## Saturday, April 6, 2013

### Learning algebra vs. "learning algebra"

In my weekly math comparisons, I've recently been exploring what today's Reform algebra texts leave out. Here's a partial list:

-Factoring polynomials of degree greater than two.
-Rational expressions with more than two terms in the numerator and denominator.
-Multiplication of more than two polynomials or of polynomials that contain more than two terms.
-Polynomial long division and other multi-step methods for simplifying multi-termed rational expressions.
-Systems of equations that involve more than two variables.
-Problems that require manipulation of polynomials in order to simplify them or convert them to special (often abstract) patterns--e.g., by adding/subtracting the same term to different parts of the expression/equation, or by rearranging terms and grouping them together in a structural hierarchy (see the second problem set in last week's Problems of the Week).
-Word problems requiring students to define multiple variables and set up multiple equations.

Not only is much of Reform algebra limited to one or two terms, one or two variables, powers of one or two, and problems involving only one or two steps; many of the problems only require students to plug in numbers or push buttons on their graphing calculators. Essentially, Reform algebra doesn't get you much beyond Reform arithmetic.

Luckily, some students are still using more traditional algebra texts. But this means there's great variation in what different students are learning in the name of "algebra"--the focus of an article in this week's Education Week. It describes a study by the National Center of Education Statistics of high school algebra and geometry courses that analyzed transcript data from 17,800 students and content data from "120 Algebra I, Geometry, and integrated math textbooks used at the 550 public schools those students attended." Its conclusion:

Students taking Algebra 1 and Geometry classes are getting considerably less substance than their course titles would suggest.
...
The study found that, on average, two-thirds of topics covered in Algebra 1 and Geometry courses focused on core content topics in each of those subjects, while the other third covered topics in other math areas. Researchers also gauged the rigor of classes based on the topics and questions covered in each book. A course categorized by researchers as beginner-level algebra had more than 60 percent of its material on elementary and middle school math topics such as basic arithmetic and pre-algebra problems such as basic equations. By contrast, a rigorous Algebra 1 courses included more than 60 percent of material on advanced topics such as functions and advanced number theory, as well as other higher-level math subjects such as geometry, trigonometry, and precalculus.
“We found that there is very little truth-in-labeling for high school Algebra 1 and Geometry courses,” said Sean P. “Jack” Buckley, the NCES commissioner, in a statement on the study.
...
For example, a student taking a rigorous Algebra 1 course covered 11 topics in advanced number theory, compared with only six for students in courses with the same name that researchers classified as beginner- and intermediate-level classes. A student in an Algebra 1 class ranked by the study as beginner-level had no exposure to advanced functions, and more than a quarter of the class was devoted to basic arithmetic and pre-algebra. A student in a rigorous Geometry class likewise covered significantly more topics in coordinate and vector geometry, and significantly fewer topics in basic arithmetic and pre-geometry, than a student in a beginner-level Geometry class.
The article turns to what this means, not for the worthiness of today's Reform books, but for the racial/ethnic achievement gap. It observes that, while differences in how many math credits appear on the transcripts of white vs. black graduates have narrowed, differences in what different ethnic groups are actually learning is another story:
There were no significant differences in the proportion of students of different racial groups who took rigorous Algebra 1 courses—roughly a third of each group—though Hispanic and Asian and Pacific Islander students were more likely than other groups to take beginner-level algebra courses. However, the NCES found that more white students in honors Geometry classes, 37 percent, covered rigorous topics, compared with 21 percent of black and 17 percent of Hispanic students in similarly titled classes.
Perhaps our math texts should be less concerned with multiculuralism, and more with multiplication, unlike what this Integrated Math textbook index (reposted from last week's Problems of the Week) suggests:

Auntie Ann said...

We had this argument with one of our kid's schools. They said they teach algebra in 8th, we said that if you look at the sequence, it made it impossible to get to calculus in high school with their 8th-grade algebra curriculum. There was an extra year in the sequence that basically spread algebra out another year. They could claim to teach it, while most parents didn't bother to notice the limitations.

lgm said...

Yep, noticed that here when I saw the quadratic formula wasn't included in Integrated Algebra 1.

Same thing has happened in elementary. Many units discarded in favor of inclusion.

It's all about gap closing by preventing anyone from 'getting ahead'.

The next 'keep 'em down on the farm' action is to reject the CC math courses that students in diverse public high schools are forced to take at their own expense in lieu of rigorous college prep courses, which were cancelled in order to provide more double period algebra one courses plus remedial courses. Many non-state colleges do not accept transfer credit from dual enrollment CC courses held on the high school campus - which in some schools is the only way to cheaply take PreCalculus, Calculus, and Diff. Equations. And rightly so, as these classes are half review of the prior class in order to be inclusive and do not cover sufficient material.

Cheryl said...

Are there any currently-published US algebra texts that you would recommend for the left-brained child? Or, failing that, a few of the least-right-brained texts from the current offerings?

Katharine Beals said...

Hi Cheryl, I'm less up on good contemporary algebra books than others are (commenters, please weigh in!), but I've heard from many people who have recommended that Art of Problem Solving.
http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra