields, alongside university faculty and recruiters, taught sessions in design thinking and tutored campers with hands-on emerging technology for STEAMERS: Science, Technology, Engineering, Arts, Math, Education, Research, Social Media. The summer camp activities started at 8:00 AM every day for 5 days, Monday through Friday. The math walk activities covered here mainly took place from 7:00 PM–9:00 PM on Monday through Wednesday of that week, with additional activities taking place on Friday morning. The math walk activities during the camp were the only activities where students were doing mathematics. Other activities focused on video game design, augmented reality, creating art, scripting in C#, and learning about the teaching profession. Thus, while it is impossible to attribute any pre-/post-changes in attitudes toward mathematics over the course of the camp to the math walks alone, they were by far the most mathematically focused and apparent activity. Each camper was given an iPad to use for the activities and took a pre-survey before the activities began and a post-survey after they ended.

The camp used a gradual release model, where campers were progressively given more freedom to determine the direction of their math walk explorations. On the first evening, campers were told to look at the campus map within the Mathfinder app and decide as a group which walk stops on the campus they wanted to visit. There were no instructions given about what the Mathfinder app did, nor what it was designed for, and they had to use the map to navigate themselves. The intention was to give them the freedom to figure out how the app worked and use it in their own fashion. The TAs were given the additional responsibility for keeping the group together on the walk. The first evening session goal was to visit two walk stop sites. The two primary professional walk stops students engaged with were the Boulevard walk stop and the Labyrinth walk stop (see

Table 2. Professional walk stops that students visited at university site.

Walk Stop	Description of Walk Stop and Its Mathematics	Images from Video
Boulevard Walk Stop	The video explores why branches tend to get smaller as you go upwards on a tree and how this relates to the amount of nutrients flowing through each cross-section. In this way, students explore how the calculated area of the branches’ cross-section is preserved, such that the sum of the circles’ surfaces remains constant.
Labyrinth Walk Stop	The video explores how to make a path that winds around itself many times, yet only has one route that takes you from the outermost point to the center. In this way, the students explored concepts from topology and graph theory, utilizing mathematical patterns and geometric structures.

). These two stops were chosen from the full set of 16 stops at the university by calling for a vote from the campers and choosing the two stops with the greatest number of votes. When students visit a professional walk stop and engage with the video, they answer several questions in the app while watching the video. They give their answers to these questions by audio-recording their voice. These questions were: (1) What are some things you notice in this space? (2) What questions do you have related to this space? (3) What questions do you have that are related to math? (4) What strategies could you use to answer your question?

For the second session on the second evening, the campers were directed to visit any location of interest across the entire campus to create one to two walk stops on their own. They were not asked to visit any professional walk stops on the second evening, although professional walk stops were still viewable in the app. Campers were split up into three groups that organically formed to walk in different directions. The campers went to different locations, and each completed creating one to two of their own walk stops at these locations. When creating walk stops, campers would take pictures, annotate the pictures, and tag the pictures with questions and answers.

The morning of the third day of camp, instructors approved the camper-made discovery walk stops in the backend server so that they would appear on the map for campers to interact with. That evening, a researcher facilitated a focus group with the students to get their feedback. After the focus group, the campers were directed to give feedback in the Mathfinder app on at least one of their fellow campers’ discovery walk stops. The morning of the fourth day of camp, the instructors reviewed all the camper-to-camper feedback for approvals and added one feedback for each camper discovery walk stop, turning on the app viewing of the feedback in the backend. The morning of the fifth and final day of camp, the campers reviewed their peer and instructor feedback and got screenshots of their work for their camp poster presentations if they desired.

2.3. Data Sources

2.3.1. Submissions Within the Mathfinder App

We coded each of the walk stops students created, as well as their responses to the four prompts they answered at the Labyrinth walk stop and the Boulevard walk stop.

Table 3. Background of each participant included in the study.

Game Name	Gender	Grade	Walk Stops Created	Professional Walk Stops Engaged with
Ailoth Scorpio	F	No response given (14 yo)	2	0
Alya Trillon	F	10	1	1
Aquarius Minima	F	10	2	2
Astrophel Fractal	F	9	2	2
Belindar Star	F	11	2	2
Moon Arctan	F	No response given (15 yo)	1	2
Rhean Barium	F	8	2	2
Thalassa Diamond	NB	7	1	2

shows how many walk stops each student created and how many of the two professional walk stops they engaged with. As the students’ responses to the professional walk stops were audio files, their speech was human transcribed. Figure 3 shows the walk stops students created, located on the Mathfinder app.

