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How Do You Stop Caring About Rating?

yes, when you play chess just to increase your score, or with the fear of it decreasing, you are not playing with the correct motivation. We only focus on the score as a number when it should instead represent our playing strength. It doesn't matter what your score is, enjoy every match and you will see that improvements will come with your commitment and passion. "Whoever is afraid of losing must be sure of defeat".
I think the best way is to just remember that it has literally no bearing on your life for >99.99% of people. I think it's a useful tool for tracking progress and improvement, but at the end of the day it doesn't actually mean anything at all. It's a number that shows how much time you've put into the pattern recognition of moving wooden pieces on a board.
It may not have a bearing on your life, but it has a huge bearing on your ego!
This was one of the best contributions I've ever read here! :)
Really helpful.
Great advice.

What I also found helpful is to recall how big natural fluctuations in playing strength are. If you improve 100 points worth of playing strength in a year, that is amazing for most adult improvers. On the other hand, strength fluctuations on the scale of 100 points aren't rare at all from event to event or even game to game, in my experience (depending for example on how you feel or how familiar or comfortable you are with the structures or type of game).

So the fluctuations are usually bigger than the long term trends and it's hard to distinguish a trend from the fluctuations. Therefore, if you too often check in with that number on your forehead looking for progress, it's bound to be disappointing and also deceptive when the fluctuations go in the positive direction. Even if/when there is actual progress. So it makes the game less fun even if improvement is a focus.
I find playing in zen mode useful, and furthermore, turning off the ratings view altogether so I don't even know my rating. Now I play more objective chess instead of thinking "I'm better than this player, I should win" or "This player is amazing, they'll crush me". I've found it freeing.
An important factor to discuss is not just the "numbers on foreheads" but the actual things people say to you and how they treat you. I've had this conversation several times:

Them: "What's your rating?"
Me: "Well, about 1050 - 1150 USCF"
Them: "Oh... Can I recommend a good book of 1-move checkmate puzzles?"

The common view seems to be that anyone below 1200 USCF is probably drooling a little bit and writing their game notation with a crayon. It's easy to say "just forget about rating and focus on process!" (btw I do that - see my Lichess blog), but you need to also account for the barrage of condescension you experience on your way up.

(BTW, a fun game when playing casual OTB is to have them try to guess your rating after a few games rather than just telling them. Usually, they guess higher - illustrating the inaccuracy / lag of OTB ratings).
Excellent self observation hygiene shared.