Table 6.3: Hannah’s strategy group data
Lowest Strategy Group Highest Strategy Group
Number in Group 6 5
Year Level 2 3
Age 6 7
Gender 1 boy and 5 girls 4 boys and 1 girl
Strategy Stage Stage 4: Advanced counting Stage 5: Early additive part-whole thinking
Achievement As expected As expected
The following excerpts and discussions illustrate how Hannah consistently positioned students in her lowest and highest strategy groups to share their own and engage with others’ mathematical know-how. Evidenced in the following excerpts is the emphasis Hannah gave to students in both groups sharing and unpacking their different, efficient, and advanced mathematical know-how. The impact the focus on students’ mathematical know-how had on progressing students’ learning is described below, first with Hannah and the lowest group, then the highest group.
6.3.1 “Oh wow, that’s different”
In the second lesson with Hannah and her lowest strategy group students were learning to count on or back to solve addition and subtraction problems. Students had been creating word and number problems for each other to solve using their counting-on strategies. Hannah confirmed with students that they were using a
counting-on strategy and not a counting-from-one strategy. As the following excerpt outlines, Hannah expressed her surprise when Portia explained her stage 5 early additive part-whole strategy and Imogen suggested a different part-whole strategy. Participant Dialogue Hannah Solomon Hannah Portia Hannah Portia Hannah Imogen Hannah
How did you work out 8 plus 5 Solomon? I went 8 — 9, 10, 11, 12, 13 [holds up 5 fingers]. Okay Portia how about you?
It’s 13 because 8 plus 2 is 10 and 10 plus 3 is 13.
Oh wow, that’s different, that’s, well that’s part-whole thinking and okay, can you tell us all about that again? Listening guys this is very interesting.
I added on 2 to the 8 to make it 10 and then I added on the 3 to the 10 to get 13.
Can anyone else explain Portia’s strategy? No but I went 5 and 5 and 3 is 13.
Oh wow another different way, okay, well these are very advanced strategies – very efficient strategies – I think we need to stop counting on and explore Portia’s and Imogen’s strategies – Portia, Imogen, explain your strategies again and I am just going to grab some equipment so that you can model your thinking for us too.
Hannah responded to Portia’s and Imogen’s explanations with oh wow and described the explanations as being different, interesting, advanced, and efficient. The girls’ explanations were appropriated by Hannah when she asked them to explain and model their know-how for their peers. To support the girls with their explanations Hannah provided equipment that included tens frames, number lines, and Slavonic abacuses. The importance of the part-whole strategies was stressed by Hannah when she commented well these are very advanced strategies – very efficient strategies.
6.3.2 Discussion: Positioning for efficient thinking
Hannah positioned herself as having the right to highlight examples of students’ different, interesting, advanced, and efficient mathematical know-how. Portia’s and Imogen’s strategies were mathematically different because they required a
more advanced part-whole way of thinking about the numbers. This group was transitioning to stage 4: advanced counting but the girls’ strategies would be considered stage 5 part-whole thinking (MoE, 2007b). Therefore, their strategies would be considered advanced or sophisticated. The girls’ strategies were noted by Hannah as being more efficient, that is, they required fewer steps than other strategies and were easier to keep track of. In assuming the right to highlight explanations that were different, Hannah simultaneously positioned students as having a duty to suggest, understand, and apply different strategies. Students’ acceptance of this positioning is evidenced by Portia and Imogen sharing their stage 5 part-whole strategies even though they had been asked by Hannah to model stage 4 counting-on strategies.
The positioning of Hannah and her lowest group created three storylines. First, students were positioned by Hannah to do more of the enquiring, explaining, and modelling. Hannah positioned herself to introduce the learning intention, pose the first two or three problems, ask questions, and provide equipment. Students were positioned to create number and word problems to be solved, model, record, and explain their mathematical know-how, and listen to, and use the mathematical know-how of others. In a parallel second storyline, Hannah expected students to become more efficient with their mathematical know-how so that they could problem solve in quicker, more efficient ways. The third storyline is that students can influence the planned direction of the lesson and progress the lesson through their more advanced mathematical know-how. The actions of the teacher and students became social acts when they were given significance or importance by the group. Efficient and sophisticated explanations became social acts with this group when Hannah highlighted their significance for students and their mathematical advancement. Portia’s and Imogen’s part-whole strategies became important when Hannah questioned students about them, reiterated that the strategies made their problem solving more efficient, and when she changed the lesson plan to explore Portia’s and Imogen’s advanced strategies.
