Feeling comfortable with working on combination shapes and solids is important, both in school mathematics and in real life. Buildings, chairs, cutlery, rangoli patterns, mosques and temples all consist of not one shape or solid, but of several put together. People are familiar with combinations of shapes, solids and volumes, but students often find it a difficult topic to deal with in school mathematics.
One of the reasons for this might be that the chapters on volume and surface area are perceived by the students to be about a series of procedures to be followed and complicated formulae to be memorised. This encourages the students to become passive learners and they might experience mathematics as something that is ‘done to them’, without any possibility for developing their own thinking and being creative. This can result in students feeling powerless, disengaged and despondent about learning mathematics.
In this unit you will focus on how to teach composing and decomposing combination solids and shapes, and the mathematical thinking involved in this process. Through activities you will also think about how to develop students’ capacity to make choices and play a more active role in their own learning.
The learning in this unit links to the NCF (2005) and NCFTE (2009) teaching requirements as specified in Resource 1.
‘Mathematical trauma’ sounds rather dramatic. However research suggests that some students experience real stress while studying mathematics (Lange and Meaney, 2011). These students feel and believe that they are unable to act or think for themselves when learning mathematics. It may seem easy to dismiss or ignore this and say, ‘Well, these students just do not get it’, or ‘They should study harder and practise more’. But there are real reasons to believe that this trauma is stopping some students understanding and then using mathematics in their everyday lives, with many negative consequences to them and society as a whole.
Mathematical trauma can have serious consequences for students who are affected. They may reject mathematics as something that they are not able to do and will never be capable of doing. Students may get into a spiral of selffulfilling prophecies, because the moment they cannot make sense of an area of mathematics, they believe it is because they simply do not, and never will, understand the topic. This can also affect their belief in themselves as being able to do other areas of mathematics as well. They begin to feel they have no choice or control.
One of the aspects of mathematics that can bring on mathematical trauma is the language of mathematics itself – both the symbolic representation and mathematical vocabulary, which can feel very alien and hard to connect to existing language knowledge and structures.
Activity 1 aims to help you address the issue of how to deal with mathematical vocabulary with your students. It requires the students to devise their own mathematical dictionary with:
Although in this case it is related to the vocabulary encountered in the chapter on surface area and volume, this approach can be taken for all topics in the mathematics curriculum.
In Part 2 of Activity 1, students are asked to reflect on their learning in Part 1. This is repeated in most of the activities in this unit. The purpose of this is for students to become more aware of what makes them learn and to become more active in their learning. This will give them a sense of choice and control over their learning.
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 the activities yourself will mean that you get insights into a learner’s experiences that can in turn influence your teaching and your experiences as a teacher.
When you are ready, use the activities with your students. After the lesson, think about the way that the activity went and the learning that happened. This will help you to develop a more learnerfocused teaching environment.
Students may do this activity individually or in pairs. It may well be an activity that is repeated with new topics, building over time, or it may be used as a revision activity. Students may even develop their own dictionaries in a separate notebook, or you may develop a class dictionary where students write entries that are then put on display and maybe reworked over time.
Tell your students to look at the chapter in their textbook about area, volume or surface area, before doing the following:
Tell your students that this part of the activity asks them to think about their learning so that they can become better at learning mathematics and feel better about it.
This is the account of a teacher who tried Activity 1 with her secondary students.
When I read about mathematical trauma, I could immediately think of several students who might be experiencing this. I also have to admit that until now I have taken the stance that some students ‘get it’ and others do not. Perhaps this is because I have never struggled with mathematics that much – which is why I became a mathematician and a mathematics teacher. So before starting on this activity, I made myself promise I would really try to support students in making their own choices.
I had expected this activity to require quite a bit of prompting by me to get them to engage, but they all got busy over their books and started finding words. It seemed they knew exactly where to look!
After a few minutes, Mina asked whether they had to identify only the ones that they did not clearly understand. Because I wanted them to make their own choices, I suggested that they could do what they felt was best and that it would be nice if we all could share their ideas, thoughts and descriptions about the words they had selected. This sharing of ideas led to interesting mathematical discussions. It also brought out some of the misconceptions that the students had and made it possible to discuss those in an informal way.
For example, we had a great conversation about the term ‘volume’: Rohit described volume as what can be put inside a figure; Sohan said volume is what a solid is made of; Rina said volume is the amount of liquid it can hold. The discussion that followed was lively with students willing to share their ideas and I was pleased to see that students did not appear to be crushed by others commenting on their ideas or suggesting other descriptions. Several concepts were talked about and clarified in the process.
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 get on and those where 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 Chadha, did some quite small things that made a difference.
Pause for thought After the lesson, think about these questions:

Working with combined shapes and solids is a good application of using analytical and logical thinking – which is a very mathematical activity in its own right. Composing and decomposing combined solids and shapes is also great for valuing students’ contributions and thinking, because there are normally many ways to get to an answer! This means that:
Activity 2 asks your students to bring their own examples from home and think about different ways to work on the mathematics involved. It requires students to exchange their ideas with other students. They could be working in pairs or small groups.
