Hello, I'd like to help you with your question.
It can be unsettling when your child behaves in such a way that it causes concern. You want him to be able to fit in with the other children and do well in school. But when he is not talking to others outside of home, it can be troubling.
If your son is able to talk well at home and with those he feels comfortable with, then it is unlikely that he has any speech or language problems. However, if you feel it would help. consider talking to his doctor about screening your son for speech and language issues just in case and for your peace of mind.
Your son does not sound like an introvert, which is someone who prefers to be alone and feels energized when not around other people. You mentioned your son is chatty at home so most likely that is his normal nature.
Most likely, your son may just be shy. He feels more comfortable talking with people he knows and since that does not include school yet, he withdrawals while he is there. He may also feel overwhelmed by the number of children in his class. Some people do not do as well with crowds as others do.
What you can do is talk with his teachers about setting up a reward system. You mentioned trying this at home which is great, but it may be very effective at school where he could get rewarded each time he speaks out. Tell the teacher(s) of your concerns and find a reward that you both feel will motivate your son to speak during school. The teacher can use this system while your son is at school to give him a reward each time he talks no matter what he says. Also, ask the teacher to make a note to you each day as to whether or not your son spoke while at school. If he did
, then reward him at home as well, praising him verbally and with the reward. That way, he is getting reinforcement from you and his teachers.
Also, practice with your son talking to others he does not know as well. Take him with you as you talk with people, letting him be part of saying hello when you do and asking him to participate in the conversations by asking him questions and making comments while including him. When he sees how you handle talking with others, he may learn how to do it on his own.
I hope this has helped you,