24. Mathematics Education Research
Paper
Rewilding the Mathematics Classroom: Embedding Global Citizenship as a Challenge to Business as Usual Mathematics Education.
Corinne Angier
University of Stirling, United Kingdom
Presenting Author: Angier, Corinne
This paper tells an overarching story emerging from an inquiry into the affordances and constraints of using global citizenship (GC) themes in secondary mathematics classrooms in Scotland. I argue that not only does mathematics, as a human tool for measuring modelling and making sense of the world, have something distinctive to offer GC education but also that GC opens spaces to rethink and potentially rewild mathematics education. The project has roots in ‘global citizenship education otherwise’ (Andreotti 2021) and critical mathematics education for social justice (Skovsmose 2014, Weist et al 2007, Stinson et al 2012). It is pragmatically connected to the current requirements of the Scottish ‘Curriculum for Excellence’ (Scottish executive 2004) where Global citizenship education is an entitlement for all learners and the responsibility of all teachers.
The secondary mathematics teachers in this project have chosen to attend subject specific professional learning courses offered by a development education centre. They engaged with GC themes and classroom materials that contextualise mathematics tasks in complex and sometimes controversial topics. Some of the teachers were able to then use these materials in their classrooms.
Two sub plots identified in the research project are drawn on here. The first describes the power and tenacity of what I have called ‘business as usual mathematics education’ where the overriding purpose is to meet performance targets in the achievement of tradeable qualifications. The second describes teachers finding and giving space, both of which require professional and pedagogic courage, in which they teach about, through and for GC.
I privilege possibility over probability to suggest that global citizenship education can help to rewild the damaged and denuded site that is the typical mathematics classroom (Ernest 2020) where mathematics is the “handmaiden of capital” (Darder 2018 pviii). GC materials (see e.g. Scotdec 2021) which echo the “spiritual turn” (Gutierrez) and the “socio-ecological ecological” turn (Coles 2022) that have been called for in mathematics education, require dispositions and pedagogies that may be unfamiliar to, but also welcomed by mathematics teachers.
Following their meta review of typologies of GCE Pashby et al (2020 p.158) “identify three different layers of analysis and intervention … methodological (the level of doing), epistemological (the level of thinking) and ontological the level of being.” Each of these maps to this story and I add a fourth axiological the level of valuing. When the teachers adapt their pedagogy to give choice or enable collaboration they are engaging methodologically. When the teachers reimagine the curriculum to embrace wider purposes that better match the use and beauty of mathematics they are engaging epistemologically. When the teachers carve out spaces within which pedagogy and curriculum can be enacted differently, they are engaging ontologically. When the teachers respect young people as more than Homus O Economicus (Valero 2018) in the making, care about their foregrounds (Skovsmose 2012) and mind about our shared world (Griffiths & Murray 2017) they are engaging axiologically.
A recovered ecosystem is not a virgin one and it is important to manage our expectations of the mathematics classroom. The social environment of individual competitive capitalism, austerity policies and powerful social media places pressure to adapt on teachers and young people. The education policy environment in Scotland seems to be very encouraging for the teacher who is committed to social justice and yet in practice the requirement to raise attainment always dominates the requirement to embed learning for Sustainability (Scottish Government 1012). This project has given some glimpses that suggest that GC in mathematics can have the strength of a plant growing up through concrete and making a crack in what seemed to be impenetrable.
Methodology, Methods, Research Instruments or Sources UsedSince autumn 2018 I have engaged with secondary mathematics teachers in GC workshops, webinars and online modules. The material collated includes online postings, which helped to craft themes for recorded conversations with teachers, and my own journal notes. Through an iterative process of writing back into the conversation transcripts using my journal notes I have gone visiting (Arendt 1992) to explore the ways in which these teachers make sense of the affordances and constraints of their contexts.
I have been guided by feminist methodology which warns against treating people as “data objects” (Stanley & Wise 2002) and has often called me “down from the attic” (Smith 1992) for some reflexive exercises to connect the lengthening string of methodological qualifiers; critical, feminist, decolonial, post critical with the activities I am undertaking. Mignolo (2011) argues that decolonisation entails the reversal of the colonial logic which translates differences into values. I have been determined not to repeat the sorting and labelling that happens in mathematics classrooms and make no attempt to categorise the teachers but rather to “form an opinion by considering a given issue from different viewpoints” (Arendt 1961 p.237). I have set out to understand something of where these Scottish mathematics teachers stand. This includes their school settings, the policy landscape, their relationship to mathematics and their conceptions of themselves as teachers of global citizenship within mathematics.
