learning about learner errors in professional learning communities

19
Learning about learner errors in professional learning communities Karin Brodie Published online: 27 August 2013 # Springer Science+Business Media Dordrecht 2013 Abstract Research on learner errors in mathematics education is beginning to focus on how teachers can learn to identify and engage with the reasoning behind these errors. Research on professional learning communities is beginning to show that they present powerful oppor- tunities for on-going teacher collaboration and learning. In this paper, I bring the two areas of research together. Drawing on data from one professional learning community in the Data Informed Practice Improvement Project, I show how teachers in this community came to understand key concepts about learner errors and shifted their ways of talking about learner errors. I identify three important shifts that the teachers made in their learning about learner errors: from identifying to interpreting errors; from interpreting to engaging errors; and from focusing on learner errors to focusing on their own knowledge. I argue that these three shifts suggest a deepening of teachersthinking in relation to learner errors. Keywords Teacher learning . Learner errors . Professional learning communities 1 Introduction Research on learner errors has a long history in mathematics education. Early work identified errors as opportunities for teaching and learning rather than problems to be avoided, particularly if the reasoning underlying errors can be identified and explored (Ben-Zeev, 1996; Borasi, 1986, 1987; Erlwanger, 1973; Nesher, 1987). The rise of constructivist theories of learning positioned errors as the performance of misconceptions, conceptual structures constructed by learners that make sense to learners in relation to their current knowledge. This constructivist position was supported by many studies that identified particular misconceptions and showed how these are reasonable and sensible to learners (see Confrey, 1990; Smith, DiSessa, & Roschelle, 1993 for reviews of the literature). The research shows that misconceptions and the errors they produce are remarkably persistent and similar across contexts, independent of curricula or teaching methods, and thus can be seen as normal and possibly necessary steps in the development of mature concepts (Smith et al., 1993). Educ Stud Math (2014) 85:221239 DOI 10.1007/s10649-013-9507-1 K. Brodie (*) Wits University, Johannesburg, South Africa e-mail: [email protected]

Upload: karin

Post on 23-Dec-2016

212 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Learning about learner errors in professional learning communities

Learning about learner errors in professionallearning communities

Karin Brodie

Published online: 27 August 2013# Springer Science+Business Media Dordrecht 2013

Abstract Research on learner errors in mathematics education is beginning to focus on howteachers can learn to identify and engage with the reasoning behind these errors. Research onprofessional learning communities is beginning to show that they present powerful oppor-tunities for on-going teacher collaboration and learning. In this paper, I bring the two areasof research together. Drawing on data from one professional learning community in the DataInformed Practice Improvement Project, I show how teachers in this community came tounderstand key concepts about learner errors and shifted their ways of talking about learnererrors. I identify three important shifts that the teachers made in their learning about learnererrors: from identifying to interpreting errors; from interpreting to engaging errors; and fromfocusing on learner errors to focusing on their own knowledge. I argue that these three shiftssuggest a deepening of teachers’ thinking in relation to learner errors.

Keywords Teacher learning . Learner errors . Professional learning communities

1 Introduction

Research on learner errors has a long history in mathematics education. Early work identifiederrors as opportunities for teaching and learning rather than problems to be avoided, particularlyif the reasoning underlying errors can be identified and explored (Ben-Zeev, 1996; Borasi,1986, 1987; Erlwanger, 1973; Nesher, 1987). The rise of constructivist theories of learningpositioned errors as the performance of misconceptions, conceptual structures constructed bylearners that make sense to learners in relation to their current knowledge. This constructivistposition was supported by many studies that identified particular misconceptions and showedhow these are reasonable and sensible to learners (see Confrey, 1990; Smith, DiSessa, &Roschelle, 1993 for reviews of the literature). The research shows that misconceptions andthe errors they produce are remarkably persistent and similar across contexts, independent ofcurricula or teaching methods, and thus can be seen as normal and possibly necessary steps inthe development of mature concepts (Smith et al., 1993).

Educ Stud Math (2014) 85:221–239DOI 10.1007/s10649-013-9507-1

K. Brodie (*)Wits University, Johannesburg, South Africae-mail: [email protected]

Page 2: Learning about learner errors in professional learning communities

More recently, research has begun to focus on teachers’ understandings of learner errors,as a route into understanding learner thinking. This body of work is less comprehensive thanthe substantial body of work on learner errors, but is beginning to identify teachers’ views onlearner errors (Gagatsis & Kyriakides, 2000) and key teacher competences required toengage with learner errors in written work and classroom interactions. Attending to learners’mathematical productions; understanding these productions from the perspective of thelearner; connecting learner errors to teachers’ own subject and pedagogical content knowl-edge; and making decisions about how to respond, often in the moment in class, have allbeen identified as important elements of teachers’ noticing of and working with learnerproductions, including learner errors (Jacobs, Lamb, & Philipp, 2010; Peng & Luo, 2009;Prediger, 2010).

An important question is how teachers learn to understand and engage with learnererrors. Gagatsis and Kyriakides (2000) show that teachers who attended an in-serviceteacher development course with a focus on understanding the mathematical reasons forlearner errors tended to explain learner errors with a focus on the underlying mathematicsrather than in relation to learners’ abilities and attitudes. Jacobs et al. (2010) show thatteachers’ expertise in noticing and responding to learner productions increases withteaching experience and through involvement in teacher development programs focusedon understanding learner thinking. This work suggests that teacher development isimportant in supporting teachers to notice and engage with learner errors but does notshow how teacher development programs might achieve such engagement. In this paper,I focus on how one form of teacher professional development, teachers working togetherin a professional learning community, supported teachers to learn about learner errors: tounderstand them as reasoned and reasonable (Ball & Bass, 2003); and to think about howbetter to engage with them in the classroom. I describe some important shifts in teachers’thinking about learner errors in relation to their practice to show how such learningmight happen.

Professional learning communities as a model for teacher learning are being increasinglydeveloped and researched internationally, both in mathematics education and more generally(Borko, 2004; Horn, 2005; Jaworski, 2008; Katz, Earl, & Ben Jaafar, 2009; Krainer &Wood, 2008; McLaughlin & Talbert, 2001). Successful professional learning communitiescreate environments where teachers can explore their strengths and weaknesses with col-leagues; develop collaborative solutions to problems of practice; and implement new ideascollectively for the benefit of learners. Data-informed professional learning communities useinformation from classrooms to identify areas of potential improvement in practice—thedata provide information on learners’ needs, and learners’ learning needs inform teachers’learning needs (Katz et al., 2009).

In this paper, I draw on the work of the Data Informed Practice Improvement Project(DIPIP), which works with mathematics teachers to create and sustain professional learningcommunities with a focus on learner errors. Learner errors are seen as evidence of learnerthinking and learner needs, and teachers can draw on learners’ reasoning underlying theirerrors to help learners to develop mathematical concepts. How teachers learn to work withlearner errors is the focus of this paper. The paper begins with a conceptualization of thenotion of “error” in the mathematics classroom as a context for the teachers’ learning,followed by the conception of teacher learning that I work with in the paper, and adescription of the DIPIP project and the working of the professional learning community.I then discuss the key aspects of the teachers’ learning as they interacted in the professionallearning community.

