Discussions and concluding remarks

In this chapter, our analysis used a social semiotic multimodal perspective (see Figure 12.1). By identifying the multimodalities that the PTs noted that the students used, we were able to discuss the PTs’ views on the students’ communication, learning potential and available identities. The PTs identified a range of semiotic resources used by the students, such as images, physical actions such as walking up flights of stairs, tools such as informal measurement units connected to the body and the skipping rope, but also oral and written representations. In some cases, the PTs’ awareness of the facility that the students had in conveying meanings about their mathematical understandings indicated that their views about how students learnt mathematics were disturbed. In other situations, those views seemed to remain undisturbed or be confirmed.

By focussing on what the PTs noticed, it was possible to identify what they considered important, as well as some of the contradictions in the stories they told. The stories of how the students used the skipping rope to measure and to find the fractional part and to calculate amounts for their own tasks are mentioned in the report presentation, the written report and the interviews.The PTs also mentioned frequently how the students’ engagement in the modelling project differed from what the students did in the mathematics classroom. Learning outside was seen as being the opposite to learning inside because outside those, who struggled inside, took up possibilities for learning and came to be seen as active learners. In this way, the PTs linked their observations of those who usually struggled to a broader understanding of learning, one of van Es and Sherin’s (2002) key aspects of noticing, “using what one knows about the context to reason about classrooms interaction” (p. 573). PTs linked the students’ choice of tools and measurement activities to learning about informal measurement units that the students created themselves, which they had been informed about in their teacher education.They also discussed the student’s counting of stairs with the number knowledge and strategies in the Grade 2 textbook.

The PTs compared what they noticed when the students worked on the modelling task with what they did when working in textbooks. It seemed that the PTs’ understandings about how mathematics could be communicated and the learning potential that the (multilingual) students gained access to, through the innovative modelling task, disturbed their expectations of what Grade 2 students could do. In particular, the PTs seemed to be surprised by how the students’ meanings could be shown through physical actions. In this way, manipulating concrete materials like the skipping rope were seen as supporting students’ mathematical capabilities rather than being an indication of them struggling in mathematics.

It also seemed that the PTs could recognise that written representations were important when they fulfilled a students’ need. For example, writing down the amount of skipping ropes to record the perimeter of the two benches was important to the students. Similarly, the drawing of the boot to show the depth of the water in the pond fulfilled the students’ need to record what they had found out in their investigation.

By comparing what the students did in the modelling activity with the expectations implicit in the textbooks about what Grade 2 students could do at this point in the year, the PTs were able to see that the students were more capable mathematically than they had thought previously. Thus, by giving control to the students to do their own problem posing and problem solving, the PTs expectations about the students’ learning potential was disturbed. However, whether this disturbance resulted in a long-term disruption or change in their understandings, we do not (yet) have the evidence to show.

With regard to the students’ identities, it seemed that those students who were considered to be struggling were re-evaluated as being more capable. Yet their identities as multilingual students were not linked to their being capable mathematics learners. The multilingual aspects of their identities were either ignored by the PTs, because almost all of the students were considered to be fluent in Norwegian, or remained linked to struggling in mathematics.The modelling task as it was formulated and then implemented by the PTs did not seem to have the possibility' to cause the same disturbances in expectations as it had for communication and learning potential.

Thus, as teacher educators and researchers, it is possible to see that innovative practicum tasks (can) disturb some aspects of PTs’ expectations about how Grade 2 students learn mathematics. However, their design and implementation of the modelling teaching task did not get the PTs to notice the multilingual aspects of the students’ identities as being connected to being capable mathematics learners. In our future work, we need to consider alternative approaches to supporting the PTs to notice how the students’ multilingual identities (can) have a positive impact on their learning of mathematics.

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