Scaling a technology-based innovation: windows on the evolution of mathematics teachers’ practices

5.1.1 How CM aligned to teachers’ existing practices

The diversity of responses reflected the mixed community of teachers and their school roles: 84 classroom teachers; 17 coordinators of the school’s KS3 curriculum; 20 deputy heads of mathematics; 17 heads of mathematics; 3 lead mathematics teachers and 10 ‘other’ school roles.

5.1.2 How CM aligned to teachers’ individual motivations

We investigated how teachers’ individual motivations to participate in the project were related to the ways they had learned about the project. These routes included: self-initiated (11); invited/nominated by Headteacher (14); invited/nominated by their head of mathematics (or another senior colleague) (113); and invited/nominated by a regional mathematics adviser (15). The self-initiated teachers stated motivations that related to the development of innovative practice and involvement in a prestigious research project. On the other hand, the overwhelming majority of the group who had been asked or nominated by a senior colleague (127) offered reasons that were highly consistent with the project’s overarching aims (i.e. to develop use of technology for mathematics learning), implying that they had, at least, understood the project aim’s and, at most, indicated an initial sense of coherence. Only a handful of nominated teachers implied a reluctance to be involved.

5.1.3 How CM PD impacted on teachers’ confidence to teach with dynamic technology

A second more pragmatic baseline was established at the end of the PD event with respect to the individual teachers’ levels of confidence to teach the unit in their classrooms (Fig. 2).

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Given that CM aims to put dynamic technologies into thousands of students’ hands, a goal that has been historically challenging to achieve at scale, this result is reassuring in that the vast majority of teachers felt that the initial PD event had prepared them to use the software and materials in their classrooms.

The prevalence of responses that related to aspects of assessing the students’ outcomes either informally (classroom-based formative assessment) or formally (using questions from the previously developed pre- and post-tests) resonates with the current English context where school inspection practices require teachers to demonstrate tangible evidence of students’ learning both within individual lessons and over time. This necessitates the differentiation of teaching to focus on the needs of all students, and also explains the high response (33 %) with reference to students with English as an Additional Language (EAL) and Special Educational Needs (SEN). As CM units are designed with an abundance of rich questions that probe deeper understanding, the teachers’ responses highlight a need to develop classroom practices in making best use of these questions (i.e. informing Process j in Table 2).

By the end of January 2014, 82 of the Focus school teachers had begun to implement the curriculum unit with a diversity of classes at the KS3 level (16 % in year 7, 46 % in year 8, 27 % in year 9 and 11 % in year groups beyond KS3).

At the same point in time, 38 (of the 82) teachers had completed their teaching of CM unit 1 and reported their final outcomes through the third questionnaire, which provided an insight into aspects of the teachers’ classroom practices.

5.1.4 How CM supported teachers’ evolving practices

The teachers reported very different pathways through the curriculum unit and all teachers completed the curriculum activities that combined all of the mathematical representations (simulation, graph, table and function). 68 % of the teachers reported that they had reached a key investigation in which students encountered negative gradient for their first time and 27 % of teachers reported that their class had completed the unit in full. The remaining 73 % of teachers all indicated an intention to revisit the unit later in the academic year and/or key stage. These teachers had given diverse reasons for delaying their teaching of the complete unit: a lack of curriculum time, a lack of continued access to suitable digital technology and the need to return to the formal schemes of work for that class due to internal (termly) assessment arrangements.

It is not possible to assume in general that any of these data are themselves indicative of changed practices. However, analysis of the teachers’ qualitative questionnaire responses highlights the nature of the mathematics that had been foregrounded during the CM lessons, providing some insight into how their personal reflections aligned (or not) with the design principles for CM. Fourteen teachers commented within the final questionnaire (n = 33) on how their students’ interactions with the software promoted students to make use of dynamic mathematical connections (a fundamental design principal of CM), for example,

The triple representations really forced the students to make the connections for themselves—if they needed to alter the animation; they were forced to engage with the graph or the table of values, which connected the ideas much better than could be done on paper.

Other responses that aligned with the CM design principles referred to aspects of the students operating (physically or cognitively) on the different representations by: creating; editing; manipulating; comparing; and linking the mathematical objects.

Conversely, in response to a question that asked teachers to describe any examples where the CM software had hindered their students from learning the mathematical ideas, 17 of the 33 respondents said that it had not. The majority of the 6 teachers who gave examples said the hindrances were technical, but there were a few teachers who cited issues relating to the particular way students were required to edit the graph. One teacher’s comment in this respect was,

The manipulation of the graphs—they did not need to be anchored where they were—it did not facilitate or increase learning.

This comment contrasted with another teacher’s view of the same ‘hindrance’,

Students found moving the lines on the graph difficult at first. They wanted to touch the endpoint and move it to the place required. They became more adept once they realised the axes moved independently.

These two examples demonstrate how a subtle design decision may or may not be noticed by teachers—and how teachers’ reflections on particular features of the CM materials might provide an insight into their personal alignment to the design principles of the innovation.

One teacher commented on how the new powerful ideas that students had accessed supported them to unlock mathematical concepts that had not previously been available to the students:

The students had the animation, graph, and table to help them describe the movement of something in relation to position and time, but his was the first time they also had the equation. One student was so excited about making the connection between speed and the coefficient of x that he went back through the earlier investigations to work out the equation of the lines.