222 K. Brodie

Page 3: Learning about learner errors in professional learning communities

2 Learner errors

In the DIPIP project, we1 define errors as systematic, persistent and pervasive mistakesperformed by learners across a range of contexts (Nesher, 1987). We distinguish errors fromslips (Olivier, 1996), which are mistakes that are easily corrected when pointed out. Errorsare more difficult to deal with: they arise even though they are not explicitly taught (Hatano,1996); they are persistent even when corrected, thus seemingly resistant to instruction(Smith et al., 1993) and they occur among learners within and across contexts—suggestingthat they are systemic rather than the result of individual learner or teacher failure.

Theoretical work to explain the persistence and pervasiveness of learner errors oftendraws on constructivism. In constructivist theories of learning, errors are said to arise frommisconceptions, which are conceptual structures constructed by learners that make sense inrelation to their current knowledge, but which are not aligned with conventional mathemat-ical knowledge (Nesher, 1987; Smith et al., 1993). Misconceptions arise from learners’overgeneralisation of a concept from one domain to another (Smith et al., 1993).Mathematical knowledge that works in one domain (e.g. natural numbers) no longer workin new domains (e.g. decimals and fractions).

An important entailment of constructivism is that misconceptions and the errors theyproduce cannot be easily “removed” or “replaced” through instruction (Smith et al., 1993).Since the concepts make sense in the light of the learners’ current knowledge and areconnected to other, correct, knowledge, they need to be restructured by learners (Hatano,1996) into more appropriate conceptual structures for the new domain. This central tenet ofconstructivism accounts for the fact that errors are so persistent—it is difficult for teachers toconvince learners that their conceptions, which make perfect sense in terms of their ownknowledge, are in fact mistaken in relation to accepted mathematical knowledge (Sasman,Linchevski, Olivier, & Liebenberg, 1998). Constructivists do not argue that there is no rolefor teaching in restructuring misconceptions, but that the teacher’s role must be far morecomplex than merely explaining the correct ideas to learners, or even creating “cognitiveconflict” (Sasman et al., 1998). This paper unpacks some of the complexity of teaching inrelation to learner errors and how teachers learn practices that support them to acknowledgeboth what is valid in the learners’ thinking and what needs to be developed further.

The notion of misconceptions proves useful both in understanding teachers’ developingideas about learner errors and in helping them to develop their ideas further. However, it hassome limitations (Brodie & Berger, 2010), which a different theoretical perspective, socialpractice theory, helps to address. Social practice theory starts from the notion of practices,which are constituted in communities, rather than conceptual structures that are constructedin the mind.

Practices are patterned, coordinated regularities of action directed towards particulargoals, and develop knowledge, skills and technologies to achieve the goals (Scribner &Cole, 1981). There are two key elements in any practice: the criteria for what count asappropriate acts within that practice, and how the community that constitutes the practicedefines what counts in the practice and holds people to account to the criteria of the practice(Brodie, Slonimsky, & Shalem, 2010; Ford & Forman, 2006). Sfard (2007) refers to such

1 Although this is a single-authored paper, I draw on the collective work of members of the DIPIP projectteam. I use the collective “we” when I talk about ideas and decisions that have become part of our collectiveways of working in the project and the singular “I” when I make claims in relation to the specific argument inthis paper.

Learner errors in professional learning communities 223

Page 4: Learning about learner errors in professional learning communities

criteria as the rules of the discourse of mathematics. Lave and Wenger (1991) argue thatpractice is always social practice and involves social and power relations among people andinterests (Lave, 1993). In the case of teaching and learning mathematics, classroom practicescan support or deny access for learners to the practice of mathematics (Lave, 1996).

In a social practice theory of learning mathematics, labelling something an error invokesthe criteria of the mathematical practice by delineating what is not acceptable, and byimplication, what is acceptable as knowledge in the practice. So errors are boundary markersfor a practice2. In this way, errors illuminate what mechanisms need to be put in place to giveepistemic access to the practice and to develop the practice further. Errors point to thedemands of the practice and are a point of leverage for opening access to the practice. Thisexplains why errors are a key area of evaluation for mathematics teachers and also gives asocial explanation of why errors are so persistent and pervasive—the newcomer to thepractice is not yet a full participant in the discursive practices of the community and is mostlikely using different sets of criteria for what counts as appropriate (Sfard, 2007).

Social practice theory supports the notion that errors are a normal part of the learningprocess, for both old timers and newcomers to a practice. Even experienced mathematiciansmake errors and in so doing often create new knowledge in mathematics, thus shifting andrecreating the boundaries of the practice (Borasi, 1996). From a social practice perspective,conversations about errors are important for establishing the criteria, or norms, of thepractice and the community. These include the mathematical norms, that is, what countsas a valid or invalid production mathematically, as well as the social and socio-mathematicalnorms (Yackel & Cobb, 1996), that is, what counts as appropriate ways of talking aboutmathematics within a community. Conversations among learners and teacher can supportlearners to communicate and act in appropriate mathematical ways and conversations amongteachers can support them to learn to engage with their learners’ errors in appropriate ways.

Both theories, constructivisim and social practice theory, suggest three important princi-ples in relation to learner errors: errors are reasonable and show reasoning among learners;they are a normal and necessary part of learning mathematics; and learner errors giveteachers access to learners’ current thinking about and ways of doing mathematics andaccess to possibilities for future growth in their mathematical thinking and practices. Thispaper addresses how teachers learn to think and talk about learner errors in these ways3. Inwhat follows, I address this question through describing what teachers learned about learnererrors in the professional learning community and the practices in the community thatsupported their learning. In the next section I develop the notion of learning used in thispaper and how it relates to the practices of professional learning communities.

3 Teacher learning in professional learning communities

Since practices are goal-directed (Scribner & Cole, 1981; Wenger, 1998), learning must becentral to a practice because as social goods and goals shift, so do the means to achievethem. According to Wenger (1998), practice entails community, meaning and learning;practices learn and people learn in practice. According to Lave and Wenger (1991), learningin a practice is defined as increased and better participation in the practice, becoming more

2 Thanks to Lynne Slonimsky for this insight.3 A second key question is not addressed in this paper but is the subject of other papers from the project: Howdo teachers learn to engage with learners’ errors in the classroom to give learners access to mathematics(Brodie, 2011; Chauraya & Brodie, 2012).

224 K. Brodie

Page 5: Learning about learner errors in professional learning communities

adept and practiced at what counts as appropriate ways of being and acting in the practice.From this theoretical perspective, learning involves building knowledgeable, skilled identi-ties (Lave, 1996). Such learning takes time, and requires learners to be able to access aspectsof the practice in ways that allow for the development of regularities of thinking and action.

A review of the literature on teacher learning through professional development suggeststhat it is a complex process and is not yet well understood (Borko, 2004; Llinares & Krainer,2006). The difficulties of understanding teacher learning arise from the complexity ofteachers’ practices and the conditions that support teacher learning. Teachers’ engagementwith the practice of teaching is influenced by a range of factors: their own personalresources; the resources made available by their schools and departments; the organizationaland institutional constraints in their school situations; and systemic issues such as standard-ized testing and curriculum developments (Hargreaves, 2005). Borko (2004) argues that“meaningful learning is a slow and uncertain process for teachers, just as it is for students”(2004, p. 6). Zawojewski, Chamberlin, Hjalmarson and Lewis (2008, p. 219) argue that“research methodologies are needed for professional development experiences where par-ticipants are not expected to converge toward a particular standard, yet teachers grow andimprove as a result of participating in the professional development experience” (p. 219).

