Creative thinking in mathematics: proportional reasoning

2 Creativity in learning mathematics

Creativity in learning is partly about allowing students to enjoy learning more and to think for themselves. Creative thinking in mathematics also importantly prepares students for their jobs in the future. In the future, jobs will rely less and less on doing things mechanistically, as this can be done by computers, and more on problem solving, and coming up with creative solutions.

It is not always easy to see how school mathematics and textbook practice can be turned into creative learning approaches. This unit aims to give some ideas for creative approaches. It builds on the perspective of creativity as ‘possibility thinking’. When you and your students can think of possibilities, or ‘what if’ scenarios, then this process will lead you to be creative in your thinking (Aristeidou, 2011).

Researchers have identified a list of teaching and learning features that are involved in possibility thinking in the classroom (Grainger et al., 2007; Craft et al., 2012). These features can be effectively addressed through the design of activities and by asking open questions and include:

  • question posing and responding
  • being imaginative
  • play/playfulness
  • risk taking.

The tasks in this unit work by developing learning activities that use these features.

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 for yourself will mean 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 and reflect again on how the activity went and the learning that happened. This will help you to develop a more learner-focused teaching environment.

Activity 1: Thinking about the physical effects of multiplication

Preparation

On the day before you have planned to conduct this activity, tell your students that they have to bring a thin dry stick of any length to the next lesson. It is important that they can break the stick easily.

This activity works best when the sticks are clearly of different lengths. If you feel that most students have sticks of approximately the same length, ask some of them to cut off a portion of their stick to shorten the length. Spare pieces could then be given to students who have not brought a stick.

At the start of the lesson, ask each student to hold the stick they have got in their hands and to raise their hands. If some students forgot to bring a stick, ask other students if they would be kind enough to share a part of their stick with these students.

Figure 1 Coconut stick brooms. (Source: Fotokannan)

The activity

Put the students into pairs or small groups.

Ask your students to answer the following questions in their groups or pairs:

  • Who has the shorter stick?
  • What is the difference between the lengths of shorter and longer sticks?
  • How many times does the smaller stick fit into the larger stick?
  • In previous questions you compared the lengths of the shorter and longer sticks in two ways. What is the difference between these two ways of comparison?

Let some groups present their findings to the class to initiate a discussion, guided to students discovering the physical effects of multiplication.

Case Study 1: Mrs Rawool reflects on using Activity 1

This is the account of a teacher who tried Activity 1 with her elementary students.

The students loved the idea of bringing thin sticks and the majority of them did get them, but again as usual there were those who forgot. They were feeling bad so they welcomed the suggestion that they could get sticks from their classmates. I put the students in groups of four. I could organise that easily by just asking two students in each group to turn around and face the other two students.

Once all of them had the sticks and were seated in their groups, they did the comparison both ways as suggested in the activity. I also asked them to note down their observations so that they could contribute to the discussions that we would be having later.

Some of them had a problem, saying that they could find the difference easily but could not tell the multiplication bit very easily. But all of them tried, and the discussions were interesting. Some of them were very quiet and had to be coaxed into giving some input. Most of them were, as usual, worried about the fact that what they said might not make sense and then they would be laughed at. So I had to convince them that there was no problem in saying the wrong things because no one is an expert and that making mistakes would help them in their learning. I had not realised how hard it is for the students to feel confident to have a go and to take risks as they might get the answer wrong.

Mona, one of the students, remarked that they were estimating. They did not have any standardised measuring equipment such as rulers or tape measures to use which meant the measurements were not accurate in any way and so they were all just guessing. It seemed the students were more willing and confident in using their imaginations and coming up with their own ideas and answers afterwards. That made me think that perhaps if I want my students to take more risks in their learning of mathematics, I should give them more activities where there are no wrong answers, but many right answers.

Pause for thought

Mrs Rawool commented on how she had to work hard to convince some of her students that it was better to give a wrong answer than no answer at all, because making mistakes would help them in their learning. How do you feel about this approach? What strategies might Mrs Rawool adopt over the longer term to create an atmosphere in her classroom in which all students feel comfortable taking risks to help them in their learning?

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?