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."
This was my initiation into the world of reform math. It is a world where understanding takes precedence over procedure and process trumps content. In this world, memorization is looked down upon as "rote learning" and thus addition and subtraction facts are not drilled in the classroom--it's something for students to learn at home. Inefficient methods for adding, subtracting, multiplying, and dividing are taught in the belief that such methods expose the conceptual underpinning of what is happening during these operations. The standard (and efficient) methods for these operations are delayed sometimes until 4th and 5th grades, when students are deemed ready to learn procedural fluency.
The idea is to teach students to "think like mathematicians." They are called upon to think critically before acquiring the analytic tools with which to do so. More precisely, they are given analytic tools for "understanding" problems and are then forced to learn the actual procedural skills necessary to solve them on a "just in time" basis. Such a process may eliminate what the education establishment views as tedious "drill and kill" exercises, but it results in poor learning and lack of mastery. Students generally work in groups with teachers who "facilitate" rather than providing direct instruction.
What I've described isn't the case in every K - 6 classroom in the U.S. but it is happening with enough frequency that it is becoming more the norm than the exception. The effect has been noticeable to high school math teachers whose algebra students do not know how to do simple mathematical procedures. At education schools, these reform techniques are taught to young and eager students who think the theory sounds wonderful. They learn that textbooks are bad and that teachers should veer from them as much as possible, supplementing with inquiry-based and student-centered assignments.