1. Maintaining Challenge at KS3
Update from the Primary Sector
Jenny Stratton
DHT Westdene, Maths SLE &
NCETM Mastery Specialist

2. Briefing will cover:
New curriculum
Primary mastery
Concrete resources
Life without levels

3. Mastery for all & the National Curriculum
“The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace.”
“Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content.”
3 aims
become fluent in the fundamentals of mathematics, including through varied and frequent practice, so that pupils develop conceptual understanding and recall and apply knowledge
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and using mathematical language
solve problems by applying their mathematics to a variety of routine and non-routine problems

4. The New Curriculum Factual & Procedural Fluency
Conceptual Understanding
INTEGRATION

5. New curriculum- new content
Add, subtract, multiply and divide fractions
Know parts of a circle
Construct pie charts
Find the area of parallelograms
Use long multiplication and long division
Understand and use more algebra
Estimate and calculate volume

6. Why?

8. What’s gone? Probability Rotation Mode, median and range
Old style mental maths paper
Level 6 extension papers

9. Teaching for Mastery – The Five Big Ideas
Number Facts
Table Facts
Making Connections
Procedural
Conceptual
Chains of Reasoning
Access
Pattern
Representation
& Structure
Mathematical Thinking
Fluency
Variation
Coherence
Small steps are easier to take

10. Key Mastery Principles

11. Paradigm Shift
a cumulative curriculum vs a spiral curriculum
longer units which build on each other (i.e. 1 term on number before applying these skills in other units); each lesson is a small step on from the one before
a focus on concepts and interconnected ideas vs isolated objectives
every lesson builds on prior concepts; use of number sense for mental strategies is made explicit
the whole class working together vs differentiation by content
all children work on the same activity; mixed ability
depth vs acceleration
‘rapidly grasping’ children do not move onto the next concept; they explain, apply and reason
‘working on’ vs ‘working through’
longer time spent picking apart an idea
intelligent practice/variation theory vs mechanical repetition
move away from ‘model’ and ‘copy and repeat’ structure
an expectation that all children can achieve vs labelling children according to ‘ability’
belief that ability is determined by effort/input – some children just need more!
White Rose, Maths No Problem, NCETM mastery booklets, Parkfield Maths Academy

12. Concrete Resources

13. Enactive
Symbolic
= 77
£ %
Iconic

14. Tens frame
8+6 = 14 +
Look carefully – how is this child adding 7+6? 10 4 4

15. Part, part whole model
Exposes the additive relationship (one relationship, rather than 2 operations – Anne Watson)

16. The Bar Model Depth/Simplicity/Clarity7 2 5 7
1.9
5.1
7.4 C
1.7
5.7 a b

17. What’s the same, what’s different?

18. Solve the following)
 + 17 = –  = 90 – 59 Consider the strategies you used?
This illustrates how a conceptual method rather than a procedural method can lead to a quicker answers. Whilst procedures are important, they do not stand alone but need to be underpinned by conceptual understanding.

19. How would find the missing numbers?=
146 – 28 =
Highlight that we can use the relationship to find the missing numbers over an arithmetic procedure 28 32
146
150

20. Variation Theory in Practice
Compare the two sets of calculations
What’s the same, what’s different?
Consider how variation can both narrow and broaden the focus
Taken from Mike Askew, Transforming Primary Mathematics, Chapter 6

21. NCETM

22. “By working on tasks which focus on the nature of the relation rather than on calculation, students’ attention is drawn to structural aspects as properties which apply in many instances.”
(Mason 2006)

23. Let’s consider fractions…
Conceptual Variation
Representing the same concepts in different ways to draw out essential features and form a fluent and deep understanding of the concept.
Concrete, pictorial to abstract
Making connections in different ways
Let’s consider fractions…

24. Children need to understand
Fractions as part whole relationship (numbers and shapes)
Fractions as a number
Fractions and division
Fractions as operators
Calculating with fractions
Fractions as ratio

25. Iterating a fraction

26. Tasks which challenge, probe understanding and provoke reasoning about concepts.
Which line is longer?
First：
Second：

27. Part whole models
“Where there is an explicit division of a whole into equal parts, children are able to determine the fraction of the part/parts indicated by counting the number of parts in the whole and the number of parts indicated. In figures where it is more difficult for the children to adopt this 'partitioning' approach, children are required to analyse the relationship of the particular part/parts indicated in relation to the entire whole.”
From “Teaching Fractions with Understanding: Part Whole Concept”, nrich
Pouring contexts provide a continuous, non partitioned model which draw attention to the ratio between the part and the whole.
In this picture…
Where is the part?
Where is the whole?

