Teaching for Mastery EMMA GALE AND SAM HICKMAN-SMITH

TEACHING FOR MASTERY SPECIALISTS

AimsTo have a collective understanding of what mastery is

To dispel myths surrounding TfM

To demonstrate TfM lessons

To explore leadership priorities

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

Post – it note questions

Questions?

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?

Teaching for Mastery and the Shanghai Exchange Programme

Evidence based practice at a local, national and global

level:

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

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

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?

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.

Mastery Means…Statement Sort

Use the statements on your table to discuss what the key aspects of a mastery approach/curriculum are

(NAMA, 2015)

5 Myths of Mastery• One single definition

• No differentiation

• Special Curriculum

• Repetitive Practice

• Text Books

Five Big Ideas – Teaching for MasteryQuality First Teaching?

Let’s talk about the:

‘Effective teaching of mathematics’

(This will be applicable to all schools at any stage of their journey.)

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?

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)

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)

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:

Representation and Structure

Key Structures•Part – Part – Whole

•Tens Frames

•Bar Model

•Language

Representation and Structure

C-P-A

Expose Mathematical Structure

Provide access and challenge

Teacher-Pupil Talk

Pupil-Pupil talk

Developing Reasoning Skills

LanguagePrecise mathematical language

Stem Sentences

Generalisations

Definitions

What is a Stem Sentence?A gap fill to support children in working with fractions.

Transferrable

Mathematically true

Precise Language

“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…

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!

“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…

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

Mathematical Thinking• Highlighting relationships

• Pattern spotting

• Reasoning

• Concept/non-concept

• Language

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.

2 strandsConceptual

Procedural

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?

Conceptual Variation

12

12

Describe an Elephant

According to your description could this be an elephant?

Concept vs. non-concept

Non-standard examples of an elephant

Standard and non-standard

Boaler, Jo. (2016) Mathematical Mindsets

a

b

c

a

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

14Take a square and fold it into 4 to show

How do you know it’s a quarter?

The whole is divided into __ equal parts and ____ of those parts is shaded.

Stem Sentence Example

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.

Non Conceptual Variation

What do you notice about these images ?

13

14

14

15

14

× × × √

Conceptual Variation

What it is

(positive)

Standard

Non-standard

What it is not

(negative)

Conceptual Variation

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)

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

Different methods

Providing Textbook Supports for Teaching Math Akihiko Takahashi https://prezi.com/s1nvam1gllv9/providing-textbook-supports-for-teaching-math/

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

What do you notice about the calculations below?

Rounding Example

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

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

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

What makes good practice?Shooting from all over the court or refining through making connections to previous shots.

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).

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.

Ping-Pong

We don’t differentiate anymore!

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’.

Break!

A Year 5 Lesson

A Year 2 Lesson

Teaching for Mastery

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:

Lunch

A Leader’s View

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

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

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

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)

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!

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

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!

WorkloadHow do the expectations on marking and writing plans support teachers in enabling them to spend time redesigning their lessons?

Lesson design is key.

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).

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

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?

Maths leadRegular release

Regular input

Demonstration lessons

Keep ahead

A voice in the school (SMT)

Team teaching

Planning support

Coaching

TIME and Money (RAP)

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

ParentsLearning Together

Parent Workshops

Parents’ evening stand

Newsletter updates

Coffee Morning

Number Facts Cards

Fluency updates

Myth Buster!

Performance Management

Increased monitoring for:

Training Needs

Consistency

Research

Learning

Evaluating

Whole School Feedback

No link to performance management in the 1st year

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

Live ‘Marking’Feedback

Policy Impact

All adults

Learning EnvironmentWorking walls (no longer laminated and up all year!)

Classroom layout

Groupings

Questions

Evaluations!Please fill in the evaluations.

Signposting

Sussex Maths Hub

NCETM