Construction and Validation of Learning Trajectories for Understanding Variables and Functions and Solving Equations

• In Event: Intensified Algebra: A Design-Based Research and Development Project for Double-Period High School Algebra

Abstract

Intensified Algebra I (IA) takes a functions approach to algebra that emphasizes the ideas of change and variation in situations and contexts, along with the representation of relationships between variables (Cai, Nie, & Moyer, 2010). This approach was selected because it better reflects the research principles described in the first paper in this symposium.

IA’s functions approach is based on hypothesized learning trajectories for variables, functions and equations. A learning trajectory is “a researcher-conjectured, empirically supported description of the ordered network of constructs a student encounters through instruction (i.e., tasks, tools, forms of interaction and methods of evaluation) in order to move from informal ideas through successive refinement of representations, articulation and reflection towards increasingly complex concepts over time” (Confrey & Maloney, 2010). While empirically validated learning trajectories have been developed for a number of K-8 mathematics topics, trajectories have not yet been validated for algebra topics. This study examines students’ acquisition of concepts and skills related to variable, functions and equations compared to the hypothesized trajectories underlying IA.

IA content was partitioned into nine large-grain mathematics competencies, including variables and unknowns, functions, linear functions and linear equations. End-of-unit assessment items were identified that provided evidence about students’ understanding of each competency. A four-level scoring rubric was developed to represent students' level of understanding (i.e., good, partial, recognition, no understanding). Three researchers independently rated all responses to these selected items, then discussed their ratings to determine a consensus rating. The consensus ratings were used to document students’ progression of understanding compared to the hypothesized IA learning trajectories.

Primary data sources were student written work on pre/post comprehensive tests, IA end-of-unit assessments, and interviews of 25 students from intensive research classrooms in which they explained how they solved selected problems on each of the end-of-unit assessments.

Preliminary results from six students in one research classroom with respect to the Variables and Unknowns competency indicate that students showed increased understanding of this competency over time; however, growth in understanding wavered between two levels, but rarely more than that. Thus, when they started with a good or partial understanding, they maintained or increased that level. However, if they demonstrated no or limited understanding on the first unit test, although students’ understanding increased, they rarely ended the year with the highest level of understanding on items for Variables and Unknowns. Similar analyses will be conducted for all students and expanded to examine growth with respect to the other competencies, with special attention to the relationship between linear equations, functions and variables.

Considerable gaps exist in our understanding of learning trajectories in mathematics, particularly in algebra. These progressions offer curriculum developers and teachers an empirical basis for making decisions about when to teach what to whom and provide a means for them to assess their students’ cognitive processes (Daro et al., 2011). This study will inform future revisions of IA, and assist curriculum developers and teachers as they make choices about presentation and sequencing of algebra content.

Authors

• Cynthia L. Schneider, The University of Texas - Austin

• Diane Briars, National Council of Supervisors of Mathematics

• Rodney L. Harris, University of Illinois at Chicago

• Mara Vanina Martinez, University of Illinois at Chicago