Investigating Children’s Structure, Interpretation and Representation of Space with an Intervention for Measuring Prism Volume

• In Event: 1-134 - Development, Refinement, and Evaluation of Learning Trajectories for Geometric Measurement

Integrative Statement

Hypotheses

In prior work (Author et al., 2011), we highlighted the difference between measuring a prism by building a model from cubic units taken separately, and conversely, measuring through a structured examination of unit cubes, taken in rows, and rows taken in layers to multiply with understanding. We sought a reliable process for prompting and documenting a transition to structured reasoning and support fluent use of multiplication. We built a computer simulation to provide a learning tool to build with unit cubes and groups of units, promoting spatial structures inherent in volume measurement. We posed these research questions:
1. How do students develop an algorithm for volume calculation through repeated experiences with a computer simulation for building prisms?
2. What are the critical features of the treatment that supported student development?

Study Population
Thirty-one Grade 3 (14) and Grade 4 (17) students at a private school in the Midwest participated. We report on those 15 students who demonstrated growth against a hypothetical learning trajectory for volume.
Methods
In a micro-genetic study, children were sampled for convenience from a nearby school to participate in a set of 11 trials, over a two-week period. Each trial included: (a) using paper and pencil calculations from a labeled drawing of each prism, (b) a computer simulation to predict and then build that same prism using repeated sets of unit cubes, rows of unit cubes, and layers of rows of cubes, also with labeled edge lengths (see Figure 1). For both cases we asked, “The volume of the small cube is one cubic unit. What is the volume of the larger solid?” We identified and coded observable behaviors and strategies.

Results
RQ1: We found that students developed understandings in three categories for volume calculation. In order to calculate volume correctly, students needed to coordinate their understanding among these three categories: (a) Structure – students developed the ability to collect units into a row, a row into a layer, and a layer into volume; (b) Interpretation –students developed an interpretation of the question of volume as cubes filling space; (c) Representation – students developed an understanding of the representations we presented, including a correct interpretation of the length labels. Students developed new understandings in each of these categories in a variety of sequences (see Table 1).

RQ2: The design of trials linking tasks between paper drawing and computer simulation enabled 15 of the 31 students to gain sophistication toward more structuring and answer correctly. We found 10 of 15 students used the computer simulation in an isolated manner, before incorporating that strategy into their work on the paper version; they reasoned from computer model to paper drawing to find volume. Four students adapted strategies on the paper (interpretation) and computer (structuring) portion of the trials simultaneously until the two strategies matched outcomes. Only 1 student developed an effective strategy on the paper portion of a trial, in isolation from the computer microworld.

Authors

• Jeffrey E. Barrett, Illinois State University
  Presenting Author

• Theodore J. Rupnow, University of Nebraska at Kearney
  Non-Presenting Author

• Jenna O’Dell, Bemidji State University
  Non-Presenting Author

• Craig Cullen, Illinois State University
  Non-Presenting Author

• Julie Sarama, University of Denver
  Non-Presenting Author

• Douglas H. Clements, University of Denver
  Non-Presenting Author