What role did Prussia play in reforming education within the United States?

They don't.

I'm assuming you meant Russia, right?

War with Prussia encouraged the United States to begin reforms after realizing illiteracy was a problem for many soldiers. Prussia sent a number of poorly educated immigrants to the U. S., which caused reforms to focus on achievement. Reformers in the U. S. adopted the idea that students should have access to the same content at similar grade levels. The Prussian ambassador to the U. S. was an outspoken education reformer who collaborated with Horace Mann.