tips for keeping hands down and quiet please!

Thank you guys! These are all amazing tips, I think the comment about my balance is correct, I think its a reaction to me trying to balance myself. In my next lesson I think I'll try to ride bareback!! 0.o (somewhat daunting after my first fall last week due to my lack of balance) I love this forum!! I'm getting so many helpful tips and tricks!!! D

Do not be afraid to fall, everyone does. The trick is to learn how to fall correctly. I used to ride on a lunge line and "practice" falling off. Learning to fall off correctly or the best way possible, is a very important thing for beginners to learn. Goodluck!
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Walk through house with cups of hot coffee in each hand, full ones. Can use water of course, but nothing like the sting of hot coffee on bare flesh to remind you to keep hands still and to give with the motion of your walking.

Also, if you are a fidgeter off a horse, as in your hands are never still? Quit that too, as it will follow you into the saddle.

When you ride you will have to work to keep hands as well as everything else in position. Concentrate harder on your hands.

Ever since I read your post, I've been trying to really make my hands still. I work in a restaurant so when I'm taking alcohol to tables I try to really keep my hands still! Thank you!

Lots and lots of work on the lunge to sort your seat out (so take away the reins all together)
Hands can only ever become quiet when you stop relying on them for your balance