How do archaeologist know the purpose of the earthworks and mounds made by the adena and Hopwell

Some were used for burial mounds , the presence of human remains would reveal that purpose.

Explanation:

When English and Scottish settlers first arrived in what was to become the United States, they encountered literally thousands of abandoned earthen and shell mounds that seemed not to be associated with occupied Indian villages. Typically, the new arrivals assumed that the “savages” were intellectually incapable of carrying out major public works. Therefore, they speculated that Europeans or advanced societies from the Middle East had once lived in the New World until they were exterminated by the Indians. It would not be for another 200 years that the public would become generally aware that about 90-95% of the societies who built those mounds had died of diseases or had been enslaved in the decades following Spanish exploration of the region.

Some tribes in the Lower Mississippi Valley were still occupying mounds when French settlers arrived, so there was no French speculation about the origin of abandoned mounds. The best known of these last mound builders were the Natchez. They also stopped building mounds after the 1720s.

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