My understanding of thinking with theory is a composting of ideas which was suggested by Andreotti et al (2018 p.10-11) in the context of dealing with the “waste” from a “modern/colonial imaginary”. The breaking down processes might describe unlearning as well as releasing key ideas from their previous constructs so they may recombine in new ways. It is an organic metaphor with a certain relinquishing of human power and control (Haraway 2015) unlike the image of dismantling a building (Stein et al 2017) in order to rebuild from the constituent parts.
Compost is made to be used and the characterisation of research as providing “input to social dialogue and praxis in society” (Flyvbjerg 2001 p. 139) echoes both Dewey’s pragmatism and Arendt’s emphasis on politics as discursive action. I have worked with the development education centre throughout this project, incorporating emerging themes back into professional learning and encouraging the teachers to share their stories and establish an online supportive community.
Conclusions, Expected Outcomes or Findings The teachers in this study know that ‘business as usual mathematics education’ has damaging consequences during what is called the senior phase [age 15-17] when young people write public examinations. They described this as “a two year parenthesis” where meaningful mathematics is “in a box” and “you have to close windows.” They also acknowledge there is a ‘backwash’ of ‘teaching to the test’ that floods the first three years of secondary school (see also Teese 2007). This focus on a narrowly conceived progress (Povey and Angier 2021) is encouraged by the inspectorate in the guise of tackling inequality.
These teachers were able to articulate the tensions they have to reconcile: “If we do not embrace these topics with our young people, we are doing them a disservice, we will not equip them with the essential skills and tools to become the critical citizens they will require to be in their futures.” [teacher B] and three of the teachers attributed talking about GC in job interviews as contributing to their success. They wanted to engage in professional learning but were often frustrated by other demands: “I need to do more. I am so frustrated I have not picked up so many of the things I learned from you in my previous training I’ve just not been doing it. It just makes me very upset I think today ... how I could have used that time for planning all the things I could do instead of planning assessments.” [Teacher B]
When GC tasks were used, they opened up the classroom to different voices, challenged the hierarchy of prior attainment and enabled young people to make authentic connections with their experiences.
ReferencesAndreotti, V., Stein, S., Sutherland, A., Pashby, K. L., Susa, R., & Amsler, S. (2018). Mobilising different conversations about global justice in education: toward alternative futures in uncertain times. Policy & practice: A development education review, 26, 9-41.
Andreotti, V. D. O. (2021). Depth education and the possibility of GCE otherwise. Globalisation, Societies and Education, 19(4), 496-509.
Arendt, H., & Kohn, J. (1961). Between past and future. Penguin.
Arendt, H. (1989). Lectures on Kant's political philosophy. University of Chicago Press.
Darder (2018) in Avci, B. Critical mathematics education: Can democratic mathematics education survive under neoliberal regime?. Brill.
Coles, A. (2022). A socio-ecological turn in mathematics education: Reflecting on curriculum innovation. Revista Paradigma, 43(Edición temática 1), 207-228.
Flyvbjerg, B. (2001). Making social science matter: Why social inquiry fails and how it can succeed again. Cambridge university press
Griffiths, M., & Murray, R. (2017). Love and social justice in learning for sustainability. Ethics and Education, 12(1), 39-50.
Gutiérrez, R. (2022). A spiritual turn: Toward desire-based research and Indigenous futurity in mathematics education. Journal for Research in Mathematics Education, 53(5), 379-388.
Haraway, D. (2015). Anthropocene, capitalocene, plantationocene, chthulucene: Making kin. Environmental humanities, 6(1), 159-165.
Mignolo, W. (2011). The darker side of western modernity: Global futures, decolonial options. Duke University Press.
Pashby, K., da Costa, M., Stein, S., & Andreotti, V. (2020). A meta-review of typologies of global citizenship education. Comparative Education, 56(2), 144-164.
