Introduction
Over the last few decades, we have experienced a number of unexpected disasters, crises, and threats such as earthquakes, hurricanes, big wildfires, and gun shootings. Each of these disasters, crises, and threats has created traumatic experiences to the members of our communities emotionally, psychologically, and financially, while making a huge impact on educating both our K-12 students and teacher candidates. Among many other crises, this unprecedented COVID-19 pandemic has disrupted, interrupted, and changed the way we normally prepare our teacher candidates in teacher preparation programs. Over the years, we have experienced a high level of anxiety, fear, and uncertainties about the current situation; losing jobs and financial difficulties; unequal access to resources; injustice, inequality, and discriminations for certain groups, to name a few.
Because of the potential risks of contamination of the virus and health issues, many schools and universities around the world have been closed and then reopened either fully and partially, while changing the mode of instruction from face-to-face to online (synchronously or asynchronously) or hybrid. The COVID-19 pandemic has changed not only the physical location where teaching and learning occur, but also has increasingly demanded technological knowledge, digital technologies, digital literacy, and digital competence for both teacher educators and teacher candidates. Especially during the COVID-19 pandemic, the integration of technologies in teacher education was not optional but inevitable. The main challenges for teacher educators are not just familiar with new technologies but the demands of reconstructing their pedagogies. In the online environment, teacher educators are required to develop, expand, and transform their specialized pedagogical content knowledge in using digital technology. The new online teaching environment demands teacher educators not only to learn new technological tools, familiarities with Learning Management System (LMS), and digital platforms (e.g., Zoom, Canvas, Blackboard, Camtasia, or Panopto) for our own instruction but it also demands teacher educators to support teacher candidates who experience lack of digital equipment, inaccessibility, and inequalities. The integration of digital tools in education has transformed the way how to plan and implement curriculum resources in methods courses (Remillard, 2005). During the reconstruction process, teachers’ knowledge and practices would significantly influence the way how to modify a set of resources (Pepin et al., 2013).
Over the past two years, the changes, issues, challenges, adaptations, and innovations in teacher education caused by the COVID-19 pandemic have been reported in many other countries such as Australia (Scull et al., 2020), Chile (Sepulveda-Escobar & Morrison, 2020), England (Velle et al., 2020), Germany (König et al., 2020), Greece (Brinia & Psoni, 2021), Portugal (Flores & Gago, 2020), and Trinidad and Tobago (Kalloo et al., 2020). The changes include course-level changes made by individual teacher educators, program-level changes initiated by individual institution, and policy-level changes. More specifically, the COVID-19 pandemic has caused pedagogical changes for individual teacher educators such as shifting from student-centered to teacher-centered and changing from group activities to individual tasks (e.g., Moorehouse, 2020). The rapid transition to the online environment made it difficult for students to engage, communicate, socialize, and collaborate (e.g., Davidson & Major, 2014). At the institution level, teacher education programs have faced additional challenges because the field experience has been removed, suspended, or replaced (e.g., Kidd & Murray, 2020). Because of the K-12 school closures, teacher candidates spent less time on practicum experience for teaching but spent more time for reading and reflection (e.g., Velle et al., 2020), replaced their on-site observation of their mentor teachers with observing publicly available classroom videos such as Teaching Channel, or virtually taught a small group of students in a breakout room. At the policy level, for example, the State of California temporarily suspended teacher testing requirements (e.g., RICA, CBEST, CSET and edTPA) for a preliminary credential but required to complete the teacher testing requirements to earn a clear credential. The previous studies have significant contributions to understand general pedagogical challenges faced by in-service teachers (e.g., Cho & Kim, 2022; Kim, 2021), individual teacher educators, alternative fieldwork experiences offered by teacher education programs, and policy changes in the requirements for earning teacher credentials, but it is not well known how teacher educators have transformed their pedagogies of teacher education for teaching specific subject-matter using technology.
In this article, we, two mathematics teacher educators (MTEs), reflect our own experiences in appropriating, transforming, reconstructing, and modifying our pedagogies of teaching elementary mathematics methods courses in making a transition from face-to-face to online environment during the COVID-19 pandemic. Using a collaborative self-study, we explored the following three themes: (1) using virtual manipulatives; (2) creating collaborative, interactive, and shared learning experiences for PSTs; and (3) making PSTs engaged in student thinking.
