By Peter Sullivan, author of Challenging Mathematical Tasks
I have asked many teachers, ‘What gives you joy when you are teaching mathematics?’
It is very common for teachers to respond with the answer: ‘Light bulbs’. I presume they mean those delightful moments when students move from confusion to clarity, when they see the pathway to a solution to a problem even if they have not yet solved the problem, and are energised by the search for a solution.
This presents an interesting dilemma for teachers. Students will not see light bulbs if they have been told exactly what to do, if the task is too easy, or if they get the answer quickly. In other words, the way to increase the chances of seeing light bulbs is for the students to work on problems they have not been shown how to do, to make those problems complex, to make sure that the problem requires students to think and that the students have the time to struggle.
But to do this, teachers need to take risks. ‘What if no one can do it?’ ‘What if students give up?’ ‘What if the students give answers that I don’t understand?’
So there is a need for teachers to be confident enough to take risks.
It seems that in mathematics, some teachers feel they need to know everything beforehand. Of course, it helps if teachers do understand the mathematics of a given question, but there are many instances in which something has come up in a lesson that I had not anticipated. The issue is having the courage to respond.
It might be that the teacher says, ‘I am not sure. Let me think about it’, or “I am not sure. Can anyone help?” Teachers might be wondering, ‘what if the students think I do not know enough?’ But even with expert mathematicians, problems can arise that they need time to think about and solve. The teacher being open to such possibilities can actually reduce the sense of risk that students feel.
One of the other risks is the potential for students to misbehave if they are posed challenging problems. But surely such students should not be allowed to subvert the education of others! It seems it is the level of effort from those students and their behaviour that should be addressed rather than adjusting the learning experiences to avoid the misbehaviour. We know that students respond well when they are told what to do and are given exercises that do not require them to think. But such low-level teaching is not much more than child minding. It is certainly not preparing students for life beyond school.
Sometimes teachers are concerned that some students might sit and do no work, even if they do not misbehave. But that inertia needs to be addressed separately and certainly is not helped by having everyone work on questions that are too easy.
A key action by teachers is to plan for the challenge and complexity and to anticipate barriers that might arise. Such planning involves:
- identifying tasks and experiences that the majority of the class does not already know how to do
- considering ways to introduce the experience without telling students how to do it
- preparing prompts for students who might experience difficulty and other prompts for students who might finish quickly
- thinking about how to use student solutions and strategies effectively
- anticipating what might be suitable follow-up experiences to consolidate the learning.
All teachers should strive to be a better teacher next year than they were the previous year. This involves taking risks, trying new things and having the confidence to improve.