In this article we regard a professional learning community as ‘a group of teachers sharing and critically interrogating their practice in an on-going, reflective, collaborative, inclusive, learning-oriented, growth-promoting way’ (Stoll & Louis, 2007, p. 2). In our study the professional learning community consisted of five mathematics teachers in one high school and the first author as facilitator. The community engaged in a number of developmental learning activities (see Brodie & Shalem, 2011, for more details), which involved conversations about their understandings of learner errors in mathematics. Our analyses of these conversations in this article sought to understand the learning opportunities that were created.

The theoretical framework that guided the study was situated learning theory (Lave, 1991; Lave & Wenger, 1991; Wenger, 1998), which regards learning as situated in human activities, inseparable from the environments in which knowledge is used, and occurring through social processes that involve the negotiation of meanings with others (Buysse, Sparkman, & Wesley, 2003). According to the theory learning is situated in communities of practice, including professional learning communities, which we regard as a special kind of community of practice (Brodie & Borko, 2016a) that occurs in the practices of such communities (Wenger, 1998). For Wenger the practice of a community of practice has three dimensions or characteristics, which are a source of coherence for the community: joint enterprise, mutual engagement and a shared repertoire. The joint enterprise is the shared basis for action with colleagues, or the ‘negotiated response to their situation’ (Wenger, 1998, p. 77) which the community strives to resolve or understand. In a professional learning community the joint enterprise is akin to a learning focus, which is an issue of concern, or a problem of practice that the teachers inquire into so as to develop new understandings that deepen their professional knowledge and practice (Katz & Earl, 2010). In this study the learning focus for the professional learning community was understanding learner errors in mathematics. The second dimension of practice is mutual engagement which refers to active collaboration involving such practices as sharing understanding, joint reflection, collective development of new meanings and joint decision-making (Katz & Earl, 2010). According to Wenger (1998) mutual engagement fosters coherence in a community of practice and is a source of new meanings and understandings for the community, also called the shared repertoire. The shared repertoire, as the third dimension of the practice of a community of practice, ‘includes routines, words, tools, ways of doing things, stories, gestures, symbols, genres, actions or concepts that the community has produced or adopted in the course of its existence, and which become part of its practice’ (Wenger, 1998, p. 83). In a professional learning community the shared repertoire consists of the meanings that are developed and ways of dealing with, or resolving, situations that are jointly negotiated and adopted as part of the community’s practice.

The three dimensions of practice are presented in Figure 1, which shows how having a shared learning focus and engaging in collective action supports mutual engagement activities which in turn lead to the development of new meanings and ways of addressing problems of practice. The three dimensions shape each other, hence the double arrows. In this article we adopted these three characteristics of practice as one of the two frameworks or analytical tools that we used in analysing the conversations of the professional learning community.

Our goal in using this framework was to understand the teachers’ participation in the professional learning community as a practice that supported learning opportunities in the teachers’ conversations, which were the data that we analysed in this article.

Another aspect of teacher learning in a professional learning community that was of interest to us was what teacher knowledge could have been learned in the conversations of the community. To analyse this knowledge we used pedagogical content knowledge and content knowledge, which are seen as part of teachers’ mathematical knowledge for teaching (Ball, Hill, & Bass, 2005; Ball, Thames, & Phelps, 2008; Hill et al., 2008). Mathematical knowledge for teaching is defined as the knowledge that mathematics teachers need to know in order to teach the subject well, and includes teachers’ knowledge of: the content of mathematics, the different representations of the mathematics so that it is accessible to learners, learners’ mathematical thinking, and how to make in-the-moment decisions that support learners’ learning. So our second analytical tool consisted of pedagogical content knowledge and content knowledge as part of teachers’ mathematical knowledge for teaching. Such knowledge is part of teachers’ pedagogical content knowledge and, as we will show, is related to their content knowledge.

The two analytical frameworks presented above have a close relationship with each other that enabled us to analyse both the teachers’ participation in the conversations and what they learned in those conversations. The joint enterprise, mutual engagement and shared repertoire framework enabled us to analyse the teachers’ participation as a practice of the community. The mathematical knowledge for teaching framework enabled us to analyse the knowledge implicated in the conversations. For us the two frameworks are related in that in each conversation episode the mathematical knowledge was part of the learning focus or joint enterprise, and through active participation in the conversation, there were opportunities for deepening the teachers’ knowledge and developing new meanings about the teachers’ mathematical knowledge for teaching.

