I think that counting approaches isn't a particularly good idea in general. It makes you focus far too much on the very start of the interaction, rather than the general meat of it. I used to do it and I actually see it as part of the reason I left conversations early. I was approaching with the intention of approaching rather than talking, so after I approached I on some unconscious level considered the job done. That meant that when it actually got into having to keep a conversation going and I wasn't comfortable with it I'd just fuck off and go do another approach to tick off on the tally.

If you want to set a goal to get you doing something a better way in my opinion is to give yourself a period of time (say 15 mins to start) where the goal is to always be chatting to someone. That way you keep the chat going as long as possible, and if you actually can't keep it going you've got to find someone else to chat up ASAP. Aim for covering this time period with as few separate conversations as possible though.

Just an idea, give that a go and see how it works. Then when you're more comfortable going up to people and keeping a chat going abandon it.