Understanding and explaining pedagogical problem solving: a video-based grounded theory study of classroom pedagogy

ABSTRACT

Background

during school lessons participants can face challenging pedagogical problems such as when science and religion topics interact. Potential types of pedagogical problem and what participants can do in response are unclear. Research designs exploring pedagogical problem solving should investigate teachers’ thinking processes and include pupil interpretations.

Purpose

This theoretical paper builds on the Pedagogy Analysis Framework by integrating it with pedagogical problem-solving theory, illustrating the resultant extended Pedagogy Analysis Framework and Pedagogical Problem Typology using data from a video-based study of one science and one Religious Education (RE) lesson. The research design also builds on previous work by exploring primary school and RE pedagogy.

Sample

One class of thirty 7-year-old pupils, and another class of 10-year-olds, each with their class teacher and teaching assistants.

Design and Method

Four research methods were used (lesson video analysis, teacher verbal protocols, pupil group verbal protocols and individual teacher interviews). Data were video recorded (managed using NVivo). Six hours of video data were analysed using grounded theory methods by two educational researchers, the class teacher and two groups of pupils (three girls and three boys). The interpretivist theoretical perspective (symbolic interactionism) was underpinned by a social constructionist epistemology (hence the methodology is Straussian grounded theory). Appropriate criteria for evaluating the emergent grounded theory were used. Data were recorded in 2019.

Results

This paper presents an extended Pedagogy Analysis Framework and a Pedagogical Problem Typology both illustrated with examples.

Conclusion

The Extended Pedagogy Analysis Framework and Pedagogical Problem Typology can help during pedagogy analysis to identify, understand and explain common and less common types of pedagogical problem. Novice-expert strategic dialogue can struggle for want of a shared vocabulary. This research design works with younger and older children.

KEYWORDS:

• Video-based
• pedagogy
• analysis
• grounded theory
• problem-solving

1. Introduction

The focus of this paper is on understanding and explaining pedagogical problem solving. This theoretical paper builds on two previous studies (Riordan, 2020; and Riordan, Hardman and Cumbers, 2021) by introducing an ‘extended Pedagogy Analysis Framework’ and a ‘Pedagogical Problem Typology’ illustrating both with examples from video-based analysis of a science and a Religious Education (RE) lesson. In school when science issues emerge during RE lessons, and when religion is discussed in science lessons, we think teachers and pupils can face particularly challenging pedagogical problems. For example, contemporary science and the key texts of the world religions (e.g. the Bible, Koran, Vedas, Sutras, etc …) give differing accounts of the origins of the universe which pupils, and their teachers, might find confusing (Hopfe and Woodward 2009). Hence, we chose to investigate pedagogical problem solving when science and religion intersect in the classroom. We argue in this paper that teachers and pupils in classrooms solve, and fail to solve, many conceptual and pedagogical problems during lessons, and that it is not obvious what a pedagogical problem is, what types of pedagogical problem there are, nor what people do to address such problems.

Understanding pedagogical problem solving is important because part of education is to teach learners to solve conceptual and pedagogical problems themselves, and part of Initial Teacher Education is to teach teachers how to analyse and solve pedagogical problems. Indeed,

the central point of education is to teach people to think, to use their rational powers, to become better problem solvers. (Gagne 1980, 85)

Understanding something as complicated as classroom pedagogical problem solving is never going to be easy as so many factors are in play and information, misinformation, and disinformation swirl (Riordan, 2020; Riordan, Hardman and Cumbers, 2021). We argue that teaching and learning in a classroom often involve such complicated interactions and explaining such experiences needs partnership between teacher, pupils and researchers.

This paper seeks to clarify some of the types of pedagogical problem that teachers and learners encounter (presented as a Pedagogical Problem Typology), and to understand and explain how participants sought to solve those problems (using the extended Pedagogy Analysis Framework). The ambition here is to make progress in constructing an understanding of pedagogical problem solving and begin to illuminate its efficacy in these subject areas, rather than provide a comprehensive solution.

2. The theoretical approach

2.1. What we know

2.1.1. What is pedagogy?

Pedagogy is a contested term (Watkins and Mortimore 1999; Leach and Moon 2008; Black and Wiliam 2018). In Riordan, Hardman and Cumbers (2021) we acknowledged that the focus of pedagogy can be on the teacher. For example:

pedagogy is the act of teaching together with its attendant discourse of educational theories, values, evidence, and justifications. It is what one needs to know, and the skills one needs to command, in order to make and justify the many different kinds of decision of which teaching is constituted. (Alexander 2008, 47)

Alexander (2008) obviously understands the importance of the learner in pedagogy and in his definition identifies many important factors. In contrast, in Riordan, Hardman and Cumbers (2021) we emphasised the behaviours of teacher and pupil, and the interactions between participants:

[Pedagogy involves] those factors affecting the processes of teaching and learning and the inter-relationships between them. (Hallam and Ireson 1999, 78, quoted in Black and Wiliam 2018, 555).

Furthermore, in Hardman, Riordan and Hetherington (2022) we argued that the interactions between people and the material objects in the classroom are also important in pedagogy analysis.

