Cracking the code: Why, how and when should students learn algebra?
By:
Kristi Garrett
In the late 1990s, Louisiana investigators were tormented by a brutal serial rapist who had eluded them for almost a decade. The Lafayette townspeople clamored for justice and the tips kept flooding in, yet the officers were stymied, faced with a thousand possible suspects.
Intrigued by the case, a Canadian detective and mathematician plotted the locations of the attacks and, factoring in other facts in the case, developed a mathematical formula that ultimately identified the area where the killer lived. The detective, Kim Rossmo, said it was like looking at the pattern created by a spray of water to determine where the sprinkler head was located. With the search area dramatically narrowed, the assailant was soon apprehended.
Rossmo’s formula—which he calls Rigel’s algorithm—got star billing during the pilot episode of the television crime drama “Numb3rs.” Interestingly, dozens of reviewers disparaged the math used in the show as implausible. Even the reputable New York Times said, “It is hard to believe the show won’t drive real mathematicians to double over in laughter,” calling the use of mystifying computer algorithms to solve crimes “problematic.”
Such widespread ignorance of the beauty and grace of advanced mathematics saddened Keith Devlin, a Stanford University math professor also known as National Public Radio’s “Math Guy.” “Maybe one day, TV critics and others will not be surprised and incredulous when they learn that math is used in different ways in many walks of life,” he wrote after the show first aired.
Unfortunately, many adults remain baffled by the string of variables, exponents and mathematical symbols in a typical algebraic equation. The fact that few people use it in their daily lives seems to validate students’ persistent objection: “Why do I need to learn algebra?”
An international trend
The need to remain competitive in a global economy has accelerated an international trend toward not only teaching algebra to all students, but doing so earlier. Most developed nations now introduce students to algebraic concepts well before high school, and California adopted standards in 1997 that incorporate algebra throughout the elementary grades.
Last year, California’s State Board of Education directed that all eighth-graders take algebra to comply with the federal No Child Left Behind Act’s requirement that all students take a test aligned with the standards for their grade level. CSBA’s Education Legal Alliance was joined by other education groups and the state superintendent of public instruction in successful legal action to prevent the SBE from implementing the plan before its impact on schools could be considered. While the state mandate is now on hold pending trial, many school districts and county offices of education remain set on a course of algebra for all in eighth grade.
It used to be that algebra classes differentiated students who were bound for college from those who weren’t. But educational opportunity is increasingly seen as a civil rights issue, so Algebra 1 is now a graduation requirement. Still, only half of California’s eighth-graders now take Algebra 1 and less than half of those “get it,” judging by proficiency rates on the state standards test.
Why learn algebra
Algebra, through its use of variables and formulas, reveals patterns and allows generalizations, making it the language of mathematics and technology. It’s even been called the Rosetta stone of nature, since algebra unlocks the mysterious proportionality that pervades the natural world.
“To say ‘You need algebra for college,’ or ‘You won’t do well on the SAT or ACT without it’ are true statements, but they don’t tell us very much,” Zalman Usiskin, director of the University of Chicago School Mathematics Project, wrote in a 1995 article, “Why Is Algebra Important to Learn?”
He expanded on his point in a recent interview.
“You can live without it, but you will not appreciate as much of what is going on around you,” Usiskin explains. “You might not be eligible for the job you would like to have or the training program or courses you would like to take. You will not be able to participate fully in our technological society.”
Because Algebra 1 is a “gatekeeper” course that opens the door to higher-level math, proponents would like to see students take it in eighth grade so they have time to complete a progression of math courses leading to calculus in the senior year.
Still, some educators insist that it makes no sense to impose a formal study of algebra on all eighth-grade students, since many young adolescents are simply not ready for abstract thinking. A frustrating experience at that age, the educators reason, could turn them off math for good.
Piaget debunked
The question often arises whether 13- or 14-year-olds are developmentally ready for abstract reasoning, based on the stages of development outlined by Jean Piaget. That Swiss researcher, a biologist by training, became fascinated by the development of his children, and during the 1920s he studied children at play, questioning and experimenting with their reactions to devise his developmental theories.
