Addressing the Reasoning and Problem-Solving demands of the new GCSE
In 2017/18, a Maths Hubs National Collaborative Project (NCP17-12), will be developing teaching approaches to match the new challenges of the new GCSE (9-1). They will be particularly addressing ‘challenging topics’ – those that students did less well with in the first exams this summer. The Secondary Magazine will be following and publicising some of their work in order to support GCSE teachers in the classroom. We begin in this edition, by considering how you might address the increased emphasis on Reasoning and Problem Solving.
This is an abridged version of a longer article which can be downloaded as a PDF. This article contains activities for the classroom to support students in the development of reasoning and problem solving skills. These are at the foot of the article, or can be found by clicking the relevant boxed hyperlink.
Many of the 2017 GCSE Mathematics questions require pupils to be willing ‘to face the unexpected and to think how to link known techniques into effective solution chains’. (Anthony D. Gardiner, Teaching Mathematics at Secondary Level, 2014).
Some questions are particularly difficult because pupils must find a starting-point for themselves. Also the construction of an adequate response depends heavily on the pupil’s ability to reason mathematically, to argue ‘if … then …’, to provide justifications for facts that they state. In the 2017 GCSE papers, questions of this kind can be sorted into three broad categories:
- ‘Work out …’ problems where something particular has to be found
- algebraic proofs about number
- geometric proofs.
When selecting or designing tasks to help pupils develop the skills needed to succeed with these questions it is helpful to be guided by some general principles that apply to all three categories:
- Challenge pupils to find as many mathematical relationships as possible between the constituent parts of situations about which it is possible to reason mathematically (see Task 1 below)
- Provide opportunities for pupils to see things for themselves. Then challenge them to try to convince other people that what they see must be true because … .
- Given a problem, challenge pupils to construct their own ‘similar’ problems by varying some aspect, or aspects, of it.
- Design tasks that will help pupils develop the habit of laying out calculations and deductions line-by-line so that a sequence of successive steps can be seen as a single chain of reasoning.
- Challenge pupils to think of different chains of reasoning to a particular result.
- Design tasks that will help pupils develop the habit of simplifying calculations and expressions wherever possible.
Before they reach Key Stage 4 pupils should be learning to use reasoning to solve multi-step word problems where they have to ‘work out’ or ‘find’ something. These reasoning skills develop into skills required in the construction of proofs. So first we consider preparation for, and approaches to, problem-solving of the first kind above - ‘work out’ problems, where something particular has to be found. (For detailed treatment of algebraic and geometric proofs, please refer to the full article).
Part A: ‘Work out …’ problems where something particular has to be found
Example 1: 2017, Edexcel, Foundation Level, Paper 3, Question 13
Example 2: 2017, Edexcel, Higher Level, Paper 1, Question 14
Example 3: 2017, OCR, Foundation Level, Paper 3, Question 19
These questions are hard because pupils have to decide for themselves where to begin. A powerful first strategy is to represent to oneself all the information that you can extract (that easily follows) from the given information. Ask pupils: ‘What does the given information tell you? What do you know? How can you show, perhaps in a sketched diagram or chart, what you know?’
In Example 1 the given information could be represented in a sketch of a triangle:
In Example 2 information given in ratio form can be converted to information in fraction form: 3/10 of all the shapes are white, 7/10 are black, and so on. These fraction facts could be represented in a diagram:
In Example 3 initial possibilities could be listed:
- the HCF is 6 means that both numbers are multiples of 6:
(6), 12, 18, 24, 30, 36, 42, 48, 54, 60, …
- the LCM is 60 means that both numbers are factors of 60:
(1, 2, 3, 4, 5, 6), 10, 12, 15, 20, 30, 60.
The process, exemplified above, of representing-to-themselves all the information that they can glean from word statements is a process that pupils can practise. Challenge pupils to represent-to-themselves all that they can derive from some information (an ‘information-list’), for example as in:
See linked Task 1
at the foot of this article
When pupils have represented-to-themselves (and shared and discussed) what any list of statements about a situation tells them, ask them to suggest what they might be asked to ‘work out’ or ‘find’. Collect and display their suggestions so that they can think about each-others’ ‘questions’. It may be that, in showing on paper what they gleaned from the stated facts, some pupils have already reasoned to the answers of some of the suggested ‘questions’. However, if they have not yet connected what they now know (have jotted down) with what they want (the answer to a suggested ‘question’), they will have to bring in to the collection of facts they are reasoning about other facts from their repertoire of known properties and general relationships. This process, of deciding what mathematical-knowledge-not-provided-with-the-given-information will reveal more facts, can also be practised, and is exemplified in:
See linked Task 2
at the foot of this article
In addition to practising …
- representing in a precise structural way, facts and relationships conveyed at first to you via word statements
- identifying mathematical knowledge and procedures that can be applied to a situation to extend what you know about it
… pupils should practise …
- writing complete chains of reasoning.
Writing complete chains of reasoning helps pupils to identify precisely where ‘if … then …’, ‘because …’ and ‘therefore …’ occur in their thinking, and so learn to apply similar thinking in new and different situations.
Pupils need to learn to lay out calculations line-by-line, with:
- given information and any symbols representing ‘unknowns’ declared
- each fresh step on a new line (and any explanation given alongside)
- the final answer clearly displayed at the end.
‘The sequence of successive steps can then be grasped as a single chain of reasoning in which each step follows clearly from those that went before.’ (Anthony D. Gardiner, Teaching Mathematics at Secondary Level, 2014)
Example chains of reasoning for a problem created from information in Task 1
If, when solving ‘Work out … problems’, pupils have had plenty of practice in thinking their own ways through to solutions and then communicating their thinking by writing-out complete chains of reasoning, the step to constructing proofs is not so great.
There is another aspect of working mathematically with which pupils need to develop competence in order to cope successfully with all sorts of ‘Work out …’ problems. It is the routine simplification of numerical expressions in a way that exploits structure (‘structural arithmetic’). (Ref: Anthony D. Gardiner, as above). When evaluating numerical expressions it is almost always quicker to simplify wherever possible than to ‘calculate blindly’. For example, in solving Example 2 of the 2017 GCSE questions shown above, a numerical expression occurs that can be evaluated rapidly like this:
Pupils need to learn through lots of practice how numerical expressions can be simplified.
Using structure to simplify numerical expressions: examples
When pupils are habitually looking for ways to use structure to simplify numerical expressions they internalise meanings, structures and procedures that they can then draw on when working with algebraic expressions while trying to construct proofs.
The full article can be downloaded as a PDF. What follows are the tasks suggested in the article
This task is about extracting and representing as much information as possible. It is not, at this stage, about finding answers to any problems. Emphasise to pupils that the task is to represent, as concisely as possible without merely writing more word sentences, what the given word statements tell them. Encourage the use of sketched diagrams, charts, numbers and symbols.
Give pupils some information-lists (such as A, B or C in Task 1, above), and challenge them with:
‘What mathematical facts could be added, and what mathematical processes could be applied, to each situation in order to reveal more information?’
Pupils could compare and discuss their suggestions.
By working on tasks as outlined above pupils could arrive at the following kind of analysis (shown for information-lists A, B, C, in Task 1):
Example chains of reasoning for a problem created from information in Task 1, List A
- Suppose List A in Task 1 is turned into a ‘Work out … problem’ by adding ‘Work out the size of angle ABD’. At least two different complete chains of reasoning to the solution of this problem could be constructed, possibly as follows:
Using structure to simplify numerical expressions: examples