Inquiry in professional learning communities is usually supported by a facilitator or ‘critical friend’, who participates in the professional learning community and supports learning by asking for critical reflection on issues, probing for justification, challenging assumptions, pushing for deeper thinking and interpretations, reminding participants of what has been achieved, and guiding the direction of the ongoing conversations (Brodie, 2016; Katz, Earl, & Jaafar, 2009). A facilitator thus plays a fundamental role in supporting teacher learning and in this article we also analysed the facilitator’s role in the learning opportunities created in the professional learning community.

Research on teacher learning in professional learning communities shows that learning can occur in some important dimensions of teachers’ knowledge and practice. In some communities teachers learned to analyse mathematical problems in greater depth (e.g. Borko, Jacobs, Eiteljorg, & Pittman, 2006). Learning was also observed in teacher reflection on classroom practices in ways that could result in changes in practice (Horn, 2010). Some studies have shown how teachers learned about their learners’ thinking in making errors and developed better understandings of learners’ thinking (e.g. Brodie, 2014; Kazemi & Franke, 2004). Other studies have shown changes in teachers’ knowledge of the curriculum (Brodie, Shalem, Sapire, & Manson, 2010), and the teaching practices (Tam, 2015). In all the studies, teacher learning was attributed to participation and collaboration in teacher learning communities.

In South Africa findings from a teacher professional development project have shown that conversations in professional learning communities provided opportunities for developing both content knowledge and pedagogical content knowledge, that content knowledge and pedagogical content knowledge conversations often triggered each other (Marchant & Brodie, 2016), and that the different project activities supported talk on different foci: learners, mathematics and practice (Chimhande & Brodie, 2016). Teacher participation in the professional learning communities supported the deepening of teachers’ mathematical knowledge (Brodie, 2014; Liebenberg, 2016), shifts in teachers’ talk about learner errors (Brodie, 2014), shifts in teachers’ practices (Chauraya & Brodie, 2017; Molefe, 2016) and shifts in teachers’ identities (Chauraya, 2016). Other studies involving teacher learning communities in South Africa have shown that teachers developed confidence that was central to their learning (Graven, 2004).

The findings cited above highlight how teacher communities can support teacher professional learning in terms of understanding and addressing problems of practice, understanding learners’ thinking, developing new teaching practices, and deepening knowledge of mathematics and the curriculum. The studies also highlight the significance of collaboration, community and inquiry to learning in such communities. Our article contributes to this growing knowledge base by presenting findings of analyses of learning opportunities created in conversations about learner errors.

Data analysis involved three stages. The first stage involved identifying the conversation episodes for analysis from the lengthy conversations of the community. The second stage involved summarising the selected episodes for presentation and analysis. The last stage involved the actual analysis of each episode using the analytical frameworks presented earlier in the article.

In identifying the conversation episodes for analysis in this article we adopted the critical incidents approach which is a qualitative data analysis technique mainly used in analysing classroom events in order to understand teacher actions and reflections in instructional situations (Angelides, 2001; Yang & Ricks, 2012). A critical incident in school situations is regarded as a commonplace event that occurs in the everyday life of a classroom but is justified as critical on the basis of its significance to the goals of the activity and the researcher’s interpretation of the event (Angelides, 2001). The criticality of an incident depends on the ‘justification, the significance, and the meaning given’ by the researcher (Angelides, 2001, p. 431). Thus a researcher looks at the underlying trends, motives and structures in the incident and ascertains whether it is critical or not depending on what they are interested in before analysing the teaching or learning effect of the episode (Yang & Ricks, 2012).

In adopting the critical incidents approach we regarded a critical episode as a conversation focused on one of the following themes: making sense of learners’ reasoning through analysis of learner interviews, understanding learners’ errors and identifying the critical concept for learners, and the teachers’ own understanding of the critical concept. These themes were deliberately chosen in line with the goals of the Data-Informed Practice Improvement Project which included developing teachers’ knowledge of learner errors and how to work with errors in their teaching.

In presenting and summarising each selected episode we identified only those conversation turns that contributed to the theme of the episode. A turn is an utterance or comment by a participant in a conversation. Turns that were not relevant to the theme of the episode were disregarded. Having identified the relevant turns in each episode, we then categorised these turns according to what was being said and the purpose that each turn served. For example a question by the facilitator was categorised as ‘facilitator question’, and then according to whether the question served to challenge the teachers’ knowledge or probe for deeper meaning. Thus our presentation summarises the selected conversation turns in each episode in tabular form, showing the nature and frequency of particular categories of conversation turns, including examples of such turns.