In brief, Piaget concluded that children progress through four stages of development: sensorimotor (in infancy), preoperational (ages 2–7), concrete operational (ages 7–11), and the formal operational stage (ages 11–adult).
During the concrete operational stage, children make sense of their world in terms of concrete objects such as people, places and things with which they are familiar. As they approach adolescence, Piaget suggests, children begin to develop the ability to think abstractly about principles and big ideas that may be represented by symbols.
Piaget’s developmental theories have had a strong influence on educational psychology and curricular design since the 1960s.
Despite his influence, Piaget has been criticized for not scientifically controlling his experiments and for the leading way he questioned his young subjects. Further, as continued scientific research sheds light on how humans learn and mature, Piaget’s four developmental stages seem unrealistically rigid to modern psychologists.
“Piaget may have made formal reasoning into a false God,” Joan Bliss, a British researcher who worked with Piaget for almost a decade before the latter’s death in 1980, concludes in the book, “Teaching Science in Secondary Schools.”
Many other researchers, including Stanislaus Dehaene of the University of Paris and Stanford’s Devlin, have found deficiencies in Piaget’s theories.
“The Piaget argument is dead in the water based on their research and a re-examination of his research,” says brain researcher David Sousa, the author of “How the Brain Learns Mathematics.” “From a brain research point of view, there’s absolutely no reason the average eighth-grader cannot handle algebraic concepts.”
With that said, many math experts say that to be successful at algebraic thinking, a student needs to develop proportional reasoning, part of the formal operational thinking envisioned by Piaget, and a fundamental part of scientific understanding. Interestingly, developmental psychologists estimate that about half of American adults never do reach the formal operational stage.
Algebra for all
In their quest to make algebra accessible to all students, educators have tried all sorts of configurations. Spreading the content over two years is one of the more popular options, but there’s no evidence that such a strategy works.
The Silicon Valley Mathematics Initiative calculates that less than 30 percent of all California students reached Algebra 1 proficiency in 2008, yet a majority had been enrolled in the course for more than one year. Consequently, the two-year algebra course ended up being the last math most of those students ever took, they found.
Teaching methods may be to blame.
“If students are taught abstract ideas without meaning, there will be no understanding,” David Foster, SVMI math program director, writes in “Assessing Mathematical Proficiency,” published by the Mathematical Sciences Research Institute in 2007. “Students need experiences with a concept to develop meaning for themselves. … When we memorize rules for moving symbols around on paper we may be learning something, but we are not learning mathematics.”
Echoing that sentiment, Jeanne Ramos, director of secondary mathematics for the Los Angeles Unified School District, says that if students are given rigorous tasks that expect them to think and reason, they step up to the plate.
“We’ve been chalking and talking, and all that demonstrates is that the teacher knows math. [Students] have to construct understanding, and that’s where we’re making our gains,” she says.
Obstacles compound
As laudable as it is to want eighth-graders to master algebra, simply enrolling all students in the course will not achieve that goal.
Some of the obstacles include teachers who themselves are uncomfortable with algebra and doubt their students can master the subject, courses and textbooks that fail to instill a true understanding of complex algebra concepts, and California’s high percentage of English learners, who struggle with a discipline dependent on technical terms like “distributive property” and “coefficient.” Students’ own lack of preparation and maturity can also thwart the goal.
During the past five years, as more districts made Algebra 1 the default eighth-grade curriculum, the proficiency rate on the California Standards Test has barely budged. Just 42 percent of eighth-graders and 18 percent of ninth-graders made the cut in 2008, although Algebra 1 enrollment in those grades jumped from one-third to more than half of students.
Simultaneously, 31 percent of middle school algebra teachers in 2007–08 were underprepared or lacked a math authorization, analysts at the Center for the Future of Teaching and Learning reported. In their analysis, schools with better student scores were more likely to have teachers with the appropriate math credentials.