2.3.2. Focus Groups

A focus group at the end of the study with all the participants lasted approximately 35 min and was video-recorded. During the focus group, one researcher asked questions according to a protocol (see

Table 4. Student focus group protocol.

Focus Group Protocol
What are some of your favorite activities? Places to go? Do you see any connections between these places and math ideas or topics?
What did you like about the math walk activities you just completed? What did you find interesting?
Here is one of the walk stops you created at the camp. Can you tell me about your process for creating this walk stop? How did you end up with this question? What else are you curious about? Do you have other questions you think would be interesting for you and others to think about? What did you learn from creating this walk stop?
What math did you do at this camp? What is “math” to you? In your opinion, what parts of the camp were not related to mathematics? What is NOT “math” to you?
Give me a few examples of math connections here at the university. How do you see math as relating to what the university is all about?
How were the math walk activities at the camp similar to or different from the math you do in school?
Now that you have completed these math walk activities here at the university, can you give some examples of mathematics you might now notice in your home or neighborhood?
Have you used any AR tools before this camp? What did you think of the augmented reality tools (e.g., measuring things and counting things in your camera)?
Tell us what you thought of: (1) being able to take and annotate pictures? (2) being able to create your own math walk stops, with questions and answers? (3) being able to see yours and other students’ math walk stops on the map?
What did you like about the Mathfinder app? What are some things that we could improve in the Mathfinder app?

). The video file from the focus group was human transcribed.

2.3.3. Pre- and Post-Surveys

The pre- and post-surveys contained six different rating scales for students’ attitudes toward and beliefs about mathematics, which were rated on a 1–5 scale. A description of each rating scale is given in

Table 5. Scales used to measure students’ attitudes toward mathematics.

Construct Measured (Citation)	Number of Items	Example Item
Mathematics Interest (Linnenbrink-Garcia et al., 2010)	8	Math is exciting to me.
Mathematics Anxiety (Bai et al., 2009)	14	Math makes me feel uneasy.
Mathematics Ability Beliefs (Perez-Felkner et al., 2017)	3	I’m certain I can master the skills being taught in my math class.
Mathematics Self-Efficacy (adapted from Bernacki et al., 2015)	1	How confident are you that you could solve math problems in the future?
Place-Based Mathematics (researcher-created)	1	I see math connections in most places I go.
Everyday Mathematics (researcher-created)	1	Math is a helpful way to understand and think about many of my everyday activities.

. The post-survey also contained four open-ended items: (1) What did you like about your experience creating math walks? (2) What do you wish had gone differently? (3) Think back to the things you did over the camp. Did how you think about math change after you did those things? If yes, give an example of how the way you thought about math changed. (4) How were the math walk activities similar to or different from the math you do in school?

2.4. Data Analysis

2.4.1. Research Question 1

We first noted the location of the walk stops that students created. We then coded the math concepts the students discussed in the walk stops they created and when answering the prompts at the Labyrinth and Boulevard using open coding techniques (Saldaña, 2021). In open coding, the data is segmented into discrete units, and labels are assigned that capture the meaning of each segment to surface potential concepts. Code definitions evolved as data were analyzed, with earlier data revisited as codes were clarified (Gibbs, 2018). Then, we coded the quality of the questions students asked in (1) the walk stops students created, (2) the students’ responses to the second prompt at Boulevard and Labyrinth, and (3) students’ responses to the third prompt at Labyrinth and Boulevard. In particular, math concepts used in both student-created and existing walk stops were defined and then refined through separation (e.g., separating surface area from volume) or combination (combining perimeter and circumference) as coding ensued. This coding was initially carried out by one coder, the author who had expertise in mathematics education. The coder formed a codebook for the different potential math concepts, organizing them by overarching mathematical strands of geometry, measurement, algebra, and number. A second coder with expertise in video game design and STEM education then independently coded the dataset; then coding was compared, and all differences in coding were reviewed and resolved through discussion. The codebook was then finalized, and a final round of coding was done.

The first coder then rated quality of each walk stop using the coding scheme developed previously in authors (date), which is given in

Table 6. Rating of students’ walk stops.

Level	Description
1	Students ask a reasonable question that is not directly about mathematics. “Why are there so many squirrels here?”
2	Students locate a simple math concept at the site and either (1) give it a math name or (2) try to measure or count it: “Where do we see obtuse angles in this tree?” “How many squirrels live in this tree?” “How tall is this tree?”
3	Students ask about a strategy they could use to measure or count something. “How can we figure out how many windows there are without counting one by one?” “How can we measure the length of the path in the labyrinth, given that it is not straight?”
4	Students ask a “why” question, a question about cause and effect, or a question about patterns that relate to math. “Why does the shape of the dome cause the sound to echo?” “What kinds of rotational patterns do we see in this roof design?” “Why does the fountain’s water slope inwards as it falls?”

