The Retraining Programme Involved in this Research Research

The participating teachers were all enrolled on the 2013-2014 Plymouth

University SKE+ programme (outlined above in 3.0). With SKE+ morphing into the current retraining programme of TSST during the lifetime of this longitudinal study, many teachers, including my participants, use the terms of ‘SKE+’ and

‘TSST’ synonymously.

Teachers were required to be teaching at least one mathematics group to be eligible for government funding. (This government stipulated criterion has subsequently been suspended for the current TSST.)The research participants attended one of two venues, at each of which I was the course tutor. Venue 1 was held on a university campus, with 14 participants; Venue 2 (with 10 participants) at a school where demonstration-lessons with pupils could be observed. The participants each experienced 100 hours of face-to-face tuition, over the course of 11 full training days (9am to 6pm), from December 2013 until November 2014. Funding was available to reimburse schools for the 11 days of

teacher release time, but travel expenses were not included. Some schools paid their participants for travel, other participants had to self-fund travel expenditure.

With autonomy to design the detail of the face-to-face sessions, the

interweaving of pedagogy with mathematical content was a key focus; subject knowledge alone is simply insufficient to develop effective teaching practices (Ball and Bass 2000). Creating a safe environment was also a priority. The social and situated learning theories (Lave and Wenger 1991) suggest

participants will contribute more of what they know and be more honest about what they don’t, if they feel ‘safe’. In any teaching scenario, I invest time in creating a cohesive group with opportunities created for social interactions and networking. Acknowledging the McKinsey (2007) report, highlighting the need for teachers to observe and reflect and identify what makes for great teaching in their subject, demonstration lessons were delivered whenever feasible. In

practice this is only possible when the face-to-face sessions are being delivered within a school setting; at Venue 2, this was the case. For a demonstration lesson, I ‘borrow’ a class of students to teach whilst the participants observe.

(The lessons are also frequently filmed – to be used by absent participants or in other training scenarios.) For the SKE+ course, this provided a vehicle for discussion and debate as we reflected on the lesson together. As the course developed, the participants were invited to also do some teaching - to teach a 10 minute segment, for example, or to team-teach a short session. Many, but not all, of the participants embraced this opportunity - as it was seen as a safe environment in which to take risks. Feedback was always given, in a non- judgmental manner - very much incorporating the HACE (Honest; Analytical;

Constructive; Empowering) principles advocated during the lectures (Magne, 2016) at Plymouth University: The teacher is always invited to reflect and

feedback first, before others are invited by the chair (NS) to contribute to the discussion. The purpose of the demonstration lessons was to give participants a rare opportunity to actually observe a mathematics lesson and to see how ideas discussed during the face-to-face retraining sessions could be put into practice.

The process of planning the lesson was not collaborative; evaluating and

reflecting upon the lessons became - over time and with greater experience and confidence - a negotiated and collaborative effort.

In addition to (but quite separate from) the demonstration lessons, Lesson Study and collaborative teaching practice were introduced to all my participants during the face-to-face sessions, and included cycles of: focus; observation;

reflective review. Lesson Study sessions were based, but much condensed, on the principles outlined in Illustration 3.1, the focus being predetermined by the topics being studied on a particular day. The whole group was split into working groups of three and a relatively short time was available during the morning (and then over lunch) for planning the micro-lesson, which would then be delivered by one teacher to the rest of us. Collective effort and knowledge (along with other resources) created the micro-lesson; the ‘efficacy of

collaborative approaches to mathematics teacher education is well-established’

(Hodgen 2011: 38). Shared responsibility was assured as no-one knew at the outset who would teach the micro-lesson. Gaps in knowledge and

understanding could be explored and teachers could receive guidance and support, thereby preventing the dissemination of poor practice (Burghes and Robinson 2010). With planning complete, straws were drawn to see who would deliver the micro-lesson. All other participants (and NS) observed.

Illustration 3.1 Lesson Study Cycle as Described by Burghes and Robinson (2010: 13)

The micro-lesson (10-15 minutes) was then reviewed, first by the teacher who had delivered the lesson, and then by others who were invited to contribute, with myself chairing the discussion; this echoing the feedback principles employed post demonstration lessons, and so already familiar to participants from Venue 2. With the limitations of time, and of invented scenarios, this collaborative experience was only intended to be an introduction to the principles of Lesson Study.

