Edited by: Brenna HassingerDas, Pace University, United States
Reviewed by: Nicole Hansen, Seton Hall University, United States; Lukas Baumanns, University of Cologne, Germany
This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology
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Informal mathematics learning has been far less studied than informal science learning – but youth can experience and learn about mathematics in their homes and communities. “Math walks” where students learn about how mathematics appears in the world around them, and have the opportunity to create their own math walk stops in their communities, can be a particularly powerful approach to informal mathematics learning. This study implemented an explanatory sequential mixedmethod research design to investigate the impact of problemposing activities in the math walks program on high school students' mathematical outcomes. The program was implemented during the pandemic and was modified to an online program where students met with instructors
Much of the research in informal math learning has examined how people use math in their everyday lives and careers (e.g., Nunes et al.,
This is contrasted with formal settings, where learners may see mathematics as disconnected from their lives and daily activities (Mitchell,
In this study, our approach to math walks draws on the successful characteristics of informal math learning, as well as on placebased education, where local communities are sites and resources for learning, and active engagement in the community is facilitated (Sobel,
One challenge of designing informal learning environments was that some individuals could feel uncomfortable knowing that mathematics was involved in the environment, and they were expected to connect the environment with mathematical topics (Gyllenhaal,
Problemposing has been described as referring “to both the generation of new problems and the reformulation, of given problems. Thus, posing can occur before, during, or after the solution of a problem” (Silver,
To contribute to the extant literature on problemposing and bridge this gap between problemposing's implementation in creating informal learning environments, this study investigated youth's problemposing performance and procedure in a math walk program called “walkSTEM.” It analyzed how this experience shaped students' dispositions toward mathematics. This study also aimed to look into youth's interactions with their peers and instructors by observing and analyzing their discussions and conversations when posing and solving math walks problems collaboratively. walkSTEM is an initiative in a large metropolitan area where youth, classes, and families take walks and find mathematical concepts and principles in the architecture, designed objects, art, and nature around them. When youth are tasked with creating their own math walks, they design “stops” on a math walk around their homes, communities, or schools, often leading their audience on the walk and explaining how mathematics is integrated into the surroundings. Since this study occurred during the COVID19 pandemic, the math walks program that was implemented during a weekend extracurricular program for high school students was modified to be fully online. Youth met virtually with the instructors and other program members to watch existing math walk videos from their local communities and design their own walks collaboratively. In terms of their selfgenerated walks, youth can create walks around not only math topics but also other STEM topics. Even though most of the walks and the selfgenerated questions were related to mathematical topics, some youth in this program created questions related to biology, environmental science, statistics, and so on. As the objective of this program was to encourage students to connect their schoollearned topics to realworld scenarios, the authors did not limit the topics to youth's selfgenerated walks. Given that remote learning has become more prevalent, this study explored the possibility of online math walks. It investigated both the advantages and challenges of implementing problemposing and math walks through virtual formats.
The purpose of this study was to (a) investigate the problemposing program's effects on youth's mathematical dispositions; (b) compare youth's problem complexity in different problemposing tasks; and (c) explore the kinds of interactions youth have when creating math walks.
Problemposing “is a feature of broadbased, inquiryoriented approaches to education” (Silver,
Extant studies suggested that integrating problemposing in students' mathematical learning can positively impact students' problemsolving skills, problemposing skills, conceptual understanding, and dispositions toward mathematics (Brown and Walter,
Problemposing activities can promote both students' metacognitive skills (Karnain et al.,
In extant literature on problemposing, researchers also analyze the complexity of studentgenerated problems to investigate the relationships among students' problemposing performance, problemsolving performance, mathematical achievement, and the type of learning tasks students are engaged in. Silver and Cai (
Unlike other learning activities, most students do not have prior experience with problemposing. Therefore, it is important to provide students with peer support and a learning environment within which they are motivated to raise various questions. Most studentcentered activelearning strategies, such as inquirybased learning, problembased learning, and discovery learning, can help to create such learning environments (Albanese and Mitchell,
We previously conducted a pilot study that investigated young children's participation in a walkSTEM afterschool program where they were asked to pose problems (Wang et al.,
This study employed a mixedmethod research design (Creswell and Clark,
This study aimed to utilize the mixedresearch design to comprehensively analyze youth's learning process and dispositions in this online math walks problemposing program with qualitative and quantitative analyses. With the quantitative analysis, this study examined the trajectories of problemposing performance throughout the program and compared dispositions toward mathematics before and after the program. With the qualitative analysis, the authors analyzed problemposing work throughout the program and youth's interviews to further analyze how problemposing shapes youth's mathematical interests and dispositions and what interactions occur among youth when they pose problems and create their own math walks. The research questions are as follows:
Participants were recruited from an existing extracurricular college preparation program in a university located in a large southwest metropolitan area. The program's objective is to help firstgeneration students from designated schools who desire to pursue college transition from high school to college. Activities were enacted during Saturday morning sessions. The program accepted students from 10 schools, where 76.45% of the students are economically disadvantaged, and 24.38% are English learners.
