In a NSF-funded longitudinal study of learning trajectories for early and elementary measurement, we began to suspect that our students had missed an early, potentially important, phase of development. Despite advances, many appeared to have particular difficulty learning more advanced concepts and procedures, or regress a bit from levels of thinking they appeared to be developing. We came to believe that this might be due to a lack of experience, specifically informal experience, with situations involving various continuous quantities and their comparison and measurement. We instituted a "Measurement Club" after school to check our conjecture. This presentation is the first report of our activities and findings.

As background, we define learning trajectories (LTs) as "developmental progressionsÉdescriptions of children's thinking and learningÉand a related, conjectured route through a set of instructional tasks" (Clements & Sarama, 2004, p. 83). Thus, LTs have three components: a goal (that is, an aspect of a mathematical domain children should learn), a developmental progression or learning path wherein children move through levels of thinking, and instruction that helps children move along that path. Research indicates that LTs help teachers focus on the "conceptual storyline" of the curriculum, a critical element often missed (Heck, Weiss, Boyd, & Howard, 2002; Weiss, 2002).

We view LTs (in particular, children's development of geometric measurement understanding) through a theoretical lens termed hierarchic interactionism (Sarama & Clements, 2009Sarama & Clements, 2009), which indicates the influence and interaction of both global and local (domain specific) cognitive levels and the interactions of innate competencies, internal resources, and experience (including instruction and available tools from the culture). In general, we expect children to progress through levels of understanding for measurement in ways that can be characterized by specific mental objects and actions (i.e., both concept and process) with the most visible progress through levels for domain-specific topics.  Along with the learning trajectories construct and the strong influence of environment and culture, relevant to this report is the hierarchic interactionism tenet of "cyclic concretization;" that is, that developmental progressions often proceed from sensory-concrete and implicit levels at which perceptual concrete supports are necessary and reasoning is restricted and informal to more explicit generalizations and abstractions that are tenuous to integrated-concrete understandings relying on internalized mental representations that serve as mental models for operations and abstractions that are increasingly sophisticated and powerful. Here we focus on the importance of those early, informal levels.

In Spring 2008, we began a longitudinal study of the development of children's measurement understanding (length, area, and volume). The project started with a cohort of eight, pre-K students in a small, private, urban elementary school; we followed them for four years, documenting their understanding of measurement, using a mixed methods approach. This approach included the use of videotaped pre- and post-assessments, clinical interviews, classroom teaching experiments, and individual teaching experiments (TEs) to document children's behaviors; transcription and analysis of these videotapes was used to assess growth in children's measurement understanding.

During the fall of the final year of the study, we began to question whether students' lack of understanding was due to inherent difficulties of learning measurement concepts per se or, alternatively, due simply to limited experience—especially more informal experience— with measurement-related situations. For example, during an individual TE focusing on volume, students were asked to identify which of two containers had a larger volume and most students could successfully identify the larger. When asked, however, what would happen if the larger container was filled with sand and then poured into the smaller container, students did not demonstrate an understanding that the smaller container would overflow. Most thought the sand would completely fill the small container to the top. Similarly, when the situation was reversed so the smaller container was filled first, most students thought the sand would completely fill the larger container as well.

Situations like the one described here directed us to provide students with a Measurement Club experience in order to explore our research questions. Measurement Club convened once a month for one and a half hours each session. Our first Measurement Club was in November 2010; the eighth and final club meeting was in June 2011 of that same school year. All students in the second grade classroom were invited to attend Measurement Club; at each meeting, six or seven students were in attendance.  

The focus of the Measurement Club was simply to provide students with a variety of informal measurement experiences. For example, during our first meeting, students were presented with 20 containers of various volumes and asked to put them in order from smallest volume to largest. Additionally, students were provided a sand table they could use. Students quickly determined a course of action that involved taking the largest container and using it as a receptacle. They then took turns filling smaller containers with sand, pouring the sand from the smaller container into the larger container, and measuring the height of the sand in the larger container; their conclusion – the higher the sand, the larger the container. In another experience, students determined how many boxes would be needed to completely fill the classroom while in another students were tasked with estimating a variety of distances/lengths.

During these group experiences, students demonstrated higher understanding of measurement concepts than they did during the individual teaching experiments (prior to the onset of Measurement Club). Individual TEs with our cohort of focus students continued during this time and students involved in the Measurement Club demonstrated transfer of those skills to tasks in the individual TEs. Further analysis of these individual TEs continues as we seek to better understand the impact of Measurement Club on student learning. Additionally, our project concluded in June 2011 with a post-assessment, data from which is currently being submitted to Rasch analysis.  Results of this Rasch analysis will provide additional data to help confirm or disconfirm our hypotheses and will be presented along with more detailed qualitative analyses.

References

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81-89.

Heck, D. J., Weiss, I. R., Boyd, S., & Howard, M. (2002). Lessons learned about planning and implementing statewide systemic initiatives in mathematics and science education. New Orleans, LA. http://www.horizon-research.com/public.htm

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research:  Learning trajectories for young children. New York: Routledge.

Weiss, I. R. (2002). Systemic reform in mathematics education: What have we learned? , Las Vegas, NV.