. This coding scheme had been used successfully in several other camps to both code the walk stop data after the camp, as well as being used during the camp to help teachers give feedback to students on their walk stops and help students give feedback to each other on their walk stops. The levels in Table 6 corresponded to colors of stars that teachers could assign to students’ walk stops as they populated the map. Once the first coder had rated the quality of each walk stop, the second coder independently coded all the data. Discrepancies were resolved through discussion to arrive at the final coding for the dataset.

We also examined the answers that students gave when posing their math questions to go with the walk stops they created and coded whether they were correct or incorrect. Many of the answers could be verified mathematically to determine whether they were correct. For example, one student asked the question, “How do you find the radius of a labyrinth?” Their answer was, “You measure the inside of it to find the diameter, and after finding the diameter, divide by 2 to get the radius.” This is a correct usage of the mathematical terms radius and diameter in terms of recognizing their mathematical relationship. Another student asked, “What is the circumference of the circle?” The accompanying answer was, “Measure from the outside of the circle to the center of it then plug the information into the equation r^2 × 3.14.” Here, the student is mixing up the mathematical concepts of circumference and area, so the answer is incorrect. Sometimes, students would give a strategy instead of a direct mathematical answer. If the strategy could plausibly work if it were applied, it was considered a correct answer.

2.4.2. Research Question 2

The focus group responses given by students were entered into a spreadsheet such that one unit of analysis was one student’s complete turn of talk during the focus group. The students’ open-ended survey responses were also entered into a spreadsheet such that a response to one of the four open-ended survey items was the unit of analysis. Students’ responses were then coded as to whether they fell into one of the two emergent categories in

Table 7. Coding scheme used for students’ open-ended responses.

Theme	Description	Count (Focus Group)	Count (Survey)
Spatial Agency	Participants’ responses reflect a sense of autonomy in their engagement in place-based settings, including authority over routes or places they visit, how they move through the environment, and what places and elements of places are important and significant.	7	8
Epistemic Agency	Participants’ responses reflect a sense of autonomy in directing their attention toward personally meaningful mathematical aspects of the environment. This reflects learners’ authority to decide what is mathematically noteworthy about a space, what questions to pose, and what strategies or representations to use.	7	8

. These categories were determined through thematic analysis techniques (Braun & Clarke, 2006). Thematic analysis involves an iterative process of coding, pattern recognition, and theme development, in which codes are clustered into broader, conceptually coherent themes that capture something significant about the research question. Careful attention was paid to the coherence and distinctiveness of themes and the alignment between data extracts and final interpretations (Braun & Clarke, 2006).

Participant responses were collaboratively reviewed four times to identify recurring ideas and patterns and to arrive at the two themes given in Table 7. In this way, the theme topics included several stages of rigorous review using thematic analysis (Braun & Clarke, 2006). During the initial open coding, segments of text responses of participant experiences were assigned descriptive codes. Two of the researchers discussed what they were noticing from the data and their initial ideas about key emergent codes. Then during theme development, the codes were grouped into two broader conceptual themes capturing recurring patterns across responses. During theme refinement, two themes advanced as construct topics, and topic categorizations were reviewed for consistency across the dataset. During theme topic coding finalization, issues with the coding scheme’s coherence were resolved through iterative comparison with the original responses and recoding. These themes represented in the responses suggest that participants exercised agency in determining where and what to observe and interpret.

The process was carried out by two coders (first and last author on this paper). Each coder first independently coded the data for important emergent themes, writing analytical memos regarding emerging ideas. Initially, both coders were focused on physical freedom as an essential characteristic of the data (which become spatial agency), while one coder was focused on use of the AR tools, and another was focused on cognitive autonomy (which became epistemic agency). After discussion, the second coder recoded the data into emergent themes a second time, writing new analytical memos. This coding and the memos were shared with the first coder and then discussed to arrive at the final coding scheme. The counts given in Table 7 are how many of the eight students mentioned each of these themes when the data was coded a final time.

2.4.3. Research Question 3

To answer research question 3, we examined students’ scores on each motivational construct shown in Table 5 over the course of the camp. Because of the small sample size, we focus on Cohen’s d effect sizes rather than inferential tests. One participant was missing their post-survey scores for math ability beliefs, math self-efficacy, place-based math, and everyday math.

