Practice Paper 4H (Calculator) Set C Higher Tier Time: 1 hour 45 minutes Materials required for examination Items included with question papers
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TESS India Teacher Education through School based Support aims to improve the classroom practices of. elementary and secondary teachers in India through the provision of Open Educational Resources OERs to. support teachers in developing student centred participatory approaches The TESS India OERs provide. teachers with a companion to the school textbook They offer activities for teachers to try out in their. classrooms with their students together with case studies showing how other teachers have taught the. topic and linked resources to support teachers in developing their lesson plans and subject knowledge. TESS India OERs have been collaboratively written by Indian and international authors to address Indian. curriculum and contexts and are available for online and print use http www tess india edu in The OERs. are available in several versions appropriate for each participating Indian state and users are invited to. adapt and localise the OERs further to meet local needs and contexts. TESS India is led by The Open University UK and funded by UK aid from the UK government. Video resources, Some of the activities in this unit are accompanied by the following icon This indicates that you. will find it helpful to view the TESS India video resources for the specified pedagogic theme. The TESS India video resources illustrate key pedagogic techniques in a range of classroom contexts in. India We hope they will inspire you to experiment with similar practices They are intended to complement. and enhance your experience of working through the text based units but are not integral to them should. you be unable to access them, TESS India video resources may be viewed online or downloaded from the TESS India website. http www tess india edu in Alternatively you may have access to these videos on a CD or memory card. Version 2 0 EM05v1,All India English, Except for third party materials and otherwise stated this content is made available under a Creative. Commons Attribution ShareAlike licence http creativecommons org licenses by sa 3 0. Asking questions that challenge thinking fractions. What this unit is about, In this unit you will think about how to introduce fractions to your students. Some students can see fractions as a very difficult topic to understand There are many reasons for this. but making sure that your students have rich and varied experiences of working with fractions will help. them to develop their understanding, In this unit you will explore the fact that a fraction only has meaning when looked at in relation to a whole. and consider how to help your students to get to know about different ways to read the symbolic. representations of fractions, Through activities you will also think about the value of asking your students interesting and challenging. questions of getting your students to ask questions themselves and talking about fractions. What you can learn in this unit, How to ask effective questions that are interesting and challenging. Some ideas to help your students construct their own understanding of fractions. Some ideas to help your students talk about fractions. This unit links to the teaching requirements of the NCF 2005 and NCFTE 2009 outlined in Resource 1. 1 What s so difficult about fractions, One of the reasons fractions can seem so difficult is that there is a lot to understand For example half of. something can be smaller than a quarter of something else An example of this is half of six is three and a. quarter of sixteen is four So learning about fractions by folding pieces of paper or by dividing circles may. mislead students especially if the paper is always the same size Students must be taught to ask A fraction. Developing an understanding of fractions is not so different from learning to understand other. mathematical concepts For example very young children are offered many different experiences as they. learn to generalise the concept of three, Despite being older when they learn about fractions elementary students similarly will need a great many. rich and varied experiences if they are to begin to develop a good understanding of fractions. Many students will have had experiences that help them to develop some understanding of fractions In her. research Nunes 2006 found that primary school students already have insights into fractions when. solving division problems, They understand the relative nature of fractions if one student gets half of a big cake and the other. gets half of a small one they do not receive the same amount They also realise for example that. you can share something by cutting it in different ways this makes it different fractions but not. different amounts Finally they understand the inverse relation between the denominator and the. quantity the more people there are sharing something the less each one will get. www TESS India edu in 1, Asking questions that challenge thinking fractions. Talking fractions using the language, Encouraging the students to talk about fractions and use the vocabulary will help them understand some of. the difficult vocabulary associated with fractions The questions you use should show the students how. important the correct vocabulary is so that everyone knows what is being referred to. First model some ways of talking about fractions and drawing attention to how words are used Then focus on. getting your students talking The more the students use the words themselves the more they will build their. understanding of fractions Asking the students to make up questions to ask one another is a good way to get. them talking Another way is to ask the students to explain the