Thinking Things Through, Problem Solving in Mathematics
Leone Burton
Introduction
Problem solving requires questions to be based in a real world context, providing students with more authentic engaging tasks.
It is only real when pupils take responsibility for their own mathematical thinking. To support this, teachers need to change their view of their own role from that of one of responsibility to one of a resource for students.
Problem solving in seen as a challenge for the students so they are more engaged and motivated in the work.
What makes problem solving different?
This chapter of the book looked at two different solutions to a problem solving questions and odd and even patterns and relationships.
Talked about problem solving not requiring a 'correct' approach to solve questions and that the students in the examples may well have achieved the same learning/understanding despite working in different ways.
Talked about problem solving not requiring a 'correct' approach to solve questions and that the students in the examples may well have achieved the same learning/understanding despite working in different ways.
Why is problem solving important?
'The ability to solve problems is at the heart of mathematics' (Cockford Report, 1982). This idea puts into concern the disconnect that students experience in a modern maths class between the skills they are learning and the knowledge to use them.
Problem solving improves pupils' attitude, confidence and achievement in class. It also embraces the spirit of inquiry which is a quintessential aspect of maths.
In a problem solving classroom there needs to be evidence of questioning. Over flow questions can be recorded on a question board to be discussed later.
There needs to be evidence of challenging. Both as a noun and a verb. Challenging questions for the students and challenging their answers between themselves.
Finally, there needs to be evidence of reflecting around the problem by the students.
Teachers need to look at the process students use, rather than the outcome they achieve.
Often teachers are surprised that there is a stark difference between what students actually know and what they use to solve problems.
Problem solving improves pupils' attitude, confidence and achievement in class. It also embraces the spirit of inquiry which is a quintessential aspect of maths.
In a problem solving classroom there needs to be evidence of questioning. Over flow questions can be recorded on a question board to be discussed later.
There needs to be evidence of challenging. Both as a noun and a verb. Challenging questions for the students and challenging their answers between themselves.
Finally, there needs to be evidence of reflecting around the problem by the students.
Teachers need to look at the process students use, rather than the outcome they achieve.
Often teachers are surprised that there is a stark difference between what students actually know and what they use to solve problems.
How to get going
Problems can easily provide differentiation. This makes problem solving an ideal technique for whole class, multi-level teaching.
Students should be offered a choice of problems to select from, although don't overwhelm them with choice either.
Problem solving could be integrated into homework.
Using representations to help solve the problems is key - teachers must make sure they have a lot available for the students to use.
Students need to understand the cyclical process of problem solving. No more 'I'm finished!'
Teachers need to promote inquiry and curiosity in their class with these questions.
Pupils should keep clear records as they go so they can easily share their thinking and present results well.
Modelling problem solving together is essential in showing pupils that the skills and process is an important one that is valued in the class.
The problem-solving process
Burton talks about the problem solving process consisting of four different phases of activity.
- Entry - where you are trying to understand the question,
- Attack - trying to find a resolution to the question.
- Review - examining and reassessing resolution to make sure it answers the question.
- Extension - investigate new problems that have arisen from the original question.
What about individual differences?
Again, student choice is essential in supporting individual differences. Students undertake the problem solving work in groups, but all children monitor their own work. After some time working together to answer problem solving questions students work out their own dynamics in groups and work together effectively.
Teachers need to focus on giving students space and acting more as a resources for students.
Have the pupils gained?
Developments and growth in the students may take a long time to become apparent. Patience is the key as problem solving covers a wide range of skills.
Students can share their understanding by writing a message to their friends to help them solve the question - they must think clearly and articulate their working well to help them understand.
Another extension activity could be to write another question that uses the students answer.
Another extension activity could be to write another question that uses the students answer.
Problems 1-30
This book also features a section with 30 problem solving questions. These are well laid out and provide teachers with prior information students need, materials required, skills covered, suggestions for solving the question and extension ideas.
Opinions
Problem solving seems to be an easy access technique to use in class that has a multitude of benefits. It not only allows students to become more articulate with their maths skills but also develops social behaviours and thinking skills.
I like how easily problem solving caters to mixed ability classes.
Using problem solving as part of weekly homework sounds like a good way to continue to develop students reasoning skills, but must be supported by in class teaching around how to approach this type of question effectively.
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