2 Working on visualisation
Research by Dörfler (1991) and Van Hiele (1986) suggests that visualisation is a powerful tool for learning mathematics. (You can find out more about visualisation in the unit Using visualisation: algebraic identities.) However, most visual representations have limitations in their mathematical use. Students have to become knowledgeable about which visual representation to use for a particular purpose.
In Activity 2, students explore these limitations for the number line. The activity asks the students to identify the properties of the different number types. It also offers choice in selecting their own examples, thereby giving the participating student a sense of ownership.
Activity 2: Representing numbers on a number line
This activity works well for students working in pairs or small groups so that ideas and opinions can be shared and challenged easily. If your students are not used to working in this way, you will need to plan how they will be organised in pairs or small groups and how they will sit so that they can easily talk to each other.
Tell your students the following:
- Plot the following numbers exactly on a number line:
- 2 – 4
- Are you happy with the results? Were you able to plot all of them in exactly the right place? How do you know?
- Can you think of other numbers that will be difficult or easy to represent on the number line?
Case Study 2: Teacher Abhay reflects on using Activity 2
I tried Activity 2 first myself and could see the potential of the learning. However, it is not the kind of activity that we are used to doing at school, so I was very apprehensive. I will describe how I went about overcoming that apprehension and how my experiences influenced my planning for this activity with my students.
Because I did not feel confident in using this activity straight away with my students in class, I decided to try it out first on my colleagues. The colleagues, not all mathematics teachers, kindly marked the numbers on the number line. I then asked them how they had done this. Their methods for representing the whole and rational-terminating numbers were the same, but they had used different methods to mark the rational non-terminating and irrational numbers. For example, for some of the numbers they used the decimal representation, while for others they used constructions. A discussion followed about whether these numbers could be plotted exactly on a number line, and what the limitations are of the number line as visualisation.
Asking my colleagues to do the task first, and reflecting on this, helped me in planning to lesson with my Grade IX students. I noticed my colleagues learned from listening to each other. I had not told them how to do it. It made me think: could my students learn in the same way? What makes me think their learning would be any different?
I decided that I would like my students to learn from their classmates as well so that they can discover which methods there are and think about their effectiveness and limitations, just like my colleagues had. This meant that I had to adapt my lesson planning to make sure there would be opportunity for a knowledge exchange to happen, i.e. I had to plan for sharing time.
So, by first doing the task myself I was convinced this was a stimulating task with great learning opportunities. Asking my colleagues to do the task helped me to overcome my anxiety about trying it out in my classroom. Hearing their responses also made me question my assumptions about learning and pushed me into thinking about different teaching approaches so that shared learning could take place. I will plan some lessons to try out my ‘new’ ideas.
Pause for thought
For more information, read Resource 5, ‘Planning lessons’.