Mathematics GRE

So I know it might be a little early to be thinking of this, but, from what I've been told, the two most important things on an application are the letters of recommendation and the statement of purpose. I can't do TOO much in terms of the letters, but I'm hoping I can write a decent statement, if I can ever get it started.

What kind of things do you think they're looking for in a statement of purpose? The stuff I've read about it is kind of contradictory: some people seem to think you should try to make yourself look well rounded, and try to humanize yourself, while some think you should focus almost entirely on your professional goals and mathematical experience (specifically, what you've done research-wise). How focused do you think the professional goals should be? I'm personally not ready to commit to a certain field, but it feels like I'm supposed to have my thesis planned or something.

The schools all seem to ask for why you want to attend their school in particular. Now, I KNOW that I'm not the only one who doesn't have really specific, academic or research based reasons for all of the schools I'm considering (some are getting considered for personal, location based reasons). How would you suggest talking about each individual school?

Long story short: What kind of things did (or will) you include in your statement of purpose?

The following is what I did, but I don't know whether it resulted in a good or bad statement of purpose, so take it with a grain of salt.

Like you, I didn't know what area I wanted to work in. I had a sentence which was effectively "I don't know exactly what I want to do, but I am interested in the areas of X, Y and Z". This in particular could well have made my application less attractive to some schools - I don't know.

I had a brief fuzzy paragraph at the start - "From a young age, I have been interested in mathematics...", but mostly my statement of purpose was an excuse to list the strong points of my application, as well as mention strengths that may not have been shown elsewhere in the application.

My statements of purpose were exactly the same for most universities, with just the name of the university changed. There were minor changes for the UK universities compared to the US, but I didn't address why I wanted to work at a particular university. Again, though, this may have been a drawback for some universities - I don't know.

Hope this helps!