Algebra in the Elementary Grades: Defining Research Priorities

1. Algebra in the Elementary Grades: Defining Research Priorities Maria L. Blanton Mathematics Dept University of Massachusetts Dartmouth Jere Confrey Frank Davis Vickie Inge Patty Lofgren Deborah Schifter Cassandra Willis

2. What we Mean by Algebra in the Elementary Grades, or “early Algebra” There is general agreement that EA comprises two central features: • generalizing, or identifying, expressing and justifying mathematical structure, properties, and relationships, and • reasoning and actions based on the forms of generalizations (Lins & Kaput, 2004; Kaput, in press). EA research typically focuses on • the use of arithmetic as a domain for expressing and formalizing generalizations (generalized arithmetic), and • generalizing numerical or geometric patterns to describe functional relationships (functional thinking).

3. …And What we Don’t Mean • It is not an ‘add-on’ to the existing curriculum. • It is not a re-packaging of algebra skills and procedures - typically taught as a ‘pre-algebra’ course in the middle grades - for elementary grades.

4. Why Early Algebra is Critical for the Elementary Grades • Historically, elementary grades has used an “arithmetic-then-algebra” approach, followed by a procedural approach to algebra from middle grades onward. • However, this approach has been unsuccessful in terms of student achievement (U.S. Department of Education & National Center for Education Statistics, 1998a; 1998b; 1998c) and has compromised the ability of US schools to compete internationally in mathematics (Heibert, et al, 2005; Stigler, et al, 1999).

5. Why Early Algebra is Critical for the Elementary Grades In addition to strengthening children’s arithmetic understanding, EA helps children learn more advanced mathematics taught in middle and secondary grades. In particular, EA helps children • develop a symbolic language system rooted in their natural language expressions. • acquire the early language of proof, including forms of inductive and deductive reasoning and an appreciation for general arguments and the limitations of empirical arguments. • learn to use representational tools such as tables and graphs and generate and organize data. • learn to express co-variation in linear, quadratic and exponential relationships and how to interpret multiple representations of functions (e.g., tabular, graphical). • learn to describe, symbolize and justify properties of number and operation - including important axioms such as the commutative, associative and distributive properties that are foundational to formal algebra courses in secondary grades. It democratizes access to mathematical ideas so that more students understand more mathematics and, thus, have increased opportunity for lifelong success.

6. What We Know from Early Algebra Research Children from diverse socioeconomic and educational backgrounds can • describe, symbolize and justify arithmetic properties and relationships; • develop an algebraic, relational view of equality; • use appropriate representational tools – as early as first grade – that will support the exploration of functional relationships in data; • identify and symbolize functional relationships; • progress from building empirical arguments to building justifications using problem contexts and learning to reason with generalizations to build general arguments; and • learn to compare abstract quantities of physical measures (e.g., length, area, volume), in order to develop general relationships (e.g., transitive property of equality) about these measures.

7. Critical Areas for Future Early Algebra Research • creating systemic change through the development of “early algebra schools”; • identifying core algebra ideas and how they are connected to the curriculum across K-12 algebra education; and • understanding the pervasive nature of children’s algebraic thinking.

8. Creating systemic change through the development of “early algebra schools” • We need schools that integrate a connected approach to early algebraacross all grades K-5 and that provide all teachers–within a school and across a district – with the essential forms of professional development for implementing early algebra. • The approach should be systemic: It should involve not only elementary teachers, but also middle school teachers, principals, administrators, education officials, math coaches, and parents. • Early algebra schools would (1) provide critical research sites for developing a comprehensive evidence base on the impact of early algebra education and (2) sustain early algebra education in schools for viability in children’s learning.

9. identifying core algebra ideas and how they are connected to the curriculum across K-12 algebra education • Early algebra needs to be deeply and explicitly connected to what children are learning so that it is not treated as an isolated topic. • We need a coherent picture of the essential early algebra ideas, how these ideas are connected to the existing curriculum, how they develop in children’s thinking across grades, how to scaffold this development, and the critical junctures in this development. • Moreover, early algebra education must be framed within the context of K-12 algebra education. Thus, critical work is needed for the development of common ideas, purposes, and goals through collaborative efforts by researchers across grades K-12.