The Choice Activity is a new kind of assignment where the student has to choose from two assignments. The drop box will not show for the student (and the assignment will not show in the grade book) until the student chooses.
The student will need to click on the Choose an Activity link, then choose one of the choices listed. When that is done, they are presented with the instructions for the assignment. Before they start the assignment, they'll need to click on the +CHOOSE ACTIVITY link (in the top right corner) again. Now the assignment will display in the grade book and the student will see the drop box.