Maths in the Staff Room – Short Professional Development Meetings
Maths in the Staff Room provides suggestions and resources for a professional development meeting for teachers, which can be led by the maths subject leader or another person with responsibility for developing mathematics teaching and learning in the school. You can find previous features in this series here
Understanding key mathematical structures. Part one: ‘Doing and undoing’
Meeting aims
 To consider the importance of structure in mathematics
 To explore understanding of a key mathematical structure, which has relevance across the curriculum
 To make explicit opportunities to embed the aims of the National Curriculum.
Timings
 Ten minutes initial input
 Ten minutes, thirty minutes, sixty minutes or ninety minutes follow up after two weeks.
Resources
 Asking Mathematical Questions Mathematically by John Mason
 Key understandings in mathematics learning. Paper 2: Understanding whole numbers by Terezinha Nunes and Peter Bryant
 Large sheet of paper for display in the staffroom with ‘Doing and Undoing’ in the middle.
Ten minute introduction
 Explain that understanding structure is an important part of mathematics and asking children to attend to structure is part of the role of the teacher in mathematics lessons. Say that John Mason explains this as the difference between ‘workingthrough’ some maths and ‘workingon’ some maths and share the following extract from the John Mason paper:
Working through exercises and working on exercises
The first describes the student who does a few questions, takes a break, does a bit more on the bus, copies a bit from a friend, and ends up with no overall sense of the exercises as examples of anything or what they are about. Contrast this with the student who in doing the exercises asks themself what is similar about the questions and what different, what it is about the context which enables the technique to work, what sorts of difficulties might the technique encounter in different situations, etc. That student is workingon the exercises.
The two states of workingthrough and workingon are completely different, and in particular they involve different energies. Workingthrough minimises effort through minimum involvement. It is unreflective and unmathematical. Workingon minimises effort mathematically, by trying to locate underlying structure and so reduce memory demands.
 Explain that you are going to begin to look what this means, by exploring one particular example, ‘Doing and undoing’. Understanding how to undo what has been done in mathematics requires an understanding of how the mathematics works and mathematical relationships and involves generalising (part of the aims of the National Curriculum). Doing and undoing includes things such as the inverse relationship between addition and subtraction which has been identified as a key understanding in KS1; reference the Key Understandings paper (page 5).
“If children are assessed in their understanding of the inverse relation between addition and subtraction, of additive composition, and of onetomany correspondence in their first year of school, this provides us with a good way of anticipating whether they will have difficulties in learning mathematics in school.”
 Introduce the large sheet of paper with ‘Doing and Undoing’ in the centre. Invite everyone to think about this theme across the maths curriculum over the next two weeks and whenever they think of somewhere it applies, to add it to the sheet, which you will come back to during a future meeting. Write ‘adding and subtracting in KS1’ as the first idea on the sheet.
Followup meeting two weeks later (you may need to prompt people to add to the sheet and model this by adding ideas during the two weeks):
 Have the large sheet which has ideas connected to ‘Doing and Undoing’. Choose from the following to explore, depending on the length of your meeting:
 Look at the mathematical ideas. Ask: Are there any other examples of doing and undoing in maths which you can think of?
Make sure the following are on the sheet:
 filling and emptying
 doubling and halving
 adding and subtracting
 multiplying and dividing, including with fractions (for example dividing one to make a unit fraction and then multiplying the unit fraction to make one)
 square and square root
 turning clockwise and anticlockwise
 reflecting and reflecting again
 translating a shape and returning it to its original position
 converting (for example from g to kg and back to g)
 sequences (such as think of a number)
 Choose one idea relevant to your children and discuss how to support understanding.
Consider:
 What might the children physically do to demonstrate this relationship?
 What contexts make sense of the relationship?
 How could it be modelled with different resources/pictures/drawings?
 How could it be recorded symbolically?
 What would you want the children to notice and be able to explain?
 What sort of questions would show if the children have understood?
 Asking the children to do things physically and modelling with resources and drawings will prompt different explanations and support the children to focus on what has changed and what has stayed the same.
An example connected to adding and subtracting:
 The children move forwards four on a large number track on the floor and then discuss what they would need to do to return to the number they started on; this would generate the language of forwards and backwards which would be matched to adding and subtracting.
 Contexts would include board games (for example, throwing four in a game, moving forwards and the square landed on says ‘move backwards four places’), saving money and then spending it, children joining a class and children leaving it, making some cakes and then eating them etc.
 Bead strings used to model adding then subtracting the same beads, which could be matched to a number line; money put in the savings box and then taken out; or a tin of counters being added to a pile of counters and then taken off the pile.
 10 + 4 – 4 = 10, x + 4 – 4 = x and 4 + x – x = 4
 It is important that the children notice that it doesn’t matter which number you start from or what you add, as long as you subtract what you have added you return to the number you started with.
 Understanding is applied in questions such as: 34 + 17 + 65 – 17 + 1=
 Ask: Does undoing what you have done always take you back to where you started? Can you think of any examples in maths where it doesn’t?
 Sometimes, doing and undoing will not always take you back to the original number. For example:
 When using some calculators 1 ÷ 3 x 3 = 0.9999999 and = 6.9999999
 Undoing to solve ‘I think of a number, subtract five then multiply my new number by itself and reach four. What number did I start with?’ which may take you to 7, when the original number was actually 2.
 Situations where it is not possible to know the original number by undoing; for example undoing multiplying by 0.
Image Credit
Page header by GerryJ10 (adapted), some rights reserved
