I have 5 children... they are all very bright. My oldest is the only one formally tested and is HG, and is recognized as gifted especially for language/writing. My second son hasn't been tested (he is a montessori school that doesn't really have a talented and gifted program, but does a pretty good job of meeting his needs, at least on the surface.
But I suspect that second son is possibly HG in mathematics, especially. He just knows stuff, intuits ideas and concepts and relationships between numbers and spaces.
What do I do with this child? I mean, how do I find out what level he is really at and how do I advocate for him at school in a montessori environment which is good for many things.
He teacher tells me that he often sits and doesn't appear to be doing anything for over an hour... and she knows that stuff is going on inside, but doesn't quite know what to do with him....
My kids go to public school. In my state, I can petition for a subject acceleration, and the school does achievement and cognitive testing to assess. The best way to advocate on behalf of a child is with solid data to back it up.
Read Developing Math Talent. Its recommendations are based on the EXPLORE test and most of the advice is for upper elementary school or older, but I found it a useful read as we were in the throws of my then 2nd grade daughter's accaleration due to boredom.
I have a 9-year-old who is probably gifted and enjoys mathematics. I'm not sure I have much helpful advice for you, though, because her educational situation is quite different from your ds's. She's unschooled, and just leads her own education, having free range over whatever resources she can find, or whatever we happen to have around. She is almost never systematic about working in a prescribed sequence, or completing an exercise or program before moving on. Led by interest she has dabbled in areas way out of the "normal" order. For the past six months or so she's done hardly any math bookwork, and that's been fine. She's grazed a bit through Ed Zaccarro's "Challenge Math" book, dabbling in probability and statistics, and and reviewing basic K-7 arithmetic ... but often goes a month or so without touching any of it. She started an art class a month ago, though, and they're doing work with spirals and mandalas, and she's got hooked on the math-art connection. She spends hours watching and then playing around with Vi Hart videos, working out Fibonacci tricks, binary trees, making hexaflexagons, and so on.
So my dd's math education has been largely interest-led, full of rabbit trails and wormholes. Same with my older kids, who all entered school at least a year or two ahead in math despite the relative lack of systematic bookwork or curriculum. Not at all Montessori-like, at least from what I know. (I honestly don't know what Montessori math looks like beyond the 3rd or 4th grade level, so I'm not sure what to suggest.)
We did use Singapore Primary Math in a loose way during the early years. Depending on what level your ds is at and what learning style he prefers, it might be suited to self-paced independent study use in the classroom. It's big on concepts, and big on mental math, and presents fairly complex word problems at early stages. It's low on repetition, and my kids managed to work through it with very little adult guidance. It only goes up to a 6th/7th grade level, though -- and it doesn't include much that's outside the [narrow] scope of a traditional K-7 curriculum.
Hopefully the book Geofizz suggested will be more help to you.
Mountain mama to three great kids and one great grown-up
Subject acceleration has worked well for us-currently ds is on track to complete 2, possibly 3 years above grade level in math. What is really working out though is using some on line resources, as well as having the opportunity to go "deeper" into some math concepts-have some fum, explore, be challenged. He also does some enrichment above grade level in terms of building, design, thinking creatively. It took a while to get here, but not meeting those needs for advancement was something that really needed to happen.
You might ask the teachers/school psychologist fpr some assistance.
Another vote for deeper (and less curricular) math. If faster also happens (child learning multiplying wants to play around with positive and negative exponents and exponents and lattice multiplying), great!
Several favorite resources that might lend themselves to supplementing: Historical Connections in Mathematics, GEMS books like Math Around the World, Math the Human Endeavor. Many great reads at the living math website. Geometry, number patterns (Fibonacci, triangle numbers), function machines, binary, code, probability seem to be big hits at that age. Some kids like Life of Fred.
If she simply wants to learn concepts, several resources that seem easy to fly through as needed are Khan Academy and perhaps the Key to series.
Yes, this is the crux of what has worked so well for my dd ... being encouraged to follow tangents laterally, rather than using forward speed through a sequential curriculum to challenge her. Often forward progress is a side effect of these lateral forays, but overall the focus is on building an incredibly robust foundation in mathematical understanding rather than on heading for college by age 12.
Mountain mama to three great kids and one great grown-up
One suggestion that I haven't seen yet is to try math competitions - mathletes kind of stuff. I'm not sure what is available for that age but it may provide him with a peer group. It also depends on how the competitive stuff is handled and how he responds to that kind of environment. My kids participated in various math competitions in late elementary/middle school years through their schools. The competitive stuff was kept pretty low key and mostly the teachers used the resources to fill out the math curriculum. The actual competitions weren't a big deal - at least for most of the kids. There's always a few who get wrapped up in the competitive aspect.
he is a weird one with competitive environments! he does lego league and the sporting part of it kind of stresses him out a bit. But I think the like mindedness, being with other kids who love math, on a team, would be really good for him.
I had a meeting with his teacher for parent teacher conferences... and wow. He is in a montessori age 6-9 nine year-olds class, and is in 3rd year, but still 8. He has completed almost all the works on the math shelf (the 2nd math shelf, not the basic one). Montessori math basically has all 1-6th grade math available to the kids, and kind of teaches it all at the same time. He is highly asynchronous-- for instance he hasn't memorized all the multiplication tables (but does intuit them rather well-- but he still thinks about it, it isn't rote) but he can rock the cubing work and is totally fluid with the algebraic binomial and trinomial. He also does racks and tubes since the beginning of last year... and will be walking to the upper el (which has some 7th grade works) to work there...
So it is super cool that he is working at 6th+ grade in mathematics. Not globally, but pretty soon. I just wonder if montessori math will translate to "tests and normal standard math."
I am glad that he is able to work on math as much as he wants and seems to be challenged, and able to work in a challenging way. I am hoping going to the other class for a bit will allow him to work with peers.