Today's fashion is to throw away the textbook and to teach kids to think like mathematicians. The problem? They're not learning how to do actual math.
In The Atlantic's ongoing debate about how to teach writing in schools, Robert Pondiscio wrote an eye-opening piece called "How Self-Expression Damaged My Students." In it, he tells of how he used modern-day techniques for teaching writing--not teaching rules of grammar or correcting errors but treating the students as little writers and having them write. He notes, however that "good writers don't just do stuff. They know stuff. ... And if this is not explicitly taught, it will rarely develop by osmosis among children who do not grow up in language-rich homes."
What Pondiscio describes on the writing front has also been happening with mathematics education in K-6 for the past two decades. I first became aware of it over 10 years ago when I saw what passed for math instruction in my daughter's second grade class. I was concerned that she was not learning her addition and subtraction facts. Other parents we knew had the same concerns. Teachers told them not to worry because kids eventually "get it."
One teacher tried to explain the new method. "It used to be that if you missed a concept or method in math, then you were lost for the rest of the year. But the way we do it now, kids have a lot of ways to do things, like adding and subtracting, so that math topics from day to day aren't dependent on kids' mastering a previous lesson."