For many mathematics educators, online mathematics instruction was new terrain when COVID-19 social distancing policies created an urgent and sudden need for its expansion. The transition to emergency online instruction was jarring for teachers, students, and their families as educators scrambled to replicate what worked for in-person instruction and, ideally, improve what did not work well (Ruef et al., 2022).
Translating brick-and-mortar classrooms to online settings revealed new technological and pedagogical opportunities and problems that called for unique solutions (Engelbrecht & Borba, 2024). Adopting and adapting technology can be expensive, both in terms of material costs and the time and energy spent learning to use it (Hodge-Zickerman et al., 2021). Time spent learning to use new technology must be weighed against time spent engaging with new material. Ideally, learning to use technology intertwines with learning course content.
The use of technology informs equity in mathematics instruction because it mediates students’ learning opportunities (Association of Mathematics Teacher Educators [AMTE], 2017; Jessup et al., 2021; National Council of Teachers of Mathematics [NCTM], 2014). Mathematics education that centers student inquiry and collaboration relies heavily on the students’ abilities to share visual representations of their ideas and arguments (Mainali, 2021; NCTM, 2014). Comparing, contrasting, and connecting visual representations is an essential part of collaborative mathematical sensemaking (Selling, 2016).
Our study examined the experiences of two cohorts of undergraduate students at a large, public university in the northwestern United States (because their program of study did not include licensure, we refer to the study participants as students). The students were completing their mathematics prerequisites for master’s-level teacher education programs during the 2020-2021 academic year. A two-course sequence on K-5 mathematics content (Grades kindergarten through 5 comprise elementary education in the United States) provided opportunities for these students to revisit the mathematical concepts and procedures they would, in turn, teach their future students. For most students, this meant learning to do math in community with pedagogy that differed from their past experiences as elementary students.
A key problem of practice arose with the switch to online instruction: how to support students in sharing visual images of their mathematical work. This study examined the role that a pivotal technology choice played in broadening the students’ opportunities for learning: distributing small document cameras to each of the students.
This report expands a framework developed by Kimmons et al. (2020) to evaluate pedagogical technology decisions, in the case of our study, distributing small document cameras to each student. This study sought to answer the following research questions:
- How did the document cameras change online instruction of the elementary mathematics content courses?
- What should be considered when choosing a document camera for online instruction?
Literature Review
The literature review is organized in terms of (a) teaching mathematics for equity, (b) the importance of sharing and connecting visual representations of mathematical thinking, and (c) a review of technological solutions to problems of practice in online teaching.
Mathematics Pedagogy and Equity
Pedagogy includes the routines, policies, and practices that govern student interactions in making sense of mathematics. Whether in-person or online, mathematics classes that focus on teaching for equity must attend to distributions of agency and authority among students. We define agency as the sanction to propose, debate, or refine a mathematical argument and, more broadly, to take action in a mathematical learning environment. We define authority as the sanction to validate a mathematical argument or condone an action (Jilk et al., 2024; Ruef, 2021). Our focus on distributions of agency and authority reflects constructivist (e.g., Vygotsky, 1978) and situated (e.g., Brown et al., 1989) theories of learning that honor both the autonomy and interdependence of learners.
Pedagogical choices reflect sociomathematical norms, the sanctioned ways that students interact with each other socially and academically in making sense of mathematics (Cobb et al., 2009; Ruef, 2021). Norms, pedagogy, and curriculum mediate distributions of agency and authority among students and instructors. When instruction centers lecture and note-taking, the bulk of authority and agency belongs to instructors, with students positioned as passive recipients of knowledge. When instruction centers students’ contributions, discussions, debates, and refinement of mathematical understanding, authority and agency are more equitably distributed across the class, so long as the instructor manages adherence to productive sociomathematical norms (Featherstone et al., 2011; Louie, 2017). Complex instruction (see Cohen & Lotan, 2014, for a full review) is a theory of pedagogy that explicitly attends to distributions of agency and authority, in part, by attending to norms and organizing students’ contributions by assigning specific group roles that mediate how students interact. Note that, because our study focused on how document cameras mediated students’ learning opportunities, we share only selected elements of complex instruction.
Equitable mathematics classes support students in collaborative sensemaking, which requires attending to the sociomathematical norms of the class (Cohen & Lotan, 2014; Louie, 2017). Broad distributions of agency and authority can foster interdependence, as students come to see each other as academic resources (Jilk, 2017). Equitable online mathematics learning environments must also maintain a humanizing aspect (Jessup et al., 2021; Ruef & Shepard, 2022; Sun et al., 2022) and foster learning cultures that are welcoming and inclusive (Featherstone et al., 2011; Louie, 2017; Nasir et al., 2014; Ruef, 2021). The instantiation of these principles (or their lack) is evident in pedagogy and pedagogical choices.
The Importance of Sharing Visual Representations of Mathematical Thinking
The NCTM (2014) has stated clearly the importance of connecting multiple mathematical representations, including (a) visual, (b) symbolic, (c) verbal, (d) contextual, and (3) physical. Selling (2016) and Mainali (2021) also made a strong case for the importance of interconnecting multiple representations. For example, a linear function can be determined from a real-world context and represented visually or verbally as an equation, a table, a graph, or a growing collection of tiles. Being able to compare and contrast those different representations of the same function allows students to build conceptual and procedural understanding (NCTM, 2014). Sharing and connecting multiple representations increases the modalities for making sense of mathematics. Increasing the ways students engage in sensemaking supports all learners, including disabled and sociologically diverse learners (Lambert, 2021; Wakefield, 2018).
Of the five types of representations (NCTM, 2014), it is easiest for most people to share verbal and contextual representations in a Zoom videoconference classroom because they can be presented orally. Visual (e.g., drawings, tables, and graphs), symbolic (e.g., equations, formulas, and algebraic expressions), and physical (e.g., models and manipulatives) representations are more challenging to share in synchronous online classes because they all rely upon visual presentations. In online settings, this requires a camera that can adequately capture both static and emergent images. Document cameras solve that intertwined pedagogical and technological problem by bridging the gap between verbal and visual representations.
