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familiar shared workspace is available, then we can ask students to draw graphs of functions. If the col laborative medium does not have a shared work space, however, we would need to pivot these questions from drawing graphs to asking questions that may be discussed verbally about a provided graph— what do you think would happen to this graph if we doubled the leading coefficient? As such, many of the same strategies of designing thinking tasks in face - to - face thinking classroom settings are therefore transferrable to the online set ting, but an enhanced attunement to the collabora tive medium needs to be considered. The BTC framework evolved from a systematic opposition and reformation of the normative struc tures that permeate classrooms around the world, towards a toolkit of strategies that together guaran tee a thriving classroom environment brimming with student thinking, and in turn, learning. The framework evolved from over 15 years of research in face - to - face classrooms. Since the COVID - 19 lockdown, however, a new evolution has taken place. One that is more sudden and even more globally challenging of various kinds of normative structures. The sudden onset of its incredibly trans formative implications shook our educational sys tems into an unanticipated and uncontrollable reor ganization where everything that was known, now became questioned. Without the comfort of regular face - to - face interactions with students and with shifting landscape of all aspects of society, no longer could the same rules and norms continue. The adaptations were born out of the protocols as sociated with the COVID - 19 lockdown, which caused many teachers to have to reorganize their ways of teaching into a mix of asynchronous and synchronous online learning approaches, at times paired with some limited face - to - face components with various degrees of physical distancing proto cols. For those who had grown accustomed to teaching in a BTC environment, it created an ur gency to find ways to transfer what BTC offered them into their respective lockdown contexts, the Conclusion

Figure 5: Image used to introduce the triangular disks task

with an option for synchronous collaborative draw ing, offering a diagrammatic medium , advantages the thinking space for students for such tasks. Stu dent work completed on Jamboard for this task is shown in Figure 6.

Figure 6: Example of diagrammatic student work on Jam board in breakout group

As may be seen, all the inputs were made with a drawing tool and the drawing required was simple enough to create with a mouse if no touch screen was available. However, it is possible that students uncomfortable with drawing digitally may find it easier to work on their own papers, which could make it difficult for collaboration to occur if they choose not to use their cameras to show each other their ideas. While the above - mentioned thinking tasks are not directly linked to curriculum (even though they may be mapped to curricular outcomes), they em phasize mathematical competencies such as reason ing, describing, formulating, predicting, justifying, and verifying. Just as in a face - to - face thinking classroom, tasks that involve curricular outcomes may be more directly introduced as thinking tasks after a thinking culture is established. When mak ing this transition, however, we need to be likewise aware of both the students ’ technological compe tencies and the affordances and limitations of our collaborative medium. Take for example identify ing tasks for the topic of graphing functions. If a

Virginia Mathematics Teacher vol. 47, no. 1

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