Mathematics Teachers' Reasoning About Fractions and Decimals Using Drawn Representations

Abstract

This qualitative study considers middle grades mathematics teachers' reasoning about drawn representations of fractions and decimals. We analyzed teachers' strategies based on their response to multiple-choice tasks that required analysis of drawn representations. We found that teachers' flexibility with referent units played a significant role in understanding drawn representations with fractions and decimals. Teachers who could correctly identify or flexibly use the referent unit could better adapt their mathematical knowledge of fractions validating their choice, whereas teachers who did not attend to the referent unit demonstrated four problem-solving strategies for making sense of the tasks. These four approaches all proved to be limited in their generalizability, leading teachers to make incorrect assumptions about and choices on the tasks.

ACKNOWLEDGMENTS

The work reported here is supported by the National Science Foundation under grant DRL-0633975. The results reported here are the opinions of the authors and may not reflect those of NSF. The authors wish to thank the Does it Work team for their support and particularly Danie Brink, Andrew Izsák, and Susan Sexton for help in conducting the interviews analyzed for this report. The authors also wish to thank three anonymous reviewers for their thoughtful—and thought-provoking—comments. Earlier versions of this report were presented at the Research Presession of the 87th Annual Meeting of the National Council of Teachers of Mathematics, the 2009 Annual Meeting of the American Educational Research Association, and the International Conference of the Learning Sciences 2008.

Notes

¹Because the assessment items are secure, the items reported here have been modified with the intent of maintaining the mathematical ideas and the complexity of the original items.

²All names are pseudonyms.

³In all the transcripts reported in this manuscript, italics have been used to emphasize the statement or phrase that aided us in classifying the strategy used.

⁴This is discussed later under teachers' inefficient problem solving strategy, Looking for a diagram that matches a solution.