Potty training varies from kid to kid, so there isn't one right way. Sounds like he's doing pretty good! IMO, I would let him go w/o the diapers when you go out. You might take a quick trip to the grocery store (or other store) and show him where the bathrooms are when he has to go. Don't be surprised if he doesn't like the flush though. (They ARE very loud and can be scary to a young kid!) Then go back home and just make your trips a little longer each time. Carry a potty chair with you in your car/van if you can too. That helps in case of emergencies.

(Don't forget some wipes and a few bottles of water to wipe out the chair!

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I also agree with your grandmother to keep the diapers off him during the day if you can help it. Just put them on at night until he shows that he can keep it dry. (That might take much longer. Usually docs don't get concerned until the boys have hit around 6-7 yrs old.) Otherwise, yer doing fine and I hope he continues to do good!