It sounds like you are already implementing your own behavioral modification program at home with the rewards and the consequences.
Behavior is generally purposeful. If he is not acting out like this at home, there must be something triggering him at school. He is only 5 so verbalizing what that is would be difficult for him. Also, at that age consequences/rewards will have to take place in a timely manner. Being consequented for something that took place while at school when he returns home, is not necessarily timely. He may not connect his school behaviors with those at home.
Some possible things that may be causing his acting out are: getting tired too easily/cranky/short term span, being bullied or being mistreated by others, not getting the attention he needs from his teacher(s), possible anxiety.
Try to find out if something takes place prior to his outbursts (you would want them to give you some examples of what had been going on, how he responded to prompts, etc) There must be consistency in his discipline there (if he is disciplined by different teachers/staff, they all must be consistent in their approach)