‘Matrices’ is a common task in my school’s scheme of work for year 8 (see Previous Post) where students investigate the effects of transforming shapes using different 2x2 matrices. The starting point is to get students to plot points A to F (shown in blue below), and to transform them with the matrix :
Thinking about how to get a class of mixed ability year 8 students to do this can be very daunting for teachers who have not taught this task before, but our department has found it to be very engaging for students once they get over the initial hurdle of being able to transform shapes. In discussing with another teacher (Miss C) how they might begin this task, I offered to teach the first lesson so that they could observe. I have previously had the opportunity to observe another teacher teach my class, and found it very influential – it is always helpful to observe other teachers, but to see what your own students do with another teacher can be fascinating and surprising! Having offered to do this, I now had the daunting task of planning a lesson for an established class of students that I’d not taught before.
In a first lesson with a new class I generally aim to get students working mathematically as quickly as possible so that I can comment on their mathematical behaviour, as this provides a way of giving students purpose and supporting them in working mathematically. I was aware that a first lesson on matrices would not necessarily be open enough for students to display many of the behaviours I consider to be mathematical; at least not for the majority of the lesson, while they need to learn (be able to reproduce) the process of transforming a shape with a 2x2 matrix. I was aware that these year 8 students were familiar with PLTS (Personal Learning and Thinking Skills) language of Creative Thinker, Self Manager, Team Worker, Independent Enquirer, Reflective Learner and Effective Participator, and would have experiences from year 7 which they would relate to these labels, so referring to these might provide strategies to support them in this lesson.
Most of my time in planning this lesson went into thinking about what the students would need in order to be successful in performing a matrix transformation, as this would provide things for me to focus my comments on in the lesson. I found it very difficult to pin down what was mathematical about transforming a shape using a 2x2 matrix. It definitely involves plotting coordinates, multiplying and adding, but there is more to working mathematically than this! I began to question what it was that I wanted these students to learn in this lesson, and while I definitely valued the learning that had taken place when I’d worked on this task in previous years, it began to feel like the main purpose of this first lesson was to set up a situation that students could go on to investigate in the following lessons. I have never really considered being able to perform a complex process to be particularly mathematical, but this seemed to be the most mathematical aspect of what the students would be doing for the majority of the lesson, and felt like something that I could comment on.
My school currently insists on having three (all, most, some) learning objectives on the board in every lesson, so I have been experimenting with these being aspects of working mathematically that I can focus my comments on. For this lesson I chose:
All: Self Manager – Be able to carry out a complex mathematical process
Most: Reflective Learner – Write a comment about what you notice
Some: Effective Participator – Share your work with other students
I was surprised that the focus of ‘Self Manager – be able to carry out a complex mathematical process’ felt completely natural in the lesson, as I could link nearly everything the students did to this! Suddenly getting stuck became part of being mathematical, as did putting your hand up to ask a question, talking to other students and checking whether answers were correct.
I was able to comment when students were demonstrating behaviours that were helpful, for example when a student put up their hand to ask for help I was able to comment that ‘when mathematicians work on carrying out a complex procedure they might ask someone for help’. Usually when I am commenting on an aspect of working mathematically I am only able to do this when I see helpful behaviours, but with the focus of being able to carry out a complex mathematical procedure I was also able to comment when students were demonstrating behaviours that were unhelpful. For example, when a student called out ‘I don’t get it’ I was able to comment that ‘it’s okay to be stuck because this is part of what we’re working on today, what to do when we’re stuck. What strategies have we got for if we’re carrying out a complex procedure and we get stuck?’
Many of these behaviours came from students’ lack of confidence and not being sure they were doing things correctly, as the process is complicated. Labelling these behaviours as aspects of being able to carry out a complex mathematical process seemed to allow the students to feel like they were making progress even when they were stuck, and provided them with extra resilience and motivation. I’m not sure about the purpose of the ‘Self Manager’ bit, but I suspect that the objective would not have been as powerful without it, as the students could recognise this as something they had worked on and thought about from ‘Learning Challenge’ lessons in year 7. My sense was that because ‘being able to carry out a mathematical process’ was an aspect of both being a ‘Self Manager’ and being a ‘Mathematician’, it gave an extra sense of purpose to what the students were doing. While the focus of being able to carry out a complex mathematical process was supportive for me in commenting on students’ behaviours, I feel that the focus of being a ‘self manager’ was more supportive for the students in finding strategies for carrying out a complex process.
I ended the lesson by displaying the image (shown above) of the shape before and after it had been transformed and asking students what had changed and stayed the same. It felt as though most students were ready and determined to have a go at performing their own transformation next lesson, and in this respect the lesson felt successful. While my sense is that most students had been challenged and had felt like they had become more at ease with working on things that they found hard, I’m still left pondering what mathematics these students learned.
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