“Instead of many arithmetic problems, the homework would contain only three or four questions, and two of those would be ‘explain your answer,’” Heather told me. “Like, ‘One bridge is 412 feet long and the other bridge is 206 feet long. Which bridge is longer? How do you know?’”

She found she could not help her daughter answer the latter question: The “right” answer involved heavy quotation from Common Core language. A program designed to encourage thought had ended up encouraging rote memorization not of math but of scripts about math.

Heather was noticing on the ground some of the same things that caused Stanford mathematics professor R. James Milgram to withhold his approval from the Common Core math standards.

Professor Milgram was the only math content expert on the Validation Committee reviewing the standards, and he concluded that the Common Core standards are, as he told the Texas state legislature, “in large measure a political document that . . . is written at a very low level and does not adequately reflect our current understanding of why the math programs in the high-achieving countries give dramatically better results.”

The Common Core math standards deemphasize performing procedures (solving many similar problems) in favor of attempting to push a deeper cognitive understanding — e.g., asking questions like “How do you know?”

In fact, according to a scholarly 2011 content analysis published in Education Researcher by Andrew Porter and colleagues, the Common Core math standards bear little resemblance to the national curriculum standards in countries with high-achieving math students: “Top-achieving countries for which we had content standards,” these scholars note, “put a greater emphasis on [the category] ‘perform procedures’ than do the U.S. Common Core standards.”

Is there such thing as deep cognitive understanding of math?