Review of Technological Solutions to Problems of Practice in Online Learning
During the COVID-19 lockdowns, online mathematics instruction was relatively new for most institutions and presented numerous pedagogical problems. The rapidity of the change to online instruction was especially challenging because instructors and students experienced multiple technology issues simultaneously that they likely did not encounter during in-person instruction. These included unreliable internet connections (Bringula et al., 2021; Pócsová et al., 2021); learning new technology (Joshi et al., 2023; Morgan et al., 2022); student engagement (Czerniewicz et al., 2001); access to childcare (Bringula et al. 2021; Calder et al., 2021); feelings of isolation (Calder et al., 2021); and equitable instruction (Ruef et al., 2022; Ruef & Shepard, 2022; Sun et al., 2022).
In online settings, equitable instruction reflects who can participate, which is mediated by the stability of Wireless-Fidelity (Wi-Fi) connections and access to reliable devices. It is also reflected in the time and effort required to learn new forms of technology. Simply put, clunky or finicky technologies are harder to use. The learning, maintenance, and use of any technology creates disproportionate demands for people responsible for providing income or for the care and support of younger siblings or their own children (Ní Fhloinn & Fitzmaurice, 2022; Ruef & Shepard, 2022). For these reasons, technology that is relatively easy to learn and use supports more equitable learning environments.
In our review of studies of technology use in emergency online mathematics classes created in response to COVID-19 restrictions, we found that many educators utilized prerecorded videos of problem solving and static images of solutions to math problems, such as PowerPoint slides; interactive software such as DESMOS; and online interactive whiteboards (Ruef et al., 2022; Lo et al., 2025). Some instructors utilized “flipped” classrooms, where students watched videos of math content lectures individually before gathering online to work on related problems (e.g., Huang et al., 2023).
As part of their analysis of 180 studies, Lo et al. (2025) catalogued studies of technology used for emergency online mathematics instruction during the period of COVID-19 restrictions. Their summary includes the following categories for technology used during emergency online instruction: (a) video conferencing platforms (e.g., Zoom, five studies), (b) hardware for remote teaching (three studies, none including document cameras), (c) mathematics applications (e.g., Desmos, five studies), (d) online sharing and collaboration applications (e.g., Google docs, six studies), (e) educational resources (e.g., video tutorials, six studies), (f) learning management systems (e.g., Canvas, five studies), and (g) social media and messaging apps (e.g., WhatsApp, five studies).
Importantly, our review of the literature revealed the prominence of relying on lecture to impart mathematical ideas to students who were positioned as passive recipients in online classes (Aloufi et al., 2021; Jukic Matic, 2021; Martin et al. 2021, Murtafiah, 2020; Ní Fhloinn & Fitzmaurice, 2022; Sonmezkale et al., 2022; Videla et al., 2022). Lecturing reflects a narrow distribution of agency and authority to students (Ruef, 2021).
We found only six studies that clearly reflected inclusive and collaborative instructional models in online mathematics classes that center teaching for equity. In contrast to the majority of the research we reviewed, Wills et al. (2021) studied mathematics courses that transitioned to online settings while retaining “rich mathematical discussions and student collaboration” (p. 61).
Three studies referenced students’ use of social media to discuss mathematics assignments in small groups (Calder et al., 2021; Haser et al., 2022; Vale & Graven, 2023). Because students in these classes created their own learning opportunities within social media spaces, these classes reflect broader distributions of agency and authority to students. Olsher et al. (2025) reported on the interactions between pairs of students and their instructor, finding that heterogeneous grouping supported learning opportunities. Driskell et al. (2025) surveyed mathematics teacher educators to determine types of technology retained after the return to in-person instruction, including those that increased student participation.
Our study joins a relative minority of studies examining the intentional use of technology to center student inquiry in online mathematics classes. Our study appears to be unique in its focus on the ways students were able to share visual images and make connections between representations of mathematical problems by using document cameras.
Conceptual Framework
To construct a conceptual framework, we begin by defining key terms. We then introduce Kimmons et al.’s (2020) framework for analyzing technology in terms of students’ engagement and pedagogy. Finally, we articulate the framework in terms of our study.
Learning, Learning Opportunities, and Positioning
Within the context of mathematics classrooms, we define learning as attaining or improving understanding of mathematical concepts or methods. We define learning opportunities as interactions between a student and the artifacts of learning, including all forms of mathematical representations, which may be examined by a student or shared with colleagues or instructors. In short, a learning opportunity is a chance to examine, critique, discuss, refine, and ultimately enhance understanding of mathematical concepts and procedures (Ruef, 2021; AMTE, 2017). Learning opportunities are mediated by the modalities of shared mathematical representations. In online settings, sharing visual mathematical representations is a nontrivial problem due to technological challenges. It is an important problem to solve because the inclusion of mathematical representations increases opportunities to learn. Evidence of learning includes statements made by students about increased understanding, making connections, or learning.
Positioning is a social process by which people are nudged, cajoled, or forced into acting in a particular manner or role (Lo & Ruef, 2020; van Langenhove & Harré, 1999). Within the context of a mathematics class, we define positioning as the ways in which students are helped or hindered in the work of mathematical sensemaking through interactions with people, artifacts, and interactions. For example, when an instructor or classmate takes up a student’s mathematical idea, they position that student as mathematically competent. When the norms of class interactions include justifying one’s reasoning, students are positioned as responsible for mathematical sensemaking. When technology limits students’ abilities to share their reasoning, they are positioned as limited in their ability to reason. Pedagogy and positioning are intertwined.
The PICRAT Framework
Kimmons et al. (2020) created a framework for evaluating how technology choices mediate pedagogy and student positioning. Originally designed for teachers to evaluate multiple technological choices in brick-and-mortar classrooms, we extend this framework to analyze students’ experiences in online classes impacted by the introduction of pivotal technology: the document camera. We have adapted their framework to reflect students’ abilities to share visual images.
The framework facilitates assessing the value of technological solutions for learning activities across two dimensions. First, the ways students are able to utilize the technology: Are they positioned as passive, interactive, or creative (PIC) learners? Students are positioned as passive when they utilize technology to observe, by listening or looking, attending to presentations of information (such as lectures, demonstrations, or podcasts). Students are positioned as interactive when they utilize technology to interact with course materials (such as online flashcards or quizzes), instructors, or other students. At this level, “learning is largely structured by the technology rather than by the student,” and learning opportunities are mediated by those interactions (Kimmons et al., 2020, p. 186). Students are positioned as creative when they utilize technology to identify and solve problems of practice, collaborate interdependently to accomplish tasks, or invent new strategies, artifacts, or materials. At this level, students are “driving” the technology to manage their own learning opportunities.
