In this section, we present two episodes to illustrate how the different stories either disturbed or confirmed the preservice teachers’ views concerning their multilingual students’ learning about mathematics. For each episode, we briefly describe what happened and how it was deemed as being noticed. We then describe the (multi)modalities identified in the preservice teachers’ stories before discussing how modalities affected the communication, potential for learning and available identities.
Episode 1: Students develop measurements units
The first episode concerned a group of students posing problems about the perimeters of two benches in the local park and the mathematics that they used to solve the problems (see Figure 12.2). Towards the end of their investigation, they wanted to know which bench base had the longest perimeter. This was deemed to have been a noticed episode because it appeared in the PTs’ oral presentation, in their written report and in the interview. In the interviews, the PTs emphasised that in modelling tasks, the students should have access to equipment without being told what they should use and this episode exemplified this point. For example, no measuring equipment was taken by the PTs to the park, but other equipment such as skipping ropes was made available.
The PTs related that the students began their investigation by measuring with their own steps, before deciding to use “Matthew’s steps”, one of the student’s own
FIGURE 12.2 Benches by the pond that the students measured
footsteps, as a measurement unit. The students discussed how to increase the accuracy of this unit, using a heel-to-toe foot measurement. However, the students still found it difficult to determine which perimeter was longer. The students then decided to use a skipping rope and found that the perimeters of the seats of the benches were the same, but the bases were different lengths. Each bench measured one whole skipping rope and most of another. One of the students suggested the left-over section of the second skipping rope length could be used to determine the number of equal parts of which this was one part. In their written assignment, the PTs wrote that the students found out that “bench number one was 1 and 4/5 ropes long, and number two was 1 and 5/6 long”.
The PTs noted that in solving the problem that the students had posed about the longest perimeter, the semiotic resources they used to convey meaning about their developing understandings about measurement and fractions were: oral language to discuss their ideas; steps of various kinds; and a skipping rope for measuring the perimeter. Writing was also used, as the PT wrote down what the students conveyed orally about the fractional part of the skipping rope. The PTs told that it was students who chose the different modalities in developing and conveying meaning. Although the PTs had provided the skipping rope, the provenance and connotational signifiers of a skipping rope are not usually connected to measuring. It was the students who decided that a skipping rope could be used in solving this problem.
The communication between the participants was affected by the modalities they used to convey meaning to each other and to the PTs. For example, the physical use of steps opened up possibilities to talk about measurement concepts of standardised units and accuracy of measurement, including consistency in starting and stopping each unit to do with measuring the length of the bench:
One of them was going to count their steps. So, we ended up calling them Matthews steps. [...] So, we had everybody try on the next bench. When we did, it became clear that the results were different with one who walked with big steps, some who were not careful enough when they went that way (placing the toes of one shoe to the heels of another). So, there was a little disagreement in the group whether the benches were really so long or not.
Naming one child’s step provided a shorthand description of their shared understanding of what was being discussed. The physical experiences of measuring with feet conveyed meaning to the students about what might affect the answer to the problem they had posed for themselves. In this way, the discussion was not about the right or wrong answer, but about the process of measuring accurately. Writing down the fractional amounts of the skipping ropes as part of representing the perimeters of the benches also provided a shared understanding:
They did not know how to write it or in a way that was meaningful to those who had seen it and had understood it. Then I could say that we can write it that way and so and then we know what it means.
By sharing the standard representation of fractions when the students had already determined its meaning, the PTs used communication to support the students’ understanding of their representation. In summarising their stories about the measurement of the bench perimeters in their oral presentation, the PTs noted that “this was the first time they had properly collaborated”.
This suggests that the preservice teachers’ ideas about communication in mathematics lessons were disturbed. The PTs noted that collaboration between the students led to them gaining new meanings about measurement and fractions and this seemed to surprise the PTs. They also seemed to have gained a different understanding about how written representations of fractions could be introduced to Grade 2 students after they had used fractions to solve a problem.
Potential for learning
In the stories, the preservice teachers compared what they saw the students doing in the park with what did in their textbooks. For example, “my group then began to calculate division even though they could only add up to 20” (oral presentation). The PTs indicated that they were impressed with how the students had worked out the fractional amount of the skipping rope as part of determining the perimeter of the base of the bench.
But then the rope was so much too long [showed with his hands how much the skipping rope was too long].What do we do then? ... He, in this way, takes the rope and divides it into several parts. So, then they took the rope between each other [see Figure 12.3aJ. If the rope - some rope was so much too long - then he took that and ... [see Figure 12.3b—c| and then we counted 1,2,3,4 and 5.And then they found out the bench was 4 out of 5 parts of the skipping rope long.
The learning potential was affected by the students’ curiosity about which bench had the longest perimeter and by their use of different semiotic resources in developing and conveying their understandings to each other and to the PT. Their steps and folding the skipping rope provided opportunities to combine insights from both physical actions to form mathematical meanings. A PT presenting the report on their practicum stated, “I thought it was so interesting to see how they actually found a solution to something they hadn’t even learnt yet”.This suggested that their
FIGURE 12.3 PTs demonstrating how the students worked out the fractional part of the rope views about the potential learning of the students were disturbed by watching what the students showed what they were capable of doing with mathematics.
By bringing a range of tools and offering to help the students write down their skipping rope measurement, it seems the PTs saw themselves as facilitators because the students had the main role as actors who could investigate, debate, collaborate, take initiatives and come up with creative solutions.The modelling activity was seen as providing the students with opportunities for showing how capable they were of investigating mathematical problems which they had not encountered before: “they understand the concept of sharing because they have physically done it on their own initiative. So, I thought it was very interesting to see how the modelling they did supported them to explore mathematics that they couldn’t (do before)”.
The PTs’ views of the students, who could only do what they have been taught at school through textbooks, was disturbed by what they saw the students doing with the different semiotic resources. The students were able to showcase themselves as actors and capable problem posers and solvers. Nevertheless, there is no discussion by the PTs about how these students used or could have used the wider range of language resources from their multilingual repertoire in the modelling task.