My oldest son has ADHD and showed signs in utero. Seriously.
I couldn't take stairs ever after the 1st trimester, because my center of gravity would shift so quickly and dramatically that I would lose my balance. Even when he was no bigger than a bean in there, when I was supposed to feel like "a butterfly landed on me", I felt like someone was playing drums inside me. At the end of my pregnancy, it felt like he was trying to beat his way out. He was so active in utero, I was worried he was having seizures inside me (he does not have seizures now EVER, so it's unlikely he was having them in utero - he was just going to town). He's a great reader now, and though he starts out sitting, he can end up on his head...reading.
When I was pregnant with my youngest son and doctors commented that he was "very active," I'd laugh - because compared to my oldest, he was Zen baby in there. So...I'd say it shows up VERY early. I'd also suggest you take determining what's up and what to do about it VERY SLOWLY.
ADHD rarely shows up alone - it can be co-morbid with OCD, ODD, and endocrine issues. So therapy is crucial. Some doctors prescribe meds without trying therapy - when in fact, many people CAN learn to moderate and manage ADHD without meds. So it sounds to me like your psychiatrist is doing the right thing - she's getting to know your son before evaluating for ADD/ADHD, and she's not in a hurry to medicate.
Our oldest son is on Intuniv. But he started after 3 months of intense therapy for OCD and ADHD related behavioral issues. I am at ease that the meds help and are the right thing. But our regular doctor wanted to start him on the usual meds (not Intuniv) right away, with little info other than the "form evaluation" - and his OCD hadn't ramped up yet.
The usual meds would have made the OCD MUCH MUCH worse. So, that would have been a TERRIBLE thing to do to him. They also would have counter-acted endocrine therapy he requires. So...yes, I suggest you go SLOWLY in getting a diagnosis and deciding what to do.
My son beat the OCD with therapy, though we have to vigilant about it. The ADHD no longer seems like a big deal. He'll probably be able to drop the meds eventually.
Just...your son is 5, so take your time. Yes, ADHD is real, but your son's care-givers don't see a problem - and I bet they are around lots of kids and have lots of experience. Teachers tend to be VERY good at spotting ADD/ADHD.
The psychiatrist knows your concerns. If your son has ADD, the psychiatrist will get there. If not, she might help you work through how to manage the issues you are having with him. I suggest talking to the therapist about how you might interact differently with your son, and for now, go from there.
Deep breaths, go slow, try to trust that the psychiatrist and doctors can help you work this out. Good luck!
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ETA: Consequence-based discipline/parenting-techniques do NOT work with kids with ADHD. "Cause and effect" wiring is...not the same for them.
I agree w/ the commenter below that hearing "he's a boy" gets to be...silly....but let's just say, it was NEVER teachers who said that about my son. No one who was with him on a daily basis thought his challenges were about "just being a boy".
ETA2: Looking at other comments, it also looks like our experiences are different, so I'd go back to going slowly and making sure what you do is right for you and your child.