Second, the ways technology impacts pedagogy: does the technology replace, amplify, or transform (RAT) pedagogy? Pedagogy frames the interactions between students, instructors, and technology. Because pedagogy is fundamentally linked to opportunities to learn, improving pedagogy (or better meeting its goals) increases opportunities to learn. When the introduction of technology does not modify learning opportunities, it replaces pedagogy. For example, lecturing online can be similar to lecturing in person. Technology that improves learning opportunities amplifies pedagogy. To amplify, technology must allow students to engage in learning in ways they could not without it. Technology that radically improves learning opportunities transforms pedagogy. To be transformative, technology must provide solutions to pedagogical problems that could not be enacted without it, and it must be driven by students for the purposes of managing their own learning activities.
Kimmons et al.’s (2020) two-dimensional PICRAT model facilitates assessing the levels of support that technology provides (Figure 1). Kimmons et al. were careful to note that it is an ongoing debate in the field of educational technology whether any form of technology can be considered transformational in isolation from other elements of learning.
Figure 1
The PICRAT Matrix
Methods
Context and Participants
The study took place in a public university situated in the northwestern United States during the winter and spring terms of 2021. Two cohorts of undergraduate students were invited to participate in the study. Jennifer Ruef was the instructor for Cohort 1 (fall and winter terms), and Madeline Ahearn was the instructor for Cohort 2 (winter and spring terms). All classes were online, synchronous, and took place on the Zoom platform. Both cohorts consisted of undergraduate students completing an Elementary Mathematics Content (EMC) course sequence (Course 1 and Course 2).
The course content attended to the mathematical concepts and procedures reflected in K-5 mathematics instruction in the United States. Course 1 included units on the Base 10 Number System, Addition and Subtraction, and Multiplication and Division. Course 2 included units on Divisibility Rules, Fractions and Operations With Fractions, and Introductory Geometry.
Each student (regardless of eventual participation in the study) was mailed a document camera during the break between Course 1 and Course 2. Given the expense and limited funds, Ruef made the decision to delay sending cameras to the second cohort while she assessed their value with the first cohort. During that term, the second cohort experienced Course 1 with Ahearn as the sole user of a document camera. Based on the preliminary findings that the cameras were indeed valuable additions, we repeated the process of providing them to the second cohort for Course 2.
At the end of the course sequence, each student was invited to participate in the study, which was overseen by the university’s institutional review board. Participating in the study meant granting us permission to repurpose existing course documents as anonymized data and, thus, did not require any additional time or effort on the part of the students. The data primarily included the students’ essays detailing how the document cameras affected their learning opportunities in the class. Eighteen of the 19 students from Cohort 1 and 10 of the 24 students from Cohort 2 agreed to participate. We intentionally limited our analysis to the students’ essays and waited until the conclusion of the course to request consent so that students could experience the course without concern over how their interactions might be analyzed.
Given the size of the study cohort, we were concerned that individual students might be identifiable within racial demographics for the class. Therefore, in Table 1 we share aggregated demographics for the entire undergraduate program (N = 402) that the participants (N = 28) were members of for the 2020-2021 academic year. We confirm that the participants’ demographics reflect the demographics of their undergraduate program cohort. We use the generic pronouns they and their to represent all students and their work, and all student names are pseudonyms.
Table 1
2020-2021 Undergraduate Program Demographics (N = 402)
| Categories | Percentages |
|---|---|
| Racial | |
| Black, Indigenous, or Person of Color | 36% |
| White | 62% |
| Unspecified | 2% |
| Gender | |
| Female | 94% |
| Male | 6% |
Course Design and Implementation
The courses met synchronously online twice weekly for 2 hours across two consecutive academic terms. English was the language of instruction. Course content comprised lesson units provided by The Elementary Pre-service Teachers Mathematics Project (2025). This curriculum was created as a research-based, open-source curriculum designed to support students who aim to teach elementary grades (K-5). Due to COVID-19 social distancing requirements, all coursework was completed online. Organization of the courses, pedagogical decisions, and learning opportunities were mediated by available technology.
For this study, we examined the students’ perceptions of how their learning opportunities changed after individual document cameras were introduced to the students between the two consecutive EMC courses. We compared the students’ experiences in Course 1 to Course 2 to articulate shifts in learning and learning opportunities.
The content across Course 1 and Course 2 provided a progression through the mathematical concepts and procedures common to grades K-5. The order of class activities (what happened in a typical class) was identical for Course 1 and Course 2. Norms for interaction included expectations for students to work collaboratively and share their thinking, unfinished or polished, in both small group and whole class settings. Liberal late assignment policies and alternative modalities for completing work reflected the Universal Design for Learning principles (Lambert, 2021; Wakefield, 2018). Paper copies of each course’s lessons and homework sets were mailed to each student before the class began, and electronic copies of those same documents were made available on Canvas, the course management system. Students also used Canvas to turn in completed work.
Each class began with a whole-class discussion designed to bridge prior learning with new content. Each student had their choice of participating with either a paper or electronic copy of the day’s lesson material. The class was organized into six groups of three or four students. During the whole class discussion of new content, each group was assigned one or two problems that they would be responsible for presenting during the whole class discussion. For each class, the groups chose a Reporter, the person responsible for sharing out on behalf of the group. The groups then met in individual Zoom breakout rooms to work on the day’s lesson. During the next whole class discussion, each group’s Reporter shared solutions for their assigned problem(s). Some classes repeated this cycle twice, others only once. At the end of the class, the instructor assigned the homework problems each student was responsible for completing and turning in before the next class meeting. The students were also able, and expected, to post images of self-selected homework problems to a discussion board on the Canvas learning management system both to ask and answer questions. For most students, this was the only method for sharing visual representations during Course 1. The only difference between Course 1 and Course 2 was the introduction of document cameras, which were mailed to students during the break between terms.
