emma gale and sam hickman-smith teaching for …• mastery is something that we want pupils to...
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Teaching for Mastery EMMA GALE AND SAM HICKMAN-SMITH
TEACHING FOR MASTERY SPECIALISTS
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AimsTo have a collective understanding of what mastery is
To dispel myths surrounding TfM
To demonstrate TfM lessons
To explore leadership priorities
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Agenda
9:30am Collective Understanding –What is Mastery?Myths of Mastery
10:30 Break 10:50 Demonstration Lessons
TRG style questionsPotential Feedback
12:45 Lunch1:30pm Whole school Implementation3:00pm Questions3:30pm Close
Agenda
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Post – it note questions
Questions?
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Where are you now?
Brand new to mastery
Working with many schools advising on mastery approaches
What are you hoping to get out of today?
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Teaching for Mastery and the Shanghai Exchange Programme
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Evidence based practice at a local, national and global
level:
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Aims of the morning:A collective understanding of TfM principles
To dispel myths surrounding TfM
To explore the 5 big ideas plus greater depth and support (keep up not catch up)
To demonstrate a TfM lesson
To explore leadership priorities and next steps
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Collective understanding• Mastery is something that we want pupils to acquire.
• So a ‘mastery maths curriculum’, or ‘mastery approaches’ to teaching maths, both have the same aim—to help pupils, over time, acquire mastery of the subject.
• That’s why we use the phrase ‘teaching for mastery.’ NCETM
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What is the mindset in the schools you visit?Pupils?
Teachers?
Support Staff?
Parents?
Governors?
Management Team?
Who is the biggest challenge to get on board?
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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- for example driving a car
• I’m really good at doing it – painting a room, or a picture
• I can show someone else how to do it.
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Mastery Means…Statement Sort
Use the statements on your table to discuss what the key aspects of a mastery approach/curriculum are
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(NAMA, 2015)
5 Myths of Mastery• One single definition
• No differentiation
• Special Curriculum
• Repetitive Practice
• Text Books
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Five Big Ideas – Teaching for MasteryQuality First Teaching?
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Let’s talk about the:
‘Effective teaching of mathematics’
(This will be applicable to all schools at any stage of their journey.)
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New terminology: don’t assume everyone speaks your language.
Prior attainmentRapid graspers and Struggling learners
Is this a group? Or is it concept and condition dependent?
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A Mind-Set Shift:“Ability labelling shapes teachers’ attitudes towards children and limits their expectations for some children’s learning. Teachers vary their teaching and respond differently towards children viewed as ‘bright’, ‘average’ or ‘less able’ ” (e.g. Rosenthal and Jacobson 1968; Jackson 1964; Keddie 1971; Croll and Moses 1985; Good and Brophy 1991; Hacker et al 1991; Suknandan and Lee 1998).
Also see Hart, S, Dixon A, Drummond MJ and McIntyre D (2004) Learning Without Limits, Open University Press (“A book that could change the world.” Prof. Tim Brighouse)
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the ‘traditional’ way we differentiate i.e. putting the children into ability grouped tables and providing easier work for the less able and more challenging ‘extension’ work for the more able has ‘a very negative effect on mathematical attainment’
‘one of the root causes for our low position in international comparisons’.
Charlie Stripp(Director NCTEM)
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It damages the less able by fostering a negative mindset that they are no good at maths
in practice it results in the less able children being given a ‘reduced curriculum’.
it damages the more able because it encourages children to rush ahead or can ‘involve unfocused investigative work’
labelling the child as ‘able’ creates a fixed mindset so the child believes that they should find maths ‘easy’ and becomes unwilling to tackle demanding tasks for fear of failure.
Charlie Stripp claims:
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Representation and Structure
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Key Structures•Part – Part – Whole
•Tens Frames
•Bar Model
•Language
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Representation and Structure
C-P-A
Expose Mathematical Structure
Provide access and challenge
Teacher-Pupil Talk
Pupil-Pupil talk
Developing Reasoning Skills
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LanguagePrecise mathematical language
Stem Sentences
Generalisations
Definitions
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What is a Stem Sentence?A gap fill to support children in working with fractions.
Transferrable
Mathematically true
Precise Language
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“To find a half we divide by 2, to find a ……we divide by…….”
When we divide by 2 we find a half, when we divide by…..we find a ……..”
Examples of stem sentences…
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What is a Generalisation?Mathematically true
A structure of their own
Should be used during the application stage of a lesson
Bridge the concrete/pictorial to the abstract
Tasks should promote discovery
To be discovered rather than told
Enable us to be fluent & efficient- we do less maths!
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“The denominator tells us how many
equal parts there are in the whole.”
(Important everyone in school owns and uses these consistently.)
