1. Maria Blanton - UMass Dartmouth Eric Knuth - UW Madison Project Associates Angela Gardiner Isil Isler Tim Marum Ana Stephens algebra.wceruw.orgkaputcenter.umassd.edu/projects/ea/alg_ready Analyzing an Early Algebra Learning Progression (EALP) for Grades 3–8 NSF PI Meeting December 1-3, 2010

2. Maria Blanton - UMass Dartmouth Eric Knuth - UW Madison Project Associates Angela Gardiner Isil Isler Tim Marum Ana Stephens algebra.wceruw.orgkaputcenter.umassd.edu/projects/ea/alg_ready Analyzing an Early Algebra Learning Progression (EALP) for Grades 3–8 NSF PI Meeting December 1-3, 2010 (part of)

3. Project Overview From 35,000 ft: Long term goal of looking at impact of early algebra: How does a sustained early algebra education develop ‘algebra-ready’ students for middle grades (and beyond)? From 10,000 ft: Preliminary work to above goals:  Develop the EALP  Develop EALP-based assessments for grades 3-8, empirically test the EALP and refine assessments and potential intervention On the ground (focus for today): Analyzing an EALP for algebra in grades 3-8

4. Prior Steps - Year 1 understand meanings for LP

5. Learning Progression (LP) A learning progression is meant to provide an ordering of concepts/understandings/skills that build toward a more sophisticated understanding of an important idea or topic (NRC, 2007).

6. Learning Progression Key Characteristics  based on the logic of the discipline, current learning research, and curricular frameworks and standards  specifies lower and upper anchor points specifies grade level competencies and connections between them  informs the design of appropriate instructional strategies and learning tasks, and grade-level assessments  identifies key transition points

7. Learning Progression Process & components (based on Shin et al, 2009)  Identify the Big Ideas  Identify associated Constructs  ….necessary to understand big ideas.  Create Claims/understandings  …that specify nature of knowledge/skills/understanding expected of students regarding a construct.  Specify Evidence  …from student work that indicates student has acquired knowledge/skills/understandings identified in claim  Identify Difficulties and Misconceptions  …that students have with a construct  Identify Research Base  …that supports the claims about the knowledge/skills/understandings expected of students

8. Prior Steps - Year 1 Understand meanings for LP gather information on –research –curricula –frameworks/standards –mathematics regarding important algebra ideas in elementary grades through Algebra 1. synthesize ideas to draft an EALP for grades 3-8

9. EALP Five content domains (Big Ideas):  Generalized Arithmetic (GA)  Equality, Expressions, Equations, and Inequality (EEEI)  Functional Thinking (FT)  Variable (VAR)  Proportional Reasoning (PR)

10. EALP Each content domain (Big Idea) is organized around the following 4 strands (based on Kaput (2008):  generalizing relationships (relationships)  expressing relationships (representations)  justifying generalized relationships (justifying)  reasoning with relationships (reasoning)

11. EALP Each content domain (Big Idea) consists of (1)tables that identify the constructs, claims/understandings, evidence, difficulties/misconceptions and research base within each strand for the Big Idea. (2)a grade-by-grade progression (grades 3-8) that details how Big Ideas develop (or can be developed) across the grades, what content would be developed at a particular grade, and what skills/understandings are expected for that content.

12. Focus: FT and EEEI Strands 1 and 2 (relationships and representations) constructs and claims algebra FTEEEIGAVARPR STRAND 1 STRAND 2 ETC GRADE-BY-GRADE PROGRESSION OF IDEAS

13. Small-Group Focus Questions Review portion of LP in your group (~10 minutes) Do the Constructs identified for each strand seem appropriate (complete, cohesive, aligned) for that strand? Are there any additional constructs we should consider? Do the Claims/Understandings seem appropriate (complete, cohesive, aligned) for the given construct? Are there any additional claims we should consider? Do the grade summaries seem reasonable and appropriate?

14. Questions for further discussion What are key transition points between elementary and middle grades for FT or EEEI? How should solving equations be addressed in elementary grades so that it leads to more formal algebraic procedures for solving equations in later grades? What algebraic aspects of proportional reasoning can be included in the elementary grades as a way to strengthen important algebra ideas in middle grades (e.g., slope)? Are there other mathematical domains (BIG IDEAS) that should be considered in the elementary grades as a route into algebra?