Ask each student to bring any one utensil to class (for example, a spoon, glass, bowl (katori), container (of any shape), bottle, serving spoon (karchi), wok, pan, etc.). It is a good idea to bring in a few examples yourself so that you have enough utensils for everyone.
Tell your students the following:
Tell your students that this part of the activity asks them to think about their learning so that they can become better at, and feel better about, learning mathematics.
You may also want to have a look at the key resource ‘Using local resources’.
This activity made me realise how much more students engage with their learning when they use something that they bring to school. It seems it gives them automatic ownership of their learning! There was excitement when the students entered the classroom, showing the objects they had brought with them and wanting to know what exactly they were expected to do with them.
When the activity was given, they were grouped in fours so that they could have a variety of objects to investigate. I told them they were to pool their objects, but that I first wanted them to think about the questions individually. They were asked to each keep notes to talk about their thinking later in the group discussion. I insisted on this individual work because I wanted them to become aware of their own mathematical thinking power, and develop and value their own ideas. I wanted them to feel in control of their own thinking. If they were stuck with thinking about one object, they could select another one.
After about ten minutes I asked them to talk to each other about their answers. I told them that at this point they did not actually have to find the areas or volumes – they just had to talk about what shapes they could identify, or decompose their object into, in order to find the area or volume. I did not want them caught up in calculations and getting stressed about not remembering formulae. I wanted them to think about the thinking process involved in working with combined solids.
The discussion over the idli maker became rather intense, because some decided that they were hemispheres and there were a few who said that they were not exactly halfspheres – they felt they were parts of spheres. I noticed that some of the students were bridging the gap between their concrete understanding about shapes, solids, volumes and area by explaining their thinking while feeling and touching the utensils.
I was especially glad to hear the way that the students listened to each other. For some of the objects, the task was more difficult. One student who is otherwise quite quiet and reticent in class offered up a helpful idea in her group about two semicircles actually making up a whole sphere, and I could see her pleasure when the other students praised her contribution – perhaps this will help her build her belief that she can do mathematics.
Pause for thought

When students look at solved examples of mathematics problems in a textbook, they can look daunting. To students, they may look like a string of alien symbols that are supposed to make sense – a feeling that can be very intimidating. This is not restricted to the chapters on calculating area, volume and surface area of combined solids and shapes! The examples do make sense once you engage with the writing and the deciphering of the mathematical symbols.
To help students overcome any sense of feeling overwhelmed by the symbolic notation of mathematics, it may help if they can identify what makes an example easy or difficult, and then make their own easy and difficult examples. Doing this can demystify the mathematical writing of symbols and offer them a gentle way into making sense of mathematical symbols. Making up their own examples also lets the students create mathematics themselves, which gives them some control over their own learning and thus creates a sense of ownership that can increase engagement and participation. Another added benefit is that, as a teacher, you end up with lots of examples to work with and exchange in the classroom!
Activities 3, 4 and 5 ask students to identify, characterise and devise easy and difficult examples. This approach works in any area of mathematical learning. The topic of combined shapes and solids has its own particular challenge of having to use rather complicated formulae for calculating the area and volume of specific shapes and solids.
To prepare and support students with the specific symbolic writing demands of this, Activity 3 asks them first to write their own formulae booklet with illustrations. The students could add to this booklet any other formulae they come across in their mathematics learning, in which case it might be good to work on loose paper sheets that can then be added to and reordered when appropriate. Having formulae to hand also reduces the stress students might be experiencing in having to remember formulae and can let them focus on the thinking process that is required for their calculations.
Advise your students that this activity is similar to Activity 1 but they are now being asked to focus on mathematical formulae rather than words. They should have a page for each formula as they will be adding pages and will want to organise the formulae over time into an order that makes sense.
Tell your students:
Organise your class into groups of three where each of them works on one example, but where they discuss what they are doing.
Tell your students in their groups to look at the solved examples and questions in the chapter in their textbook about volume and surface area of combined solids, and to do the following:
Bring the class back together and discuss the last two points to find out how far students have been able to articulate what factors make an example easy or difficult, and what inventive ideas they have about making an example even harder. You could get the class to vote on which example is the most difficult and then set that for homework!
Tell your students that this part of the activity asks them to think about their learning so that they can become better at, and feel better about, learning mathematics.
What did you learn about how you (could) learn mathematics?
You may also want to have a look at Resource 2, ‘Involving all’, to find out more.
Ask the students to imagine that they are a writer of questions for mathematics examinations and they have been asked to devise three questions on the topic of surface area and volume of combined solids: one easy, one average and one difficult question. Give them the following instructions:
Tell your students that this part of the activity asks them to think about their learning so that they can become better at, and feel better about, learning mathematics.
Activity 3 was given to the students as an independent exercise and I went around observing how they were able to do it. They identified almost all the formulae well and wrote down the shapes that they represented, but when it came down to drawing and writing down in their own words what it meant to them, they had some problems. To help them become more aware of their own learning, and to be able to pinpoint what it was they were stuck on, I asked the students to note down their thoughts about their problems at whatever point they were, so that they could contribute to a discussion. The main issue appeared to be about drawing threedimensional solids. Because I wanted the students to know that there is not just one correct way of doing this, I called the students who had been able to draw a certain figure to come and draw it on the blackboard.
