For this tool you will need the following resources:
- IE-3 PowerPoint
- Handout 1: Characteristics of problem-solving tasks
- Handout 2: Skills required for problem solving
In this tool we further explore the meaning of problem solving and what a problem in mathematics may look like. Firstly, we examine four tasks of different types to compare their main features and identify their particular characteristics. (The tasks you use do not have to be those provided in the toolkit. You can select resources relevant to your curriculum but the samples should include a range of different features, for example a word problem or scenario-based task with some structure; an open-ended and less structured investigation; an assessment task; a structured task with a single correct answer). These samples are used to stimulate discussion about what is meant by a problem-solving task and then to consider the range of mathematical and personal skills that might be developed using such tasks.
Firstly, ask the teachers to read through the sample tasks and then work in pairs to compare them, to identify the main features of each task and discuss the differences between the tasks. Suggested sample tasks:
As a whole group, share the outcomes of your discussions and try to list the main features of each task. Discuss the demands on students of each task and how you might put the tasks into order according to the problem-solving demands on students.
For the next activity teachers will need to work in small groups. Provide each group with a set of the ‘problem solving’ cards to share (or use Handout 5: Features of problem-solving tasks). The cards show some of the features commonly associated with problem-solving tasks. Ask the teachers to discuss these and try to place the cards into the following categories:
- Always – the feature will always be present in a problem-solving task
- Sometimes – the feature will sometimes be present in a problem-solving task
- Never – the feature would never be present in a problem-solving task.
(Note: Teachers may find it difficult to identify any cards to place in the Never category. The activity is designed with this in mind so teachers are prompted to think further about what is involved in a problem-solving task). Teachers may also have other ideas of features that they want to add and they should be encouraged to do this.
Ask the teachers to share their thinking with the whole group. This should stimulate discussion about the varied nature of problem-solving tasks and how some tasks may present more opportunities than others for students to develop problem-solving skills. Note that such limitations are not always undesirable, since students may sometimes need to focus on particular aspects of problem solving.
- The mathematical knowledge and skills required
- Any other knowledge required (e.g. work-related or vocational knowledge)
- Any other skills required (e.g. personal skills).
They should use Handout 6: Problem solving skills to record their thoughts. Teachers often find it easier to think about the mathematical knowledge rather than the skills needed. The following (incomplete) list of skills can be used to prompt ideas:
- Developing and using a range of different strategies
- Recognising and replicating patterns (in number or shape)
- Confidence in personal mathematical ability
- Reading and interpreting a word problem set in an unfamiliar context
- Identifying the mathematical processes required to solve the problem
- Self reflection
- Critically reviewing progress
- Team working.
Ask the teachers if they agree with the items in this list. What other skills might students need?
As a group, select a task (from the mascil classroom materials or from other sources) and discuss how you might use this in the classroom. Alternatively, the group might consider how they could develop one or more of the sample tasks to increase the opportunities for problem solving.
Before next time, ask each teacher to try the task with a class. They should reflect on how effective it was; what progress students made with the development of problem-solving skills and be prepared to share their thoughts with the group at the next session.