Oh, for the days of the sleeping babe in the sling. It must have driven more seasoned parents mad when I waxed lyrical about how easy, how special, it was to bring my son with me to a gallery or museum. Now he’s a toddler who can change direction like a squirrel and is rather taken with the word “no”. And I’m beginning to wonder: should we be sticking to exhibitions and events targeted at children?

I’m not a joiner, and the thought of stay-and-plays and singalongs is enough to send me running. And yet, here I am, parking the buggy, unbuckling my son and walking with him, with some trepidation, towards a “family-friendly drop-in” at the Fitzwilliam Museum in Cambridge.

Five stars, he cries! Or he would if he could, ogling a knight on a horse, gleaming in silver armour. I gently coax him into the next gallery – Arts of the Near East – where we find a few other parent-child duos and a couple of smiley staff members. I brace myself, anticipating instruction, but it quickly becomes clear that these events – free to attend; no need to book – are child-led and oh-so-loosely structured (to my relief, no organised fun). There’s a table with colouring pencils and paper. Big, squashy cushions and stripy rugs. Books and building blocks, musical instruments and gauzy fabrics, which the children are welcome to play with, or not.

When I think of kiddy-specific sessions in museums, I picture cordoned-off spaces far away from the art and the adults appreciating it. Special zones where children can be noisy and run around and show their enthusiasm. Not so at the Fitzwilliam. Surrounding us in the gallery are glazed cabinets packed with ceramics, pottery, glassware and metalwork. The space we’re in opens on to another, and another, and another, each one lined with floor-to-ceiling displays. With a maraca in each hand, my son wanders freely between the rooms, delighting in the ceramic dogs, horses, deer.

“We’re absolutely committed to welcoming young children, but we recognise that museums aren’t built for them, they’re built for adults,” says Kate Noble, assistant research professor in museum participation and practice. Which is why they pay attention to placing objects at child level and providing activities that are rooted in the collection. At the drop-in are stencils for the children to make tiles inspired by those on display (free rein scribbling is also encouraged) and foam blocks that echo the decorative abstract shapes.

“Part of our argument is that you don’t necessarily need to put on a special exhibition for children, because children are really interested in the same kind of art that adults are interested in,” says Noble. “They might see it and understand it in different ways, and obviously there are some objects which speak more readily to them.”

Crouching in front of a tureen resembling a carp – “shark!” my son yells – it occurs to me that I probably wouldn’t have noticed this bizarre table setting (or the crayfish salt pig beside it) were it not for him. That if he hadn’t shaken his maracas at a cabinet of plates, I wouldn’t have picked out the rabbits snuffling among the crockery.

Michael Corley, deputy director of learning and public programmes, tells me that he recently saw a child looking at a portrait while playing with a swatch of fabric from a trolley filled with resources for families to use in the museum. “I genuinely think it’s one of the longest times I’ve seen anybody in the museum observe a painting,” he says. “They were there for minutes, just gradually, consistently staring at the portrait while adjusting the material without looking down. I don’t think we could have ever designed that. It’s just about giving enough to a child to play, to explore, and they’ll do that well.”

To my surprise, we – and I mean we – are having so much fun at the drop-in that I almost forget about my plan to slink off early and check out the exhibition on ancient Egypt upstairs. When we finally emerge from the lift, I’m glad we’ve made the effort. The staff here are friendly, too: a woman takes one look at my son and offers to show him a dog on a 3,300-year-old papyrus. There are tomb paintings and mummy cases. It’s beautiful and fascinating. But the feeling of freedom in the downstairs galleries is hard to beat, and it’s not long before my child is in the carrier. When he pronounces our visit “all done”, I’m with him.

The monthly family-friendly drop-in was born out of research that Noble and former colleague Nicola Wallis have been conducting for almost a decade. “If you weren’t brought to a museum as a child, it can be really intimidating to do that as a parent, when you’re learning about your own child and about how to be out and about with them in public,” Noble says. “This session is just an introduction to being in a museum with your child. It’s about giving people a different route in to developing that confidence.”

Don’t expect to find us solely at child-focused sessions from now on, but this experience has made me more open to them. Oh, and if I had one complaint, it would be that the drop-in is dangerously close to the shop. You try steering a squirrel away from brightly coloured books and toys and crayons.

Three more kiddy-specific sessions within gallery spaces

Toddle Tours at MK Gallery, Milton Keynes

Mini Masterpieces at Dulwich Picture Gallery, London