Research question 3 also involved the explicit integration of quantitative data on students’ changes in attitudes with qualitative data about their reflections on their math walk experiences. Initially, the qualitative and quantitative data were analyzed separately, with integration occurring primarily during interpretation. Findings were linked by participant pseudonym and merged at the participant and case levels. For each participant, we compared app-based mathematical activity and qualitative descriptions of spatial and epistemic agency with the participant’s descriptive pre-to-post pattern on the mathematics attitude and belief measures. We examined areas of convergence, complementarity, and divergence, with the quantitative data used to further deepen the qualitative analysis and more explicitly detect change over time.

3. Results

3.1. Research Question 1

The coding revealed that students made math connections across seven different locations (Figure 4). At these locations, they made math connections to five math concepts related to measurement (green text in Figure 4; diameter/radius, height, perimeter/circumference, volume, and time), three math concepts related to geometry (red text in Figure 4; 3D geometric structures, 2D geometric shapes, and parallel lines), three concepts related to algebra (black text in Figure 4; variables, coordinate grid/graph, and speed), and one concept related to number and operations (blue text in Figure 4; numbers/counting). The most connections were made between math concepts and the two locations that all the students visited on their walks: the Labyrinth and the Boulevard. In addition, the math concepts that were most likely to be connected to places included perimeter and circumference, diameter and radius, 3D geometric shapes, time, counting/numbers, and area. Thus, measurement concepts, as well as standard geometry concepts, were the most common connections the students made. More advanced geometric connections (e.g., volume and surface area) and algebra connections were both rarer. Overall, both the math connections made and the places visited were somewhat diffuse, which corresponded to the open-ended nature of the camp. Many of the math concepts corresponded to middle school-level mathematics, and students connected a wide variety of concepts to a variety of different places on the college campus.

In addition, returning to Table 3, we see this issue manifesting in a slightly different way. There was variance in how many students chose to actually engage with the professional walk stops and with how many walk stops that students themselves created. In addition, had the focus and structure of the camp been different, the students may have been able to create more than one to two walk stops. This demonstrates how the approach to give students freedom and agency may have resulted in changes in both the products students created and mathematical answers and explorations engaged in.

The results for the coding of the quality of the questions students asked are given in

Table 8. Levels of questions in walk stops created by participants.

Level of Question Asked	New Walk Stops Created at This Level	Boulevard Questions at This Level	Labyrinth Questions at This Level	Example
1	0	1 (0, 1)	5 (4, 1)	What is the main purpose of the Labyrinth, and why was it designed?
2	5	2 (0, 2)	5 (0, 5)	I just want to know how old these trees are.
3	6	2 (0, 1)	1 (1, 0)	How would you find the volume of this building?
4	2	1 (1, 1)	1 (1, 0)	Why do we build up bricks by overlapping and not straight?

Note. The numbers in parentheses separate out whether the question was posed in response to the first prompt, “What questions do you have related to this space?” (first number in parentheses) or the second prompt, “What questions do you have that are related to math?” (second number in parentheses). As can be seen from the numbers, the second prompt was not in general associated with higher question levels.

. As can be seen from the table, most questions students posed were at levels 2 or 3, especially when the students were making their own crew discovery reports. Level 2 involved labelling math concepts or asking direct measurement or counting questions. Level 3 involved thinking about strategies for measurement or computation. There were a few examples of level 4 questions that asked about causative relationships or why questions. Overall, when given agency to explore mathematical concepts and places on their own terms, the participants posed mathematics questions at a variety of levels of mathematical sophistication, building from simpler to more complex questions. When learners determined new places to ask mathematical questions, these questions tended to be relatively complex at levels 2 and 3. When students asked math questions about existing locations highlighted in videos, these questions tended to be less complex at levels 1 or 2. Although level 4 questions were somewhat rare, they did sometimes emerge from students’ work, despite the lack of explicit guidance to create this kind of question.

For the 13 walk stops students created as crew discovery reports in the app, they gave the correct answer to the question they asked 10 times, the incorrect answer one time, and answers that were uninformative or could not be verified two times. Although the goal of the activities was not to provide accurate answers to math problems, students largely engaged in mathematically valid reasoning when considering the questions they posed, with occasional issues that went unaddressed.

3.2. Research Question 2

During theme development to describe key elements of the participants’ experiences with math walks, the camper responses were grouped into two broader conceptual themes capturing recurring patterns across responses: spatial agency where students could move and navigate around outdoor spaces and epistemic agency to notice and follow curiosity examples when making mathematical connections.

