Discussion: Positioning for mathematical difference

Delphi positioned herself in the three lessons with her lowest strategy group as having the right to assist students to determine and apply mathematically different strategies and to build connections between their existing and new learning. In Excerpt 6.4.1, Delphi helped students to come to an understanding that for a strategy to be different it needed to tell a different story. To be able to identify mathematical difference, students need to understand the explanations that have already been discussed to be able to judge the extent of the similarities and differences. Delphi positioned students to share their different number stories and strategies and reflect on the differences and similarities between them. This positioning provided additional learning opportunities for students and extended their cognitive activity beyond simply solving the task to comparing the different ways the task could be solved.

Delphi’s positioning of herself and students in these three lessons created two storylines. In the first storyline students in Delphi’s lowest group were expected to share, explain, critique, compare, defend, model, and record their own and their peers’ mathematical know-how. The second storyline stressed the importance of being able to identify and describe mathematical difference in explanations. In Excerpt 6.4.1, Delphi provided the opportunity for students to discuss what it meant to be different and develop their own definition that highlighted the numbers telling a different story. Students’ explanations became significant social acts when they were promoted by Delphi. Graham’s idea of the numbers telling a different story became important when Delphi referred to his strategy, when she directed Wiki to reflect on her example of mathematical difference in consideration of Grahams’ idea, and when she reminded students to remember and apply Graham’s excellent idea to their future problem solving.

To summarise, Delphi had a right to expect students to understand mathematical difference and suggest mathematically different strategies, and a right to position students to reflect on and use existing know-how to advance their mathematics. Students had a duty to share their number stories, defend the mathematical difference of their number story, determine what constituted mathematical difference, reflect on their definitions of difference, and apply their new understandings. These positions were readily accepted by students and there

were no examples in the lessons of students refusing the positioning or trying to position someone else to do the work for them. Delphi’s teaching and positioning decisions with her highest group are described in the following sections.

6.4.3 “Now that is very interesting”

In the second lesson with Delphi and her highest strategy group, students were learning how to “add by counting on when the larger number is given first” (MoE, 2007e, p. 18). In this lesson Delphi drew students’ attention to interesting strategies they used to solve their addition problems. The first problem Delphi asked students to solve was 3 + 63.

Pio and Joseph shared their strategies and Delphi asked students to consider whether they thought Pio’s and Joseph’s strategies were different and why. Connor thought the two strategies were different and he used the names of the strategies to explain the difference – Pio doubled and Joseph counted on. Delphi confirmed Connor was correct and presented the next problem for students to think about and then discuss: 5 plus 91 jellybeans. In the following excerpt Delphi highlights Pio’s really interesting strategy of swapping the numbers.

Participant Dialogue

Delphi

Pio

Delphi

Pio

Okay who wants to go first?

Oh me Miss I went 95 plus 1 equals 96

Wow that’s a really interesting strategy to use Pio — you need to tell us how you knew to work it out that way.

I swapped the numbers; I swapped the 5 and the 1 around so I didn’t need to go 91 plus 5, I just go 95 plus 1 and its 96.

Participant Dialogue Delphi Pio Delphi Joseph Delphi

Okay who has an answer and a strategy for working that out? Easy 3 and 3 is 6 so it’s 66.

Oh interesting Pio — could you record that for us please? Did anyone else solve that differently?

Yes [holds up 3 fingers] 64, 65, 66.

Very clever Joseph. Can you write that down for us please? 64, 65, 66. Now can you have a think for a minute — are Pio’s and Joseph’s strategies different — and why are they different?

Delphi Delphi Delphi Tyson Pio Delphi Students Delphi Rohan Delphi Aroha Tyson Pio

That really is very clever — have a talk to someone about Pio’s strategy — how did his strategy make his adding easier?

Students discuss Pio’s strategy in pairs.

Great discussing guys. Okay I have a question for you. If I solved 91 plus 5 jellybeans by counting on [holds up fingers] 92, 93, 94, 95, 96 — would that be different to how Pio swapped the numbers and then counted on? Have a think and then we will have a talk.

Students discuss Delphi’s question in pairs.

Okay what do we think — would those two strategies be different? No because Pio and you counted on.

Yes because I swapped before I counted on — I made the 1 the 5 and then it was 96 — I done it easier.

Oh now that’s interesting — Pio says his way made the problem easier to solve. So Pio is saying that 95 plus 1 is easier to work out than 91 plus 5. Do we agree?

Yes.

So maybe Pio’s strategy is a bit different because he swapped the numbers but what is really interesting is that he made the sum easier to solve. I wonder if there is another way that would make our adding easier.

[giggles] Eat some of the jellybeans before we start adding!

[laughs] Yes well that would make it easier — but how about a way where we

didn’t have to eat the equipment! Okay I am going to give you five different sums to talk about and solve. What strategies have we discussed today and you can use?

Counting on. Doubles.

Swapping numbers.

Delphi drew students’ attention to Pio’s ‘swap-the-numbers’ strategy first by asking him to repeat his strategy and then by asking students to decide if Pio’s strategy was different to the counting-on strategy she modelled. Tyson did not think the strategies were different because both Pio and Delphi counted on. Pio defended his strategy and claimed it was different because he swapped the numbers before he counted on and had therefore done it easier. Delphi highlighted for students that using an easier strategy was also interesting. Students were asked to review the strategies they could bring to their addition problems – and students shared counting-on, doubles, and swapping numbers.