If you look at many kinds of forms and applications, you will probably see either of two instructions. Either a list of races will include the instruction to "choose," or it will ask "how do you identify yourself?" One may think "political correctness" has simplified the process, but in fact it can make it much more complicated.
During my growing-up years, the subject of race was rarely brought up at all. The only exception was being told people who carry more than one race are usually not accepted by any of those races. Although I spent the longest part of my youth in a neighborhood and school that were 100% "Anglo," it was never an issue. In fact, while some members of my family encountered racial discrimination in the past, I never encountered it myself until recent years.
My experiences have been similar to what I was told as a youngster. The first experience did not involve me personally; it was about an acquaintance talking about someone she knew. With a combination of rudeness, sarcasm, and disrespect, she was laughing behind her friend's back, saying since her friend was "only 1/4 Puerto Rican," she "had no business calling herself Puerto Rican." More recently, residing in an area where most people are rabidly racist, I have experienced this form of racism myself.
When it comes to "how do you identify yourself?" previous generations in my family had less difficulty then than I have today. Hyphens were not used; the word 'part' was not used; and the word American described the American population in general. Equally important, how people identified themselves was all there was to it-- no one felt the need to find out if other people agreed with their self-identity. To me, it is partly about my own heritage-- not being willing to dismiss any as unimportant; and partly about the experiences previous generations went through. If a person is to be the sum-total of their entire heritage, every bit of it counts.