Problem solving
Overview
This unit considers how, as teachers, we can work with students so that they become better at solving problems. As a result of contributions from the four GCSE awarding bodies, the unit addresses identified issues that have an impact on pupil progress in solving problems. The unit considers a problem solving scenario and then gives opportunities to compare this to solving mathematical problems in the classroom. Working through these activities as teachers may provide you with some additional strategies to use when working with students in the classroom. The aim of this unit is not to provide you with definitive answers but rather to provide you with an opportunity to continue to develop the understanding and the answers that you already have.
Where are you now?
Individually consider your responses to the comments on Resource sheet 1 and write a comment that you would contribute to the conversation. Share these comments within your department.
Activity 1: Problem Solving Skills
Watch the video: Problem solving by a clever crow
It is a short video so you may want to watch it twice – the solution happens quite quickly! Recreate together, as a group, the sequence of events from the video.
If you do this activity with pupils, you may wish to cut up and sequence the cards in Resource sheet 1a – table 1 at this point.
Working as a pupil trying to achieve a C grade, consider the problems on Resource Sheet 1c and Resource Sheet 1d. Solve the problems. Then, working in pairs or threes, match your actions in solving the problems to those of the clever crow.
If you do this activity with pupils you may wish to cut up and sequence the cards in Resource sheet 1a – table 2 at this point, and lay them alongside those from table 1.
As a group, think of some words or phrases to describe the characteristics of the crow’s behaviour as he tries to solve the problems in the video(eg resilient)
If you do this activity with pupils you may wish to ask them to pick out words from those supplied on Resource sheet 1b and explain how the crow showed that characteristic
Activity 2  Considering alternative approaches
NCETM would like to thank Edexcel for providing this examination material.
Pupils who are good problem solvers will be able to find different methods to solve a problem – sometimes using one method to solve the problem and another to check their answers. This activity models an approach for encouraging students to consider different ways of solving a problem.
Working on your own, answer the examination question on Resource sheet 2a.
At this point it may be useful to share that the answer is Dougies, but acknowledge that there are different ways of explaining and justifying this answer
Now join with another person  compare your solutions and the way you have justified them. If you have both used the same justification, try to work out a way of providing a different one.
As a pair, now join with another pair and compare your work. Decide if you, as a group, have got all the possible ways of explaining and justifying your answer?
The methods seen by the examiners can be seen on Resource Sheet 2b
You may wish to consider which of the words or phrases to describe the characteristics of the crow’s behaviour, that you generated in Activity 1, might apply to your behaviour in this task
Activity 3  Working on an examination question
NCETM would like to thank OCR for providing this examination material.
Students often find it hard to get started on solving a problem and might not realise how much they can do. This activity models a way in which students can be encouraged to work on a problem.
Look at the diagram on Resource Sheet 3a.
Write down as many things as possible that you know about the diagram. Think about the sorts of things that students might write, such as facts about the sides and angles of a parallelogram, an expression for the sum of the angles of the parallelogram, 5x – 3y = 2x + y + 20 etc
Write down some questions could you ask about the diagram? Share the questions you have written and decide which ones could be answered.
Share the examination question ‘Work out the size of each angle in the parallelogram’. How can you use the statements you have written down to help answer the questions?
Share the student scripts on Resource sheet 3b. Decide whether the students have answered the questions correctly.
You may wish to consider which of the words or phrases to describe the characteristics of the crow’s behaviour, that you generated in Activity , might apply to your behaviour in this task
Activity 4 – The move to algebraic solutions?
NCETM would like to thank AQA for providing this examination material.
The featured question (below) was Question 4(b) on 93701H (GCSE Applications of Mathematics Unit 1 Higher tier) June 2011 and Question12(b) on 93701F (GCSE Applications of Mathematics Unit 1 Foundation tier) June 2011
Examiners have identified that pupils are slow to make the transition into algebraic solutions, preferring to make extensive use of trial and improvement methods; this question shows how algebraic thinking (no algebraic symbols need to be seen) can provide a concise solution. This activity models a way for students to consider the usefulness of algebraic thinking in problem solving
Work through the examination question on Resource sheet 4a. Compare your solutions.
Working in pairs, consider the student scripts provided on Resource sheet 4b. Rank order them according to what you consider to be the best solution. Compare your ranking with the examiner’s comments for these answers which is provided on Resource sheet 4c.
You may wish to consider which of the words or phrases to describe the characteristics of the crow’s behaviour, that you generated in Activity 1, might apply to your behaviour in this task
Activity 5 – Gaining ownership of the problem
NCETM would like to thank WJEC for providing this examination material.
Making sense of a problem is an important step in finding the way in which you are going to solve it. Students seem reluctant to gain ownership of some problems, especially if this means drawing diagrams or sketching possible scenarios. This activity models a way in which students could draw diagrams to make sense of a problem.
Working on your own, answer the question on Resource sheet 5a. Now working with someone else, draw different diagrams to represent your solutions to the different parts of the problem – this could be scale diagrams or sketches. Particularly highlight the word explain in part b – a mathematical explanation is clearly given by a diagram in this situation. Share your diagrams as a group and compare them. Are any of the diagrams more useful than others? Is it always better to draw a scale diagram or is a sketch useful?
You may wish to consider which of the words or phrases to describe the characteristics of the crow’s behaviour, that you generated in Activity 1, might apply to your behaviour in this task
Reflection
It is unlikely that you will do any of these activities in your classroom if you do not believe that they will make a difference to your pupils’ problem solving abilities. As a department – what do you believe? You may have worked through this workshop thinking about Year 10 and 11 students – but ideally students in Year 7 onwards need to continue to solve problems as part of their everyday mathematics lessons. Spend some time as a group identifying problem solving characteristics and tasks that you can prioritise in Year 7 to enable pupils throughout your school to become better at solving problems.
Implementing and continuing to learn
Ask members of your team to consider what they might aim to do :
Tomorrow
You could identify an objective or characteristic that relates to the way students solve problems to focus on next lesson. For example  You could say that you will tell pupils that you will spend a lesson answering their questions with questions of your own (as a way of starting to develop independence)
Next week
You could collect some examples of student answers to problems that you could use as a department resource in the ways suggested in this workshop
Next year
You might want to look at the Scheme of work and identify tasks that will give pupils experiences solving problems as part of their everyday mathematics lessons
Further reading
NCETM Department workshop: Preparing to teach the LinkedPair Pilot GCSEs
NCETM Department workshop: Mathematical Processes & Applications
NCETM Department workshop: Developing functionality in mathematics
Using contexts and models to support mathematical development by Frank Eade and Sue Hough, Manchester Metropolitan University and related resource materials
What's in a task? Generating mathematically rich activity by Susan McDonald and Anne Watson, University of Oxford
Connecting mathematics with reality: connecting reality with mathematics by Geoff Wake, University of Manchester.
The elephant in the classroom (Chapter one)
Starter problem selection
Bloom’s taxonomy
Working Effectively with All Learners
Cooperative problem solving
Bowland Trust Professional Development modules
Developing higher order mathematical skills (Welsh Government )
This module has been coordinated by NCETM with contributions from AQA (Sandra Burns), Edexcel (Rob Summerson), OCR (Janet Sheaf) and WJEC (Betsan Jones).
