Discussion: Positioning for self-regulating

Jenna positioned herself with her lowest strategy group as having the right to ensure students were explaining, monitoring, and proving their own and peers’ mathematical know-how. Students were positioned by Jenna to use Xiang’s correct model and to critique and justify Duncan’s correct counting strategy. Esmee was positioned to self-correct her counting strategy and when she was unable to, Jenna called on others for help. Jenna noticed Duncan’s self-correction and asked him to explain his action. Explaining, monitoring, and proving their own and their peers’ mathematical know-how positioned students as having a duty to provide counting models, represent their thinking on materials, pay attention to others’ explanations, help each other, and justify their correct answers and self- corrections. Jenna expected students to share their mathematical know-how and attend to peers’ mathematical know-how. Students in the lowest group readily accepted their duties and actively participated in their group’s know-how.

A prevalent storyline within Jenna’s teaching with her lowest group is the importance of students becoming self-aware about their own and others’ mathematical know-how. Jenna ensured students had opportunities to notice

Participant Dialogue Jenna Duncan Jenna Duncan Jenna Duncan Jenna

What have you done Duncan? Why did you take one off? Um.

Why did you have to take 1 off?

[shakes his head side to side] Because it not right, it’s not 4.

Oh wasn’t it 4? What did you have to do then to make it right? Take 1 off and it is 4.

their own and others’ mathematics through the questions she posed and the expectations she held. This was done by asking students to check they were correct, explain how they knew they were correct, and positioning students to help each other. The awareness students were expected to have, regarding their own and peers’ mathematical know-how, was a prevalent storyline with this group. Jenna set the expectation that students would participate in each other’s know-how by positioning them to provide models, explanations, and assistance as required. Students accepted this positioning and on many occasions were observed volunteering their examples and help. The correcting of models such as Xiang’s model of three fingers, errors such as Esmee’s miscounting, and Duncan’s self-correction became significant social acts for this group when they were appropriated by Jenna and other students.

In summary, Jenna’s positioning decisions with her lowest strategy group supported students to share their own and experience others’ mathematical know-how. Know-how was shared and experienced through students providing counting models and representations, and checking their own and others’ correct and incorrect models and representations. In positioning students to monitor and reflect on their own and others’ learning, Jenna assisted students to develop self- regulated learner skills. The similar positioning choices Jenna made with her highest group are illustrated through the following excerpts.

6.5.4 “How do we know who is right?”

The learning intention for all three lessons with Jenna’s highest strategy group was learning how to “add tens to a number by counting on in tens or adding the tens together” (MoE, 2007e, p. 22). Jenna began the first lesson with her highest group by scattering three jellybeans and asking students to decide how many jellybeans. After answers and explanations had been shared, Jenna scattered seven, then 11 jellybeans, and again answers and strategies were shared. The fourth scattering contained 87 jellybeans and Jenna asked students to predict the number of jellybeans. Suggestions of 20, 40, and 100 were offered but students were unable to give a mathematical reason for their answers. Ainsley suggested counting the jellybeans in groups of 10 and Jenna directed students to work together to count the jellybeans into sets of 10, and put the sets of ten into a

canister. The following excerpt illustrates how Jenna positioned students to determine who was right and why.

Participant Dialogue Jenna Students Students Jenna Shane Jenna Students Jenna Students Jenna Students Jenna

Okay so how many have we got altogether? 87.

No 78!

Is it 87 or 78, how do we know who is right? Count them.

Good idea — let’s use Ainsley’s idea to count in tens.

[pointing to the canisters] 10, 20, 30, 40, 50, 60, 70 80.

Okay so what would we keep counting in now – what would be next?

[pointing to the single jellybeans] 81, 82, 83, 84, 85, 86, 87

So how many jellybeans? 87

Who thinks Ainsley's idea of counting in tens is easier than trying to work out that great big muddly pile of jellybeans? Was it easier that way?

Jenna gave the responsibility for determining the correct number of jellybeans to the students by asking: Is it 87 or 78, how do we know who is right? Shane suggested counting the jellybeans and Jenna agreed counting was a good idea and specified the type of counting by stating let’s use Ainsley’s idea to count in tens. By counting in tens and ones students determined there were 87 jellybeans. Jenna further engaged students with the counting in tens strategy by asking students Who thinks Ainsley's idea of counting in tens is easier than trying to work out that great big muddly pile of jellybeans? This lesson concluded with Jenna putting students into pairs, giving each pair a pile of fewer than 100 jellybeans and asking them to count in tens to work out how many jellybeans they had.

6.5.5 “Who can tell us or show us how they know that they are right?” The second lesson began with students sharing the total number of jellybeans they were given at the end of the first lesson and explaining their strategy for knowing how many they had. Jenna asked students to explain how they knew

three pots would contain 30 jellybeans and in doing so elicited different examples of mathematical justification. Students explained how they knew they were correct and they provided explanations that included repeated addition, skip counting, and multiplication. Shane, Lilly, and Ainsley referred to their written recordings and modelled their thinking on materials to ensure their explanation was clear to others. By asking students to further explain their strategies, Jenna provided the opportunity for all students to hear examples different to their own. In the same lesson, Ainsley claimed there were 100 jellybeans because they had 10 pots of 10 jellybeans, and 10 times 10 equals 100. In the following excerpt Jenna positioned Ainsley and her peers to explain her multiplicative thinking.

Participant Dialogue Ainsley Jenna Ainsley Jenna Ainsley Students Jenna Lilly Jenna Lilly Jenna Candace Jenna Candace Jenna Candace Jenna Bianca

And there's 100 altogether.

Why do you think there are 100 altogether? Because 10 times 10 equals 100.

But why do we have 100 altogether — are you sure?

Yes — we have 10 pots and 10 jellybeans and 10 times 10 is 100 so 10,

[holds up 1 finger] 20, [holds up 2 fingers]

[chant and hold up fingers] 30, 40, 50, 60, 70, 80, 90, 100.

Do you think Ainsley is right? Yes.

Why do you think she is right Lilly? Because she counted up in tens.

What do you think Candace? Do you think she might be right? Yip.

Why do you think she might be right?

Because if there is 10 packets and they all have ten there is 100 [counts each

pot] 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 and 10 times 10 is 100.

[places 10 packets of 10 in front of Candace] So there's your ten packets of

jellybeans so are you saying that's 100? Yes because 10 pots of 10 is 100. Bianca — is she right?

Yes it’s the same as plussing them [points to each pot] 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10.

Jenna positioned Ainsley to share her multiplicative know-how and positioned students to make sense of, and explain, Ainsley’s strategy. Ainsley discussed the relationship between skip counting and multiplicative thinking, Candace reiterated this, and Bianca described an additive relationship. By expecting students to explain each other’s correct answers and strategies, Jenna positioned students to engage with other ideas at a higher cognitive level. As well as unpacking and explaining their own ideas, students had to understand others’ strategies so they could also unpack and explain them. Students had access to their group’s mathematical know-how, explanations, and understandings.