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Assignment kokkaliari

  1. 1. Uninterested, hard-to-reach pupils in secondary school mathematics: Teacher’s puzzleABSTRACT: The aim of this assignment is to claim uninterested student’s attitude towardsmathematics as a central concern within mathematics classroom in secondary education, notin the traditional sense usually teacher’s use it as an unchangeable, stable one, that causestheir students failure in mathematics, but as a powerful and vital aspect of both teacher andresearcher for further collaboration. Drawing on research evidence, a case is made that it isnever too late to change students’ attitude towards mathematics and this implies that it isnever late to change a teacher’s attitude towards his own teaching practices. The article alsoraises concerns about the ability of the education system to positively address collaborationbetween researchers and teachers in the classroom in a permanent basis for developmentlearning environments for students in mathematics classrooms with an identity of self-efficacy and values, happy citizen of the future world.Keywords: identity of uninterested learner, students’ attitudes towards mathematics,engagement, motivation, teacher, researcherIn the fourth age BC Aristotle states: ‘those who educate children well are more to behonoured than parents, for these only gave life, those the art of living well.’, and his gratefulstudent Alexander the Great some years later declares: ‘I am indebted to my father for living,but to my teacher for living well’. Of course what is not mentioned so frequently is that,Philippe, Alexander’s father searched a lot for his son’s teachers among the most famousacademics.Since then a lot of things changed but families have still the same concern for their children’seducation, looking for the best schools and teachers to guarantee their well-living. Education
  2. 2. has become a human right for almost every child all over the world and Mathematics ishonoured to be - after pupils’ native tongue - the second compulsory subject for most yearsof schooling. If we consider a Mathematics classroom as ‘a social space in which gender,ability and other social identities are played out’ his role as a teacher is, nevertheless, crucial.Educators, researchers and teachers work continuously to make sense of school, classroom,mathematics, curriculum, children, knowledge, learning, teaching, tasks, activities, etc. Forany one of these notions a mass of information has been gathered, a lot of theories have beendeveloped, most of the time in contradiction to each other, in the best case complementary.But how can theories, derived from psychology, sociology, philosophy and mathematics, helpthe teacher solve the problems he faces in a mathematics classroom? It is well known thatevery theory includes some and excludes others. Hence no matter which of them, consciouslyor not, he adopts, a not belonging-marginalised student identity is formed, who is the blacksheep of every educational system, including the teachers who attribute the responsibility tothe student’s attitude towards mathematics, the family background and the previous teachers’attitudes. Uninterested, in mathematics, students and sometimes whole classes are one of themost popular topics discussed among teachers in professional meetings. Every teacher feelsuncomfortable with these students who most of them cause problems, that make teacher’s lifedifficult, behaving in ways that teaching learning processes do not include. They ask theircolleagues how they handle these difficult cases, especially the less experienced ask foradvice. They express emotions of disappointment, when they try to apply strategies ofclassroom management unsuccessfully. They ask the head teachers’ help and they, in turn,try to make them be aligned with school’s rules, they punish them and they ask their parents’help. In this unpleasant teaching condition, the teacher expects urgently material assistance inthe form of innovations, as Brousseau states: ‘there is a controversy involving innovators andthe defenders of action research about what didactique is, what it can do and what it should
  3. 3. do’ (Theory of Didactical Situations in Mathematics Didactique des Mathématiques, 1970–1990). Is there any way in which mathematics education. Why during their pedagogicalmeetings at school they talk about the same difficulties as their colleagues did many yearsago, in spite of the large development of Maths Education? Are they right when they say:‘Yes, this is a fantastic idea, but how can I implement it in my class?’ winking at the teachernext to them? Or when they complain that the research should provide them with ready-to-use tools for their problems in the classroom? Even though mathematics classrooms have similarities, they also have differences.As Anna Sfard and Anna Prusak claim ‘different individuals act differently in the samesituations and differences notwithstanding, do different individuals’ actions often reveal adistinct family resemblance’. Hence the teacher drawing on evidence of a previous researchshould make his own research for his own unique classroom which ‘like every humancommunity, is an individual at its own scale of organization. It has a unique historicaltrajectory, a unique development through time. But like every such individual on every scale,it is also in some respects typical of its kind’ (Lemke 2000).The teacher should take into account the complexity of the interactions between knowledge,students and himself within the context of a particular class. His particular pedagogy whichpositions makes available for his student? ‘Ideas, emotions and actions of participants areshaped by the dynamic of interactional practices, and how positions available in discoursecan be realised as positioning in practice. It provides evidence of excitement and anxietiesfelt by these pupils, showing how they are associated with their positioning in differentdiscursive practices. By analysing the positions occupied by each pupil in interaction, weunderstand how hierarchical positions are (re)produced, as well as the role that emotionsplay in adopting, modifying, ‘submitting to’, or claiming, a position’ (Jeff Evans, CandiaMorgan, Anna Tsatsaroni, 2006). And how these positions interact with his? To which extent
  4. 4. it is responsible for ‘the uninteresting, hard-to-reach student position’ which some studentsoccupy’ and what are the emotions he experiences adopting the position of not participating,of the follower who expects his classmate’s help, who is not consistent with his homeworkand his achievement is not as he expected?Does he feel he wants to change his students’ attitude towards mathematics? This is theturning point. What does he want to do? What about should he be aware? The first thing he isaware of is that behind all these questions is hidden a whole mass of information thatMathematics Education has gathered. Usually disappointed he stops trying under the shelterof ideas putting the blame on family, previous teachers, not existing prerequisite knowledge,lack of time, difficulty of Mathematics.As Marshall and Drummond (2006) argue, teachers’ conceptions of learning are central tounderstanding and enacting these practices. Or he starts questioning everything he does,starting from the tasks; whether they are meaningful, the teaching methods; whether they areengaging, in a few words he tries to find ways to change student’s attitude, to engage him inworking, to motivate him. But what are finally attitudes, engagement, motivation; thesecontroversially discussed multifaceted and mutually interacting notions?We can say that in everyday life by the term attitude we mean someone’s liking or dislikingof a familiar target. The term ‘attitude’ is often used by teachers as a negative one to attributetheir students’ difficulties or failure in mathematics. The last forty years researchers makeexperimentally studies on the construct ‘attitude’ giving it a multidimensional definition.Everyone has his own point of you but it seems that they agree on the fact that there is arelationship between students’ attitude towards mathematics and achievement in that.
  5. 5. For example, Frost et al., 1994; Leder, 1995 state that girls tend to have more negativeattitudes towards mathematics than boys,McLeod, 1994, states that attitudes tend to become more negative as pupils move fromelementary to secondary school. Tony says: ‘In the primary school maths was my favouritesubject, but later I could not get it, I should work hard without success…... so disappointing’and Haladyna et al., 1983, that the general attitude of the class towards mathematics isrelated to1) the quality of the teaching ‘since the day this new teacher came, mathematics hasproved to be the most boring subject, because he speaks continuously and we do nothing elseexcept coping what he is writing on the blackboard’and to2) The social-psychological climate of the class ‘although the teacher tries hard, all theboys try to make noise and they don’t attend the lesson. They are afraid of being teacher’s petand when a student says he likes the lesson they laugh at him during the break’.In this way theory justifies practice: As Ruffel, Mason and Allen (1998) state ‘Teachersattitude to mathematics is increasingly put forward as a dominant factor in children’sattitudes to mathematics’. When the teacher feels that his students’ attitudes to his lesson arenegative and decides to intervene, the first thing he does is to be problematized for his ownattitude and practice. He tries to find out what is wrong, starting from his own positioning inthe class: his emotions and methods. ‘I always like teaching older students. I cannot standvery young pupils.’Markku Hannula has developed a framework for analysing attitude and can be used by theteacher as an analytical tool for exploring his students’ attitudes. She builds a foundation
