It would be great if we could limit the live chat function to specific groups of students (e.g., 4-5 students working on a group exercise together), as well as to specific sections.
email@example.com, we've opened this idea for voting. In the meantime, consider using the Conferences feature to hold a chat with selected students. You'll find a comprehensive list of resources for Conferences in https://community.canvaslms.com/docs/DOC-18572-conferences-resources
I believe this fits closely with your idea. I would like the ability to turn off the public chat and shared notes options during conferences. As a 5th grade teacher my students love to use this feature to be distracting. I'm sure with so many elementary teachers using this feature for the first time that many would agree.
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