Povey, H., & Angier, C. (2021, July). Against 'progress'. In Forum (Vol. 63, No. 2, pp. 20-31). Lawrence and Wishart.
Scottish Executive. (2004). A curriculum for excellence. Edinburgh Scottish executive.
Scottish Government. (2012). Learning for Sustainability the report of the One Planet Schools Working Group.
Skovsmose, O. (2012). Students' foregrounds: Hope, despair, uncertainty. Pythagoras, 33(2), 1-8.
Smith, D. E. (1992). Sociology from women's experience: A reaffirmation. Sociological theory, 10(1), 88-98.
Stanley, L., & Wise, S. (2002). Breaking out again: Feminist ontology and epistemology. Routledge.
Stinson, D. W., Bidwell, C. R., & Powell, G. C. (2012). Critical pedagogy and teaching mathematics for social justice. The International Journal of Critical Pedagogy, 4(1).
Teese, R. (2007). Time and space in the reproduction of educational inequality (pp. 1-21). Springer Netherlands.
Valero, P. (2018). 2018--Human Capitals: School Mathematics and the Making of the Homus O Economicus. Journal of Urban Mathematics Education, 11(1-2).
Wiest, L. R., Higgins, H. J., & Frost, J. H. (2007). Quantitative literacy for social justice. Equity & Excellence in Education, 40(1), 47-55.
24. Mathematics Education Research
Paper
Towards Equitable Mathematics Classrooms: Enhancing Learners’ Engagement
Alprata Ahuja
Dr B R Ambedkar University, Delhi, India, India
Presenting Author: Ahuja, Alprata
India’s National Curriculum Framework 2005 (NCERT, 2006a) posited an epistemic change towards learning. Learning was seen through a constructivist lens with the learner being focused on in the policy discourse (Sarangapani, 2007; Saxena, 2006). The Position Paper in Teaching of Mathematics proposed the guideline of ‘Math for all’ (NCERT, 2006b, p. 12). Curriculum was proposed to be coherent, ambitious and having depth rather than being tall and spindly (NCERT, 2006b, p. 7). It was recommended that mathematics learning should cater to the minority of learners who have the potential to do well in mathematics as well as include the majority of learners who usually find mathematics challenging (NCERT, 2006b, p. 12). The proposed shift raised an expectation on teachers’ role in classrooms. They were expected to make the classroom space more inclusive in order to make mathematics learning equitable. The teachers were assumed to have a role of implementing these envisioned changes in their practice by implementing reform-based curriculum and textbooks (Chowdhuri, 2020). NCF 2005 and earlier policies as well as curricular frameworks have been criticised to follow the route of top-down approach in bringing any change (Anand & Lall, 2022; Ball & Cohen, 1996; Batra, 2005; Choppin, 2009; Remillard, 2005).
It has been well researched that teachers negotiate with reform-based textbooks and curriculum in varying ways (Remillard, 2005). Teachers’ negotiations with any proposed reform are a product of their identity and beliefs (Heyd-Metzuyanim, 2019, Thompson, 1984). The proposed research explores the relation between teachers’ identities, their beliefs and choices they make in their practice. For the given research, case study of seven teachers teaching mathematics in primary grades were done. These teachers were employed in a non-governmental organisation (NGO) but were teaching in state-run government primary level schools. The site of research presents a unique case of partnership of an NGO with the government. Open-ended interviews were carried ahead with each of them. Classrooms of five of these teachers were observed for a period of a week to see how their choices translate into practice.
The conclusions derived from the study discusses how teachers viewed themselves as an agent of change. Working within the protocols followed in the school and the expectations set by the NGO, they kept the learner in focus while making any decisions related to their practice. These practices encouraged both individual and collective participation of learners. Collaboration through group work was encouraged to make learners interdependent. These collaborative practices shared the onus of learning within the learners. Teachers also talked about challenges and limitations they usually faced.
This proposed research will add to the discourse of teachers’ engagement with mathematics education reforms in India. At global level, teachers’ engagement with the reforms has been researched upon from teachers’ identity, teachers’ beliefs, teachers’ agency perspectives and this research will enrich the existent discourse from the India context (Anand & Lall, 2022). It is imperative that theorization of teachers’ engagement with reforms is evolved from ground and this research will help take steps in that direction.