Example of a generalisation…
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FluencyEfficiency - Accuracy - Flexibility
• Deep understanding of low number
• Composition of numbers to 10
• Repertoire of facts to draw upon
• Solid knowledge of 10 and 0 and their relevance in the place value system
• Clear understanding of the 4 operations
• Relationships between operations
• Variety of calculation methods
• Solid understanding of equality
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Mathematical Thinking• Highlighting relationships
• Pattern spotting
• Reasoning
• Concept/non-concept
• Language
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VariationIn the late 1970s, mathematics teaching in China came across big
challenges. Regarding students’ mastery of mathematics:
• Understanding of mathematical concept was ambiguous and vague.
• Pupils could not identify the mathematics when its context was
slightly changed
• When pupils encountered mathematical problems with slight
variation, they did not know what to do.
• What they had learned in mathematics was inflexible and
unconnected.
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2 strandsConceptual
Procedural
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Conceptual Variation• Varying the representation to extract the essence of
the concept
• Supporting the generalisation of a concept, to recognise it in any context
• Drawing out the structure of a concept – what it is and what it isn’t
• To find out what something is, we need to look at it from different angles – then we will know what it really looks like!
• What’s the same and what’s different?
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Conceptual Variation
12
12
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Describe an Elephant
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According to your description could this be an elephant?
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Concept vs. non-concept
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Non-standard examples of an elephant
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Standard and non-standard
Boaler, Jo. (2016) Mathematical Mindsets
a
b
c
a
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Over half of eight year olds did not see these as examples of a right angle,
triangle, square or parallel lines
Boaler, Jo. (2016) Mathematical Mindsets
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14Take a square and fold it into 4 to show
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How do you know it’s a quarter?
The whole is divided into __ equal parts and ____ of those parts is shaded.
Stem Sentence Example
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15
Non Conceptual Variation
The red part is , True or False?
× √ ×16
What is the concept, what is not the concept?
Use the stem sentence to help you decide.
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Non Conceptual Variation
What do you notice about these images ?
13
14
14
15
14
× × × √
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Conceptual Variation
What it is
(positive)
Standard
Non-standard
What it is not
(negative)
Conceptual Variation
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Conceptual VariationThe aim of variation is to develop a deep understanding of the concept. An important teaching method ... It intends to illustrate the essential features by demonstrating different forms of visual materials and instances or highlight the essence of a concept by varying the nonessential features.It aims at understanding the essence of object
and forming a scientific concept by putting away the non-essential features
(M Gu 1999)
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Procedural Variation
• Procedural variation occurs within the process of doing mathematics.
• Provides the opportunity to focus on the relationship (not just the procedure)
• Small steps are made with slight variation
• There is a connection as you move from one example to the next
• Make connections between problems, using one problem to work out the next
• Recognition of connections needs to be taught
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Different methods
Providing Textbook Supports for Teaching Math Akihiko Takahashi https://prezi.com/s1nvam1gllv9/providing-textbook-supports-for-teaching-math/
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Procedural VariationWhich of these do you think is a
better example?
Set A120 – 90235 – 180502 – 397122 – 92119 – 89237 – 182
Set B120 – 90122 – 92119 – 89235 – 180237 – 182502 – 397
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What do you notice about the calculations below?
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Rounding Example
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VariationVariation: What is it?
‘A well-designed sequence of tasks invites learners to reflect on the effect of their actions so that they recognize key relationships’
Teaching with Procedural Variation: A Chinese Way of Promoting Deep Understanding of Mathematics
Mun Yee Lai http://www.cimt.org.uk/journal/lai.pdf How do we do this? ‘Influencing the way children think through what we keep the same and what we change’ Debbie Morgan
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Procedural VariationProcedural Variation
the questions that are asked are important… Providing the opportunity:
- for practice (intelligent rather than mechanical); •
-to focus on relationships, not just the procedure;
-- to make connections between problems;
- to use one problem to work out the next;
- to create other examples of their own.
-The questions that are asked are important as they develop mathematical thinking
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BenefitsSupport deep learning by providing rich experience rather than superficial contact
Provide the necessary consolidation (in familiar and unfamiliar situations) to embed and sustain learning
Focus on conceptual relationships and make connections between ideas
Support pupils’ ability to reason and to generalise
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What makes good practice?Shooting from all over the court or refining through making connections to previous shots.
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Coherence
• A comprehensive, detailed conceptual journey through the mathematics.
• A focus on mathematical relationships and making connections
The smaller the distance from the existing knowledge and the new learning, the greater the success (Gu, 1994).
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Whole Class Teaching
Inclusion is essential but it must be thought about in a different way to
allow ALL children an equality of access to quality teaching and learning in
mathematics.
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Ping-Pong
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We don’t differentiate anymore!
