A method of fieldwork where someone studies a culture by doing what the people of that culture do is called

The very definition of Participant Observation is a method for data collection in research where the researcher introduces themselves into the field of study in order to collect more detailed and meaningful data for a prolonged time period (months to years). All of the other methods mentioned don't require being a participant in the culture being investigated.

Anthoprology

Explanation:

Anthropologist: A scientist who studies societies and cultures.

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that was the type of government the spanish colonists were used to.