The third work, is a 2010 article "What Community College Developmental Mathematics Students Understand about Mathematics." Its an excellent study of the mathematics proficiency (or lack thereof) of students entering Community Colleges. MathAMATYC Educator ~ Vol. 1, No. 3 ~ May 2010. Like Liping Ma's book, this article strongly suggests that the American approach to elementary mathematics is failing to provide a concept-based foundation in math, and that this approach is crippling students ability in later life to understand the math that they need to function in high school, post-secondary education, and in their careers.
The "Globally Challenged" report by Harvard's Kennedy School is one of many that warns that American mathematics performance is not up to par:
U.S. students in the Class of 2011, with a 32 percent proficiency rate in mathematics, came in 32nd among the nations that participated in PISA (Program for International Student Assessment). Although performance levels among the countries ranked 23rd to 31st are not significantly different from that of the United States, 22 countries do significantly outperform the United States in the share of students reaching the proficient level in math. In six countries plus Shanghai and Hong Kong, a majority of students performed at the proficient level, while in the United States less than one-third did. For example, 58 percent of Korean students and 56 percent of Finnish students were proficient. Other countries in which a majority—or near majority—of students performed at or above the proficient level included Switzerland, Japan, Canada, and the Netherlands. Many other nations also had math proficiency rates well above that of the United States, including Germany (45 percent), Australia (44 percent), and France (39 percent). Globally Challenged, Are US Students Ready to Compete....Minnesota has ranked relatively high among the American states on measures of mathematics performance. Of all the states, only Massachusetts had a majority of its students (51 percent) scoring at or above the proficiency mark on PISA. Minnesota, the runner-up state, had a math proficiency rate on PISA of just 43 percent.
What is the reason for our sub-par proficiency rates--what can we do to improve? Some people want to argue that its poverty, or language barriers, or other social and environmental problems. Others want to argue that the problems are structural: unions, lack of competition, the seniority system, or our inability to fire bad teachers. But what if, instead, the weight of the evidence compels the conclusion that we in the United States approach the teaching of elementary mathematics in the wrong way. What if so-called "good" teachers and "bad teachers", private school teachers, charter school teachers, good schools and bad, are all approaching math education in the wrong way, because we Americans have just plain adopted the wrong approach to learning mathematics? What if the focus on these non-academic solutions is a symptom of the underlying problem, that we Americans don't really understand the discipline mathematics, and so we try to find a solution that can accomplished without a deep understanding of mathematics?
In short, what if we can't fix American shortfalls in mathematics unless we fundamentally restructure how we teach mathematics?
In this post, and the post that follows, I'm going to summarize the arguments that if we want to close the global mathematics gap, we need to discuss mathematics itself, not the structure of education.
One of the most persuasive arguments for fundamentally changing the teaching approach in the United States is an older book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. The author, Liping Ma, studied the teaching approach and mathematical knowledge of Amercan and Chinese teachers. She argues, anecdotally, that well trained Chinese elementary math teachers are significantly better prepared to teach mathematics than their American counterparts. She argues that in the United States:
- Many elementary teachers are poorly prepared in the mathematics of arithmetic. In other words, they have a poor, or even an incorrect, conceptual understanding of the foundations of arithmetic.
- Elementary arithmetic in the United States is taught with an emphasis on learning procedural rules, rather than an understanding of the mathematical meaning of those procedures. The result is that when students progress to algebra or more complicated arithmetic (e.g. fractions and problem solving) they lack the building blocks that they need to succeed
- That the teaching of arithmetic rushes through basic concepts so rapidly that students don't have time to understand what they are doing. That most elementary teachers don't have strategies to develop mathematical understanding or to make connections between the inter-related building blocks of mathematics.
- That teachers in China spend more time preparing, more time collaborating with other teachers.
- That many teaching elementary mathematics in the United States cannot themselves solve simple math problems, and cannot explain the reasoning behind the solution of those problems.
If Liping Ma's book is correct, our problem can't be solved with a quick-fix or with a structural reform. We have a deep-seated structural problem in the way that students are prepared in the United States, and that problem is infecting our schools of education, our supply of elementary teachers both good and bad, and indeed our entire culture. The problem is our approach to mathematics, not the structure of our public education system iteself. I'll give some specific examples of how this problem manifests itself in elementary school in a future post. But I want to mention that this is not an issue that should be characterized as good teacher versus bad teacher: arguably, the American approach to teaching mathematics in elementary school permeates public and private education alike, and the American approach is used by both good teachers and bad.
Most of the popularly discussed reforms involve changes in compensation rewards, implementation of accountability systems, or the introduction of choice and competition. But what if Lipping Ma's insight is telling us that if we want to improve performance in mathematics, we need to focus on what we are teaching, how we are teaching it, and the preparation and qualifications of those who are teaching it?
A recent conference on the status of teacher development in mathematics is reported in the National Academies Press article that I've listed at the bottom of this post. The section on Teacher training states:
....most elementary teachers in the United States are trained as generalists and do not have [appropriate] mathematics training. In most states they have taken 6 credit hours of mathematics, with that number rising to 12 credit hours in some states. “There’s not a lot of math there.....and many are very uncomfortable teaching mathematics.”" It's not just the number of mathematics courses that teachers have taken: its also the content and the focus on concepts and understanding. For those teachers who don't like mathematics and would really rather not teach it, taking mathematics in college preparatory to teaching may be more about overcoming a hurdle necessary for certification than preparing to teach." See National Academies Press Teacher Development Forum
Charter schools, breaking unions, changing seniority rules -- whatever the merits of these ideas -- don't speak to the great American failure in mathematics, because it is a systemic problem that is passed along in schools of education and in public and private schools alike. We have decided in America, to try to learn mathematics without understanding, by rote, by learning procedures instead of by learning concepts. We've decided that anyone can teach the foundations of elementary mathematics whether they themselves understand it. And as long as this belief permeates our culture, no matter what framework we choose for delivery of education, we are bound to fail.
Blog Post Part II
National Academies Press Teacher Development Forum
Liping Ma, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States.
Globally Challenged, Are US Students Ready to Compete
What Community College Developmental Mathematics Students Understand about Mathematics."