Course 1. During Course 1, only the instructor had a document camera, and most of the students were limited in sharing visual images. Because the pedagogy and sociomathematical norms conveyed expectations of broadly distributed agency and authority, the instructor and students attempted to bridge the problem of limited ability to share visual images by having the instructor scribe as students described the visual images they had created on their lesson packets. Students sometimes held their own papers up to laptop cameras, which was generally frustrating because the images were both difficult to frame and to parse. Two students (one of whom was a participant in the study) solved the problem of sharing visual images by learning to use drawing programs on tablet computers that could be linked to Zoom videoconference calls.
During small group work, which took place in Zoom breakout rooms, the students were left to their own devices, literally and metaphorically, to share mathematical representations. Because this meant holding papers up to cameras or verbally describing representations for most students, groups sometimes labored silently in parallel until they returned to the whole-class discussions on Zoom. Groupwork in Zoom breakout rooms also differed from brick-and-mortar classrooms in that it was not possible for the instructor to attend to each group in real time without serially “popping in” to breakout rooms, which some students found startling. The instructor often waited outside the breakout rooms until called in to respond to a group’s request for help.
In practice, whole-class discussions functioned like a talk show, with the instructor interviewing one student as they scribed visual representations based on verbal representations of a problem. While other students could ask questions, the “main show” was the conversation between instructor and students. During groupwork, which took place in breakout rooms, most groups had to call the instructor in to gain access to the talk show version of a scribed visual representation shared by document camera, unless they were fortunate enough to have a self-taught group member who could share visual representations by using a drawing application on the device they used for Zoom classes.
Course 2. By the beginning of Course 2, all students had their own document cameras. The format and expectations for participation remained the same, but with an important addition: Students were able to share visual images by plugging their document cameras into the devices they used to attend Zoom classes, switching the visual input to their document camera, and displaying their written work, diagrams, and use of manipulatives to solve problems during whole-class and small group work. The pedagogy shifted from the instructor scribing and students explaining verbally to students regularly sharing their own representations as they asked questions about where they were stuck or shared new understandings and connections.
Students’ ability to share more fluidly in breakout rooms promoted more interaction during groupwork. Groupwork became more inclusive. It could better center students’ mathematical arguments, as more students shared their thinking and invited their colleagues to do so. The instructor or another student would scribe for the few students who were unable or unwilling to operate their document cameras. In Course 2, whole-class and small-group discussions could include multiple participants, shifting discussions to round table experiences.
Positionality of Authors
Lisa Jilk identifies as a White, middle-class, English-speaking, cis woman who grew up in a small town in the US Midwest. She was a high school mathematics teacher and now works with math teachers to create humane learning spaces in which everyone is valued. Because she was not an instructor for either course, Lisa was essential in providing outside eyes in data analysis and constructing claims.
Jenny Ruef identifies as a cis-gendered, white, English-speaking, North American, and middle-class woman. She taught secondary mathematics for 20 years before completing her doctorate and transitioning to humanizing teacher education. She was one of the instructors for the EMC course sequence and took care to consider her positionality as both researcher and educator.
Information Technology Support
Having experimented with multiple modalities for sharing visual images online (e.g., interactive whiteboards and sticky-note “jam boards”) and in consideration of the primarily paper-and-pencil nature of the EMC curriculum, the instructors concluded that providing document cameras to each student was an equitable and efficient solution to the central problem of practice of broadening who could share visual images during synchronous online classes. To enact this plan, the instructors were supported by information technology experts (ITEs) during and after the switch to online instruction.
The college’s head ITE was consulted before the lead instructor purchased document cameras for both EMC instructors and again after grant funding was secured to purchase and mail document cameras to each of the students. The ITE requested that we survey the students to determine (a) the bandwidth of their Wi-Fi internet access and (b) if they required an adapter to connect a document camera to the device they used for Zoom calls. This information, along with mailing addresses for each student, was collected by Qualtrics survey.
We also consulted with the head ITE to choose a document camera. Our criteria included that it mimic, as much as possible, the in-class document camera we used for in-person instruction, including (a) clarity of image, (b) durability, and (c) zoom and focus features. The preferred model, based on the experiences of the instructor and a cohort of six preservice secondary math teachers who compared two different models, was the InSwan document camera. This shared preplanning was essential in increasing the likelihood that students would be able to use their document cameras.
Data and Analysis
The body of data included the students’ final essays comparing learning opportunities in Course 1 and Course 2 and the instructor’s memos and course records. At the end of Course 2, the EMC students were asked to write an essay answering the following prompts:
- If you were not able to make your camera work with Zoom, you can write about that.
- If you were able to use your camera in small groups or whole class discussion, you can write about that.
- If you used your camera to help someone learn math via video call, you can write about that.
- If you were able to use the camera, how did it support you in learning mathematics?
- How did others using their cameras support your learning?
The students had the option to answer some or all of the prompts, and their essays varied in length from 113 to 961 words, with a mean of 301 words. The unit of analysis was an entire essay.
Kimmons et al.’s (2020) PICRAT framework provided a priori codes. Because of the unique context of the study, which compares the students’ experiences in the online course sequence and is informed by the instructors’ experiences of both online and in-person versions of the course sequence, we felt the need to differentiate between two levels of interaction, which is reflected in the code book.
Both authors coded the 28 essays a total of three times. After the first round of coding, we met to compare and combine coding schemes. Jilk then coded all essays and both authors met to discuss, collapse codes, and refine the coding scheme. Both authors then co-coded 11 essays, discussing any differences in coding until agreement was reached. Of the remaining 17 essays, Ruef coded eight, and Jilk coded nine, with each verifying the other coder’s agreement with the coding scheme.
Four essays required adjudication, which consisted of both authors discussing the coding until agreement was reached. Coding examples and initial coding are shared in the appendix. We used the instructor’s memos and course records to develop course descriptions and for context to support data interpretation. We coconstructed the codes based on these analyses. Following is the content of the final code book for analysis of the essays to answer Research Question 1.
The Students’ Relationship to Technology
Creative. Students drive learning opportunities rather than the technology driving students’ interactions. Students are agentic and inventive in choosing how to use the technology to construct opportunities to learn. Students use the technology to generate artifacts from and for their learning. Students work collaboratively, which may include evidence of (1) interdependence, (2) comparing and contrasting multiple solution strategies, or (3) use of different modalities (e.g., manipulatives). This description reflects a “round table” discussion with multiple participants sharing, comparing, and contrasting multiple ideas.
