Theorising community of practice and community of inquiry in the context of teaching-learning mathematics at university

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<ul><li><p>This article was downloaded by: [Eindhoven Technical University]On: 22 November 2014, At: 00:27Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK</p><p>Research in Mathematics EducationPublication details, including instructions for authors andsubscription information:</p><p>Theorising community of practice andcommunity of inquiry in the contextof teaching-learning mathematics atuniversitySimon Goodchildaa Department of Mathematical Sciences, University of Agder,Kristiansand, NorwayPublished online: 13 Jun 2014.</p><p>To cite this article: Simon Goodchild (2014) Theorising community of practice and community ofinquiry in the context of teaching-learning mathematics at university, Research in MathematicsEducation, 16:2, 177-181, DOI: 10.1080/14794802.2014.918352</p><p>To link to this article:</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (theContent) contained in the publications on our platform. 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Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &amp;Conditions of access and use can be found at</p><p></p></li><li><p>Theorising community of practice and community of inquiry in thecontext of teaching-learning mathematics at university</p><p>Simon Goodchild*</p><p>Department of Mathematical Sciences, University of Agder, Kristiansand, Norway</p><p>Introduction</p><p>Biza, Jaworski, and Hemmi (2014) make a valuable contribution to theorising universitymathematics teaching through the lens of community of practice theory (CPT) and byjuxtaposing community of practice (CoP) and community of inquiry (CoI) withinuniversity settings. The paper has stimulated my thinking around the theoretical issuesand I am pleased to have the opportunity to write this reaction. Ideally, this reactionwould be just one link in a longer dialogue with the authors of the paper; appearing as itdoes as an afterword it could seem evaluative and judgmental I do not want to engagein this type of critique. My purpose is to set out how the paper has provoked me to reflecton both development and application of theory. I do this by posing three questions:</p><p>First, is it possible for an adapted community of practice theory (CPT) to offer all theconstructs necessary to provide a theoretical underpinning of community of inquiry (CoI),without violating some of the a priori planks of CPT? Secondly, is it possible to create agenuine CoI with university students who are constrained in alignment to the teaching-learning practice chosen by their teacher within the context of strong institutional forces?Thirdly, does a community of inquiry theory (CIT) need to include constructs fromcognitive and constructivist theories, requiring the swallowing of some paradigmaticcamels?</p><p>In the paper, Biza et al. first explain why they adopt sociocultural theory and morespecifically CPT. From the introduction and second section it is evident that the authorsintend the paper to focus on theory: Our aim in this paper is to give more theoreticalinsight and to take this theorisation forward (p. 161). They show how Wengerstheorisation of CoP might be applied to university teaching and learning. The first of twoResearch Cases (Case 1) described in the second half of the paper demonstrates how thisis achieved within an empirical study. Wengers mode of belonging, alignment, isspecially scrutinised and questioned Biza et al. explain that it is possible to be criticallyaligned to practice, through adopting an inquiry stance. Many of the constructs orcategories of Wengers theory are then re-examined and explained in the context ofinquiry; hence the notion of CoI emerges. The second Research Case (Case 2)demonstrates how these adapted constructs are applied in developmental research. In</p><p>*Email:</p><p>Research in Mathematics Education, 2014Vol. 16, No. 2, 177181,</p><p> 2014 British Society for Research into Learning Mathematics</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Ein</p><p>dhov</p><p>en T</p><p>echn</p><p>ical</p><p> Uni</p><p>vers</p><p>ity] </p><p>at 0</p><p>0:27</p><p> 22 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p><p>mailto:simon.goodchild@uia.no</p></li><li><p>this reaction I will try to enter into the authors intended discourse and focus ontheorisation.