Here in Japan, there is a textbook for every subject including maths, and most lessons are textbook based, including in elementary schools. The Japanese education system has plenty of faults, but it does do a damn good job of teaching maths.
By contrast, many of the international schools herethe "western" ones, anyway (American, Canadian, Australian as opposed to Korean or Indian ones)don't use textbooks and seem to practically boast about this fact--like it means their teaching is somehow more imaginative or exciting or what have you. "Ah, we don't use textbooks here. Our teachers craft lessons from a rich library of resources," I was told by headmaster of one particularly oddball American-style school.
By the way, all these international schools are selective at some level, and have privileged, middle-class intakes consisting of well-motivated parents and kids. So I'm not buying this thing about "Oh, British schools can't use textbooks because they have such difficult and unmotivated student populations and can't trust kids with textbooks and the ability of students varies too widely for textbooks." I really think it's a cultural difference. English-speaking countries' teaching traditions just seem to be so anti-textbook--it really seems to be seen as a lesser way of teaching.
A friend of mine has a son at one of these schools, and the lack of textbooks has been a bit of a disaster, quite honestly. Endless grotty, dog eared worksheets coming home crumpled up at the bottom of the school bag, cluttering up the house, getting lost. She has no easy way of keeping track of what he has been doing or revising past content, so it was a long time before she was able to figure out that he was lacking in a lot of basic skills that he ought to have. Nor can she look ahead to see what will be coming up next. The whole curriculum exists only in the teacher's head, which is very disempowering and frustrating.
What I think is weird is that the anti-textbook people tend to be the same people who are always going on and on about "independent student-centered learning" and how teachers shouldn't be the fount of all knowledge and how students should develop their own strategies for this and that and so on. Okay, fine, but wouldn't an obvious first step to be to have a textbook that students and parents could use and refer to and check up on and revise from and do self-studying from? It is so unnerving for students to have so little sense of what is going on in the maths curriculum.