Conceptual Limitations in Curricular Presentations of Area Measurement: One Nation’s Challenges

ABSTRACT

Research has found that elementary students face five main challenges in learning area measurement: (1) conserving area as a quantity, (2) understanding area units, (3) structuring rectangular space into composite units, (4) understanding area formulas, and (5) distinguishing area and perimeter. How well do elementary mathematics curricula address these challenges? A detailed analysis of three U.S. elementary textbook series revealed systematic deficits. Each presented area measurement in strongly procedural terms using a shared sequence of procedures across grades. Key conceptual principles were infrequently expressed and often well after related procedures were introduced. Particularly weak support was given for understanding how the multiplication of lengths produces area measures. The results suggest that the content of written curricula contributes to students’ weak learning of area measurement.

Acknowledgments

We acknowledge the significant contributions of Kuo-Liang Chang, Dan Clark, Hanna Figueras, Patrick Greeley, Jia He, Leslie Dietiker, KoSze Lee, Aaron Mosier, Alexandria Musselman, and Matthew Pahl to the development of our coding scheme and to the coding of textbook content. The second and third authors presented parts of this analysis at the 2011 Meeting of the International Group for the Psychology of Mathematics Education (PME 35) in Ankara, Turkey.

Funding

The research reported in this article was carried out by the Strengthening Tomorrow’s Education in Measurement (STEM) project team at Michigan State University with support from the National Science Foundation (REC-0634043 and DRL-0909745). The opinions expressed here are the authors’ and do not necessarily reflect the views of the Foundation.

Notes

¹ The underlying issue concerns whether subtracting two equal areas from two larger equal areas leaves two equal areas remaining.

² EM also included a Student Reference Book and a collection exercises, Minute Math, for grades 1 through 4. We coded the all pages of these materials that contained area content.

³ That each square meter contains 10,000 square centimeters is not arbitrary once a meter has been defined in terms of centimeters, but the chosen length of the standard meter is mathematically arbitrary.

⁴ One graduate student had experience tutoring in a community college math lab; a second had no mathematics teaching experience.

⁵ SFAW’s large grade 1 total (n = 492) was due to the large number of instances (n = 208) where students were asked to compare the size of two or more shapes or objects.

⁶ Worked Examples, by definition, appeared only in student materials, where Demonstrations (by the teacher or teacher-designated student) appeared only in teacher materials.

⁷ This result reflects, at least in part, how we defined and coded Problems and Questions.

⁸ In grades K through 2, the most frequent “other” procedure was Partition in Half with Support; it accounted for more than half of all “other” procedural codes for those grades in all curricula. In grades 3 and 4, the most common “other” procedures involved reading and making circle graphs and spinners (in probability lessons), solving problems involving sums and differences of area, along with Partition In Half with Support.

⁹ The curricula often provided visual support for partitioning in half via folding actions and dotted partition lines, especially in the early grades. We separated instances where support was provided from those where it was not and only counted the latter in this procedural group, as only those instances required qualitative judgments of area.

¹⁰ We distinguished procedures for covering with different units (e.g., all pattern blocks) from procedures for covering with the same unit.

¹¹ This work expanded to include regions with fractional units, primarily in grade 4.

¹² We placed this procedure in the arrays group because it involved reading the geometric structure of rectangular arrays and eventually counting of squares, not multiplying lengths.

Additional information

Funding

The research reported in this article was carried out by the Strengthening Tomorrow’s Education in Measurement (STEM) project team at Michigan State University with support from the National Science Foundation (REC-0634043 and DRL-0909745). The opinions expressed here are the authors’ and do not necessarily reflect the views of the Foundation.