3 Key issue 3: How can you teach so that students are happy to be lifelong learners and users of mathematics?
For students to be lifelong learners and users of mathematics they will need to be able to do more than merely answer questions that are just like the ones they have practised with a teacher. They will need to feel sufficiently confident to be able to tackle unfamiliar problems in unfamiliar contexts using the mathematical ideas and skills at their disposal. After all, when they are working and are asked to solve a problem, it can be no excuse to say, ‘I can’t do that because I haven’t done things like that before.’
Reflection point
What does being a lifelong user and learner of mathematics mean to you?
In the English Key Stage 4 National Curriculum Programme of study (DfE, 2014), students are required to solve problems that:
- develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
- develop their use of formal mathematical knowledge to interpret and solve problems, including in financial contexts
- make and use connections between different parts of mathematics to solve problems
- model situations mathematically and express the results using a range of formal mathematical representations, reflecting on how their solutions may have been affected by any modelling assumptions
- select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret their solution in the context of the given problem.
The implication of the above is that the curriculum must be extended beyond structured questions on a single theme to making connections, choosing appropriate mathematical skills, modelling situations and interpreting their outcomes. The ‘problems’ often found at the ends of exercises in textbooks are unlikely to fit the demands made in the above list. Instead, students must be exposed to problems that are unfamiliar and where they could not reasonably be expected to know a path to the solution. In other words, they must find themselves in situations not unlike those they will experience in their lives and careers.
The problem solver will have a great deal of autonomy, looking at various possibilities and deciding what to try. This mixture of unfamiliarity and autonomy is also likely to result in students being stuck, which is also important. Thus students will face difficulties when working in this way and some will say that this is not ‘maths’. They will need support and to learn that lifelong learners and users of mathematics problem solve, helping them feel OK about being stuck is just one part of this.
The Cockroft report (DES, 1982, para. 249) states that:
the ability to solve problems is at the heart of mathematics. Mathematics is only ‘useful’ to the extent to which it can be applied to a particular situation and it is the ability to apply mathematics to a variety of situations to which we give the name ‘problem solving’.
Activity 5 Changing practice, changing roles
Using problem-solving as a learning experience means that the role of a teacher will change. Make a list of at least five aspects of a lesson that you see as typical that may have to change in a problem-solving lesson. Think about the changes in your role that each change may necessitate.
Now think about the students. How would their role change? Think about what they would be expected to do and how they would be expected to work. Again, list at least five changes in what they would do or how they would act. Does thinking about the students and their role suggest even more changes in your role as a teacher?
If you were to help your students take on more autonomy and work more collaboratively, then you as a teacher should avoid making suggestions or offering explanations wherever possible. Instead you could keep your comments at a strategic level, discussing how the students are tackling the problem rather than the mathematical details of the problem itself. You might reply to students who ask for help as follows:
- ‘What do you think …?’
- ‘What if you …?’
- ‘Did any other approaches occur to you?’
Add at least three more questions that you might use to this list, possibly thinking in terms of making connections and stimulating mathematical thinking, as discussed earlier in the course.
Make yourself an aide memoire of ‘good questions to ask students’. You could laminate a card to keep in your pocket or a large size card to attach to the classroom wall.
When you have the opportunity, share these with your students and encourage them to ask each other these questions.