28. CPA 1 1 Concrete “Fill the glass about one fifth full.” Pictorial
“Imagine this beaker is one fifth full of water. Draw a line to show approximately where it would come up to.”
Abstract
“Can you estimate where the number one fifth sits on this number line?”
And on this one… 1
“If I now want the glass one quarter full, should I pour more rice in or take rice out?
“If I fill the glass one millionth full, will there be a lot of rice or a very little bit of rice?” 1

29. Building a concept of non unit fractions
The side of the glass/vase becomes a proxy for the number line: 19 1
About there 17
About there 17 19
Therefore…

30. How much squash is in the bottle which is 3 4 full?
Year 3
Depth
Here I have two 2 litre bottles. One of them has litre in it [fraction as number in itself], and one of them is full [fraction as operator on a set]. Which is which?
How much squash is in the bottle which is full?
What fraction of the bottle which has litre in it is full?
An Anne Watson activity:
‘Measure is the way into fractions’
One really important bit of subject knowledge which it is useful to know is that fractions can be:
operators on sets (“can you pass me half of the apples?”) or
number in themselves (“can you pass me half an apple?”)
We tend to be not very good at embedding an understanding of the latter in children, and if asked to mark the number
1 2 on a 0 – 5 number line, children will often incorrectly place it at 2 1 2
We used these bottles to help embed the understanding of fractions as operators on sets vs numbers in themselves in Year 4.

31. Deepening learning 17 4 Write as a mixed number Calculate 17 ÷4
What is the same, what is different?

32. Connection to division
Children will have explored these relationships:
12 ÷ 4 = 3 ¼ of 12 = ¼ = 3/12
24 ÷ 4 = 6 ¼ of 24 = ¼ = 6/24
48 ÷ 4 = 4 ¾ of 48 = ¾ = 36/48
before doing equivalent fractions

33. The Fraction Ruler
Take your strip of card

34. The Fraction Ruler
Take your strip of card

35. Fractions as Numbers

36. Making connections through representations
multiply simple pairs of proper fractions,
writing the answer in its simplest form e.g.
× = 1 4 2 8
Animated talk through using the bar
Talk through two ways of looking at it.
Start with ¼ and scale it by ½
OR Start with a ½ and find a quarter of it
Talk through steps
Need to start by focusing on a ¼ of the bar so divide it into 4
Then halve one of the quarters
Split the rest of the bar into equal amounts to check the proportion of the whole that the product is equal to.

37. Y6 Multiplication and division of fractions
divide proper fractions by whole numbers (e.g. ÷ 2 = ) 1 4 8
Note that the example is carefully chosen to mirror the multiplication
Draw out that x ½ is equivalent to division by 2 – just as a fraction with 2 as the denominator means to divide by 2.
Finding a half – which happens in KS1 is actually scaling by ½

38. What is depth? Factual and Procedural Fluency Conceptual Fluency
Reasoning
Sustained learning
Connected Learning
Simplicity and Clarity

39. How is depth achieved? Longer time on topics
Intelligent practice (variation)
Detail in exploring the concept – all aspects exposed and linked (conceptual variation)
Small steps
Questioning and activities develop reasoning and make connections

40. Interim teacher assessment frameworks
Life without Levels
Interim teacher assessment frameworks vs
new SATs scaled scores

41. NB. We cannot record any child who is working at greater depth.
Place value (large numbers and decimals)
Mental calculation
Multi-step problems
FDP equivalents
Calculate with FDP
Formula (area, volume etc)
Measures (calculate)
Missing angles P
NB. We cannot record any child who is working at greater depth.

42. Scaled scores

43. Greater depth?
Expected standard

44. Any questions?