(Re)Thinking TPACK in the Context of Teacher Education
Teachers are expected to have integrated knowledge between content knowledge (CK) and pedagogical knowledge (PK). Shulman (1986) coined the term pedagogical content knowledge (PCK). As digital technology has become more accessible and incorporated into classrooms, researchers have broadened the knowledge teachers need for teaching with the effective use of technology. By incorporating technological knowledge (TK), Mishra and Koehler (2006) introduced a conceptual framework, the Technological, Pedagogical, and Content Knowledge (TPACK). TPACK model consists of seven domains: TK, PK, CK, PCK, technological pedagogical knowledge (TPK), technological content knowledge (TCK), and TPACK (see Figure 1). They described TPK as “knowledge of the existence, components, and abilities of various technologies as they are used in teaching and learning settings, and conversely, knowing how teaching might change as the result of using particular technologies” (p. 1028). For example, to enhance collaboration, students can share and communicate what they learned with multimodal presentation tools such as PowerPoint or Google Slides that allow students to present their ideas. Mishra and Koehler (2006) defined TCK as “teachers need to know not just the subject matter they teach but also the manner in which the subject matter can be changed by the application of technology” (p. 1028). For example, dynamic mathematical software for geometry can provide students an opportunity to engage with geometrical concepts and relationships by dragging and measuring (Hollerbrands, 2007). This type of knowledge is integrated with technology and fundamentally changes the subject matter students can learn. TPACK refers to knowledge where teachers engage in teaching of subject matter and in guiding student learning with technologies as a foundational intersection between TK, PK, and CK (Koehler & Mishra, 2009). For example, teachers revealed knowledge of how to use Wiki as a communication tool between teachers, students, and even parents as a collaborative learning site for students’ mathematical learning (Bos, 2011).
Although many studies have examined PSTs’ TPACK such as a developmental model (e.g., Niess et al., 2009), instrumental measures (e.g., Zelkowski et al., 2013), and progression in a teacher education program (e.g., Gill & Dalgarno, 2017), teacher educators’ TPACK has not been well known yet. Recently, researchers have identified teacher educators’ practices and competencies to integrate technology into their teacher training programs (e.g., Voithofer & Nelson, 2020). However, unsurprisingly, much little study has been done under the context of COVID-19 pandemic. To bridge this gap, this study aims to identify two MTEs’ practices, challenges, experiences, and lessons learned responding to the current situation through the meaningful use of technology. To be specific, after identifying themes of teaching practices related to the use of technology, we examined what type of knowledge was associated with TK, PK, CK, PCK, TPK, TCK, and TPACK. By providing the examples of embedded MTEs' knowledge, it would be possible to understand better how PSTs' TPACK could be potentially improved by MTEs' TPACK. Clearly, before making PSTs technology-integrated instructional activities engaged, MTEs require to be assessed their own TPACK and to have metacognitive practices to increase PSTs' understanding and design of activities (Kramarski & Michalsky, 2010)
Methods
Collaborative Self-Study
To explore how we made a transition to the new online environment and redesigned our courses accordingly during the COVID-19 pandemic, we employed a collaborative self-study design. The self-study design involves “a thoughtful look at texts read, experiences had, people known and ideas considered” (Hamilton & Pinnegar, 1998, p. 236). According to LaBoskey (2004), self-study has five features: self-initiated, self-focused, improvement-aimed, interactive, and qualitative. Especially, among teacher educators, self-study of teacher education practices (S-STEP) has widely been used in different settings to understand practices and self who engaged in those practices more clearly (Loughran, 2004). This integration between self and practices are critical in S-STEP to develop more vivid educational theories. If a study focuses only on practices without consideration of the self, it would be considered as action research. This study also has a collaborative feature between two MTEs. Teacher educators can collaboratively acquire and develop their knowledge of teaching and learning about teaching from diverse perspectives (Hamilton & Pinnegar, 2013). The collaborative self-study has been used and adopted by many teacher educators to identify fundamental problems and understand essential commonalities and differences (e.g., Clift et al., 2005; Kim et al., 2021; Moorhouse & Tiet, 2021). This method not just examines participants' practices but also can be used as a form of professional development with a supportive manner (Bullock & Ritter, 2011). Using a collaborative self-study, we aim to share our experiences, challenges, pedagogical decisions, and reflections on making a transition from face-to-face to the online environment.
