1. TRG 1: Introduction to Mastery
Thursday 25th February 2016

2. Welcome! Rationale of TRG Structure of TRG sessions
Share current practice
What is Mastery?
The Five Big Ideas
Concept 1: Part-part-whole
Concept 2: Bridging 10
Interim task

3. Rationale of TRG Teacher Research Group Share knowledge and ideas
Explore understanding together
Try things out and discuss findings
Move forward our own practice
“I don’t know all the answers!” – will try and answer any questions and will endeavour to find out for you if I don’t know

4. Structure of TRG sessions
TRG 1: Introduction
TRG 2: Lesson observation and discussion
TRG 3: Lesson observation and discussion
TRG 4: Twilight: Evaluate and going forwards
Interim tasks
Lesson observations – Year 1 and 6
Post-it notes on tables – any questions that you want answering now or as thoughts come up during the session – questions section at the end, ask during and any “long term” questions, areas, issues that need addressing we can build into future sessions

5. Current Practice
Teaching using Mastery concepts from NCETM residentials in Years 1 and 6
All staff using NRich for deepening understanding
Aim to spread Mastery concepts throughout school from EYFS to Year 6

6. What does it mean to master something?
I know how to do it
It becomes automatic and I don’t need to think about it
I’m really good at doing it
I can show someone else how to do it
Think of something that you do that you would say you have “mastered”? How would you explain that idea of “mastering” it? What does it mean about that activity?

7. Mastery in Mathematics
Achievable for all
Deep and sustainable learning: combat superficial learning
Build on something that has already been sufficiently mastered
The ability to reason about a concept and make connections
Fluency: calculate with confidence
Thinking mathematicians
Superficial: “ a mile wide and an inch deep”: previously sped through curriculum, too wide, children could “do” but not necessarily explain “how” or “why”
WARNING – NCETM based on Shanghai – Cultural questions – Professor Mertten – eg memorization

8. Mathematical Thinking
Teaching for Mastery
Number Facts
Table Facts
Making Connections
Procedural
Conceptual
Chains of Reasoning
Access
Pattern
Representation
& Structure
Mathematical Thinking
Fluency
Variation
Coherence
Small connected steps are easier to take

9. Representation and Structure
How to show the concept to pupils
CPA (concrete – pictorial – abstract)
Bring structure to surface
Identify patterns
Structure – why do the patterns work in that way?
Resources: which ones, available to all (child’s choice to opt out)

10. Mathematical Thinking
Develops and holds mathematical learning together
Reasoning process behind process – the “how” and “why”
Looking for patterns and relationships
Logical Reasoning
Making Connections
Language
Reasoning processes – how answers are obtained, why the method/strategy worked, what would be the most efficient method?
Identify patterns – why does that pattern happen? (structure) – generalise
Language – precise, in full sentences, modelled by teacher, pupils supported to do the same when discussing their maths

11. Pairs of Numbers (Stage 1)
If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Can you use them all? Say how you got your answer.  
Now put the counters into pairs to make 12.
Can you use them all? Say how you got your answer.
Now put the counters into pairs to make 13.
What do you think would happen if I wanted to make 14?
What Maths are you using? Are you consolidating? Are you just using calculation?
Generalising; hypothesising; reasoning (use facts/computational strategies children know to efficiently determine unknown answers)
“Value added” – some children will just practise skills of adding; others work on reasoning (LOW CEILING, HIGH THRESHOLD)

12. Fluency Factual knowledge: number bonds, times tables Speed of recall
Bringing together different elements to produce a coherent response
Draw on relevant facts (connections)
Flexibility: knowing different strategies but selecting the most efficient for the task in hand
More than memorization of facts or just one procedure – the meaning of operations and their relationships, a large repertoire of number facts and their relationships eg 4 X 5 and 4 X 50, understanding base 10 number system including structure of numbers and number behaviour using different operations eg 24 X 20 = 240, = 34
Eg long multiplication: 36 X 27 – what do pupils require in terms of fluency?
Factual knowledge – times tables, knowledge of number relationships eg 20 x 6 from 2 x 6 and then x 10, multiplying by 10
Flexibility – children generate own calculations – 13 x 2 – lay out as long multiplication – efficiency???? Or 42 x 10!

