Oddly, with regard to the well respected "Freedom Of The Hills" most of us in the east, myself included' tend to use poles, then crampons when needed & then ice axe in rare circumstances. FOTH, shows the axe should come out first, not so much for self-arrest but for self belay. A pole, or even two make a lousy belay. You can put weight on them when under control but not on ice, the tips just slide.
I use poles, too, but they are not at all helpful on even a little bit of ice and I am very conscious of not depending on them, which is hard if I have planted them! Habits are difficult to break. Strangely, in recent years I've found myself using them as a counter-balance, not just a resource for placing weight. Crossed poles help me get up in deep snow, if they haven't prevented me from falling, as well as stream crossings, or getting someone's attention.
So, I wonder how many people who have ice axes have learned proper techniques for using them or even practice using them? There are so many scenarious and right and wrong ways that I've found refreshers are very helpful for helping in situations we don't experience very often.
My favorite route is the Lions head winter route and I always use an ice axe when i climb it. I have always made it a point train with all my gear in many techniques. Self arresting, french technique, steep snow and ice, you cant just buy a tool like an ice ax and charge up a route. You practice in a controlled enviorment and hone your skills before you try them on a steep route. I had to tech myself, books I used where basically 2 "Freedom of the Hills" and yvon Choinards(sp) climbing ice, buy those to, study them, practice and your good to go.