Marc P. answered 09/21/19
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The quotation sets up a contrast between the "common past" and "common future." Let's examine each of those.
First, think about the common past: Did most Americans in the 1930s share a common past? Did they come from the same place? Did they have similar experiences in their homelands? Did they have similar experiences in this country? Did they enjoy similar liberties? (Remember that the Statue of Liberty has that name for a reason and that it was the first thing that many immigrants saw when entering this country through New York Bay.) It may be helpful to think of some major groups of Americans, including recent immigrants (Irish, Italians, etc. who immigrated from the late 1800s to early 1900s), descendants of the original colonists, and African-Americans.
Now, think about the common future: Did most Americans in the 1930s look forward to similar things? Was it likely that their futures would be (relatively) similar? Did they expect to have similar experiences (living in the same place, shared culture, a common relationship to the government, etc.)?
I think you'll find that the answers to the questions about a common past are mostly negative, while answers to the questions about a common future are mostly positive.
To determine whether Roosevelt really agreed with this, look not only to his words but also to his actions: Were his domestic initiatives (such as those included in the New Deal) consistent with a "common future"?