We use a forked stick. You flip over a rock underwater and when the silt settles, sneak up behind it and capture it in the fork. Actually I use my fingers, the same as a stick. Trick is, if you get the fingers too far forward on the carapace, you are in pinching range. Ouch. Splash.
I think you can catch em with a line and a piece of bait,that they grab hold of, and flip them into a bucket if you're fast, I never had much luck doing it that way. But maybe that's what the song is about, really. An excuse.
There's also a lovely version of Froggy's Courting with a chorus:
"Here's to Cheshire, here's to cheese
Here's to the pears and the apple trees,
Here's to the lovely stawberries,
Ding, dang, dong go the wedding bells."

I don't see it in DT, but it's in Rise Up Singing.
Blessings,
Barbara