La observación estructurada no participante y de laboratorio

This review of the literature has helped me address the two research questions that had begun as an interest into why all mathematics lessons feel and look the same in both style and design. The link between professional language

terminology as a component of lesson design features and the mathematics shared with pupils is poorly understood, but given that lesson design has a relatively recent research history this is not surprising. The interchangeable nature of terminology does not make for clarity of understanding even though most teachers would admit they profess to understand what colleagues are expressing. The professional vocabulary that teachers use to communicate with each other about learning and teaching is often fuzzy but paradoxically generally well understood by those working in the profession.

This review of the literature did help me with highlighting the interconnections between the research questions but more significantly the literature did confirm my view that a clear shared pedagogical language for lesson design is of importance for both the teacher and the learner. The literature also gave me a deeper insight into the number of ways in which the division of fractions is taught and confirmed my view that some ‘standard’ ways of teaching particular topics are not necessarily the most cognitively effective for the learner. The literature also raised the question of how else mathematics could be planned and taught. It also made me consider how the pupil outcomes from a change in design of a lesson might be measured and how the change in a lesson design might

influence participating teachers. The social constructivist, situated- and problem- based learning literature enabled me to consider alternative approaches to teaching and pointed me towards a way of conceptualising the different aspects of the pedagogical terms under investigation.

Certainly the literature helped me to formulate working definitions for the four terms (activity, skill, exercise and task) which I consider to be fundamental learning episodes when designing a mathematics lesson. In this respect the

112 literature around pedagogical language was helpful in identifying the problems

and care in the use of language to design lessons that I would need to take when talking with teachers in the participating school.

Trying to synthesise exactly the individual characteristics of each of the four pedagogical terms (activity, skill, exercise and task) led me to the belief that there are inevitable overlaps in their meanings. The review of the literature did point towards the fact that one of many reasons for studying mathematics is for learners to acquire problem solving expertise. With the recent debate about fluency and mastery in mathematics (Foster, 2017; Howard, 2018) being essential to mathematical development then, these four terms are inevitably intimately bound to the ways in which the subject is presented to learners. The purpose of defining these four pedagogical terms was to promote a structure for the way in which the subject is presented by teachers to encourage competent, fluent mathematical learners. In this structure a task is seen to be where fluency and mastery are demonstrated with the role of a skill being to present new knowledge, an exercise as an opportunity for practise and a degree of self- exploration of the subject; and an activity as a means of encouraging knowledge recall. The definition of a task is heavily reliant on the work of Ainley’s (2008) view of purpose and utility together with Swan’s (2005) views of richness and collaboration.

I acknowledge that the separations in meanings of the four pedagogical terms as defined early are neither watertight, nor exclusive conceptual categories. However these have been developed in the interests of trying to improve the pedagogical language of mathematics teachers with the aim of coming to a common shared understanding. There is inevitably some conceptual fluidity between aspects which I recognise and that others may question in the definitions proposed.

The definition of an activity, presented here, as not introducing any new learning is slightly problematic. Social constructivist theory tells us that learning takes place when learners interact, and this is a fundamental aspect of an activity, hence

learning is likely to occur. The difference between the definition of a skill and that of an exercise lies in pupils posing their own questions when engaged with an aspect of learning defined as an exercise. The definition of a skill has the development of

113 mathematical fluency at its core (Taleporos, 2005; Ofsted, 2012; Foster, 2013). Contrastingly the definition of an exercise promotes the opportunity for pupils to demonstrate mastery (Hewitt, 2015; Foster 2017). Whilst the distinctions between the four terms are subtle, with inevitable overlaps, this just serves to exemplify the problems teachers have when using pedagogical language. Trying to clearly and unambiguously define commonly understood pedagogical terms in some respects replicates the view that learning is a “very complex matter, and there is no generally accepted definition of the concept” (Illeris, 2018, p. 1).

Trying to establish an optimum order in which these four pedagogical terms should be used to structure a lesson is not part of this research, but it would seem logical to start a lesson with an activity to recall prior learning. The order in which the other three are used is open to debate and further research, however, I decided to use the order activity, skill, exercise and task for the study lesson in this research. Finally for completeness and as a recap I will be using the definitions for the four pedagogical terms under investigation in this study as detailed in appendix 35. These were shared with the teachers involved in the study.

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