Should my 3rd grader still be counting on her fingers for math?

As a math teacher and tutor, I've found that students who don't memorize their math facts usually end up struggling in math. Some kids try to get by with figuring them out each time, but that takes so much time and effort that they have trouble understanding the rest of more complicated problems. Figuring the facts on your fingers may work for simple multiplication, but it makes it very hard to get your head around long division, or fractions, or algebra.

You don't need a tutor. You just need to have her work on her facts. There are computer games that you could have her use. Or have her drill with flashcards daily. Or my dad used to make me write them 100 times each. Or there are lots of games and books and gadgets at bookstores and educational stores; however you have her work on it, the main thing is she needs lots of practice. Lots and lots of practice. Until she knows them without having to stop and think about them.

For multiplication she should not be counting on her fingers.

If it is for basic addition such as 20-8, that is completely normal and even my math teacher does it.

I'd make flash cards for all the multiplication facts up to 12 and then quiz her on each one and for any she gets wrong or doesn't answer within a set time 1-3 secs then put it in a separate pile and make her repeat those until she gets them right in a timley fashion

This is what taught me the multiplication, subtraction, and addition facts

Well, I know I always counted with my fingers, through elementary and now in high school. I am in tenth grade taking eleventh grade honours math. I would not worry about it because as she does her homework and assignments and all that she will pick it up as she goes... She'll start to remember the numbers and all. You could also try to help her by quizzing her.

Get her to memorize her multiplication tables through constant teaching. Only way to getter better in math is to practice.

I believe that it is perfectly alright. I am in the eighth grade and still count on my fingers when adding.