I also put my sons into martial arts. People will tell you that whatever style they are studying is the best one. It's better if you visit each school, talk to the instructors, the students, the parents of the students. Watch a few classes in each dojo. With children as young as yours are, you need to see if the class is structured to hold their attention, if what is being taught is in their physical abilitiy (motor skills are a bit better when the children are older, and you don't want them to start out frustrated).
Two things to watch out for, never sign a contract that says you have to pay even if your children don't come to class, and guarantees that they will have their black belt in a short period of time. If you hear either, walk out. Ranks should be given on ability, not on the calender. Good luck. PS, I took karate with my sons, started when I was 38, and loved it!!!
I'm sorry about this, but I also have to disagree with a previous post. It's obvious that he has not taken a karate class, and probably has never seen one. I hold the rank of nikyu, brown belt, 2nd class. I have been taught the moves that he was talking about, one finger jabs and two finger jabs to eyes, elbows to many different parts of the body, knee to groin, groin grabs, many kicks below the waist. I have also been taught the vital areas and the fatal areas. My senseis teach not only the moves but where they can go, what they will do, and how much damage they will inflict. You must learn the difference between pushing someone away, maiming them, or killing them. You don't want to kill a drunk in a bar that grabbed your arm, it's enough to push him off. I have learned anatomy, and how the body moves, the physics of the blows. I do agree that the high spinning kicks are very showy, but not a lot of good. Basic physics. Too much energy going up, and not enough going forward, depletes the power of the kick. He needs to apologize to all of the martial artists that he has disrespected because of his lack of knowledge. (If he had been taught in a traditional dojo, he would know not to put down another's teaching or sensei)