Productive struggle in the middle-school mathematics classroom
Keywords: Education.

It may sound strange, but in today's mathematics classroom, teachers are encouraged to have students struggle. This struggle, however, is not what might come to mind upon first hearing the word. The struggle I refer to here is what researchers have termed "productive struggle, " which is present when students must persevere in solving challenging problems for which solutions are not immediately apparent (Hiebert & Grouws, 2007). Opportunities to struggle productively are essential if students are to develop rich understandings of foundational mathematics concepts (Carpenter et al., 1989; Hiebert & Grouws, 2007; Kapur, 2010; Stein & Lane, 1996). Such opportunities, however, remain more the exception than the norm for many students (Carter, 2008; Stein et al., 1996; Stigler & Hiebert, 2004). While students increasingly have access to the kind of challenging problems necessary to elicit struggle, merely providing students with such problems does not guarantee that they will struggle productively. After assigning a challenging problem, teachers must keenly observe students as they work in order to determine if their struggles are productive or unproductive. Specifically, they must "notice" struggle, attending to evidence of students' struggles then interpreting that evidence in assessing the magnitude of students' struggles. Only then can they respond as needed, standing back when students' struggles are productive or stepping in when they are not. Little is known, however, about teachers' noticing of student struggle or those aspects of this practice with which teachers might most benefit from support. Moreover, few tools are currently available to support teachers with the complex practice of noticing struggle and few studies have examined the relative proportions of the various types of struggle students encounter and that teachers need to notice. This dissertation contributes to addressing these research needs. The dissertation consists of three separate, but related studies rooted in a collection of classroom videos portraying student struggle during collaborative problem-solving. In the first study, four mathematics education researchers, myself included, sought to develop an understanding of struggle, what it looks like and sounds like, and how one can tell if students' struggles are productive or unproductive. We did this by viewing and discussing a collection of eight 10-minute video-clips portraying different groups of students struggling to varying degrees, and in varying ways, while solving challenging problems together. Initially, we sought to identify examples of "struggle, " yet found this very challenging to do, as essentially everything in the videos seemed like struggle to us (e.g., student questions, comments, gestures). Over time, we decided to instead look for instances of students becoming stuck with "roadblocks, " which we defined as an obstacle or impediment that slowed students' progress. In looking for roadblocks, we found it much easier to spot examples and only considered a question, comment, or gesture as evidence of struggle if it indicated that students had encountered a roadblock. This notion of roadblocks, as well as several other insights we had in our work together, were incorporated into a framework describing different levels and types of struggle students encountered in the videos. In the second study, a second researcher and I conducted a detailed analysis of the different types of struggle, or roadblocks, students in two 45-minute videos encountered in solving challenging problems together. We entered this study guided by the assumption that, while students would encounter individual struggles with the mathematics, they would also encounter struggles germane to the collaborative context in which they were solving problems. We identified the start and end times for every roadblock each individual student encountered. For each roadblock, we then determined the type of roadblock encountered, whether or not it was overcome, and which resources the students called upon in striving to overcome the roadblock. Contrary to our expectations, we found that only one-fourth of students' struggles consisted of individual struggles with the mathematics, while three-fourths of their struggles were related to collaboration (e.g., struggling to reach consensus, struggling to have an idea heard by others). Students not only encountered a large number of struggles related to collaboration, but overcame only about 25% of these struggles. By contrast, they overcame over 50% of their individual struggles with the mathematics, doing so in almost every case by turning to their peers for support. Students rarely turned to the teacher for support, nor were offered such support, unsolicited. Finally, in the third study, I examined the evidence of struggle that 10 middle-school teachers noticed when viewing videos of students struggling while solving problems together. Specifically, I examined the types of struggle teachers attended to, their interpretations of these struggles, and their overall assessments of students' struggles as productive, unproductive, or something else. Additionally, I implemented a brief pilot intervention designed to support teachers' noticing of struggle. The intervention consisted of multiple components: 1) a revised prompt for viewing the videos, 2) a compilation of different types of roadblocks students encounter, and 3) a framework listing and distinguishing different levels, or magnitudes, of struggle. The purpose of the intervention was to broaden teachers' conceptions of struggle such that they would consider not only individual, mathematical struggles, but also struggles related to collaboration. Results showed that teachers attended most often to students' individual struggles with the mathematics and, to a lesser degree, their struggles to engage with their peers' thinking and have their voices heard and ideas acknowledged. Teachers' interpretations of the evidence they attended to appeared guided by multiple, distinct narratives about teaching and learning (e.g., that struggle is productive if it is devoted to understanding the mathematics). Following the intervention, teachers attended to less evidence of students' individual, mathematical struggles and slightly more evidence of their struggles regarding group dynamics (e.g., struggling to be heard). There were few notable shifts, however, in teachers' guiding narratives, although teachers did become slightly more likely to assess students' struggles as productive. A key takeaway from this dissertation is that there appears to be a misalignment between the types of struggle students encounter most and the types of struggle teachers tend to notice. While the second study showed that students encountered few individual struggles with the mathematics, teachers attended most to this type of struggle. As students encountered mostly struggles related to collaboration and had the most difficulty overcoming these, it might have been preferable for teachers to have been particularly attuned to these, yet the third study showed that this was not the case. I discuss this key takeaway and future directions in this line of inquiry in the concluding chapter