1. Inquiry based learning IBL in mathematics
Tool IE-3T: Characteristics of problem solving tasks in mathematics
In this tool we further explore the meaning of problem solving and what a problem in mathematics may look like.
© 2016 mascil project (G.A. no ). Lead partner University of Nottingham; CC-NC-SA 4.0 license granted. The project mascil has received funding from the European Union’s Seventh Framework Programme (FP7/ ).

2. Overview
Aim:
To understand the key features of problem-solving tasks and the skills that students may develop by using these.
We will:
Discuss a sample of tasks;
Identify key problem-solving features;
Consider the skills that students need to be successful with these tasks.
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.

3. Comparing the tasks Read through the sample tasks provided. In pairs:
Compare the tasks;
Identify the main features of each task;
Identify the differences between them.
Firstly, ask the trainee teachers to read through the sample tasks and then work in pairs to compare them, identify the main features of each task and discuss the differences between tasks. Suggested sample tasks: Mixing paint, Fencing, Magic V investigation, Prisms.

4. Characteristics and demands
Share the outcomes of your discussions.
As a group try to list the main features of each task.
What demands do these tasks make on students?
As a group place the tasks into order according to the problem solving demands on students?
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.

5. Prioritising the key features
In small groups, discuss the set of cards provided and sort them into categories to show which of these features would be present in a problem-solving task in mathematics? Always Sometimes Never
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). 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 – this 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.

6. Discussing the tasks
Share your thinking about the card sorting activity together as a group.
Which features would you always expect to be present in a problem solving task?
Which would you expect to be sometimes present?
Which would you never expect to be present?
What other features might you expect?
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.

7. Problem-solving skills
In small groups now revisit one or two of the sample tasks discussed earlier and consider:
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)
(You can use Handout 6 to record your ideas)
Ask the teachers to work in small groups again and revisit one or two of the tasks from earlier. This time ask them to discuss:
- 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 to record their thoughts. Teachers often find it easier to think about the mathematical knowledge rather than the skills needed.

8. Skills for problem solving
The following skills have been suggested. Do you agree? What else would you add?
Skills in developing and using a range of different strategies;
Skills in recognising and replicating patterns (in number or shape);
Resilience;
Confidence in their mathematical ability;
Skills in reading and interpreting a word problem set in an unfamiliar context;
Skills in identifying the mathematical processes required to solve the problem;
Self reflection;
Skills in critically reviewing progress made;
Skills in team working.
The following (incomplete) list of skills can be used to prompt ideas:
Skills in developing and using a range of different strategies
Skills in recognising and replicating patterns (in number or shape)
Resilience
Confidence in their mathematical ability
Skills in reading and interpreting a word problem set in an unfamiliar context
Skills in identifying the mathematical processes required to solve the problem
Self reflection
Skills in critically reviewing progress made
Skills in team working.
Ask the trainee teachers if they agree with the items in this list. What other skills might students need?

9. Finishing off
Before next time think about how you could develop one or more of the sample tasks to increase the opportunities for problem solving. You might also read one of the related articles provided.
Ask the trainee teachers to consider before next time how they could develop one or more of the sample tasks t increase the opportunities for problem solving. They might also read one of the related articles listed below.
References
ACME (2016) Problem solving in mathematics. Available from
assessment of problem solving report - june final.pdf
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. Handbook of research on mathematics teaching and learning, Available from
Learning to Think Mathematically.pdf