Leikin and Zazkis (2010) develop the concept of learning through teaching (LTT), whereteachers’ own teaching practices provide sources and supporting mechanisms for teachers’learning of new mathematics and new mathematics pedagogy. They present a model forthese sources and supports and show that LTT can happen when teachers interact withlearner thinking, through learner questions, unexpected correct answers, or learner errors.They also suggest that teachers need to be deeply reflective and have a profound under-standing of the subject matter in order to identify these moments and do the work required tolearn through practice. Tirosh, Even and Robinson (1998) show that experienced teachersunderstood and found ways to engage with the common learner error of conjoining algebraicexpressions (e.g. 2a+5b=7ab), while less experienced teachers noticed the error, but did notunderstand why it occurred nor how to engage with it. None of these authors addressed howteachers can learn to engage with learner thinking and hence learn through their teaching.

A number of teacher development projects attempt to develop the orientation andcapacity to learn through teaching (see for example Borko, 2004; Borko, Jacobs, Eiteljorg,& Pittman, 2008; Jaworski, 2006, 2008; Kazemi & Hubbard, 2008). These projects,including our own, tend to focus on teachers’ interactions with learner thinking in thecontext of teachers’ own practices, but slightly removed from the classroom in order toallow for teachers to support each other’s enquiry and reflections into their learners’ thinkingand their practices. In our case, we work to develop and sustain professional learningcommunities, where teachers learn to engage more deeply with learner errors.

The key characteristics of professional learning communities are that they provide a safeand challenging environment for teachers to engage in collective enquiry about learner needsand how to address these (Boudett, City, & Murnane, 2008; Katz et al., 2009; Stoll & Louis,2008). All four aspects are important: enquiry, collectivity, safety and challenge. Enquiry isimportant because the community must engage seriously with learners’ difficulties with aview to improving practice and learning. Genuine enquiry into learning and teaching issupported by a quest to understand what the classroom data tell the community about whatthey need to learn and do to address learner needs. Enquiry supports teachers to questiontheir taken-for-granted assumptions (Jaworski, 2006, 2008; Katz et al., 2009) and thinkabout ways to see recurrent problems in a new light—in our case, learner errors as potentialsources for mathematical growth, rather than as problems to be avoided by teachers. Enquiryprevents communities from merely recycling current practice and provides opportunities to

Learner errors in professional learning communities 225

Page 6: Learning about learner errors in professional learning communities

interrogate and re-invigorate practice (Hargreaves, 2008; Jaworski, 2006, 2008; McLaughlin& Talbert, 2008).

Collectivity is important because teachers can support each other in coming to understandwhat the data show about their teaching and their learners’ learning and because collectivelygenerated shifts in practice are likely to be more sustainable and can provide learners withmore coherent experiences than changes in individual teachers’ practices (Horn, 2005;McLaughlin & Talbert, 2008). Safety is important because in order to learn, teachers haveto admit to weaknesses in their practice and knowledge, and this can only be done honestlyin a safe environment. Challenge is important because if we always feel safe, we cannotlearn. So teachers have to be challenged to move outside of their comfort zones to create newways of thinking about and interacting with their learners. Brodie and Shalem (2011) showhow safety and challenge work together in a professional learning community, to produce anexperience of solidarity among members, which in turn supports feelings of safety and awillingness to challenge. A key element in any professional learning community is theleadership of the community, which has two main functions: first, to establish a safe andchallenging environment for collective enquiry; and second, to ensure that the communityhas the appropriate resources for learning. A community that cannot draw on outsideresources and expertise can become solipsistic and stagnant (Jackson & Temperley, 2008;Jaworski, 2006, 2008; Katz et al., 2009).

In the DIPIP project we added to the above by explicitly positioning teachers as bothexperts and learners—experts in their classrooms and current teaching practices, but withsignificant possibilities for learning about engaging with learner errors and improvingpractice. We chose a focus—an engagement with learner errors—that we thought teachershad engaged with differently in the past and we found ways to position teachers as bothknowing and not knowing about learner errors. Teachers clearly know a lot about learnererrors: they are faced with them every day and have practised ways of dealing with them,often through correcting or avoiding them. However, they may not yet look for what is validin learner errors and therefore may not see where learners need further support.

An important question relates to the role and responsibility of teachers in producingerrors. Errors are seldom taught directly by teachers and yet all learners, even high achievinglearners, make errors. A related question is what supports some learners to transform currenterrors into appropriate mathematical thinking and whether teachers can amplify this supportby engaging with the reasoning in learner errors. Teachers may sometimes exacerbate errorsthrough taken-for-granted use of language and concepts, and, at another level, through notbringing errors and the reasons why they are errors into the public domain. At yet anotherlevel of complexity, a deeper understanding of errors suggests that teachers cannot deal witherrors quickly or easily because transforming errors requires a shift in the criteria of whatcounts as mathematics. Errors arise in the interactions between the features of mathematics,of learning, and of social practice. So a focus on errors as a learning tool allows teachers todevelop nuanced understandings of the nature of mathematics, of teaching and learning andthe relationships between them. Our goal was for teachers to come to see learner errors innew ways and to challenge their taken-for-granted and often automatised ways of dealingwith them.

The literature on teachers’ work with learner errors suggests a number of analyticprocesses that teachers need to engage in as they work with learner errors (different termsare used by different authors, I have synthesised them here). First, teachers need to identify(Peng & Luo, 2009), attend to (Jacobs et al., 2010) or show an interest in the error (Prediger,2010). Next, teachers need to interpret or evaluate the error, and finally they need to decidehow to engage the error (Peng & Luo, 2009; Prediger, 2010). All of these processes will

226 K. Brodie

Page 7: Learning about learner errors in professional learning communities

occur simultaneously in practice, but a breakdown in working with the error will most likelybe attributed to a breakdown in one or more of these processes. In the DIPIP project, wework with teachers to identify and notice where they have used these processes successfullyand where they have broken down and the reasons for the breakdown.

The research also argues that teachers need to employ subject matter knowledge andpedagogical content knowledge in order to interpret learners’ thinking and their errors, andestablish the reasoning behind the errors (Jacobs et al., 2010; Leikin & Zazkis, 2010; Peng &Luo, 2009). Jacobs et al. (2010) argue that knowledge of particular learners over time and ofthe research on learners’ mathematical development is required, while Prediger (2010)argues that, in addition, general knowledge about learning and the role of errors in learningis important. In this paper I show that conversely, a focus on learner errors can lead to thefurther development of teachers’ mathematical and pedagogical content knowledge.

4 The empirical site: the DIPIP project

This paper reports on Phase Two4 of the DIPIP project where teachers from different schoolscame together at our university to work on understanding and engaging learner errors.Teachers came from a diverse range of government schools including township schools,many of which are under-resourced with poor learner achievement, and suburban schools,most of which are well resourced and achieve good results. Teachers worked in small grade-level groups (three to four teachers and a group leader) over a period of three years toanalyse common learner errors in particular mathematical concepts and to design lessons toengage these errors. Some teachers in the group volunteered to teach the lessons and werevideotaped. The small groups then reflected on the videotaped lessons and chose twoepisodes, one where they thought the teacher had done well in engaging with learner errorsand one where they thought the teacher had not done so well. They presented these twoepisodes for further discussion and reflection to a larger group, made up of the smallergroups. This paper reports on the conversations in the larger group of grade 7–9 teachers,made up of two small groups in each of the three grades: 7, 8 and 95. Most of the videoanalysis work took place in 2009, and this paper focuses on conversations in the large groupduring 2009, the second year of the project.