2.1.2 What are pedagogical problems and decisions?

We situate this present study in the pedagogical problem solving and teacher decision-making literatures. Teachers face challenging pedagogical problems and decisions in the classroom regularly (de Simone 2008). Problem solving and decision making are sometimes treated as synonymous according to Weitzman and Weitzman (2014), who then argue it can help to distinguish the two to clarify how these concepts are complementary and how they relate. In this approach problem solving involves diagnosing the issue and identifying alternative potential solutions, whereas decision making is concerned with evaluating and choosing among these options then committing and implementing this choice. This is not a linear process, so for example, when faced with a decision a participant may choose to consider more options (i.e. further problem solving). A ‘problem’ itself is a contested concept according to Jonassen (2000), who argues a problem is a known unknown where finding a solution has value to a participant and where finding the unknown is the process of problem solving (Jonassen 2000, 65). In the present paper, we extend previous work (the Pedagogy Analysis Framework; Riordan (2020) and removed 2) by incorporating Jonassen’s (2000) understanding of problem solving (i.e. the grounded theory process of ‘theoretical integration’ to be discussed later).

Research in the 1970s by Alan Bishop, Lee Shulman and Richard Shavelson among others, identified teachers’ decision making in a certain context as critical to understanding pedagogy (Borko, Roberts, and Shavelson 2008).

[The teachers’ decision-making is] at the heart of the teaching process. (Bishop 1976, 42)

Additionally, Bishop and Whitfield (1972) and others recognised the importance of the learner, but the focus of this research strand remained the decisions of teachers.

the teacher is not the builder – the child is (Bishop and Whitfield 1972, 16)

Seminal work in this field by Shavelson and Stern (1981) recommended developing a taxonomy of decisions:

This taxonomy would identify critical decisions made by teachers, the major alternatives considered in the decision, the information used making the decision, and the consequences of the decision on teachers’ instructional planning, on interaction with students, and on student achievement. (Shavelson and Stern, 1981, 488)

2.1.3 What type of problem is a pedagogical problem?

A number of such taxonomies were developed, such as the teacher decision-making framework (Bishop and Whitfield 1972, 6) where background information, participant values and aims, and decision schema in some context led to an action. A revival in interest in teacher decision making occurred in the 2000s (according to Borko, Roberts, and Shavelson 2008). This present study builds on previous work (Riordan, 2020; Riordan, Hardman and Cumbers, 2021; and Hardman, Riordan and Hetherington, 2022) where a taxonomy, as envisaged by Shavelson and Stern (1981), called the Pedagogy Analysis Framework emerged through using the processes of grounded theory on video data of small groups led by experienced teachers, lessons, teacher verbal protocols and group pupil verbal protocols. We acknowledge the importance of teacher decision making in pedagogy, but our approach differs from the teacher decision-making literature in two ways. Firstly, we understand pedagogy, as discussed earlier, as involving the interactions of any participant (be they teacher or pupil) with any other, and have found in those earlier works (Riordan, 2020; Riordan, Hardman and Cumbers, 2021; and Hardman, Riordan and Hetherington, 2022) that participant interactions with the material are also critical for understanding pedagogy. Secondly, our previous model (the Pedagogy Analysis Framework) was an emergent grounded theory underpinned by an interpretivist approach rather than a hypothesis to be tested like in the positivist branch of the teachers’ decision-making literature (e.g. that of Shulman and Shavelson according to Borko, Roberts, and Shavelson 2008). What type of problem is a pedagogical problem?

Thinking can involve ‘problem solving’ (i.e. the search among potential options to achieve a goal), or ‘reasoning’ where logical rules are used to derive conclusions. Problem solving may be further subdivided into algorithms (in which a procedure is certain to lead to a solution if followed correctly) and heuristics.

[A heuristic is] an informal, intuitive, speculative procedure that leads to a solution in some cases but not in others. (Berlyne, Vinacke, and Sternberg 2021)

Different sorts of problems lend themselves to either heuristic or algorithmic problem-solving strategies. Classroom pedagogy is generally too messy for logical rules to help much and complex social relations between participants mean it does not involve problems that can be solved with algorithms, but we argue pedagogical heuristics can help teachers, and pupils, with the problem solving they do before, during and after lessons. Cognitive psychologists Shavelson and Shulman explored teachers’ decision making and found teachers use heuristics according to Borko, Roberts, and Shavelson (2008). Pedagogical problems do not have clear solution paths (they are ‘wicked’, and ‘ill-structured’; Rittel and Webber 1973; Simon 1973; Coyne 2005; Reed 2016), often occur simultaneously and are cyclical (i.e. output from one problem becomes part of the input to the next; Pretz, Naples, and Sternberg 2003). Pedagogical problems in areas which may be disputed or controversial, such as in origins, where different disciplines address the matter in different ways (narrative/mythic meaning or cosmology) and possible learning outcomes contested (reaching certainty around atheistic or theistic evolution or achieving an agnostic possibilitarian conclusion for instance) are particular territories of sensitivity for pedagogical problems. This is why science and religion lessons, on these topics, were selected for this present study.

