thats odd tori as most swimming teachers i know and life guards etc actually say you are better NOT to use arm bands, they are uncomfy and a child cannot learn to swim properly with them on, they cant move their arms to swim correctly.
i think there was a thread on this recently where a mother from australia said that they NEVER use arm bands all children just start of in shallow water and gradually learn to swim, if they use anything at all it is a float that they can hold to whilst they kick, you can also get them to hold on to the edge of the pool (lots have a bar to hold) and practise kicking like that.
swimming lessons where they have fun and practise splashing and playing are the way to go, sing songs and bob up and down, your child doesn have to get their face wet but if they keep seeing you do it then they will gradually realise it is an ok thing. another thing is to try and get them to blow bubbles or take ping pong balls to the pool and let them blow them across the pool, most little ones love to do this and it encourages them to put their face near the water
my eldest (now 8) went swimming with us regularly as a baby and never wore arm bands, then he started lessons at four and now is a confident swimmer, tho is better swimming on his back.
ds2 who is 6, didt go as often when he was little but he also started lessons at 4 and was wary to begin with but his teacher was great, held him, let him take a fave toy into the pool with him (obviously not a cuddly, but he took little playmobile people or plastic animal type things) just so he had something familiar from home, and gradually his confidence has built up and he is getting there, he loves to swim with one of those foam 'noodle' things which can go under each arm, across the chest and gives them a bit of buyoncy
btw i never had armband and could swim before i could walk pretty much, but we lived in sardinia and i was at the beach every day and then we lived in cyprus and i swam in the beach and in a swimming club a couple of times a week, i dont remember any of my friends having arm bands and we all learnt to swim fine