Stigler's article provides evidence that this problem still persists two decades after Liping Ma's book was first published. Stigler points out that way too many American students arrive at Community Colleges unprepared to perform basic arithmetic. As a result, they are placed in remedial math programs, and many must take two years of basic arithmetic and math before they reach the point that they can take an entry level math course in their Community College:
A student placed in basic arithmetic may face two full years of mathematics classes before he or she can take a college-level course. This might not be so bad if they succeeded in the two-year endeavor. But the data show that most do not: students either get discouraged and drop out all together, or they get weeded out at each articulation point, failing to pass from one course to the next (Bailey, 2009). In this way, developmental mathematics becomes a primary barrier for students ever being able to complete a post-secondary degree, which has significant consequences for their future employment.As I have said, Stigler's article suggests that the problems identified in Liping Ma's 20 year old research continue to dog us today. Stigler says:'
the fact that community college students, most of whom graduate from U.S. high schools, are not able to perform basic arithmetic, pre-algebra, and algebra, shows the real cost of our failure to teach mathematics in a deep and meaningful way in our elementary, middle, and high schools. Although our focus here is on the community college students, it is important to acknowledge that the methods used to teach mathematics in K-12 schools are not succeeding, and that the limitations of students’ mathematical proficiency are cumulative and increasingly obvious over time.I strongly recommend that you read Stigler's article. That's why I've posted two links to the article, one at the beginning of this post and one at the end. What Stigler has done is to analyze more than just the rate of math proficiency, as most articles do. He has drilled into the test results and used them to analyze why students seem to be making errors, and then discusses what those errors disclose about the way that we teach mathematics in the United States. He is telling us, really, that although we have known for decades that we desperately need to change the way we teach mathematics in the United States, we still haven't made meaningful progress in implementing that task.
Our problem is that Americans learn arithmetic and problem solving by rote...by learning a series of procedures, as if we could learn math like learning to recite poetry. We might trace this problem to the victory of the fundamentalists in the math wars of the 70's and 80's in which we were bullied into believing that children who learned to do their sums and times tables and long division were somehow adequately prepared in mathematics. That people who urged that students must learn to reason in mathematics were part of a socialist conspiracy of some kind. Or, this problem might derive from our commitment in the education community to the concept that the same person can teach art, music, reading writing, social studies and science, without actually having demonstrated strength in all of those subjects. Liping Ma's book drills into what American elementary teachers really know, or don't know, about basic arithmetic, and it is painful reading. But Stigler displays the consequence of being taught under the American system of drill and kill.
Perhaps some examples will whet your appetite to actually read Stigler's entire article. When asked to place four numbers (two fractions and two decimals) in size order, only 22 percent of students entering community college got the right answer. Their answers suggested that students believed that 0.53 is smaller than 0.345 because the number with more digits must be larger than the number with fewer. How did they get through elementary school still believing that to be the case? Only 19 percent of students could add a fraction to a decimal (as in 0.5 + 1/4 = 3/4). Only 24 percent of students could add two improper fractions, and the most common error was to add the two numerators and the two denominators. Many students don't understand that a fraction is a division because they were allowed to get by in fifth grade if they could memorize temporarily how to manipulate fractions, without understanding what they are. They have been taught the procedures to use when confronted by pesky little problems like 1/4 plus 3/5, but because they don't have a clue why the procedures work, they soon forget how to apply them, and if they remember the rule incorrectly, they have no way of checking to see if they are correct.
Students had great difficulty solving any problem that needed two steps to arrive at the correct solution. We know, for example, that in many classrooms in the United States, teachers despise having to teach solving what we used to call "word problems." Solving problems with mathematics successfully requires building skills over time, learning the different approaches to problem solving (make a table, working backwards, drawing a diagram, estimating the answer, creating a number sentence) and so on. When some teachers skip over the "word problem" curriculum, because they despise it, then later teachers don't have anything to build on. "I hate word problems," the students say. They learn distance problems (Dirt problems we used to call them) by trying to figure out from the word problem which number in the problem is D, what R probably is and what number in the problem is T, and then put them in the D=RT formula.
We asked students, “Which is larger, 4/5 or 5/8? How do you know?” Seventy-one percent correctly selected 4/5 and 24 percent selected 5/8. (Four percent did not choose either answer.) Twenty-four percent of the students did not provide any answer to the question, “How do you know?” Those who did answer the question, for the most part, tried whatever procedure they could think of that could be done with two fractions. For example, students did everything from using division to convert the fraction to a decimal, to drawing a picture of the two fractions, to finding a common denominator. What was fascinating was that although any of these procedures could be used to help answer the question, students using the procedures were almost equally split between choosing 4/5 or choosing 5/8.I want to conclude by arguing that this issue -- how to teach mathematics -- is as important to improving our American education system as any of the issues that so -called reformers want us to focus on. We can fire teachers who have unacceptable results. We can abolish the seniority system. We can ramp up the voucher system, or expand charter schools without limit. We can rate schools, rate teachers, or berate them. But none of these solutions promise to change the current American approach to elementary mathematics, because frankly, the labor force available for hire have all been indoctrinated into the American way of learning mathematics. If we want to reform the teaching of mathematics, we have some hard work to do and that means we need to start wading into mathematics. The reform movement, or should I call it the reform fad, is trying to sell the idea that we can get something for nothing: improve nationwide mathematics through strategies devised by people who don't understand mathematics or the teaching of it.
Abolishing something isn't good enough. Turning something upside down won't make it happen. We need a strategy to identify what needs to be done, figure out how we are going to deliver to American schools a program of teaching and a labor force capable of teaching a concept based curriculum, and then we need to deliver that program not just to a few lucky kids, but to all of them.
Blog Post Part I
Blog Post Part III
Stigler, What Community College Developmental Mathematics Students Understand about Mathematics."