I am an M.Sc. student in mathematics. I was recently invited by some Ph.D. students and Post-Docs (a group of 5 people, including myself) to join their study group. We are reading a specific text, which should get us ready to read some more advanced work. This work is relevant to the research of some of the other people in the group, but my main goal is to experience this learning methodology with the added advantage of getting to know a few aesthetic results in mathematics.
We met around 6 times, and it is not what I'm used to from courses in the sense that we don't completely understand all the details. Nevertheless, we go on reading.
We encounter a definition and we can't find out its exact meaning. In this case we usually know of an example of a mathematical object satisfying this definition (because it's mentioned in the text) and we just try to see how the propositions in the text apply to the specific example.
A proof is given with very few details - we manage to fill in some of the gaps, but not all of them, so we just take an example again and simply accept the statement of the theorem so we can use it later.
An excercise is given in the text and we only solve part of it.
We allow ourselves to skip some details because this text is only meant to get us ready for some more advanced, but more specific, material. My question is how we can find out whether or not we are gaining anything, and how we can gain more given the fact that we are all busy and don't want to invest much more time in this specific reading (we have a 3 hours meeting every week).
I have a feeling that I'm "getting used" to some ideas and facts while reading this text (in contrast with "completely understanding"), but I'm not sure if I'm really gaining anything or whether it's just an illusion and I'm not sure how to test my gain of knowledge. The exercises in the text allow us to test our understanding of the details, but not of the general ideas.
EDIT: I will clarify what the question is, in response to aeismail's comment:
As Charles and Nunoxic say, the question of whether shallow-reading is useful is separate from the fact that we are studying in a group. So, the 2 separate questions are:
When reading without understanding all the details, how can I find out whether or not I'm gaining anything?
How can we make the process of studying in a group for 3 hours a week most efficient?
These 2 may have better been asked as 2 separate questions, but I did not notice that (in my mind they were related because the group study was the first time I encountered shallow-reading). To summarize the answers I got so far:
It is possible, for some people, to gain knowledge from shallow-reading and one way to test it is to see if you understand why each topic is being developed and why the text is structured the way it is.
When studying in a group, one should test his ability to work out the details himself after the group sessions.
I think the answer I got for (1) is excellent and the answer for (2) is somewhat lacking so far.