3.2.1. Spatial Agency When Choosing and Moving Through Spaces

All eight of the participants reported positively on the spatial agency provided by the math walk activities, several with specific comments about how it differs from their school math experiences. For example, Astrophel Fractal described how “you get to walk around while, as in school, you’re instructed to stay seating and listen to the teacher.” Several other responses described spatial agency with comments about being outdoors and exploring while walking. For example, Rhean Barium described how math walks allowed them to “explore, just walk around, and see new stuff around out of school.” And there were also more general positive comments about the interactivity of being immersed in outdoor settings. Astrophel Fractal described how “I like how it encourages you to there’s a question before you start the video. That’s what do you see that is interesting in this space. It encourages you to look around and enjoy your surroundings.” Belindar Star described how “I liked how you really got to see nature, like yesterday, we went to the park. We saw the amphitheater. It was Yeah, so that was cool.”

3.2.2. Epistemic Agency When Noticing Potentially Mathematical Features

The participants’ descriptions of their epistemic agency connected to their agency and engagement with the mathematics concepts through the place-based activities. Epistemic agency was often reported positively in connection with spatial agency, raising the possibility that these two constructs tend to be mutually supportive. For example, Alya Trillon described how “I like the math walks, because I know sometimes math can be, like, a little not fun, but I like doing the math walks, because you can, like, take pictures and like, walk around and like, also do math at the same time, so it makes it more fun,” connecting spatial agency outdoors to epistemic practices of doing mathematics. There were also similar connected comments about participants noticing mathematics through nature. For example, Belindar Star described how they “saw how math is in trees outside,” and Thalassa Diamond said they “can see math connections better” when engaging with informal learning spaces.

3.2.3. Epistemic Agency When Transforming Features into Mathematical Objects

Students described epistemic agency with respect to transforming features they notice into mathematical objects, with Aquarius Minima describing how “I feel like it made it easier to visualize the math in front of me, rather than just being handed a paper with some word problems, and I have to, like, either draw it on my page or just visualize it in my mind. It’s easier to have it just right in front of me.” Belindar Star described how “creating math walks truly made me truly think about what I was seeing and think deeper into math concepts.” Spatial and epistemic agency was also sometimes combined with students acknowledging their value of control over timing as they transformed features into mathematical objects, with Alya Trillon describing how “I was able to make my own math connections without rush, and I was able to have fun.” Finally, Ailoth Scorpio brought up the nature of the math connections students were making, expressing some disappointment that they largely tended to be related to geometry rather than algebra: “I feel like at this camp, we’ve done, a lot of like geometry and like you said that this whole math walk was supposed to be, pre-algebra, and algebra, and math, to me, is like she said, getting to know things logically with, numbers, functions and things like that nature.”

3.2.4. Posing Their Own Questions Through Epistemic and Spatial Agency

There were also reflections from campers about spatial and epistemic agency that related to the structure of the camp itself and how it allowed them to pose their own questions. On the first evening of doing math walk activities, students were asked to visit existing walk stops that were on the Mathfinder map. However, the second evening of activities removed any restrictions on visiting existing walk stops on the map, and the campers chose to visit different locations that were of interest in smaller groups. The majority of the campers reported positively on the epistemic agency provided by the activities on the first and second evenings, with several reporting on the experiential and environmental affordances of having agency to physically explore spaces when learning about mathematics. For example, one Astrophel Fractal reported that “anything that I saw interesting or had some sort of number entwined with it, like any type of counting object. I took a picture of it because I knew that would make for a math question.” Students especially appreciated the freedom of the second night of activities, with Rhean Barium describing choosing to go to the meditation garden, “Yesterday, we went to the meditation garden. And like, just like a bunch of trails you could go on, you could go on, you could see a bunch of different stuff. And I took a picture of the building, and like, you could see from the distance. And the question was like, what would be the, how would you find, like, the volume of the building? Like, what would the formula be? And then, like, the answer would be, like, volume equals length times width times height.”

3.2.5. Developing a “Math Lens” for the Future

The students also described how this epistemic agency and spatial agency that they experienced while doing math walks helped to develop their “math lens” that allowed them to see math everywhere they went. Ailoth Scorpio described how “seeing more problems and equations that you can see, because I try to go on walks throughout my neighborhood, you know. So, you know, seeing all the different types of houses around and seeing the different geometric shapes that are in those houses.” Aquarius Minima said, “I liked how it made math more interactive, rather than just there are some digital assignments that I do with math, but it’s just staring at a screen. There might be some black and white images, but I like how with math walks, you’re able to walk around your environment and see what’s in front of you, and you’re making more connections as you know, as you see them with the assignments. So, I was even just walking around today, and I noticed some math, having to do with math, I’ve noticed some things around the college.” This shows how students might be able to leverage their experiences with mathematics that involve both spatial and epistemic agency to continue their learning once the camp is over.