2.1.4. What was the pedagogy analysis framework?

This present study builds on previous work (Riordan, 2020; Riordan, Hardman and Cumbers, 2021) from which emerged a grounded theory called the Pedagogy Analysis Framework. This was a model for how participants in a lesson respond to events. To explain briefly, any lesson consists of participants (p₁, p₂, …) where each, at a particular time, has a certain set of information and a set of means available. Means can be things, other participants, or oneself. For a particular means there is a set of ways it can be used at this particular time termed strategies which have meaning for a participant or set of participants. It can be helpful to distinguish three categories of strategy. Firstly, actions are a simple use of a means (e.g. the arm (a means) of participant₁ could be raised in the air to attract attention). Secondly, tactics are a sequence of actions familiar to participants in the context (e.g. ‘question and answer’ is where a participant₁ asks a question, another (p₂) indicates willingness to respond (e.g. by the action of raising a hand), the first (p₁) indicates somehow that a participant (e.g. p₂) should answer, and this participant₂ communicates information somehow. Finally, a grand strategy is a set of incidents where a rich narrative (a thick description) is necessary to understand what has occurred. For this same ‘means’ the imagined outcome of a strategy or set of strategies is termed an end. Any means, strategy or end can be considered by a participant to be real or imaginary (and participants may agree or not about these classifications). Means, strategies and/or ends, at any particular time can be anticipated by a participant or participants (the intended curriculum), occur (enacted curriculum) or be reflected upon (evaluated curriculum). The significance to a particular participant of any means, strategy and/or end is part of the information set of that participant, and this can be shared or not with other participants. For a participant, a subset of their information set is misinformation (accidentally untrue information) and another subset is disinformation (deliberately untrue information). The present study develops this theory by integrating ideas from problem-solving literature (i.e. Jonassen 2000) into the Pedagogy Analysis Framework (Riordan, 2020; Riordan, Hardman and Cumbers, 2021) with the intention of understanding and explaining why means, strategies and ends are used.

2.2. What we do not know

Participants in any school lesson, be they staff or pupil, encounter and sometimes solve pedagogical problems, but what types of pedagogical problem there are, and what solving such problems entails, is unclear.

2.2.1. Research questions

1. What types of pedagogical problem are there?

2. How are pedagogical problems addressed by participants during one primary science lesson about the Big Bang and one primary RE lesson about ‘creation’?

This paper has outlined our theoretical approach, including what we know, and what we do not, and now explains the research design.

3. Research design

3.1. The theoretical perspective and epistemology

In this study idiographic and nomothetic interpretations entwine, moving backwards and forwards between the particularity of the complex social context of real classrooms, and tentative general conclusions.

[Sociology] is a science which attempts the interpretive understanding of social action in order thereby to arrive at a causal explanation of its course and effects. (Weber 1994, 228)

Hence, following Weber, this study adopts an interpretivist theoretical perspective (symbolic interactionism) to understand and explain a context in which learner, teacher, objects and researcher interact. The research design was underpinned by social constructionist epistemology.

With deep roots in symbolic interactionist sociology and pragmatist philosophy, the grounded theory method can be viewed as a theory/methods package with an interpretive, constructionist epistemology. (Clarke 2003, 559)

Corbin and Strauss (2008, 2) acknowledge symbolic interactionism and pragmatism as the philosophies which inform Straussian grounded theory. Symbolic interactionism can be seen in the research methods. For example, participant verbal protocols from video-prompts allow some access to the meanings ascribed to incidents which underpin actions (Blumer 1969). The research methods and how grounded theory was used will be explained briefly next.

3.2. Data collection and analysis

The four research methods used were lesson video analysis, teacher verbal protocols, pupil group verbal protocols, and an interview with each teacher. Briefly, the advantages and disadvantages of using these methods are as follows (please see Riordan, Hardman and Cumbers, 2021, for more detail). Firstly, lesson video analysis allows passages of a lesson to be viewed multiple times, by multiple participants (including during teacher verbal protocols and pupil group verbal protocols) which facilitates deep analysis. Secondly, verbal protocols by the teacher and a group of the pupils allows several types of triangulation making the data more trustworthy (discussed below in section 3.4). Finally, the research methods generate large amounts of rich video data, but processing and analysing this is laborious. The two lessons, each lasting approximately one hour, occurred as they would normally in the school year, the only changes being the presence of three video cameras (one with a 360-degree microphone attached, and another connected to a lapel microphone worn by the teacher), and the attendance of one researcher at the back. Two of the cameras were positioned either side at the front, and one was at the back in the middle. The teachers were encouraged to plan and teach as normal. Analysis began after the first lesson and continued after the end of data gathering (the grounded theory process called ‘concurrent data collection and analysis’; Birks and Mills 2011). Delay between each lesson and the follow-up teacher and pupil group verbal protocol interviews allowed time for analysis and pupil group verbal protocol video clip preparation (see section 7.3 ‘limitations’). The teachers were video recorded for about two hours each making verbal protocols whilst watching the video of their lesson. Pupils analysed their lesson during pupil group verbal protocol interviews during which the teacher was present. Video clips from the relevant lesson were used for pragmatic reasons. The theme of problem solving emerged during analysis (after the data had been collected from the pupil group verbal protocols), so clips could not focus on problem solving. Video clip selection criteria were: where a pupil had expressed something that a researcher, or the teacher, wished to be clarified; where we wished to compare a researcher/teacher interpretation with that of a pupil; and/or where we were unsure how to understand an interaction. As findings emerged, these influenced the selection of clips used in the subsequent pupil group verbal protocol interview (a grounded theory process called ‘theoretical sampling’). The rationale for the selection of the clips evolved during the study (another grounded theory method called ‘constant comparative analysis’). Individual teacher interviews were also video recorded and analysed. With these research methods, insights from the multiple perspectives of pupil, teacher, and researcher into the same lesson could be considered and compared. Hence, findings could be corroborated using triangulation from different sources, methods, and investigators (Carter et al. 2014). Combining verbal protocols with interviews was originally proposed by Taylor and Dionne (2000), and such an approach can give rich data (Leighton 2017). All six hours of video data were managed using NVivo. Lewins and Silver (2007) argued NVivo is the most suitable software, among those available, for a grounded theory study.