Study Context: Stories from Two Mathematics Teacher Educators
This study was conducted by two MTEs at two public universities in the United States. After the COVID-19 pandemic, both MTEs taught elementary mathematics methods courses in the real-time online environment but offered different modes of instruction according to the regulations, guidelines, and policies of each university. During the 2020-2021 academic year, MTE 1 offered synchronous sessions via a Zoom platform for 15 weeks, whereas MTE 2 offered hybrid sessions (both in-person and synchronous sessions via Zoom platform) for 12 weeks.
MTE 1 had taught an elementary mathematics methods course in a face-to-face mode before the COVID-19 pandemic but made a quick transition to online teaching because of the campus closure in the mid-March of 2020. Because MTE 1 did not have any prior experience of real-time online teaching before the COVID-19 pandemic, it was not easy to rapidly switch the mode of instruction in a week and adapt to the new online environment for the second half of Spring 2020 semester. After the first eight weeks of online teaching in Spring 2020, MTE 1 took a number of professional development courses of online teaching offered by the university during summer 2020. For a two-and-half hour weekly session, MTE 1 utilized Zoom tools such as poll to check PSTs’ reading reflections and to keep track of their understanding and misunderstanding, breakout rooms for small group discussion and peer run-throughs, screen share and annotate (both by the instructor and PSTs), chat to ask questions (some of the questions were addressed and answered by the instructor, whereas other questions were answered by their classmates), raise hands and reactions function as well as other interactive working spaces and various virtual manipulatives sites. In addition, MTE 1 uploaded PowerPoint slides and resources on Canvas (the main Learning Management System used by the university), used quiz and discussion on Canvas, and recorded the sessions if necessary.
MTE 2 had taught an elementary mathematics methods course in a hybrid mode since Fall 2020. Prior to teaching this course, MTE 2 took several online courses about educational technology as a student and supported online master’s degree programs as a teaching assistant. Due to the COVID-19 pandemic, the elementary mathematics methods course was already assigned as a hybrid mode that PSTs weekly rotated instructional modes between in-person and online. That is, if a half of students come to the campus, the other half would come in the following week to maintain the social distance. MTE 2 provided the same resources, activities, and tools not only for students who attended the course in person but also for students who attended the course via Zoom. MTE 2 utilized the Zoom breakout room, virtual manipulatives, interactive working spaces, and Blackboard (the main Learning Management System used by the university). To facilitate PSTs’ engagement, MTE 2 designed a digital shared place (Google Slides) to make students engaged in the activities in order to connect both PSTs on campus and PSTs in a Zoom simultaneously.
Both MTEs teach an elementary mathematics methods course using the same textbook (Van de Walle et al., 2019) and share similar pedagogical visions for teaching PSTs. Because the PSTs in both sites need to pass the state-mandated performance assessment to get a teaching credential in each state, both MTEs incorporate analyzing student work samples and designing re-engagement lessons to address errors, misconceptions, and partial understanding. Despite the similarities addressed above, we are different in terms of teaching experience, geographical location, regulations and guidelines of each university and affiliated school districts during the COVID-19 pandemic. At the time of conducting this study, MTE 1 in California had four years of teaching experience of teaching elementary methods courses for undergraduate students and post-baccalaureate students in the credential program, whereas MTE 2 in Alabama was his first year as a faculty member in mathematics education for teaching undergraduate students in the credential program. Because the online environment was new to both of us at the beginning of COVID-19 pandemic, we have experienced a steep learning curve during the first semester of online teaching and then refined our pedagogical approaches over time.
Data Collection and Analysis
From the beginning of Fall 2020 to the end of Spring 2021, we have had regular weekly video conferences via Zoom to discuss challenges of teaching an elementary mathematics methods course in the online environment and to share resources to improve our teaching practices. Each meeting lasted about two hours and video-recorded. The video conferences were transcribed by voice typing feature in Google Docs and the transcriptions were reviewed individually to revise errors after the meetings. The first part of discussion was to reflect on our teaching practices through reflective dialogue with guided prompts to facilitate our reflection on key parts of teaching (Rosaen & Gere, 1996). For example, we discussed the following prompts: How have your teaching practices been changed from face-to-face to online? To what extent have your students’ engagement, participation, and performances been changed from face-to-face to online? To what extent would this new online environment impact on your teaching practices after the pandemic? Then, the second part of discussion was open to any issues, challenges, or concerns raised or experienced both in the course-level and in the program-level. In addition, artifacts from each MTE were collected: syllabi, class activities, assignments, artifacts of teaching, PowerPoint slides, weekly reflections, class discussions, feedback from students, and recorded sessions. To increase trustworthiness of the data analysis, we both worked as critical friends to increase clarification and validation by providing constructive feedback and support (LaBoskey, 2004; Mishler, 1990).
