[jarvist]

Practically speaking (running a little maths circle in the UK for my children, and some of their friends from nursery and primary school), I have found the Nrich website to be the single best source of resources: https://nrich.maths.org/about-nrich

There is also https://parallel.org.uk/ by Simon Singh, but this is aimed at ages ~10+.

The book by Rozhkovskaya has some really nice activities in it. https://www.amazon.co.uk/gp/product/1470416956

The book by Zvonkin described in the article is a very good motivator, particularly for the honest descriptions of lessons gone badly wrong, and staying up late cutting out pieces of cardboard! But it's quite difficult to use as a teaching resource.

[hugetim]

A nice source of middle- and high-school aged problems: https://www.usamts.org/

One of the joys of high school was discussing these with a friend (after submission---but one could forego the contest and just do the circle thing with them for fun).

[dfeng]

I would love to hear more about your experiences running a maths circle here in the UK. My two daughters are a little young (3.5 and 0.5), but this article has inspired me to get the ball rolling.

[jarvist]

It's been really rewarding. I definitely recommend jumping in. I started with my reception-age child (+ school friends), and have just extended it to their younger sibling (+ friends from nursery). Your 3.5 year old will have started the EYFS (Early Years Foundational Stage) syllabus at nursery if they attend (which is also what they do in the 'Reception' year at primary school, before starting the national curriculum in the first year), so they will now be exposed to counting and comparisons. The perfect time to get started, in other words!

The NRICH material is really good: https://nrich.maths.org/teachers/early-years

There's some NRICH funded research that showed that exposure to symmetry and reasoning at this level was much more predictive of future abilities than numbers and counting. I think when parents try and help at the early stages, they often try to e.g. get their kids to count to 100, which is conceptually identical to counting to 10.

For number fluency there is the free White Rose '1 minute maths' app, which does a very nice job of gamifying subitising & etc. A lot of primary schools in London seem to have adopted the White Rose teaching resources. https://whiteroseeducation.com/1-minute-maths