ColinsMum, could you recommend some sources of problems in addition to Alcumus. We've almost never had DS7 solving (non-routine) problems at all, yet, as we were just getting him through the basic K-5 maths so he'd have some basic knowledge, but now he desperately needs to be challenged.

You've had some good answers while I was away, but here's mine anyway. We've mostly used past papers from maths competitions. Early on the ones which are multiple choice worked much better than anything where he had to "show his working"; that's less of an issue now, and ymmv anyway. The three families of UK contests we've used are:

The Mathematical Association's Primary Maths Challenge, aimed at children up to the age of 11. A few papers are available

here (February is the follow-on round, so harder than November) and there are also books, usefully sorted into challenge levels, e.g.

this.

The UKMT ones - very usefully wide range, starting from the Junior Maths Challenge, aimed at children up to 13. One of each paper available from their website

here, and again you can buy collections. The mentoring scheme papers,

here, are also worth knowing about (perhaps for the future more than now, or if not I'm seriously impressed!)

The Scottish Mathematical Council ones, see

here. In the actual competitions, these require full written solutions.

School have used scholarship papers (for entry to senior schools at 13, e.g.

these or

these) too. Unfortunately the ones that are on the web are the exception.

Another question, what's your approach to assessing "mathematical maturity" and readiness for certain mathematical activities? For example, I haven't mentioned at all to DS7 about theorems and proofs, since he's not ready for that. He can understand and explain things, but I'm just happy for him to think about things without worrying about rigor at this time. Know what I mean? What other stages of "mathematical maturity" should I be thinking about?

Interesting question, not least because it isn't something I've been in the habit of thinking about at all!

I suppose I suck it and see. I can think of three occasions when he's worked at something that, with hindsight, he wasn't ready for:

- Probably the first time I ever saw him have what looked like conceptual difficulty with something was when, at 7, he came to a section on cumulative frequency graphs in the textbook he was working through in class. He had trouble getting what it was they were showing and how. He could do the questions mechanically, but didn't like it and made weird mistakes. I helped him through the material in the book's (short) section as best I could, and didn't linger, because it really felt like something he wasn't ready to learn yet, rather than like something where he was having trouble that would be productive to work on. This - and especially the fact that I don't really understand what the problem was (given that he has had no problem with any other kind of graph or data handling) - is probably why I've been holding off on him doing more stats, actually.

- That same year he did the (now defunct) ALEKS course on high school chemistry. He'd been desperate to do this for a while, and I'd insisted on his getting at least to the end of ALEKS's grade 6 maths before he did so. I hadn't realised, though, how far the course would take him, or I might have tried harder to put him off starting it! It started fully appropriate, but got much harder towards the end. I did suggest fairly forcefully that he stop and come back when he was older. He insisted on finishing the course, but there was a lot that didn't stick. (He's now doing their introductory college chemistry, which has a lot of the same material, and it's a different story entirely.)

- The other thing he'd been very keen on learning for years before I let him at it was calculus. (I had a short thread

here, actually.) I subscribed to the eIMACS course and put it on the list of things I let him pick from to do on the bus (we have a long bus journey together to school in the mornings and that's where he gets most of his mathematical input from me!) He started it with great enthusiasm and did a chapter on limits (that was useful, I think) and one or two chapters on differentiation, but just gradually asked for it less and less often. The year's subscription lapsed without his getting to integration and I didn't renew it. In this case, it wasn't really that anything was hard - it was more that he'd satisfied his curiosity, perhaps. I didn't attempt to keep him interested, because I thought it'd go better after he had more trigonometry, anyway.

[ETA and to answer the later question about eIMACS, I thought this course was OK, but not great. Understandably, not many of the exercises were automatically graded, so he definitely needed someone there to check he was understanding; well, there are answers available, but since other correct answers are possible, that's of limited use. Pedagogically it was not bad, but there was a fair bit of "in the AP exam you can expect" to be ignored.]

I suppose my general attitude is that he has time, and it's fine to be guided by his interest. If he isn't ready for something, he can leave it for later, no problem, or if he wants to have a go anyway but it doesn't stick, he can do it again later, equally no problem.

With DS I wouldn't, incidentally, have picked proofs as an area to delay for mathematical maturity reasons - he's been very interested in proof, and proof theory, for a long time (there's a resource I would love to recommend, but can't till it's finished and published!). Writing proofs longhand has been another matter as until recently handwriting was a problem; I did have to push back against a teacher who wanted him to focus strongly on "showing his working" for this reason.

Another question, anyone know of a good resource (especially online) for learning very basic logic (and, or, not, quantifiers) and the same for set theory. These topics are totally absent from the school curriculum, so this void needs to be filled.

As kcab said,

Language Proof and Logic - for the software (more than just Tarski's World here, though that's good), more than for the book, which obviously isn't written to be appealing to children and isn't particularly so to DS. And this other resource that I can't point you at yet!

HTH!