The data were analyzed through a constant comparative method (Corbin & Strauss, 2007). After reading the transcripts several times, we sought for similarities and differences between our practices and rationales for those practices. We identified explanatory accounts of these similarities based on the evidence from the transcripts, course materials, artifacts of teaching, student work samples, and recorded sessions. Then, we compared and discussed these explanatory themes together until we reached consensus. In such a process, three common themes emerged: (1) choosing and using virtual manipulatives; (2) creating collaborative, interactive, and shared learning experiences for PSTs; and (3) making PSTs engaged in student thinking. However, some themes were not further analyzed due to somewhat vagueness to make connection with or unpack our own TPACK. For example, both MTEs experienced the importance of practicing, rehearsing, and simulating the core practices of teaching in the elementary mathematics methods course. Because of the limited or restricted field experiences during the COVID-19, we used microteaching, peer run-throughs, simulated student interaction, and using digital simulations via zoom. Although we changed in-person activity to online activity (where PSTs did an activity in a breakout room during the class meeting or in a separate zoom meeting beyond the course meeting) and collected the artifacts of PSTs’ learning from the recorded session, it is less clear for us to make connection with or unpack our own TPACK. For example, when MTE 1 implemented a number talk activity in a face-to-face setting before the pandemic, PSTs rehearsed a number talk in groups and recorded the elicited methods in a poster attached to the wall of classroom. MTE 1 circulated the classroom to monitor the number talk activity of each group but was not able to observe the entire verbal exchanges because all groups did a number talk activity simultaneously. On the other hand, during the pandemic, MTE 1 asked PSTs to schedule an individual group zoom meeting, record the number talk activity, and write the elicited methods on the virtual whiteboard. Because of this, MTE 1 was able to observe each group’s number talk activity in depth. However, we interpreted this as affordance of using technology rather than mapping into the TPACK framework. The next section discusses each of these three common themes in detail.
Results
Choosing and Using Virtual Manipulatives
To develop PSTs’ pedagogical content knowledge related to students’ learning mathematics, both MTEs incorporate a variety of manipulatives, models, drawing, and visual representations in the elementary mathematics methods courses. Before transitioning to the online environment, PSTs mostly utilized physical manipulatives throughout the course and were often introduced virtual manipulatives (e.g., National Library of Virtual Manipulatives developed by Utah State University) toward the end of course to let them know potential options for their future classrooms. The virtual manipulatives website (http://nlvm.usu.edu/) organizes the resources by content domain and grade-level but often has problems with loading virtual manipulatives with Java. Although PSTs appreciated the use of physical manipulatives to explore mathematical ideas, the physical manipulatives had some limitations. For example, there is little access to those physical manipulatives outside of the class because of the cost. Some PSTs purchased a set of base-ten blocks to use in the future, but were not able to afford to equip a full set of different manipulatives for their personal uses.
After making a transition to the online environment, one of the biggest challenges was to provide the same level of learning opportunities for PSTs to use manipulatives. We explored various virtual manipulative websites, analyzed the major features (both technologically, pedagogically, and mathematically), compared the functions from different websites, and analyzed the affordances and limitations of virtual manipulatives compared to physical manipulatives. The virtual manipulatives used by both MTEs included unifix cubes, two-color counters, number frames, number line, hundred chart, Cuisenaire rods, fraction bars, fraction circles, pattern blocks, tangrams, geoboard, algebra tiles, partial product finder, angle, to name a few. Some physical manipulatives used in the face-to-face mode (e.g., rational number wheel, balance scale) are not available to use in the online environment, whereas other physical manipulatives unavailable in the face-to-face mode (e.g., dynamic geometric tools) are now available to use in the online environment. In the online environment, we provided opportunities for PSTs to interact with the virtual manipulatives. In this section, we introduce examples that require a different level of transforming pedagogies in teaching elementary mathematics methods courses by switching from physical manipulatives to virtual manipulatives.