  6. 6. from the background of psychology of emotions and separates the observable category‘student’s attitude towards mathematics’ into four different evaluative processes:1. the emotions the student experiences during mathematics related activities; It is remarkable that emotions in the mathematics class are not stable, but mayinclude both pleasant ‘well multiplications, additions, subtractions it was fun…..the besttime’ and unpleasant ones ‘it started being a little bit confusing, difficult and could not get it,so disappointing,’, as John comments.2. the emotions that the student automatically associates with the concept mathematics;(‘all these letters instead of numbers….and geometry, with all these proofs…..was soconfusing and embarrassing’ )3. evaluations of situations that the student expects to follow as a consequence of doingmathematics;4. The value of mathematics-related goals in the students global goal structure. ‘Me,now, I am going on with History I don’t need any maths.’Goldin and DeBellis 1997 suggest four facets of affective states: emotional states, attitudes,beliefs, and values/morals/ethics, which provide insight into the development of attitude andHannula uses to reconceptualise attitude with emotion and cognition as two central conceptsso intensely mutually interacting that can be seen as two sides of the same coin.Although there is not simple recipe for the teacher, this framework of emotions,associations, expectations, and values seem to be useful in describing attitudes and theirchanges. The most important think is the way it is constructed as a theoretical view point for
  7. 7. an accurate interpretation of students’ behaviour, capable of steering future action. As DiMartino and Zan say (2009) ‘The relationship is rarely told as stable, even by older students’and, in contrary to what mathematics teachers in higher secondary classes think, it is nevertoo late to change students’ attitude towards mathematics.Researchers have provided teachers with a powerful instrument for analysing, their studentsattitude, but mainly theirs. They can take the responsibility of their teaching methods,question their classroom culture. They first have to be aware of every student’s needs, inorder to help him reach his own potential. Mary is a competent mathematics teacher. Shenever forgets the day her new mathematics teacher came to school. ‘She was a fantasticteacher’ she says. ‘I used to hate mathematics before . I thought it was so boring andnobody used to care about it in my class. It was a really terrible class, with many problemsdue to the multi ethnicity of the students. She never discriminated anyone and in a few weeksshe gained the class’s respect, which had an influence on student’s attitude, they weremotivated and soon we turned to be the most creative mathematics class in the school.Almost everybody was engaged and nobody could understand how she managed it.’, asBerstein (1990) states: ‘If the culture of the teacher is to become part of the consciousness ofthe child, then the culture of the child must first be in the consciousness of the teacher.’But now how he can go along with that? What can be the trigger in students’ development?What is engagement and how can the teacher engage his students with mathematics in a waythat mathematical learning takes place?Again he is not alone in this attempt. There is a mass of research about engagement theteacher can use to find his ‘unique’ way. A mass of information and interpretations isavailable. Finn (1989, 1993) proposed the “participation-identification model” that describes
  8. 8. students identification with school. In addition he suggested that students’ academicengagement comprises three constructs: cognitive, affective and behavioural engagements.1. Affective engagement implies a sense of belonging and an acceptance of the goals of2. Cognitive should be:Flexible vs. Rigid Problem SolvingActive vs. Passive Coping with FailureIndependent vs. Dependent Work StylesIndependent vs. Dependent JudgementPreference for Hard Work vs. Preference for Easy work3. EmotionalIn the form of Anger Interest NervousnessHappiness Sadness CuriosityBoredom Discouragement ExcitementBehaviouralClass Participation vs. UninvolvementOn-task vs. Off-task BehaviourExtra-curricular Academically Oriented vs. Extra-curricular Non-academically OrientedCareer PlansClasses Skipped
  9. 9. TardinessA framework for conceptualising and measuring engagement in mathematics was developedby Kong, Wong, and Lam (2003) through research and validation, identify significantmarkers of engagement. In this study they adopt these markers as a framework forinvestigating, categorising and interpreting student engagement, and are as follows:Affective engagement (Interest and achievement orientated by Anxiety and frustration)Behavioural engagement (Attentiveness, Diligence, Time spent on task, Non-assigned timespent on taskCognitive engagement and strategies that could be drawn to succeed in effective learning, inthe form of:• Surface Strategies (memorisation, practising, and test taking strategies)• Deep Strategies (understanding, summarising, making connections, justifying)• Reliance (on teachers/parents)One would state that it is not realistic to say that a teacher can use all these constructs. Butevery experienced teacher finds in all this research ways he acts, consciously and/or, most ofthe time, unconsciously in his everyday practices. Such constructs are results from teachers’practices and are powerful up to the day that a more viable idea can emerge: BecauseMathematics education is ‘a theory in my work, or, better, a set of thinking tools visiblethrough the results they yield, but it is not built as such … It is a temporary construct whichtakes shape for and by empirical work.’ as Bourdieu claims in Wacquant, 1989. And this isthe only way teachers can use it in their attempt to interpret their students’ negative attitudes,consequently to trigger them in mathematics learning. The means they use is the pedagogic
