"Many times we are in a hurry and are ready to move on to the next lesson, subject or project. I wonder what would happen if we allowed students to linger longer on problems? Students might solve less problems, but my guess is that the quality of work will improve dramatically."
I too am guilty of pushing my students to "get to the next problem". After all, I have SO MUCH curriculum to teach and so little time. If I don't get through it all, how will my students be able to perform well on standardized tests? Or, what will happen next year when they are in "so and so's" class and I haven't prepared them for the new mountain of material they will have to master there? And so I play into the same pattern of thought that we have had for many years: teach mile wide and inch deep (at best) of content. Hope that the students pay attention the first time because there won't be time to go over it again. It frustrates me on the days when the students ask good questions that keep us from completing all the problems I had prepared for them to do. If I answer the good questions, I won't get through all my notes (and many times I don't). Then they won't be able to answer the questions I gave them for homework. Aaaaggghhhhh!!! It's a vicious cycle because I often digress and answer their questions!
So as I get ready to try more flipped activities (being slow to implement because I want to have everything PERFECTLY ready), I am realizing that I will have to wait and see (being a newbie) about the impact on the learning. I THINK it will be more powerful for the students. I HOPE that it will get into their long term memory instead of the short term memory (that doesn't even last until they get home and try their homework). But I am realizing that this class practice won't allow them to get through ALL the great problems that are currently on their homework worksheets, because the time that discussion takes is much LONGER than the time I imagine it takes them to do their homework. However, I have come to the realization as well that (a) they don't do all the problems because they either get frustrated or see a word problem (and by the very nature of that, it must be skipped), and (b) they miss the nuances in the problems that they really need to understand if they are to continue getting new problems correct (i.e. test/quiz questions, or AP test questions). So fewer questions, but giving them more time to linger might be just the thing I'm looking for to deepen their understanding of the content.
I will have to ponder this more, for sure! I can see I have to put much more thought (and I know that I've been doing that already) into that face-to-face time so as to make sure they get through the "harder" problems and tackle those meaningful word problems (no more skipping them!).