When my kid was three I encountered this lovely book and thought I'd read it to attempt to replicate something like it (despite all the warnings that it's a journal and not a guide); unfortunately I never got around to finishing the book (nothing against the book, just got distracted) and now the kid's already past seven :) But what little I read was delightful; thanks for posting this summary!
I have a similar experience. The book skips the first 20 weeks. Many of the activities require materials and I could have benefited from a more prescriptive curriculum. Some other math circle curricula are not as inspiring but are much easier to follow.
The book skips the first 20 weeks for the boys but covers those activities for the girls (around the end of the book). So I'd suggest reading the girl circle and then coming back to the boys circle... I've absolutely loved that book and I've been trying to incorporate some of the ideas in games with my son.
That said, do keep in mind that despite being from 3 to 7, his son is actually 3 years old and 10 months when he started (and I think the same for the daughter, I forgot). Personally, I've noticed that my just turned 3 years old son is not developmentally ready for a lot of the activities yet (but he's been surprisingly good at other activities), so I do think that you need to adapt depending on the children you try this on.
A couple of other books that I think are useful and easier to execute than Zvonkin's books are the books by Christopher Danielson. Which One Doesn't Belong? is great to play with a child and Talking Math with Your Kids has some good ideas (way too short though)
