I have heard these questions so many times and felt it was time to respond. First of all, I don't understand why this is so hard to understand? The people who go to college for teaching are not really learning EVERYTHING in EVERY SUBJECT. They each have their weak subjects just like the rest of us. Also, they spend more time learning how to manage a classroom FULL of students.
I only have my children to teach -- not a classroom full of students. Therefore, my children receive A LOT MORE INDIVIDUAL, PERSONAL ATTENTION. If any of my children struggle with anything I can give them aid and time to learn it or I may become aware that they may need something more. My youngest son had two years of speech classes and later, a year and a half of Vision Therapy. Had he been in a public or private school, he most likely would have been mis-diagnosed and would have been medicated instead of receiving the vision therapy. Many people do not know much about vision therapy yet. It is not related to "eyesight". It has to do with how the two eyes work together. If both eyes do not work together as they should, it can cause printing on a page in a book to go in and out of focus or to look like they are moving. It may cause other issues as well, and it affects the child's behavior due to their frustration or headaches, and they are unable to tell what the problem is because they don't know it's not normal for their eyes to do whatever they are doing that is causing the problem.
Also, no one knows my children as well as my husband and I do. I know their strenghths, weaknesses and interests. I can allow them to pursue their interests mor in depth than they would be able in a public or private school setting. I can let them work ahead in the things they excel in and give them the extra time and aid they need in the things that are more difficult and challenging for them.
Most importantly, my husband and I can teach them our morals, Christian beliefs and lifestyle and know that they will not be constantly exposed to morals, beliefs and lifestyles that we do not agree or approve with. We do not hide them in a closet or away from the world. They have plenty of interaction with others in church, in the community and in sports programs, but we get to choose what to expose them to and when it is appropriate to do so. We also get to decide how to present the information.
Finally, we are also not just given complete freedom in our homeschooling. We are held accountable. Here in the state of Pennsylvania, we have some of the most stringent homeschool laws in the United States. We must report to our local school district annually, and we must have a certified school teacher evaluate our children's progress at the end of each school year. We go to see our evaluator once a year and must present him/her with samples of our children's work from beginning, middle and end of the year to show their progress. We must also present the evaluator with a log showing that we completed at least 180 days of schooling. (We do not get to log less, even for illness). Our evaluator also talks to the children and asks them questions about things that she sees that they have studied over the year.
So, we are not just homeschooling and flying by the seat of our pants, and our children receive a good education. I will show more evidence of this in future posts, one such example is my weekly post of "Famous Homeschoolers".