One of the benefits of Agile is that each team can determine what works best for them, and go with it.
However: should you take notes? No; in my experience, the minute a team decides to start backtracking on some of the core principles such as:
- individuals and interactions over processes and tools (documented scrums that begin to be non-scrum-like)
- working software over comprehensive documentation (scrum minutes one day, who knows what would be next)
then that team will begin to move away from Agile and toward a more documentation-heavy and low-velocity approach to work.
In Martin Fowler's "It's Not Just Standing Up", there's nary a mention of note-taking or documenting the "minutes" of a stand-up meeting. All you should take away from that stand-up meeting are GIFTS:
- To help start the day well
- To support improvement
- To reinforce focus on the right things
- To reinforce the sense of team
- To communicate what is going on
As a mnemonic device, think of GIFTS:
Good Start, Improvement, Focus, Team, Status
However, if someone has a blocker that you need to help him or her work through and you take a note on what they are saying, that's totally different. -- knock yourself out with that.
As a point of reference, I'm a Product Owner and am mentoring the ScrumMaster right now in my company, and of all the Agile meetings we have (scrum, sprint planning, sprint review, sprint retrospective), the only one we take official minutes of is the retrospective, because that gives the team something concrete to work toward and refer to in the next sprint (and those "minutes" are a couple sets of short bullet points).