Aquarius Minima also talked about the transferability of the social skills that students learned in the camp, describing how “I like that while we are doing math, and while we are like educating ourselves, we’re also having team bonding and being able to get to know each other. Because I like whenever I’m going to be in a workspace in the future, I want to be able to familiar, familiarize myself with the people that are around me, so I’m not like in an uncomfortable situation to where I don’t really have anybody to talk to or anything.”

3.3. Research Question 3

The results from the quantitative ratings on the pre- and post-surveys are given in Appendix A. The students had the largest directional changes in terms of standardized effect size in their beliefs about everyday math (d = 0.63), math self-efficacy (d = 0.38), and math interest (d = 0.31). This suggests that examining the impact of math walk activities on students’ math attitudes and beliefs when activities are implemented in a self-directed manner may be a promising direction for future research. Overall results here are limited by the very small sample size and thus should be interpreted in an exploratory and non-causal manner.

Table 9. Pre-to-post changes of individual students.

	Pre-to-Post Change in Math Interest	Pre-to-Post Change in Math Anxiety	Pre-to-Post Change in Math Ability Beliefs	Pre-to-Post Change in Math Self-Efficacy	Pre-to-Post Changes in Place-Based Math	Pre-to-Post Changes in Everyday Math
Ailoth Scorpio	−0.67	−0.38	NA	NA	NA	NA
Alya Trillon	0.78	−0.35	−0.33	0.00	0.00	1.00
Aquarius Minima	0.00	0.00	0.00	0.00	0.00	0.00
Astrophel Fractal	0.56	0.08	NA	0.00	0.00 *	1.00
Belindar Star	1.11	−0.69	0.33	1.00	1.00	1.00
Moon Arctan	0.44	−0.15	−0.33	0.00	−1.00	2.00
Rhean Barium	0.00	0.15	0.00	1.00	0.00	0.00
Thalassa Diamond	0.00 *	0.10	0.00 *	0.00 *	0.00 *	0.00

Note. Entries indicated with a * meant the student was already at the maximum (5) at pre- and remained at the maximum (5) at post-. A negative value for mathematics anxiety means that anxiety decreased, thus a positive outcome.

shows how individual students in the sample changed in their math attitudes and beliefs from pre- to post-. Some students, like Belindar Star, saw large and uniformly positive changes over the course of the camp. Others, like Ailoth Scorpio, had more mixed profiles, with some positive and some negative changes. And still others, like Aquarius Minima, saw small to no self-reported change from the camp. Overall, individual changes were heterogeneous and may depend on individual characteristics of how each student experienced math walks, as well as their overall attitudes toward and goals for the camp experience.

Our integrated, mixed-methods analysis suggests that although students’ qualitative accounts of the math walk experience were generally positive, these accounts did not correspond to a uniform pattern of quantitative change. Integrating the strands revealed several forms of convergence and divergence across participants. When considering Table 9 in conjunction with the qualitative data, we see several important trends. For Belindar Star, the uniformly positive quantitative changes we see correspond to the qualitative data presented showing that this student (1) liked the element of exploring the outdoors and nature, especially being able to choose the amphitheater and experiencing spatial agency and (2) acknowledged that the activities allowed her to think deeper about math concepts, linking this spatial agency to the learning of mathematics and epistemic agency. For Ailoth Scorpio, we saw mixed quantitative findings, with decreases in both math interest and math anxiety. Qualitatively, Ailoth Scorpio was the student who made the important comment that she had hoped for more connections to be made to algebraic concepts rather than just the more perceptually salient geometry concepts. This mismatch between her expectations for the epistemic work she would be engaging in and the actual emerging focus of the camp could be one possible explanation for the loss of interest, with the reduction in anxiety potentially occurring because the math in the camp seemed too easy for her. With Aquarius Minima, we saw no quantitative changes on any measure, while her quotes emphasized her dislike of school mathematics and the strengths of the math walks over school mathematics. One possible explanation is that this may be an example of a student not seeing math walks as “counting” as disciplinary mathematics and, thus, it not changing their attitudes when asked about the math walk experience using this terminology, although more data would be needed to substantively make this claim. Aquarius Minima also was the student who discussed the importance of the social elements of engaging in math walks, which may have not been well-captured in the survey items.

4. Discussion

Our analyses showed how open-ended math walks may distribute agency in two dimensions: spatial agency over where learners move and epistemic agency over what they consider as mathematics. We first discuss our results along each of these two dimensions and then discuss the central tension that arose in our study between student agency and traditional goals of structured mathematics instruction.