Interactive 2. The technology implicitly defines what the students should be doing to learn and the students are following those implicit instructions to share visual images. In the case of the document camera, the implication is to “show and tell.” In the case of our study, the connection between visual and verbal, as mediated by either the student’s use of the document camera or their peers’ use, is evident. This description matches the “talk show” context of Course 1, but because students can share widely in small groups (break out rooms), it also marks an increase in students’ interactions. This is true even for students who did not use their camera (or another method of sharing visual representations).
Interactive 1. The technology implicitly defines what the students should be doing to learn and the students are following those implicit instructions to share visual images. The student’s learning opportunities reflect the use of document cameras in Course 1, with students primarily interacting verbally or by holding images up to an embedded computer camera.
Passive. The student does not interact verbally or visually with their colleagues or instructor. The student participates by watching and listening to others presenting their work.
The Instructors’ Use of Technology
Transforms. The pedagogy evolves from “talk show” to “round table” discussion. Learning opportunities, such as comparing and contrasting multiple representations, are now possible and happen because of the technology. The pedagogy, supported by the technology, enables students to use the technology themselves in ways that afford collaboration and co-constructing understanding. This happens in small group/breakout rooms. Students are able to interact in new and enhanced ways.
Amplifies. The instructor uses the technology to improve students’ learning outcomes, however the pedagogy is not radically different from Course 1. The pedagogy “refines” or strengthens a pre-existing practice: whole class or small group discussion reflect a “talk show.” The students use their cameras to “show and tell” their solutions but there is no evidence of comparing and contrasting representations.
Replaces. The pedagogy of Course 2 replicates the pedagogy of Course 1.
The appendix includes the students’ essays, which are highlighted to show the text that indicated the code cell to which it was assigned. Our initial summary descriptions indicating how we applied codes follow that table. To answer Research Question 2, we looked to the instructors’ memos, course records and references in the students’ essays.
Findings
We begin the findings with the code counts for the participants’ essays. The findings that answer the first research question are organized in terms of applying the PICRAT (Kimmons et al., 2020) model for choosing and analyzing the utility of the chosen technology. The findings that answer the second research question are organized in terms of why some students chose not to use their cameras, followed by key features and considerations. A checklist for choosing and implementing the use of document cameras is provided in the appendix.
Codes and Code Counts
Figure 2 reports the code counts that represent the participants’ experience of the differences between Course 1 and Course 2.
Figure 2
Code Counts, Students’ Reports of the Shift From Course 1 to Course 2 (N = 28)
Sixteen of 28 students reported their experience of Course 2 rose to the level of Creative interactions with colleagues and found that the use of individual document cameras transformed the pedagogy of the class. Two students found that their experience of Course 2 rose to the level of Interactive 2 (sharing visual images) and that the cameras transformed the pedagogy of the class. Ten students described the level of Interactive 2 interactions and that the cameras amplified the pedagogy of the class.
Of note, none of the 28 students experienced Passive or Interactive 1 levels of interaction with colleagues or artifacts. None of the 28 students indicated that Course 2 replicated the pedagogy of Course 1, despite five students not using their cameras. Those five students’ experiences are described later in the findings. Figure 3 shares example text for each code cell in Figure 2.
Figure 3
Example Text for Codes
The following sections provide additional text and analysis to explain and support the code counts.
Research Question 1
In the following section, we consider how the distribution of document cameras affected students’ interactions and the pedagogy of the class, both of which mediated the students’ opportunities to learn. First, we provide context and comparisons between the in-person version of the course sequence, Course 1, and Course 2. Next, we present an analysis of the data in terms of the PICRAT framework.
Comparing In-Person Courses With the Online Course 1 in Terms of PICRAT
It was challenging to enact this comparison because the study participants never experienced in-person instruction. However, the instructors had memories and records that allowed a comparison of in-person to online Course 1. The instructors’ initial adaptations, as the shift from a brick-and-mortar, in-person classroom to synchronous online instruction happened within two weeks, were by necessity rushed. In particular, the brick-and-mortar classroom featured a document camera that the instructors and students used to share visual representations with fluidity. Sharing with the camera also created impromptu rehearsals for students to practice their evolving teaching skills. The in-person classroom featured multiple modalities for sharing, including large and small whiteboards and markers, but the document camera was the most fluid way for anyone in the class to efficiently share thinking accompanied by visual representations. There was enough time in the two-week transition for the instructors to secure and test small document cameras, but not enough time to instantiate new ways for the students to share their visual representations.
Based on the instructors’ reflections and course records, the students’ interactions in Course 1 were limited to Passive and Interactive 1 levels of engagement, with two students rising to Creative levels by seeking out and learning to use drawing programs to share visual images. This is not to say that the students were not creative in finding ways to engage — it is to say that the technology limits were a hindrance. The instructors’ use of a single document camera to scribe students’ verbal descriptions was an attempt to replace the communal document camera of the in-person courses. However, given that access to the document camera was restricted to the instructors, the pedagogy actually regressed and, thus, is not reflected in the PICRAT framework.
During Course 2, the document cameras provided opportunities for students to share their own visual images in online classroom settings. At a minimum, the students’ document cameras were meant to replace the large central document camera of the in-person classroom. The instructors had taught the course series across multiple terms and, thus, had memories, practices, and records of how previous cohorts of students had interacted in brick-and-mortar classrooms where everyone had access to a large, central document camera. During pre-COVID-19 restrictions, the in-person versions of Course 1 and Course 2 featured students comparing, sharing, debating, and refining mathematical arguments by comparing and contrasting their work on paper in small groups. Individuals or groups also presented visual images of their work to classmates during whole-class discussions. These whole-class presentations often featured the in-class document camera projecting onto a large central screen.
For Course 1 of the online version of the course, only the instructor had a small document camera to project images to the students in the Zoom classroom. Students verbally shared their thinking in whole-class discussions as the instructor scribed their ideas for the whole class. When the class was divided into small groups of three or four students in breakout rooms, it was more difficult for students to share their thinking because of the limited options for sharing visual images. Course 2 was markedly different: By the time it began, all students had received their own document cameras and were suddenly able to share their own images both in whole-class discussions and during small-group work.
