Length Measurement in the Early Years: Teaching and Learning with Learning Trajectories

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

Integrative Statement

1. Introduction
A microgenetic study of a learning trajectory for length measurement (Table 1) assessed the efficacy of a semi-structured intervention from the End-to-End Length Measurer (EE) LT level to the Length Unit Relater and Repeater (LURR) level. In particular, we aimed to identify specific instances of the development of conceptual understanding.
2. Hypothesis and Theoretical Perspective
We hypothesized that for students to transition from the EE to LURR LT levels, procedural understanding and skills for determining length and measurement must be constrained by particular conceptual understanding.
3. Study Population and Methods
Participants from a grade 1 elementary classroom in a large suburban included four females and four males (6.85 to 7.67 years) at the start of the intervention. The previously validated intervention protocol (Fig. 1) included six total one-on-one sessions over a two-week period: one pre-assessment, four instruction, and one post-assessment. Instructional sessions ranged from 15.00 to 43.50 minutes (M = 21.36); total instructional time ranged from 66.50 to 103.25 minutes.
4. Results and Discussion
All students pre-assessed at the EE LT level. Although instruction was brief (M = 85.44 minutes), all students demonstrated mastery of the LURR level and one student began transitioning to the next level (i.e., Length Measurer) by post-assessment. Throughout instruction, many students were found to lack conceptual understanding of length measurement that would appropriately constrain their existing procedural knowledge, including the use of rulers as length measurement tools. For instance, students initially relied on the second endpoint on the ruler to determine the length of a strip regardless of the location of the first endpoint, resulting in an inaccurate, longer measurement as the item was moved across the ruler. Furthermore, when the same strip extended past the end of the ruler, some students provided a new measurement that neglected the portion of the strip hanging off the end of the ruler. Lastly, when using a Lego ruler to measure a strip, all students initially utilized incorrect units (e.g., “length”), inappropriate standard units (e.g., feet and meters), or no unit at all to describe the length.
Post-instruction, we found students built a conceptual understanding of these LURR-related skills and ability to accurately enact these skills to measure the length of an item. In particular, they benefitted from instruction targeting misconceptions, such as asking them if the strip grew longer as they moved it across a ruler—an activity that created cognitive dissonance. With conceptual understanding undergirding students’ use of procedural knowledge, they were both able to identify and correct their error(s) and apply correct use of rulers as tools to transfer items. We find that it would seem that conceptual understanding of length measurement is necessary to constrain students’ ability to enact LURR-related procedural skills with accuracy, further indicating that instruction should not solely focus on the procedural use of measurement tools. Instead, we suggest instruction incorporate opportunities for students to build conceptual understanding of the meaning inherent in measurement tools and units (e.g., what the space on a ruler represents).

Authors

• Julie Sarama, University of Denver
  Non-Presenting Author

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

• Ellen Joseph, University of Denver
  Non-Presenting Author

• Candace Joswick, University of Denver
  Presenting Author

• Jeffrey E. Barrett, Illinois State University
  Non-Presenting Author

• Craig Cullen, Illinois State University
  Non-Presenting Author