A Closer Look at Students' Weaknesses in Algebra (Updated)
I'm neither a mathematician nor math teacher. Plenty of the readers of this blog do fall into those categories, however, and today I'm seeking them out.
A new report by Achieve, released today, shows students in 13 states struggling, big-time, with algebra, both at the introductory and advanced level. More than 80 percent of students in each of the states (which took part voluntarily in the exam) were not prepared for college-level math in Algebra 2, by the standards of the test. Those results won't strike a lot of people as surprising, given the fact that students are flummoxed by algebra, and that this exam was designed to be an especially tough one.
Yet the Achieve report also includes breakdowns of where students struggled the most, by algebra topic. In Algebra 1, it was in data, statistics and probability. They did better, on the other hand, in non-linear relationships. In Algebra 2, students had difficulties with polynomials (a math expression with three or more terms) and rational functions. They fared a bit better on exponential functions.
Here's a snapshot of the percent of students reaching "mastery," as defined by the test, by category:
—Non-linear relationships, 26.5 percent reached mastery
—Linear relationships, 24.6 percent
—Operations on Numbers and Expressions, 22.5 percent
—Data, Statistics, and Probability, 18.9 percent
—Exponential Functions, 24.3 percent
—Function, Operations, and Inverses, 22.7 percent
—Equations and Inequalities, 21.8 percent
—Operations on Numbers and Expressions, 20.2 percent
—Polynominals and Rational Functions, 18.8 percent
A couple questions for readers who are tasked with explaining these math concepts to students every day—either at the K-12 or college level: Are these results what you would have expected? Do you find that your students tend to flail in data, statistics, and probability, and polynominals, more than other math topics? Or could these results simply be a function of this test's content?
UPDATE: Here are some thoughts on the question I posed from William McCallum, who directs the mathematics department at the University of Arizona. I wasn't able to get his comments about students' specific algebra shortcomings in my original story. While he notes that his interpretation would depend on knowing more about the test items, he also says:
"[P]olynomials and rational functions are a topic that many students struggle with because they require a real proficiency in algebraic manipulation that goes beyond just being able to perform the steps." That type of problem-solving "really requires an ability to step back from a calculation," he added, "and foresee which way it's going to go, and maintain some supervision of the calculations to detect error...This is a higher level of proficiency in symbol manipulation than many students acquire."
The Achieve test also found that students struggled most on constructed-response math questions, as opposed to multiple choice. Said McCallum:
"[Of] course [these] are always going to be more difficult, because they require an independent ability to plan a solution and marshal techniques, rather than just perform the techniques. But I have to believe that the large number of students who got zero on those is partly (perhaps largely) the result of the test not having any consequences, so that students would have just blown those off."