Some children are more dependent on their parents than other. Some parents promote this dependence because of their own needs for power over their children. They use intimidation and bullying to remain bossy and overbearing in the lives of their children. Independence produces confidence and ability which gives children the opportunity to avail opportunities that come their way which gives them a better chance of survival and making it in on their own in the real world.
You must teach your children to solve problems for themselves. Instead of responding immediately (unless it is an emergency) stop to think if this is a problem that they could solve on his/her own.
Children have more skills than we give them credit for, but they usually try to take the easy way out by calling out to parents or someone older than them to do things for them instead of making an effort themselves. Children need encouragement and support when they are learning a new skill or doing something for the first time. Sometimes they need to do the thing a few times before they get the hang of it. At times like these, children need your time and patience. It might not be possible to spare time all that often, but when we do have the time do give it.
Your children will remember you more for that time than for anything else you might give to them. More than anything else, children are happier if you give them quality time. Children need to be taught words, what to say and when to say it. When I taught children to recite the alphabet, I did not expect them to learn to read words on their own. The language develops by hearing others talk. What the child hears is what the child is going to say. So, if you want to child to be more independent, the child must learn the social language.
The child must learn how to communicate with friends, or to tell the teacher that he/she did not understand something. Learning to solve their problems or conflicts will give children the needed words to speak with one another. Another way that your child can be independent is to make lists of things to be done and prioritize the work in order of importance and cross out what has been completed. This will give the child the capabilities of doing things himself and will make the child less dependent.