1. Session 2 – Representation and Structure
Mastery Unlocked
Session 2 – Representation and Structure

2. The 5 big ideas

3. Representation and structure
Mathematics is an abstract subject, representations have the potential to provide access and develop understanding.
A representation needs to pull out the concept being taught. It exposes the underlying structure of the mathematics.
The purpose of a representation is to reveal the underlying structure of the mathematics.

4. Creating a conjecturing community…..
Jot down a 2 digit number.
Reverse the digits.
Add them together.
What do you notice? Can you make a conjecture?
How many examples do you need to do to spot a pattern?
Why do we get multiples of 11?
= 55
= 22
= 99
John Mason – create a conjecturing community. Need to accept all answers without judgement.
Patterns start emerging after 3 examples.
Can you prove why using dienes or pictures? Eg
Every ten matches a unit therefore it will always give you a multiple of 11.

5. The stages of representation...
Famously, the educational psychologist Jerome Bruner recognised that children have difficulty accessing abstract concepts – like the value of a number compared to the written numeral. Success can be achieved by experiencing three stages of representation:
Concrete / enacting
Pictorial / iconic
Abstract

6. Concrete, Pictorial (iconic), Abstract

7. Constructing meaning
Mathematical tools should be seen as supports for learning. But using tools as supports does not happen automatically. Students must construct meaning for them. This requires more than watching demonstrations; it requires working with tools over extended periods of time, trying them out, and watching what happens. Meaning does not reside in tools; it is constructed by students as they use tools”
(Hiebert 1997 p 10) Cited in Russell (May, 2000). Developing Computational Fluency with Whole Numbers in the Elementary Grades)
The tools themselves don’t construct meaning – the meaning is constructed by the child using them.

8. Progression through the stages
Struggling learners often get “stuck” in the concrete stage.
It is important that we move them on and show them how to move through each stage and onto the abstract.
Fast graspers often view the concrete as unnecessary or even ‘beneath them’.

9. Resources and Representations of Mathematics
Resources to help build concepts
Counter - representation activity……..5 counters…..show 3, show 50, show 41, show 64….in a different way?
Draw attention to the point made that resources help build concepts.
If children are able to understand and manipulate representations and structure in mathematics it helps them problem solve and apply their learning.
Different representations and resources draw attention to different things – you need to pay attention to which you use.
Ofsted 2013

10. Here are two representations of numbers to 10
Teachers need to select the representations the select depending upon what it is they want to draw attention to.
Tens frames draw attention to the difference to 10 whereas the numicon highlights the odd and even property of number.
These are both very helpful representations of number but, crucially, they are representing different structures.
When would you choose each one?

11. + 17 = =
If children were taught using structure which encouraged them to look for relationships and patterns we could move towards algebraic rather than arithmetic thinking making children efficient rather than procedural mathematicians.

12. Which is the most useful representation of 7 X 6? Why?
Each have specific jobs.
An array is a common representation of multiplication but think about which representation is the most useful?
Children need to see multiplication as a binary operation with 2 inputs: the first input represents the size of the set and the second input represents the number of replications of that set. The array representation of the structure of multiplication is very useful for children.
Consider the 2 different arrays:
The red array is useful for drawing attention to the commutative property of multiplication, whereas the blue array shows not only commutivity, but also why the distributive laws applies to multiplication:
7x6 is equal to 7 x (5+1) or (5+2) X (5+1)
7 x 6 = 42
6 x 7 = 42
7 x (5+1) = 42
(5+2) x (5+1) = 42
7x6 = 42
6x7 = 42

13. Arrays extend into upper KS2 and beyond 4/7 x 7/8 =
The overall array has been divided into 8 x 7 smaller parts, hence the denominator is 56.
The shaded part is 7 x 4 = 28. So the answer is 28/56 or 1/2.
“Models in Mind,” Mike Askew nrich.maths.org

14. Traditional way
Representation and structure is not just something to be used with young children. Consider introducing algebra to children in year 6. The example above is an arithmetic example of solving an algebraic equation. But if algebra was taught with an emphasis on representation and structure what model could we give children to begin to develop a conceptual understanding?

15. The representation of a balance scale is incredibly helpful.
Consider the flexibility that this model gives children. They can apply this conceptual understanding to any algebraic problem.

16. Part 2 The tens frame The bar model
Familiarisation with 2 different representations…
The tens frame
The bar model
There are many other representations which help children visualise
the structure of mathematics, these are just 2 of them.

17. The ten-frame
Ten-frames are two-by-five rectangular frames into which counters are placed to illustrate numbers less than or equal to ten.
Ref: Nrich – “Number sense series: A sense of “ten” and place value.” Jenni Wray
Multiple tens frames
The tens frame is both powerful and cheap! Schools can draw up their own and use counters – there is no need to buy pre-manufactured frames.
The tens frame can help children develop a strong sense of ten that needs to be developed as a foundation for both place value and mental calculations.

18. How many dots? How do you see them?
Number talks – Sherry Parish

19. “There are 8 because 2 are missing.”
Although the tens frame is no longer needed for whole numbers to 10, it can be introduced again where the whole frame represents the number 1 and the counters are tenths or decimals 0.1.
This child has a strong sense of ten and its subgroups, an essential stage from counting to calculating.

20. 2 digit numbers and place value
1 (ten) 3
Ten frames can be a useful tool to build an understanding of place value by the introduction of a second frame.
Notice how the labels make a connection with the abstract.

21. Using the structure of the tens frame
Using your tens frames illustrate this calculation:
There are 7 daffodils and 5 roses
How many flowers are there altogether?