</p><p>Theorisation of community of inquiry</p><p>The transformation of a community of practice (CoP) into a community of inquiry (CoI)can prove theoretically troublesome. Biza et al. point to some of the difficulties. Inparticular they draw attention to individual agency in CoI and explain that agency isnot covered in CoP theory. I want to expand this a little. Wenger does in fact write aboutagency (1998, pp. 11, 13, &amp; 15); however he appears to align himself with theories ofsubjectivity that do not:</p><p>[take] for granted a notion of agency associated with the individual subject as a self-standingentity, they seek to explain how the experience of subjectivity arises out of engagement in thesocial world (1998, p. 15).</p><p>Agency is related to goal directed activity, which is also problematic in CPT (Lave,1988, 1996; Nardi, 1996). Thus it is not that agency is not considered; it appears that thenotion is problematic from the CPT perspective. Elsewhere, I have argued that CPT andCoI are inconsistent and incompatible, and belong within different paradigms: interpret-ive and critical respectively (Goodchild, 2011). The difficulty of explaining CoI throughCPT is also evident as Biza et al. cite a paper by Jaworski, Robinson, Matthews, andCroft (2012) which applied activity theory to gain further insights in the contradictions[which] occurred in an inquiry-based lesson for mathematics to engineers (p. 25,emphasis added); thus implying that CPT may not be adequate in theorising CoI.</p><p>Biza et al.s discussion is helpful because it draws into sharp relief the differences inkind between CPT (as a theory) and CoP and CoI as sociocultural constructs and entitiesopen to empirical study. They outline CPT as a theorisation of CoP based on the writingsof Wenger (1998) and Lave and Wenger (1991). The theory is applied within ResearchCase 1: proof in the process of entrance to the mathematical community. However,Research Case 2: seeking conceptual understanding of mathematics is explained usingthe constructs of CoI, which are adapted from CPT. I want to suggest that CPT does notprovide a theoretical underpinning of CoI.</p><p>Community of practice theory (CPT), through the work of Wenger (1998) and Laveand Wenger (1991) especially, offers a theorisation of CoP. CoI is a development usingadapted constructs from CoP/CPT, but the adaptation introduces new constructs such asagency, goal directed actions, contradictions and tensions. It seems that CPT does not,and cannot, offer a theorisation of CoI. The juxtaposition of Case 1 a researchersexploration of university students experience of proof, and Case 2 a university teacher/researchers developmental activity within her own practice (albeit as part of a largerresearch team) is helpful. The research in Case 1 appears to accept CPT as fundamental,and examines teaching and learning proof in university mathematics through the lens ofCPT. This entails the explanation of university mathematics teaching-learning using thelanguage and constructs of CPT. In Case 1, the hermeneutic challenge is to apply andinterpret the theoretical constructs from CPT to the practice of teaching and learningproof in university mathematics. The case provides an analytical description, which offersinsights about the complexity of how to deal with proof. The case is not used to criticallyexamine CPT.</p><p>178 S. Goodchild</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Ein</p><p>dhov</p><p>en T</p><p>echn</p><p>ical</p><p> Uni</p><p>vers</p><p>ity] </p><p>at 0</p><p>0:27</p><p> 22 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>Case 2 is described as involving developmental research that introduced innovationsto the practice aimed to engage students in mathematics in ways that encouraged them tothink mathematically (p. 20, emphasis in original). The description of the study feelslike a form of action research, although the action includes more than just the teacher asresearcher. The study is not purely interpretive because there is a clear effort to developpractice both the teachers teaching and the students approach to learning. Thedevelopmental approach is goal driven (aimed to engage ) and founded on aprinciple of human agency, from which the teacher chooses to develop her practice.Further, it emerges in the description of the case that some tensions arise because of theapparent contradiction between the approach, which students recognised as favouring abetter understanding of mathematics, and the need to practise for examinations. CPT doesnot provide a principled explanation of agency, goals, tensions and contradictions. Itseems that a theorisation of CoI, that is a Community of Inquiry Theory (CIT) may benecessary. Alternatively, as Jaworski et al. (2012) demonstrate, a new theory may not berequired because Cultural Historical Activity Theory (CHAT) offers all that is necessaryfor a full analytic explanation.