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I've recently quit my job as a maths teacher after doing it for 3 years and I have been incredibly lucky to be offered a fully funded doctoral training course at a top university. This is an extraordinary opportunity for me, especially given that I previously withdrew from a PhD, and this time I really want to make it work.
I am quite rusty on some things that many other starting students will not be, not because I can't do them but because it's been a very long time since I needed to apply them. I am currently a bit worried that I may be the weakest in the cohort or that it'll take me a very long time to understand simple things and that I may feel incompetent and like I don't deserve to be there.
What are some good ways to overcome these feelings of idiocy?
You have to accept that you are rusty and need to brush up on things. It is what it is, but there are others, myself included, who started a PhD after a few unproductive years.
The number one thing to do is not to take more classes than you're required to. If you are in the US system, there will be tons of homeworks and class work, so you need to make sure you will not fall too far behind. Your first objective is to pass the qualifier exam if your university has one.
But, you will also need to get an adviser, who will want you to do research. You need to pick wisely. Your adviser must be able to support you, especially if you are not allowed to teach. It is better to find someone who is a good planner and doesn't expect a student who takes classes and probably teaches to also do 60+ hours of research work. Not that it wouldn't be beneficial.
If teaching is included in the time balancing act, you have to be careful that your students do not abuse your time. Be strict with office hours, and be hard to find especially when preparing for your own exams.
The part with getting up to speed is harder, but be open and let people help you. If you have questions, do not be shy to ask your younger peers, or the course instructors. Also, participate in group study, especially if you feel it helps you and check homework results and discuss them with colleagues. You will learn a lot, but it will also help you understand more accurately where you stand in terms of knowledge compared to your colleagues. Even if you are more of an introvert and work a lot better alone, you still need the interaction with your peers.
Another thing is to plan alone time far from work. You have to have a more relaxed day, at least, when you don't have any work planned or you will end up burnt out and depressed. Also, if you have a family already, you have to understand that the PhD will take you from them more than a normal job would. Make sure you have a discussion with them where you set the expectations, and also try to leave the work at work as much as you can.
For starters you aren't an idiot. The evidence is against that hypothesis. However, there is a phenomenon called Imposter Syndrome that is fairly common among even well accomplished people. "I can't possibly be good enough to work with these people."
Well, yes you can.
But being rusty on lots of details is also common, but I'll guess that you know how to deal with that so that it doesn't become an issue. Working groups of like-minded folks is a good way. Just throwing ideas around can help. But so can re-doing old exercises.
From my experience, asking a lot of questions is the best way. I've said similar things here in answer to other questions. Don't be too cautious in asking questions in class, if the class size is reasonable. Very often if you have a question about something in lecture or otherwise, others will have the same or similar issue, but be afraid to ask.
Also, it is ok to start out slow. Don't be concerned that others are faster. That is always the case.
Work with others. Ask questions. Take advantage of office hours or tutors. Solve small problems on the way to bigger ones. Take a lot of notes (by hand). Never. Give. Up.
I am currently a bit worried that I may be the weakest in the cohort or that it'll take me a very long time to understand simple things and that I may feel incompetent and like I don't deserve to be there.
If this is the mindset you're carrying into the doctoral study you're going to have a very bad time. Because even evidence is not pointing to that direction, you can still be subconsciously looking too hard into clues that match your expectation. I think you may have some introspection to do.
While competitions exist, doctoral study is closer to an individual backpacking than a 5k race. You chart your route, pack your stuff, get onto the track, deal with contingencies, take rest, enjoy, get lost, find new way, and perhaps enjoy again. Your supervisor would be like that shop keeper at a travel gear store giving you different levels of advice/service, or be like your GPS; most of the time, you're responsible for yourself.
But what of those horror stories about adviser pitching their students to fight over resources? About those shaming experiences for those who didn't get student grants? Or corridor gossips about so and so being dumb? Yes, they may happen as well, but these are largely out of your control. What I found useful to deal with most of the adversity are two things: the will to grind and move forward, and a set of internal personal values.
