I use Saxon with my daughter. But I'm also really lazy and don't always check to see if they're introducing a new concept--so I just give her the worksheets, let her go to town, and then step in a 'teach' if it turns out there's something she can't handle.
On her own, she 'discovered' that 4*10 is just like 4 dimes, and that multiplying is really just adding over and over, really quickly. No tables, no group work, no guesstimating....
The same sort of thing has happened with the fractions in second grade Saxon, and with almost every concept that has come up. (I had to step in to explain the meanings of 'symmetry, oblique, horizontal and vertical' Everything else, she just figures out when she hits it.
In a sense, this is what 'discovery' math is supposed to be--kids teaching themselves and wrestling with new concepts until they understand them--- The problem is that Discovery Math in the classroom is often about 'fun!!!' instead of learning. However, I'm seeing that 'guide on the side' DOES work..
1. You have a very small number of students
2. The students are fairly bright
3. Your curriculum has a good, orderly structure.
My son has struggled all year with math -- his weekly math homework consists of a packet of worksheets, which we realized is where any and all direct instruction is occurring, in the form of me teaching him. He has also decided now in first grade that he "isn't good at math" and that "math is too hard."
I've just recently discovered, via Barak Rosenshine's article in American Educator, the idea of 'elaboration' in learning, and it makes perfect sense to me. In fact, it's more or less the 'missing piece' in my own thinking about all this.
Elaboration means (as I understand it) that we learn best when we .... manipulate or develop or grapple with the content we're trying to learn (remember) in some way.
That's what writing does for me: I learn via writing because it is through writing that I 'elaborate' on the content I'm trying to master.
That's also got to be what a good class discussion does; a good class discussion allows everyone to elaborate on the novel or poem or history text etc. they are studying.
"Elaboration" also goes a good long ways towards explaining why watching a video of a lecture doesn't work. In a real lecture students interrupt and ask questions or make remarks, or the lecturer self-interrupts and asks questions, or the lecturer simply sees from the looks on student's faces that he/she needs to take another tack, etc.
I'm thinking that progressive educators may have confused "explain your reasoning in words" with "elaborate upon the content you are trying to learn."
Because we all speak English, for most of us 'elaboration' is probably verbal -- or, at least, language is the medium in which we engage in shared elaboration (via class discussion, writing papers, etc).
But offhand, I don't see any reason why elaboration in math wouldn't take place IN MATH.
I don't know what the relationship of math to language is for math specialists; a lot of the 'math people' at ktm seem to be pretty verbal. I'm guessing that 'math people' can engage in useful elaboration via language --- BUT elaboration per se does not mean 'put into words' as far as I can tell.