Making Sense of Double Number Lines in Professional Development: Exploring Teachers’ Understandings of Proportional Relationships

Oct. 01, 2012

Source: Journal of Mathematics Teacher Education, Vol. 15, No. 5, October 2012, p. 381-403.

(Reviewed by the Portal Team)

This study aims to understand how teachers used their existing knowledge about proportions to make sense of a representation that was new to them and the ways in which their existing knowledge proved to be helpful or unhelpful.

Context and methodology
Data for this study were collected as part of a larger project focused on teachers’ learning in professional development course.
The NSF-funded InterMath program was designed to enhance middle-grade teachers’ knowledge by engaging them in learning important mathematical concepts through tasks that typically had one right answer, but many ways of approaching the solution.

The course of interest was taught in a large, urban school district in the United States.
The class included 13 participants—three substitute teachers and 10 full-time teachers—in grades four through eight who taught mathematics all or part of the day.
The authors analyzed videotaped sessions of a group of urban middle-grade teachers across five class meetings.

Most of this analysis focuses on the six participants who were active in the class discussions and who allowed themselves to be videotaped as they worked on their own or with a partner.


The authors observed that participants began with disconnected, but useful, ideas about proportions and were able to coordinate some of those ideas.
They identified two knowledge components that were important to the participants’ sense-making activities.
The first necessary component of knowledge for making sense of the DNL was coordination of the units, which they see as evidence of the need for some level of composed unit reasoning.
Partitioning (Confrey and Maloney 2010) was the second critical concept for reasoning with the DNL.
The DNLs generated by looking at the whole and dividing it into pieces better supported composed unit reasoning.

The authors also found three knowledge components participants invoked in these tasks that prohibited effective reasoning with the DNLs.
One of these was scale and the idea that the scale should be the same for the two lines.
A second component that impeded reasoning with the DNL was estimation.
This highlighted the teachers’ reluctance to engage in proportional reasoning rather than focusing on finding a solution.
Finally, the authors noted that consistently drawing on knowledge of calculation strategies— particularly cross multiplication—limited the value of the DNL.


The analysis showed that the ways in which these participants approached the tasks was logical even when it was flawed.
Further, the authors saw evidence that participants were selecting only a portion of their knowledge components in engaging with the DNL.
Over the three phases, however, we saw productive pieces of knowledge being applied to the tasks to reason with the DNL with different levels of success.

The findings suggest that incorporating the DNL into professional development may require tasks and discussions explicitly aimed at helping participants organize their knowledge so they can efficiently determine which mathematical understandings are applicable in specific situations.


This analysis suggests three implications.
First, it suggests that content-focused PD needs to simultaneously build new pieces of knowledge and support teachers’ construction of coherent understanding to allow applicability to a wider range of situations.

Second, it suggests that the traditional view of teachers having significant deficiencies in content knowledge is both overly simplistic and, possibly, inaccurate.
The authors' experience here showed that the participants had more content knowledge related to proportions than we saw them implement in the DNL tasks.
Finally, this study suggests that careful consideration needs to be put into the tasks selected to support teacher reasoning.

Confrey, J., & Maloney, A. (2010). The construction, refinements, and early validation of the equipartitioning learning trajectory. In K. Gomez, L. Lyons, & J. Radinsky (Eds.), Learning in the disciplines: ICLS 2010 conference proceedings (Vol. 1, pp. 968–975). Chicago: University of Illinois at Chicago.

Updated: Apr. 23, 2014