reasoning they used to arrive at their answers. The first activity is for you to think about issues of learning fractions in your classroom. Activity 1 Thinking about your students learning fractions. Think about what your students need to know about fractions and make some notes on the different. ideas Use your textbooks If you have a multigrade class you will need to think about what different. students need to know about fractions,how to find out a fraction of a quantity. what fraction one quantity is of another,how to add fractions together. For each of the ideas associated with fractions write down how the vocabulary associated with those. ideas and the way it is used to express ideas For example half of ten means divide 10 by 2 but it can also. mean multiply 10 by The students might also see which has the same outcome and is thus equivalent. in meaning but which may also be expressed as 10 divided by 2 or 10 shared between 2 people. Think about some specific students in your class What activities might help them to understand the. different ways that fractions can be expressed and the different meanings given to those interpretations. 2 Developing an understanding of fractions, The second activity focuses on students physically representing the concepts of fractions This is also. called embodiment You will ask them to use their bodies to represent mathematical ideas If the students. move themselves to make fractions of a whole they will begin to develop their concept of what a fraction is. and how they can work with fractions, Before attempting to use the activities in this unit with your students it would be a good idea to complete all. or at least part of the activities yourself It would be even better if you could try them out with a colleague as. that will help you when you reflect on the experience Trying them for yourself will mean you get insights into. a learner s experiences which can in turn influence your teaching and your experiences as a teacher. Activity 2 Physically representing fractions,Preparation. First create a space and ask eight students to come to the front of the class or somewhere where the rest. of the class can see them,2 www TESS India edu in, Asking questions that challenge thinking fractions. The activity, Ask your students to arrange themselves into a rectangle. Ask someone else to divide the group into half, Reform the rectangle then ask another student to divide the group in half in a different way. Ask the students what is the same and what is different about the new half of the group. Now ask another student to divide the eight students into quarters fourths Again ask whether. there is a different way to do this division and what is the same and what is different about the. new way of dividing into quarters, Now change the number of students and go through the process above again It may be that. dividing into quarters is difficult but depending on the chosen number continue asking for and. so on until a fraction that cannot be done is reached Ask the students why you cannot find that. fraction of these students Dividing one student into bits is not allowed. Ask the students to work in groups of 12 You could appoint a leader in each group to note down. ideas if the class does not split evenly into groups of 12 Ask them to work out all the fractions. they can divide 12 students into,Video Using questioning to promote thinking. http tinyurl com video usingquestioning, Case Study 1 Mrs Rawool reflects on using Activity 1. This is the account of a teacher who tried Activity 1 with his elementary students. First I invited eight students to come to the front of the class and to form themselves into a rectangular. shape where the rest of the class could see them I then asked student Anoushka to come and divide these. eight students in half which was easy to do, I then asked the class if the group of eight students could be divided in half in another way This proved to. be a little challenging as the students were used to mathematics questions having just one answer so they. wondered at first if Anoushka was wrong They needed clarification about what different meant here Of. course whichever way they divided the students in half there were always four students in each half. Since this was the answer I was looking for I gave them time to talk about these ideas. Next I asked student Nita to come to the front and divide the group into quarters This time the students. were able to suggest different ways to achieve this and they were happy there would always be two. students in each part, I then asked another group of students to come to the front this time with six students This time I asked. them to divide themselves into half in two ways I asked Do you always get the same answer Yes sir. they said Then I asked What other fraction can you divide yourselves into They tried to divide. themselves into quarters but they could not but what they did find was that they could divide themselves. into three parts and discussed what this fraction was called. I then put the class into groups of 12 and asked them what fractions they could embody in their groups. One group came up with twelfths but most worked happily on halves quarters thirds and sixths. www TESS India edu in 3, Asking questions that challenge thinking fractions. Reflecting on your teaching practice, When you do such an exercise with your class reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and being able to progress and those you. needed to clarify Such reflection always helps with finding a script that helps you engage the students to. find mathematics interesting and enjoyable If they do not understand and cannot do something they are. less likely to become involved Use this reflective exercise every time you undertake the activities noting. as Mrs Rawool did