  10. 10. task which they should plan taking into account the fact that it should be useful andpurposeful. But how can this be feasible? In Connecting engagement and focus in pedagogictask design Janet Ainley, Dave Pratt and Alice Hansenb (2004) warn the teachers of theproblems they can face in planning tasks. The central idea is defined as ‘the planningparadox’ in their one word: ‘If teachers plan from objectives, the tasks they set are likely tobe unrewarding for the pupils and mathematically impoverished. Planning from tasks mayincrease pupils’ engagement but their activity is likely to be unfocused and learning difficultto assess.’ The point they make is how the teachers can produce tasks, which give theirstudents the chance to be engaged in essential content set out by the curriculum in focusedand motivational ways. These two latter can be afforded by carefully selected tools whichconnect the knowledge that students must gain with their everyday experiences.Motivational ways raise an issue of what motivation is. For one more time we can find a hugeamount of research on this new construct, which is created to help us explain, predict andinfluence behaviour. Within psychology one important approach to motivation has been todistinguish between intrinsic and extrinsic motivation (Deci & Ryan, 1985).Intrinsic motivation has emerged as an important phenomenon for educators— a naturalwellspring of learning and achievement that can be systematically catalysed or underminedby parent and teacher practices. In students’ narratives as story tellers we can find witness ofthis kind. ‘I like solving mathematical problems. It reminds me of my holidays spendinghours with my father playing cards, answering puzzles and solving strange problems, whichderived from mathematics, as I understood later. Our family’s habits….’ Intrinsic motivationmust not be undermined because it results in high-quality learning and creativity. However,equally important can be the different types of motivation that fall into the category ofextrinsic motivation and is present in students’ narratives about their experiences. ‘I am notsure that I really like maths, but I study hard because a good grade is important for studying
  11. 11. medicine which is my deepest desire’. In the classic literature, extrinsic motivation hastypically been characterized as a pale and impoverished (even though powerful) form ofmotivation that contrasts with intrinsic motivation. In mathematics education not manyresearchers have focused on motivation (See Evans & Wedege, 2004; Hannula, 2004b), andonly a few researchers have distinguished between intrinsic and extrinsic motivation. Holden(2003) makes a distinction between intrinsic, extrinsic and contextual motivation. Shesuggests that the students’ motivation always is governed by some kind of “rewards”.According to her, students who are extrinsically motivated engage in tasks to obtain extrinsicrewards, such as praise and positive feedback from the teacher. The students’ intrinsicmotivation is governed by intrinsic rewards, which concern developing understanding,feeling powerful and enjoying the task. Students who are contextually motivated are doingsomething to obtain contextual rewards, such as acknowledgement from peer students,working with challenging tasks and seeing the usefulness of the task. Goodchild (2001)relates extrinsic and intrinsic motivation with ego and task orientation and with performanceand learning goals. According to him a student is extrinsically motivated when he is doingsomething because it leads to an outcome external to the task, such as gaining approval orproving self-worth. A student is intrinsically motivated when he considers the task to have avalue for its own sake; he is engaging in the task in order to understand. Evans and Wedege(2004) consider people’s motivation and resistance to learn mathematics as interrelatedphenomena. They present and discuss a number of meanings of these two terms as used inmathematics education and adult education. In Hannula’s dissertation his approach tomotivation involves needs and goals, rather than intrinsic and extrinsic motivation (Hannula,2004a).Kjersti Wæge in Intrinsic and Extrinsic Motivation Versus Social and Instrumental Rationalefor Learning Mathematics (2007) discusses the relation between two different concepts of
