Aside from getting the amount right, the “correct” way is to give coins first so they are secure in the person’s palm of their hand, and then the bills, which they can hold onto in their fingers. If you give the bills first, the change can too easily slide off of the bills.
I’m a math person so to speak, and I never could do the count backwards thing, I just did the subtraction in my head. The way I do it is just figure the change from 100. So, for instance if the change on the bill was 35¢, I know I owe 65¢ in coins. Then I figure the rest. So, if the bill is $3.35, then I know it’s the 65¢ then I do 10–3=7, assuming they gave me a $10.00, but they get back one less, so their change is $6.65. That’s probably not the best way though, because for me I don’t really have to think about the steps I just see it all at once.
I just tell the customer, “$6.65 is your chance.”
When I am the customer I have the change amount in my head, and when a cashier counts the money back to add up to what I gave them it’s unwanted noise to me. It makes it hard for me to count the change amount. If that makes sense.
Most registers make change though, I’m surprised you need to do the subtraction. Keep an inexpensive calculator handy.