Methodology, Methods, Research Instruments or Sources UsedQualitative research is undertaken to understand the meaning the actors make of the “events, situations and actions they are involved with” (Maxwell, 1996, p. 17). The role of context in which the actors participate and the influence of this context on their meaning making is also focussed in qualitative research. Qualitative research is undertaken to understand “the process by which events and actions take place” (Maxwell, 1996, p. 19). Methods of data collection in the qualitative research “in natural setting is sensitive to the people and places under study” (Creswell, 2007, p. 37). Analysis of data in qualitative research is inductive in nature. Patterns and themes are generated from rounds of data analysis in qualitative research. Here the conclusions emanating from this qualitative research will make contribution towards the theorisation of teachers’ engagement with reforms.
For the given research, case study of seven teachers teaching mathematics in primary grades were done. These teachers were employed in a non-governmental organisation (NGO) but were teaching in state-run government primary level schools. The site of research presents a unique case of partnership of an NGO with the government.
The following parameters were kept in mind while making sample selection:
• All the teachers should be teaching any of the primary grades.
• Across the sample of teachers, varied grades were covered.
• Each teacher had at least three years of experience any of the grades so that they have some experience at hand before participating in the research.
• The teacher should be comfortable participating in the interviews, getting her classes observed. The schools should also provide necessary support for the same.
• Requisite permissions and purpose of the proposed research were explained to the all the concerned personnel. This contributed towards trust building with schools and the teachers. Trust and rapport building are crucial element of a qualitative research (Maxwell, 1996).
Further, purposeful sampling (Maxwell, 1996, p. 70) was deemed suitable technique for this research. The reason for choosing this sampling technique was that the teachers were be identified as per the criteria listed above.
Methods of data collection were open ended interviews and classroom observations. Open-ended interviews were carried ahead with all the seven teachers. Classrooms of five of these teachers were observed for a period of a week to see how their choices translate into practice.
Conclusions, Expected Outcomes or FindingsThe conclusions derived from the study discusses how teachers viewed themselves as an agent of change. Working within the protocols followed in the school and the expectations set by the NGO, they kept the learner in focus while making any decisions related to their practice. These practices encouraged both individual and collective participation of learners. Collaboration through group work was encouraged to make learners interdependent. These collaborative practices shared the onus of learning within the learners. Teachers also talked about challenges and limitations they usually faced.
This proposed research will add to the discourse of teachers’ engagement with mathematics education reforms in India. At global level, teachers’ engagement with the reforms has been researched upon from teachers’ identity, teachers’ beliefs, teachers’ agency perspectives and this research will enrich the existent discourse from the India context (Anand & Lall, 2022). It is imperative that theorization of teachers’ engagement with reforms is evolved from ground and this research will help take steps in that direction.
ReferencesAnand, K., & Lall, M. (2022). Delhi’s Education Revolution: Teachers, agency and inclusion. UCL Press. https://doi.org/10.2307/j.ctv2f4v5nz.
Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is – or might be – the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–14.
Batra, P. (2005). Voice and Agency of Teachers: Missing Link in National Curriculum Framework 2005. Economic and Political Weekly, 40(40), 4347–4356. http://www.jstor.org/stable/4417232.
Choppin, J. M. (2009). Curriculum-context knowledge: Teacher learning from successive enactments of a standards-based mathematics curriculum. Curriculum Inquiry, 39(2), 287–320.
Chowdhuri, M. N. (2020). Teachers’ use of reform-oriented mathematics textbooks: A multiple-case study of Delhi government primary school teachers (Doctoral thesis). https://doi.org/10.17863/CAM.52111.
Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches. Sage.
Heyd-Metzuyanim, E. (2019). Dialogue between discourses: Beliefs And identity in mathematics education. For the Learning of Mathematics, 39(3), 2–8. https://www.jstor.org/stable/26854425.
Maxwell, J.A. (1996). Qualitative research design: An interactive approach. Sage.
NCERT. (2006a). National curriculum framework 2005. NCERT.
NCERT. (2006b). NCF 2005 position paper on teaching of mathematics. NCERT.
Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.
Sarangapani, P. M. (2007). Re-evaluating constructivism and the NCF: An examination of the arguments. Contemporary Education Dialogue, 4(2), 238–242. https://doi.org/10.1177/0973184913411144.