Assessing Students’ Interactions (PIC) and Pedagogy (RAT) in Course 2
The following sections are organized by the three PICRAT categories represented in our findings. They provide a detailed analysis of excerpts from the students’ essays to explicate and support our coding.
Interactive 2 and Amplified. Ten of the 28 students experienced the shift from Course 1 to Course 2 to as rising to the level of Interactive 2 (sharing visual images) interactions and amplified pedagogy, meaning that the document camera mediated the students’ interactions. For these students, the pedagogy improved but did not dramatically change from Course 1 to Course 2. Hetty said:
I was able to use my document camera in small group and in class discussion. It was a lot easier to share thinking in small groups with the doc cams. We were able to sketch out our thinking, pointing out what we were doing as we solved the problem instead of holding it up to the [built-in face] camera and trying to talk everyone through it. I found myself using the doc cam a lot to express my thinking in small groups. In the main class I would say it wasn’t as big of a difference to be honest. Because you scribed for those who didn’t have doc cams, we saw everyone’s thinking anyway.
Hetty’s reflection captures the relative ease of using a document camera as opposed to trying to hold a piece of paper in front of a face-capture camera or verbally describing one’s thinking. This quote also represents a related theme — the four students who did not, or could not, share visual images were still able to share their thinking with the class because the instructor could still scribe for them, as was done in Course 1. Because Hetty described their learning opportunities as a version of “show and tell,” which is similar to the pedagogy of Course 1, their essay indicates Interactive 2 levels of interactions and an Amplification of the pedagogy. Their colleague Baylor said,
I was able to use my doc camera a few times in the breakout groups near the end of the term, and I actually really liked it. I unfortunately did not have a desk until about a week ago though, so trying to get it to sit up before then was really hard. I actually found myself paying closer attention and learning more through my peers using the doc cameras as well. It made it a ton easier for me to understand the steps that they were taking and how their thinking was working. I think using the camera helped me get my point across as well. I found myself working harder on the problem I was working through on the doc cam as well. I also don’t mean to sound weird, but it almost made the problems easier? I don’t know if it was because I just was more in the right set up or what, but I really enjoyed it.
Baylor’s experience with the document camera, while riddled with unanticipated technological and furniture problems, indicates that they were able to show and tell their work and learn from others doing the same, indicating Interactive 2 levels of interaction. Their description of camera use indicates amplification of the pedagogy in Course 2 because this ability to share visual images was strengthened from Course 1 to Course 2.
Interactive 2 and Transformative. Two of the 28 students experienced the shift from Course 1 to Course 2, rising to the level of Interactive 2 (shared visual images) interactions and Transformative pedagogy. Maddy said,
I mostly ended up using my document camera during our whole-class discussions, when I was the reporter for my group. I found it to be especially useful when talking about problems that had more visual elements, when actually showing my paper was essential to getting my point across. With remote classes, I often get bummed out, because it feels like I’m missing out on what could be some really great learning experiences if the class was in person, but the addition of document cameras definitely helped me feel like I was getting as much as possible out of our class. Being able to share work with my classmates and see their work made the experience of the class much more personal and engaging. I would absolutely say that the camera supported my learning in more ways than one.
Because Maddy mostly used a document camera to share their work and see their peers’ work, much like a talk show interview, their level of interaction was coded as Interactive 2. However, Maddy’s heartfelt description of interpersonal engagement in the class and the role the document camera played in replicating in-person instruction rose to the level of Transformative pedagogy.
Neecy said,
In terms of how the camera supported my learning in mathematics, being able to see the variety of ways in which my classmates represented their work visually was really valuable and helped open my eyes to ways of thinking about math that I wouldn’t have thought of on my own. I also found it quite helpful when my small group members used their cameras to share their own work, especially if there was a particular problem we were confused about and wanted to work through together.
Neecy’s essay was one of four that required adjudication. After discussion, we determined that there was insufficient evidence to confirm a Creative level of interaction, but ample to determine Interactive 2. The use of cameras to work through moments of confusion indicated a Transformation of pedagogy because the students were using the document cameras to coconstruct mathematical understandings, an option not available in Course 1.
Creative and Transformative. Sixteen of the 28 students experienced the shift from Course 1 to Course 2 as rising to the level of creative interactions and transformative pedagogy. The following excerpts indicate the role their document cameras played in that shift. Kaycee said,
Another aspect I appreciated was being able to see my peers’ work and be exposed to other ways of thinking. In [Course 1] we would share ideas and methods, but mostly we would not show our work visually in our groups. Having that added layer in [Course 2] was so helpful in comparing my work and also making sense of problems I didn’t understand. I appreciated it when someone offered to turn on their document camera so that I could see the way they were thinking about a problem. It felt like we were actually working together and collaborating, albeit virtually.
Kaycee’s description of changes between Course 1 and Course 2 reflects that the personal document cameras did not simply replace the function of the in-class document camera, largely because the students had not experienced in-class instruction for this course sequence. Shifting from an instructor scribing for students to students being able to share their own visual images increased learning opportunities, indicating that the groups’ interactions rose to the level of Creative. This description indicates that the addition of personal document cameras went beyond replicating the function of the instructor’s use of a document camera in Course 1, rising to the level of Transformation.
Cece said,
I really enjoyed using the camera in the small group. It was nice to be able to see and follow along with my classmates’ work. It made the class much more inclusive and supportive. This was a great addition to the class because I also feel like it allowed everyone in my group to be on the same page and share similar ideas. This was a challenge in [Course 1], because a lot of time my group members would work on their [own], and it made it difficult to understand what we were learning in class and made it hard to work as a group. Having the document cameras made everyone more engaged and people collaborated even more. … This made it easy to see what people were explaining and actually work through problems with my classmates.
Small group breakout rooms in online classes differ from in-person small group work in an important way: The instructor cannot glance around the online “room” to visually and auditorily assess how groups are progressing. Thus, in online classes, students are pressed to become interdependent in different ways, especially in a classroom focused on interaction. As Cece made clear, the document cameras supported students in supporting each other and collaboratively making sense of mathematics. Through the shared work of inventing new ways to compare and contrast mathematical representations with their colleagues, Cece’s experience rose to the level of Creative and Transformative.
