Answering "Why do you want to become a research mathematician?"-type questions in a statement of purpose
I can't speak for all mathematics graduate admissions everywhere, but when I was on the admissions committee for my department (UGA) we generally did not take the statements of purpose all that seriously.  Most statements of purpose are indeed a bit dull and similar to each other.  The particular question "Why do you want to become a research mathematician?" is certainly not explicitly asked across the board (somewhat crankily I could ask "Why are you asking that for a master's program?  The degree needed to become a research mathematician is a PhD.  Depending upon your local academic culture, getting a master's degree could either prepare you to enter a PhD program or it could be totally unnecessary for that.").

It's not a bad question, but to me a very acceptable answer -- really the best answer -- is because I really enjoy mathematics, I am good at it, and I am eager to learn and do more of it.  Not to brag too much, but I am in fact a research mathematician.  Why did I become one?  Please see the above bold-faced text: even with the benefit of many years of retrospective, I don't really see a better answer than that.  In fact, honestly I think the bold-faced answer is probably better than the wordier answers I gave in my early 20's: how mathematics is the best field because it has an amazing internal consistency and a level of rigor and certainty unmatched by any other intellectual endeavor, how mathematical theorems are eternal so proving them is a form of immortality, and so forth.  Yikes.  It's not so much that I don't believe these things anymore but rather that I don't find these beliefs very interesting or distinctive: I think they fall nicely under "because I really enjoy mathematics".

The problem with taking the personal statement too seriously is that the ability to write an excellent personal statement -- especially an unusually interesting or insightful one -- is only weakly correlated to success in a math PhD program.  I looked at personal statements to see whether they were adequate, e.g.:


Do they use flawlessly correct grammar, syntax and punctuation? Do they express their (not so complicated) ideas clearly?  Do they convince me that this student has strong enough writing skills not to be dragged down by this in the future?  (Many STEM-types have alarmingly weak writing skills, and for sure it drags them down.  I take the GRE Verbal score very seriously.)
Do they avoid gross misconceptions about what a math PhD program and a math research career will be like?  I don't expect an incoming math PhD student to really understand either one: most don't, but they learn eventually and adapt accordingly.  However some students are clueless so far beyond the norm that it becomes a risk that as soon as they arrive they'll think "Wait, what is all this??" and drop out in their first semester.
Do they seem like a responsible person and not a snowflake who thinks that graduate school is a summer camp that they get paid to attend?  (It happens...)
Do they avoid sounding too weird? Yes, mathematicians can be weird, like many people, and the amount of allowable weirdness in the mathematical profession is probably greater than most.  But it is still only a finite amount, and I know people who have the intellectual skills to be a mathematician but not the requisite level of socialization.  It's not pretty but it's true.


If you avoid these and similar negatives in a personal statement and mostly just evince a sincere, not terribly pretentious or twee "I really like math", then I suspect you'll do fine with this part of your application.  You would in my department, certainly.