The Failure of Diagrams on Standardized Testing to Assess Students’ Understanding: What Teachers Need to Know to Use Diagrams Appropriately

Casey Hawthorne, Charles Michael, Mikayla Miley


Many standardized tests have begun to broaden their assessment beyond the use of symbolic representations and now include more concrete diagrams. This paper reports on the conceptual resources 5th grade students leveraged when attempting to answer such questions in the domain of fraction operations. Results demonstrate how such questions failed to assess students’ understanding of the diagrams or of fractions, as the vast majority of participants used symbolic procedures or superficial qualities of the diagrams to navigate their answer choice. However, student responses, when pressed to explain their thinking about the diagrams, revealed rich thinking, illustrating how such questions can serve as productive formative assessment tasks.

Full Text:



Battista, M. (2012). Cognition-Based Assessment Teaching of Multiplication and Division: Building on Students Reasoning. Portsmouth, NH: Heinemann.

Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics. Retrieved from

Huinker, D. (2015). Representational competence: A renewed focus for classroom practice in mathematics. Wisconsin Teacher of Mathematics, 67(2), 4-8.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Smith, M. S., & Stein, M. K. (1998). Reflections on practice: Selecting and creating mathematical tasks: From research to practice. Mathematics teaching in the middle school, 3(5), 344-350.

Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76(3), 265-280.

Webel, C., Krupa, E., & McManus, J. (2016). Using representations of fraction multiplication. Teaching Children Mathematics, 22(6), 366-373.


  • There are currently no refbacks.

Copyright (c) 2021 Charles Michael

The Ohio Journal of School Mathematics is published by the Ohio Council of Teachers of Mathematics and hosted by The Ohio State University Libraries.

If you encounter problems with the site or have comments to offer, including any access difficulty due to incompatibility with adaptive technology, please contact

ISSN 2472-5986 (print)
ISSN 2472-601X (online)