Some groups pressed their use of the document cameras further, finding ways to use the technology to create, compare, and contrast new mathematical representations. Taylor said,
The good old [document] camera. … I believe it to have been super beneficial to my learning and an easy way for me and my classmates to collaborate. I loved seeing other people use their cameras in class, because it was easy to see exactly all their work and thinking on paper. I know personally in all my breakout rooms we used the heck out of these cameras. We messed up and could erase our work and being able to see that was awesome. We would use colors, so it was easy to define and see everything and we even would doodle and draw pictures as for some of us that worked best. It really gave me a sense of how each person learned visually.
Taylor’s essay reveals how her group’s work shifted dramatically in Course 2. They invented dynamic and interdependent ways to share their thinking with each other. In these ways, the document cameras supported these students in Creative problem solving that Transformed the pedagogy for Course 2.
Bridging Language Barriers. English-dominant and multilingual students both appreciated the facility of the cameras to bridge language barriers. Delia, an English-dominant student, said:
One of my groups had an ESL [English as a second language] student in it, and I could feel the student’s hesitancy when explain concepts and, yet, when asked to explain via document camera the student was able to show really cool, new ways to look at the problems.
Delia’s appreciation for their colleague’s mathematical insights is evident in their framing of “really cool new ways to look at the problem.”
Acadia, a multilingual student, said,
In addition, document camera could help me to avoid the idea that I can’t express. Sometimes I might forget some words, or I don’t how to pronounce the specific words, I could use document camera to help me with that. I just need to write down the idea that I want to express then my classmates and professor will see it.
The document camera provided Acadia with new ways to be mathematically fluent and for their work to be utilized and appreciated by their colleagues. The document cameras provided essential communication bridges for both English-dominant and multilingual students, supporting their ability to make sense of mathematics collaboratively. These poignant uses of the cameras to bridge language challenges were Creative and Transformative.
Research Question 2
The following sections share the frustrations and workarounds shared by five of the 28 students and a summary of key features that students and instructors appreciated in the document cameras.
What Stopped Some Students From Using the New Technology?
Of the 28 students, all but four were able to make use of their document cameras. One student was afraid of damaging the camera, believing that they would need to return it (all students got to keep their cameras). One student lacked a device to plug their camera into. One student simply never got around to learning to use their camera. One student lacked a desk or table to set the camera on.
Additionally, one student chose to continue using the drawing program they became fluent with during Course 1 rather than switch to the document camera. Most (20) of the students reported some fear, frustration, or anxiety (hindrances) with the sudden ability to share their thinking visually, learning to use new technology, or connectivity issues (Zoom would sometimes fail when they switched the visual input to the document cameras). Upon reviewing this data, one instructor made the note that in future online instruction, devoting part of a class period to playing with and practicing using the cameras to normalize sharing visual images might alleviate some stress for students. Despite the challenges, 23 students were able, and chose, to use their cameras during class.
Key Features and Cost Considerations
The students reported that the zoom and autofocus features helped in showcasing visual images that revealed their thinking. They also appreciated the ability to show solutions using manipulatives (three-dimensional mathematical modeling tools). While some students downloaded the application software made available with the document cameras, most simply plugged them into their computers and switched the visual input on the Zoom call. In some cases, the application software was an impediment to using the camera. In general, the students most appreciated the ease of plug-and-play technology.
The actual cost of technology is an important consideration. The document cameras for this study were purchased with funds from a grant earmarked to support researchers during the COVID-19 restrictions that, in addition to hampering instruction, also restricted researchers’ access to educational settings. Acadia pointed out that
there is also a problem, because this time the document camera was sent to us by the teacher rather than purchased by ourselves. In the following courses, unless the teacher has a sponsor, the students may not be willing to buy this camera. Since students might cost a lot of money on their textbook, I think this is the most serious problem that teacher needs to consider.
Acadia made an excellent point: Monetary costs must be considered and factored into the decision to require all students to purchase technology, just as the cost of a textbook would be considered.
Discussion
We first recap the findings, then discuss how the students’ document cameras supported more equitable online learning. Next, we discuss key pedagogical and practical considerations in choosing technology and how they might apply to classes in other content areas. We then address theoretical and methodological concerns by discussing how our study expands Kimmons et al.’s (2020) PICRAT framework, followed by limitations and next steps. We close with a final word from the students who participated in our study and summarize the cost-benefit analysis for adopting document cameras for online instruction.
Summary of Findings
The document cameras broadened participation opportunities for all students, whether by showing their own visual images, benefiting from seeing images shared by colleagues, or developing innovative ways to use the camera as they coconstructed understanding. For the instructors, who had experienced previous in-person versions of the course sequence, the cameras replaced an important feature of brick-and-mortar classes. However, for the students who had no prior experience with the in-person version of the class and its communal document camera, the distribution of personal document cameras expanded learning opportunities when comparing Course 2 to Course 1.
Based on our analysis, we conclude that the document cameras were a good choice for solving the problem of students’ need to share visual images. The reflections from the students’ essays indicated that the introduction of the cameras Amplified (12) or Transformed (16) the course’s pedagogy. Collectively, the students’ engagement with the cameras improved the quality of their interactions with each other and with artifacts that conveyed visual images, such as drawings, written work, or manipulatives.
While all students reported an improvement in their interactive experiences of Course 2 in comparison to Course 1, 16 of the students’ interactions included Creative solutions that extended beyond the implicit use of a document camera. Sixteen of 28 (57%) students reported inventing new ways of comparing and contrasting their work after the introduction of the cameras, even though there was no formal instruction provided on how to do so.
Technology and Equity in Online Mathematics Education: Practice and Pedagogy
Instructors’ Equity Considerations
Faced with the rapid transition to online mathematics instruction, two university instructors worked with their students to craft a technological solution that honored the bandwidth, both literal and metaphorical, of the learning community to adapt their collaborative classes to online settings. When considering technological choices for reshaping classes in online spaces, Hodge-Zickerman et al. (2021) remind us to “let instructional theories, not technology, guide instruction” (p. 7). Kimmons et al. (2020) articulated that goal and process in the decision flow chart featured in Figure 4.
Figure 4
PICRAT Technology Decisions Flowchart (Kimmons et al., 2020)
The instructors experienced the loss of access to a communal document camera during online instruction as a barrier to equitable instruction, because of the limits imposed on students’ abilities to share their thinking. We were gratified that the decision to use research funds to distribute individual document cameras to the students reflected principles of teaching for equity.
