The school asked for my ds1 to be seen by a psychologist because they could see that his obvious intelligence was not matched by what he was producing in the classroom. He was absolutely fine in one-to-one sessions: chatting away, answering everything, writing stories, enjoying talking about maths problems etc, but in the classroom would seem incapable of doing anything unless constantly pushed and guided. He was also pretty silent in small groups for the more able children. Part of this appears to have been sheer anxiety, since once he got over his inability to express his needs to anyone, he became a lot more confident and proactive in class time.
The point of seeing a psychologist is to help work out what it is that is holding a child back - it could be a whole number of things, from sensory problems (they can't cope with the excessive stimulus of having lots of other children around them), an autistic spectrum disorder, anxiety, poor attention, boredom, physical problems etc, etc... If a school is not convinced it can overcome whatever is holding the child back in a school setting using the normal strategies it has in place (and this in itself may take time with some children, as they need to be seen by the SEN co-ordinator, have one-to-one time, have IEPs put in place, etc), but agrees that something is holding the child back, then the next step would be seeking outside help and advice. It is unlikely to be simple boredom that stops a child from reading all the words in a book properly or refusing to try something he is worried he might not be able to do properly. Sometimes children who behave in this way grow out of it with the right support and sometimes it turns out there is something going on in the background that they need a bit (or even a lot) of help with. So, if the school isn't willing to investigate what the problem is, then maybe you need to start pushing for them to take his behaviour a bit more seriously (and not in terms of saying you think he's just bored, but asking for help in finding out why your ds is not able to produce work in the classroom that he has no trouble in producing at home).