As indicated earlier, in the actual analysis we used two frameworks or analytical tools. The dimensions of practice in the community as depicted in Figure 1 enabled us to analyse: what formed the focus of each conversation episode (the joint enterprise), how the teachers engaged in the conversation (mutual engagement), and the nature of the developing new meanings or practices (the shared repertoire). The mathematical knowledge for teaching framework enabled us to identify the nature of mathematical knowledge in each conversation episode and any new meanings that could have been developed about that knowledge.

The relevant university research ethics committee cleared the project from which the data and findings reported in this article were drawn. The ethics clearance number was 2009ECE171. The Gauteng Department of Education approved the project through a letter detailing the conditions for the conduct of the project in schools in the province. To maintain anonymity all the names of participants in this article are pseudonyms and the name of the school is not mentioned anywhere in the article.

As part of the analyses of learners’ errors, each teacher interviewed selected learners on specific errors that they had made. Learner interviews were intended to support the teachers to understand learner errors more deeply, in particular what was both valid and not valid in the learners’ reasoning. The interviews were also the initial step for the teachers in trying to make sense of the learners’ errors and were central to conversations about learner errors.

Table 1 shows an extract of one teacher’s interview with a learner on the error: 8005and8003. All the teachers’ and learners’ names are pseudonyms. Janeth is the teacher in these extracts.

Interviewing learners about their reasoning was not part of teachers’ practice prior to the professional learning community. In preparation for learner interviews, the professional learning community had discussed the need to ask and follow up on questions that would elicit the substantive learner reasoning behind the errors. The teachers role-played interviews and learned how to distinguish between questions that tried to guide the learners to the correct answer and questions which served to access the learners’ reasoning. In the episode in Table 1 the teacher started off with a general question ‘Can you explain how you solved this problem?’ After the learner explained his strategy, the teacher probed further for the learner’s reasoning, which the learner struggled to provide – he kept talking about his strategy. Even the teacher’s question ‘Why did you do that?’ produced the answer ‘Because I wanted to get the final answer’. Our data is replete with such examples, with learners struggling to make their reasoning explicit and teachers struggling to support them to do this. The teacher was responsive to the learners’ difficulties, and shifted her question to ‘But this idea of dividing, where did divide come from?’ This question supported the learner to say that he was guided by the word ‘divide’ in the problem, which shows the teacher where some of the difficulty lies.

This interview extract shows how the learner interviews were learning opportunities for the teachers to learn how to elicit, in more detail, learners’ thinking behind observed errors. Through the interviews the teachers also had opportunities to begin to understand why learners make errors in mathematics, and how the errors are reasonable for learners (Brodie, 2014). This is an important aspect of teachers’ pedagogical content knowledge, which they can work with in their teaching as they respond to learners’ errors and learners’ learning needs. However in the interviews the teachers often missed opportunities for getting more insights into the learners’ understandings of some mathematics concepts. In the episode above the teacher did not ask the learner to explain his understanding of the ratio ‘5:3’, thereby missing the opportunity to get the learner’s understanding of the concept.

In the professional learning community initial conversations about learner errors focused on understanding the learner errors, giving the possible reasons for the errors, and identifying the critical concepts for the learners, also known as the learners’ learning needs. The learner interviews served to provide evidence for the learners’ thinking in making errors. Understanding learners’ reasoning in errors formed the learning focus or joint enterprise for the community in these conversations. The conversation presented in Table 2 shows how the teachers mutually engaged in trying to make sense of the learners’ thinking in making the observed errors.

In our analysis below we use an ordered pair notation, for example (T5, Sibanda); the first value Ti refers to conversation turn number and the second value refers to the pseudo names of the participants.

The episode shows certain patterns of interactions in the community conversations which were also evident in other and similar conversations. The episode was characterised by the facilitator’s questions that served to initiate and probe for deeper thinking, the teachers’ interpretations, explanations and justification of learner errors, and elaborations by the facilitator as a way of summarising the meanings being developed. These patterns of interaction characterised the mutual engagement by the teachers in the conversations. The conversation drew from the learner interview presented earlier on, which illustrates how learner interviews were useful data that supported the joint enterprise. The larger part of the episode consisted of turns in which the teachers interpreted the error and proposed reasons for the error. Interpreting the error involved identifying what the learners did in making the error. Explaining the error involved suggesting the possible reasoning behind the error, and in this case the explanation was that the learners took ‘divide in the ratio …’ to mean ‘divide by …’. One teacher’s comment that this was correct according to the language suggests that she was beginning to see the reasonableness of the error from a learner’s perspective, which is an important step for teachers in learning how to view errors as integral to learning mathematics. However, the same teacher’s suggestion that the error could be justified in terms of learners forgetting what they had been taught was an example of blaming learners for errors, something that shows limited understanding of the nature of errors in mathematics learning.

