Because historical paths to algebra have been largely unsuccessful in preparing students for formal algebra (e.g., Hiebert et al., 2005), it is now argued that students need long-term algebra experiences in school mathematics, beginning in the elementary grades (e.g., NCTM 2000; CCSSI, 2010). As a result, the emergence of early algebra research has provided us with critical evidence regarding how elementary grades children think algebraically. Given that this is a new field of research, much of this work has necessarily focused on the use of qualitative research paradigms to explore critical questions regarding the nature of children’s algebraic thinking. However, a fundamental—but largely untested—premise of early algebra education is that it will increase children’s algebraic understanding as well as their success with more formal algebra in later grades. This promise of improvement, coupled with the call for experimental research “to investigate the effects of programs, practices, and approaches on students’ mathematics achievement” (US Dept of Education, 2008, p. 84), points to the need for experimental research to examine the impact of early algebra interventions in elementary grades and beyond.

Objectives

This symposium coordinates the recent work of three long-term early algebra research programs that have begun to explore impacts of early algebra interventions on children’s algebraic thinking, both within elementary grades and between elementary and middle grades. The programs represent contrasting approaches to early algebra research across diverse populations, with two oriented towards explorations in number as a path to algebraic thinking in diverse content domains (e.g, generalized arithmetic, functional thinking) and the third based on an approach that begins with the study of generalized quantities and moves towards particular instantiations of those generalizations. Each presentation is described below.

Presentations

The first presentation examines within-grade impacts of one-year classroom interventions at grades 3-5. The intervention design, based on the presenters’ grades 3-8 early algebra learning progression [EALP], was organized around the content domains of generalized arithmetic; functional thinking; equality, expressions, and equations; and variable. Experimental and control classrooms were selected from two schools within one district. All students (n=300) were given a 60-minute assessment prior to and after the year-long intervention. The assessment was developed to possess good content validity. Items were developed using the EALP, with only minor modifications made to items that had performed well psychometrically in previous work. All items underwent extensive review, piloting, and revision to ensure they would function well for their intended audience. Results indicate that the experimental group (n=126) scored significantly higher on the post-test than the pre-test, while the control group (n=174) did not score significantly higher pre-to-post. Specifically, with the control group 18 of the 20 sub-items were not statistically significant. For the experimental group, 17 of these 18 identical sub-items were statistically significant. Experimental statistical significance resulted with chi-square values ranging from ² (1) = 3.93 to ² (1) = 55.273 and p-values ranging from p < 0.0001 to p < 0.05, suggesting that, overall, the intervention led to a statistically significant improvement in the experimental group’s algebraic understanding.

The second presentation reports on immediate (at the end of the intervention) and medium (two years after the intervention ended) term outcomes of an early algebra classroom intervention in which 26 grades 3-5 students who attended an urban public school participated in weekly early algebra lessons, designed and implemented by the project team.  The approach taken towards early algebra by the project team can be characterized as functional, without ignoring aspects of the generalized arithmetic or modeling quantitative relations approaches.  In terms of the immediate impact, at the end of fifth grade, experimental group (n=20) results were significantly higher than those of a control group (n=19) for items related to the early algebra lessons. Regarding medium term outcomes, two years after the intervention ended six students from the experimental group were compared with 16 control group students, all of whom participated in a summer camp.  The experimental group outperformed the control group in both before and after camp assessments, but the difference between the two groups increased after summer camp participation, supporting the claim that students who had been exposed to early algebra benefited more from the more advanced algebra lessons.

The last presentation considers the impact of a non-traditional elementary mathematics program on fifth and sixth grade students’ (n=19) ability to work with variables as critical indicators of their preparedness to handle the concepts of a secondary algebra course. The program’s design was a response to the Vygotskian approach to the development of algebraic thinking—initially from the most generalized conceptual base, with ascent from the abstract to the concrete. The introduction of literal symbols was rooted in the use of scalar quantities—length, area, mass, and volume—to provide the context in which young children could study the mathematical structure of the real number system. The fifth and sixth grade students who had experienced the program (n=19) were compared with sixth grade control students (n=41) on the Chelsea Diagnostic Mathematics Test: Algebra (CDMT1) (Hart, Brown, Kerslake, Küchemann & Ruddock, 1985). A regression analysis showed a significant positive correlation between the level of achievement on the CDMT1 and experience in the non-traditional program. A closer examination of students’ responses revealed that students who had the experimental program were as prepared as students who typically received three to four more years of mathematics preparation.

Session Organization:
After the three research presentations (45 minutes), a discussant will use his expertise in quantitative methods and large-scale studies to reflect on what the combined work addresses by way of impact, promises and challenges of the work, and possible future directions (15 minutes). Finally, audience discussion (30 minutes) will be organized around the following questions:

(1)  What are the curricular implications for early algebra nationally?

(2)  How might promising results from the work presented influence policy regarding early algebra?

(3)  What additional research is needed to convince relevant stakeholders that early algebra is a worthwhile investment?

Priority Areas:

As detailed above, this session focuses on results of instructional interventions in early algebra.