**Educational Significance of the Session**

The mathematics curriculum of the U.S. has historically focused on arithmetic in the elementary grades followed by a superficial treatment of algebra in the eighth or ninth grade. Poor student performance with this “arithmetic-then-algebra” approach has led the National Council of Teachers of Mathematics (2000) and others (e.g., Kaput, 1998, 1999) to argue for the treatment of algebra as a K-12 strand in order to give students the necessary time and space to develop depth in their understanding. Several mathematics education researchers (e.g., Author, 2008; Brizuela & Earnest, 2008; Carpenter, Franke, & Levi, 2003) have investigated how elementary students’ informal intuitions can lead to formalized ways of mathematical thinking. While the foci of such efforts vary (e.g., some focus on students’ functional thinking while others focus on students’ understanding of mathematical equality), consensus exists that algebra in the elementary grades (i.e., early algebra) should not involve an exclusively symbolic focus typical of a traditional eighth- or ninth-grade course, but rather should introduce students to algebraic forms of reasoning that are accessible to them now and that we believe will benefit them in future years as they begin a more formal study of algebra.

This assumption that early algebra will have a positive impact on students’ preparation for and, ultimately, successful performance in algebra in the middle grades and beyond has not been empirically tested. To do so will require an instructional intervention that can appropriately coordinate algebra learning in the elementary grades with that in the middle grades. However, to date, algebra research in elementary grades and middle grades has focused largely on questions that, while of great significance in and of themselves, are often within and unique to these respective grade levels. As a result, little is known about how the curricular, pedagogical, and mathematical distinctions of each grade domain are connected to and influence one another (RAND Mathematics Study Panel Report, 2002).

As such, we see an initial step in developing a systematic approach to Early Algebra to be the design of an instructional intervention that would help establish the efficacy of such an approach. This entails identifying an early algebra learning progression [EALP] that coordinates research, curricular, and mathematical perspectives to identify core algebraic concepts and their progression in children’s thinking from elementary through middle grades. The EALP would then provide the critical basis for designing assessments to help measure the efficacy of early algebra.

Objectives of the Session

The purpose of this working session is to critically examine EALP-based middle grades algebra assessments for their alignment with a proposed EALP and for their consistency with research, curricular (e.g., Common Core, NCTM Standards), and mathematical perspectives. The EALP and associated assessments were developed by the presenters as part of an ongoing project examining the impact of Early Algebra on students’ algebra-readiness for middle grades. The EALP-based assessments are organized around the following five core ideas: (1) Equality, Expressions, Equations, and Inequalities; (2) Functional Thinking; (3) Generalizing Arithmetic; (4) Variables; and (5) Proportional Reasoning. The proposed session will critically examine project assessments for grades 6-7, including their alignment with the EALP, their connections to the 5 core ideas, and their fit with previously developed assessments for grades 3-5. We view this session as invaluable to our work as it will offer the opportunity to elicit critique and analysis from a broad audience, thereby strengthening our overall assessment design. Using the five central questions identified below to focus our discussion, this working session is expected to provide the presenters with essential feedback regarding the alignment of middle grades assessments with the proposed EALP.

**Organization of the Session and Audience Participation**

Audience participation is the central component of the proposed session. After an introduction framing the objectives of the session, presenters will briefly outline the proposed EALP (15 minutes). Presenters will then lead small group discussions (45 minutes), with each small group critically examining assessment items for a given core idea. The small group and whole audience discussions will be guided by the following five questions:

- What are important algebra ideas connected to arithmetic and how do these ideas progress across grades 3-8?
- What are important algebra ideas in the study of functions and how do these ideas progress across grades 3-8?
- Do the assessment items for a given core idea address appropriate grade-level content?
- Do the assessments, overall, capture the essence of what students should know in algebra for the given grade level? If not, how might the assessments be revised?
- What are some of the critical elementary-to-middle-grades transition points and do the assessments capture these transition points?

Small group activity will include thinking through the algebra concepts identified in the assessment items in terms of fit, completeness, and clarity. The remaining 30 minutes of the session will involve whole-audience discussion of the five focus questions in order to summarize participants’ thinking. Results of the session will then be used to refine the proposed EALP-based assessments.

**References**

Brizuela, B. M., & Earnest, D. (2008). Multiple notational systems and algebraic understandings: The case of the "best deal" problem. In J. J. Kaput, D. W. Carraher & M. Blanton (Eds.), *Algebra in the early grades* (pp. 273-301). New York: Lawrence Erlbaum.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). *Thinking mathematically: Integrating arithmetic and algebra in the elementary school*. Portsmouth, NH: Heinemann.

Kaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by "algebrafying" the K-12 curriculum. In S. Fennel (Ed.), *The nature and role of algebra in the K-14 curriculum: Proceedings of a National Symposium* (pp. 25-26). Washington, DC: National Research Council, National Academy Press.

Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. A. Romberg (Eds.), *Mathematics classrooms that promote understanding* (pp. 133-155). Mahwah, NJ: Lawrence Erlbaum.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics.

RAND Mathematics Study Panel Report. (2002). Mathematical proficiency for all students: A strategic research and development program in mathematics education. Washington, DC: U.S. Department of Education.

This working session will engage participants in critically examining grades 6–7 assessment items designed to measure students’ understandings of some of the “big ideas” of early algebra. Conjectured early-algebra learning progressions for grades 3–8 will frame the examination and subsequent discussion.

Session Type: Work Session