Straussian grounded theory was used to analyse these data. The present study built on previous work with some similarities in research design (Riordan, 2020; Riordan, Hardman and Cumbers, 2021) and used the following grounded theory methods:

[Initial] coding and categorization of data; concurrent data generation or collection and analysis; writing memos; theoretical sampling; constant comparative analysis using inductive and abductive logic; theoretical sensitivity; intermediate coding; selecting a core category; theoretical saturation; and theoretical integration. (Birks and Mills 2011, 9).

Initial coding involves labelling potentially significant words or phrases (‘incidents’), a pragmatic approach to coding recommended by Bryant and Charmaz (2010). We grouped, ungrouped, and renamed categories (intermediate coding), before identifying key themes (selective coding). After recording each set (i.e. a lesson, a teacher verbal protocol interview, a group pupil verbal protocol interview and a teacher interview), that data was analysed before the next set was conducted (concurrent data collection and analysis). Theoretical researcher ideas (memos) were used to build the grounded theory (i.e. the extended Pedagogy Analysis Framework and Pedagogy Problem Typology; ‘theoretical integration’). Hence, the coding grid became the extended Pedagogy Analysis Framework and Pedagogy Problem Typology. Our abilities, as researchers, to identify instances and relations deepened as the study progressed (‘theoretical sensitivity’). We continued using the grounded theory methods until no new insights emerged and we had many examples of our categories (‘theoretical saturation’; Bloor and Wood 2006). For more detail on the research design please see Riordan, Hardman and Cumbers (2021).

3.3. Participant selection

One class of 30 7-year-old pupils took part, and another class of 30 10-year-old pupils, each class with their teacher and teaching assistants. The teachers were recruited by contacting nearby primary schools (a convenience sample). As the theoretical perspective is interpretive, a small sample was used so that data could be examined in considerable detail. The school used is a Church of England school. About 34% of state schools in the UK are faith schools, of which 68% are Church of England schools (DfE 2012). RE is compulsory in all state-funded schools in England. This present small-scale qualitative study is part of a large research project which incorporates several research designs (e.g. large-scale quantitative work) focussing on science and religion encounters in primary and secondary schools. Hence, the decision to analyse a science lesson with a topic where religion was likely to come up, and an RE lesson where science was expected to be discussed. The research methods generated over six hours of video in total, so practical considerations like time available also influenced the sample size. One group of 7-year-old pupils, and one group of 10-year-old pupils participated (each with three girls and three boys) in each pupil group verbal protocol interview (all volunteers).

3.4 Trustworthiness

Criteria for evaluating qualitative research like this present study were defined by Lincoln and Guba (1985).

The four terms ‘credibility’, ‘transferability’, ‘dependability’ and ‘confirmability’ are … the naturalist’s equivalents for the conventional terms ‘internal validity’, ‘external validity’, ‘reliability’ and ‘objectivity’. (ibid., 300)

We used the same techniques (proposed by Lincoln and Guba 1985, 219) for establishing trustworthiness in this study as in Riordan (2020) and Riordan, Hardman and Cumbers (2021). These techniques include triangulation (sources, methods, and investigators), thick description (Geertz 1973), auditing the study in an NVivo file (available on request), using memos, etc. For further detail please see Riordan (2020). We argue that the techniques of Lincoln and Guba (1985) used alongside grounded theory methods, can give trustworthy evidence.

3.5. Ethics

As pupils and teachers participated, we followed the British Educational Research Association guidelines (BERA 2018). Teachers were invited to participate directly. Formal permission was then requested by letter to the headteacher. Pupils were first informally invited to participate by the teacher, then given letters with reply slips for their parents or carers. Pupils also filled in a consent form, to ensure informed consent. Regarding confidentiality, pupils and teachers are referred to using only letters.

4. Findings (part 1): the extended pedagogy analysis framework

The extended Pedagogy Analysis Framework (Figure 1) and Pedagogy Problem Typology (section 5: Findings (part 2)) are grounded theories which emerged using Straussian grounded theory (section 3.2) with the lesson, teacher verbal protocol, and pupil group verbal protocol video data.

Figure 1. The extended pedagogy analysis framework.

Next, we explain the extended Pedagogy Analysis Framework. Firstly, at time t₀ a participant₁ has some situational knowledge of the context (c; including in a limited way about past and potential future contexts) and prior conceptual, procedural and strategic knowledge₀. Secondly, participant₁ at t₁ generates known₂ knowns₁ or known₄ unknowns₃. Thirdly, participant₁ at t₂ may, or may not, want (i.e. have volition; v₁ or v₂) to do something (which may or may not include seeking a solution, i.e. trying to find the unknown₄). If participant₁ wants to do something this may or may not lead to some intention at time₃ (using whatever means_i, strategies_i and ends_i judged appropriate). Finally, at time₄ this participant₁ acts, or not, using actualised means_a and strategies_a, resulting in changes in knowledge and context (ends_a). The events during, and outcome, of one iteration of this cycle, can become part of the prior knowledge₀ and context (c) of participants in the next cycle. This argument is illustrated in Figure 1 above. We suggest this extended Pedagogy Analysis Framework matters because pedagogy analysis needs models that help understand and explain pedagogical problems and how participants act on what they think they know (including sometimes seeking solutions). The same extended Pedagogy Analysis Framework can also be used to understand and explain unknown knowns and unknown unknowns. For example, another participant₂ might know₁ that participant₁ does not know₃ something (i.e. a p₂ known unknown or a p₁ unknown unknown). To illustrate, Bob sees a picture of a whale (c). He knows₀ what a mammal is, whilst not knowing₀ that a whale is a mammal. He then realises₄ that he does not know₃ what type of animal a whale is and decides (v₂) he would like to find out. He thinks of finding out (e_i) by looking this up (s_i) in a book (m_i) but ends up doing an internet search (s_a) on his mobile (m_a) to find out and is rather surprised to learn (e_a) that the whale is a mammal. The next iteration of the extended Pedagogy Analysis Framework might be Bob’s teacher (p₂) explaining how a mammal like the whale came to live in the sea.

