First, let me make it clear that the question in the post title does not refer to cultural contexts involving student-teacher interactions. If a teacher raised in an Italian neighborhood in New York City finds themselves teaching in the rural Midwest, they should probably keep in mind that some of the people might be a bit more reserved than people back home. Rather, I'm referring to the cultural context of the educators, officials, and members of the public debating over the purposes and methods of mass education.
The modern education reformers mostly make a genuine and sometimes very successful effort to be scientific in their approach. That is to be applauded. If they have reproducible data from carefully-controlled experiments, one might argue that it doesn't matter if they happen to be cultural heirs of Great Awakening preachers rather than Puritan essayists. I nonetheless maintain that it does matter for 3 reasons:
1) Truly double-blinded, single-variable experiments are often impossible in social science. The person who walks out of the workshop speaking of how "transformed" they are is not going to simply adopt a new teaching technique. They are going to adopt it with enthusiasm and passion and come into class with a degree of energy and preparation that they might not otherwise project if they were doing the same-old-same-old. The effect of the experimenter's interests on the experimental subjects is of old vintage in social science, being named the Hawthorne Effect after an experiment on work conditions at a factory in the 1920's. It is probably similar in mechanism to the placebo effect in medicine.
I maintain that there are cultural factors intimate that determine which interventions will resonate with the instructor, cultural factors that are entwined with American views on democracy, equality, the role of the expert, the practical value of knowledge, and even religion. You cannot understand the enthusiasm that American educators will have for a teaching method, and the way that they will transmit that enthusiasm to their American students, unless you understand American cultural history.
So far I've argued that most educational experiments are two-variable experiments (method and enthusiasm) rather than 1-variable experiments (method only). But I would go farther and argue that there is often a third variable, one that is also laden with cultural baggage:
2) Adoption of a new instructional method is often accompanied by a change in what is taught and tested. That is not a priori bad from a normative standpoint, but it certainly complicates interpretation of experimental data. In physics, for instance, there are different schools of thought on the relative importance of "conceptual" understanding and more traditional emphases that might, for lack of a better term*, be termed "mathematical", "calculation", or "problem-solving." The precise nature of the difference is less important than the fact that if two classes emphasize different things AND use different instructional methods, and then both groups of students take the same test, a test that happens to better align with what was emphasized in one class but not the other, you cannot be sure what to attribute differences in performance to.
Now, the Puritans and the Awakeners were certainly not debating electromagnetic theory (excepting Puritan descendant Benjamin Franklin, of course), but the different approaches in physics do roughly correspond to a difference in how mathematics is used or emphasized, and in American culture mathematics is a subject that is often perceived as carrying heavy intellectual baggage. Americans believe that you have to be a real Einstein to "get" math. It's a subject for "smart kids" (and is often lamentably stereotyped as being for smart boys but not smart girls). If a pedagogical divide happens to include a divide over the aspects of the subject that carry the heaviest intellectual and cultural baggage, it is fair to examine the ways in which these pedagogical schools might emulate different American traditions concerning experts, intellectualism, etc.
3) However scientific the founders of a movement might be in their scholarly approach, however empirically-driven their proposals might be, no movement can scale beyond small cohorts of enthusiasts without escaping the control of its founders and getting caught up in broader cultural currents. If reformed pedagogy ultimately enjoys mass adoption it will become its own thing, subject to the biases and quirks of adopters, the pecuniary interests of publishers, the agendas of school administrators and public officials, and the wider culture of the Academy, to name just a few among a multitude of factors. Indeed, Hofstadter's penultimate chapter notes that Dewey, like many a Great Man before and after him, ended his career shaking his head at the excesses of his followers, people who tried to institutionalize the Revolution.
Finally, one might note that I keep speaking of American culture, even though many college professors are immigrants and many of our students are immigrants or first generation Americans, and mostly do not have any ancestors who were either Puritans or participants in the Great Awakening. My first answer is that there is nobody more American than an immigrant. Whenever I examine cultural strains that have been repeated again and again in American history I am examining events that included the participation of innumerable immigrants. My second answer is that culture is not transmitted solely by parents. If America is good at anything, it is good at assimilating the children of immigrants into our culture. Schools, entertainment, churches, neighborhood kids, employers, and other factors all conspire to take children with parents from every corner of the globe and make them into bog-standard Americans (for good or for ill).
I am thus comfortable in saying that American cultural history is an absolutely crucial background to educational debates, even if we should have a Middle Eastern immigrant professor teaching the theories of German physicists to the children of Latin American immigrants (a common enough occurrence in my department).
Indeed, let me gaze at my own navel: To my knowledge, I have no
Puritan roots in any meaningful sense. What English blood I have is
from people who either moved to the South or came to Canada in the
1800's. To the extent that I identify with any culture other than
Midwestern White, I identify with southern Italians because of my very
close bond with my grandfather, who was born in the Midwest but was
raised in a family and community of southern Italian immigrants. I
attended a Franciscan grade school that was founded primarily by Slovak
immigrants. I attended college and grad school in California, not at a
New England school of Puritan derivation. Even a lot of my academic
mentors did most of their training at schools not founded in the Puritan
tradition. (Though there is nonetheless a bit of New England in my
academic family tree.) Despite that, through my interest in books and
science, and my subsequent sojourn in higher education, I internalized
many cultural traits of America's intellectual classes, and thus there
is a strong dose of Puritanism in the cultural patterns that I have
internalized. Much of this Puritanism comes from my teachers and relatives and friends who also have no direct ties to New England but have likewise absorbed those values through American culture.
*We traditionalists have, to our discredit, been bad at articulating precisely what we value and the ways in which it differs from the approaches termed "conceptual."