Chelsea positioned herself with her highest strategy group to explain and model students’ correct and incorrect strategies for them and to expect students to apply the strategy that was the focus of the lesson. Applying one strategy per lesson was not a requirement of any other teachers in my study. In Excerpt 6.8.4, Chelsea explained the difference between a 4 by 5 array and a 5 by 4 array, clarified why she had modelled 4 sets of 6 and not 6 sets of 4, and explained how she thought students solved 4 times 6. Chelsea did not acknowledge or appropriate the different doubling and place value strategies students suggested in both excerpts or the use of basic fact knowledge in Excerpt 6.8.5. Instead, she emphasised the importance of students applying the compensation strategy. Students had a duty to follow Chelsea’s model and explanations and apply the strategy selected for the lesson. In these excerpts there was no evidence of students being provided opportunities to flexibly use their mathematical know- how.
Eliza Luke
But it’s still right.
There were two storylines occurring in Chelsea’s teaching with her highest strategy group. In the first storyline, applying the strategy that is the focus for the lesson appeared to be more important than applying the strategy that could be more efficient. Students were not given opportunities to trial different strategies and test them for efficiency; instead, they were directed in each lesson to use a particular strategy. In a conflicting storyline some students continued to challenge the need to apply only one strategy in each lesson. Eliza and Luke struggled to understand why they could not apply the strategy of their choice when their answer was correct. The strategy focussed on in each lesson would have held significance as a social act for the group.
To conclude, Chelsea’s positioning decisions in regard to herself and students in her highest strategy group limited opportunities for students to share their mathematical know-how. Restrictions to students sharing mathematical know- how occurred through Chelsea’s positioning of herself as the one to model and explain the mathematical thinking behind students and incorrect answers. The emphasis Chelsea placed on correct answers as evidence of understanding and her procedural approach to learning about different strategies may have constrained students’ opportunities to share their own and experience each other’s mathematical know-how.
6.9 Summary
This chapter illustrated the consistent teaching and positioning decisions of seven of the 12 teachers in my study. Six teachers – Greer, Hannah, Delphi, Jenna, Kendra, and Sheridan – positioned themselves and students in their lowest and highest strategy groups to ensure students in both groups had opportunities to share their mathematical know-how. Chelsea was also consistent in her positioning decisions regarding her lowest and highest groups but this positioning did not appear to provide opportunities for students in either group to share their mathematical know-how.
The six teachers whose positioning decisions afforded the sharing of mathematical know-how positioned themselves in five key ways: by providing
models and representations; highlighting mathematical connections; emphasising the importance of different, efficient, and sophisticated explanations; stressing the need for students to check their own and others’ answers and strategies; and incorporating students’ questions and advanced strategies into the learning. Teachers positioned students to share their mathematical know-how in six important ways: by providing opportunities for students to explain, model, and record their thinking; consider peers’ thinking; notice mathematical connections; provide and evaluate explanations for difference, efficiency, and sophistication; review and critique their own and others’ work for accuracy; and inquire about their mathematics learning. These decisions show how teachers gave themselves facilitative positions and gave their students dynamic positions, which appeared to enable students to realise their teachers’ expectations for them to engage in their own and peers’ mathematical know-how. Using a facilitative position, it appeared teachers gave the responsibility for undertaking and completing the mathematics tasks to the students. Being positioned to do most of the mathematics work could ensure the mathematics learning of these students progressed more confidently and competently than had the teachers chosen to do the work for them.
Chelsea’s decisions appeared to constrain opportunities for students to share their mathematical know-how as she positioned herself to undertake most of the mathematical thinking and modelling. Explanations and models of mathematical know-how were shared more by Chelsea than by students. The expectation of students in both groups was to provide correct answers and apply specific strategies. Not being positioned to do most of the work may have inhibited the depth of mathematics understanding that could enable students to move forward. In meeting Chelsea’s expectations of them, students’ opportunities to share their know-how may have been constrained.
Chapter Seven presents the case studies of the five teachers in my study who were inconsistent with their positioning decisions. Inconsistent positioning decisions were applied to individuals within a group and to the whole group.