Meaning of 'scholarly approach' for graduate award nomination
Actually, I don't think "scholarly approach" is obvious even to a native anglophone. Consider the following a guess. Others may supply additional suggestions, I hope. 

If you have rationalized how it is that you find (have found) interesting research problems, then you can say something about that. How do I go about deciding what is worth exploring. That is a useful question for any researcher to be able to answer. It also seems related to things about your research experience. 

Since you are being considered for an award, I assume that you don't need, anymore, to rely heavily on your advisor(s). So I suspect that you may have given that some thought already.