Using the idea of ‘constructive alignment’ (Biggs and Tang 2007: 50), I teach and organise retraining sessions in a way that incorporates and highlights the

‘Knowledge Quartet’ (Turner and Rowland 2006: 2, see 2.3.2). With a focus on conceptual understanding, common misconceptions are deliberately highlighted and addressed and precise use of mathematical language and accurate board

skills, demonstrated and encouraged. ‘Problem solving’ is a key teaching

strategy I employ. The aim is to use guided rediscovery to drive deeper thinking and to see and make use of the many mathematical connections. Making connections, and developing one’s own understanding, resonates with Piaget’s belief that we only truly understand what we have created, invented or mastered for ourselves (Hunt and Chalmers 2013). This is supported by the much more recent work with neuroscience and the work done by Dweck (2012) highlighting the elasticity of the brain and the significance of having a ‘growth mindset’; the learner as the creator of understanding, underpinning the rationale for teaching by ‘problem solving’. Problem solving is a principle teaching strategy in Japan, and one I observed first hand during a visit in 2012. The most proficient

Japanese practitioners, with Level 3 competency (APEC 2013), employ

methods of teaching which incorporate ‘guided rediscovery’, enabling students to think for themselves and cognitively construct ideas. It is this, Level 3 type teaching, I strive to model, in the quest to develop effective teachers of mathematics.

3.2 Summary

Clearly the aim of the government funded SKE+ programme was to deliver the DfE’s objective to develop competent and effective teachers of mathematics.

But once again we come up against the universally unresolved question: What is effective mathematics teaching? During the meeting to launch TSST

(Watterson 2015), Watterson defined this to be teaching which leads to improved student achievement using outcomes that matter to their future success, thereby referencing the definition agreed upon by Coe et al (2014).

But as discussed previously (in 2.2.1), it is difficult to interpret both what

‘outcomes matter’ and ‘future success’.

In terms of outcomes, there is agreement (Hunt and Chalmers 2013) that superficial rote learning, often linked with behaviourist learning theories, can only support short term ‘outcome-based’ targets (such as exam success) and do not promote long-term learning (2013: 5). Cockcroft (1982) reported similar conclusions, and one paragraph in particular, shown in Table 3.2, provoked much analysis and discussion; this summarisng the need for so much more than rote, or procedural type teaching and learning.

Mathematics teaching at all levels should include opportunities for:

• exposition by the teacher;

• discussion between teacher and pupils and between pupils themselves;

• appropriate practical work;

• consolidation and practice of fundamental skills and routines;

• problem solving, including the application of mathematics to everyday situations;

• investigational work.

Table 3.2 Cockcroft 1982 Paragraph 243

Ingram et al (2018) highlight the ‘complex relationship between teaching and learning’ and focus on several different observation frameworks to examine this interrelation. The frameworks vary in terms of the levels of low inference

recordings and high inference judgements required, and were designed for varying intent, but share several similar indicator items, many of which are broadly encased in the ‘Framework to identify effective mathematics teaching’

(see Table 3.3) (adapted from ‘Guidelines to identify effective teaching’

(Burghes 2005)). An example of which is: ‘The teacher demonstrates genuine warmth towards all students’ (ISTOF system) which could be seen to be similar to ‘[teacher] addresses the children in a positive manor’ and ‘reacts with

humour, and stimulates humour’ (QOT framework) (Ingram et al 2018, 17-19) and this links with ‘Teacher likes being with learners’ (Framework to identify effective mathematics teaching, Table 3.3.)

For the purposes of this study a working definition of effective mathematics teaching is required. Drawing on all the research discussed in Chapter 2 and above, I will define effective mathematics teaching to be that which promotes or encapsulates the attributes outlined in the ‘Framework to identify effective mathematics teaching’ (see Table 3.3); this was disseminated to all participants on the retraining programme. This framework does demand high inference judgements. To manage this, a system was developed in which the actual lesson observations were recorded as non-judgmental narratives and the post-lesson reflections were co-constructed with the teacher with reference to this framework.

In summary, for the purposes of this study, effective teaching is described as that which promotes active participation and deep thinking amongst learners, to provoke deep understanding.

Teacher is a good communicator, loves mathematics and likes teaching Teacher orchestrates activities and can respond to unexpected outcomes Teaching is aspirational and challenging

Teacher gives clear explanations; can select and instruct efficient and effective methods Teacher can see the ‘big picture’ and promotes mathematical content connections

Teacher promotes deep thinking (For example: Why? How? What if? questioning technique) Teacher encourages creativity and discovery

Teacher listens to learners Teacher likes being with learners

All mathematics written by teacher clear, correct and precise; mathematical language embedded throughout

Considered interactive questioning techniques to involve all pupils, and to reflect and evaluate progress

Teacher has control of the class

Non-confrontational ethos in the classroom

Learners keen, enthusiastic and motivated to learn

Ownership of ideas encouraged and active participation expected: including for example, demonstrating and articulating at the board

All learners feel encouraged and are able to make progress Learners cooperate and collaborate with peers

Learners on task

Table 3.3 Framework to Identify Effective Mathematics Teaching (adapted from

‘Guidelines to Identify Effective Teaching’ (Burghes 2005))

Following eight teachers, over a period of 4 years, I have undertaken an

explorative study to consider the impact this retraining may have had. Chapter 4 looks at the research questions in detail and considers the methodology and methods for conducting the study.

Chapter 4: Research Questions, Methodology and