At first glance the concern about socialization seems valid. First of all, we have an active family life and communicate freely with one another. The image of the children in the Laura Ingalls Wilder series rises to mind frequently, as well as all the pioneer and missionary children in the world. They tend to do well, even though they may only see other children who share their own values and culture infrequently. The concern about our ability to educate our children is also valid, but the history of home schooling shows the home to often be a healthier learning environment than the classroom.
Every parent is a natural teacher.
However, parents cannot teach all things. We, personally, certainly cannot do all of it. For math it has been the same. Of the more involved academic skills, we cannot really say from personal experience since our oldest child just turned twelve this year, but home school literature shows that home schoolers in general achieve their academic goals using a variety of resources other than the public school classroom that are available to the average family. There are the books available in nearly every subject.