  12. 12. motivation for learning mathematics: intrinsic and extrinsic motivation as defined in SelfDetermination Theory and Mellin-Olsen’s concept of rationale for learning mathematics inactivity theory’s point of view. Within Self Determination Theory, as she claims, onesuggests that extrinsic motivation varies considerably in its relative autonomy and thus caneither reflect external control or true self-regulation, comparing to Mellin-Olsen’s tworationales for learning mathematics in school; an S-rationale (Social rationale) and an I-rationale (Instrumental rationale). Both points of views are very interesting for the teacherand can help him decide in which way he could motivate his student by analysing his attitudeand at what extend he could use them depending on his teaching practices and his ownideology.What is undoubtedly clear is the fact that mathematics teachers cannot always rely onintrinsic motivation to foster learning. Many of the tasks, their students should perform,despite their efforts, are not inherently interesting or enjoyable. That’s why knowing how topromote more active and volitional forms of extrinsic motivation becomes an essentialstrategy for successful teaching.When the teacher first meets all these information about these meaningful, for his work,constructs usually has the same feeling with his uninterested in mathematics, unwilling tocooperate student. How can he cope with all this information? Is it worthy to go on or thetime available is so short that the results are doubtful? The most interesting thing that isrevealed by Alexander’s and Aristotle’s quotes in the beginning of this assignment is the hintthat students and their teachers talk about the same stories. Their attitudes towards themeaning of teaching and learning are shaped in a strange interaction. The teacher who givesup in front of this difficulty putting the blame on other factors beyond him (‘I can do nothingbut wait’) is honoured with the most difficult and failing students. Others find some of theseideas charming and decide to use it. ‘Collaborative learning is very interesting’ or ‘Tasks
  13. 13. with computers are very motivating for students’ ‘The curriculum is very demanding, Ishould expect less from them’ are some ideas they have, especially those who decide tointerview their students and take for granted everything they say about what they think that isdiscouraging for them in being more successful.Every attempt is important and gives the teacher insight for parts of the work. A teacherwhose teaching method is traditionally oriented in teacher centred methods finds interestingthe change in some students’ positioning when he decides to ask them work collaboratively.He likes it, although he faces problems by others who prefer working individually, or byways students’ new positionings interact in a disharmonic way: ‘He tells me what to do anddoes not listen to me’, ‘she behaves as if she were my teacher, I cannot stand it!’. Workingwith digital technology is very popular among students (‘maths seems to have a differentdimension, it is a fantastic experience to see how functional and easy to understand aregraphs when using computers’) but new problems appear for the teacher who is not familiarwith it like the ‘planning paradox’, which we have mentioned before when talking aboutAinley and Pratt’s work. Some in front of these difficulties give up and others begin to realisethat teaching is a complicated process, in which sometimes a teacher needs to be a student innew learning processes. The latter goes on trying new methods, enjoying insight and skills hehad not before. This is the moment that becomes aware of the difficulties that all, withoutexception –hopefully- students face in different fields of the teaching learning process. Everystudent can be uninterested in different ways and under some special conditions can shape ahard-to-reach identity.This is the moment we can state that in turn, it is never late for a change in a teacher’sattitude towards teaching mathematics. In Brousseau’s sense, the didactic contract is brokenthis time for the teacher. The situation reminds him of The Nine Dot Puzzle, when he shoulddraw no more than four straight lines (without lifting the pencil from the paper) which will
  14. 14. cross through all nine dots, that shape a square. A solution was difficult to find until he waswilling to ‘think outside the box’. Again Mathematics education research is going to give himthe ‘tool’.How his student from a curiosity machine, as every child is, turned to a mathematical idiot?Brouseau gives an interesting theoretical framework for what he calls situation didactic: itconsists of the learners, the teachers, the mathematical content and the classroom ethos, aswell as the social and institutional forces acting upon that situation, including governmentdirectives such as a National Curriculum statement, inspection and testing regimes , parentaland community pressures and so on (milieu).He states that learning takes place when the didactic contract is broken. For every student itcan happen in different ways, for which is a teacher’s duty to look individually. When heneglects, an uninterested, hard-to-reach identity is shaped that sounds loudly, even in silence.Every experienced teacher can identify it watching, for example, video clips from classroomspractices all over the world in the most different didactic contacts, even conducted by the bestteachers. The ways depend on his pedagogy, ideology, and teaching practice. Ways derivefrom psychology and can be applied even in short time.What is in question is how every student can reach his potential and become the moremathematically literate he can. He must be the researcher of his own practice, of his students’special needs, gathering information from other colleagues, the parents/carers, the students. Anew learning contract among teacher and student must be signed. He must take into accounteverything he is said, starting from questioning his own expectations (individualised), tasks(adapted to everyone’s potential- giving the chance everybody to be motivated), guaranteethat the contract is kept by both parts so that new habits could be formed with the hope newengaged –not marginalised- identities are possibly shaped. Even if it does not happen the
