You can look at this two ways, and either way you get the right answer. 'while' could be seen as a conjunction here, in which case the phrase is an ellipsis of 'while I was reading', in which case it's a participle. 'while' could be seen as having the function of a preposition here, and as prepositions are generally followed by the gerund, then you could consider it to be a gerund Either way, you need the 'ing' form after 'while'. 99% of native speakers don't have a clue what a gerund or a participle is, and they don't worry about it. They get it right because they've heard it, practised it, and know that it 'sounds' right. So (by way of example to illustrate your previous question about 'try') you should try to do the same. Try listening to lots of English songs, try talking to native speakers on skype, try watching films with and without subtitles - try to get a 'feel' for what sounds right.