The above range of quotations indicate how teachers may have individually ‘bought into’ the principles of CM, and how some may be well placed to become advocates of CM, one of the key characteristics for sustainability emerging from the prior USA studies. In addition, we would hypothesise that individual teacher ‘buy-in’ is a necessary pre-cursor to school buy-in and a pre-requisite to Products b, d and f and Processes b, d, f, i and j within Table 2.

Finally, with respect to the development of teacher community (Process b in Table 2), 23 of the 38 respondents to the third questionnaire indicated that they had used the CM online community in the following ways: keep up to date about the project (n = 15); read questions/comments (n = 21); post questions/comments (n = 6); access the e-version of the Teacher’s Guide (n = 11); access the e-version of the Student’s Workbook (n = 9); and one teacher uploaded her own adapted lesson resources to the community’s shared resource area.

5.2.1 Case study 1: Emma

Emma was the Head of Maths at a newly formed ‘Free school’⁷ that had a strong philosophical vision to promote creative, exploratory approaches to learning, student independence and leadership. Emma attended the one-day PD event in June 2013 where she engaged enthusiastically in the PD tasks and activities. She had no previous experience of the CM teaching units. Emma’s responses to PD tasks and surveys revealed her desire to ‘give students control over variables’ and encourage them to predict-check-explain, implying an allegiance to the CM pedagogic approach (see Table 4).

Emma chose a class of twenty-three, 12–13-year-olds, a homogenous group that she described as having ‘higher ability’ within the year group, whom she felt would be challenged by the curriculum unit. They accessed the software using shared laptops (2–3 students) in their normal mathematics lessons. Emma commented that she would co-plan the lessons with a colleague and that they would be ‘building in questions’ to additional resources they would create to support the teaching (Processes f, g, i and j). She also planned to develop some ‘tasks designed to assess what they had done’. Emma dedicated the first half term’s mathematics lessons (approx. 25 h) to her teaching of the CM curriculum unit. She had also considered how the school would evaluate the project, which involved: feeding back to colleagues in department meetings; a systematic assessment of the students’ outcomes; inviting senior school to observe lessons; and reporting the final outcomes to school leaders (Process e).

Emma’s legitimate lesson adaptations, which were most evident through the presentation slides she displayed in the classroom, concerned: enhancements to the unit’s context that strengthened to students’ personal connection to the project; the sharing of explicit learning objectives; the reiteration of the ‘predict-check-explain’ pedagogic approach; ‘review’ questions; and directions to students concerning where they should start and stop and what would be discussed during the whole class plenaries.

Emma completed Unit 1 (14 investigations) and in her response to the third questionnaire reported that it had been very easy for her to integrate the unit within her current scheme of work and that the school planned to teach the unit in the future (Process j). She made the following comment about how she would extend the students’ work,

We are now going to develop our own stories and put together a cross curricula project where we create a story of a journey and then we have to represent the story with a drawing, graph, equations and table of values.

Emma’s overall response to the project evidenced her enjoyment of CM, but it also suggests how the design of the course resonated with her ideas about the mathematics, as she felt compelled to build upon the idea of the narrative with her students.

The students spent a further 8 h within ‘project based learning’ curriculum time developing their written stories and creating travel graphs that represented their journeys. In total, Emma dedicated 33 h of her teaching time to the topic—considerably longer than the 12 h we had expected. In their final project work students demonstrated high levels of transfer of the mathematical ideas they had encountered within the CM unit and the more able students extended the mathematical content, for example identifying the equations of piece-wise linear functions that did not intersect with the y-axis (Process h).

5.2.2 Case study 2: Robert

Robert became involved in the CM project in summer 2011 as one of two teachers from his school that were to pilot the first adaptation of the Unit 1 software and materials that had been originally developed for the Texas randomised control trial. His school was also in East London and, although in an area with a history of educational underachievement, it was a school that was reversing this trend, particularly with respect to mathematics. The school had higher than average numbers of students with EAL and the organization of students within mathematics was unusual in that students were taught in heterogeneous classes in KS3 (11–14 years).

In Robert’s summary of his experience during the English pilot project, he commented,

Initially I was very sceptical—I was really not sure—but now we’re really excited about the unit and the project, I’m very hooked on it because I can see the impact it has had on my students. We’re definitely going to use this unit again—it’s going to be something that we embed. For us it’s not a trial and we’re very excited about it. (Product c; Process g).

For scaling, Robert’s change of heart, and later commitment to expand the project in his school are crucial to Themes 2 and 3 within our elaborated framework.

Whilst the pilot was still in process, Robert expanded the teaching of CM throughout his school by organizing other teachers to observe CM lessons and by leading a 2 h PD session for the department in their faculty meeting time, during which they worked through the initial investigations alongside the student workbook (Products d, f; Processes f, g, i, j). Once the department had agreed where the curriculum unit would be located in their KS3 schemes of work, Robert organised the teachers in pairs to plan and evaluate their lessons, justifying this by saying ‘this leads to greater collaboration and discussion’ (Processes f, g, i, j). Their resulting schemes of work included Unit 1 (split into two parts) and this included ‘assessment and review’ lessons at the end of each unit, which had been hand-written by the member of the department with responsibility for the KS3 (Process e).

Finally, Robert was aware of the need to maintain an ongoing dialogue with his Headteacher, particularly as Robert saw it as his role to ensure that students and teachers had access to the technology (to include classroom support from a dedicated technician) as designated by the scheme of work.