This episode illustrates how conversations about learner errors created opportunities for learning how to interpret learner errors and explain the thinking behind the errors, as well as identifying learners’ learning needs from their errors – all examples of opportunities for developing pedagogical content knowledge mutual engagement through practices such as facilitator questions and teachers’ responses and explanations which supported opportunities for learning about learners’ thinking in making errors. The teachers’ explanations of the error served as examples of developing new thinking or understanding about learner errors, which could be part of the teachers’ developing shared repertoire.

The facilitator’s central role in this episode was that of initiating and probing through questions, as well as clarifying and building on the teachers’ contributions, all of which supported mutual engagement in the conversation. In the next section we show how the conversations shifted to the teachers’ subject content knowledge of ratio.

One of the errors was attributed to learners not understanding what it meant to divide a quantity in a given ratio. In the following episodes we analyse how the facilitator shifted the conversation from the identification of learners’ learning needs to teachers’ learning needs. Two critical episodes in the conversation focused on the teachers’ understanding of ratio as a concept and the conceptual meaning of the steps in the algorithm for dividing quantities in given ratios. Both episodes were on the teachers’ subject content knowledge of ratio. Table 3 shows a summary of the critical turns in the conversation and how we categorised the turns according to how they contributed to the theme of the episode.

The facilitator’s questions served to support the teachers to share their individual understandings of ratio. The teachers’ responses showed an algorithmic understanding of ratio, in which they described ratio in terms of the steps in the algorithm for dividing a quantity in a given ratio. The ensuing funneling questions by the facilitator were a way of getting the teachers to the concept, upon realising that the teachers were not likely to give the conceptual meaning. In the last turns of the episode the facilitator elaborated the concept further as a way of supporting the teachers’ understandings.

Although there was no evidence that the teachers learned the conceptual meaning of ratio, the episode was an opportunity for the teachers to develop a new shared understanding of the concept of ratio, thus deepening their subject content knowledge. The data above show that in an effort to shift the teachers’ algorithmic understanding to an understanding of the concept of ratio, which includes the algorithm, the facilitator deliberately used funneling questions. We acknowledge the limitations of funneling questions, and in this case there was no evidence that this move by the facilitator resulted in deeper learning of the concept by the teachers.

The conversation episode summarised in Table 4 focuses on the teachers’ understandings of the conceptual meaning of the steps in the algorithm for dividing a quantity in a given ratio. Table 4 presents a summary of the conversation and our categorisation of the critical turns.

The facilitator’s challenging questions and the teachers’ expressions of their own understandings were the main forms of mutual engagement in this episode. The teachers’ responses to the initial question by the facilitator showed knowledge of the steps in the algorithm, but they could not explain the conceptual meaning of the algorithm as a whole. The teachers’ difficulties in explaining the algorithm could be attributed to how they normally teach the algorithm. Mathematics teachers in South Africa often teach the algorithm without worrying about its conceptual meaning, and knowledge of the steps in the algorithm is considered to be sufficient and relevant for the curriculum and learners’ examination needs (Mogari, 2014). In not regularly thinking about the concepts behind the algorithm, the teachers come to see the algorithm as the concept. Brodie and Sanni (2014) give similar examples in which Nigerian teachers’ pedagogical content knowledge and knowledge of the curriculum constrained conceptual reflection on their content knowledge.

The episode culminated with the facilitator sharing his own knowledge of the algorithm, a move that was necessitated by his realisation that the teachers could not explain the conceptual meaning of the algorithm as a whole. This move by the facilitator was necessary for purposes of building professional knowledge, an important aspect of learning in a professional learning community and a practice that was also evident in other conversations that we analysed. After the facilitator’s explanation, remarks by Mandla and Bongiwe, while not necessarily showing their own conceptual understanding of ratio, indicate that they saw the conversation as developing their knowledge of teaching the ratio algorithm differently.

In terms of their mathematical knowledge for teaching, the episode was an opportunity for the teachers to deepen their understanding of the algorithm, and hence their pedagogical content knowledge. We acknowledge that there was no evidence to indicate that they had learned this knowledge, but two teachers’ expressions of their willingness to work with this knowledge in their teaching could be indications of new and shared understandings that could influence their teaching.