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Background This comparative case study examined the use of math walks with middle grade youths and adult facilitators in an informal STEM learning space. Math walks are place-based walking tours where youths and facilitators critically examine and ask math-related questions about their environment. Method Drawing on situated theories of learning and frameworks for understanding group participation, we examined how facilitators constrained or supported youths’ mathematical thinking as they participated in math walks at the local zoo. Results Using interaction and stance analysis, we identified, analyzed, and compared three contrasting cases: In the first case, the facilitator may have overly constrained youths’ mathematical thinking by asking leading questions and not providing time for youths to discuss their personal interests. In the second case, the facilitator may have underly constrained youths’ mathematical thinking by allowing youths to ask too many new questions without refining or developing any one specific question. In the third case, the facilitator supported mathematical thinking by praising youths’ work, layering on mathematical terminology, and providing clear and actionable instructions for how youths could refine their mathematical questions. Conclusions Findings support efforts to understand how adult facilitators can support youths in seeing mathematics within and asking mathematical questions about the world around them. 

This qualitative case study examines the learning that occurred when a small group of middle grade youths embarked upon a personal excursion during a game-based math walk. Math walks are an informal learning activity where learners and facilitators explore mathematical concepts embedded in everyday spaces. The MathExplorer app is a location-based mobile game designed to enhance and gamify math walks. In our broader research, we investigated a group of 18 middle grade learners who used MathExplorer to engage in math walks at a local nature preserve. While most youths in this study used the game as planned by the researchers, one group deviated from the plan and devised new ways of playing the game and participating in the math walks. We see this deviation, or personal excursion, as a source of insight for research on game-based math walks. To understand the learning that took place during this personal excursion, we draw upon sociocultural and self-directed theories of learning. Using methods of interaction analysis and embodied action conversation framework, we analyzed the small groups’ discussion, movement, and game-use to understand: (1) the point at which the students departed from the planned use of MathExplorer; and (2) the learning that took place after this departure. The findings include how the youth explicitly incorporate mathematics into game play through an activity-as-planned, and how the youth embark on a personal excursion relating to game mechanics and gamification, with an implicit focus on mathematics. We discuss the importance of personal excursions for designing informal mathematics learning experiences.