How can a PhD student defend their correct ideas or claims to their supervisor?

How to defend your correct ideas or claims in front of supervisor?

If your advisor is anything like me, they care much less about right or wrong in these meetings, but they want to hear what your thought process is and which angles you have covered. Suggestions to try something else should not usually be understood as "you did something wrong, do this other thing instead" but more like a probe whether you have also investigated this other angle, and if you have not chosen it, how you justify not doing that.

As such, the central element in "defending" your idea is to, briefly but clearly, outline why you have done what you have done, and not done what you have not done. Proving stuff in front of your advisor should help, as long as you do it correctly and clearly. If it doesn't, you are either not being as clear as you think you are, your advisor isn't competent in your topic, or you have a much deeper problem (e.g., you and your advisor are not on the same page about what problem is to be solved). However, note that even an objectively "correct" approach is not immune to questioning - maybe there are other approaches that are easier, faster, or preferable in some other way? You need to have an argument at the ready for why you have not chosen them.

If looked at in that light, situations where they suggest approach A in the first meeting and then probe you on your usage of A in the second start making more sense - they have not suggested A because of some secret advisor-tech that makes them know the right answer in advance, and telling in the second meeting that they suggested the idea is not a convincing enough reason to go forward with it by itself. Even if they suggested it, you still need to explain why A was better than other approaches.

Too many times, I have to defend my ideas or claims in front of my supervisor.

Frankly, I can hardly imagine a supervisor who questions your ideas "too much". As long as they are respectful and helpful, questioning your work and suggesting alternatives is pretty much all that good advisors do. It's kind of the job description.

My research supervisor has told me that I need to defend my ideas and claims. There have been many times where he asked me to change something about the idea; I did it; and then he asked me why I changed it.

My initial reaction to this is that it sounds like you are trying to please your supervisor (and others) when defending your own ideas. This is wrong.

Either you believe in the ideas you state (whether you change them or not) or you don't.

Your supervisor's job is to probe for failure of logic, but also to probe for failures to defend and shifts in position based not on logic, but on a desire to please or confusion.

Question: How to defend your correct ideas or claims in front of your supervisor?

Explain the logic and answer the questions. Either the logic holds or it does not. It's simply a matter of being open to your "correct" ideas being flawed or imperfectly explained (i.e. the idea can be correct but you can fail to properly justify it).

Edit : Many times I have tried to defend a wrong idea

... which will never work. I cannot think of any reason to do this.

and there were some times in which I have changed my right idea ( based upon his feedback) and after I changed my idea he told me "your original idea was right, why did you change it?". How to deal with this situation also?

That's precisely the point.

Why did you accept a change ?

If you accept a change you should be able to justify that change.

This suggests you are simply saying "yes" to your supervisor instead of thinking about what they say more deeply and deciding if it's really true or not. Your immediate reaction is to try and please your supervisor (wrong !) instead of considering their questions and ideas properly.

Changes should be things you believe in, not things they say.

It's your thesis, not theirs.

Nothing you say should be something you do not understand and believe.

Ultimately, the supervisor is saying that your idea is not clear. Since we are assuming your idea is correct (premise of the question), then the problem boils down to how to be a good teacher. It sounds like you already know the material well (you are going over it multiple times in advance of your meeting and proving it step by step) -- now you just have to work on your explanatory skills.

This is also something I struggled with. Ultimately, getting more experience teaching and presenting helped me because I was able to begin to think of my meetings as opportunities to teach my advisor about my claims and ideas.

When teaching your advisor about your claims or ideas, you want to be able to explain the same concept multiple ways. I learned that my supervisor really grasped ideas more readily when I could give a real-life example for him to mentally grasp onto. Another method was drawing diagrams to support my ideas. This especially helped with multi-step processes.

The final trick is to make sure that your tone comes off as neutral, rather than arrogant. I find that phrases like "As you know..." and "You probably already see that..." can be used liberally while reminding the advisor of fundamental facts without talking down to him.