Developing Mastery
and Solving Problems
Review
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 of Mathematics is more…..
• Achievable for all
• Deep and sustainable learning
• The ability to build on something that has
already been sufficiently mastered
• The ability to reason about a concept
and make connections
• Conceptual and procedural fluency
A Mastery Curriculum
• A belief that all pupils can learn
mathematics
• Keeping the class working together so
that all can master mathematics
• Development of deep mathematical
knowledge
• Development of both factual/procedural
and conceptual fluency
• Longer time on key topics
Teaching for Mastery
A connected small step journey
Teaching Concepts
Choose a curriculum that supports conceptual
knowledge. If conceptual knowledge is indeed
so difficult to learn, it makes sense to
(1 ) study just a few concepts each year, but
study them in depth so there is sufficient time to
comprehend one concept before the next one is
introduced and
(2) sequence topics so, as much as possible,
the mental distance between concepts is
small and the previously learned concept will
help in learning each new one.
Knowing to …
I can know-that something is true as a fact. I can knowhow to do something. I can know-why those techniques
work or why those facts are true.
But these “knowings” when taught are hard to move
beyond book-knowledge, knowledge-about. The
knowing-that can be memorised; the knowing-how can
be routine, and the knowing-why a collection of
“incantations” and learned phrases.
What really matters in our increasingly problem-oriented
culture is knowing-to use this or that technique, this or
that way of thinking, this or that approach, in a given
situation as and when it arises.
(From “Questions and Prompts for Mathematical Thinking”, Watson and
Mason, ATM, 1998).
8
Mathematical Structures
and relationship
A focus on the relationships within the
structure of the mathematics, is key to solving
Problems
Children’s difficulty is not in doing the maths but
knowing what maths to do
9
Attention to Structure
We observe that children from quite early
ages are able to appreciate structure
to a greater extent than some authors
have imagined. Initiating students to
appreciate structure implies, of course,
that their appreciation of it needs to
be cultivated in order to deepen and to
become more mature.
John Mason Appreciating structure for all
Ralph posts 40 letters, some of
which are first class, and some
are second.
He posts four times as many second class
letters as first.
How many of each class of letter does he
post?
11
He posts four times as many second class letters as first.
How many of each class of letter does he post?
1st
2nd
Class
8
8
40
8
40 ÷ 5 = 8
8 x 4 = 32
1st Class 8 letters
2nd Class 32 letters
8
8
Ralph posts 40 letters, some of
which are first class, and some
are second.
He posts four times as many second class
letters as first.
How many of each class of letter does he
post?
13
Developing a connected coherent journey
through the mathematics.
Mastery is when the mathematics is
sufficiently deep in order for it to be built
upon and able to be linked to the next
Concept.
A Joiurney with Year 1
The First Problem
There are 5 cars in the car park
How can you represent it?
5 Cars in a Car Park
3 cars drive
away
How can you
represent it?
3 Cars Drive Away
What does one
counter represent?
Connecting to the Part
Part Whole Model
Introduction to difference
There are 5 red cars and
3 blue cars
What is the
difference between
the red cars and
the blue cars?
Making a connection to
the Part Part Whole model
5
3
2
There are 7 children and
4 dinner tokens
Representing difference
with part part whole
Finding the Difference
The Star Challenge