Early Childhood

Kiddie Algebra

By Sean Cavanagh — February 10, 2009 7 min read
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Melissa Romano grew up attending school in classrooms that were quiet and orderly. And she liked it that way.

Today, as a 2nd grade teacher, Ms. Romano has learned to tolerate and even encourage more spirited discussion among her pupils, in the hope of cultivating their mathematical skills and, specifically, their algebraic thinking.

As educators and policymakers search for ways to prepare students for the rigors of algebra, Ms. Romano, at Broadwater Elementary School, and other teachers in the Helena, Mont., school system are starting early. They are among the teachers in a number of schools who are attempting to nurture students’ algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.

Melissa Romano works with 2nd graders Konnar Smith and December Frost on their algebra assignment at Broadwater Elementary in Helena, Mont. Teachers are introducing such math concepts in the early grades.

To accomplish that aim, Ms. Romano says, it’s not enough that she simply present pupils with a problem, collect their answers, correct their mistakes, and move on.

She takes relatively simple problems, then expands them, integrating algebraic thinking along the way. She changes the conditions and calls for class discussions. And she asks individual children to explain aloud: How do you know that?

As that give and take unfolds in class, “they’re talking,” Ms. Romano said, and “it’s loud.”

That process was “a huge, huge change for me,” said the educator, who began using the algebraic reasoning and number-skill model in 2005. “The first two years were a big learning curve on my part.”

When she was in elementary school herself, teachers gave her a math problem “and let me work on it,” she said. “And I didn’t understand half the math.”

Young, But Skilled

A 2nd grader at Broadwater Elementary School in Helena, Mont., reaches into a bin for visual aids to help him solve a math problem.

Ms. Romano and her colleagues in the 8,000-student Helena school district began to rethink math instruction, and the connection between basic arithmetic and algebra, four years ago, when they attended an institute offered by the Northwest Regional Educational Laboratory, or NWREL, a nonprofit research and evaluation organization in Portland, Ore.

Those sessions focused on improving math instruction in the early grades. They placed a heavy emphasis on the principles of “cognitively guided instruction,” an influential approach to math teaching and professional development first developed in the 1980s by Thomas P. Carpenter and other researchers at the University of Wisconsin-Madison.

A core idea of cognitively guided instruction is that young children arrive at school with a surprisingly strong set of intuitive math skills in areas such as understanding numbers and problem-solving. Teachers who understand those skills, and how students’ math knowledge develops, can greatly improve their instruction, proponents of the methodology say.

Rather than emphasizing drill and procedure, cognitively guided instruction encourages teachers to nurture students’ broader understanding of the relationships between numbers, patterns, and the fact that symbols can be used to represent numbers—skills that prove essential in algebra.

Long-Term Payoff

Education policymakers at all levels are grappling with the question of why students have difficulty in algebra and what can be done to help them. They reason that students who overcome challenges and complete introductory algebra, or Algebra 1, relatively early in school have a jump on advanced math and, presumably, a broader array of skills they will need in college and in the job market.

A 2nd grade pupil put pencil to paper to work through a math set.

That rationale has led two states, California and Minnesota, to phase in requirements that students take Algebra 1 in 8th, rather than 9th grade, though the California measure has been held up in court. Many districts, meanwhile, are scrambling to improve teacher training in algebra and create intervention programs for students who cannot keep up.

Such struggles can be traced in part to schools’ narrow conception of algebra, Mr. Carpenter said. Many teachers present arithmetic as a tool for “getting answers” and, separately, algebra as a more complicated study of relationships between numbers. A better approach, he said, is to zero in on the “big ideas” in arithmetic that will help students conquer algebra down the road.

“If you learn these big ideas early, there’s a lot less to learn” later in algebra courses, Mr. Carpenter said. “Learning with understanding pays off in the short run, and it pays off in the long run.”

If students, by contrast, approach math only by memorizing steps and procedures, he argues, their instruction leads to misconceptions that haunt them when they reach a full-fledged algebra course.

One common misconception Mr. Carpenter often cites involves the “equals” sign. Many students are mistakenly taught to regard the equals sign as signifying “the answer comes next,” as in 8 + 4 = __.

