Third grade is not too early to begin teaching algebra, nor is early algebra instruction incompatible with the regular arithmetic curriculum, says a new study in the *Journal for Research in Mathematics Education*.

Third-graders who received 19 one-hour algebra lessons on 5 big algebraic ideas for about 10% of math class time significantly outperformed controls on an algebra assessment, indicating that algebra is well within reach of the typical 3^{rd}-grader, according to the study.

“It might not be surprising that students who received the early algebra intervention performed as they did,” the researchers write. “However, the performance of these students highlights a particularly compelling aspect of our findings about early algebra: As early as Grade 3, children can successfully develop** critical components of algebraic thinking skills** that are foundational to the successful study of algebra in the later grades.

“The performance of these students highlights another compelling aspect of our findings: Typical arithmetic-based elementary school mathematics curricula and instruction does little to prepare students for the successful study of algebra in the later grades.”

### 5 big ideas

Students who received the intervention **improved their ability to think relationally about the equal sign,** recognize fundamental structures and properties in equations and to generalize, represent unknown quantities in meaningful ways with variable notation and symbolically represent functional relationships between covarying quantities, according to the authors.

The 5 big ideas for the lessons were a) equivalence, expressions, equations, and inequalities; b) generalized arithmetic; c) functional thinking; d) variables and e) proportional reasoning.

Participating in the study were 106 3^{rd}-grade students from one school district in the northeastern United States. A total of 39 students from 2 classrooms received the intervention and 67 students from four classrooms served as controls. Both groups received the same amount of math instructional time covering the same topics. Students from the two groups were comparable academically, based on their performance on an algebra pretest and on the state’s standardized mathematics assessment administered at the end of 3^{rd} grade.

Each algebra lesson began with small-group discussions on previously taught concepts. **New concepts were then introduced through small-group problem solving and whole-class discussion.** The problem-solving tasks engaged students in the algebraic practices of generalizing, representing, justifying and reasoning with mathematical relationships. All lessons were taught by one member of the research team, a former elementary school teacher.

“That third-grade students exhibited a clear development in their ability to think structurally about algebraic tasks after a 1-year intervention (with only about 20 hours of instruction) suggests that a sustained, multiyear approach to algebra instruction beginning in the elementary grades could have significant effects on students’ ability to generalize, represent, justify, and reason with mathematical structure—all fundamental practices of algebraic thinking,” the authors write.

This study is part of broader, ongoing work funded by the National Science Foundation to develop an early algebra learning progression (EALP) for grades 3-7.

* “The Development of Children’s Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade,” by Maria Blanton et al., Journal for Research in Mathematics Education, 2015, Volume 46, Number 1, pp. 39-87.*

functional relationships between covarying quantities

I am an English teacher in Mexico, so my mother tongue is Spanish, I am having trouble understanding what covarying is. I have looked up in the dictionary but there is no definition of it, could you help me please. Thank you in advance.

Lucia