Students’ Equitable Enactments
Teaching for equity requires attending to sociomathematical norms and culture building (Cohen & Lotan, 2014; Louie, 2017). Relatedly, solving technological problems of practice requires attending to the pedagogy, curriculum, and assessment goals of a class (Hodge-Zickerman et al., 2021; Jessup et al., 2021). Our data and analysis support the conclusion that the document cameras supported the pedagogical goals and norms that included broad distributions of agency and authority (Nasir et al., 2014; Ruef, 2021). One happy surprise that arose from this analysis is the level of creativity with which students utilized their cameras. It is clear from the data that, rather than the cameras driving learning opportunities, more than half the students found ways to drive technology to support their individual and collective learning. As they shifted from “talk show” to “round table” discussions, the students metaphorically moved themselves from the periphery to the center of pedagogical choices. In doing so, they established Course 2 as a shared and coconstructed learning space.
Beyond Online Mathematics Classes
Our study featured the use of a document camera in online mathematics classes, which addressed a specific technological need: the ability to share, compare, and contrast visual images (Mainali, 2021; NCTM, 2014; Selling, 2016). Technology exists to create electronic images of mathematical drawings and solutions, and two students found drawing application software that they implemented on their own during Course 1. The cameras were readily adopted by the students, perhaps because their mathematical sensemaking was largely a paper-and-pencil endeavor. Any content area facing similar challenges would benefit from the use of document cameras.
Additionally, the students found other uses for the document cameras in their daily lives that solved similar problems created by COVID-19 restrictions. Beyond the online EMC class, students creatively extended their use of the document camera in the following ways: sharing a document with their advisor during an online meeting; tutoring their own students; meeting with instructors for office hours; playing Magic the Gathering (a role-playing card game); facilitating a training meeting; and making an international call to their Mum.
Expansion of the PICRAT Framework: Theory and Methods
The PICRAT framework was originally conceived by Kimmons et al. (2020) as a tool for preservice teachers to analyze and decide the pedagogical value of multiple technologies for teaching. We expanded their framework for post hoc analysis and evaluation of a key pedagogical choice. As shared in the methods section, we introduced a bifurcation of the Interactive level to better articulate the context of our study: comparing the students’ experiences with Course 1 and Course 2 in reference to the instructors’ experiences of both online courses in comparison to previous in-person version of the course sequence. Our study shares this expanded use of the PICRAT framework (Kimmons et al., 2020).
Kimmons et al. (2020) shared a key concern in the field of educational technology research: whether any technology can be considered transformative.
Of all the processes affected by PICRAT, transformation is likely the most problematic, because it reflects a longstanding debate on whether technology can ever have a transformative effect on learning (e.g., Clark, 1994). … Many researchers and practitioners have noted that transformative uses of technology for learning may only refer to functional improvements on existing practices or greater efficiency. A tipping point exists, however, where greater efficiency becomes so drastic that new practices can no longer be distinguished from old in terms of efficiencies alone. … Uses of technology that transform pedagogy should be viewed differently than those that merely improve efficiencies, even if the transformation results from functional improvement. (p. 188, bold emphasis added)
We agree that claims of transformational technology should be weighed with caution. We agree with Kimmons et al.’s careful assertion that the use of technology must be considered in the context of pedagogical implementations, including the norms, culture, and social practices of educational settings. We report that, even with this careful weighing and consideration, a majority of our students found the use of their document cameras to be Transformative because they were better able to create collaborative learning opportunities.
Limitations and Future Research
Our study was limited in scope to the two cohorts of students who took the class. Enactment of the course sequence, including pedagogical choices, was heavily influenced by the instructors who shared common theories and practices of teaching mathematics for equity. Though our students’ experiences may not generalize to nonsimilar populations, this was not a goal of the study. Our goals included assessing the document cameras for their utility in teaching for equity and to expand existing theory and methods in educational research.
When adopting any new technology, instructors need to consider the costs, in terms of finances and the resources of time, energy, and patience to adapt to new ways of communicating. While this study utilized a small grant to provide the cameras to students, the financial cost of a document camera parallels a textbook or course charge and might be well worth the expense to students in exchange for the rich opportunities to connect, build community, and enhance opportunities to learn in online mathematics classes (Engelbrecht & Borba, 2024; Wills et al., 2021).
Future studies could refine the measures of students’ experiences with online instruction, beginning with explicit instruction on how to set up and use the technology, methods for solving technological problems, and opportunities to practice using the technology with low-stakes outcomes. A checklist for implementing document cameras in online classes is shared in the appendix. Data collection for future studies can be more purposive in comparison to our ad hoc analysis of existing course documents.
Conclusion
The students made excellent use of the cameras both in class and in their personal lives. The extended uses that the students found for the document cameras speak to their creativity in making good use of technology during a time of social distancing. As Lloyd said,
Lastly, I used this document camera to make a video call with my mum. It would be impossible for the desktops to have a video call without a front camera. With this camera, it not only helps me with my [Course 2] math work, but also convenient my life of connecting with my family and friends.
In addition to the creative uses that students found for their cameras, all technology choices come with costs, in terms of money, access, and time. All three costs present equity considerations. Instructional time must be spent acclimating students to any new technology. The relative ease of a plug-and-play document camera minimized that instructional time, reserving it for mathematical sensemaking. The cameras also proved to be relatively accessible in the face of internet access and bandwidth considerations. The monetary cost, covered in this case by grant money, is comparable to a textbook. We were able to provide an open-source, free, curriculum to our students. For that reason, should we teach this course online again we would request students purchase a camera in lieu of a textbook. Recognizing that each course and content area will have unique demands and limitations, we maintain that small document cameras can mitigate a key problem of practice: the ability to fluidly and efficiently share visual images in online settings.
Acknowledgements
The authors wish to thank Madeline Ahearn and Shareen Springer for their contributions to this study. We thank the Hope Baney Fund and the University of Oregon’s COVID-19 Impact Grants for funding this research and the purchase of the document cameras. We also thank our participants, many of whom became first-year teachers amidst the challenges of emergency online instruction. Their dedication to their own students provides us hope and joy.
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