...Beyond the testing difficulties, one of the criticisms of the Common Core, in New York and elsewhere, is that it can be too demanding for young grades. Diane Ravitch, an educational historian, has said that very little of what is taught to first graders about ancient civilizations will stick with them; Mr. Hirsch and other defenders of the Common Core say children in early grades need lessons in history, civics, science and literature to build vocabularies and thrive.

In interviews with a range of teachers in New York City, most said their students were doing higher-quality work than they had ever seen, and were talking aloud more often. But it has not come without sweat. Homework is more complex and takes longer, several said, and in some cases is frustrating parents.

Teachers also said that pupils who were already struggling, particularly those who speak limited English, were facing greater challenges. Nonnative speakers are having a harder time in math because the new curriculums require greater use of word problems....