Most potatoes produce flowers - when they do I fudge around in the ground and see if I can feel some potatoes, if I can and most are about the size of a hen's egg then I'll dig some up for a meal.
When digging put the fork in some way away from the plant and ease it up gently and hopefully you won't spear any spuds. Then when you think you've got all the potatoes turn the soil over again quite deeply and /or scrabble around with your hands, and you'll probably find some more potatoes hiding down there
Occasionally there are some varieties of potato that don't flower - not sure why - then they ought to be ready 10 weeks from planting - if the size of the potatoes is disappointing give them another week or so.
Oh, and the green things that look like tomatoes and that sometimes appear on the the potato plants after the flowers, are the seeds - make sure the children know they are not tomatoes as they are poisonous if eaten.