3 Coping with daunting mathematical writing

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 re-ordered 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.

Activity 3: Making your own formulae booklet

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:

  • Look at the chapter in your textbook about area, volume and surface area.
  • Design a page with at least four sections. (Read through this activity before you decide on the layout of the page.)
  • Identify any formulae you come across and write them near the top of the page.
  • Above this, write down what this formula is for.
  • In the second column, write down the explanation that the book or your teacher gives for why or how this formula works.
  • Now write down your own explanation that makes sense to you in the third column. Use language and examples that make sense to you in the third column. It does not have to be complete yet, or entirely correct, because you will be able to make changes to it as your understanding grows.
  • Now make a drawing or sketch to give an illustration of what the word means that makes sense to you in the fourth column. Again, it does not have to be complete yet, or entirely correct, because you will be able to make changes to it as you gain confidence.

Activity 4: What makes a question easy, average or difficult?

Organise your class into groups of three where each of them works on one example, but where they discuss what they are doing.

Part 1: The mathematical activity

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:

  • Identify and agree on one easy, one average and one difficult solved example.
  • Draw the objects you chose. State in your own words what shapes and solids this object consists of (that is, decompose the combined solid into single solids).
  • Look at your formulae booklet, your dictionary and the drawing. Can you identify which parts of the worked out example relate to the parts of your drawing?
  • When you have all discussed and recorded your thoughts about the three worked out examples, think about what is the same and what is different between an easy, an average and a difficult example. What is it that makes an example easy or difficult? Make a note of your thoughts.
  • Look at your difficult example. Work together to make it even harder by adding or changing something.

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!

Part 2: Reflecting on your learning

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 find easy or difficult about Part 1 of this activity?
  • What did you like about this activity?
  • What mathematics did you learn from this activity?

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.

Activity 5: Making your own mathematical examples

Part 1: The mathematical activity

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:

  • Write the questions. Remember you have to provide solutions as well!
  • Exchange your examination questions with another student in the classroom and solve each other’s questions. Check the answers against the solutions.
  • Discuss with your partner what makes a question easy or difficult. Discuss with your partner good methods to tackle such questions. Write these methods down.

Part 2: Reflecting on your learning

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 find easy or difficult about Part 1 of this activity?
  • What did you like about this activity?
  • What mathematics did you learn from this activity?
  • What did you learn about how you (could) learn mathematics?

Case Study 3: Mrs Meganathan reflects on using Activities 3–5

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 three-dimensional 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 three-dimensional 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 impossible-to-remember 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

  • 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 Mrs Meganathan did? If so, what was your reasoning for doing so?

2 Many ways to get to an answer