In total, 35 students were recruited (26 Hispanic, seven African American, one Asian, and one student who identified as two or more races). Among the 35 students, there were 24 female and 11 male students. All participants were high school students, and there was one freshman, 13 sophomores, four juniors, and 17 seniors. The 13 instructors (11 females and two males) in this program were tutors in the college preparation program, who were all undergraduate students from this university. Of the 13 instructors, seven were Hispanic, three were White, two were Asian, and one was African American.
In the virtual math walks program, there were three main problemposing activities for students: watching walkSTEM videos and posing their own problems based on those videos, taking #STEMlens photos and posing problems based on those photos, and creating virtual math walks and presenting the walk in small groups.
The walkSTEM videos were short videos in which prior youth or informal STEM educators discussed STEMrelated problems in their surroundings. The STEM problems could be based on a place (e.g., a museum, a shopping mall, and a park), an activity (e.g., playing basketball and playing music), or a STEM topic or concept (e.g., geometry and biology). After watching the videos, students were asked to complete a videowatching questionnaire (see
Students met with their instructors nine times for the program during the semester, including three longer sessions (one 90min session and two 120min sessions), five 30min checkin sessions, and one final presentation session. The researchers, the program coordinators, and the college preparation program staff met with the instructors for training purposes before implementing the program. More descriptions of the instructional activities in each session are listed in
Student activities in each math walk session.
Session #1  Students completed the presurvey. Instructors introduced the walkSTEM program, the gameboard, and the #STEMlens photos. Students watched one walkSTEM video and completed the videowatching form 
Session #2  Students watched three walkSTEM videos and completed three videowatching forms. Instructors checked in with students regarding their #STEMlens photos 
Session #3  Instructors checked in with students regarding their #STEMlens photos. Students submitted at least one #STEMlens photo. Students who finished earlier would watch two more walkSTEM videos and complete the forms 
Session #4  Instructors introduced the Final Walk project to students by watching previous studentcreated Final Walk videos. Each student completed a Final Walk project planning sheet and started to work on the first two math walk stop design worksheets 
Session #5  Students completed the first two math walk stop design worksheets and finalized at least one math walk stop, including the question, the photo/video, and the response to the question for the stop. Students who finished early would watch one more walkSTEM video and complete the form 
Session #6  Students started to work on the third math walk stop design worksheet, watched one walkSTEM video, and completed the form 
Session #7  Students worked in groups to each select one math walk stop from their projects to form a group Final Walk. Students gave feedback to each other, wrote the script for their Final Walk, and created the slides for the presentation on STEM day 
Session #8  Students finalized their group's Final Walk presentation and rehearsed 
Session #9  Students presented their group's Final Walk to their parent's peers. Students completed the postsurvey after the presentation 
Research data were collected through six sources: the student pre and postsurvey, the instructor pre and postsurvey, the instructor mid and postinterview, the student postinterview, the students' problemposing work, and the video recordings of all of the meetings.
The students' pre and postsurveys are presented in
Students who participated in all three problemposing activities were selected to be interviewed using the interview protocol in
Studentgenerated problems' content complexity and students' ratings in the mathematical dispositions survey were the main quantitative outcome variables in this study. The content complexity was coded with the criteria adapted from Liu et al. (
Content complexity scoring examples.
