If I had to give this post a subtitle, it would be... "And Why the Schools Do It Wrong." It isn't because the teachers are bad (just as homeschooling parents aren't bad teachers), but it's the way things are done. The "system" doesn't work.
For example... Typically a kid learns a new concept in class, and is then sent home with practice problems to work. If he did every single problem wrong at home, he won't find out until the next day (if he's lucky), right before the teacher moves on to the next concept. This makes no sense.
The reason one on one education is superior to classroom teaching is because the teacher is available for immediate feedback. If a child doesn't do something correctly or misunderstands a concept, the teacher sees it, corrects it, and the child is back on the path. Well, usually...
Because a lot of parents use the same school system to teach their kids at home, their kids are not benefiting from the one one one instruction in the best way. If your child is doing 20 problems on his own and then you are grading it later, you have just wasted your child's time if he didn't work the problems correctly. And he has just re-enforced the incorrect method to solve the problems. I actually figured this out through working with my own kids. For one full year, I worked with both of my kids every day, with every problem. If they had 20 problems to work, we did them on the white board together. So if they got stumped, I was there to guide their thinking correctly. They were never able to "think wrong" and think they were right. I was able to stop and re-teach a concept if necessary, or explain it in a different way.
When my kids hit 6th and 7th grades, I decided to give Teaching Textbooks (TT) a try. I love teaching math, but it was getting overwhelming. The TT method works really well. They learn a concept, and then do some practice problems (as well as review problems). If they miss the practice problem, they know right away. Additionally, they can click a button to learn how to work a problem correctly. I love this.
If you use a different curriculum but have a lot of kids and not enough time, an alternative is to allow your kids to have the answers to their problems so they can self check as they go. You can't do this with younger kids, but you will know when your kid has reached the age where this will work.
The way to teach math is by providing immediate feedback, one problem at a time. Whether you do it, or the computer does it, or the answer sheet does it, a kid needs to know if he's on the right track WHILE he's on it. Following a system at home that doesn't work for the public schools is probably creating unnecessary frustration in your home.
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