Introduction

This study aims to provide insight into teacher knowledge by examining how teachers drew from their prior understandings in making sense of proportional situations as they interacted with a single representation, the double number line (DNL), in a professional development course.  

Theoretical Framework

We build from the knowledge in pieces framework (e.g., diSessa, 2006; Smith, 1995; Smith, diSessa, & Roschelle, 1993) to understand not only what teachers struggle with but also what they understand. This moves beyond focusing only on teacher deficiencies. Knowledge in pieces posits that learners have understandings that they use to make sense of new situations. While these understandings may not be robust or appropriate in different settings, the learner is still using a rational process for invoking them. We view teachers as learners who have a tremendous amount of knowledge, but recognize that it may not be organized in ways that allow them to draw from their knowledge base in ways that support meaning making. Further, we recognize that teachers have more understanding of a concept than we see demonstrated because some aspect of the presented task cues them to engage in certain kinds of reasoning (see Lobato, Ellis, & Muñoz, 2003).

We also build from prior work with representations that roots mathematical understanding in a leaner’s ability to “flexibly manipulate the idea within given representational systems”, and the ability to “accurately translate the idea from one system to another” (Lesh, Post, & Behr, 1987, p. 36). Given this as a definition for understanding, a teacher needs to be able to support students in thinking flexibly through representations, which means the teacher must be able to do this.

This work contributes to the emerging literature on mathematical knowledge for teaching (e.g., Ball, Thames, & Phelps, 2008; Hill, Ball, & Shilling, 2008; Silverman & Thompson, 2008). We are interested in not only moving beyond studies that only highlight deficiencies in teacher knowledge (e.g., Armstrong & Bezuk, 1995; Ma, 1999; Post, Harel, Behr, & Lesh, 1988) but also contributing to a the dialogue about how to support teachers in building a coherent understanding about mathematics (e.g., Thompson, Carlson, & Silverman, 2007).

Methods & Data Sources

This study was part of a larger project in which one of the research questions focused on what teachers learn in professional development. The current study focuses in on a single representation (DNL) of a single concept (proportional reasoning) to understand how the teachers reasoned about DNLs, what knowledge they invoked in making sense of them, and the struggles they had with them.

The professional development met three hours per week in a school computer lab for14-weeks. There were 15 middle grades teacher participants. The facilitator was a middle school teacher with a strong mathematics background. The course focused on operations with fractions and decimals, ratios, and proportions. The explicit goals were to increase attention to referent units, reasoning with drawn representations, and improving proportional reasoning.

Data analyzed included the videotapes of the five three-hour class meetings in which participants used DNLs. The videotapes used picture-in-picture technology so that drawings being discussed were visible along with the participants’ faces or bodies. We focused on the six participants who actively contributed during these five sessions.

Data were initially coded to highlight areas for further analysis. Once we identified some issues teachers had with DNLs, we conducted an analysis grounded in the research on students’ reasoning about proportions (e.g., Lamon, 2007). For this analysis, the first author watched video of all discussion focused on the DNL tasks in the five sessions and memoed it to characterize teachers’ reasoning (Strauss & Corbin, 1998), then classified the thinking into categories that aligned to key themes in the proportional reasoning literature such as reasoning with composed units or iterating units. The second researcher considered the plausibility of the arguments being developed. That is, in each analysis the second researcher considered only the key data identified by the primary researcher, but determined whether the codes assigned seemed plausible or whether additional data were needed. In this way, all reported codes and data were the results of an inter-rater agreement process that allowed triangulation of data across researchers (Denzin, 1989).

Results and Conclusions

Our participants struggled to make sense of DNLs. They found them confusing and were vocal about their dislike of DNLs, which were new to them. We identified five key knowledge pieces that were the most prevalent in the teachers’ reasoning. Two of these led to productive reasoning while three led to confusion. The two productive strategies were using the DNL to think about the coordination between the two units (one on the top line and one on the bottom line) and partitioning that allowed the coordination of values in the DNL to happen. The less productive strategies included trying to treat them as plots on the one number line with the same scale for both units, using the cross multiplication algorithm to get an answer that was then drawn on the representation, and using estimation strategies.

Importance of the Research:

The findings suggest two implications. First, professional development needs to build new pieces of knowledge while supporting teachers’ construction of coherent understanding. Second, it suggests that the traditional view that teachers have significant deficiencies in content knowledge is overly-simplistic. The teachers certainly had some gaps in their understanding, but they also knew a lot about proportionality that they were initially unable to use with the DNL.

Our poster will highlight the findings with more elaboration on the affordances and limitations of the knowledge the teachers used to make sense of the DNL. It will also include a brief introduction to DNLs because they are a lesser-known representation.