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.
For example:
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.
Answer
My answer may not be completely relevant but I still thought it was worth putting in. I have had a few such sessions and I realized a few things. Not all of them could be true in general and I might have been a bit extreme with what I treat as knowledge. Here goes:
Your level of understanding is directly proportional to a few things:
Your ability to frame and ask grammatically correct sentences as an individual. This also encompasses communication of ideas/questions to experts of the field. For instance, suppose you are learning Linear Algebra in a group. You have a few gaps which are filled by others in the group. However, unless you can form sentences using Linear Algebra "handles", its unlikely you'll get far in research. Literature is way to dense in keywords. Unless you can talk in terms of Column Space, Rank and Eigenvectors (Rather than Linear Combo of all column or the vector which only scales) you are far from knowledgeable.
Your ability to participate in discussions. Its not difficult to lag and be left behind in a group of impatient, overachieving academics. Further, it can be a bit demotivating at times when the senior students who read the same content (owing to their heightened intuition) seem to grasp more. As junior students, it is often necessary to substitute the lack of intuition by more work.
Your ability to appreciate the nuances of the field. I cannot stress this enough from my experience. If you cannot appreciate the subtleties yourself, you didn't learn much. It doesn't take much to learn Elementary Fluid Mechanics per se. But, IMO, you actually "learn" when you go OMG when you see the transport equation and play with it till you are satisfied. An extension of this is motivation. When some concept is developed by an author, he doesn't write in random order. One of the most important aspects is to be able to understand why something is being developed. For instance, most Aerodynamics books start off with Euler Angles and then move on to Quaternions. Its simple to understand what Quaternions do and how to solve equations based on them. However, unless you know that they are used to prevent gimbal locks, there is no point of knowing about them.
The usual: Your ability to write alternate proofs, write codes (if possible), interpret the results of these codes and the usual other-than-textbook stuff.
One ending comment : Your level of knowledge is a function of how good you are in the group and how good you are without them. If you are able to develop proofs in those 3 hours together but aren't able to get started on your own later, you need to investigate whats wrong.
If you want to know how good these group sessions are, find out how good you are getting at that field as an individual
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