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
Teaching for Mastery
• Access
• Pattern
• Making
Connections
• Chains of Reasoning
Problem Solving
• Making Connections
Representation
& Structure
Mathematical
Thinking
Small steps are
easier to take
Coherence
Variation
• Procedural
• Conceptual
• Making
Connections
Fluency
• Number Facts
• Table Facts
• Making
Connections
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
Representing the Part - Part Whole
Model
Attention to Structure
Shanghai Textbook Grade 1 Semester 1
10
10
10
1
5
2
8
3
9
5
2
3
3
4
8
10
2
5 apples and
2 apples
Amy
Developing Depth/Simplicity/Clarity
2
7
5
1.9
7
5.1
1.7
7.4
5.7
C
a
b
19
It is generally perceived as one of the
most valuable experiences within
Chinese mathematics education
community (e.g. Sun, 2011).
Conceptual
Variation
An aspect of variation –
developing depth through dong nao jin
2
3
1
4
4
9
2
3
True or False?
3
8
3
9
2
14
+
-
2
8
2
9
1
7
5
= 16
1
= 9
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?
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.
Daniel Willingham
Is it true that some people just cant do math?
Adding 1 and 2
Bonds to 10
Doubles
Bridging/
compensating
Adding 10
Near doubles
Adding 0
Y1 facts
Y2
facts
Claire Christie
+
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
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 (12 )mushrooms.
The whole is divided into (2
each part is (
1
2
each part has (
) equal parts,
) of the whole,
6
) mushrooms.
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