But that’s a mistaken belief, which is compounded when students reach algebra, where variables such as x and y can appear on both sides of the equation. The equals sign, in fact, connotes a relationship between numbers—that both sides of the equation are of equal value, or in balance. Yet for many students, Mr. Carpenter said, “the misconception continues all the way through algebra.”

Working together, Mackenzie Bright and Caleb Holm sit on the floor with paper cutouts to solve math problems.

Linda Griffin, the director of the mathematics education unit at the Center for Classroom Teaching and Learning at NWREL, has led institutes in which she introduced teachers to principles of cognitively guided instruction, or cgi. Her organization has trained about 700 teachers in algebraic reasoning and number sense since 2004, typically over four or five days. NWREL also encourages districts to arrange for mentors and coaches to provide on-site support in those methods.

The idea of building algebraic reasoning in the elementary grades is a major departure for many teachers, Ms. Griffin said. Many were taught through their own experiences in school, and their professional coursework, to emphasize procedural knowledge, as opposed to “making sense of mathematics,” she said.

Ms. Griffin and others who promote cgi strategies emphasize that they are not attempting to “teach algebra” to elementary school students in the strictest sense, so much as to “algebra-fy’ early-grades math in ways that carry forward.

Schools tend to “act like algebra’s a whole new world,” Ms. Griffin said. “Done well, it shouldn’t be that way.”

Tutoring Parents, Too

Ms. Romano tries to make the shift from arithmetic to algebra in subtle ways.

During one recent class, she gave her 2nd graders a problem about geese flying in a V formation. She used flocks of different sizes—three geese, five geese, seven geese. If the geese fly in a perfect V, she asked them, how many end up flying on each side?

Over the course of the class period, the pupils learned that with odd-numbered flocks, they could solve the problem by dividing the total number of geese by two, and subtracting one, for the lead goose. Ms. Romano moved on to ever-larger numbers: 49, 103. Eventually, she and the children worked out a formula they could use to solve the problem:

X - 1

Some students ended up making calculations into the hundreds. When others struggled, Ms. Romano encouraged them to draw pictures to illustrate their thinking. All told, the teacher spent 60 minutes working on variations of that single problem.

As Ms. Romano has been forced to change the way she thinks about math, she’s asked parents at her school to adjust their way of thinking, too.

Some mothers and fathers, when they see their children’s math homework, are eager to jump in and provide the answers for them. Ms. Romano urges them to hold back, and let children sort through problems and provide explanations on their own.

Just because pupils can spit out an answer, that doesn’t mean they understand what they’re doing, the teacher tells parents. When parents understand the process, she said, “they are amazed at what their kids can do.”

Talking It Over

The principles used in cognitively guided instruction have played a significant role in shaping curriculum and the overall thinking about children’s math skills in the early grades, said Douglas H. Clements, a professor of learning and instruction at the State University of New York at Buffalo.

2nd graders at Broadwater Elementary are taught algebra through an approach that relies heavily on cognitively guided instruction.

Mr. Clements has developed an early-grades curriculum that builds math skills through games and other means—work that is based on research about how young children learn math tasks. The scholar, who says his own work has been shaped by principles of cgi, served on the National Mathematical Advisory Panel, a White House-commissioned group that last year issued recommendations on how to prepare students for introductory algebra.

While Mr. Clements supports having students think and talk through math problems, he also says teachers face a challenge in knowing when they have to step in and correct children when they make clear mistakes, so that students are not led astray mathematically.

“You can get so caught up in the talk, talk, talking,” Mr. Clements said. “Sometimes, a teacher has to say, ‘That’s wrong.’ ” Teachers’ time in class in limited, he noted. “You’ve got to use it for students’ advantage.”

Marla Ernst, an elementary teacher in Oregon’s Lebanon Community School District who uses the algebraic-reasoning strategies discussed by NWREL, tries to correct students’ errors in ways that encourage further discussion. She will ask them to explain a correct answer and diagnose where a wrong answer went awry.

“We don’t leave a lesson until it’s clear,” Ms. Ernst said. “Math is about finding a right answer, but it’s also about a process. You don’t want to lose either part.”

Coverage of mathematics, science, and technology education is supported by a grant from the Ewing Marion Kauffman Foundation, at www.kauffman.org.
A version of this article appeared in the February 11, 2009 edition of Education Week as Kiddie Algebra


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