First, the virtual number frames in various websites differ in terms of available frames (e.g., five-frame, 10-frame, 20-frame or creating their own frame), using two-color counters or choosing different objects with different colors, rotating ten-frame vertically or horizontally, moving five or 10 counters/objects at the same time, annotating number sentences/equations, and saving the screen as a file. PSTs were given the printed ten-frame card in the face-to-face mode, whereas they were able to represent numbers by creating their own ten frames and decomposing numbers in different ways in the online environment. As seen in Figure 2, the physical ten-frame does not accompany the number sentence, thus the additional question needs to be asked to make connections between pictorial representation and numerical representation (e.g., Can you write it in a number sentence?). On the other hand, the virtual ten-frame offers the option of inserting a math text tool, so it is much easier to make a connection between pictorial representation and numerical representation. Because the virtual ten-frame offers the option of choosing different objects and colors, however, PSTs often represent the number 8 with different objects and different colors. In such a case, using less colors (up to two colors) is more effective for subitizing, finding a pattern, and decomposing the numbers to two addends.
The virtual base-ten blocks in different websites offer different features such as place value mat, switching from whole numbers to decimals, break/join pieces, availability of multiple colors to differentiate each place value (ones, tens, and hundreds) or to differentiate between minuend and subtrahend, grouping objects to join or delete pieces, magnetic function, annotate function, availability of number cards, and saving the screen as a file. As shown in Figure 3, the physical base-ten blocks can be customized by adding the place value mat, inserting ten frame in ones place, and adding number cards in the face-to-face mode, but the virtual base-ten blocks can be limited to the functions provided by the websites. One of the major benefits of using virtual base-ten blocks is that they offer a regroupable model by just clicking “break pieces” or “join pieces” on the bottom of the menu, compared to a pre-grouped model of physical base-ten blocks which requires the manual trading process.
The virtual dynamic angle measure tool fundamentally transforms the way of introducing angles in the elementary mathematics methods course. In the face-to-face mode, PSTs were introduced to make a nonstandard patty paper protractor by folding it multiple times, to measure the angles using the wedges created, and to estimate the angles. This approach offers to conceptualize the angle as “the space between two rays sharing a common vertex” (Browning et al., 2007, p.285) from the static view of angle. However, the virtual dynamic angle measure tool (https://www.visnos.com/demos/basic-angles) allows to turn one ray from another, to show the dynamic relationship of complementary angles and supplementary angles, and place the protractor on the bottom of the virtual dynamic angle measure tool.
At the beginning of transitioning from face-to-face to online, we have faced with challenges of using physical manipulatives in the online environment and searching for the available virtual manipulatives. The virtual manipulatives are not just a simple replacement of the physical manipulatives. A certain feature of virtual manipulatives changes the model of manipulatives (e.g., from pre-grouped to regroupable) and offers different ways to conceptualize mathematical ideas (e.g., from the static to the dynamic view of angle). As it requires substantial technological, pedagogical, and content knowledge to compare physical manipulative and virtual manipulatives, PSTs would have benefits with discussing the affordances and limitations of virtual manipulatives to teach and learn specific mathematical ideas.
Creating Collaborative, Interactive, and Shared Learning Experiences for PSTs
In the online environment, one of the challenges is to set up a collaborative learning environment for group activities and discussions. If PSTs were in a face-to-face mode, they were able to collaborate in a group table. On the other hand, the online environment requires a different shared work space for PSTs to collaborate, communicate, and interact. In our elementary mathematics methods course, we utilized various shared collaborative learning spaces such as Google Slides, Web Whiteboard, Zoom Whiteboard, Jamboard, Padlet, PollEveryWhere, and Canvas Discussion (See Table 1).
As an interactive working space, we used various real-time collaborative tools. First, Google Slides allow for PSTs to work collaboratively. Aligned with the instructor’s slides for presentation, PST shared their golden quotes from reading assignments and recorded their group’s opinions after solving problems or watching videos. PSTs also used Google Slides to share individual reactions. For example, after watching three short video clips of formative assessment strategies, one of the groups recorded the features of each strategy and then discussed the most preferred strategy to implement.