Familiarize yourself with the school/program expectations. Attend workshops and read up about opportunities. Build professional networks. Work with your advisor, lay out plans and milestones, and work towards completing them. If things go wrong, come up with some suggestions and consult the advisor.
Don't outsource all your sense of success or accomplishment to others. Think larger goals: focus on what career you'd like to craft, what kind of students you'd like to produce; not if you're the top pick of the faculty among your peers. Draft and keep working on a personal mission statement, and be very clear to base them on your own values: leave the world a healthier place, make education more equitable, etc. And not "people would think that I am the best this or that..." I can't give a whole introduction here, but do search online on how to craft a personal mission statement. Do that soon if you don't have one; it's super useful as a personal compass.
Up play your strength: e.g. as you have worked as an educator, you may also consider applying for teaching assistantship. Being a TA would allow you a great chance to review the materials, pocketing some money, and enriching your post-graduate teaching portfolio.
There will be stellar candidates, and plenty of people brighter than you (which, is exactly the reason you should be there to begin with.) If they are nice, befriend them and perhaps you just made the first step of a life-long research collaboration; if they are not nice, learn from their strengths and move on. Be aware that the playing field will be unfair: some will have more experience, more resources, more popularity; and that's exactly how the rest of the system works. You worked before, you'd understand.
While I don't want to trivialize your worry, the fact is that this move opens a lot more opportunities to development, expansion, enrichment as it does disappointment, anxiety, and frustration. In other words, you have a lot more important things to plan and be prepared for. Take your head out of the birdbath, stop drowning yourself, because if you turn around, you'll see that you have a sea to chart.
I'd recommend a couple titles that you may find useful:
"The Professor Is In" by Karen Kelsky: This title discusses many things on how to be academically marketable. Kelsky, a seasoned admission director, shares a lot about getting ready to be a professional scholar.
"F*ck Feelings" by Michael Bennett and Sarah Bennett: This, or any other similar title that had recently conquered the Self-help book shelves, sometimes provides useful approach to deal with feelings. While it did not change my life, I did find seeing feelings through a comedic lens soothing.
"The 7 Habits of Highly Effective People" by Stephen Covey: Classic oldies that you can probably get a used one for 25 cents. It serves to give a good framework of seeing what success is. It's also one of the earlier titles (that I know of) which mentions the idea of "personal mission statement".
"Networking for People Who Hate Networking" by Devora Zack: I hate networking or pitching myself. This book puts much of my anxiety into perspective.
I wish you best of luck. Take the dive and really enjoy the time. Move forward.
Congratulations on your great offer but...
... for starters, you should try to stop using/thinking the bit in bold (quote yours, emphasis mine):
"I have been incredibly lucky to be offered a fully funded doctoral training course at a top university".
Someone (multiple people) looked at your application and chose to offer you a place. Your name was not drawn at random; someone with experience in admitting worthy students thought you would be a good fit for the program and you will have that in common with your fellow students :)
Another thing you can do that might help convince you you belong there is be clear about what you are interested in. Be able to tell people why you picked this program, what you hope to get out of it, what you plan to do research on. There are bound to be brilliant people in your cohort and you don't have to do better than them - that's not really a Ph.D. mentality since your aim is to contribute to knowledge rather than to come top of a course. It may help you to feel like their equal if you can speak with confidence about what your research interests are.
There's no quick-fix for being rusty. I guess you already know that you'll have to put in the necessary hours to brush up on things you've forgotten so certain projects might take a little longer at first. Don't let it isolate you - graduate projects are much more manageable if you talk to other people in your cohort / attend office hours (for courses) and talk to your supervisor frequently (for thesis work). Don't let the fear of appearing silly stop you from asking for help (this advice comes from recent experience in my own Ph.D.)!