22. How can we use 10 to solve the addition problem?
Bridging 10
How can we use 10 to solve the addition problem?

23. Number facts: Doubles
Step 1: Representation and structure – draw out the idea that doubles are always even numbers. Reference the inverse, e.g. double 3 = 6, so half of 6 is 3.
Step 2: Quick fire - supported by model
Step 3: Can you imagine?
Step 4: deriving calculations from a given model
Step 5: Missing numbers

24. Representation and Structure: Doubles
Get the children to say the full fact aloud: 1 plus 1 equals 2.
What does this number represent? The red/yellow counter/s. The total number of counters.
What do these counters represent?
What is the value of the whole? What is the value of the red/yellow part?
Can you make in a different way in a tens frame?

25. 1 + 1 = 2 1 2 1 1 1

26. 2 + 2 = 4 2 4 2 2 2

27. 3 + 3 = 6 3 6 3 3 3

28. 4 + 4 = 8 4 8 4 4 4

29. 5 + 5 = 10 5 10 5 5 5

30. 6 + 6 = 12 6 12 6 12 6 6

31. 7 + 7 = 14 7 14 7 14 7 7

32. 8 + 8 = 16 8 16 8 16 8 8

33. 9 + 9 = 18 9 18 9 18 9 9

34. = 20 10 20 10 20 10 10

35. Can you imagine…? Doubles
Can you imagine double the number of counters?
Are the children visualising the extra counters? If not, encourage them to.

36. Can you imagine double the number of counters?
2 + 2 = 4

37. double 7 = 14
7 + 7 = 14

38. double 9 = 18
9 + 9 = 18

39. double 8 = 16
8 + 8 = 16

40. Doubles What can you see…?
Can you write a calculation using the word double/half?
Can you write a calculation using the + sign?

41. 3
double = 6
half of 6 = 3
= 6

42. 5
double = 10
half of 10 = 5
= 10

43. 6
double = 12
half of 12 = 6
= 12

44. 9
double = 18
half of 18 = 9
= 18

45. Missing Numbers: Doubles
Imagine the grid is filled with 2 equal groups making _____ in total. How many counters would be in each group?

46. 3
double = 6
half of 6 = 3
= 6

47. 6
double =12
half of 12 = 6
= 12

48. 9
double = 18
half of 18 = 9
= 18

49. The bar model
“The bar model is used in Singapore and other countries, such as Japan and the USA, to support children in problem solving. It is not a method for solving problems, but a way of revealing the mathematical structure within a problem and gaining insight and clarity as to how to solve it. It supports the transformation of real life problems into a mathematical form and can bridge the gap between concrete mathematical experiences and abstract representations.” NCETM
The bar model should be preceded by and used in conjunction with a variety of representations, both concrete and pictorial, all of which contribute to children’s developing number sense. It can be used to represent problems involving the four operations, ratio and proportion. It is also useful for representing unknowns in a problem and as such can be a pre-cursor to more symbolic algebra.

50. 8 3 ? 3 3 Addition and subtraction model
Identification of relationships and making Connections supports depth and sustainable learning and paves the way for later learning + =
3 3 + = - = - =

51. Bar modelling for scaling
Peter has 4 books Harry has five times as many books as Peter. How many books has Harry?
                 
           
4 × 5 = 20 Harry has 20 books 4 4 4 4 4 4
Notice how each section of the bars in the problem below has a value of 4 and not 1. This many-to-one correspondence, or unitising is important and occurs early, for example in the context of money, where one coin has a value of 2p for example. It is also a useful principle in the modelling of ratio problems.

52. Bar modelling with division
Mr Smith had a piece of wood that measured 36 cm. He cut it into 6 equal pieces. How long was each piece?
                
              
36 ÷ 6 = 6 Each piece is 6 cm
When using the bar model for division it is the image of sharing rather than grouping which is highlighted in this representation.

53. Bar modelling with division and fractions
Find 1/5 of 30
                         
The same image can be used to find 2/5 or 3/5 of 30 etc.
The bar model is valuable for all sorts of problems involving fractions. An initial step would be for pupils to appreciate the bar as a whole divided into equal pieces. The number of equal pieces that the bar is divided into is defined by the denominator. To represent thirds, I divide the bar into three equal pieces, to represent fifths I divide the bar into five equal pieces. A regular routine where pupils are required to find a fraction of a number by drawing and dividing a bar, using squared paper would be a valuable activity to embed both the procedure and the concept and develop fluency.

54. He posts four times as many second class letters as first.
Ralph posts 40 letters, some of which are first class, and some are second.
He posts four times as many second class letters as first.
How many of each class of letter does he post?
Have a go

55. He posts four times as many second class letters as first.
How many of each class of letter does he post?
1st 40
2nd
Class
40 ÷ 5 = 8
8 x 4 = 32
2nd Class 32 letters
1st Class 8 letters 8 8 8 8 8

56. GCSE higher paper 2012!
Ralph posts 40 letters, some of which are first class, and some are second.
He posts four times as many second class letters as first.
How many of each class of letter does he post?

57. Bar modelling to support solving word problems
Decision tree to solve word problems
Many schools use (RUCSAC) Read, understand, choose the operation, solve, answer, check.
Bar modelling can help children unpick and make sense of the understand and choose the operation stages.

58. The 5 big ideas
Recap on 5 big ideas and that this session has focussed on representation and structure and how important it is to allow children to develop conceptual understanding in mathematics.
Key messages are:
Choose the representation carefully. Does it draw attention to the structure you want to reveal.
Good models enable the children to “think with them” without eventually needing to make marks on paper. The model is imagined and has become a tool for thinking. “Models in Mind,” Mike Askew Nrich article.
Meaning does not reside in the resources themselves – children construct meaning by using them in a way guided by the teacher to help them gain insight into the underlying structure.