Final answer:
Shiao and Tuan used an ethnographic research methodology to study Korean Adoptees and their ethnic exploration. Ethnography involves observing and interacting with subjects to understand cultural phenomena from their perspective, without prioritizing hypothesis testing.
Step-by-step explanation:
The methodology used by Shiao and Tuan in their research on Korean Adoptees and the Social Context of Ethnic Exploration was ethnographic. This type of methodology is characterized by qualitative research that aims to explore cultural phenomena where the researcher observes society from the point of view of the subject of the study. An ethnographic study typically focuses on how subjects view themselves in relationship to the community, and does not prioritize hypothesis testing.
Given the nature of the Shiao and Tuan research, which likely involved detailed observations and interactions with the subjects to understand their ethnic exploration within their social contexts, an ethnographic approach would be most suited to capture the nuanced experiences of Korean adoptees. This is in line with other social research methodologies which may include surveys, case studies, interviews, and secondary data analysis, all having their own strengths depending on the research question at hand.