But in the past 30 years or so, there have appeared immigrants with a new attitude, who want to remain Puerto Ricans, or Cubans, or whatever. This is - in the 300-odd year history of immigration - something new, and opens up the promise, but also the problems, of a multi-cultural society.
Today the "new Asians" are graduating (at the top of their class) from high school or college. Asians who came to the United States without a word of English, who fled with their parents when the North Vietnamese moved down to conquer the South, and are now moving easily into the professions. These doctors of several specialities - lawyers, professors, scientists - some of them in their 30s, are already getting ahead of the best in their field.
They are the children of refugees, from Korea, from Vietnam, and from Pol Pot's dreaded Cambodia. The most striking thing about them is that they do better in school and better in college, and when they move into a profession are superior to the other two big minorities: the blacks and the Latin-Americans.
This very day, the first big story I read in a San Francisco paper carried the bleak headline: "Grand jury calls San Francisco school integration a failure. " I don't know how I can convey the truly shocking impact of that headline. It would have been an impossibility 40 years ago - shortly after the Supreme Court handed down its historic ruling which abolished the old rule (for blacks) of "separate but equal" in education and required the integration of all public services: theatres, restaurants, buses, jobs and so on. The most dramatic ruling of my time, and the most promising.
We guessed that within 10 years - say 20 at most - blacks and whites would go to the same schools together, and we should see, if not the millennium, then a newer, freer America that had shaken from its society the stigma of race.
What an inhuman hope! It developed that integration of schools required a minority - sometimes of whites, sometimes of blacks - to be bussed five miles, 10 miles, or 30 miles, to the common school.
We know now the appalling social complications of doing this: of taking children out of their neighbourhood, their life, for a part of the day - and then back; for the whites usually to the comfort of a better neighbourhood, for the black children too often a return from an undreamed-of outer world to something they now saw as a ghetto.
It would take until Christmas to detail the social and legal problems that cities encountered in trying, in good faith, to maintain integrated schools. We can now admit - sadly - that in many if not most places it has not worked. Why? Not yet has there been born what the Supreme Court might call a "colour-blind generation".
Anyway, the conclusion of the report by this civil grand jury, which had been sitting for a year reviewing the state of integrated public education in this city, is that while there is a racial balance in the 64,000 district students, it has failed in the ultimate purpose: there is no equality of scholarship or achievement between the whites and the black and the Latin-American students. They are still way behind.
The most striking thing about this report is what isn't there. No mention of Asian Americans, who do better than anyone in school.
I've thought about this and talked about this in this city with many sorts of people. And to the question "Why do Asian children excel in school?" the answer - which I'm afraid is not susceptible to statistical proof - comes from the students themselves, and often quite spontaneously: "We study hard to have our parents proud of us."
I find that this desire for family pride, is paramount in young Asians. Perhaps their children - the first Americans - will lose it. But there it is. It cannot be legislated. I doubt it can be taught. It is part of an Asian tradition, as a child said, "to be worthy of my family. That is why I work hard".
This is an edited extract from Alistair Cooke's Letter from America first broadcast on BBC Radio 4 on August 8