The extended Pedagogy Analysis Framework therefore integrates problem-solving literature with the Pedagogy Analysis Framework (Riordan, 2020; Riordan, Hardman and Cumbers, 2021). Pedagogy Analysis is clearly complicated (and time consuming), so partnership between teachers (both novice and experienced) and educational researchers will always be necessary. We acknowledge similarities between parts of this model and others (e.g. schemas; Gick 1986; Reed 2016).

5. Findings (part 2): a pedagogical problem typology

The following Pedagogical Problem Typology (Table 1) emerged from this study alongside the extended Pedagogy Analysis Framework.

Table 1. A pedagogical problem typology.

	Pedagogical problem type
5.1	Prior knowledge problems
5.2	Context problems
5.3	Knowledge about knowledge problems
5.4	Volition problems
5.5	Intended means, strategy and/or ends problems
5.6	Enacted means, strategy and/or ends problems
5.7	Solver problems
5.8	Solitary or social problems
5.9	Time and/or space problems
5.10	Grand strategy problems

No suggestion is made that any teacher would consciously categorise pedagogical problems as they occur during a real lesson. Teachers are far too busy teaching in class, and expertise with pedagogical problem solving is, we argue, largely tacit. Teacher educators make the tacit explicit for novices, so the Pedagogical Problem Typology may help this dialogue.

We now explain each problem before illustrating them with a thick description of excerpts from the data. The ways in which prior knowledge₀ can cause problems has been much discussed in the literature:

The name ‘conceptual change’ embodies a first approximation of what constitutes the primary difficulty: students must build new ideas in the context of old ones; hence, the emphasis on ‘change’ rather than on simple acquisition. Strong evidence exists that prior ideas constrain learning in many areas. (Di Sessa 2006, 265)

A prior knowledge₀ problem can occur when a participant (p₁) knows₀ something (or does not know something) that influences behaviours (5.1). Prior knowledge₀ can have a negative effect, like a spanner in the works, but can also bring benefits and resemble a grain of sand in an oyster, gradually being transformed into a pearl. For example, a teacher might ask themself, ‘Does Ann know about numbers?’ before a lesson on addition.

A participant₁ will have some situational knowledge which can lead to context problems (5.2). This participant₁ may not know that (or how) some factor external or internal to p₂ (and/or p₁) influenced, is influencing or potentially might influence another element of the extended Pedagogy Analysis Framework. For example, the teacher might reflect on the pupil’s emotional state with respect to her capacity for attentiveness, ‘How does Ann feel about doing some addition?’.

Jonassen’s (2000) definition of a problem (i.e. that a problem is a known unknown where a solution is perceived to be of value) begs the question of unknown to whom. In ‘knowledge about knowledge problems’, a participant₁ may know or not know that (or how) p₂ (and/or p₁) knows something or does not know it (5.3). For example, a teacher might consider, ‘Can Ann add?’. This is the infamous Rumsfeld conundrum of what types of known and unknown there are, further complicated by the fact that participants may be correct or incorrect about these understandings. Ramasesh and Browning (2014) argue that it does not necessarily follow that something unforeseen is unforeseeable (i.e. some ‘unknown unknowns’ can be identified and rectified). A subset of knowledge about knowledge problems is ‘information about information problems’. Here, the status of information (i.e. whether something is information, misinformation, or disinformation) is unknown to a participant, who may seek to clarify this.

What each participant wants, or does not want, matters in pedagogy analysis. Motivation is the setting or selection of goals, in contrast to volition which is goal realisation (Heckhausen and Heckhausen 2018, 9). Jonassen (2000) argued (see 2.1.2) that one key element of problem solving is how a participant values finding the unknown. Hence, we included volition in the extended Pedagogy Analysis Framework. A volition problem is when p₁ may know or not know that (or how much) p₂ (and/or p₁) wants something or does not want it (5.4). For example, the teacher might contemplate, ‘Does Ann want to learn to add?’. Misperception of who knows what and who wants what in the classroom is compounded by the propensity of all participants in real classrooms to engage in deception (Riordan, 2020). For example, a pupil may hide (‘mask’) from others that they want something through embarrassment.

An intended means_i, strategy_i and/or ends_i problem occurs when a participant₁ may know or not know that (or how) p₂ (and/or p₁) intends to use some combination of the available set of means_i, set of strategies_i and/or set of ends_i (5.5). For example, a teacher might think, ‘Could Ann use (s_i) Cuisenaire rods (m_i) to show that 2 plus 2 equals 4 (e_i)?’ An enacted means_a, strategy_a and/or ends_a problem occurs when the intended means_i, strategies_i and/or ends_i of a participant₁ may not correspond to the enacted means_a, strategies_a and/or ends_a (5.6). For example, the teacher may reflect ‘How did Ann use Cuisenaire rods (m_a) to show incorrectly (s_a) that 2 plus 2 equals 8 (e_a)?’. Here perhaps Ann used blocks of two instead of single units because the single units were lost.

