I am a Ph.D. student in theoretical computer science. I meet my research supervisor once a week for around 2 hours. I prepare some of my research and try to present in front of him. I always try to make a list of theorems and claims given in the paper to discuss with him in a certain flow but most of the time he starts asking questions and then we go in other directions (which is okay from a research point of view). Due to this, it sometimes takes 2 months to read a single paper. I always try to follow my designed plan but it rarely works. Also, I don't feel satisfied most of the time because many times I am only able to discuss 50-60 percent of what I have planned. Please help.
Speaking from experience, both as a student and as a Phd supervisor in theoretical computer science. The meetings with your supervisor serve many purposes. To narrow the scope a little bit, let me only discuss the type of meetings you mention - where you read a research paper and discuss it to a high level of detail with your advisor.
In TCS (and maths, I would assume), the primary goals of having the student read a paper and present it to the supervisor are:
So that the student learns the contents of the paper, especially the main technical tools used and developed in the paper, to a degree where the student can recognize a setting where these tools can be applied, and apply the tools in relevant settings. Because this is research the tools will have to be modified at least slightly every time they are applied. Hence, in order for the endeavor (reading the paper) to be any useful at all the student needs to achieve a very deep understanding of the paper she is reading.
So that the student learns to read, understand, and present (and eventually write) technical proofs. This is a big one - it is incredibly important that whenever you feel that you understand a proof, that you are able to present it at various levels of detail. On one end you should be able to give a “big picture” view of what the Lemma statement actually means, what it gives you, why it should be true, and why this is a non-trivial statement. It is equally important that when probed, you are able to re-create to full detail a presentation of the proof. By full detail I really mean full detail - are you saying “let x be an element of S” you better have proved that S is non-empty first. Are you invoking Lemma 3.5? You have to point to precisely which objects you are applying Lemma 3.5 to, why these objects satisfy the premise of Lemma 3.5, and which objects this invocation of Lemma 3.5 generates for you to use further in the proof.
The level of understanding that I describe above is really the only level that will be useful for you in your research. It also takes a lot of work to get there, this goes way beyond the level of understanding you need to get A’s in your classes. For this reason it is perfectly normal to take 2 months to read a 10 page paper. This is also a great cause of misunderstandings and frustration between students and advisors (the student thinks “I already know this to a level where I would ace an exam on this paper”, while the advisor thinks “the student has no idea what is going on”).
I can not speak for your advisor, but in such discussions with students I often ask follow up questions in order for the student to herself come to the realization that she needs to understand this or that part of the proof better.
Also, professors are not gods. In fact, for me, the vast majority of follow up questions that I ask is because I am struggling with understanding what it is the student is talking about, and I’m trying desperately to figure out whether I missed something obvious, or whether the student is talking nonsense. Doing this (for 2 hours) is mentally exhausting to the point of the infeasible. For this reason the conversation will often naturally after an hour or so shift to lighter topics.
I can’t actually know whether any of the above applies to you, but I do find that the above is a pretty common problem. Let me end with some pieces of advice (much of which has already been said in other answers).
Talk to your supervisor about this. This might well not bring about the change you are looking for, but you will better understand each other’s motivations, frustrations, etc.
If you are worried about your productivity, show initiative. Ask your supervisor about what are the next natural papers to read, with or without discussing these papers with her. Then read them, to the best of your abilities. Ask your supervisor for research questions, and think about them. Talk to your fellow students, and initiate research projects with them. Remember that in the last year or two of your Phd. you should be able to complete a research project from A to Z without any help from your supervisor at all. This involves obtaining the relevant background, asking feasible and interesting research questions and answering them. The best way of getting there is to try to do this as much as possible from day 1 of your Phd studies, and then using your supervisor to help you with your weakenesses. As an example, when I was a student I had no problems with solving problems, but had no idea how to find relevant research questions, and so I would lean on my supervisor and collaborators for this to a large degree. For you there might be another aspect of research that you will need extra support for, and that is what supervisors are for!
Your advisor doesn't need to understand all the details of the paper, so long as you understand them. The point of meeting with your advisor isn't to go point-by-point through the paper, it's for your advisor to help you out if you're confused or stuck, to give you suggestions of research directions to go in, and to supplement the material in the paper with related information. Your advisor listening to you explain the details of something that you already understand is a waste of both of your time.
The only thing you've said in your question that seems like it might be a problem to me is "it takes 2 months some time to read a single paper." You should discuss this with your advisor, but I'd suggest that you do more to condense the point of the paper so that you focus your discussion on the key points and what your advisor wants to discuss, and then you move on to the next paper rather than spending week after week on something you already understand.
Disclaimer: this is based on my own experience of both sides of this but in life sciences, not computer sciences. And individual supervision styles vary; there is no single "right" way to do this (although some ways are right and some are just wrong).
I am not completely sure that what you're currently getting is unproductive; you might just have a different idea of the 'product' than your supervisor. The function of a doctorate is to equip you to function as an independent researcher. If you have technical/theoretical questions and have not been able to answer them yourself, then you can by all means bring them to the meeting - but your supervisor may feel (and this is an entirely valid perspective) that these meetings are not for answering a long list of preprepared specific questions which could be dealt with via email (or that he might expect you to be able to resolve independently). It sounds like he sees your plan as a springboard to get into fairly open-ended discussion of the area of research. This is a great opportunity, and you seem to partially recognise this, as indicated by your quote below.
he start asking question and then we went to other directions (which is okay from research point of view )
As a supervisor myself I would prefer to bounce ideas around with a student than go through a paper item by item (unless there was a really good reason for it). There is however a balance - in the early stages you want to make sure a student has a firm understanding of the basics, in the middle stages you want them to start developing big ideas, and by the end you want to make sure they'll submit on time :)
Meet with your supervisor and discuss your concerns with him. Tell him that you are not sure whether your current style of meeting is productive enough. Try to find a solution together. The goal of your discussion is not that you just spend an hour or two with him. Your discussions should advance your research and help you, but that is not happening right now. So it is your own responsibility to address this issue with him. All in all, this looks like something that can be solved quite easily after a short discussion with your supervisor.
Please don't confuse advisory meetings with Journal Club.
If you don't belong to a Journal Club, I suggest you start one. In Journal Club, the members take turns presenting. When it's your turn to present, you can "make a list of theorems and claims given in the paper" and present them to the club "in a certain flow." In Journal Club, you can expect to be allowed to finish what you've prepared.
For your work with your advisor, the initial stage is about finding a topic. Some advisors propose a specific topic. At the other extreme, the student proposes it. There are lots of collaborative ways in between of finding a topic. The advisor should provide guidance, though, especially in the sense of avoiding getting stuck in a cul-de-sac (a topic that's not going to go anywhere).
Try this: when you bring in a paper as material for discussion, and your advisor starts asking you somewhat tangential questions, write those down (or record them), and spend the next few days working on one or more of those questions. In your next meeting, report on that work. Remember, the PhD is about doing original work, not just retracing someone else's steps.
As you get more into it, you will probably start to ask yourself some of that type of open-ended questions, and then you can show up to a meeting with your advisor to report on what you asked yourself, and where that took you.
(Fast forward after some time working in this way)
If you become concerned about the amount of time that has passed without your having settled on at least a preliminary area of inquiry, you can express your concern to your advisor, e.g. "I'm concerned because I don't have a research topic defined yet." Then give your advisor some time to think about your concern -- maybe one to four weeks. Hopefully he will have some ideas for how to get things moving toward a topic definition in a more efficient way.