Second, A Web Whiteborad, Zoom Whiteboard, and Jamboard were employed to provide online space for problem solving or to share their strategies. A Web Whiteboard, an unlimited digital canvas with drawing and annotating tools, is helpful for remote students to collaborate online in real time. The instructor could monitor individual progress as a single digital canvas and facilitated further discussion based on their products. Similarly, Jamboard also provides an online collaborative work place. However, one distinct feature of Jamboard is to use individual pages. PSTs were assigned to a specific page or they chose any page and recorded their strategy about the given problems. For example, each group of PST solved a minibus problem with Jamboard. The problem and prompt were already inserted by the instructor and a group of PST used pictorial representations with written explanations. Zoom Whiteboard is also used for drawing and texts between group members during video conferencing in real time.
Lastly, Padlet, PollEveryWhere, and Canvas Discussion were also utilized to enhance online communication. In Padlet, PSTs can stack sticky notes as an online bulletin board and contain texts, links, videos, images, and files. For example, each PST wrote story problems for 52-13 for two different interpretations of subtraction (take-away interpretation vs. comparison interpretation) and represented each subtraction interpretation using a number line. In Padlet, MTEs were able to monitor PSTs’ real-time responses and highlight the differences between take-way interpretation of subtraction and comparison interpretation of subtraction.
At the beginning of transition to the online environment, we concerned that it would be difficult for PSTs to collaborate, interact, and share their thinking with peers. Unlike our initial concerns, however, the shared collaborative learning spaces make it visible PSTs’ thinking in a public space which allows MTEs to monitor PSTs’ thinking more explicitly. In a face-to-face environment, PSTs wrote their ideas, reactions, and responses in their lecture notes or personal devices individually and then shared them with the peers. Even though the MTEs tried to carefully monitor PSTs’ thinking, we had challenges with making all PSTs’ initial thinking available in a public space before opening to the group discussion, selecting PSTs who can contribute to the whole-group discussion, making connections among their thinking, and even recording or saving the artifacts of teaching and learning in a face-to-face mode. As we have tried to figure out the features of each shared collaborative learning spaces and to strategically use each of them for instructional activities in the online environment, it allows PSTs to use multimodalities beyond their verbal contribution and construct their knowledge with others.
Making PSTs Engaged in Student Thinking
The COVID-19 pandemic not only changed the learning experience of PSTs in the university-based courses but also changed the quantity and quality of their field experience. It is not just a change of physical locations for field experience (from face-to-face to online/hybrid) but it faces challenges of providing high-quality and authentic clinical experience for PSTs. Among many other challenges that PSTs face in a new online environment, we aim to support PSTs to engage in student thinking during the COVID-19 pandemic.To provide a rich opportunity to understand how children think and are engaged in mathematical tasks, we deliberately choose video clips, purposefully sequence the video clips according to the central focus of each session, and facilitate the discussion using the prompts for watching each video clip. Before the pandemic, although PSTs have sufficient opportunities to observe their mentor teachers’ teaching and analyze students’ thinking during the field experience, their experiences would be still limited to the particular teaching practice of their mentor teachers and particular student thinking in the context of their field placement. In addition, their observations, analysis, and reflections are often left in a private space, rather than publicly discussing with colleagues. By using video clips—one of the widely used materials to support PSTs’ learning to teach mathematics (Jacobs et al., 2010; Van Es & Sherin, 2008)—in an elementary mathematics methods course, we were able to make the analysis of student thinking in a public space, to vary the grain size of teaching practices, to decompose the complex practice of teaching, and to zoom in and zoom out the particular aspects of the complex teaching practices. Table 2 illustrates the video clips used by MTE 1 in her first eight sessions of the elementary mathematics methods course and Table 3 illustrates the video clips used by MTE 2 in his first eight sessions of the elementary mathematics methods course.
More specifically, MTE 1 selected the video clips from the whole class for the first two sessions (sessions 1-2), the video clips from individual interviews for the next three sessions (sessions 3-5), and selected the video clips from the whole class again for the next four sessions (sessions 5-8). By varying the grain size of the video clips and sequencing them in this particular order, MTE 1 provided opportunities to see the whole-group interaction and to see individual student thinking. The length of video clips ranges from 22 seconds to 14 minutes and 9 seconds, but none of the video clips exceed 15 minutes. MTE 1 posted the video clips on Canvas so that PSTs were able to watch the video on their own paces before the class and PSTs shared their observations either synchronously during a Zoom session or asynchronously on Canvas. In the online environment, playing the video clip from the instructor’s device was challenging for PSTs because of the unstable internet connection issues for some PSTs. Rather, PSTs can watch the video clips from their own devices which make it easy for them to pause, tag, reflect, and rewatch the video clips. For example, PSTs watched Gretchen’s subtraction video from session #5 and discussed their posts on Canvas (see Figure 5).
