1 ‘Compare and contrast’ tasks to learn about mathematical properties
‘Compare and contrast’ is a technique to make students aware of mathematical properties and applications. It is effective for learning about subtle samenesses and differences. When you compare you identify what is the same, when you contrast you identify what is different.
The actions of comparing and contrasting force us to think about the properties of mathematical objects and to notice what is the same and what is different. While doing so, students may make connections they might not normally consider. They are prompted into mathematical thinking processes such as generalising, conjecturing about what stays the same and what can change (called ‘variance’ and ‘invariance’), and then verifying these conjectures. This is an example of the national curriculum requirements of helping students use abstractions to perceive relationships, to ‘see’ structures, to reason things out for themselves, and to argue the truth or falsity of statements.
Volume and capacity are properties of three-dimensional objects. Volume is the space that a three-dimensional object occupies or contains; capacity, on the other hand, is the property of a container and describes how much a container can hold. Students often get confused by these two concepts (Watson et al., 2013). Activity 1 will help your students to become aware of the characteristics and measurements of three-dimensional shapes. The activity also requires the students to start thinking intuitively about the difference between volume and capacity.
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 learners’ experiences which can, in turn, influence your teaching and your own experiences as a teacher.
When you are ready, use the activities with your students and, once again, reflect and make notes on how well the activity went and the learning that happened. This will help you to develop a more student-focused teaching environment.
Activity 1: Exploring three-dimensional objects
- Ask your students to name any object that they have used during the previous day. As they name objects, write them on the black board. You will end up with a list of objects, e.g. glass, tube of toothpaste, plate, book, pen, pencil, coins, ruler, paper, bowl, knife, spoon, bottle, eraser, chalk, telephone, television, bucket, mug, towel, ball, etc.
- Once your students are done, circle some of these objects and ask them if they can find something that is common to all the circled objects. Choosing objects for which the students could easily estimate the three dimensions will save time.
Now arrange the students into small groups or pairs. Ask the students the following questions:
- For each circled object, estimate the following:
| Object | Length | Width/breadth | Height |
|---|---|---|---|
| Glass | |||
| Tube of toothpaste | |||
| Book | |||
| Pencil | |||
| Coin | |||
| Bottle | |||
| Television |
- If all these objects were made of gold, which would be the most expensive (or least expensive)? Then, arrange these objects in the increasing order of their worth.
- In comparing the worth of each object above, which measurement was most useful? Why?
Ask the students to present their findings to the whole class. Not all students have to agree. As long as their reasoning is based on mathematical properties and on logic, then all arguments are acceptable.
Case Study 1: Mrs Meganathan reflects on using Activity 1
This is the account of a teacher who tried Activity 1 with her elementary students.
To get ideas for a list of objects the students had used the day before, I started with writing ‘glass’ and ‘book’ on the blackboard and said I had used these yesterday. Because asking them what objects they had used yesterday would be rather unusual for me to do, I thought it would help focus their minds.
They came up with lots of examples, which I wrote on the blackboard. To be honest, some of them would be really awkward and complicated to calculate the volume of, for example a bicycle! I could have left such examples on the blackboard for them to work with later in the activity but I was not sure how I would handle that as a teacher. So I said that I would now pick six of these objects, and picked the ones for which the three dimensions were easier to estimate. I think next time I actually will feel more confident in leaving the ‘awkward’ examples on the blackboard as well.
I put the students in groups of four or five – this I can do easily by asking every other row of students to turn around, so putting them into groups does not take too much time or hassle.
I drew the table on the blackboard as it says in the activity and wrote all the questions at once on the blackboard. I had been thinking whether to do that one step at a time but thought that having them all on the blackboard at once would:
- give the students an idea of how the activity would develop
- give more learning time to the students because they would not have to wait for others to finish each question.
This worked well, apart from at one stage, where I felt I was really running from one group to another to answer their questions, like ‘How do we do this?’ or ‘What do we do next?’ So I did stop the class after some time and said that if they had a question, please first check with the neighbouring group whether they knew. It became much more manageable for me after that!
The question about the worth of the object if it was made of gold did make them think about ideas to do with volume and capacity, without actually using those terms. The presentations and the discussions that followed developed these ideas further and proved very good scaffolding to lead to the thinking required in Activity 2.
 Pause for thought What do you think about Mrs Meganathan’s solution to finding herself running between groups? What additional strategies might she have used to make this part of the lesson more manageable? You may already have some good ideas for this, but if you are fairly new to working in this way have a look at Resource 2, ‘Managing groupwork’. |
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 a reflective exercise every time you undertake the activities.
Pause for thought Now reflect on how your class got on with Activity 1:
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What you can learn in this unit