The group leaders of the small groups were experienced mathematics teachers, many ofwhom have done extensive professional development work. The project coordinating teamconsisted of four people: myself and another project leader, and two of the small groupleaders who managed the project6. This team met regularly to discuss the on-going academicand administrative progress of the project, including the training of the small group leaders. Itrained the small group leaders to keep channels of communication open; build trust amongthe group; keep the group on track; help the group get to the heart of the mathematics in theactivity; and support patterns of constructive critique and rigorous inquiry by acknowledgingand working with the strengths of teachers’ ideas (as experts) and at the same time pushingteachers to move beyond their taken-for-granted assumptions into uncharted territory (aslearners). I was the facilitator of the larger group and I worked to do all of the above, with the

4 Phases One (2008) and Two (2009–2010) involved the same teachers. The project is currently in PhaseThree (2011–2013) where we work with a different group of teachers in communities located in schools.5 See Brodie and Shalem (2011) for more details on this project design.6 There are a number of publications across all phases of the project, some of which are referenced in thispaper. The project team also works with the teachers in the professional learning communities to present theirwork at teacher conferences.

Learner errors in professional learning communities 227

Page 8: Learning about learner errors in professional learning communities

additional challenges of supporting each small group to make its thinking clear to the othersand to hear and take account of feedback and challenges from the other groups7.

At the beginning of each large group session, I reminded the group that all contributionsshould be in the spirit of constructive critique, which included challenges and positivecomments, always with justifications and always with the aim of taking our learningforward. Then, the teacher presented the two episodes, contextualizing them in the broaderset of lessons and justifying why each one was chosen as an example of good or not so goodengagement with learner errors. Once the presentations were finished, I asked for commentsor questions from the floor. For the most part, I allocated speaking turns, although sometimesthe passion of the discussion made my chairing role difficult. I chose teachers in preferenceto group leaders to speak, and other group leaders in preference to the presenting group’sleader. I also made decisions as to when to allocate my own turns, when I felt it wasimportant to bring in my own expertise. Questions and comments were usually directed tothe presenter, but we established the practice that any group member could respond.

The main data for this paper come from videotapes of eight large group sessions, where eachof two Grade 8 and 9 groups presented episodes of teaching to the group of grade 7, 8 and 9teachers8. The eight sessions ranged from about 40 to 70minutes long, with an average of about50 minutes. I analysed the conversations in these eight sessions by: writing detailed summariesof each session; looking for how the teachers spoke about learning and teaching to each otherandwhat counted for them as important to speak about at each point; noting all the learner errorsthat were identified, how the teachers spoke about them and the extent to which the teacherstried to understand the learners’ thinking; and noting all explanations that the teachers gave forthe learners’ errors, the kinds of language that they used to speak about learner errors and theteaching strategies they suggested to deal with learner errors. In addition, in order to make surethat I understood the context of each group discussion, I looked at the lesson plans for each setof lessons presented, watched the full set of lessons on videotape and looked at the teachers’written reflections on their lessons. In some cases, the additional data provided confirmations ofmy emerging hypotheses about what the teachers might be learning and in some cases they ledme to rethink my ideas or develop them further. The analysis of the lesson videotapes confirmedthat the groups did choose important episodes to present to the group in relation to the teachers’engagement with learner errors.

Using the method of constant comparison (Strauss & Corbin, 1998), I identified threeareas where teachers’ learning is visible in the group’s conversations: identifying andexplaining the reasoning behind learner errors; dealing with learner errors; and identifyinglimitations in teachers’ own mathematical thinking. I identified insights expressed by theteachers as indications of learning. In some cases, teachers spoke explicitly about their ownlearning in the project, and in other cases, what they learned is more implicit but still visiblethrough the analysis. Each of the above areas suggested a significant shift made in theconversations, and I will discuss the data in the following sections in terms of the three shiftsmade by the teachers. As social practice theory predicts, the shifts occurred and re-occurred,suggesting that they do not of themselves form a developmental trajectory and that they needrepeated practice. The shifts occurred across the year in different ways in different conver-sations, showing, as Zawojewski et al. (2008) suggest, that where teacher growth can beidentified, it does not occur evenly or in particular trajectories.

7 A detailed analysis of facilitation is beyond the scope of this paper but will be analysed in other papers of theproject.8 There were usually about 30 member of this larger group, both teachers and group leaders from the smallergroups.

228 K. Brodie

Page 9: Learning about learner errors in professional learning communities

A limitation of focusing only on the conversations is that it does not capture the growth inteaching practices through the project. This growth has been discussed elsewhere (Brodie, 2011;Chauraya & Brodie, 2012), and we are continuing to do this analysis. In the end, what teacherslearn has to include a combination of how they talk and what they do, and this adds to thecomplexities of capturing teachers’ learning. My purpose in this paper is to tell a part of the story,the part where teachers come to think and talk to each other in new ways about learner errors.

5 From identifying to interpreting errors

In the teaching episodes, a number of learner errors were identified and for the most part, thesmall groups chose interesting errors to present to the large group. A pattern of teacherresponses to the episodes that became apparent in the first few sessions was to suggest to thepresenting teacher that s/he could have clarified or simplified the language of the task or thetask itself (Stein, Grover, & Henningsen, 1996) and thus avoided the learner error. The goalof these responses seemed to be that the teacher should try to elicit the correct answer fromthe learner, and should shift her/his moves in order to do so. Any ambiguity in the task wasseen as problematic, even if it did help the teacher to get at underlying concepts.

When teachers suggested avoiding the error, I would regularly ask a number of questions such as:what could the learner be thinking in order to make the error; how can we see the error from thelearner’s perspective; and howmight the error make sense to the learner, even if not to the teachers.In this section I present two of many examples where such questions focused the teachers oninterpreting and understanding the learner’s error. In the first example, the teacher did not understandthe learner’s error and so had ignored it in class, and in the second, the teacher had recognised theerror in class but had not interpreted it in a way that supported him to engage with the error.

In Tebogo’s9 Grade 9 lesson, he called on a learner to label the axes of the first quadrantof the Cartesian plane and the learner wrote the following:

The discussion in the professional learning community began as follows:

Kathy: That little boy who put the minus signs (indicates the y axis with her hands),what was that about, how do we understand that, how do we explain that, you said youwere completely surprised, you’ve never seen it before

9 All the teachers’ and the facilitator’s names are pseudonyms. Kathy is the facilitator. The discussion of thislesson took place in August 2009.

