4 Dealing with misconceptions and mistakes
When you learn to ride a bicycle you have to learn to gain balance. It is accepted that in that process of learning to ride a bike you will fall off a couple of times, make the wrong movements, pull the handlebar too briskly, forget where the brakes are, or have the wrong idea about how gears work. It is only by making mistakes like these that you will develop and refine your cycle skills. Making mistakes and being confronted with your misconceptions makes you learn better in the future.
Learning mathematics is no different from learning to ride a bike or anything else. Making mistakes is part of that process, and is actually a good thing, because it interferes with your routine thinking patterns and makes you reflect on what you are doing. If there is a classroom ethos that learning happens from making mistakes, and hence, making mistakes is part of effective learning, then students will also become more willing and more able to experiment and try out new ideas, less afraid of getting it wrong, and consequently be better equipped to explore and enjoy mathematics.
Establishing such a classroom ethos where students are willing to share their mistakes and misconceptions does not happen overnight – it requires nurturing and careful coaching. However, there are methods you can use to promote this ethos:
- Use work from fictitious students to expose possible misconceptions. Because it is not their own work that is being scrutinised, there are no emotional reactions and/or feelings of embarrassment.
- Encourage reflection on mistakes and misconceptions rather than dismissing an answer as simply ‘wrong’ – this provides a fruitful way for learning.
- Promote sharing of ideas, even if they are half formed or ‘wrong’.
The next activity is designed so that it is very likely students will make mistakes. Discussing these explicitly will help elicit misconceptions and turn these into opportunities for learning mathematics.
Activity 2: Triangles challenge – learning from misconceptions and mistakes
Part 1: Playing the triangle challenge
Divide your class into groups of three students.
Explain the following to your students:
- One student in each group, the leader, has to draw a triangle, without showing the triangle to the other two students. The leader describes their triangle, one fact at a time. The other two students each need to draw the same triangle from the facts that the leader gives. They have to concentrate on getting the size and shape right.
- If the two students cannot complete their triangles using the facts shared with them, they say, ‘Tell me more’. The leader then reveals another fact about their triangle. After the two students have drawn their triangles, the leader gives them one point if their triangle is the same; otherwise they get no points. The original triangle and the ones that get awarded a point are kept in a class pile of triangles. The others are discarded.
- The group repeats the activity two more times, with other students taking turns as leaders.
After they have done the activity above, ask the students the following questions:
- Which triangle was easier to make? Why?
- Why did some of you not get a congruent triangle while others did? What mistakes were made and why?
Part 2: Continuing the challenge
Explain to your students that the class pile of triangles created in Part 1 now becomes the focus. Each group of three collects one triangle from the pile. One student volunteer picks one of the remaining triangles at random from the pile and then describes it to the class one fact at a time. Students check their triangles in their groups to determine if they have an ‘exact match’ or a ‘similar match’. Again, an exact match gets a point.
Repeat this exercise until all the remaining triangles have been used up.
After each round, answer the following questions:
- After how many facts was a match found?
- Could you have gotten a match with fewer facts?
- Did you find an ‘exact’ match or a ‘similar’ match?
- Why do you think this is so?
Part 3: Reflecting on your learning
This part of the activity asks the students to think about their learning so that they can become better at and feel more comfortable about learning mathematics. Ask your students the following.
- What did you find easy or difficult about Part 1 of this activity?
- What did you like about the activity?
- What mathematics did you learn from?
- What did you learn about how you (could) learn mathematics?
Case Study 2: Mrs Nagaraju reflects on using Activity 2
I distributed pieces of paper to the groups for writing down their answers. I then explained the activity to them. I modified the activity slightly by asking them to also write down the facts given by the leader that they used to draw their triangle and told them this would help them to see why they could draw a triangle and in just how many steps.
The students were very excited about doing this activity. They seemed to like the sense of competitiveness and also that they had a chance of all being the leader. There were some complaints about how one student had cheated and been given an extra point. I told them that it was a discredit to them if they cheated and they would have to explain how they had got to where they were anyway.
The discussion about learning from mistakes was interesting. I had expected the students to be reluctant to talk about their mistakes but I was wrong – they talked about them with gusto! Perhaps it was because they were able to talk about them in groups first, which was less threatening. I then asked the students to share their mistakes with the whole class, and wrote them on the blackboard. We discussed these in terms of: ‘This is interesting. Why would this mistake have been made? What were they thinking?’ All the students in the class learned much from these misconceptions. We also discussed whether we did learn from making and discussing these mistakes – and we all agreed that we did. Some students mentioned that they had now finally discovered that they had a misconception. Others who did ‘get it right’ from the start said it was good to hear and find out why it was they were thinking correctly because often they just did the activities and questions without thinking.
 Pause for thought
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3 Developing mathematical language