  15. 15. commitment will give emotions worth experiencing and insight for facing new undesiredsituations.It sounds utopic and perhaps it is, but doing something is always more than doing nothing.All the students have something to learn in Bernstein’s ‘totally pedagogised society which isshaped through pedagogy rather than productive processes’. Mathematics is a very importantsubject which supplies skills of competence, especially for those who finish compulsoryeducation mathematically illiterate (‘I have learned some maths (generally) which is goodenough, but then in Year 10-13 you have to learn some advanced maths, which I don’t thinkthat a lot of people are going to use in their everyday life. I am going on with IT, Computersetc. The only thing I need to know from maths is how many wires you need 1,2,3,4,5,6,7,8nothing else, you only need practical skills’, says George who after 10 years of schoolingcannot recall anything else). Of course it is a difficult process for the teacher and sometimesunsuccessful but he is not alone in this attempt. In the era of globalisation new conditions areformed that need to be interpreted. Researchers from all over the world in collaboration withthe teachers exchange aspects of every facet of the educational systems, using qualitative,quantitative and mixed methods they measure students’ performance in mathematics. Therole of parents, teachers, students and curriculum in countries where students have a highattainment is researched. On the other hand, digital technology provides new teaching toolsand environments that create new mathematics. Constructivism gave birth to constructionismand other learning theories in community of practice. A very interesting aspect ofparticipants’ engagement , imagination and alignment in the activity and practice of thisspecified community with its own purposes and goals, is stated by Wegner 1998 and Lave1991 in a community that sounds perfect. In universities the creation of inquiry communitiesbetween didacticians and teachers in a co learning inquiry, a mode of developmental researchin which knowledge and practice develop through the inquiry activity of the people engaged
  16. 16. (Jaworski, 2004a, 2006), to explore ways of improving learning environments for students inmathematics classrooms. Research both charts the developmental process and is a tool fordevelopment learning environments for students in mathematics classrooms.This seems to be the greatest hope for the teacher, who expects research of his own uniqueenvironment by experts to provide him and his students with powerful and vital aspects ofunderstanding the ways in which this ‘complex site of political and social influences, socio-cultural interactions, and multiple positioning involving class, gender, ethnicity, teacher–student relations, and other discursive practices in which power and knowledge are situated’’(Lerman, 2001a, p.44), for helping with shaping an identity of a citizen of the world withvalues and competence.Until then professional mathematics teachers will share with researchers their inquiries andfears in MA Mathematics Education classes in the same strange, charming and uniquerelationship of the kind of teacher and student.ACKNOWLEDGEMENTSI should like to thank most sincerely Candia Morgan, Cathy Smith, Melissa Rodd, EiriniGeraniou, Dave Pratt for their interesting stories I heard during the sessions of ‘Issues inMathematics Education’ and last, but not least I want to thank all my students of so manyyears who trusted me for their mathematics educations.MAGDALINI KOKKALIARIKOK11094464MMAMAT_04MA STUDENT ()
  17. 17. MATHEMATICS EDUCATIONINSTITUTE OF RDUCATIONUNIVERSITY OF LONDONFEBRUARY 2012REFERENCESAinley, J., Pratt, D. and Hansen, A. (2006). Connecting engagement and focus in pedagogictask design, British Educational Research Journal. 32(1), 23-38.Sfard, A. & Prusak, A. (2005). Identity that makes a difference: Substantial learning asclosing the gap between actual and designated identities. In H.L. Chick & J.L. Vincent (Eds.)Proceedings of the Twenty-ninth Meeting of the International Group for the Psychology ofMathematics Education (Vol. 1, pp. 37-52), Department of Science and MathematicsEducation, University of Melbourne, Victoria, Australia.Haladyna, T., Shaughnessy, J., & Shaughnessy, M. (1983). A causal analysis of attitudetoward mathematics. Journal for Research in Mathematics Education, 14(1), 19–29Hart, L. (1989). Describing the affective domain: Saying what we mean. In D. Mc Leod & V.M. Adams (Eds.), Affect and mathematical problem solving (pp. 37–45). New York:Springer.Kulm, G. (1980). Research on mathematics attitude. In R. J. Shumway (Ed.), Research inmathematics education (pp. 356–387). Reston, VA: NCTM.
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