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That's generally accurate. The heavy hitters, with a few exceptions like Faraday, were of noble ancestry or otherwise privileged. We really need to look to the Americans, to Horace Mann and John Dewey who understood the wider non-functional ends of education; nation building, supporting democracy, living a happy life etc. That fed back into British society mainly post-war, driven by the need to rebuild a devastated society. Today, in our coddled complacency, that seems far away and 'optional'.