«In a recent PME paper Askew (Askew, 2007) drew our attention to a need to reconsider the viability of Vygotskian notions of scaffolding within ...»
Considering this lesson it appears that both sets of individuals have been scaffolded to play specific roles. The teacher used the script to play out a role in which he can address a specific and narrow learning outcome, which he has detailed at the start of the lesson. The role he took was to show and tell. The students in turn adopted the role he cast them in and played out their role to acquire the specific piece of mathematical knowledge. They were ‘talking and doing’ mathematics but the question is, “What knowledge of themselves as mathematicians were they developing?” Moreover, “What were they learning about talking and doing mathematics?” The following section contrasts the preceding lesson episode, which occurred in the first week of the study with a mathematics lesson which took place towards the end of the study.
Scaffolding as a ‘Tool and Results’.
As outlined in the Research Design section, extensive scaffolding was provided by the teacher to support students to use a range of proficient mathematical practices including reasoned mathematical explanations, justification and generalisations. Scaffolding was also used to support students to develop a repertoire of questions and prompts to use to inquire into the sense-making of others. In addition, the teachers also paid specific attention to establishing group norms to ensure interthinking occurred. In relationship to the New Zealand Numeracy Development Project, the teacher continued to draw on the curriculum material to provide guidance for his lessons. But then now he no longer followed the script and he wrote problems, which better matched the interests of his students.
In this lesson the teacher wanted the students to explore the strategy of partitioning but he had selected numbers, which support emergence of multiple ways of reasoning towards a solution strategy. The lesson consisted of two components; small group problem solving and then a large group discussion. This episode describes the first section of the lesson in which the students had been placed in groups of three and without teacher-led discussion they were given a problem25 and asked to discuss and develop a number of solution strategies.
Saawan: What about five times 700 and then… Hine: Five times fifty, and then five times six.
Sonny: Hey mine’s the same but mine’s starting from the six, fifty, and then seven hundred.
Hey all our ways are the same, well kind of, because you can start both ways.
Saawan: Well let’s see if that right…so you say we can start both ways, yeah that’s cool it works.
The students began immediately to work together, interthinking, and they constructed a solution strategy using the distributive property. They continued to discuss and explore whether the order of how the factors were distributed affected the solution as they recorded them in the different ways. As Sonny studied the recordings he introduced the group to an alternative idea. This strategy was one that drew on distributing the factor of five rather
than the factor of 756:
Sonny: I have just thought and I know another way. Can you do seven hundred and fifty six times two and then plus it so the times two becomes…becomes times four…equals… Saawan: What? Let’s write it down.
Sonny was playing with the idea of the generalisation the group had collectively constructed. He introduced it as he thought out loud and Saawan’s answer indicated that although he had not yet made sense of what Sonny was saying he was open to the new contribution. Sonny showed that his thinking was still being formed when Hine recorded it
vertically as 756 + 756 and he told her:
Sonny: No times two is easier.
Bart Simpson had five different coloured marbles. He had 756 of each colour and Lisa wants to know how many he has altogether. Can you help him tell Lisa how many he has? Lisa might challenge him to prove he has more than her so can you work out some different strategies he could use?
Saawan followed Sonny’s reasoning closely and his argument indicated that he was making a link to their previous reasoning. He then extended his reasoning and that of his
peers when he argued that multiplication was repeated addition:
Saawan: Times two yeah but doing it that way is the same way really, you can say it as a plus because that’s the same as times like before when we went the other two ways not just one way.
Hine, listening to the exchange crossed out the recording, replacing it with 756 x 2. Then Sonny continued with the new thinking as Hine and Saawan tracked closely and examined
the reasoning section by section:
Sonny: Seven hundred and fifty six times two equals one thousand six hundred and twelve… Hine: Wait, one thousand… [Lapses into silence as she records 700 x 2 then writes 50 x 2 and 6 x 5].