A solver problem is when a participant₁ may consider or not a problem to be their own, and/or to consider or not a problem for another participant₂ to be theirs (i.e. p₂’s problem) (5.7). For example, ‘Why does Ann want Bob to do her sum?’. Solitary or social problems occur when a participant₁ considers that a problem should be solved by themselves (p₁), or solved by someone else (p₂), or solved by a group (e.g. p₁ & p₂), or not solved by a group (e.g. p₁ & p₂) (5.8). For example, ‘Why does Ann want to do the sum with Bob?’

When to start and stop activities (and how to do this) are perennial problems for teachers. A ‘time problem’ is when a participant₁ may know or not know when some element of the extended Pedagogy Analysis Framework has or has not occurred, is or is not occurring, or will or will not occur (5.9). Other subtypes of time problem include judging what is enough time but not too much, and identifying solved, solvable (with the subset of solvable now) and insolvable problems. Furthermore, in some time problems a sufficient solution is a better outcome than a perfect one (e.g. dishwasher packing), whereas in others a perfect solution is necessary (e.g. nuclear power station control rod design).

Physical position of intended and/or actualised means (either human or non-human) can cause pedagogical problems. A ‘space problem’ occurs when a participant₁ may know or not know where a means has been or has not been, is or is not, or will be or will not be (5.9). For example, ‘Where did Ann put the rods?’. Other subtypes of space problem include how to get a means from one position to another, and how to keep a means in one place. Cataloguing the varieties of subtypes of pedagogical problems is beyond the scope of this paper.

A thick description is sometimes necessary to explain a complicated pedagogical problem (here termed a ‘grand strategy problem’; 5.10). Several types of pedagogical problems may be entwined. Participants may disagree about aspects of a pedagogical problem. The elements of the extended Pedagogy Analysis Framework may interact and/or not be sufficient to understand a problem. For example, ‘Ann wants to know if numbers exist or not (and insists on solving this problem before proceeding)’ could be a metaphysical problem, and/or Ann using deception (of a type called ‘dazzling’; Riordan, 2020) to avoid doing a sum.

6. Thick description

Next, we analyse the pedagogy in an incident involving Duplo Lego, using short extracts from lesson, teacher verbal protocol and pupil group verbal protocol data, and a thick description that demonstrates the extended Pedagogy Analysis Framework and the pedagogical problems in the Typology. We chose the following extracts as together they provide examples of all the types in the Pedagogical Problem Typology. Pedagogical problem types are frequently combined like this in these data. There is no suggestion that the pedagogical problems in the following examples were particularly challenging for this teacher, but problem density is tricky in real lessons and some problems are more complicated (5.10). We acknowledge that different interpretations of the same data are possible. The point here is not to provide definitive interpretations, but to suggest theory to give a shared vocabulary for pedagogy analysis.

L1:5.50 – 7.51PT1 (Primary Teacher 1): [Pupils and teacher are sitting in a circle on the carpet in their classroom] I would like you to just watch me. I’ve got some Lego [Duplo], and I’m having a think about … maybe … building something [PT1 starts to build something]. I’m not sure what I’m going to be building yet. I could put that there. Mmm. I don’t know. I might change that. And then I want to put this here.

6.17P1: I can’t see yet.

6.19PT1: Let’s have a little look. Hmmm. Maybe I want to put something here.

6:29P2: Oh, I know what that is. I know what that is.

6:31PT1: Now do I want to put something that side as well? Maybe. Oh! I’ve got … Oh! I could put a face on and maybe have something red.

6:49P2: Oh, I know what that is. A train.

6:56PT1: Oh. Do I like what I’m making? Oh. It is a bit like an octopus, with different arms. [Holding the model up] Oh! I really like [7:09; PT1 throws the model dramatically on the carpet and breaks it. Several children gasp.] Oh. [7:12; PT1 puts her finger to her lips; a pupil₃ goes to pick up the pieces. 7:13; PT1 puts her hand out with the palm facing this pupil₃] Just leave it for now. Just leave it. I was just building something, and I was taking my time, and I was thinking about what I was making. I was thinking through the choices, but then what did I do? [7:30; Several pupils put their hands up] What did I do [P₃ name]?

7:32P4: You smashed it.

7:35PT1: I dropped it and smashed it all over the floor. I have destroyed my creation [said with emphasis]. [PT1 stands up] Oh. How do you think I’m feeling now?

Getting reluctant pupils to sit in a circle is a ‘space problem’ (5.9). A pupil unfamiliar with how to build with Duplo would be a context problem (5.2). How to stop one pupil from clearing up, perhaps to avoid this becoming a distraction, is an intended means_i (the hand), strategy_i (a gesture) and end_i (the pupil stops clearing up) problem (5.5). The teacher clearly did not want this pupil₃ to pick up the pieces (i.e. a solver problem; 5.7), so this is an example where the intended end_i (the pupil listening) did not correspond with that actualised end_a (the pupil clearing up; 5.6). The line, ‘Just leave it for now’ indicates a time problem (5.9). The way the teacher indicates that she does not want the pupil to help her clear up (at that moment) illustrates a ‘solitary or social problem’ (5.8). In the corresponding pupil group verbal protocol interview a pupil questions this teacher about her prior knowledge₀.

PVP1 3.33 – 3.51P1: Did you just like not think about it [i.e., what to do with the Duplo model once constructed] and then build it and then think about what it could be?

3.41PT1 (Primary Teacher 1): I was thinking about what it could be.

