Niadalla, I don't know if that's such an easy question. I think for many parents they "start" because there is something wrong - many gifted kids have ASD traits, and/or ADD traits, some have physical things going on - for example many gifted kids have low muscle tone - (as do many ASD kids) and this makes sitting still really hard for them - there are the behaviours that go along with being bored/frustrated/sad because they feel different and don't know why and can't connect with their peers, and with lack of mental stimulation etc - defiance, acting out, hitting out for some and more. So I think for many of us it's the other things going on because of the giftedness that starts us on the path of finding out what is going on. Many parents will know that they looking for giftedness when they start, others, like us, start out looking for a reason for the behaviour and not knowing what they will find. That is not to say all gifted kids have other things going on. And often, particularly with younger siblings of gifted kids, the giftedness is not picked up until much later, or even at all.
I think that if there are signs that a child is gifted and that the child needs more than they are getting, or will need more, then seeing a psychologist who specialises in identifying gifted kids would be a good start (not cheap though!). I wouldn't necessarily start with a full assessment (really, really expensive!), but a consultation would give you an idea at least. If your child seems really smart, but is happy and doesn't have anything else going on that is a sign that things aren't quite "right", then I probably wouldn't do anything until they are nearly at school at least. Then I'm not sure - either a test so you know one way or the other, or see how the child goes at school. There are some questionnaires around the place that parents can answer to see if it is likely, and sites like the QAGTC have lists of signs that are common in gifted kids and that can be helpful too.
In your case, there do sound like some signs. The large vocab can be a sign, the early maths ability is also a sign. Family history is usually pretty strong too (although many parents of gifted kids were never as identified as gifted because it really wasn't understood well and was rarely identified when we were kids, looking back often parents can see the signs in themselves). The lack of concentration is a sign, and the not sitting still sounds like it could be from lack of concentration or also from low muscle tone which as I said does seem to be more common in gifted kids. Another really big clue for parents is what they are hearing from others - kindy teachers generally recognise "difference", and also what your friends say - other parents also recognise when kids are advanced in some way. As you say, with your first child, your benchmark about what kids do at an age is what your child is doing. So listening to others when they say "that's not what kids this age do" is helpful.
I guess your next step is to decide whether he needs more now, or whether he seems happy now, but it's something to keep an eye on when he starts school. Also, if you suspect the low muscle tone and want to look into that further, a physiotherapist can do an assessment.
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