2 Using paper folding as a manipulative to learn about angles
Using manipulatives such as blocks, plates, rods or counters in the learning of mathematics requires the students to use their hands to work out some mathematical problem. The idea is that this physical involvement helps the students to develop mental images of the concepts. The physical movements help students to think more actively about the mathematical processes involved and make mathematics more concrete and practical instead of a merely theoretical subject.
One of the challenges with using manipulatives to learn mathematics can be how to move from the manipulatives to a pen and paper approach without losing the imagery that has been developed by the use of manipulatives. The students must eventually move on to the symbolic representation of mathematics that they will encounter in textbooks.
Bruner (1966), an eminent educator, labelled these different ‘worlds’ as enactive, iconic and symbolic. Activity 2 aims to make that transition, and take the learning from ‘doing’ the mathematics in a physical way (Bruner’s enactive phase), to making or drawing pictures that represent enactive learning (Bruner’s iconic phase), to using mathematical symbolic notation and making sense of it (Bruner’s symbolic phase), as is found in textbooks and examination papers.
The second part of the activity addresses this challenge. Students bring their folded angles to the blackboard and trace them. By then asking the students to annotate this traced shape with symbols such as the angle symbol, the right-angle symbol and the size of the angle, the students move to the symbolic representation of mathematics.
Activity 2: Folding paper to learn about different angles
Old newspapers are relatively easy to obtain in large quantities. If you don’t have newspaper, try to find other paper. The first part of the next activity uses newspaper as a resource for paper folding. The students are asked to fold their paper to construct angles of the same measurements as in Activity 1.
Part 1: Constructing angles of different measures by paper folding
Show the students how to fold the paper so they get an angle of 180°.
Show using your fingers where the straight angle on a straight line would actually be (otherwise students sometimes cannot make the link between a straight line and a straight angle of 180°).
Ask the students to fold their paper, one at a time and holding it up so all can see, so they construct angles of the following measures:
- an angle of 90°
- an angle of 0°
- an angle of 180°
- an angle of 45°
- an angle of 135°
- a straight angle
- an obtuse angle
- a right angle
- an acute angle
- (in groups or pairs of students) several different right angles.
As in Activity 1, different examples are possible for the acute and obtuse angles. Discussing the validity of these different examples offers a good opportunity to talk about definitions and possible variations within these definitions. Doing this will again help the students to refine their understanding, refine the mathematical language they use and help them to memorise the ideas. The aim of the last point is for students to realise that right angles are the same even when the shape is enlarged.
Part 2: Moving on to iconic and symbolic representation of the angles
Using paper folding, ask the students again to now quickly construct, folding, one at a time, angles of the following measures:
- an angle of 90°
- an angle of 0°
- an angle of 180°
- an angle of 45°
- an angle of 135°
- a straight angle
- an obtuse angle
- a right angle
- an acute angle.
After each constructed angle ask a student to come to the blackboard to trace their constructed angle. Ask the student to annotate this traced shape with symbols such as the angle symbol, the right-angle symbol and the size of the angle (for example, 90°).
 Video: Involving all |
Case Study 2: Mrs Nagaraju reflects on using Activity 2
The students had never done a paper folding exercise before, or at least not for a long time, and therefore they seemed to be a little confused at the start. Somehow they seemed to find the concept that you can ‘half an angle’ unusual.
I had asked them to each construct their own angles, but they were also allowed to talk to a nearby student. I think that helped them in understanding more quickly what to do, without me telling them or demonstrating how to do it. As suggested, I asked them the same questions as I did in Activity 1, and deliberately asked students who normally never volunteer answers. They did it well, and it must have been reassuring for them that they could ‘talk mathematics’.
I very much liked the step from enactive to iconic to symbolic, and I can see the potential of this. Hopefully, when the students now encounter a symbolic representation of an angle in their textbook they will think back to how this is the same as the angle they constructed using paper, and that is the same as the angle they can depict using their own bodies. Perhaps they will not do this at first, but I will try and remind them from time to time.
What I also liked about that part of the activity is that we really paid attention to the symbols used, how to write them correctly, and how to ‘say’ them. I think I too often forget that knowing the meaning of symbols is not something we are born with and that needs to be learned.
 Pause for thought
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1 Using embodiment of angles to make learning about angles more meaningful