How did the spartans educate there children?

Spartans believed in a life of 'discipline, self denial, and simplicity,' and so the purpose of education was, simply, to produce an army. When babies were born, soldiers came to check the child. If it appeared healthy and strong, they would be assigned to a 'brotherhood' or a 'sisterhood,' however if the baby appeared weak and small, the infant would be left to die on a hillside or taken away to be trained as a slave. It was 'survival of the fittest' in Ancient Sparta.

Boys
Male Spartan children were sent to military school at the age of six or seven. They lived with their brotherhood.School courses were very hard and painful for boys, and school was described as a 'brutal training period.'
Between the age of 18 and 20, Spartan males had to pass a fitness test that consisted of fitness, military ability, and leadership skills. If he didn't pass, he becams a person who had no political rights and was not even considered a citizen called a perioidos. If he did pass, he would continue to serve in the military and train as a soldier until he was 60, when the soldier could retire to live with his family.

Girls
Girls were trained in their sisterhood, and were taught physical education. They also started school at the age or six or seven. It is unknown as to whether their school was as rough and hard as the boys', but some historians believe the two schools were very similar in their objectives, to produce a strong group of women. 
At age 18, the Spartan girl also had to pass a fitness test. If she passed, a husband would be assigned to her, and she would be allowed to go home, however if she failed she would also become a perioidos. A woman in Sparta things were very different for citizen women than they were in other Greek cities, where women would stay home most of their lives and be controlled by their husband. In Sparta, women had a lot of free will and were almost as good fighters as the men.