Notrelevant or incomprehensible  0  All circles together. (Prompt A)  Prompt A 
Relevant statement  1  This could be a probability question. (Prompt A)  
Relevant problem, but with ambiguity  2  Why were they built like that? (Prompt B)  
Relevant problem without any ambiguity  3  From just looking at the picture, how many circles can be calculated by each color? (Prompt A)  
Nonroutine relevant problem without any ambiguity  4  If the real estate agency wanted to renovate and deduct 10 meters in the living room to give more space to both Terrace and kitchen, what would be the area of the Living room? (Prompt B)  
Nonroutine relevant problem without any ambiguity; problem allows for multiple solutions  5  How do the color and space between each color make this picture pleasing to the eye? (Prompt A) 
We compared students'
The linear mixedeffects model was fit using the linear mixedeffects regression (
The qualitative analysis portion of this study employed a singlecasestudy design (Creswell,
Descriptive statistics of all measures.
Presurvey interest in mathematics  35  3.63  0.75 
Postsurvey interest in mathematics  18  3.88  0.64 
Presurvey posed problem content complexity  31  2.77  1.15 
Postsurvey posed problem content complexity  16  3.41  1.08 
Videobased problems content complexity  18  3.13  0.20 
#STEMlens content complexity  15  3.15  0.39 
Final walk content complexity  12  3.83  0.33 
Pretest procedural fluency score  35  2.73  1.12 
Pretest conceptual understanding score  35  2.84  0.89 
Pretest problemsolving score  35  1.39  1.24 
The average complexity of studentgenerated problems in the pre and postsurvey and the different problemposing learning tasks are included in rows 3–7 of
The ShapiroWilk's test for the difference between presurvey and postsurvey interest mean indicated that the difference was normally distributed (
Following the quantitative analyses, we used thematic analysis to analyze the transcripts of the postintervention student interviews, and the following themes emerged from the analysis.
Eight out of the 10 students being interviewed mentioned that they started to think more deeply and positively about mathematical concepts. One student (female, grade 10) explained as follows:
[the program] actually gives you a reflection of yourself that you did not know. Because something as a student you just ask like, why would the teacher ask me this kind of question. And when you do this kind of project you actually understand what situation the teacher was in and why did she ask this question. … In this kind of program, I think you'll actually understand and have more, more understanding, and more clarification on questions.
The same student also described her experience with the #STEMlens photo activity to further demonstrate a similar idea. The picture and questions she mentioned are presented in
#STEMlens activity student work—The window. 1How many tables do we need to fill the whole window? 2How many rows are we going to create? 3How many columns are we going to create?
So one of the picture I took was the picture of my window. So I think, I like the creativity because when you create the question sometimes can't get that type of question… But I have multiple questions, I have other things we can actually put on the thing that were kind of complicated. So I was proud of myself because that makes me think I still remember I still have that kind of … the capacity, memory, how you can interpret reallife problems … I found myself asking questions that the teacher doesn't even ask.
Five students expressed that they became more interested in mathematics to some extent. One female student in grade 12 stated:
Just slightly more it's not like I really got into math or I really got into science but I really like it increased my like interest on it. Just to think about like why doesn't it happen or how is this related with stuff that I've learned before but I've never paying attention to it.
Three students mentioned that they were more patient and perseverant when solving mathematical problems after the intervention. In this program, students were only required to solve their selfgenerated problems in the Final Walk project, and students' Final Walk problems were the most complex according to the coding manual. That is to say, students spontaneously chose to pose and solve problems that were more complex and required more effort to answer. Students described the problemsolving process here as research and highlighted that it was different from the textbook problems they were used to
It was a good experience and then I get I got to learn more about it how it really is to do a research most importantly because I think it's good … it help me like think more about how they kind of research really goes and I mean, it's not a full research. It's not a full research but I got like a glimpse of it (female, grade 12).
Yes, Because I think I learned more I gain more experience on how to solve stuff, having patience, because it can be hard at some point, but having patience, take it easy … we can find a solution (female, grade 12).
Thus, the quantitative and qualitative results were not consistent.
The mixedeffects model was employed, and the regression results and Cohen's
Mixedeffects linear regression model comparing problems' complexity—Presurvey problemposing task as reference group (No. of observations: 261).
Student ID  0.44  0.66  
Fixed effects  95%CI  Sig.  
(Intercept)  0.97  1.41  [−1.80, 3.74]  0.50  
Presurvey problemposing task  (ref.)  
