In what manner should one speak in her statement of purpose for phd applications?
Should she speak factually or slightly overly self-praising?
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There is a difficult tension to balance here: one the one hand, you don't want to hide your light under a bushel. On the other hand, the people reading the application will have seen a lot, and will probably be pretty good at detecting somebody puffing themselves up and will not be impressed.
Two useful strategies for approaching this:
Follow the writing strategy of "show, don't tell". If you've done something awesome, don't tell that you were awesome: "I made important contributions to the study of newts." Instead, show the thing that you've done: "I studied newts under Prof. Coolness as a junior, and our paper on how newts prefer turn clockwise under the full moon was recently accepted to the Journal of Arbitrary Herpetology." Give the reader enough evidence that they can conclude awesomeness for themselves.
Once you've got a draft written, ask a trusted professor to look at it and check whether you've left anything important out or are overinflating yourself.