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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:

Algebra 1
—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

Algebra 2
—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."


I am not surprised by some of these results from the student body. And I do agree that some of the polynomial and rational functions do require a greater deal of proficiency in manipulation of equations. However, I think we, as teachers, are expected to cover such a broad range of topics by our state standards that I am not surprised that most students lack this ability to perform well on test involving polynomial equations. We need to spend more time mastering manipulation and even more time on WHY and HOW to use these skills that they should acquire

I wouldn't worry too much about an area where only 19% of the students get it, when the best any area does is 26%. Should we put a great emphasis on the 19% areas and push to get them up to 26% also? Would that be success? I don't think so. Some possible explanations. Some schools might really try to teach students to be college ready and know that there will be no questions about statistics or probability on their student's placement exam in college, and, thus, not spend too much time on statistics and probability, leading to lower scores there. As for "non-linear" functions, whatever that means for algebra I students (I know what it means: 1/x, exponential, and maybe, if lucky, a dab of quadratics, to use the phrase "non-linear" is nonsense), and exponential functions, there aren't too many serious questions you can deal with in high school about exponential functions, so they might seem more proficient just for that reason. Steve

We need to rethink how math is taught from K - 12. Too many elementary teachers are math phobes and functionally illiterate in math. They operate on a low math level and communicate their dislike for the subject to the kids, who are only too happy to find excuses for not having to think.
There are reading specialists in elementary school -- why not math specialists? Kids spend 3 hours daily on language arts in some schools, leaving about 45 minutes for math. The kids are sometimes grouped by ability but by the time they travel to another classroom and take out their books, maybe 20 minutes of instructional time is left. And have you seen the books they use?? Calling them useless is a compliment.
So what's the solution? Better teacher education in how to teach math, and a huge attitude change. How many adults would proudly say that they are just no good at English? Yet admitting to math deficiencies is socially acceptable, even among educators. Ever notice how math and science teachers end up sitting together at lunch because no one else will talk to them? Algebra is not a dirty word and it is not a mystery. Treating the subject as a huge deviation from the mathematical continuum only opens the door to excuses for not working at it.

Completely agree with both Steve Wilson and Carolyn Melo.

Merriam Webster has this definition of algebra: "al·ge·bra: a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic". Algebra is defined according to the rules of arithmetic, but so many students see no connection whatsoever between arithmetic and algebra. They most often learn those rules of arithmetic as factoids, somewhere in between the math facts and algorithms that they memorize. Teaching memorization and calling it math leaves students unprepared for algebra, where memorization is not so useful. Instead, students need to learn to USE those rules. They are buried in the standard algorithms, and they are the same rules that are used to solve algebraic equations.

In a nutshell, start with "Operations on Numbers and Expressions". Everything else follows from that.

I consider myself both a mathematician and a math teacher - although my title is engineer, my work (and my education) are based in math; and although I formally taught only briefly I continue to teach both within my technical work and with educational outreach.

National math test scores continue to be disappointing. This poor trend persists in spite of new texts, standardized tests with attached implied threats, or laptops in the class. At some point, maybe we should admit that math, as it is taught currently and in the recent past, seems irrelevant to a large percentage of grade school kids.

Why blame a sixth grade student or teacher trapped by meaningless lessons? Teachers are frustrated. Students check out.

The missing element is reality. Instead of insisting that students learn another sixteen formulae, we need to involve them in tangible life projects. And the task must be interesting.

A Trip To The Number Yard is a math book focusing on the building of a bungalow. Odd numbered chapters cover the phases of the project: lot layout, foundation, framing, all the way through until the trim out. The even numbered chapters introduce the math needed for the next stage of building and/or reviews the previous lessons.

This type of project-oriented math engages kids. It is fun. They have a reason to learn the math they may have ignored in the standard lecture format of a class room.

If we really want kids to learn math and to have the lessons be valuable, we need to change the mode of teaching. Our kids can master the math that most adults need. We can’t continue to have class rooms full of math drudges. Instead, we need to clandestinely teach them math via real life projects.

Alan Cook
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If we really want kids to learn math and to have the lessons be valuable, we need to change the mode of teaching. Our kids can master the math that most adults need. We can’t continue to have class rooms full of math drudges. Instead, we need to clandestinely teach them math via real life projects.

This view seems to prevail in the never ending dialogue about how to teach math. There is a presumption that math is taught in a rote fashion with no conceptual understanding. In fact, many of today's textbooks do just what the commenter is suggesting; lots of "real world" applications. The problem is that the mastery is left out. What students need is proper instruction (not left to discover things on their own, with time to practice and master the skill.

Students enjoy being successful and being able to solve problems. Most math problems in the K-8 range are directly applicable to everyday situations, so it's relevance is rarely disputed. As far as algebra, if students are successful at doing what is taught, it is not necessary to bend over backwards finding "real life" problems that are for the most part contrived, more tedious than rote learning, and frequently amount to "data mongering".

As an elementary teacher of math, I see value in teaching fundamentals in depth, with a heavy dose of practical play where math is not drudgery but fun. I realize this removes the percieved sweat from the work, however, it may reduce the anxiety from those who may find it difficult to grasp the symbology.

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