Standards Based Assessment is a bit of a misnomer. Yes, it involves using standards to guide your assessment choices, but it does not dictate any particular assessment methods. You can still give regular homework assignments, quizzes and exams. Or you can use oral exams, projects, or play darts. (Well, you can play darts if one of your standards involves throwing darts.)
Standards Based Assessment is really a communication scheme. The basic idea is to clearly delineate the desired student learning outcomes (these are your standards), make these public, tie them explicitly and publicly to whatever assessments you use, and communicate with students about their progress on these outcomes in place of recording and reporting some sort of abstracted grade.
I am experimenting with using SBA in my courses these days. It feels like it should be a good thing. But I am a too far away from a reasonable amount of experience to say that it works great. I believe in the idea, and now I am trying to see if I can make it work.
If I can find some good references, I will post them here. For now, you can look at these slides from my part in a panel discussion at the 2013 Joint Mathematics Meetings, and you can run through some of the writing of Shawn Cornally. It was in reading Shawn’s stuff that I got excited to try this.