MTE 2 selected classroom videos to provide a holistic picture of our current classrooms in the first three sessions (session #1-3). From the next five sessions (sessions #4-8), most of the videos were one-to-one interview videos, which intended to look for details of problem solving and to identify mathematical thinking with evidence. With regard to content areas, the selected videos covered most of the content areas in elementary mathematics. The selected videos provided not only an opportunity for PSTs to explore students’ mathematical thinking but also an entry point to discuss how they pay attention to various details of videos related to learning and teaching mathematics. For example, PSTs watched two contrasting videos of the fifth grader’s using standard algorithms of subtraction. The fifth grader used the algorithm with an error in both videos but her belief in using algorithms changed over time. After watching those videos together, they discussed a student’s mathematical thinking about the subtraction algorithm in each video and compared them (see Figure 6).
Although we chose different video clips in our methods courses, we have shared goals of providing rich experiences to learn from practices, making the analysis of teaching and student thinking in a public space, and citing the supporting evidence from artifacts, which are somewhat limited in their student teaching during the pandemic. To address the gaps between university coursework and field experience, a number of teacher education programs across the United States implemented a practice-based teacher education program and designed instructional activities so that PSTs can learn teaching in, from, and for practices (Lampert & Graziani, 2009). The limited access to the field placement during the pandemic has increased attention to the practice-based teacher education program so that PSTs learn teaching practices from the use of actual artifacts of teaching in the elementary mathematics methods course.
Discussion
During the COVID-19 pandemic, many teacher education programs suspended their face-to-face mode and then transitioned to either online or hybrid modes. The change of instructional mode requires not just simply to integrate technology in the course, but also requires transformative knowledge and extended skills for teacher educators (Peachey, 2017). Integrating various digital tools, teacher educators have reconceptualized instructional materials related to what and how they were using such resources (Pepin et al., 2013). From this documentational genesis approach, we identified how two MTEs responded to this change and used their specialized knowledge (e.g., TPACK) and practices in the context of online/hybrid elementary mathematics methods courses. In the study, we employed collaborative self-study (i.e. S-STEP) and found three key pedagogical transformations: (1) using virtual manipulatives; (2) creating collaborative, interactive, and shared learning experiences for PSTs; and (3) making PSTs engaged in student thinking.
These findings indicated that online teaching requires transformative knowledge for teacher educators. Transferring face-to-face to online is not a simple matter of putting the existing content to online (Bakia et al., 2004); it should focus on pedagogical improvement in teaching mathematics rather than technology’s sake (Lederman & Neiss, 2000) or how it can be repurposed in a new online environment in a way that students’ learning is optimized (Mullen et al., 2021). For example, when introducing virtual manipulatives, two MTEs decided to ensure the similar level of learning opportunities for PSTs through dragging and clicking just as hands-on manipulation (TPK) and to explore mathematical features related to number and operations (TCK). Beyond those knowledge, PSTs were asked to investigate potential students’ misconception in using those manipulatives and to consider prior knowledge of children’s arithmetic operations (TPACK). Clearly, both MTEs showed a progressive transformation of the pedagogical and/or knowledge to TPACK, which incorporates PCK with TK. In addition, PSTs’ engagement in children’s mathematical thinking and their collaborations via digital tools are involved in the MTEs’ intentional practices with knowledge of technological, pedagogical, and content in teaching mathematics. For example, the MTEs selected and sequenced a series of videos with specific pedagogical goals. PSTs were also asked to analyze student thinking on the particular mathematical topic by citing the evidence to support their claims and inform their pedagogical decisions.