some quite small things that made a difference. Pause for thought,Good questions to trigger such reflection are. How did it go with your class,What responses from students were unexpected Why. What questions did you use to probe your students understanding. Did you feel you had to intervene at any point,What points did you feel you had to reinforce. Did you modify the task in any way If so what was your reasoning for this. 3 Asking questions effectively, Teachers ask a lot of questions in their work some research suggests that teachers ask up to 400 questions. every day when they are teaching The better the questions that teachers ask the better their teaching will. Much research has been undertaken about good questions for example by Wragg and Brown 2001 and. Hattie 2008 The research concludes that effective questions. are firmly linked to the learning of the lesson,build on students previous knowledge. involve interest and motivate the students, are sequenced to encourage higher order thinking but not too soon. enable the students to build their own knowledge,reveal misconceptions and misdirections. prompt and challenge thinking and reasoning,Pause for thought. Reflect on the questions you asked in the last lesson you taught. Did they challenge the students to think, Could a small change have uncovered more about the students current learning. Could your questions have encouraged the students to build their own learning more. Activity 3 asks you to first prepare for asking effective questions and then to try out these questions when. teaching your students,4 www TESS India edu in, Asking questions that challenge thinking fractions. Activity 3 Asking effective questions about fractions. Part 1 Preparing to ask effective questions, If you can do this part of the activity with another teacher you may find that it is easier. Think about the next lesson in which you will teach on fractions What is it you want the students to know. Write some notes about that now, What previous knowledge do you think they will need in order to understand the ideas you want them to. learn Write a question which will enable you to know whether or not they have that prior knowledge For. example you could ask your students Can you give me an example of And another And another And. another And another Asking for more examples could help you to find out the extent of their knowledge. and some of the students misconceptions, Think about some of the ways that fractions are used in the real world Write a question that might. interest or engage the students because it is based on something they know about and use. Now write an easy question for the particular topic you have to teach and then write a hard question Write. a sequence of questions that will challenge your students but not too much. Think about all the ways that misconceptions can happen in fractions Write two or three questions that. will help you check whether or not your students have these misconceptions You can find some examples. of such questions in Case Study 2 It is also important to think ahead about how you might respond to your. students answers in the best way to reinforce learning and extend their thinking You can use Resource 2. to help you think about some ideas for how to receive your students responses. Now write a question that will encourage your students to reason their way to a solution For example. Your big sister never believes what you say How will you convince her that your method works. Part 2 Using your effective questions in the classroom. Now you have written these questions use them with a class. Did you think that the class learned more because they used these questions. Don t forget to use real objects to allow your students to work with ideas on fractions and to approach. challenging questions through a process of reasoning. Video Planning lessons,http tinyurl com video planninglessons. Case Study 2 Mrs Mohanty questions the students to check their. understanding of fractions, When thinking about Part 1 of Activity 3 I decided I would use my normal introduction to fractions by. demonstrating fractions on the blackboard as usual but to be very precise and repetitive in the questions. and instructions I was going to use I wrote them down on a piece of paper and put them on my desk so I. would not forget them,www TESS India edu in 5, Asking questions that challenge thinking fractions. These are the questions and prompts I prepared, Show me how you divide this circle into halves quarters eighths. o How do you know this is correct,o Please describe your method clearly. o Would anyone do it another way, Show me on this circle one half one quarter one eighth. o How do you know this is correct,o Please describe your method clearly. o Would anyone do it another way, Show me on this circle one third one sixth one twelfth. o How do you know this is correct,o Please describe your method clearly. o Would anyone do it another way, Show me on this circle one third one fifth one seventh. o How do you know this is correct,o Please describe your method clearly. o Would anyone do it another way,Show me on this circle three quarters six eights. o How do you know this is correct,o Please describe your method clearly. o Would anyone do it another way, I drew the circle using chalk I then invited students to come to the blackboard and asked them the. questions Having the questions written down really helped me to focus and helped to avoid diversions. from what I had intended to do I also noticed that as a result there was less teacher talk and more. student talk and student work,Pause for thought, What questions did you use to probe your students understanding. Did you feel you had to change your planned questions at any time Why. How effective did you feel your responses to the students answers were in. reinforcing learning and helping you to understand the way your students think. 