Why might some believe... the earth is round?

Really ancient Greeks

...Such as Pythagoras, believed, for aesthetic reasons, that circles and spheres were perfect forms, and that therefore the earth was most likely spherical.

But they also considered observations that supported a round earth...

Ships dropping below the horizon

Go south and see different constellations

Aristotle travelled from Greece south to Egypt, and noted that different constellations could be seen, which could not be seen from Greece.

Demonstrate this with two student: Most of the class are the traditional constellations. As a student moves parallel to the front of the classroom (to simulate being on the flat earth).

The "new constellation" student is positioned slightly behind the moving student, and cannot be seen for parallel motion. But if the student pivots while moving...

20th century -- Pictures from space

Here's a picture snapped by the astronauts from the Apollo 17 mission to the moon...

21st century -- Using logic...