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the cheques (see Figure 3) rather than crossing them out in order to try their attempts at beating the tax collector. This makes it an object - oriented me dium rather than simply a writing medium , offering a form of interaction for students not automatically available on a whiteboard in a face - to - face thinking classroom. However, the disadvantage was that students be came less inclined to take note of the order of each move they made, nor of the brainstorming of fac tors available for each number. As may be seen in Figure 4, a whiteboard as a writing medium for this same thinking task engendered rather different trac es of mathematical thinking, which perhaps oc curred only verbally in the online setting.

Figure 2: Example of student work on a thinking task in a face - to - face BTC setting

the shared workspace. This illustrates the shift in modes of communication as determined by the type of workspace used.

Another example of a task used in a Jamboard is as follows:

Start with a collection of paychecks, from $1 to $12. You can choose any paycheck to keep. Once you choose, the tax collector gets all paychecks remaining that are factors of the number you chose. The tax collector must re ceive payment after every move. If you have no moves that give the tax collector a paycheck, then the game is over and the tax collector gets all the remaining paychecks. The goal is to beat the tax collector.

Figure 4: Example of student work on a whiteboard in a face - to - face BTC setting

Another example of a thinking task used in a Jam board is one that relies mostly on students drawing to communicate their thinking, and may be intro duced as follows: You have a triangle of numbered disks as shown on the left of Figure 5, and we want to move the disks so that this triangle transforms into what is shown on the right of Figure 5. What is the least number of disks that need to be moved to achieve this transformation? Can you predict the number of disks you will need to move for larger triangles? In this problem, students were primarily communi cating through drawings in a way that would have been difficult to communicate about strictly verbal ly or textually. Therefore, a collaborative space

This problem in particular was advantaged by the online medium because it allowed students to drag

Figure 3: Example of student work on a Jamboard with an object - oriented setup

Virginia Mathematics Teacher vol. 47, no. 1

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