At the two-week camp, the boys spent their time dressed in a military-style uniform and were trained by officers with the Hamas National Guard and militants with Hamas' armed wing, the al-Qassam Brigades.
Mohamed said that as well as karate and other street-fighting techniques, they were taught how to throw hand grenades.
Hamas has denied that live weaponry is used in the course or that militants are involved in the training.
But Samar Zakout, of the Gaza-based human rights organisation Al Mezan, described it as "unbelievable" and likely to encourage Israel to see schools as targets during conflicts.
Mohamed Syam heads the Education Ministry in charge of the Futuwwa programme, which may be extended to girls' schools next year.
"We are not conducting military training in our schools, we are providing information," he said.
Yet an article written in Arabic on the Hamas Ministry of Education website credits the al-Qassam Brigades for its contribution to the course and notes their attendance at a graduation ceremony, also attended by Syam.
A YouTube clip showing a military demonstration in a Gaza school also appears to contradict the Hamas official line.
Posted on April 5, the video shows a mock Israeli military post erected in a school playground, where Palestinian militants enact a mock battle during which a supposed Israeli soldier is killed and another captured.
Syam said the video, which he claimed not to have seen, was not representative of the new initiative, and that the training course was designed mainly to school Gaza's youth in discipline and respect.
"The military aspect takes up only 1 per cent of the course," he said.
"On the other side, in the occupied territory, [Israelis] are teaching the youth to hate and to kill Arabs and are training their teenagers in the military."
Israeli teenagers are drafted into an obligatory three-year military service after graduating from secondary school.
There is no official Palestinian army.