</p><p>Alignment to a practice is not purely a matter of choice, as Biza et al. recognise: Inmost practices, alignment of some kind is unavoidable (p. 164), and critical alignmentmay encounter strong resistant forces within the practice. This is evident in Case 2:</p><p>Thus although there was evidence of student understanding, and some appreciation of howaspects of the innovation contributed to understanding, the various influences on the practice,and especially the assessment by examination (despite the more formative projectassessment) proved overwhelming (p. 172, emphasis added).</p><p>The case also describes how students prior experiences and expectations resulted inresistance to the innovation. Using the language of CPT, examinations reify mathematicalknowledge and competence in the form of grades and certificates. The transition fromschool to university entails a transformation of the scholarly nature of the discipline, butnot institutional aspects such as classes, timetables, teachers, examinations, rules andregulations. Educational structures that reify knowledge in the form of grades andcertificates lie beyond students agency to change; they exert an overwhelming force toalign with practice. The question I ask is whether CHAT offers an adequately principledexplanation of institutional forces and constrained alignment.</p><p>Critical alignment and students practiceThe second case described by Biza et al. provokes another question. Were the studentsparticipating in a genuine CoI? They were certainly components, or subjects withinthe teachers CoI. Also, I do not doubt the teachers intention to engage the students in aform of practice that would model a CoI, but was it genuine? If the situation is examinedfrom what might be termed as the foundations of CoI, were the students critically alignedto the practice? It appears that, with some resistance, the students engaged in the activitiesdesigned for them, activities that were intended to establish an inquiry stance towardsmathematics.</p><p>I think it could be argued that the students were aligned (not critically) to the practiceof teaching-learning the teacher defined the approach, the students did what it takes toplay their part (Wenger, 1998, p.179). The students were provided with tasks that reified</p><p>Research in Mathematics Education 179</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Ein</p><p>dhov</p><p>en T</p><p>echn</p><p>ical</p><p> Uni</p><p>vers</p><p>ity] </p><p>at 0</p><p>0:27</p><p> 22 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>university mathematics, the tasks were intended to stimulate wonder, curiosity, andinquiry and possibly they did achieve this. In the paper it is claimed that the students arecritically aligned:</p><p>The community of mathematical practice transforms into a Community of Inquiry wheninquiry becomes a part of the practice. All participants engage in critical alignment: ratherthan expecting to be told by the teacher, students are encouraged to ask mathematicalquestions and seek their own way of expressing mathematical ideas; the teacher lookscritically at her own practice. (p. 171, emphasis in original)</p><p>It is not clear, however, whether the students were stimulated to inquire into the nature ofuniversity mathematics as a practice and critically align themselves to the practice ofteaching-learning university mathematics. Consider the example provided by Biza et al.,where students were required to explore straight lines intersecting with graphs ofquadratic functions (p. 170), or the other example about constructing triangles to a given(ambiguous) specification. The questions are designed to provoke inquiry and engage-ment in mathematical thinking. If this is critical alignment, I suggest it is in the form of ametacognitive disposition towards the relationships between the mathematics, mathem-atical representations and their own thinking. It appears that the students appreciated therationale for the approach. However, a university mathematician will be as concernedwith identifying the interesting questions as with finding answers. At what stage dostudents become the authors of mathematically interesting questions, and not just thereceivers?</p><p>I do not mean the above to be in any way critical of the developmental activity of theteacher or researcher in Case 2. The evaluation of practice is not the intention of thisreaction. My purpose is to raise the question as to whether it is possible to design a CoI inwhich students can be genuine participants; and, if it is, what are the conditions thatensure such participation?</p><p>Constructs from cognitive science and community of inquiry theory</p><p>University mathematics education embraces many practices: mathematicians doingmathematics, researching mathematics and mathematics...</p></li></ul>