Hey!!! I have a green 6yrold mare who doesn't understand how to canter!! I've seen her breeder video, and basically how they got her to do it was trot so fast eventually she broke to canter. I don't want to do this: it's dangerous and obviously not the right thing to do. But she doesn't understand the command at all. All she does is trot faster!!! What can I do without being overly harsh (because she's not being bad)

Sounds like a case of her not being balanced in her flatwork. If the horse gets strung out and on the forehand, they can't make a good transition to the canter. Doing flatwork that encourages her to rock back onto her hind legs and use her back will enable her to make a better transition up to canter.
You can also try lunging her at the canter, this will help her find her balance and get the rythm of the canter. You can also train her to the voice aid "Canter" so then when you start riding her if she doesn't understand the leg aid, you can reinforce with the voice aid, this way she will understand what you want her to do.
Something to remember is that not every horse has a good canter. Some breeds and conformation faults makes it harder for a horse to canter than it is for them to trot fast.

I did try to lunge her...and no go for that either. The flatwork thing I am working on. She did a lovely series of changing rein at trot. I could really feel her using her hind legs easily. Also what might be helping is that we don't have an indoor arena, so what snow there is is what we got to ride in!! Pick up those pretty feet! Is there anything that would really help those hind legs? She is learning NOT to lean on my hands, so I think she's getting it.

Well its going to be very difficult because you are riding in the snow, but, if its a case of needing flatwork its not something that you are going to be able to fix in 2 weeks.
Doing some dressage will help working her back on her hind legs, things like:
- lots of transitions to walk/trot/walk LOTS!
- small circles, and leg yeilding out to a bigger circle, then back in to the small
- leg yeilds on a quarter line
- serpentines and figures

If you do these kind of exercises it will encourage her to build up muscle in her hind legs (provided of course that you are making sure she is using her body properly)
I would also really suggest that you keep trying the lungeing at the canter. Even just to get a couple steps out of her, (not 3 rounds at first) canter/trot transitions are also very good. Lunger her at the trot and ask for canter get a couple steps, then let her to back to trot.
She needs to build up the muscle and get the balance, she can't just jump right into it.
Hopefully you will notice her being able to keep the canter longer, and to start to learn to voice aid.
Again as I said before its going to take some time to develop this with her. You need to create a workout program so she gets consistent training. (ie lunge everytime before your ride) but make sure to not do the same things every time you ride, or she will start to anticipate.

thank you very much! SO...plan is to leave canter until spring when we can lunge again, and work on flatwork NOW. This girl is pretty green, so except for transitions, those other things you talked about I will have to teach her! This will be an interesting winter....

The good part is she LOVES being ridden! She was pretty wild when I started, then she got laminitis (long story) and now I've been able to ride her consistently. She was wild to begin with, but seems to have settled amazingly quickly. Good girl!

I have very exciting news!! My mare cantered for the first time today!! What worked was letting someone work on canter while I waited my turn. THen when I tried her, I used normal aids. It didn't work, so next corner i tapped my crop against her shoulder (never used crop with her...doesn't need it!). She immediately started trotting faster, so I slowed her down. Next corner, I asked, and at first trotted fast, but suddenly broke into it! I hung on for the ride, and when I managed to walk her, made a HUGE fuss over her. I will definitely work more on flatwork, but now she's done it! She'll understand a whole lot better now! YIPEEEEE

YAY! Congrats! You must be happy!