#STEMlens photo  0.70  0.99  0.19  [0.33, 1.08]  0.002  ** 
Final walk project  1.22  1.72  0.19  [0.85, 1.60]  < 0.0001  *** 
Videowatching activity  0.37  0.52  0.16  [0.05, 0.69]  0.02  * 
Postsurvey problemposing task  0.45  0.63  0.22  [0.006, 0.89]  0.048  * 
Presurvey math interest  0.06  0.26  [−0.46, 0.57]  0.83  
Pretest procedural fluency score  0.11  0.16  [−0.20, 0.43]  0.50  
Pretest conceptual understanding score  0.33  0.23  [−0.12, 0.79]  0.17  
Pretest problemsolving score  0.04  0.18  [−0.30, 0.39]  0.79  
Gender female  (ref.)  
Gender male  −0.68  −0.96  0.33  [−1.33, −0.04]  0.05  * 
9th Grade  (ref.)  
10th Grade  0.33  0.46  0.77  [−1.17, 1.83]  0.68  
11th Grade  −0.13  −0.19  0.84  [−1.77, 1.51]  0.88  
12th Grade  −0.03  −0.04  0.73  [−1.47, 1.41]  0.97 
Adjusted
Cohen'
^{*}indicates the correlation is significant at the.05 level (twotailed),
^{**}indicates the correlation is significant at the.01 level (twotailed),
^{***}indicates the correlation is significant at the.001 level (twotailed),
Pairwise comparison results of student—Generated problems.
One student's problemposing work is presented in
Eric's problemposing work.
Presurvey  From just looking at the picture, how many circles can be calculated by each color? 

What type of measurement is used to determine that each part is equal? 

Videowatching  talkSTEM Videos: 
How many toppings can I add to my drink? 
#STEMlens  Student submitted 19 #STEMlens photos.  
#1: What is the radius and/or the diameter of this lamp's circular form?  
#2: What could be the area of the degree of the squaresize tablet?  
#6: In my backyard, there is a huge tree, bigger than my house, and I have noticed that the smaller branches are usually pulled down because of the spider webs. Question: Does the size of the spider's web really affect how the smaller branches are pulled? And is the spider's webbing good enough to catch prey?  
#8: From the picture, I have speculated that the wooden walls in the backyard are falling. Question: What would be the cause of the wood falling? Metal bars have been added to support it, but even so, they still fall. Is there a logical explanation for the wood getting weaker?  
#10: Can the size of the bag or box affect the amount of chips inside it? Or, to be more specific, can you say a cylindrical shape holds more chips than a box or a bag?  
#13: Do the fans work more effectively if they are far apart from each other to a certain degree?  
Final walk  
Postsurvey  I see all of the circles on top of each other, and I would ask the question, What could the radius of all the circles be, and could they all be the same? I describe this picture as a way to figure out what the size of each circle could be. What could be the radius of each circle and are they all the same? From this picture, it makes me think about what could be the radius of each circle and which formula could help with that? And if each circle is the same size as each other  
What could be the cm of each room of this house, and how you turn it into an m? 
We analyzed students' participation during the online meetings and identified one key type of interaction: students giving each other feedback and collaborating to create themebased problems.
In the #STEMlens and the Final Walk problemposing activities, students were asked to pose problems based on the provided rubrics (
I will rate the question as a four I think. Because it is not that specific, it's just in the details. The markups, I think a four because you cannot see the complete image of the cone.
#STEMlens activity student work—The birthday hat.
Once students became familiarized with problemposing, they started to work on the Final Walk project. An added layer to this project compared to #STEMlens photos was the presence of a theme. Each group had to choose one theme, which could be a STEM topic, a place, or an interesting area. As a result, when students worked together in groups to create the Final Walk, they had to collaborate with each other to make sure their problems shared the same theme. In this excerpt, Abby (grade 12) started with a problem more related to geometry than biology, and she managed to modify her problem based on some feedback she received from Gina (grade 12) and the instructor. Abby's photo is presented in
Abby: My photo was a tree like a tree branch in the form of a triangle. And I was going to ask, what is the space between both of the branches if I'm given a squared plus b squared equals c squared?
Instructor: So I guess my question to you is, would that be more related to biology or geometry with that question?
Abby: Geometry.
Instructor: Geometry, because you're talking about Pythagorean Theorem, a squared plus b squared plus c squared. So you kind of want to think about it in a more biological lens, if that makes sense. So other than Aurora, thank you for sharing, Jennifer and Nathalie. Anybody? What kind of questions can we ask about a tree that is in a that forms a triangle? What kind of questions we ask about it from a biological or environmental science lens, rather than a lens of geometry?
