5 Learning to convince
The formal rules of argumentation and deduction that are used in mathematical proof are not easy to teach at school level. The language used is alien and the concepts are often considered to be far removed from the world of the students’ experience. To some extent this is surprising, because reasoning and deduction are skills and powers that we all have, along with knowing about different levels of convincing, which is the same as rigour.
The next two activities aim to use the skills and experiences that students have from everyday reasoning to make their reasoning more structured and formal. In this way it builds on what students know and do already. The first one uses the task design of ‘always true, sometimes true or never true’ to find out whether and when a statement is valid.
The results of this activity are then used in the next one, where students work on different levels of rigour in their reasoning and argumentation. The task design here uses the approach of ‘convince yourself, convince a friend, convince Ramanujan’.
Distinguishing who needs to be convinced helps students to become more precise. Convincing yourself is often easy: you may be happy with empirical evidence and the language used can be sloppy and vague, as there is no need to explain it aloud. Convincing a friend requires more precision in the use of language, as the thoughts need to be organised and verbalised for communication and the validity of the arguments used. However, your friend might still be willing to accept what you are saying because they are your friend, although they might challenge you more than you would challenge yourself. To convince Ramanujan, the great Indian mathematician, requires students to come up with solid mathematical justification, because he will try and pick holes in their reasoning. To convince Ramanujan, argumentation will have to be based on accepted or established mathematical statements because only those are indisputable.
Activity 4: Always, sometimes or never true
Preparation
The statements underneath require the students to have some knowledge about these topics. Choose the ones that are relevant to your classroom.
The activity
Tell your students the following:
- Read the following statements. Which of these are always true, sometimes true or never true? Discuss your reasoning with your partner.
- a.If p is a prime number, then p + 1 is a composite number.
- b.N chords of a circle divide the circular region into N + 1 non-overlapping regions.
- c.If the volumes of two spheres are equal, then the spheres are congruent.
- d.The centroid of a triangle lies inside the triangle.
- e.There are infinitely many prime numbers.
- Make up some more statements of your own and give them to your partner. Which of these are always true, sometimes true or never true?
Activity 5: Convince yourself, convince a friend, convince Ramanujan
Tell your students:
- Remember that Ramanujan will try and pick holes in your reasoning.
- Redo Activity 4, but instead of simply discussing your reasoning with your partner, you now have to come up with justification that will:
- convince yourself
- convince a friend
- convince Ramanujan.
- Share your most convincing justification with the class. Are the other students convinced by your argumentation?
Case Study 4:Mrs Nagaraju reflects on using Activities 4 and 5
I decided to do Activity 4 – discussion whether a statement is always, sometimes or never true – by putting students in groups of three and four, because I thought this would give them the opportunity to generate more ideas. I did notice that, in this case, the groups of three seemed to work better – they were more engaged with each other – while in a number of the groups of four there were students not engaging and only listening. For Activity 5, I asked them to work in pairs to give everyone a chance and the time to practise their justification. A good thing about these tasks is that it forces students to have an opinion and to make them come up with justification even if they only had been listening previously.
When I asked the students whether there had been a difference in their argumentation to convince oneself, a friend or Ramanujan, they said that indeed in some cases it made them become more precise and thorough. However, in other cases some students felt they could not come up with reasons in their argumentation because they could not think of any – they just knew the statement was correct because they had learnt it from me! To help them think about steps that might help them to develop a logical argument I thought that someone in the classroom would have some suggestion and maybe they would be more critical when it came from another student instead of me. I asked the groups to point out which statements this had happened with and asked other students if they had come up with some justification. The justification was explained to the whole class and I asked the all the students to be critical of the justifications just as real mathematicians would be. After this I went back to those students who had been stuck and they said that the discussion had helped them to get unstuck.
For more ideas about using questioning and talk for learning in your classroom, see Resources 2 and 3.
 Pause for thought
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4 Task design to help students think about mathematical properties