Geoff Petty - Teaching students to speak maths

Peer-to-peer dialogue and whole-class discussion can really get your students thinking, says Geoff Petty. Geoff is the author of Teaching Today and Evidence Based Teaching and has trained staff in more than 300 colleges and schools.  

Teachers begin their career worried about what they will do in the lesson. Later, they worry what their students will do. If they become good teachers, they worry what the lesson will make their students think about. 

One answer is to get students into a dialogue where they can correct many of each other’s errors. Another is to get students arguing about alternative solutions in a whole-class discussion. In each case, it’s by arguing about maths that they learn how to make sense of it. And making sense of maths, is all there is to learning maths. 

Managed well, peer-to-peer dialogue and whole-class discussion are greatly enjoyed by students. They get everyone curious, arguing, thinking, learning and understanding. Remember that students are much more interested in each other’s mathematical ideas than they are in yours. It doesn’t matter that these ideas could be right, wrong or just plain odd, it will make students puzzled and curious. Not knowing makes them think. 

Here are two methods to try. As always, try them for a short time and try them often. Do this at least five times, making improvements each time, by then you’ll know whether the methods can work well for you and how. 

Snowballing solution 

In recent studies with underachieving youngsters in low-performing schools, this method greatly increased students’ interest and enjoyment of maths, and improved their average standardised test results from 45 per cent to 79 per cent. 

Interestingly, this degree of improvement took time – approximately six years – as teachers developed their facilitation skills, but you can expect some improvement quite quickly. 

Explain your no-blame ground-rules, then give students a challenging maths question: one they can attempt or get started on, but which they often won’t be able to finish by themselves. Then… 

1. Individuals write down their own answer, or their attempt, working alone.
2. Students pair up and show each other their work. They give constructive criticism to each other, and decide on their best method.
3. Pairs form fours, which look at each pair’s method and again give pros and cons and decide on the best method. Leave time for this dialogue.
4. The teacher chooses individuals by name from each, or at least some of the groups, to present their group’s solution/work to the class, and to justify it to the class.
5. The class as a whole nowscrutinisesthe work presented, and decides which method(s) is/are best and why. 
6. Only when discussion has finished do you give your views. 

Student demonstration 

This method is used after a teacher has demonstrated how to do a certain type of maths problem. The aim is to check and correct understanding of a skill before all students practise it alone. It is initially daunting for students but they will enjoy the method if you insist on a blame-free ethos. 

Use snowballing (above) first, as this prepares them for student demonstration. Begin by asking for volunteers to show their solutions, then nominate those who will show their answers. Give them fair warning if you are going to nominate. 

The basic procedure for student demonstration – and it works for all subjects – is as follows. 

1. You set a task:

“Working in pairs, factorise 6x2 - 6x - 8.” 

“Okay, in pairs, punctuate this paragraph.” 

“Working by yourself this time, can you see any personification or metaphors in the third or fourth verse of the poem?” 

2. Students work on the task. This can be done in pairs initially, but after a bit of practice they do tasks individually, perhaps checking each other’s completed answers in pairs.

They strive to get the answer, with any justification such as necessary reasoning or working etc. If students are in pairs they make sure that either of them can provide this justification. 

3. You monitor the work. You check attention to task and occasionally ask:

“Can everyone do this one?” 

“Can you all explain your answer?” 

Students who can’t answer the question are required to own up and get help at this stage, otherwise they are ‘fair game’ for the next stage. 

4. You choose a student to demonstrate their answer to the rest of the class. If students are in pairs you choose one student at random to give the pair’s answer. The student gives their answer on the board, explaining each step and its justification to the class. You ask questions to clarify, but do not yet evaluate the answer.

“Why didn’t you use six and one as the factors of six?” 

“Why did you choose a full stop and not a comma?” 

“So how did you choose between personification and metaphor?” 

5. You ask for a ‘class answer’. You ask the class if they agree with the student’s answer and its justification, or whether either could be improved. The aim is not tocriticisethe student’s answer, but for the class to agree with a ‘class answer’. The student who did the demonstration becomes the class scribe, writing up any changes the class agrees to. You again facilitate without evaluating the answers or the arguments. 

“Why do you think it should be plus four and not minus?” 

“How many think it should be a comma? Why is that?” 

“So why exactly is it not a metaphor?” 

6. You comment on the class answer. Praise any useful contributions and confirm any correct reasoning, and correct any weak reasoning.
7. The process is repeated with another task; after sufficient practice the students can do stage two as individuals rather than in pairs.

Students are often initially resistant to doing a demonstration to their classmates if they are not used to it. So you could make use of volunteers to begin with, but try to move on to students nominated by you as soon as you can. They will be more confident of answers that they have produced in pairs than answers produced in isolation, so when you first start nominating students do it after pair work. 

The best maths teaching, according to a systematic review of research** can be summarised by 10 principles. Here are two of them: 

• Effective teachers use a range of assessment practices to make students’ thinking visible and to support students’ learning 
• Effective teachers are able to facilitate classroom dialogue that is focused on mathematical argumentation 

*Effective pedagogy in mathematics. By Glenda Anthony and Margaret Walshaw, IAE (2009) 

Further reading 

• Why Don’t Students Like School? By Dan Willingham, Jossey-Bass (2009) 
• Improving Student Achievement in Mathematics. By Douglas A. Grouws and Kristin J. Cebulla, IAE (2000).