Learner errors in professional learning communities 229

Page 10: Learning about learner errors in professional learning communities

Tebogo: Actually I was amazed by it

Linda: But you know what, the same thing happened in Sarah’s lesson, a kid came upand did exactly the same thing, he labelled the positive part of the y-axis withnegatives, all the way down

Kathy: Okay, so now we need to take it seriously, what’s going on … how do we,cause remember our aim here is to understand why it is that our learners make thesemistakes, particularly if more than one of them make it, what’s going on

Both the teacher and the facilitator acknowledged surprise at the error and did not knowhow to understand it. The fact that the error came up more than once indicated that it mightbe pervasive rather than idiosyncratic and was therefore worth thinking seriously about. Twograde 7 teachers, Nadine and Tarryn, made suggestions as to what the learner was thinking.

Nadine: I think that the kids are understanding that the number line has no starting andend point and they know that there are positive and negative numbers, so when it goesfrom nought to twenty or whatever, then they know that after zero comes negativenumbers, minus one, minus two and so on, so they going minus one, minus two, minusthree but regardless that its supposed to be a straight line like this (Tarryn nods), its justthe lines, these are the positive ones, (indicates horizontally) and these are the negativeones, (indicates vertically), so its just, it’s a line

Kathy: Ohhh, so that one becomes flipped over that way (indicating with her hands)

Nadine: Ja (yes)

Tarryn: I’m nodding my head because I had a grade seven who did that last year andwhen I asked him about it, he said, mam you told me zero must be in the middlebetween the positive and the negative, and he had positive numbers (indicateshorizontally), a zero, and then he had negative numbers (indicates vertically), so theway that I have explained it about zero being in the middle between the positive andthe negative, I obviously didn’t make it clear enough that it must be on one straight,horizontal, one line, as far as he’s concerned, it is still a line, its just got a bend in it.

Nadine and Tarryn each present a case for why the learner’s error made sense to him. Nadineexplained what the learner did understand about number lines, that there are no end points, thatthere are positive and negative numbers and that the negative numbers precede the positivenumbers and zero. She also explained what the learner might not understand, that the numberline had to be a straight line, from his perspective a bend in the line was acceptable. Tarrynconfirmed that a learner had provided a similar explanation to her. The assumption, stated byTarryn, and shared among the three participants is that such an explanation is sensible, althoughincorrect, and we could see the sense in the learner’s thinking.

A second example comes from Tawana’s Grade 8 lesson10 on equations and the equalsign. A learner had written:

p − 28 ¼ 4þ 1¼ p−28 ¼ 5¼ p ¼ 33

10 The discussion on this lesson took place in March 2009.

230 K. Brodie

Page 11: Learning about learner errors in professional learning communities

The equal signs at the beginning of the second and third lines suggest that the learner usesthe equal sign to indicate “and the next step is”, rather than equivalence. From the readingsin the project, Tawana recognized this as an indicator of operational thinking with the equalsign. The learner’s operational thinking led him to write an incorrect mathematical expres-sion in the last line, although from his perspective he presented the correct answer. Inpresenting the episode to the larger group, Tawana explained why he did not engage with thelearner’s error in class, even though he understood it:

So we were now solving those equations, and I moved around in the groups, and realisedthat some of the children were, were repeating a misconception that I had dealt with …even some of the best students were doing that, and one of the students actually justifiedwhy he was doing that…. I actually found it annoying… you can hear from my tone ofvoice, that I was now annoyed because I explained it to great depth

Tawana shows that he had identified the error in class; in fact, he had explicitly taught learnersto avoid the error. However, precisely because he had taught them, he became annoyed that theydid not seem to understand. On reflection in his small group, he had come to realize that in factthe first two lines that the learner wrote were correct mathematically, and he could have validatedthat part of the learner’s work, which may have helped the learner, and other learners, to see whythe third line was incorrect.What he did not yet see is that certain practices are established amonglearners, and these are difficult to challenge, which is why errors are so persistent11.

These two examples illustrate how the professional learning community supported shifts in theteachers’ learning about learner errors from identifying the errors and possibly avoiding them, tointerpreting the errors. First, we see systematic enquiry into the learners’ thinking from the learners’perspectives. This enquiry is framed and guided by the facilitator’s questions, which keep thecommunity focused on the object of enquiry—the reasons for learners’ errors. In both examples, theteachers in the community supported each other to interpret both what was valid and not valid in thelearner’s thinking. Second, we see members of the professional learning community, both theteachers and the facilitator, admit to lack of knowledge, where they did not understand the error, as inTebogo’s case, or where, in Tawana’s case, he partially understood the error but did not know how todeal with it and became impatient and annoyed. Third, we see knowledge of learners’ errors beingbuilt collectively. In both cases, at least two or three community members contributed to interpretingthe learners’ errors. It would not have been possible to develop these interpretations without thecommunity’s support. These three features of the community’s practices supported the teachers andthe facilitator to interrogate taken-for-granted assumptions about why learners make errors and totake the perspectives of others, other teachers in the community and the learners. A key assumptionthat was challenged in these conversations was that an error is always entirelymistaken. In the errorspresented to the community, members of the community came to see that there is often both validand invalidmathematics in an error and that there are differences between the error and the reasoningbehind the error. In this way, teachers could begin to articulate the inner logic of learners’ errors.

6 From interpreting to engaging

As the teachers interpreted the errors, they began to re-think what they might do about them.A common suggestion was to re-teach the concepts because the learners had not understoodthem fully. In the first example above, Tarryn argued:

11 In a subsequent episode, Tawana showed that he had learned from this conversation and listened morepatiently to a learners’ contribution, trying to identify what was and was not valid in the learners’ thinking.

Learner errors in professional learning communities 231

Page 12: Learning about learner errors in professional learning communities

I obviously didn’t make it clear enough that it must be on one straight, horizontal, oneline, as far as he’s concerned, it is still a line, its just got a bend in it.

This was followed by Brenda’s comment:

The challenge that we’ve got in front of us is how do we remedy the situation, becauseby just telling the learner this is wrong and this is right, when the learner draws thenext Cartesian plane, that problem might pop up again and to remedy this problemyou’ll need to start from the beginning and tell them and move on, a quick fix won’twork here and that is a challenge for us.

Brenda recognized that merely correcting an error might not deter learners from theirlong-held concepts and that a “quick fix won’t work”. Her solution is to re-teach the concept“from the beginning”, the implication being to make sure that the ideas are explained“properly” the second time around. Jacob argued similarly:

Besides that I think again that to the learner, when we introduced the concept of theCartesian plane, it was not well explained to them about the quadrants, whichquadrant, which axis is positive and which axis is in which quadrant.

