# Making sense of double number lines in professional development: exploring teachers’ understandings of proportional relationships

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## Abstract

This study presents a qualitative analysis of the knowledge teachers in one professional development course used to reason about proportional relationships with double number lines. We work from the knowledge-in-pieces perspective to consider the existing knowledge the participants did or did not invoke when learning to reason with this new-to-them representation. We analyzed videotaped sessions of a group of urban middle-grade teachers across five class meetings. Our findings include discussion of the two pieces of knowledge that emerged as important for reasoning about proportions with the representation and three knowledge pieces that deterred meaning making. Implications for professional development are discussed as are the implications for conceptualizing teachers’ understanding through a knowledge-in-pieces lens.

## Keywords

Middle grades Professional development Proportional reasoning Drawn representations Teacher knowledge## Notes

### Acknowledgments

Work on this project was supported by the National Science Foundation (DRL-0633975 and DRL-1036083). The opinions expressed here are those of the authors and do not necessarily reflect the views of NSF. The authors wish to thank Gunhan Caglayan for his work on the data analysis as well as the Does it Work team, the participating teachers, and the facilitator for their support with data collection.

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