vmt-award-2024_47-1-_blue

innovative and creative ways, and to engender a culture of collaborative thinking in the classroom. However, online settings offer certain constraints that need to be considered. Since thinking tasks involve communicating and representing ideas about a problem, the most prominent aspect for de termining which tasks are appropriate for an online setting has to do with the modes of communication it requires from students as well as their technolog ical competencies . In face - to - face BTC environments, students have ubiquitous access to a variety of proximally located shared workspaces which, together, allow them to employ a variety of modes of communication. With little to no effort, students in these face - to - face set tings naturally employed a combination of commu nicative tactics including gestures, verbal utteranc es, diagrammatic presentations, and various written notations for communicating their mathematical ideas around a thinking task. However, these taken for - granted affordances are hindered in an online space. To communicate ideas in an online setting, students are not only limited by the technological tools they have at their disposal (e.g., cameras, microphones, writing inputs, etc.), but also by their technological competencies and willingness to engage while con necting remotely (Roddy et al., 2017). The effec tiveness of a thinking task is therefore intimately interwoven with the modes of communication it requires of students and their ability to express their ideas through technological means. As such, choosing thinking tasks for online collaborative settings and identifying ways to use them with stu dents in such settings needs to take into account not only the nature of the task itself, but also the stu dents ’ technological competencies. For example, if the state of students ’ technological competencies are unclear, then it is best to begin with tasks that invoke verbal or gestural modes of communication without the need for notation heavy or diagrammatic output. Such tasks may in vite students to visualize, explain, predict, conjec ture, justify, or verify (Van de Walle et al., 2015). In a synchronous environment, students may be

collaborating on the task in breakout groups using a microphone and camera, while in an asynchronous setting, they may be using basic text functions in a discussion board dedicated to collaborating about the thinking task. Either way, the thinking task needs to allow for multiple ways of thinking and be focused on emphasizing discussion of various viewpoints rather than on solving to get a single solution. An example of a thinking task that achieves this and that does not require students to have high technological competencies for commu nication of ideas may be introduced as follows: A bicycle crosses a freshly painted road line, and the smears on the bike ’ s tires from the road line continue to make imprints on the pavement as the bike continues down the road. What do these imprints look like? (Mason, Burton, & Stacey, 2011) Given a limited technological space that does not offer written or diagrammatic output, but that simp ly relies on communication through a microphone and possibly a camera, this task then becomes heavily reliant on students ’ ability to visualize pos sible solutions. The limitation of technology here provides an opportunity for emphasizing visualiza tion, followed by verbal and possibly gestural com munication. In using this task in multiple instances with different groups of students, we observed that students debating heatedly about their predictions, often using gestures through their video cameras while explaining their ideas verbally through their microphones. Even when we provided students with a collaborative writing surface, very little was drawn on it and the primary mode of communica tion was verbal as well as gestural. The nature of this task lends itself to verbal communication, and therefore, by using it we may harness the potential ities of the online setting rather than being limited by its constraints. While verbal and gestural communication about thinking tasks offers a plethora of possibilities for both non - routine and curricular thinking tasks, the need for written and diagrammatic communication is bound to arise. In such cases, another tool is needed for students to use for communicating their

Virginia Mathematics Teacher vol. 47, no. 1

10