Dax Mitchell asked a very good question in his June 8 posting. I'm paraphrasing, but it was essentially, when using the spiral approach, when a topic is revisited at a more sophisticated level, do the students have to relearn what they studied on the previous visit?
If they really learned the material the first time, a reminder-in-a-similar- context is usually sufficient at the elementary level. If they did not really understand the basic concept on the previous visit, they will surely have yet to learn it--but one hopes they will, given a second opportunity when they are somewhat more mature in their thinking and have had additional related mathematical experiences--which is possible in a unified spiral. But if not, there will be yet a third exposure. And, remember, these discrete points represent only the specific curriculum lessons targeted to the concept. If, in addition, a program utilizes, say, a student workbook which provides periodic practice with ideas encountered in lessons over, perhaps, a two-week period, there will be still other chances to absorb concepts.
The best way I know to communicate this is by example. So, if interested readers will identify ONE specific grade level of particular interest to them [anywhere, K-6], and will give me a POSTAL address, I will send them a representative sequence of spiraled lessons on some topic.
Ron Ward/Western Washington U/Bellingham, WA 98225 firstname.lastname@example.org