Solving mathematics problems and working with designed student responses: the role of social metacognitive regulation

When students work productively together on a mathematical problem they are required to simultaneously manage their own efforts, while understanding what others are doing, and saying. Consequently, it is argued that monitoring and regulating the construction of meaning involves individual metacognitive processes, operating socially. This can take three forms: learners regulate their own cognitive processes; scaffold their partner’s understandings by taking on the role of a tutor; or mutually regulate their joint understanding. Engaging in these ways when solving unstructured non-routine problems can substantially extend students’ understanding and raise levels of achievement as measured by standard assessment criteria. Unfortunately, extant literature suggests that these processes of social metacognitive regulation (SMR) can remain elusive for many students. The aim of this study was to develop an intervention to support students in this endeavour. In so doing, local theory concerning the intervention was established. Theoretically guided classroom resources were created for teachers to use in UK secondary school mathematics classrooms. They were designed to provide opportunities for students to work with solutions to unstructured non-routine problems, authored by others. These solutions were either those constructed by peers or provided by the teacher - I designed the latter. These were worked-out solutions to problems, in the form of designed student responses (DSRs). The intention was that features such as their coherence, anonymity, and accuracy would engender a less demanding situation than when students jointly-solved problems. Rather than focusing on performance, students would attempt to comprehend and evaluate the DSRs. Integral to this undertaking was an opportunity for students to practise social metacognitive regulation (SMR). These developed practices could then be applied to the more challenging environment of constructing a joint solution to a problem. Working within a design research paradigm, three intervention studies were undertaken to establish the veracity of this claim and, simultaneously, develop theory. In each study, a structured sequence of activities was enacted in a mathematics classroom of 13 and 14 year old students. The scope of the sequence varied from study to study. Students tackled between one and seven unstructured problems and then worked with DSRs to each of the problems. Through theory-driven iterative cycles of design, implementation, and analysis, the intervention was refined. A rich range of instruments to measure SMR processes through a duel focus on what students did and said were also developed. Accordingly, the class’s written work and the dialogue between one or two pairs of students as they worked together were analysed. In general, the results supported the efficacy of the intervention. They revealed that between the start and end of an intervention, students’ problem solving improved. There was a modest shift from students producing concrete solutions towards better quality, more abstract ones. The shift, however, also coincided with an increased demand inherent within the problem. It was found that the way students formulated the problem and the strategy they employed was key to their success. The results from the studies also indicated that DSRs acted as a mediating tool to support students to practise SMR. Two important phenomena underpinned this finding. Firstly, compared to constructing a joint solution, when students worked together with DSRs, SMR manifested itself more frequently in their conversations. Students also generated a higher proportion of utterances that included reasoning. Secondly, DSRs that incorporated a visual strategy prompted better quality conversations than those that did not. When students worked with these DSRs, SMR episodes were characterised by extended interactions, in which contributions were acknowledged and built on rather than ignored. Furthermore, such DSRs provoked more student evaluations that considered specific mathematical criteria as opposed to non-specific, personal criteria. The research also considered the factors that determined when students produced a better joint solution than their initial, individual conceptions of a problem. The findings suggested that when there was little divergence in initial conceptions, students more frequently introduced a new, improved strategy into their joint solution. In these cases, students more effectively adopted the role of the tutor to support their partner’s understanding. Students were less successful in the role when their initial conceptions were highly divergent. When students worked with the DSRs, however, they were consistently successful as tutor. Although the research on SMR is growing, there is a pronounced lacuna in relation to unstructured problems used in a classroom setting, and none with regard to DSRs. This study contributes to this research agenda. The findings support the notion that DSRS can act as a mediating tool for social metacognitive regulation (SMR). They also illuminate the circumstances under which students productively collaborate when solving a problem