4.1. Spatial Agency

Students in our study highlighted the opportunities that the approach to math walks gave them to exercise spatial agency, determining where they would explore and allowing them to spend time outdoors and with nature. They contrasted this kind of agency with what is typical in their school life. The movement that students engaged in sometimes shaped which places and features of those places were given mathematical attention. This builds on research on the interest-driven and voluntary nature of informal learning (Rogoff et al., 2016) and research on place-based learning as a facilitator of students’ STEM engagement and learning (Gruenewald, 2003; Sobel, 2004). We also extend work on the role of mobility in learning (Leander et al., 2010), highlighting how giving youth navigational and spatial agency in outdoor environments can provide a contrast to increasingly adult-controlled, indoor, and digital interactions. We show how youth perceive and act upon environmental affordances (Kyttä, 2004) and how this in turn can shape the nature of the kinds of mathematical knowledge that were pursued in the social system (Leander et al., 2010).

4.2. Epistemic Agency

We further found that students’ movements around the campus may allow them opportunities to participate in mathematical knowledge construction. They engaged in practices where they noticed features of environments, determined that these features were mathematically significant, transformed the feature into mathematical terms by posing a question and then answering, and then shared their knowledge artifact with others. Both peer and instructor feedback, as well as the knowledge that their walk stops would be shared by others, had the potential to add accountability and a sense of communal knowledge building to students’ activities. This was suggested by most answers to questions being mathematically accurate and by the sophistication of the questions stretching across our different levels. This builds on research detailing how physical spaces can be connected to academic concepts during place-based learning (Zimmerman & Land, 2014), as students notice disciplinary concepts and engage in representing these concepts, producing knowledge artifacts to share like math walk stops. This also builds on research on epistemic agency in science education (Miller et al., 2018; Stroupe, 2014), showing how this kind of agency may arise in the domain of mathematics, and what implications it has when enacted in place-based informal learning environments. This also builds on prior research showing how math walks can support mathematical thinking (Barbosa et al., 2022).

4.3. Implications of Spatial and Epistemic Agency

The integration of spatial and epistemic agency (see Table 1) in math walks has the potential to enhance students’ attitudes toward and beliefs about mathematics. Although our study was not designed to support causal inference, some of the important areas for future research might be to examine the development of mathematics interest, mathematics self-efficacy, and students’ ability to connect math to everyday activities, as potential positive impacts of math walks. This builds on research examining the development of interest (Hidi & Renninger, 2006), particularly in mathematics contexts (Bernacki & Walkington, 2018), but showcases a more open-ended environment where student agency is more structured into the task. It also builds on research on the benefits of math walks in terms of increasing student engagement in mathematics (Buchholtz, 2021; Milton et al., in press; Wang et al., 2021). However, it may be important to take into account that students might not always recognize the mathematics they do in informal settings as counting as mathematics (Pattison et al., 2017), especially if explorations are unstructured and student-led. This also builds on research looking at the impact of place-based learning on student outcomes (Powers, 2004; Yemini et al., 2025).

Spatial and epistemic agency can become realized by students through several different layers. The design of the activities and the opportunities structured for students represents potential for students to choose places and mathematical methods or representations (Letourneau et al., 2025). Students needed to perceive these affordances and perceive that they were able to move and pursue mathematics in ways that they found meaningful (Kyttä, 2004). Students then had to actually enact this agency (Letourneau et al., 2025), negotiating routes and leveraging physical affordances, while also initiating mathematical questioning and constructing and critiquing chains of mathematical reasoning.

The combination of spatial and epistemic agency (Table 1) may shape the kinds of mathematics connections that students made. The physical environments students chose, and students’ prior knowledge and experiences, could make some mathematics concepts more perceptually available, including concepts related to shape, number, length, area, and perimeter. Alternately, mathematical connections to functions, covariation, and generalized relationships were more rarely made by students, and they may require additional representational or facilitative support. This builds on prior research. A common practice during math walks is “laminating,” which is taking a concrete element of a real-world space and laminating a math term or idea on top of it (Sherard & Walkington, in press). Laminating does not explicitly require causal, transformational, or generalized forms of reasoning that go beyond specifically visible instances. Having spatial and epistemic agency also potentially related to the ways in which students talked about and represented their mathematical ideas; while some students chose to use school-based notations, others were more free-form in communicating their ideas, even those that were not explicitly mathematical. This builds on prior work describing low-to-no notation informal learning activities and the unique kinds of mathematical thinking and different time scales of consciousness that they entail (Williams-Pierce et al., 2024). This led to an important tension around maintaining the rigor and coherence of the mathematical concepts as students are given epistemic and spatial agency.