13. Variation
The central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features.
Gu, Huang & Marton, 2004
Allowing children to see connections by deciding what to vary in your examples
PROCEDURAL
CONCEPTUAL

14. Procedural Variation
changing the numbers within a procedure to highlight patterns and structures

15. Procedural Variation
The child is carrying out the procedural operation of multiplication, but through connected calculations has the opportunity to think about key concepts involving multiplication and place value
Provides the opportunity: To focus on relationships, not just the procedure; To make connections between problems; Using one problem to work out the next

16. Conceptual Variation
different ways to represent the same concept to draw attention to its key features

17. Conceptual Variation
What do I want to draw their attention to? What do I need to vary to do this?

18. Coherence Small, logical steps Sequence of lessons Time!
Keeps the class moving together – nobody gets left behind
Another cultural issue – time?!

19. Mathematical Thinking
Teaching for Mastery
Number Facts
Table Facts
Making Connections
Procedural
Conceptual
Chains of Reasoning
Access
Pattern
Representation
& Structure
Mathematical Thinking
Fluency
Variation
Coherence
Small connected steps are easier to take
BREAK! Discuss so far. Post-its for questions
After break – Teaching, planning and 2 concepts!

20. Mastery Lessons
Short but intense sessions: minutes then additional sessions during the day for practice and intervention
Sharp and concise focus (objective)
Teacher modelling – CPA
Embed concept in context early on
Pace: teacher led is interspersed with pupil discussion/short activities
Questioning to elicit misconceptions early
Accurate language: teacher and pupils
Focus: Builds directly on previous step from day before
CPA: continual interchange between representations
Embed concept: eg Tom has 7 pencils and Sally borrows 2, how many does he have left?
Pace: teacher led, with partner, discuss together, practice – cyclical
After first session – totally different session in between then come back – 30 minutes practice time (can be time where you have support groups, but all children should have time to practice independently)

21. Planning Lessons Overarching objective – work backwards in small steps
Identify key concept of first lesson
Examples: Variation – what do you want to the children to notice?
Representation: resources, CPA
What may be the difficult points?
Language models
Challenge question
Challenge question – gain more depth without accelerating into new material

22. Concept 1: Part-Part-Whole
Understanding of number bonds
Addition and subtraction relationship
Used in Year 1 and in Year 6

23. Part-Part-Whole (Pictorial)

24. 1 4 ? ? Before then now At first Left? 4 – 1 = 3 Go away
Pictorial – language, whole class repetition, children see abstract alongside pictorial
Move to all abstract - Show example of work from pupils (year 1, adapted for year 6)
4 – 1 = 3

25. 10 10 5 10 1 8 2 3 5 9 2 3 10 2 8 4 1 – 4 = procedural variation
1 - 5 = conceptual variation
6 = challenge question – thinking mathematically, fluency 8 4

26. Concept 2: Bridging 10
Pictorial – could be 7 flowers in one vase and 5 in another – how many altogether?
Number bonds to 20, addition and subtraction
Year 1 and Year 6 (use for decimals)

27. Mathematical Thinking
Teaching for Mastery
Number Facts
Table Facts
Making Connections
Procedural
Conceptual
Chains of Reasoning
Access
Pattern
Representation
& Structure
Mathematical Thinking
Fluency
Variation
Coherence
Small connected steps are easier to take
BREAK! Discuss so far. Post-its for questions
After break – Teaching, planning and 2 concepts!

28. Teaching for Mastery The belief that all pupils can achieve
Keeping the class working together so that all can access and master mathematics
Development of deep mathematical understanding
Development of factual, procedural and conceptual fluency
Longer time on key topics, providing time to go deeper and embed learning

29. Interim Task Next date – before Easter (TBC) In school time
Before the next TRG:
Plan a sequence of lessons using the big 5 mastery ideas
Try using one of the concepts discussed within your sequence of lessons
Bring your own reflections, pupil work and thoughts from the pupils

30. Questions and Discussion
Mastery for older years
Vulnerability
NRich
Thank you for coming!