Gina: Maybe why the tree took that form? Like is there something else? Like if it got trapped between something or just why does it has that shape?
Abby's final work problem photo. Problem: what caused the tree to grow in that shape or form? Does it have to do with doil?
In this online program, students were not able to collaborate with each other in the same ways as they usually do in inperson meetings. Naturally, the peer collaboration rate decreased significantly as some students did not even turn on their cameras. However, once students started to work on the Final Walk project, they were more likely to critique each other's problems and discuss how they could pose different problems so that their problems could be integrated into a themebased walk. In this online program, the Final Walk project was implemented last and fewer students participated in this Final Walk project than the #STEMlens activity due to the high attrition rate. However, instances in which students collaboratively pose problems only occurred during the Final Walk project. The two examples above showed how students interacted differently when evaluating their peers' problemposing work in #STEMlens and the Final Walk project. In the first example, the student's comment only focused on the criteria in the #STEMlens rubric (e.g., the markup and the clearness of the photo). However, in the second excerpt, Gina proposed some new ideas and questions about the tree in Abby's photo, and Abby was able to connect her question to the group's theme (i.e., biology and environmental science) with Gina's suggestion.
According to our quantitative analyses that investigated students' mathematical dispositions, there was not enough evidence to conclude that math walk activities enhanced dispositions. One explanation for this insignificant result is the small sample size. A recent metaanalysis calculated the average weighted effect size of students' dispositions after attending problemposing interventions and reported an effect size of 0.54 (Wang et al.,
As introduced earlier, students participated in both semistructured and free problemposing. The results suggested that students were able to pose more complex problems by the end of the program in the postsurvey than in the presurvey, which validated the positive effect of this online program. In addition, students posed more complex problems in the Final Walk project than in the videowatching activities and the pre and postsurvey, which resonated with the finding from the metaanalysis introduced earlier (Wang et al.,
In short, students tended to pose more complex problems in a free problemposing task than in a semistructured problemposing task. Moreover, collaborating with peers to pose and solve problems and the requirement to present the problems to the audience also was associated with more complex problems. This result provides evidence for the authentic audience effect discussed in Crespo (
The limitations of this study were discussed from three perspectives. First, when generalizing the research findings to other students or other problemposing interventions, caution should be taken. All of the meetings in this program were delivered through virtual online meetings. In addition, this program was implemented during a pandemic, and the majority of the students were already attending online classes all day from home. As a result, it could be difficult for students to be fully engaged in all of the activities and meetings, and the instructors were not able to monitor students' learning progress. Second, the small sample size was relatively small for quantitative analyses. As suggested above, these were the challenges and limitations caused by the online format and the special time of the program. The researchers employed this mixedmethod research design and used various data sources to triangulate the findings and results to address this limitation. Finally, we acknowledge that our positionalities (as an international doctoral student and a faculty member interested in mathematics education and problemposing) impact analyzing data and interpreting results and findings in this study.
This study tested and established the possibility of implementing a purely online math walks program. In prior studies, math walks were mostly implemented through inperson programs where children and youth meet with their facilitators at the learning sites (Lancaster,
This study employed a mixedmethod research design to investigate an online math walks program's effects on students' mathematical dispositions and problemposing performance. The online math walks program created an informal STEM learning environment for youth and engaged them in a series of problemposing activities. The results partially validated how the math walk informal learning environment and the problemposing activities youth participated in influenced youth to develop more positive mathematical learning dispositions. Through posing problems in their homes and communities, youth were able to think deeper and differently about mathematical concepts and make connections between school math and realworld applications. This study also compared youth's problemposing work in different learning activities. It concluded that youth posed more complex problems in free problemposing tasks when they were instructed to collaborate with each other to create problems and present their selfgenerated problems to the audience.
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
The studies involving human participants were reviewed and approved by Southern Methodist University Institutional Review Board. Written informed consent to participate in this study was provided by the participants' legal guardian/next of kin. Written informed consent was obtained from the individual(s), and minor(s)' legal guardian/next of kin, for the publication of any potentially identifiable images or data included in this article.
Both authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
This research was funded by the Caruth Institute of Engineering Education at SMU. This material is also based upon work supported by the National Science Foundation under Grant DRL 2115393.
The authors thank Koshi Dhingra, Rick Duschl, and David Deggs for their expertise and guidance throughout the planning and implementation of the online math walks program.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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