4 Effective questioners give students time to think. Mary Budd Rowe 1986 researched the wait time that teachers allowed after asking a question Wait time. is the length of quiet time that teachers allow after asking a question before they expect a student to. answer or before they rephrase the question or even answer the question themselves Her team analysed. 300 tape recordings of teachers asking questions over six years They found the mean wait time was 0 9. If you ask a question that requires the students to think are you really giving them enough time to think or. are you only giving them time to instantly react, The teachers in Budd Rowe s research were trained so that they became able to increase their wait time to. between three and five seconds The increased wait time resulted in. 6 www TESS India edu in, Asking questions that challenge thinking fractions. an increased length of student response, an increased number of unsolicited but appropriate replies. a decreased failure to respond,an increased confidence of response. an increase in the incidence of students comparing their answers to those from another student. the number of alternative explanations offered multiplying. In other words the students had more time to think and that increased the level and quality of discussion. that went on in the classroom which in turn meant the teachers learnt more about their students thinking. and were able to act on any misconceptions Increasing wait time is not easy to do and can feel odd when. you start but if your students are to think they must be given sufficient time. Activity 4 asks you to experiment with increasing the wait time in your classroom in a similar way. Activity 4 increasing the wait time, Like the teachers in Budd Rowe s research in your next lesson increase your wait time for students to. respond to five seconds After the lesson reflect on whether you observed. an increased length of student response, an increased number of unsolicited but appropriate replies. a decreased failure to respond,an increased confidence of response. an increase in the incidence of students comparing their answers to those from another student. the number of alternative explanations offered multiplying. The next activity links together many of the ideas that have been discussed so far It suggests that you. ask the students to work with concrete objects to answer some challenging questions. ask the students to work together so that they can support one another. give them more time to think,Activity 5 Learning about fractions. Preparation, This activity is an example of the kinds of rich activity that students need to build their understanding of. fractions For this task you will need a quantity of paper plates or card cut into rectangles of the same. Arrange the students to work in groups of three or four and give them a pile of paper plates or cards You. may want to look at the key resource Using groupwork http tinyurl com kr usinggroupwork to help. you prepare for this,The activity, Ask your students first to show you half of a plate then a quarter of a plate It is important not to. tell them to take one plate here let them think this out for themselves. Then ask each group to show you half of six using the plates. Make sure everyone is able to do this before carrying on. www TESS India edu in 7, Asking questions that challenge thinking fractions. Ask the students to suggest several fraction problems that they can solve using the plates Each. time ask the class to show you the solution using the plates If they don t suggest questions using. just one plate then prompt them, The idea is to give everyone time to play a little with fractions and to think about what fractions are. Ask the students what is the same and what is different about the two types of questions they. have worked on so far This will help the students recognise some of the different ways that. fractions are used in mathematics, Figure 1 A group of students using plates to learn about fractions. Now move on to problems that mix the two ideas, Ask the class to find of 12 then ask them to find of 13. When the class have had time to solve this ask one group to explain the process they went through. in order to solve the second problem, Now ask the students to suggest how they could record the problem and the answer using. mathematical notation Spend time on this as it is important that the students recognise the. relationship between the way the problems are written and what they are doing with the plates. Now ask the class to solve other hard fraction problems using the plates for example of 12 or. of 10 which require two plates to be shared out, Again ask for suggestions and discuss how these ideas might be written down. Ask each group to make up an easy problem and a hard problem with fractions to give to another. group to do Ask each group to record their answers. Case Study 3 Mr Bhatia reflects on using Activity 5. I gave each group 12 paper plates The plates were to help to support the students thinking that finding. fractions is about sharing out equally, First I set them the task of dividing the plates into quarters I asked several of the groups to talk about. the process of dividing into quarters Then I asked them to divide their 12 plates into thirds When they. had done this I once again asked the students to explain how they did it I made sure that everyone was. 8 www TESS India edu in, Asking questions that challenge thinking fractions. comfortable sharing out the objects that they were working with in this case the plates The students. enjoyed working and collaborating together in groups and completing the task. I then decided that the class was ready for a more challenging question I gave each group one more