Once the students had some ideas about how to draw a threedimensional solid, and had practised it themselves, we went on to discuss the explanations given. I asked all those who had different explanations about a certain formulae to share their thoughts and ideas so that all of the students could hear ideas and think about what makes a question easy or difficult.
We did Activity 4 over two periods because they got so engaged with the activity. They worked on their own but talked to a classmate about the choices they had made. They used their dictionary and formulae booklet without prompting and I did notice students using their fingers to point and keep track of what part of the example related to which formula. I also saw students covering up part of the drawings in order to ignore the bits they were not working on, so that they could focus on the parts the calculations were about.
Mona said if only they could take such a dictionary and formulae booklet into the exam! We then had a discussion about how to try to remember the formulae using logical thinking. Sushant suggested that it could be helpful to think of a cylinder whose lateral surface area would be the circumference (that is, the perimeter for a circle) multiplied by its height and its volume the base area multiplied by its height. Sushant then said that you could then think about how the solid you were working with was different from the cylinder and adapt the formulae accordingly. We also discussed how this relates to going from two to three dimensions, and why some questions were difficult for some and why others were easier. Ramona said they were all easy, so I asked her to work on the last question to try to make it harder.
I gave them the first part of Activity 5 as a homework assignment and I told them that they had to prepare the test questions for their classmates, although who would get to solve whose would be a mystery. They came back the next day enthusiastically with their questions, happy that they – rather than me, or an examination board – were setting the test. The next day I distributed their questions randomly, although I did try to match the difficulty level to individual students’ attainment. I had to swap two papers when I got to the end as I found that I was giving Mona her own question back, and a couple of students had to double up as that day there were more students in class. They settled down to work on the problems. The class especially liked that the test was marked by the originator of the questions and they enjoyed doing the marking.
The activity allowed them to pinpoint what made a question hard. Instead of saying the whole topic was difficult to do, they agreed it was only the questions involving a frustum – and not because of its shape but because of its complicated formula, which is nearly impossible to memorise! We discussed how we could avoid memorising the formula, especially as so many mistakes are made with writing it down from memory. We talked about how the formula was derived from other, simpler formulae, and that working through that logical process means you do not have to memorise the impossibletoremember formula. I think this helped some students, but there still were those who insisted on learning the formula.
Doing all these activities did take a considerable amount of time, but I think it was well worth it. The students learned a lot of mathematics, seemed more relaxed and in control of their learning, and engaged actively with the tasks. All students, whatever their attainment, could do the task and could learn at their own pace and level. They had to think, be creative and make their own decisions. They really seemed to enjoy doing the mathematics and there were smiling faces and even laughter in the classroom – which I absolutely loved. I think they will also remember the mathematics they have learned more, which will save me time in the long run because I will not have to revisit the topic as often!
Pause for thought

This unit has asked you to explore how your students find the volumes of combinations of solids. It discussed ways for you to use to make students feel more involved in the process of learning mathematics, and how they can understand that mathematics is about reallife ideas, not just formulae in a textbook. You have learned how to support students in making choices in mathematics that can allow them to feel in control of their learning: choices about how to solve problems and how to explain ideas in their own words. Making choices means that they have to think through these ideas, which makes them learn more effectively and own that learning. They no longer feel that they are doing something that really has nothing to do with them.
These approaches are important, because many students find learning mathematics so traumatic that they simply do not want to think about it. They worry so much about using the one right process to get the one right answer that they are not able to think about the mathematics. They worry that they will look foolish if they give the wrong answer so they would rather not try. Overcoming these widely held beliefs will take time and persistence, but making sure that your students are involved in their lessons using the ways described in this unit will help them believe that they can do mathematics.
Pause for thought Identify three ideas that you have used in this unit that would work when teaching other topics. Make a note now of two topics you have to teach soon where those ideas can be used with some small adjustments. 
This unit links to the following teaching requirements of the NCF (2005) and NCFTE (2009), and will help you to meet those requirements:
The diversity in culture and in society is reflected in the classroom. Students have different languages, interests and abilities. Students come from different social and economic backgrounds. We cannot ignore these differences; indeed, we should celebrate them, as they can become a vehicle for learning more about each other and the world beyond our own experience. All students have the right to an education and the opportunity to learn regardless of their status, ability and background, and this is recognised in Indian law and the international rights of the child. In his first speech to the nation in 2014, Prime Minister Modi emphasised the importance of valuing all citizens in India regardless of their caste, gender or income. Schools and teachers have a very important role in this respect.
We all have prejudices and views about others that we may not have recognised or addressed. As a teacher, you carry the power to influence every student’s experience of education in a positive or negative way. Whether knowingly or not, your underlying prejudices and views will affect how equally your students learn. You can take steps to guard against unequal treatment of your students.
There are several specific approaches that will help you to involve all students. These are described in more detail in other key resources, but a brief introduction is given here:
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