All three students examined the recordings and checked the total. Then Saawan took the
pen, from Hine and he recorded 1512 x 2 as he continued to explain:
Sawaan: And then we times, no we add them together then times it by two and add seven hundred and fifty six on to it [Records 3780].
Hine: But hang on how did we get that?
Sonny: [Directs her attention to the recording as he explains] By timsing this by two, and this by two, and then adding.
Hine: [Nods her head] Yeah I get it now.
The teacher had been sitting silently listening and observing the interaction. Then he observed Hine’s continuing uncertainty and so he prompted her to question, emphasising that she needed to do so until she had complete understanding Teacher: You look like you are still a bit puzzled. Look at what he has explained and if you need to, ask more questions. Make sure you are convinced that it works. Think about a good question and ask it.
Hine: Why did you times one thousand five hundred and twelve by two?
Saawan: Because it’s like…because then when we times that by two [he points at the second two] it is like that will be like four and then we only have to add seven hundred and fifty six. It’s just doubling.
The teacher’s prompt for further questioning left the mathematical agency with the students. After closely listening to the student provided explanation he then pressed them
to further explore the reasoning:
Teacher: By adding this [He points at + 756] what’s another way of saying that because I think maybe that…how could you say it differently instead of saying adding seven hundred and fifty six?
Now Sonny and Hine indicated that the reasoning Saawan introduced had become
integrated within their collective understandings:
Sonny: You could multiply it by one… Hine: Okay, I get it now so multiply by one yeah so when we times two, times two, times one because the whole thing is seven hundred and fifty six times five, so times five yeah, [she laughs then refers to the context of the problem] huh that’s a good one Lisa better understand from Bart.
In this second lesson scaffolding took a different form from that reported in the first lesson. Scaffolding had become a tool, which mediated the mutual engagement of all participants in the collective reasoning. The use of problem solving groups where mathematical expertise was more evenly distributed across the members changed their interactions. This resulted in each individual’s role emerging and changing minute by minute in the discussion, as they were pulled into a shared communicative space. The different contributions scaffolded the group members being extended beyond their own capabilities. Importantly, the mathematical understanding they were developing was of equal importance to what they were learning about acting as mathematicians and ‘talking and doing’ mathematics.
Conclusions and ImplicationsThe paper sought to explore and examine scaffolding used in two different ways in classroom episodes, and the learning, which emerged as a result. The paper illustrated that when scaffolding is used as a tightly controlled tool within what Askew (2007) describes as a “technical-rationalist view of teaching and learning” (p. 239) the roles the teacher and the students hold and the mathematical talk they use and the knowledge they develop is limited. Likewise, the students’ learning to ‘talk and do’ mathematics in ways mathematicians do, are restricted. However, when scaffolding is used within a widened dimension that affirms both the importance of the construction of mathematical knowledge and the manner in which it is constructed, the learning potential for all participants is enhanced.
This paper confirms the results in Askew’s (2007) PME paper but extends these results to show the learning potential available when teachers scaffold students to work together to construct a collective mathematical view within zones of proximal development. As other researchers (Goos, 2004; Goos et al., 1999; Lerman, 2001; Mercer, 2000) have illustrated, the act of interthinking and developing a collective view was a key factor which scaffolded how these students learnt to talk and do mathematics. Of importance too, was careful teacher preparation, which drew on the New Zealand Numeracy Project as a tool for classroom activities rather than a rigidly followed formula. The use of grouping and the careful selection of numbers allowed the lesson to unfold and the students to improvise and play with the numbers, in a form of mathematics, which was generative.
Implications of this study suggest the need for mathematics educators to consider not only the importance of the development of mathematical knowledge but also how it is constructed. In this form scaffolding needs to be metaphorically viewed as a ‘tool-andresult’. National projects such as the New Zealand Numeracy Project (Ministry of Education, 2004) have an important place as a professional development tool but teachers need to develop their own script rather than use the materials rigidly.
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