Integrating the qualitative and quantitative findings complicated an overall positive interpretation of the experience. Although nearly all participants described opportunities for spatial and epistemic agency, changes in mathematics attitudes and beliefs varied substantially. The relationship between student agency and mathematics attitudes may depend on factors such as alignment between students’ expectations and the mathematical content that emerges, initial attitude levels, and whether the measured constructs capture the interactive, social, and place-based qualities students valued.

4.4. Tensions Between Spatial and Epistemic Agency and Explicit Mathematical Goals

In this case study, students’ freedom to select locations may have opened opportunities to notice and mathematize personally meaningful features of the environment. However, such agency did not reliably produce particular, predetermined mathematical content outcomes. Students generated mathematical questions and, in many cases, made broad connections between the environment and mathematics. Many of these connections involved readily visible geometric, measurement, and counting-related ideas, with algebraic and relational ideas less common. In addition, occasionally answers to mathematics questions were inconsistent or unverifiable, and students asked few mathematical questions at the highest level of our question taxonomy. This kind of tension is reflected in the research on personalizing learning to students’ interests (C. Walkington & Hayata, 2017) and the research on students’ mathematical funds of knowledge (Civil, 2007). When students’ rich experiences with the world must be linked to explicit and predetermined mathematical goals and structures, this can compromise the depth and meaningfulness of students’ situated experiences.

And finally, while the activities may have been associated with some descriptive changes in math interest and self-efficacy, mathematics anxiety and ability beliefs showed little descriptive change. This tension is important considering prior research suggesting that highly structured math walks can be effective for students’ intrinsic motivation, internal regulation, and mathematics performance (Cahyono & Ludwig, 2017). Research also suggests that layering on even more structure in terms of scaffolding tools and prompts may further enhance positive outcomes when considering augmented reality tools (Cahyono et al., 2020) and hints and feedback (Jablonski, 2024). While students in the present study had access to both AR tools and feedback from instructors and peers, using the AR tools effectively and meaningfully interpreting and acting upon the feedback they received may have been challenging in a less structured environment.

Whether these tensions are significant depends on the goals for implementing math walks in an informal learning setting. Open-ended place-based exploration where students exercise significant epistemic and spatial agency may broaden what students regard as mathematical, spur interest, and allow students to apply mathematics to place-based contexts in ways they normally would not. However, if there is an interest in broadening the math that students notice and consider new, less perceptually salient topics or in deepening the nature of the questions they ask to include less visible relationships or formal explorations of math phenomena, more and different types of disciplinary scaffolding may become increasingly important. These could include strategies like (1) prompts that steer students to think about algebraic relationship, like “what changes and what remains constant?” or “how are those two quantities related?”, (2) encouraging students to use AR tools to collect multiple measurements of different associated phenomena, (3) integrating representational supports like tables, equations, and graphs into students’ math walk processes, and (4) providing more opportunities to verify and discuss answers to walk stops and iterate on submitted materials. This builds on research examining the kinds of just-in-time scaffolding that students need to engage effectively in math walks in informal learning settings (Sager et al., 2024; Sherard & Walkington, in press).

4.5. Limitations

This study had several important limitations. The sample size was small with eight students; these students had heterogeneous ages and grade levels, and the math walk experiences were relatively short in duration. However, all three of these limitations are typical of how math walks tend to be implemented at real informal learning sites. Further, the students in the study self-selected (or were placed by their parents) into a STEM-branded residential camp and may have been predisposed to find activities like those in the present study interesting or engaging; thus, it is not clear the degree to which results would hold in a general population of students. We additionally point to the lack of a comparison condition as a weakness, as well as possible influences on student outcomes from the larger STEM camp they were participating in.

5. Conclusions

This study operationalized the concepts of spatial and epistemic agency in the context of math walks, showing how they may arise as students engage in open-ended explorations of place-based settings. We found that math walks can involve both forms of authority, with students selecting which locations to visit, what to notice, and, ultimately, what mathematics to construct. While this agency can, in some cases, support ownership, question posing, and the potential for personally meaningful mathematics, it may also make the mathematics content less predictable and more perceptually driven.

Future research could examine the ways in which scaffolding could be added to such an agentic environment to preserve agency while also supporting both the depth and the breadth of the mathematical explorations. The small, heterogeneous sample and brief camp implementation limit the generalizability of the findings, and the exploratory survey patterns cannot be interpreted causally. Future studies should examine how spatial and epistemic agency interact in mutually reinforcing ways, the kinds of places and collaboration structures that spur rich place-based mathematical reasoning, and the ways in which spatial and epistemic agency arise in different types of learning settings. Math walks need not simply relocate school mathematics outdoors; they can redistribute authority over where mathematics is found and what mathematics becomes possible.