In response to these comments, one of the subject advisors, Erna, noted:

… if you visit a school and you find that that is what’s been happening, it’s the thirdterm and you’re still on LO one12, you ask the teacher why, then they say because thechildren don’t understand so I have to go back and explain that’s exactly whathappens, you can go back in the fourth term, they’ll still be on LO one

Here, she reflects on the situation in many South African schools where teachers, in trying tobe responsive to what learners do not know, re-teach certain topics that are necessary to movefurther. However, they spend so much time re-teaching concepts, often concepts that they feelmore comfortable with, that they do not makemuch progress with the curriculum. A key aspect ofpedagogical content knowledge that might be missing for the teachers here is that sometimes, asyou progress with the curriculum, prior concepts become clearer in the light of new knowledge.Instead of working with a dynamic conception of how different concepts influence each other, theteachers work with a somewhat linear concept, that each piece needs to be in place beforesubsequent knowledge can be built. The notion of errors that we try to build in the project is that itis not necessarily learners’ prior knowledge, or teachers’ prior teaching that is to “blame”, but thaterrors are a normal part of learningmathematics and can and should be addressed as they come upin class. In relation to these understandings, the facilitator responded as follows:

Okay, I’m not sure that you have to start from the beginning because remember… justbecause there’s a misconception doesn’t mean that the learner hasn’t understood whatthey’ve learned before, remember misconceptions are taking what they’ve learnedbefore and applying it in a way that makes it wrong, okay like they say multiplicationmakes things bigger but suddenly you multiply by fractions … you don’t have to goback and explain to them, you have to connect with the misconception at the point thatthey’re making the misconception … I think its much more important to engage thelearner on the misconception, maybe engage the whole class and try to understandwhere it comes from and talk about it, then you’ve got much more chance ofremedying it … you can’t remedy it unless you understand why its happening, that’s

12 Learning outcome 1 in our curriculum—number and algebra. There are five learning outcomes, but manyteachers focus on learning outcome 1 at the expense of the others.

232 K. Brodie

Page 13: Learning about learner errors in professional learning communities

what you have to try to understand, that’s the challenge, that is the biggest challengefor every teacher.

My intervention at this point served to validate and support the teachers’ expressed views that“quick fixes” don’t work, and that errors do need to be “remedied” in some way. I challenged theprevailing idea that the best way to remedy was to re-teach, and the explanation underlying thisidea that errors arise from incorrect prior teaching or learning. I used the opportunity to arguefurther that errors and misconceptions arise in the interconnections of different ideas, andtherefore, teachers’ best hope of dealing with them is to understand and deal with theseconnections, as we had done with this example, rather than to re-teach concepts.

The teachers’ response to the second example above was slightly different. Here, thegroup did not think about re-teaching: rather they were annoyed because the teacher hadtaught the ideas and the learner did not seem to remember them. Tawana said:

I actually found it annoying because I had actually worked out a question, you can hearfrom my tone of voice, that I was now annoyed because I explained it to great depth …

In response to Tawana’s comment, another teacher said quietly “we used to it”,suggesting that Tawana’s is a shared experience. Renee (in Tawana’s group) further sup-ported Tawana’s argument by saying:

In the very first lesson … Tawana had gone through something very similar and hademphasized and asked, is it correct for me to say that this step is equal to the step aboveand he went through it and he showed them very carefully why part of it was correct andwhy part of it was incorrect. And that was in the first lesson. Now we are five lessons onand then this happens and I see why he’s saying he was just annoyed …

The facilitator then asked why it might be that many learners, who had been taught somethingwell, would still struggle to understand it. Nadine and Andrea both made similar contributions:

Nadine: The kids are so used to writing, they get taught that every step they must write,when they in the younger grades, equals and then whatever and then the next step equals

Andrea: I find that because in junior school they’ve been told that on pain of deaththey don’t put equal signs on every line, they now put equal signs on every lineregardless of whether the lines are actually equal or not

Nadine and Andrea both made the point that in primary school, the operational meaningof the equal sign tends to dominate, and this is what learners are probably working with.Their explanation relates to practice: this is what learners are used to doing, it makes sense tothem, it has been previously evaluated as correct and it has become part of their functioningin mathematics classrooms. Earlier Andrea had argued:

Themisconceptions are so deeply ingrained in them that actually they take them for grantedand they don’t think about, and it doesn’t even cross their minds to change their thinking.

And here she argues:

I mean some of the stuff that I said, its pretty deeply ingrained in me as well. If I listento some of the language that I use, I use things like the answer …

These kinds of explanation, together with understanding the mathematical reasoningbehind learners’ errors as discussed in section 1, show the deepening of systematic enquiryin the community and consequently the beginning of a powerful shift in the teachers’

Learner errors in professional learning communities 233

Page 14: Learning about learner errors in professional learning communities

thinking. Having interpreted the learner errors, they now talk about teaching practices thatmight explain the widespread and systematic errors that we see, and why they are so resistantto change. Andrea coined the word “ingrained” which we continued to use in the communityto describe what was happening when learners struggled to shift their thinking away fromtheir errors. The teachers’ more powerful understandings supported them to shift fromtalking about their current practice of re-teaching to thinking about engaging with thelearners’ errors in new ways, a real engagement with what the learner might be thinking,rather than a simplification of the task or language. As Andrea said:

I said to them, the equal sign on the left doesn’t mean that there’s nothing on the left, itmeans that we carrying down the first line on the left but we just push it to the left a littlebit, so there is something on the left, its there, just be aware that its there, like a ghost

Here, she reconciles for herself how the equal sign at the beginning of the line (whenworking with expressions) can have a relational rather than an operational meaning and shefinds a way to explain it to learners.

In the episodes presented above, we again see the characteristics of professional learningcommunities supporting the teachers to shift their thinking. The systematic enquiry de-scribed above was supported by challenges to assumptions about errors and how they arebest engaged with, and the focus on errors supported new ways of thinking about learning,and the interactions between learning and teaching. The challenges, which were supportedcollectively by the community, helped to bring to the surface a number of taken-for-grantedassumptions and practices and helped the teachers to shift their thinking from identifyingand interpreting learner errors to engaging with them.

7 From learners’ knowledge to teachers’ knowledge

Interpreting learner errors also supported the teachers to re-think their own knowledge. Thefollowing example13 shows how investigating what seemed to be a learner error turned intoinvestigating teacher errors and teacher knowledge.

The task was for learners (Grade 9) to write an equation with variables in it. Each groupof learners wrote their equation on the board, and the teacher discussed them with the class.One group wrote: 4x×5x=20x, and the teacher asked the class: what is wrong with thisequation. The following discussion ensued among the teachers:

Shoriwa: The one which is four x times five x is twenty x, is it wrong?

Chomane: Is it wrong?

Shoriwa: It’s correct, that one

Sebolai: It is wrong

(many comments)

A number of the teachers agreed with Sebolai (who was the teacher in the episode), thatthe equation is incorrect, while a few agreed with Shoriwa and Chomane that it is correct. Inthe next extract, Shoriwa and Lorraine explain how the statement is correct, when it is seenas a quadratic equation rather than an identity.

13 This was discussed in the community in September 2009.

234 K. Brodie

Page 15: Learning about learner errors in professional learning communities

Shoriwa: Right let me come again, the people now what you are looking now, you aregiving a limited view of that, they are looking at it as an identity, but as an equation, itscorrect

Lorraine: It’s a quadratic equation

This explanation convinced some teachers but not others. For example, Eunice argued:

I just wanted to comment on that four x times five x, I think it was to be correctedimmediately, I’m supporting Linda that now it should be twenty x squared, they mustknow …

While Chomane reflected:

That one was very interesting, in that when I saw it first, I was also rigid and I onlythought of exponentials and forgot that that can be a quadratic, which can be solved

This example brings up a subtle distinction within mathematics, the difference betweenan equation and an identity, which pushed a number of teachers to the limits of theirmathematical knowledge. Until Shoriwa pointed out the difference, many teachers agreedwith Sebolai that the statement was incorrect, indicating that they did not usually make thisdistinction. Once pointed out, some teachers still thought that it was incorrect, althoughothers could see their error. Chomane suggested that many teachers (as well as learners) maycompartmentalise their knowledge—had they seen the equation in a section on quadraticequations, it is highly likely that they would have solved it as a quadratic, but in this case, thelaws of exponents seemed to over-determine their responses. Linda explained this in relationto her teaching:

Just say that five x times four x equals twenty x, I understand the, what if you weresolving for x to make this true, in grade nine we’re not at that level, we’re doingexponents, so if you are saying that is correct you are throwing out your exponentialrules, you are just creating one huge misconception, they’re not at that level.