plate. so that they then had 13 plates and again asked them to divide the plates into quarters and then thirds. This time the students discovered that they needed to subdivide the extra plate in order to share out the. plates equally into quarters and thirds, This time I spent more time on the feedback session in order to make sure that everyone understood the. reason why one of the plates had to be subdivided I then asked the class to divide the plates into thirds. and this time I offered them scissors as well Several students gave some good reasons why they needed. to divide up the extra plate but working in groups helped them all to try out their ideas first before telling. the whole class,Pause for thought, What questions did you use to probe your students understanding. Did you feel you had to intervene at any point,What points did you feel you had to reinforce. Did you modify the task in any way like Mr Bhatia did If so what was your reasoning. This unit has focused on teaching fractions but you have also looked at how to ask questions that require. students to think and the importance of giving students sufficient time to think. In studying this unit you have thought about how to enable your students to develop their ideas about. fractions and about the necessity of providing rich and varied activities if students are to learn understand. and use ideas about fractions, You have also seen how reflecting on learning and how learning happens is important in becoming better at. Pause for thought, Identify three techniques or strategies you have learnt in this unit that you might use in your. classroom and two ideas that you want to explore further. Resource 1 NCF NCFTE teaching requirements, This unit links to the following teaching requirements of the NCF 2005 and NCFTE 2009 and will help you. to meet those requirements, View students as active participants in their own learning and not as mere recipients of knowledge. how to encourage their capacity to construct knowledge how to shift learning away from rote. www TESS India edu in 9, Asking questions that challenge thinking fractions. Let students see mathematics as something to talk about to communicate through to discuss. among themselves to work together on, Let students learn important mathematics and see mathematics is more than formulas and. mechanical procedures,Resource 2 Receiving answers from students. Your response matters, The more positively you receive all answers that are given the more students will continue to think and try. There are many ways to ensure that wrong answers and misconceptions are corrected and if one student. has the wrong idea you can be sure that many more have as well You could try the following. Pick out the parts of the answers that are correct and ask the student in a supportive way to think a. bit more about their answer This encourages more active participation and helps your students to. learn from their mistakes The following comment shows how you might respond to an incorrect. answer in a supportive way You were right about evaporation forming clouds but I think we need to. explore a bit more about what you said about rain Can anyone else offer some ideas. Write on the blackboard all the answers that the students give and then ask the students to think. about them all What answers do they think are right What might have led to another answer being. given This gives you an opportunity to understand the way that your students are thinking and also. gives your students an unthreatening way to correct any misconceptions that they may have. Value all responses by listening carefully and asking the student to explain further If you ask for further. explanation for all answers right or wrong students will often correct any mistakes for themselves you will. develop a thinking classroom and you will really know what learning your students have done and how to. proceed If wrong answers result in humiliation or punishment then your students will stop trying for fear of. further embarrassment or ridicule,Improving the quality of responses. It is important that you try to adopt a sequence of questioning that doesn t end with the right answer Right. answers should be rewarded with follow up questions that extend the knowledge and provide students with. an opportunity to engage with the teacher You can do this by asking for. a how or a why,another way to answer,a better word. evidence to substantiate an answer,integration of a related skill. application of the same skill or logic in a new setting. Helping students to think more deeply about and therefore improve the quality of their answer is a crucial. part of your role The following skills will help students achieve more. Prompting requires appropriate hints to be given ones that help students develop and improve. their answers You might first choose to say what is right in the answer and then offer information. further questions and other clues So what would happen if you added a weight to the end of your. paper aeroplane, Probing is about trying to find out more helping students to clarify what they are trying to say to. improve a disorganised answer or one that is partly right So what more can you tell me about how. this fits together,10 www TESS India edu in, Asking questions that challenge thinking fractions. Refocusing is about building on correct answers to link students knowledge to the knowledge that. they have previously learnt This broadens their understanding What you have said is correct but. how does it link with what we were looking at last week in our local environment topic. Sequencing questions means asking questions in an order designed to extend thinking Questions. should lead students to summarise compare explain or analyse Prepare questions that stretch. students but do not challenge them so far that