Saxena, S. (2006). Questions of epistemology: Re-evaluating constructivism and the NCF 2005. Contemporary Education Dialogue, 4(1), 52–71. https://doi.org/10.1177/0973184913411131.
Thompson, A. G. (1984). The Relationship of Teachers’ Conceptions of Mathematics and Mathematics Teaching to Instructional Practice. Educational Studies in Mathematics, 15(2), 105–127. http://www.jstor.org/stable/3482244.
24. Mathematics Education Research
Paper
Using Mathematical Concepts in the Schoolyard for the Well-being and Development of ALL Children
Catarina Cravo Martins, Carlos Gonçalves
MEC, Portugal, Portugal, Portugal
Presenting Author: Cravo Martins, Catarina;
Gonçalves, Carlos
The following paper comes from one of the dimensions that make up the Yard4All project, built under the Erasmus+ program. To reform conventional education, escape the traditional four walls and get in touch with reality, this document exposes the importance of working mathematical concepts in concrete, bringing this curricular area closer to children's daily lives. According to Feille (2021, 2013), the schoolyard is an essential pedagogical tool in the holistic development of any child, that helps them achieve several skills.
When students are able to touch their learning, experience it in real life, their understanding seems to emerge like snap peas peeking out of the fresh dirt. Their green tips poking above the nourished and moist earth. Nurtured with care and attention, they stretch and reach for fresh air and sun. (Feill, 2013, p.1)
The theoretical component presented here is linked to a research question: What practical activities, using mathematical concepts, are necessary to promote the global development of children, while exploiting the schoolyard?
Consequently, this project essentially defines three main objectives: (i) Develop and improve, in every child, cognitive, social and emotional skills, throughout non-formal learning and teaching environments, using outdoor activities in the schoolyard, that follow the principles of permaculture, while using mathematical concepts; (ii) Develop practical and useful activities that motivate all children to have a positive attitude towards Mathematics, where the entire school community is included and can actively participate; and (iii) Create a guide for teachers and other educational actors, specifying some activities, within the scope of Mathematics, for children with special educational needs and children without impairments, between six and twelve years old.
Firstly, it is important to define permaculture, since all the activities created are based on it. Permaculture is a form of agroecology and an alternative paradigm of production, based on ecological principles such as recycling waste, minimizing energy and water use, maximizing genetic diversity, regenerating soil, and promoting other beneficial biological synergies (Hathaway, 2016). Permaculture school gardens offer particularly valuable opportunities for a sustainable education, crucial for the XXI century learner. This practice is innovative in the school context and can provide children with practical experience of sustainable production means (Yard4All, 2022). But to work in a learning environment with these principles, children need to go outdoors and get to know nature, sense the biosphere, and explore their motor, cognitive, emotional, and social abilities. In fact, while discovering the schoolyard, children find opportunities to make decisions that encourage problem solving and creative thinking (Sá, 2016). Mathematics is often seen by children as a “bogeyman”, being considered not very humane and useful for everyday life. But Mathematics is in any day-to-day circumstances, and it is possible to relate it to various topics, formally and informally. The Yard4All (2022) project focuses on one of the most appropriate ways to work with Mathematics nowadays, using it with context. Mathematics in Context aims to use mathematical concepts in significant real contexts, moving from concrete and informal situations to the formal understanding of abstract concepts (Martins, Fernandes, & Guedes, 2020). This allows children to gain confidence and feel predisposed to learn Mathematics, in a place where they are curious, safe and, at the same time, have unpredictability (Feille, 2013) – the schoolyard. As a pedagogical tool, the schoolyard allows children to participate in the development of “their social and cognitive abilities, their sense of connection with other people, with nature and their natural environment” (Tsevreni & Bentenidi, 2013, p.39), including children with special needs. The schoolyard is a place where everyone can have their sense of belonging and participate in decision-making, characteristics of an inclusive education.
Methodology, Methods, Research Instruments or Sources UsedWithin the European area, activities involving the use of mathematical concepts in the schoolyard were developed by four different countries: Portugal, Spain, Romania, and Hungary. In each school, the methodology was adapted according to the educational context, in specific, according to the characteristics of the space, the children (individually), and the work group (child-child and child-teacher relationship).