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OfstedDifferentiation should therefore be about how the teacher helps all pupils in the class to understand new concepts and techniques. The blend of practical apparatus, images and representations (like the Singaporean model of concrete-pictorial-abstract) may be different for different groups of pupils, or pupils might move from one to the next with more or less speed than their classmates. Skilful questioning is key, as is creating an environment in which pupils are unafraid to grapple with the mathematics. Challenge comes through more complex problem solving, not a rush to new mathematical content. Good consolidation revisits underpinning ideas and/or structures through carefully selected exercises or activities. Mastery calls this ‘intelligent practice’.
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Break!
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A Year 5 Lesson
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A Year 2 Lesson
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Teaching for Mastery
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Teaching the whole class together
Small steps approach
Precise use of mathematical language
Speaking in full sentences
Opportunities for children to go deeper
Analysis of strategies
Discussion
Conceptual variation
Useful context
Link to relevant life-experience
Key facts
Procedural variation
Small focus
Misconceptions at the forefront
Review at the start of lessons
Opportunities to make connections
Generalisation found and used
Colour-coding
C-P-A
Maths not ‘clouding’ learning
Features of a lesson:
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Lunch
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A Leader’s View
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Action PlanningA helpful order for implementation:
Mindsets – common language, clear vision, clear expectation
Keep up and not catch up sessions
Number Facts
Coherence
Representation and Structure
Mathematical Thinking
Fluency
Variation
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Keep up, not catch upBenefits:
• Live assessment
• Quick intervention
• Provided by an expert
• Closing the gap instantly
• Available to all learners
• Less marking
• Effective use of teacher time
• Perceptions of intervention
• Data – Target groups
• Pre-teach
• Greater depth
Challenges:
• Timetabling
• Staffing
• Repeated use by same children
Maths Mentors
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CoherenceSupporting staff with the sequence of teaching
Longer time teaching key topics
Whole class teaching
Small steps approach to lesson planning – reduce distance between old and new learning
SKE
Lesson Crafting
Teach with low number
Beware of the Golden Cloak
Tasks and resources are used appropriately by the teacher
Assessment of pupils’ strengths and weaknesses informs choice of task and how these address misconceptions
Tasks build conceptual knowledge in tandem with procedural knowledge
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Representation and StructureAn expectation that all children reach the abstract phase
Structures used to expose structure and not help get to answer
Contexts used to support understanding
Key resources purchased/dusted off! (double sided counter, tens frames, Cuisenaire)
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Mathematical ThinkingHow will you get a whole school, consistent approach to the use of language?
Do staff understand the importance of repetition?
Is reasoning understood as linking calculations, identifying non-examples, solving questions through use of structure (not over-calculating) – It’s not always lots of writing!
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FluencyIs handwriting taught in the writing lesson?
Spelling? Reading?
How can extra fluency sessions be provided to children?
-Parents?
-Homework?
-Early Morning Work?
-After lunc
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VariationThe hardest to implement
Which resources could help them?
Text books
NCETM professional development spines
White Rose
Nrich
None used exclusively, just like we wouldn’t teach reading with one reading programme!
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WorkloadHow do the expectations on marking and writing plans support teachers in enabling them to spend time redesigning their lessons?
Lesson design is key.
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Greater DepthShow us your sparkle
Flamingo challenge
Zoom
Variation
Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content (DfE, 2013).
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Subject Knowledge!Audit
Professional Dialogue
Whole school planning
Knowledge of what comes before and after to prevent mathematical untruths!
NCETM – subject knowledge section
Well timed ‘Mastery’ inputs from maths lead
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Assessment - PPMAutumn PPM
Coverage vs. progress
Depth of understanding
Diagnostic assessment identifies specific difficulties and whether interventions are appropriate
How assessment informs planning
Use of specific and clear feedback
Teacher knowledge of misconceptions and how to address them
How does the assessment system provided support teachers is spending more time on key topics?
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Maths leadRegular release
Regular input
Demonstration lessons
Keep ahead
A voice in the school (SMT)
Team teaching
Planning support
Coaching
TIME and Money (RAP)
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Most Successful CPD Whole school planning support
Focus year groups
Team teaching
Bi-weekly monitoring
Lesson study
Talent shifting
TRGs
Constant drip feeding (at least a staff meeting every 2 weeks)
Demonstration lessons
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ParentsLearning Together
Parent Workshops
Parents’ evening stand
Newsletter updates
Coffee Morning
Number Facts Cards
Fluency updates
Myth Buster!
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Performance Management
Increased monitoring for:
Training Needs
Consistency
Research
Learning
Evaluating
Whole School Feedback
No link to performance management in the 1st year
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Lesson ObservationWhat has changed in lesson observations?
Prerequisites in place
Ping Pong Introduction
Worked Examples
Less examples
Language
Active adults
C-P-A
Repetition
Pace
Concept driven – no maths clouding the learning
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Live ‘Marking’Feedback
Policy Impact
All adults
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Learning EnvironmentWorking walls (no longer laminated and up all year!)
Classroom layout
Groupings
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Questions
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Evaluations!Please fill in the evaluations.
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Signposting
Sussex Maths Hub
NCETM