In review, Hannah positioned students in her lowest group to share their mathematical know-how by creating, modelling, and solving problems. Alongside from the expectation that mathematical know-how would be shared was an agreement to understand that the difference, efficiency, and sophistication of the
know-how were important. Hannah’s positioning of herself and students in her highest group are explored in the following sections.
6.3.4 “Which strategy do you think is more efficient?”
Within each lesson with her highest group Hannah asked students to “pretend I don’t know” and suggested “if I didn’t understand this how would you help me to understand?” In doing so she positioned students to ensure they were being understood by her and their peers and she promoted them as co-teachers within the lesson.
Students in Hannah’s highest strategy group were learning how to use tidy numbers to add to and subtract from 100 in their third lesson. Tidy numbers are numbers that end in at least one zero. Students were discussing how to solve the problem of Garry Grasshopper visiting at number 56 and needing to get home to number 100 quickly, recorded as 56 + = 100. Hannah facilitated the opportunity for students to strategise more efficiently by bringing a less efficient approach to their attention. She began the lesson by claiming she only knew how to count in ones and asked students for some help.
Participant Dialogue Hannah Coral Hannah Coral Hannah Students Hannah Lee Hannah Jin Hannah Students
Should he [referring to Garry Grasshopper] go like this? [counts slowly] 56, 57, 58, 59, 60…
No.
Or is there a faster, easier way? He could hop 40.
He could hop 40? — How would that sound? 56 - 66, 76, 86, 96.
Then? 96 — 100.
Could Garry have jumped another way to 100? 56 to 60, then 60 to 100 — 44.
Oh is that faster or easier? Yes.
Hannah Okay pair up and write the number sentence for the two ways Garry could jump from 56 to 100. Talk about which strategy you think is more efficient, which strategy would get Garry home quickest.
Hannah provided a less efficient model of counting on in ones for students to review and critique. The purpose of the counting model was to illustrate that jumping to and with tidy numbers was a more efficient strategy than counting on in ones. Strategies that make problems easier to solve are both easy to understand and easy to manipulate. When discussing the efficiency of the two strategies 56 + 4 + 40 = 100 and 56 + 40 + 4 = 100, students decided that both ways were efficient, but according to Lee it depended if you wanted to add the big (56 + 40 + 4) or the small (56 + 4 + 40) numbers first. Students agreed that being efficient meant attending to the numbers in the task and applying the most efficient strategies to the task.
6.3.5 Discussion: Positioning for efficient thinking
Hannah’s positioning of herself and students in the highest group was similar to her positioning with the lowest group. With both groups Hannah positioned herself to draw students’ attention to quicker, faster, and easier ways of problem solving. Hannah deliberately modelled an inefficient counting strategy to encourage students to look for a faster, easier way to solve the addition problem in Excerpt 6.3.4. Students in Hannah’s highest group were also positioned to do most of the work. Using a similar sequence to that of her lowest group, Hannah introduced the learning intention, asked the first two or three problems, and provided the equipment. Students had a duty to share answers, strategies, models, and written recordings to create number and word problems, and to listen to and use the mathematical know-how of others. As with the lowest strategy group, these duties were readily accepted by students in the highest group.
The storylines and social acts were similar in Hannah’s two groups. In related storylines students in the highest group were expected to model and explain their mathematical know-how and listen to the mathematical know-how of others. Similarly, they were expected to become more efficient with their mathematical know-how so that they could problem solve in quicker ways. Using different, efficient, and advanced explanations became important social acts as they featured strongly in students’ discussions about their mathematics. Students
discussed the efficiency of applying their number knowledge of patterns and of counting on in tidy numbers. Jin’s pattern became socially significant for the highest group when Hannah questioned students about the pattern and assisted students to connect their knowledge of patterns with quicker problem solving.
In conclusion, Hannah positioned students in her highest group to share their mathematical know-how by creating, modelling, and solving problems. She ensured students had opportunities to do more of the mathematics work than she did. As with her lowest group, there was also an expectation that students would enhance their mathematical know-how by applying strategies that were different, efficient, or sophisticated.