CFTL estimated that the state would need as many as 1,900 more certificated math teachers to teach all eighth-graders Algebra 1—a high hurdle when recruiting and retaining qualified teachers is already a stubborn challenge.
Teaching the teachers
To help students develop algebraic reasoning, and thus a true understanding of the concepts behind the formulas, the San Diego Unified School District partnered with San Diego State University to improve math instruction.
The university has designed a two-year, 12-unit math specialist certificate program that supplements an elementary teacher’s multiple-subject credential. Delivered over the summer and throughout the school year, the program aims to help teachers learn more about how to deliver math instruction and deepens their understanding of the subject.
Nadine Bezuk, director of San Diego State’s Improving Student Achievement in Mathematics program, says many student teachers are surprised to find that each student may solve a problem in a little different way. “A teacher has to understand the math to know if the way the student did it will always work or if it’s just coincidental,” she explains. “Sometimes a teacher will cut off that type of thinking and reasoning, which we really want to foster.”
One teacher, Bezuk recalls, asked her students to list all the fractions they could think of equal to one-half. After the class offered some typical options, one fourth-grade boy called out “two-and-a-half fifths.” The teacher rejected his answer, contending a fraction can’t be piled on top of a fraction. Actually, as Bezuk points out, a compound fraction is an acceptable answer.
“She gave him the wrong answer and probably may have had a negative influence on his outside-the-box thinking in math,” Bezuk says.
“Frankly, it’s not surprising that math teaching is not as strong as it could be,” she continues. “Teachers tend to teach math the way they learned math, in a very procedural way. That works for some kids, but it doesn’t work for all kids. Teachers need to learn new teaching strategies to help all students be successful.”
Ramona Unified School District, 35 miles from San Diego State, was one of the first districts to participate in the San Diego State math specialist credential program. The university agreed to come to the district. (Courses are also offered online.) Over a two-year period, half the teachers in grades 4–6 spent 30 full days building up their knowledge of the subject and learning strategies for teaching math.
The district has since had all those teaching algebra in grades 7 and up go through a similar 12 days of professional development. Subsequently, the participants brainstormed a series of new classes the district hopes will keep most students in some type of math throughout high school. Ten new semester electives include courses that apply math problems to the real world, computer math, personal finance, trigonometry, and probability and statistics, which Superintendent Bob Graeff says kids love because of the application to gaming and what-if scenarios.
“We’ve given kids a way to expand math throughout their graduation day without scaring them,” says Graeff, noting that historically more than 40 percent of juniors take no math, and the percentage avoiding math is even greater in the senior year. Students who take math throughout high school are much less likely to need remedial math courses when they go on to become college freshmen.
Proceed with caution
Despite the district’s emphasis on math, Graeff does not advocate enrolling all eighth-graders in algebra. Ramona has already tried that.
A few years ago, in an attempt to avoid the penalty the state assesses on its Academic Performance Index for not testing students in algebra, the district mandated that all eighth-graders take the subject. They tried everything, according to Graeff: double and triple dosing, pull-out classes, male only, female only, after school, Saturday morning sessions. But after two years, scores remained dismal, “so we came to the conclusion that not all kids in Ramona are ready for algebra,” Graeff says. “Some had internalized a feeling of failure.”
The district still offers Algebra 1 as an option for eighth-graders who are ready for it, but now focuses on preparing students to succeed in ninth grade.
“That doesn’t mean we’re copping out or that we’re surrendering or lack confidence,” he insists. “It’s just that some kids at age 13 are not ready. If you push [kids] faster than they’re ready, it has devastating consequences for the rest of the kid’s school career and from which they may not recover.”
All this talk of student self-esteem is not just psychobabble. Evidence suggests that students who fail repeatedly may simply give up and slip away, dropping out before talking to anyone about the problems they’re having.
Graeff has some advice for districts heading down the path toward algebra for all in eighth grade: “Be cautious. In our experience, in a middle-class community, we just don’t think this is a one-size-fits-all program. … Just imagine in a more challenged community what the fight might be.”
Kristi Garrett (kgarrett@csba.org) is a staff writer for California Schools.
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