Five Big Ideas to

Develop Mastery

Debbie Morgan

Director for Primary

What is Mastery?

Means that learning is sufficiently:

Embedded

Deep

Connected

Fluent

In order for it to be:

Sustained

Built upon

Connected to

The Mastery Specialist Programme

• 140 primary teachers developing as

mastery specialist teachers

• The programme brings together all that

we have learnt from:

- Visiting China

- Interacting with our colleagues from

Shanghai

- Observing teaching

- Reading research

• Number Facts

• Table Facts

• Making Connections

• Procedural

• Conceptual

• Making Connections

• Chains of Reasoning Problem Solving

• Making Connections

• Access

• Pattern

• Making Connections

Representation

& Structure

Mathematical Thinking

Fluency Variation

Coherence

Teaching for Mastery

Small steps are easier to take

A comprehensive, detailed conceptual

journey through the mathematics.

A focus on mathematical relationships

and making connections

Ping Pong

Provides a clear and coherent journey through

the mathematics

Provides detail

Provides scaffolding for all to achieve

Provides the small steps

Pupil Support

One of the most important tasks of the teacher is

to help his students…

If he is left alone with his problem without any help

or insufficient help, he may make no progress at

all…

If the teacher helps too much, nothing is left to the

student

(Polya 1957)

The Planning S

Key conceptual

ideas on post its

Representations

Difficult points

Variation

Going deeper

Making

connections (idea adapted from Devon

Advisory Service )

Mathematics is an abstract subject,

representations have the potential to provide

access and develop understanding.

Representation and structure

“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

http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm

Part Part Whole Models

Part whole relationships

7 is the whole

3 is a part and 4 is a part

Shanghai Textbook Grade 1 Semester 1

Representing the Part - Part Whole

Model

Attention to Structure

10

5

10

1 8 2

10

9

2

3

5

8 4

3

3

10

2

5 apples and

2 apples

Amy

Developing Depth/Simplicity/Clarity

19

7

5.1 1.9

7.4

5.7 1.7

7

5 2

C

b a

It is generally perceived as one of the

most valuable experiences within

Chinese mathematics education

community (e.g. Sun, 2011).

Conceptual

Variation

3

2

4

1

9

4

An aspect of variation –

developing depth through dong nao jin

3

2

True or False?

3 8

2 8 + =

5 16

3 9

2 9 - =

1 9

2 14

1 7 - =

1 7

Year 1 January 2016

3 + 2 = 5

6 + = 8

+ 7 = 9

9 = + 7

7 - = 4

6 = - 2

4 + 3 = 5 +

动脑筋 (dong nao jin)

A regular part of a lesson

In general ,this part is not from the textbook.

Sometimes it is:

• A challenging question for students,

• A “trap” for students.

• Very “tricky” which may let the students

“puzzle” again

• It is an opportunity help student think about the

knowledge in another way.

动脑筋 (dong nao jin)

There are two parallelograms, the areas are same or not?

Can you draw other parallelograms which have the same area?

(Let the students pay attention to the bottom and height, it is the key point of

the whole lesson.)

Why Facts and Procedures?

Daniel Willingham

Is it true that some people just cant do math?

In teaching procedural and factual

knowledge, ensure the students gets

to automaticity. Explain to students

that atomicity with procedures and

facts is important because it frees

their minds to think about concepts.

+ 0 1 2 3 4 5 6 7 8 9 10

0 0 + 0 0 + 1 0 + 2 0 + 3 0 + 4 0 + 5 0 + 6 0 + 7 0 + 8 0 + 9 0 + 10

1 1 + 0 1 + 1 1 + 2 1 + 3 1 + 4 1 + 5 1 + 6 1 + 7 1 + 8 1 + 9 1 + 10

2 2 + 0 2 + 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 + 8 2 + 9 2 + 10

3 3 + 0 3 + 1 3 + 2 3 + 3 3 + 4 3 + 5 3 + 6 3 + 7 3 + 8 3 + 9 3 + 10

4 4 + 0 4 + 1 4 + 2 4 + 3 4 + 4 4 + 5 4 + 6 4 + 7 4 + 8 4 + 9 4 + 10

5 5 + 0 5 + 1 5 + 2 5 + 3 5 + 4 5 + 5 5 + 6 5 + 7 5 + 8 5 + 9 5 + 10

6 6 + 0 6 + 1 6 + 2 6 + 3 6 + 4 6 + 5 6 + 6 6 + 7 6 + 8 6 + 9 6 + 10

7 7 + 0 7 + 1 7 + 2 7 + 3 7 + 4 7 + 5 7 + 6 7 + 7 7 + 8 7 + 9 7 + 10

8 8 + 0 8 + 1 8 + 2 8 + 3 8 + 4 8 + 5 8 + 6 8 + 7 8 + 8 8 + 9 8 + 10

9 9 + 0 9 + 1 9 + 2 9 + 3 9 + 4 9 + 5 9 + 6 9 + 7 9 + 8 9 + 9 9 + 10

10 10 +

0

10 +

1

10 +

2

10 +

3

10 +

4

10 +

5 10 + 6

10 +

7

10 +

8

10 +

9

10 +

10

Adding 0

Adding 1 and 2 Bonds to 10

Doubles

Adding 10

Near doubles

Bridging/

compensating Y1 facts

Y2

facts Claire Christie

The role of repetition

I say, you say, you say, you say, we all say

This technique enables the teacher to provide a sentence stem for

children to communicate their ideas with mathematical precision and

clarity. These sentence structures often express key conceptual ideas or

generalities and provide a framework to embed conceptual knowledge and

build understanding. For example:

If the whole is divided into three equal parts, one part is one third of one

third of the whole.

Having modelled the sentence, the teacher then asks individual children to

repeat this, before asking the whole class to chorus chant the sentence.

This provides children with a valuable sentence for talking about fractions.

Repeated use helps to embed key conceptual knowledge.

https://www.ncetm.org.uk/resources/48070

There are ( )mushrooms.

The whole is divided into ( ) equal parts,

each part is ( ) of the whole,

each part has ( ) mushrooms.

1 2

2

6

12

Fill in the sentences

Memorisation

Memorisation does not necessarily need to

be devoid of understanding.

It can be developed first with limited

understanding and then used as a

framework to deepen understanding

The understanding can be developed first

and then the facts memorised

Abby

Thinking about relationships

Intelligent Practice

In designing [these] exercises, the teacher is advised to avoid mechanical repetition and to create an appropriate path for practising the thinking process with increasing creativity.

Gu, 1991

How do we teach for it?

• Representing the mathematics in ways that

are accessible

• Whole Class Ping Pong style

• Avoid cognitive overload

• Repetition and stem sentences

• Learning facts to automaticity

• Variation and intelligent practice

• Dong Nao Jin