At the end of third grade, our daughter's teacher approached us (and our school's principal) and suggested that with her test scores (high-school level) and level academic mastery, we should put her in a private school with a gifted and talented program (our public school system has, for I think good reasons, done away with that), or else have her skip a grade. She felt like our daughter's level was such that differentiation within the fourth-grade classroom wouldn't be sufficient.
We aren't really interested in private school, and were a little concerned about social issues, and eventually a deciding factor was that the ELA curriculum for fourth grade focuses on writing and composition — an area that's always been one of our daughter's rough spots. So, with the help of the principal and teachers, we came up with a plan where she divides her time between 4th grade (homeroom, ELA) and 5th (math).
This has worked reasonably well (she's continued to excel at math and has really caught up in the writing skills), but had some problems as well — the daily classroom switch was stressful, and since our state has mandatory testing, that was done at the 4th grade level for all subjects (she did just fine, but again, it was a stressor). So now, as this year comes to a close, we're evaluating what to do next year — whether to continue the split pattern, to move her entirely into 6th, or even to stay in 5th and work with the math teacher on more advanced / in-depth education this year. (The principal has indicated that all of these are options).
I've read What are the pros and cons of having an advanced student skip Kindergarten?, and found the answers helpful in general, but I think the situation at the start of the school system is different from that at later grades — see for example this answer.
I recognize (and again see the prior question) that social groupings might be a factor, but in our case, our daughter really only has a few close friends in her grade (and we'll continue to be friends with those families anyway), and already has friends in the higher grade as well (from the math class, and just from the neighborhood).
What are other factors we might consider, and particularly, ones that might be especially important at this grade level?