Linda was one of the teachers who initially agreed with Sebolai. She was convinced byShoriwa and Lorraine that in fact there are two ways to see the statement; however, she alsoargued that in the context of a Grade 9 classroom, it was important to focus on one aspect,the identity and the laws of exponents. She was not yet ready or willing to admit that learnersmight benefit from her new insight. Chomane was able to take his learning one step further:

I wanted to say that that was a good example, that the teacher could have takenadvantage of the misconceptions, extending it to say that equations can have more thanone solution and again put an example of that identity, where this one is a specialequation with many solutions

He argued that the difficult mathematics presented an opportunity to teachers, once theyhad realised their own error, to work in more mathematically sound ways with learners andto bring two usually disparate elements of the curriculum into connection with each other.

In this example, we again see the professional learning community supporting teachers’learning, this time about their own mathematical knowledge and their pedagogical contentknowledge. We see teachers feeling safe enough to challenge each other and themselves inconstructive ways, risking some professional conflict (Katz et al., 2009) in order to deepentheir understanding of their own knowledge. Through enquiry into learners’ knowledge,teachers begin to enquire into their own knowledge, and we see their enquiry shiftingbetween their own mathematical knowledge and their pedagogical content knowledge. In

Learner errors in professional learning communities 235

Page 16: Learning about learner errors in professional learning communities

this session, a number of teachers shifted their mathematical positions and deepened theirunderstanding of mathematics, and we see at least one teacher deepening his pedagogicalunderstanding based on his new insight into the mathematics. In this case, we see theboundary of acceptable mathematics expanded by discussion of the “error”; teachers cameto see familiar mathematics in new ways. So the what seemed to be a learner error served asa boundary marker for two sets of practices, teaching mathematics to learners and theteachers’ own learning practices in the community.

8 Conclusions

Social practice theory argues that communities of practice contain the seeds for both growthand resistance to growth (Lave, 1993). The regularities of practice support the continuedexistence of both generative and non-generative practices. Moves to change non-generativepractices require re-organisation of the practices into new, more powerful practices with newregularities of action. Such re-organisation of practice requires learning among practitioners,and this learning is itself a practice, which sets up its own regularities. In this paper, I haveargued that a focus on learner errors can be a powerful mechanism for teacher learningprecisely because working with learner errors is a regularity of all mathematics classroomsand at the same time can be shifted into more generative practices for teachers and learnersthrough teachers coming to see errors as boundary markers, giving access to the practice forboth themselves and their learners. Teachers can come to see errors as reasonable, aspartially valid mathematically, as giving access to learners’ thinking, and as points ofdeparture for new mathematics learning, for themselves and their learners.

The literature on teachers working with learner errors suggests three important aspects ofthis work: identifying, interpreting and engaging with learner errors. In this paper, I haveshown how teachers can and do shift from identifying errors to interpreting them, and frominterpreting them to engaging with them. These shifts do not necessarily occur in chrono-logical order because the nature of shifting practices is that they need to be revisited, in orderto take hold. I have argued that making these shifts requires teachers to question somefundamental assumptions about teaching and learning mathematics and to see recurrentlearning and teaching occurrences in new ways, in this case learner errors as potentialsources for growth rather than problems to be avoided. I have also shown that a focus onlearner errors can sustain teachers’ enquiry into their assumptions about their own mathe-matical knowledge and their pedagogical content knowledge.

I have argued that a professional learning community can be a mechanism for supportingteachers to challenge fundamental assumptions about how they work with learner errors and togrow towards interpreting and engaging with learner errors and their own knowledge. The keyfeatures of professional learning communities—enquiry, collectivity, safety and challenge—supported the teachers’ growth.We saw enquiry into the valid reasoning behind learners’ errorsand how teachers began to understand the role of their own practices and knowledge in relationto learner errors. We saw teachers collectively support each other to make sense of learnererrors, their current teaching practices and knowledge, and to think about possible new practicesand new ways of thinking about mathematics. We saw teachers safe enough to admit toweaknesses in their own practices and knowledge: in getting annoyed and losing patience, innot always understanding learner errors in class and in acknowledging the limits of their ownknowledge. We also saw teachers safe enough to challenge and support each other in enquiringinto their strengths and weaknesses. Through this support and challenge, the teachers were ableto distance themselves from their current ideas and take new perspectives.

236 K. Brodie

Page 17: Learning about learner errors in professional learning communities

The facilitator was central in creating possibilities for enquiry, collectivity, safety andchallenge in the community. In addition to chairing discussions and creating an atmospherewhere teachers could support and challenge each other—difficult tasks in themselves—thefacilitator is key in bringing in expertise about teaching and learning mathematics, andlearning more generally. In this case, the facilitator introduced more general ideas abouterrors and the sources of errors, and could respond to the teachers’ emerging ideas abouttheir knowledge and practices. The facilitator could recognise that teachers wanted to avoidand correct errors, or re-teach content, and worked with them to move beyond these teachingactions to clarify or simplify the task or re-teach the section, towards teaching actions basedon a deeper understanding of learner thinking.

Research on learning through teaching mathematics (LTT) suggests that teachers can anddo learn much from interactions with learner thinking in class, if they are reflective enoughto notice and work with important interactions in the classroom and in their planning. In thispaper, I have argued that professional learning communities can help teachers to developthese reflective qualities, which might support their learning through teaching.

References

Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D.E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44).Reston: National Council of Teachers of Mathematics.

Ben-Zeev, T. (1996). When erroneous mathematical thinking is just as correct: The oxymoron of rationalerrors. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 55–79).Mahwah, NJ: Lawrence Erlbaum Associates.

Borasi, R. (1986). Algebraic explorations of the error 16/64=1/4. Mathematics Teacher, 79(4), 246–248.Borasi, R. (1987). Exploring mathematics through the analysis of errors. For the Learning of Mathematics,

7(3), 2–8.Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Norwood: Ablex.Borko, H. (2004). Professional development and teacher learning: Mapping the terrain. Educational

Resarcher, 33(8), 3–15.Borko, H., Jacobs, J., Eiteljorg, E., & Pittman, M. (2008). Video as a tool for fostering productive discussions

in mathematics professional development. Teaching and Teacher Education, 24, 417–436.Boudett, K. P., City, E. A., & Murnane, R. J. (2008). Data-Wise: A step-by-step guide to using assessment

results to improve teaching and learning. Cambridge: Harvard Education Press.Brodie, K. (2011). Professional learning communities and teacher change. In L. Paditz & A. Rogerson (Eds.),

Proceedings of the 11th international conference of The Mathematics Education into the 21st CenturyProject: On turning dreams into reality. Transformations and paradigm shifts in mathematics education.Grahamstown. South Africa.

