Question: Do you give each team member a user story
A see an anti-pattern here. As a Scrum Master, you don't have to assign stories & tasks to team members. It's their job (under your servant leadership) to self organize and pick sprint backlog items. They can use swarming techniques, they can pair program, they can breakdown technical tasks into granular level to work as a team and deliver their commitment - the Sprint.
plan according to the velocity and then the team should work together to deliver a potentially releasable sprint rather than a half completed sprint?
This is better. Remember that the estimates are a guess after all. So velocity is a more accurate measure against which you can plan your upcoming sprint. Don't fall into the trap of ensuring that everyone is 100% utilized. Instead focus on 100% optimization. In team retrospective meetings, the current velocity can also come under discussion. Does the team think that their velocity is low? Maybe they have become efficient enough so that the velocity can be go up by planning for more stories in the next sprint.
On a related note, there are two views on assigning tasks to individual team members during the sprint planning meeting. One argument is to have names against all sprint tasks when you end the sprint planning meeting. The second point of view is, task assignment should not be done in the sprint planning meeting. Let the team pick tasks as they go, with sprint completion as a common objective.
Mike Cohn mentions in his blog post:
If a team walks out of sprint planning with a name next to every task,
individual accountability will definitely be increased. I will feel
more responsibility to finish the tasks with my name or initials next
to them. And you will feel the same for those with yours. But, this
will come at the expense of team accountability.
My recommendation is that a team should leave sprint planning without
having put names on tasks. Following a real-time sign-up strategy will
allow more flexibility during the sprint.
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