I found DS Investigations very troublesome to use as a teacher, because there didn't seem to be enough opportunity for pencil and paper practice. I don't really like "drill and kill", but there should be at least a little bit of it. There were pages like this with Investigations that you could use, but it was really left up to the teacher to decide whether or not to print them out.
A teacher who really knew what she was doing could make Investigations works in a fabulous way. A teacher who didn't know what she was doing, could really bomb with it.
Singapore would be a really good choice to help augment a program like Investigations at home.
From the posted references it appears that the Investigations material is seriously defective as a math course. Perhaps it could be used to supplement Singapore Math or something else, but not the other way around.
Having bought Singapore Math, in my opinion it's not perfect but it doesn't have any big weaknesses either. It does plenty to strengthen conceptual understanding, with lots of different types of illustrations to make sure that students have a strong conceptual framework
before getting into any pencil-and-paper drill. One can add extra reinforcing practice with the workbook, which has problem solving interspersed with more traditional drill on what's been called "algorithms" in this thread.
I think Singapore Math is a good example of how a good basis for understanding can be taught without necessarily needing to be discovered by a student first, and the emphasis on problem solving would tend to keep most students from being too passive or bored. The Singapore Math approach in third and fourth grade seems to be to illustrate a concept graphically; when appropriate to have a student do some discovery (e.g. in the 3B materials introducing fluid volume, students are given small explorations with real-world / household materials to get a feel for combining volumes, etc.); and when a conceptual basis has been established, get into working problems. Regarding problems, it starts with simple non-word problems first, then gets into simple word problems, then multi-step word problems. In addition Singapore Math seems to intentionally remove explicit instruction on how to do the problems as the student progresses, so that by the end of a section the student is (hopefully) fully self-sufficient.
