Before I get to the work of Dr. Loewenberg Ball and others, I want to begin with a parenthetical point. Whenever one advances this ideathat we need to change our approach to elementary mathematics  immediately someone steps forward to say that, yes, we need to use "manipulatives" and teach math to elementary students more concretely. Using math manipulatives is often the centerpiece of reform, because installing manipulatives allows us to give the impression that we are doing something really different, and because children enjoy working with blocks, leggos and other concrete things. But adding baseten blocks, set cards, fishcounters, and cuisenaire rods, whatever their benefit in the classroom, completely misses the point. When manipulatives are used by teachers who lack "mathematics for teaching", the manipulatives are not likely to solve the fundamental problem in American elementary mathematics....which is that we aren't teaching for understanding.
The argument for fundamental reform begins with the this simple idea: that in order to teach mathematics well, a teacher must know mathematics, but to know mathematics in a special way that makes teaching for understanding possible. It is true that just about anyone can teach, and just about anyone can learn, tbasic math facts  sums and times tables, how to subtract integers and do long division. It is possible to learn these simple chores with a deck of flashcards and lots of repetition. There was a time when a person who could do his sums, who could make change at the cash register, was considered to have mastered enough arithmetic to get by. But these days, cash registers can do the arithmetic for you, and so being able to make change is barely enough to keep a minimum wage job. More and more of us need to use mathematics for applications in the world of work that require understanding. If we do not make the transition to concept based learning in mathematics, other countries are going to pass us by in engineering, computer sciences, telecommunications, architecture, banking, insurance, and a host of other areas that require application of mathematics to solve problems.
As I've been collecting information to write these posts, I ran across a great power point, online, that makes the points I've been advancing way better than I could ever do. Its called "The Case for Elementary Mathematics Specialists", authored by HungHsi Wu of the University of California at Berkeley (Cal) and Deborah Loewenberg Ball, Dean of the University of Michigan school of education. Dr. HungHsi Wu, a research mathemetician and Ball, a teacher and now Dean, argue that its asking too much of traditional elementary teachers to attempt to teach mathematics the way it needs to be taught. The slideshow summarizes ideas expressed in Ball's article in the American Educator, "Knowing Mathematics for Teaching." (Fall 2005).
Ball and Dr. HungHsi Wu point out that a lot of people in the United States believe that it should be really easy to teach math  that math is just a matter of learning to add, subtract, multiply and divide. If you're one of those people who believes that we just need to "get back to the basics", or even if you are not, I urge you to spend a few minutes reading through the power point, or the "Knowing Mathematics for Teaching." Both are great supplements to the book by Liping Ma, that I discussed in my first post on elementary mathematics. The point is, that"in order to really teach mathematics successfully, a teacher needs to explain the mathematical foundation of basic arithmetic, and "explaining mathematics...is not all that easy." The point here is that you can do cash register math without understanding the mathematics behind elementary mathematics, but failure to understand elementary mathematics in a deeper way disables the teacher from teaching, and the a student from learning, enough math to serve as an appropriate foundation for high school and college coursework using math skills.
Here is an example from the slideshow mentioned above.
Many teachers can teach children to do multidigit multiplication...that is the procedure commonly used to multiply with paper and pencil...but how many elementary teachers can explain why his computation correct, and why it work?
82
x 59
738
410
4838

In her book, Lipping Ma, discusses interviews of both American and Chinese teachers, in which she asks elementary teachers to explain multidigit multiplication, and the answers are tremendously revealing. When they were asked whether the answer would be different if a zero were added as a place holder after the 410 in this problem, many American teachers answered that adding a zero would be improper, because it would change the answer, or create mathematical confusion. Some American teachers said that if a student wanted a place holder after 410, an X should be used, not a zero, to make sure that students were not confused into thinking that 4100 was the same as 410. In contrast, her interviews with Chinese teachers suggested that many had a much deeper understanding. They saw multidigit multiplication as using the distributive law: (82)*(59) = (82) * (50+9) = 9*82 + 50*82.
"Clearly, being able to multiply correctly is essential knowledge for teaching multiplication to students. But this is also insufficient for teaching. Teachers do not merely do problems while students watch. They must explain, listen, and examine students' work. They must choose useful models or examples. Doing these requires additional mathematical insight and understanding.An elementary teacher should be able to:
 Analyze student work and understand what misunderstanding has contributed to an error in the answer.
 Represent the meaning of mathematical algorithms or solutions that illustrate why they work.
 Use the language of math in ways that are understandable to students, yet do not distort the fundamental mathematical principles.
 Answer student questions like "why does this work," and "why can't we do it this way".
 Find alternative ways to explain procedures, so that students have multiple windows on mathematical understanding.
Those of us who want to "reform" education need to recognize that reforming education means reforming the way we deliver instruction. Whether we organize the delivery education into charter schools, traditional public schools, or some nonunionized Walmart  style education factory, if math education is delivered by teachers who don't have "mathematical knowledge for teaching," the results are going to be the same. Too often, we believe that changing the organizational structure of education will magically transform what students learn. But in mathematics, and in other fields, putting old wine in new bottles is not the solutoin. When we discuss the reform of education, we need to focus, I believe, on the mechanism by which those reforms are going to actually translate into better math, better writing, stronger literacy skills, and better understanding of history, economics and civics.
Blog Post Part II
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