3.44P1: But you didn’t know what. So, so, you just built it and then you thought you liked it and about what it could be.

Here, this pupil (p₁) is asking and resolving his own prior knowledge₀ problem (5.1).

The following teacher verbal protocol transcript excerpt illustrates a ‘knowledge about knowledge’ problem (5.2).

TVP1 7.11-7.15PT₁ (Primary Teacher 1): {PT₁ is watching the L1 video “I’ve got some Lego, and I wonder … ” [PT₁ pauses the video and comments]} I’m just going to be interested to see their faces [PT₁ points at the laptop] because they don’t know what is coming, do they.

This teacher thinks that the pupils in the class at this time do not know what is to come (an ‘unknown unknown’ for the pupils). The intakes of breath, shocked expressions, and comments by the pupils during the lesson support this teacher’s interpretation.

How to use deception to get pupils thinking in an RE lesson about how humans and God might feel about nature being destroyed carelessly is an example of a grand strategy problem (5.10). In the following extract from the pupil group verbal protocol interview, the pupils have just watched the clip. One pupil (p₁) at first does not know why the teacher would destroy the model after explaining that she liked it (k₄unk₃), then another pupil (p₄) exclaims that this was deliberate (k₂k₁):

PVP1 3.04 – 3.23P₁: … when she [PT₁] like destroyed it was like saying, “Oh I really li … I really like … ” And then boom [P₁ mimes throwing the Lego model on the floor.] So, I don’t get it because um you say you like it and then you destroy it. Why did you destroy it when you just said you liked it [PT₁ is smiling as are some of pupils we can see on the video]?

3.19P₂: I liked …

3.20P₃: You did it on purpose!

3.21PT₁: I did do it on purpose [with a smile].

Participant₁ is unsure why the teacher would break a valued construction (i.e. a volitional problem for p₁; 5.4). Deception has been defined as a deliberate distortion of perceived reality (Whaley 1982; Riordan, 2020) and in this example one pupil (p₁) does not (at first) realise the teacher was deceiving her, whilst another (p₃) has rumbled what was going on. The information problem is concerned with participants trying to identify like this if something is information, misinformation or disinformation.

In real time, the passages above lasted 2 minutes and 42 seconds in total. The data set consisted of 6 hours and 28 minutes in total, so these excerpts represent about 0.7% of the entire data set. In total 12,001 incidents were coded in all six sources (see Table 2).

Table 2. Number of references coded, and number of codes used, in each data source.

	Codes	References
L1	182	4822
PVP1	88	881
TVP1	63	355
L2	151	4077
PVP2	77	1241
TVP2	66	625
Total		12,001

Figure 2 below shows how many times each element of the extended Pedagogy Analysis Framework and Pedagogical Problem Typology was coded overall.

Figure 2. A graph of how often each element of the extended Pedagogy Analysis Framework was coded in all data sources for each teacher (PT1 and PT2).

Figure 3 below shows how many times each type of problem in the Pedagogical Problem Typology was coded for all data sources.

Figure 3. Number of times each type of Pedagogical Problem was coded for all data sources.

7. Discussion

We have illustrated how the extended Pedagogy Analysis Framework (Figure 1) and Pedagogical Problem Typology (Table 1) can help during pedagogy analysis to identify, understand and explain pedagogical problems and solutions. This work builds on previous research conducted with secondary science teachers (Riordan, 2020; Riordan, Hardman and Cumbers, 2021; Hardman, Riordan and Hetherington, 2022). Concepts within the literature that inform the extended Pedagogy Analysis Framework are contested, so selecting and piecing together the elements is challenging (like a jigsaw of pieces made from set honey on a warm day). However, without the extended Pedagogy Analysis Framework novice-expert strategic dialogue can struggle for want of a shared vocabulary. This study adds to the current state of knowledge by integrating problem-solving literature (Jonassen 2000) into the Pedagogy Analysis Framework (Riordan, 2020; and Riordan, Hardman and Cumbers, 2021) to make the extended Pedagogy Analysis Framework, thereby expanding our previous work. We introduced the Pedagogical Problem Typology and related it to the extended Pedagogy Analysis Framework. In addition, these two grounded theories contribute to teachers’ decision-making literature (section 2.1.2). Firstly, we have argued here and elsewhere that pedagogy analysis needs to incorporate both teacher and pupil perspectives (Riordan, 2020; Riordan, Hardman and Cumbers, 2021) allowing exploration of the decision processes of any participant, not just the teacher, and what effect that may have in a classroom. Secondly, pedagogy analysis must explore the processes of teaching and learning, but also the interactions between teaching and learning (Hallam and Ireson 1999) and the extended Pedagogy Analysis Framework can help understand and explain such interactions, even when they get complicated. Our interest is in teacher and pupil decision making; how, when and where such decisions are actualised (or not); and how interactions between participants (and between participants and things) influence events. Next, we discuss how pedagogical synthesis and analysis skills interact.

7.1. Pedagogical analysis skill and pedagogical synthesis skill

In a previous grounded theory study with a similar research design (Riordan, 2020), from which the original Pedagogy Analysis Framework emerged, one of us (JR) asked an experienced teacher:

5c:9JR: […] Are there particular strategies that you’re consciously employing?

5c:10Teacher: … Um. [Shaking his head] Probably no.

5c:11JR: Or is it intuitive?

