For seven semesters, our research team has explored children’s development and learning of measurement concepts. We have conducted a longitudinal teaching experiment (Steffe & Thompson, 2000) with sixteen students from Grade 2 to Grade 5. Our research team used previously developed Hypothetical Learning Trajectories (HLTs) for length, area, and volume (Sarama & Clements, 2009) to inform assessment design, task development, and data analysis. In this poster session, we will present data from three of sixteen students as a representative sample of the participants in a longitudinal teaching experiment to illustrate how HLTs are useful tools for teachers and researchers.

Theoretical Framework

We explore children’s learning of mathematics through the lens of hierarchic interactionalism (Clements & Sarama, 2007). This term refers to a cognitive theoretical framework that synthesizes empiricism, (neo)nativism, and interactionalism. Hierarchic Interactionalism has 12 tenets that describe the “influence and interaction of global and local (domain specific) cognitive levels and the interactions of innate competencies, internal resources, and experience (e.g., cultural tools and teaching)” (Sarama & Clements, 2009, p. 20). Tenet 11, learning trajectories, is the most relevant to our discussion here because the field of research on measurement needs a comprehensive structure for learning, teaching, curriculum development, and assessment. A learning trajectory has three components: a learning goal in a mathematical domain, a developmental progression describing a hypothetical route for levels of thinking and mental actions with observable characteristics, and instructional tasks used to support and promote progress through the levels (Clements & Sarama, 2004).

Research Question

How do students develop coherent knowledge and integrated strategies for measurement from Grade 2 through Grade 5 in the context of a teaching experiment?

Methods and Data Sources

The sample consisted of sixteen students from a Midwestern public school.  Each participant has been regarded as a case within a four-year longitudinal study, focused on investigating children’s thinking and learning across length, area, and volume measurement from grades 2 – 5.  Our team conducted a four-year teaching experiment with eight of the students. The research team generated sets of tasks and predictions for students’ responses over an extended series of research sessions. Each session was designed in relation to the outcome of prior sessions (Steffe and Thompson, 2000). 

Here we consider data collected from three open-response assessments and sixteen teaching episodes with three of the eight students. Each teaching episode was 15 to 30 minutes in length and conducted in the form of a semi-structured individual or pair interview. The assessments were administered in March 2008, April 2010, and May 2011. They were designed to serve as tools for checking and refining the trajectory, and therefore, contained items that probed students’ thinking within each trajectory level for length, area, and volume. Although the structure of each assessment supported both qualitative and quantitative methods of analysis, this report utilizes the qualitative data of student responses to describe individual student’s actual trajectory for reasoning and solving area measurement tasks.

After the teaching episodes were videotaped and transcribed, the research team used the HLTs to analyze them.  The first step in the data analysis for each teaching episode involved using the HLTs to describe student thinking and mental actions. Next the team considered each teaching episode as a collection of tasks in order to describe student thinking and mental actions holistically. We described the scope and sequence of all exhibited strategies in terms of the HLTs. These records were tracked and charted per strategy, per task, per interview, per semester, per year to demonstrate each student’s trajectory from Grade 2 to Grade 5. As a second step in the data analysis, the research team retrospectively looked for differences and commonalities among students’ actual trajectories over the four-year period.  Due to space and time restrictions, the data presented in this poster will address a four-year account of three students’ actual trajectories along the area measurement HLT.

Summary of Findings

Our results suggest that some students can progress through a level per year, but it takes longer for other students. We have also observed students demonstrating large variations in thinking and mental actions, alternating between demonstrating sophisticated levels of thinking and more naive levels of thinking.

We summarize the actual trajectory account of three students, Arielle, Abby, and Drew.  Both Arielle and Abby followed consistent and steady trajectories through the area trajectory with minor fluctuations.  These fluctuations appeared within or between individual teaching episodes in which Abby or Arielle occasionally demonstrated fallbacks or scaffolded growth in their thinking or strategy choice.  In contrast to Arielle and Abby, Drew exhibited a trajectory of long-term growth with many inconsistencies within and between consecutive teaching episodes.  The actual trajectories of Arielle, Abby, and Drew will be illustrated and discussed during the presentation.

These results support Vygotsky’s work and description of his zone of proximal development.  Depending on the day, the task posed (tasks did not always proceed as expected), and the supports provided, students can follow the supports and reach beyond their present level of thinking, or they can “fall back” to simpler levels of thinking in order to cope with the complexity in the task and/or situation.  Drew’s variability in his actual trajectory corresponds to a wider zone of proximal development than was observed in Arielle and Abby’s trajectories.

Educational Importance of the Research

In a recent Research Report from the Consortium for Policy Research and Education, Daro, Mosher, & Corcoran (2011) recommend that learning trajectories should be translated into useable tools for teachers.  Researchers must first lay groundwork to illustrate how learning trajectories can be used as diagnostic and analytical tools for teachers.  With this presentation, we intend to continue that conversation and belay misunderstandings regarding what it means for a student to be “at a level.”  We also seek to confront several dialectical issues, including 1) how instructional tasks can be designed to support and promote progress through the levels but cannot necessitate or demand results or conceptual change, and 2) how variability across students involving fallbacks and scaffolded growth is consistent with the learning trajectory construct.