From my research, it seems pretty clear: Educators should expect parent involvement to be fundamentally different from school instruction.
Then I read this load of crap from the U.S. Department of Education (via KTM II):
Try to be aware of how your child is being taught math, and don't teach strategies and shortcuts that conflict with the approach the teacher is using. Check in with the teacher and ask what you can do to help. Ask the teacher about online resources that you can use with your child at home.
In case anyone (like the DOE) needs the obvious stated, parents have been successfully passing on low-order and high-order cognitive skills to their offspring long before there were schools, lesson plans, professional development seminars, research grants, or instructional philosophies.
This fact, in and of itself, makes the DOE's recommendation sound annoyingly pompous. But there's another point here. Parenting behavior—specifically as it relates to learning—is socially malleable. But it is not completely a product of social forces. It has evolutionary roots. In other words, there are certain characteristics that are common to ALL parenting at ALL socioeconomic levels and in ALL cultures. (I'll happily make an exception for the Bongo Bongo tribe though.) Schools and classroom instruction, on the other hand, are social creations. We had to invent those things. Therefore:
If a school wishes to engage parents in academic involvement, then it is the school's burden to provide ways for that involvement to occur by meeting parents' needs. And it is the school's job to create parent involvement activities and programs that fit with the way parents are naturally involved with their children.
There are three general characteristics of natural parent involvement, but I'll have to get to those in another post.