5c:12Teacher: [Pause] I think probably if I wound it back to different times the answer would be yes. … But I’m really conscious now that when I try to model stuff for […] [s]tudent teachers and Newly Qualified Teachers - that I can’t put [it] into words, or I can’t any longer [TY smiles] do what I was making [the pupils in the pupil group verbal protocol interview] do, which was put into a logical sequence why I do things. I just know […] that I do them. And I do them because it works.

This experienced mentor and advanced skills science teacher might be finding it harder to provide analysis for novice teachers because the gulf between expert and novice may increase with mentor age and/or experience. In addition, pedagogy is like a fractal; the closer one looks the more one sees. We have argued that teachers and pupils in the present study combined (synthesised) the elements of the extended Pedagogy Analysis Framework (Figure 1) during the lessons to try to solve pedagogical problems of 10 types (Table 1). Furthermore, participants (teachers, pupils and researchers) also untangled (analysed) these same elements during the teacher verbal protocol and pupil group verbal protocol interviews. Next, we present a model (Figure 4) which illustrates four ways in which pedagogical synthesis skill and analysis skill can combine (Figure 4, ‘a, b, c and d’) and 12 ways such skills can change. Grounded theory involves concurrent data collection and analysis (sometimes spanning several studies like this), and the integration of theory like this (see section 3.2).

Figure 4. A model for how pedagogical synthesis skill and pedagogical analysis skill can relate.

Hence ‘d’ in Figure 4 represents a novice unskilled in pedagogical synthesis (i.e. they struggle to combine the elements of the extended Pedagogy Analysis Framework so that pupil learning emerges) and in pedagogical analysis (they need help untangling what is going wrong). ‘1ʹ in Figure 4 then illustrates the path to better teaching and better pedagogical analysis (i.e. towards ‘a’). Eleven other options are possible. The transcript above exemplifies how an experienced teacher maintained pedagogical synthesis skill but perceived their own pedagogical analysis skill to be regressing (i.e. ‘10ʹ in Figure 4). We argue that the model in Figure 4 could help strategic dialogue among pedagogy researchers, teacher educators and teachers about changes in pedagogical analysis skill and/or synthesis skill during a teaching career. We think better analysis of pedagogy can improve teaching.

7.2. Originality

We argue, using the criteria for originality identified by Wellington (2012) that this study has the following innovative aspects. Firstly, the extended Pedagogy Analysis Framework and Pedagogical Problem Typology are new knowledge. Secondly, this study synthesises the findings of an earlier study involving six teachers working with small groups of pupils (Riordan, 2020) and one including three whole class secondary science lessons (Riordan, Hardman and Cumbers, 2021) with this one (two whole class primary lessons). Furthermore, this work synthesises the Pedagogy Analysis Framework with research about problem solving (Jonassen 2000). Finally, we replicate the research design from previous work (Riordan, Hardman and Cumbers, 2021) in a new context (primary science and RE lessons).

7.3. Limitations

In grounded theory, a substantive theory explains phenomena in one context. If substantive theory achieves ‘theoretical saturation’, a formal theory is possible which explains a phenomenon in a wider context (Glaser and Strauss 1967). This present study has addressed some of the limitations of previous work (Riordan, Hardman and Cumbers, 2021) but the reasons why we argue this is not yet a formal theory will now be outlined. Firstly, two teachers teaching different age groups are included, but more teachers working with a greater range of age groups (including secondary age pupils) are still needed. This was planned for this study, but we have been unable so far to collect the secondary school data because of Covid-19. Secondly, the Pedagogical Problem Typology and extended Pedagogy Analysis Framework represent significant yet tentative adaptations to the Pedagogy Analysis Framework. Hence, we argue the theory in this paper remains a substantive theory with some aspects that may have wider applicability.

The most significant limitations remaining are as follows: Influences of culture and history on pedagogy, as explored by Alexander (2001), are beyond the scope of this work. Pupil verbal protocol interviews could not be done with all pupils for practical reasons. Lesson clip selection for the pupil group verbal protocol interviews pre-empted participants’ choice. The time delay between the interviews may have influenced the findings. Interpretation in qualitative research like this always has elements of subjectivity. Deception by participants, part of the Pedagogy Analysis Framework, necessarily has implications for the credibility of findings. We acknowledge researcher influence on these data, though we sought to minimise this impact. Grounded theory methods were used carefully, but with so many codes and lots of video data, we acknowledge that some mistakes in coding are likely. This paper can only illustrate how the video data supports the findings using a tiny proportion of the full data set (see the data availability statement below). Many technical challenges remain, particularly with audio quality from noisy classrooms.

7.4. Next steps

Video is already used in a variety of ways during many Initial Teacher Education courses (Santagata and Guarino 2011). Research could investigate if video stimulated Problem-Based Learning using a typical range of types of pedagogical problem during ITE improves the pedagogy of novice teachers. The utility for novice teachers and teacher educators of the extended Pedagogy Analysis Framework and the Pedagogical Problem Typology could be investigated. The influence of volition and emotion on pedagogical problem solving both need further study. Tracing how Pedagogical Units of Analysis link, or not, throughout a lesson may be of interest. The research design could be used to investigate university andragogy and pedagogy for children with Special Educational Needs. A longitudinal study with a similar research design could investigate how the pedagogy of novice teachers develops over time. These findings may help teachers and teacher educators understand and explain pedagogy, but how to help someone become a competent pedagogue is beyond the scope of this paper. Knowing the rules of chess is a prerequisite for play but does not make one a good player.

Acknowledgments

We thank [Removed4].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available in [Removed5].

Additional information

Funding

This work was supported by the Templeton World Charity Foundation [TWCF0375].