Blog updated to include additional partners in the project.
When it comes to algebra, new research suggests you can’t start too early.
Algebra is considered the keystone of students’ understanding of higher mathematics, and districts across the country have moved to introduce the course in middle school rather than high school. But studies previewed at the American Educational Research Association meeting here suggest students can understand and benefit from being introduced to algebraic concepts even in elementary school.
The studies are part of Project LEAP, for Learning Early Algebra Progression, a program developed at the University of Wisconsin-Madison, TERC, University of Texas at Austin, City College of New York, and Merrimack College to boost math concepts for low-income elementary students.
“We’re not talking about taking a traditional [algebra] curriculum and pushing it earlier in the sequence,” said Rena Stroud, a senior researcher for Project LEAP at Merrimack College. “We’re talking about what algebraic thinking looks like in early grades.”
“Say I give our 3rd graders a number sentence. If I say, ’49 + 3 = 50 + 2. True or false?,’ how could you consider solving that? You could use a purely computational approach, right? You could add on each side and say, yes, that’s true and you get a correct answer. But you could also think about the structure of that relationship. So either way you could be correct or incorrect ... but the way that you get to the answer is important. And so we are not only interested in whether or not students thought an item correct, but also the type of strategy that they use to solve it,” Stroud said.
The three-year curriculum includes 18 weekly hour-long lessons, in each of grades 3-5, in which teachers introduce students to generalizing, representing, justifying, and reasoning about numbers. In the process, they correct common misconceptions that often trip up older students. For example, students learn that the "=" sign means that the numbers on either side are equivalent, not simply that one side is the “answer” to a problem.
Researchers randomly assigned some 440 incoming 3rd graders in nine high-poverty urban, suburban, and rural schools to either receive the algebraic lessons or standard grade-level arithmetic. Both groups of students were tested at the beginning of grade 3 and the end of grades 3, 4, and 5.
By the end of 3rd grade, students participating in Project LEAP were growing in math skills 9 percent faster than their peers who were taking standard arithmetic. The Project LEAP students continued to grow 2 percent faster than their control group peers at the end of grades 4 and 5, and by the end of grade 5, they performed 13 percentage points higher in math than students who had not participated in the program.
“We’re looking at these schools where we have 100 percent of students who are receiving free or reduced-price lunch. ... and so we did see that pattern of improvement,” said Stroud. “This is encouraging for us, given what we know about underperformance of low [socioeconomic status] students, the importance of algebra in the schools, and the specific importance of algebra for these grade levels.”
A Box of Candy Concepts
Project LEAP researchers also previewed a second longitudinal study of even younger students.
In one early experiment, researchers presented 10 kindergartners with a closed box of candies, with two placed on top of the box. Researchers asked the children what they knew about the unseen candies in the box. Eight of the kids shook the box, or tried to guess how many were inside, but two gave more straightforward answers: “I don’t know” or “some.”
“We know from previous work that students even this young are capable of using a variable representation. ... And sometimes among 3- to 5-graders, we found that to represent relationships, for example, students prefer to use variables,” said Ana Stephens of the University of Wisconsin-Madison’s Wisconsin Center on Education Research.
Rather than a mental shrug, “some” is a sophisticated concept for young children. After the first experiment, students went through a lesson discussing this early concept, using everything from coins in a piggy bank to the teeth in—and sometimes recently out of—the students’ heads. Teachers turned the physical boxes into mental ones, introducing letters (such as “a” or “x”) to represent uncertainty.
By the end of the lesson, all of the children could describe the basic equation: All of the candies together would be what was in the box, plus the number on top. Four of them correctly chose a letter to represent the unknown quantity of candies in the box.
“We see this as just an initial introduction, not as something that the kids can immediately pick up on and become extremely comfortable with in one lesson,” Stephens said, adding later, “We believe that students as young as kindergarten did show evidence of structural thinking.”
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A version of this news article first appeared in the Inside School Research blog.