Brodie, K., & Berger, M. (2010). Toward a discursive framework for learner errors in mathematics. In V.Mudaly (Ed.), Proceedings of the Eighteenth Annual Meeting of the Southern African Association forResearch in Mathematics, Science and Technology Education (SAARMSTE) (pp. 169–181). Durban:University of KwaZulu-Natal.

Brodie, K., & Shalem, Y. (2011). Accountability conversations: Mathematics teachers learning throughchallenge and solidarity. Journal for Mathematics Teacher Education, 14, 419–439.

Brodie, K., Slonimsky, L., & Shalem, Y. (2010). Called to account: Criteria in mathematics teacher education.In U. Gellert, E. Jablonka, & C. Morgan (Eds.), Proceedings of Mathematics Education and Society 6(MES6) (pp. 180–189). Berlin: Freie Universitat Berlin.

Chauraya, M., & Brodie, K. (2012). Mathematics teachers change through working in a professional learningcommunity. In D. Nampota & M. Kazima (Eds.), Proceedings of the Twentieth Annual Meeting of theSouthern African Association for Research in Science, Mathematics and Technology Education(SAARMSTE) (Vol. 1, pp. 19–31). Lilongwe: University of Malawi.

Confrey, J. (1990). A review of research on student conceptions in mathematics, science and programming. InC. B. Cazden (Ed.), Review of research in education (Vol. 16, pp. 3–56). Washington: AmericanEducational Research Association.

Learner errors in professional learning communities 237

Page 18: Learning about learner errors in professional learning communities

Erlwanger, S. H. (1973). Benny’s conceptions of rules and answers in IPI mathematics. Journal of Children’sMathematical Behaviour, 1(2), 7–26.

Ford, M. J., & Forman, E. A. (2006). Redefining disciplinary learning in classroom contexts. Review ofResearch in Education, 30, 1–32.

Gagatsis, A., & Kyriakides, L. (2000). Teachers’ attitudes towards their pupils’ mathematical errors. Educa-tional Research and Evaluation, 6(1), 24–58.

Hargreaves, A. (2005). Educational change takes ages: Life, career and generational factors in teachers’emotional responses to educational change. Teaching and Teacher Education, 21, 967–983.

Hargreaves, A. (2008). Sustainable professional learning communities. In L. Stoll & K. S. Louis (Eds.),Professional learning communities: Divergence, depth and dilemmas (pp. 181–195). Maidenhead: OpenUniversity Press and McGraw Hill Education.

Hatano, G. (1996). A conception of knowledge acquisition and its implications for mathematics education. InP. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 197–217). New Jersey: Lawrence Erlbaum.

Horn, I. S. (2005). Learning on the job: A situated account of teacher learning in high school mathematicsdepartments. Cognition and Instruction, 23(2), 207–236.

Jackson, D., & Temperley, J. (2008). From professional learning community to networked learning commu-nity. In L. Stoll & K. S. Louis (Eds.), Professional learning communities: Divergence, depth anddilemmas (pp. 45–62). Maidenhead: Open University Press and McGraw Hill Education.

Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematicalthinking. Journal for Research in Mathematics Education, 41(2), 169–202.

Jaworski, B. (2006). Theory and practice in mathematics teaching and development: Critical inquiry as amode of learning in teaching. Journal of Mathematics Teacher Education, 9, 187–211.

Jaworski, B. (2008). Building and sustaining enquiry communities in mathematics teaching development:Teachers and didacticians in collaboration. In K. Krainer & T. Wood (Eds.), Participants in mathematicsteacher education: Individuals, teams, communities and networks (pp. 309–330). Rotterdam: SensePublishers.

Katz, S., Earl, L., & Ben Jaafar, S. (2009). Building and connecting learning communities: The power ofnetworks for school improvement. Thousand Oaks: Corwin.

Kazemi, E., & Hubbard, A. (2008). New directions for the design and study of professional development.Journal of Teacher Education, 59(5), 428–441.

Krainer, K., & Wood, T. (Eds.). (2008). The international handbook of mathematics teacher education (Vol. 3:Participants in mathematics teacher education). Rotterdam: Sense Publishers.

Lave, J. (1993). Situating learning in communities of practice. In L. B. Resnick, J. M. Levine, & S. D. Teasley(Eds.), Perspectives on socially shared cognition (pp. 63–85). Washington, DC: American PsychologicalAssociation.

Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture and Activity, 3(3), 149–164.Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge

University Press.Leikin, R., & Zazkis, R. (2010). Teachers’ opportunities to learn mathematics through teaching. In R. Leikin

& R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge andexpertise. Dordrecht: Springer.

Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teacher educators as learners. In A.Guitierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp.429–459). Rotterdam: Sense Publishers.

McLaughlin, M. W., & Talbert, J. E. (2001). Professional communities and the work of high school teaching.Chicago: University of Chicago Press.

McLaughlin, M. W., & Talbert, J. E. (2008). Building professional communities in high schools: Challengesand promising practices. In L. Stoll & K. S. Louis (Eds.), Professional learning communities: Divergence,depth and dilemmas (pp. 151–165). Maidenhead: Open University Press and McGraw Hill Education.

Nesher, P. (1987). Towards an instructional theory: The role of students’ misconceptions. For the Learning ofMathematics, 7(3), 33–39.

Olivier, A. (1996). Handling pupils’ misconceptions. Pythagoras, 21(10–19).Peng, A., & Luo, Z. (2009). A framework for examining mathematics teacher knowledge as used in error

analysis. For the Learning of Mathematics, 29(3), 22–25.Prediger, S. (2010). How to develop mathematics-for-teaching and for understanding: The case of meanings of

the equal sign. Journal of Mathematics Teacher Education, 13(1), 73–93.Sasman, M., Linchevski, L., Olivier, A., & Liebenberg, R. (1998). Probing children’s thinking in the process

of generalisation. Paper presented at the Fourth Annual Congress of the Association for MathematicsEducation of South Africa (AMESA). Pietersburg.

238 K. Brodie

Page 19: Learning about learner errors in professional learning communities

Scribner, S., & Cole, M. (1981). The psychology of literacy. Cambridge: Harvard University Press.Sfard, A. (2007). When the rules of discourse change but nobody tells you: Making sense of mathematics

learning from a commognitive standpoint. Journal of the Learning Sciences, 16(4), 565–613.Smith, J. P., DiSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: a constructivist analysis of

knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163.Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical

thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. AmericanEducational Research Journal, 33(2), 455–488.

Stoll, L., & Louis, K. S. (2008). Professional learning communities: Elaborating new approaches. In L. Stoll &K. S. Louis (Eds.), Professional learning communities: Divergence, depth and dilemmas (pp. 1–13).Maidenhead: Open University Press and McGraw Hill Education.

Strauss, J., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developinggrounded theory (2nd ed.). Thousand Oaks: Sage Publications.

Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness andteaching approaches. Educational Studies in Mathematics, 35, 51–64.

Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge: CambridgeUniversity Press.

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation and autonomy in mathematicsclassrooms. Journal for Research in Mathematics Education, 27, 458–477.

Zawojewski, J., Chamberlin, M., Hjalmarson, M., & Lewis, C. (2008). Developing design studies inmathematics education professional development. In A. Kelly, R. Lesh, & J. Baek (Eds.), Handbook ofdesign research methods in education (pp. 219–245). New York: Routledge.

Learner errors in professional learning communities 239