they lose the meaning of the questions Explain how. you overcame your earlier problem What difference did that make What do you think you need to. tackle next, Listening enables you to not just look for the answer you are expecting but to alert you to unusual. or innovative answers that you may not have expected It also shows that you value the students. thinking and therefore they are more likely to give thoughtful responses Such answers could. highlight misconceptions that need correcting or they may show a new approach that you had not. considered I hadn t thought of that Tell me more about why you think that way. As a teacher you need to ask questions that inspire and challenge if you are to generate interesting and. inventive answers from your students You need to give them time to think and you will be amazed how. much your students know and how well you can help them progress their learning. Remember questioning is not about what the teacher knows but about what the students know It is. important to remember that you should never answer your own questions After all if the students know. you will give them the answers after a few seconds of silence what is their incentive to answer. Additional resources, A newly developed maths portal by the Karnataka government. http karnatakaeducation org in KOER en index php Portal Mathematics. National Centre for Excellence in the Teaching of Mathematics https www ncetm org uk. National STEM Centre http www nationalstemcentre org uk. National Numeracy http www nationalnumeracy org uk home index html. BBC Bitesize http www bbc co uk bitesize, Khan Academy s math section https www khanacademy org math. NRICH http nrich maths org frontpage,Art of Problem Solving s resources page. http www artofproblemsolving com Resources index php. Teachnology http www teach nology com worksheets math. Math Playground s logic games http www mathplayground com logicgames html. Maths is Fun http www mathsisfun com,Coolmath4kids com http www coolmath4kids com. National Council of Educational Research and Training s textbooks for teaching mathematics and. for teacher training of mathematics http www ncert nic in ncerts textbook textbook htm. AMT 01 Aspects of Teaching Primary School Mathematics Block 1 Aspects of Teaching. Mathematics Block 2 Numbers I Block 3 Numbers II Block 4 Fractions. http www ignou4ublog com 2013 06 ignou amt 01 study materialbooks html. LMT 01 Learning Mathematics Block 1 Approaches to Learning Block 2 Encouraging Learning in. the Classroom Block 4 On Spatial Learning Block 5 Exploring Numbers Block 6 Thinking. Mathematically http www ignou4ublog com 2013 06 ignou lmt 01 study materialbooks html. www TESS India edu in 11, Asking questions that challenge thinking fractions. Manual of Mathematics Teaching Aids for Primary Schools published by NCERT. http www arvindguptatoys com arvindgupta pks primarymanual pdf. Learning Curve and At Right Angles periodicals about mathematics and its teaching. http azimpremjifoundation org Foundation Publications. Textbooks developed by the Eklavya Foundation with activity based teaching mathematics at the. primary level http www eklavya in pdfs Catalouge Eklavya Catalogue 2012 pdf. Central Board of Secondary Education s books and support material also including List of Hands on. Activities in Mathematics for Classes III to VIII select CBSE publications then Books and. support material http cbse nic in welcome htm,References bibliography. Bell A 1987 Diagnostic teaching 3 provoking discussion Mathematics Teaching vol 118 pp 21 3. Budd Rowe M 1986 Wait time slowing down may be a way of speeding up Journal of Teacher Education. vol 43 pp 44 50 Abstract available from http jte sagepub com cgi content abstract 37 1 43. accessed 3 February 2014, Hastings S 2003 Questioning TES Newspaper 4 July Available from. http www tes co uk article aspx storycode 381755 accessed 22 September 2014. Hattie J 2008 Visible Learning A Synthesis of Over 800 Meta analyses Relating to Achievement New. York NY Routledge, Hattie J 2012 Visible Learning for Teachers Maximising the Impact on Learning Abingdon Routledge. National Council for Teacher Education 2009 National Curriculum Framework for Teacher Education. online New Delhi NCTE Available from http www ncte india org publicnotice NCFTE 2010 pdf. accessed 3 February 2014, National Council of Educational Research and Training 2005 National Curriculum Framework NCF New. Delhi NCERT, Nunes T 2006 Fractions Difficult but Crucial in Mathematics Learning Teaching and Learning Research. Brief Economic and Social Research Council UK Available from. http www tlrp org pub documents no13 nunes pdf accessed 3 February 2014. Watson A Jones K and Pratt D 2013 Key Ideas in Teaching Mathematics Oxford Oxford University. Wragg E and Brown G 2001 Questioning in the Secondary School London RoutledgeFalmer. Zack V and Graves B 2001 Making mathematical meaning through dialogue Once you think of it the Z. minus three seems pretty weird Educational Studies in Mathematics vol 46 pp 229 71. Acknowledgements, This content is made available under a Creative Commons Attribution ShareAlike licence. http creativecommons org licenses by sa 3 0 unless identified otherwise The licence excludes the. use of the TESS India OU and UKAID logos which may only be used unadapted within the TESS India. 12 www TESS India edu in, Asking questions that challenge thinking fractions. Every effort has been made to contact copyright owners If any have been inadvertently overlooked the. publishers will be pleased to make the necessary arrangements at the first opportunity. Video including video stills thanks are extended to the teacher educators headteachers teachers and. students across India who worked with The Open University in the productions.
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