In this study, we do not claim that we are implementing the most effective pedagogies of teacher education in the online environment. Rather, we aim to explore our own journey to adapt to the new environment, make a transition of our hands-on, inquiry-based, and collaborative course to online or hybrid, share challenges in such a transitional process, and reflect on our practices to further improve our courses. It was not just substituting or relocating activities and resources from face-to-face mode to online or hybrid mode, but required us to think of what needs to be modified, transformed, reconstructed, redefined, or modified to provide the similar level of learning experience for PSTs in the online environment. We are still in the progress of adapting and refining our courses based on our own reflections, informal and formal feedback from students, and their needs or concerns in the online environment. Because various technological tools are available and new technological tools are emerging, one of the challenges is to learn these new technological tools and to choose an effective or optimized tool to facilitate student learning. We experienced that introducing every new technological tool and switching tools from one session to another was not very effective both for MTEs and PSTs. For example, MTE 2 tried to employ different digital tools with specific intention. MTE 2 decided to use Blackboard Ultra as a main video conferencing tool. However, in the first session, many PSTs struggled to use this new tool with accessibility issues because they were not familiar with the environment and interface yet. From the second session, MTE 2 decided to use Zoom rather than Blackboard Ultra for the rest of the course. Having similar experiences, MTE 2 planned classroom activities by considering PSTs’ comfort zones first. When PSTs were familiar with a new digital tool, they were introduced to another new one. Another challenge for us is to be familiar with the learning management system, tools, and gamification used in K-8 schools and to develop PSTs’ competences in using those learning management systems for the content-specific course.
We experience that the online environment is very challenging for teacher educators and PSTs, but we also believe that there are certain benefits of the online environment. In the face-to-face mode, the use of technology can be perceived as an optional and considered as content-specific tools (e.g., dynamic geometry tools such as GSP, Geogebra, Logo, Tessellation) in the elementary mathematics course. For example, MTE 1 mainly used physical manipulatives throughout the course and introduced some virtual manipulatives sites to PSTs toward the end of course before the pandemic. During the pandemic, MTE 1 used virtual manipulatives more strategically and deliberately by exploring different features and functions available on each virtual manipulative site and comparing the affordances and limitations of using them. In the face-to-face mode, PSTs utilized physical manipulatives during the class, but did not utilize those manipulatives outside of the class (either at home or at school sites) unless they, their mentor teachers, or their school sites purchased them. However, PSTs have increased access to those manipulatives virtually during the pandemic. Second, the online environment makes every PST’s thinking visible by using shared collaborative work spaces, prevents the whole-group discussion dominated by few students, increases PSTs’ participation, randomly assigns PSTs in groups so that they can collaborate with other students each time, develops arguments by citing the specific evidence, reduces the anxiety of public speaking, and creates the records of our own practices and PSTs’ thinking which made us to reflect on our own teaching practices.
For the first few weeks of the online environment, we had concerned that the elementary mathematics course during the COVID-19 pandemic might not bring the same level of interaction, participation, and engagement from PSTs as we did in a face-to-face setting before the COVID-19 pandemic. Some of the anecdotal advices from other faculty members include shortening instructional time for synchronous classes because of zoom fatigue and making the recorded session of lecture no more than 15 minutes for asynchronous class. However, our experiences in the online environment were not negative ones. As being forced to teach the elementary mathematics methods course in the online environment, we experienced the transformation of our pedagogies of teacher education in several different ways beyond increasing our knowledge, skills, and sensibility to use technology in the context of teacher education. In a contrary to substantive and collective efforts made to integrate new instructional tools, implement innovative instructional activities, reflect on teaching, and transform pedagogies in K-12 settings, teacher educators had less experiences with explicit conversations about their pedagogical choices, decisions, changes, adaptations, and transformations and then made collaborative efforts to improve teacher education programs before the pandemic. As previous studies reported that teacher education programs experienced adaptations or changes forced by the COVID-19 in the online learning environment, the COVID-19 pandemic created opportunities to integrate technologies while exercising their own pedagogical content knowledge; make all PSTs’ thinking available in a public space in the shared collaborative learning spaces and make connections among PSTs’ ideas instead of eliciting few responses; and address the gaps between university coursework and field placement by analyzing the artifacts of teaching in the elementary mathematics methods course. Besides these three themes we identified in this study, we, as MTEs, have participated and engaged in the community of professional learners by observing the recorded session of each other’s elementary mathematics methods course. Recently, many teacher education programs have returned to the face-to-face mode of instruction at the university level and have provided the same level of field experiences for teacher candidates before the pandemic. However, our pedagogies of teacher education do not remain the same as before the pandemic. As we move into the post-pandemic, we need to reimagine how our pedagogies of teacher education could be further transformed by being forced to use technology. The lessons we have learned over the last two years can be still applied and further expanded to the (new) normal.