The activities used in this project contained: concept development periods, that focused on revealing student’s prior knowledge of mathematics; moments for developing students' understanding of import mathematical ideas, linking concepts to their previous mathematical knowledge; episodes of problem solving, assessing, and developing students’ ability to apply the mathematical knowledge in real and significant learning contexts; and stages of reasoning development, in a flexible way for non-routine and unstructured problems, in the field of mathematics, using situations linked to the school’s garden. The activities were also designed for an average time of 90 minutes, but the teacher can evaluate and adapt these activities to the desired duration, with a minimum time of 45 /50 minutes.
The methods used while applying the created math activities were various. One of them is called peer learning. Peer learning refers to “the use of teaching and learning strategies in which students learn with and from each other without the immediate intervention of a teacher” (Boud, Cohen & Sampson, 1999, pp. 413-414). The teacher was responsible for forming the groups, according to each activity purpose, so each group had between two and four children. The learning contexts incorporated children with special educational needs and children without impairments, from six to twelve years old and, because of that, teachers were able to do a pedagogical differentiation, based on observation in class and through reports prepared by Special Education teachers.
In conclusion, the instruments for collecting information included direct observation, photographic records, written records by the children and the analysis of questionnaires given to the children, their caregivers/family, and their teachers.
Conclusions, Expected Outcomes or FindingsIn Yard4All (2022), a math guide was prepared, with enriching outdoor activities, namely in the school garden/ schoolyard. The pilots were carried out in the schools of the four countries stated and, the collected data provided final reports that reveal several findings: 1. The Mathematic guide from the project should be recommended to other schools and Mathematic teachers; 2. The guide will be improved so it can provide a greater variety of activities, including different approaches, innovative changes and new ways to work in and with the schoolyard; 3. Children showed enthusiasm and were motivated during the activities carried out within the framework of the project; 4. The existence of a permaculture garden in the schools was important, due to the different situations it provided for the school's service and because it was an inclusive environment. It also proved the following: “(…) schools and local communities have a symbiotic relationship. Permaculture offers an ethical system to reintegrate the care and edification of the next generation” (Raynolds, 2022, p.2); 5. Children said they learned and felt good when they were performing the tasks at the schoolyard; 6. Collaborative work and social skills were developed, which also demonstrates the benefit of a peer learning approach; 7. Children also liked to be in contact with vegetable gardens, comparing them with their owns at home, and they loved to take care of them and to be in contact with the plants and the soil; 8. It is a simple and unique way and opportunity to develop skills such as observation, planning, cooperation, communication, decision-making and the possibility of exploring their own interests; 9. The families, who took the pilot and responded to the survey, rated the project very highly. The comments that children made to their parents about the project were also very positive.
ReferencesBoud, D., Cohen, R., & Sampson, J. (1999). Peer learning and assessment. Assessment & evaluation in higher education, 24(4), 413-426.
Feille, K. (2021). A framework for the development of schoolyard pedagogy. Research in Science Education, 51(6), 1687-1704.
Feille, K. (2013). Getting outside: Three teachers' stories of using the schoolyard as an integrated tool for elementary teaching. The Electronic Journal for Research in Science & Mathematics Education, 17(3).
Hathaway, M. D. (2016). Agroecology and permaculture: addressing key ecological problems by rethinking and redesigning agricultural systems. Journal of Environmental Studies and Sciences, 6, 239-250.
Martins, C. B., Fernandes, D., & Guedes, T. M. (2020). Observar, manipular e comunicar sequências e regularidades da Ribeira do Porto. Indagatio Didactica, 12(5), 369-392. https://doi.org/10.34624/id.v12i5.23490
Raynolds, M. (2022). Permaculture and Sustainable Educational Systems. Holistic Education Review, 2(2).
Sá, Â. F. G. D. (2016). Espaço exterior como promotor de aprendizagens: Brincar e Aprender (Doctoral dissertation).
Tsevreni, I., & Bentenidi, K. (2013). Space as a pedagogical tool for children with additional educational needs participation and empowerment. Education in the North.
Yard4All (2022). Yard4All - Using School’s yard for ALL child’s wellbeing and development. Link: https://www.yard4all-project.org.
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