Mathematics Mastery
head teachers, senior leaders
&
mathematics mastery school leads
Introduce yourself to a person on your table who
is not from your school.
Introduce this person to the rest of your table.
Name, role, and three facts.
Pioneer partner schools
Manchester (1)
Birmingham (4)
Peterborough (1)
Northants (1)
London (20)
Suffolk (6)
Essex (2)
Reading (2)
Portsmouth (1)
Kent (1)
Sussex (2)
“Mathematics Mastery looks like it has all of the attributes that we are seeking
to transform mathematical learning and success in our school.”
A belief and a frustration
ARK Schools wanted a new maths curriculum to ensure that their
aspirations for every child’s mathematics success becomes reality,
through significantly raising standards.
• Success in mathematics for every child
• Close the attainment gap
The connections
Best practice – national and
international
Research findings and evidence
ARK
Schools
Mathematics Mastery
Curricular principles
• Fewer topics in greater depth
• Mastery for all pupils
• Number sense and place value come first
• Problem solving is central
Feedback from 2012-13
Problem solving and investigations give pupils the
opportunity to demonstrate an in-depth understanding of
the topic.
Since teaching in a mastery style,
I have really had to think about
my questioning which has
improved my subject knowledge.
Agenda
10.00am
What's really possible?
11:15am
Break
11.30am
What does ‘mastery’ look like in the classroom?
12.30pm
Lunch
1.00pm
Mastery in your school
2:45pm
Action planning
3:30pm
Questions and answers
4:00
Close
Why are we here?
Why have you and your school chosen to join
the Mathematics Mastery community?
What do you want/need to know by the
end of today?
Why are we here?
“We know that no child is limited by their background
and that by working hard all children can become
excellent mathematicians. ”
Research shows:
• The gap at age 10 between our strongest and weakest maths performers is
one of the widest in TIMSS - with fewer of our pupils overall reaching the
very highest levels
•
The 10% not reaching the expected level at age 7 becomes 20% by age 11
and, in 2012, almost 40% did not gain grade C at GCSE
• Girls are less likely than boys to study maths beyond 16 and less confident
about their ability overall
• Lower income pupils are falling behind in maths
International Trends
2009 PISA
Nationally, what are we
doing well? What are
we not doing so well?
Maths is not a measuring tool
“Mathematics education should be so much more than
just passing exams and Mathematics Mastery will help
us achieve this. We want every child to not just pass
GCSE mathematics but pass with top grades and to leave
our school with a love of mathematics. ”
Our shared vision
• Every school leaver to achieve a strong foundation in
mathematics, with no child left behind
• A significant proportion of pupils to be in a position to choose
to study A-level and degree level science, technology,
engineering and mathematics-related subjects
What is necessary to make this vision a reality?
Who are our students and what are they
capable of?
“Many of our students come from disadvantaged backgrounds
and arrive at secondary school not fully equipped mathematically
and so there is a necessity to close the attainment gap.”
“We have a key stage two score on entry that is below the national
average. This means that in order for our young people to achieve
A and A* grades at GCSE mathematics we have to work harder for
them and develop a mathematics curriculum that is beyond
outstanding.”
A/A*
B
C
Above
expectations
Meeting
expectations
Below
expectations
Maths achievement in your school
What is your expectation of your students’ achievement in mathematics?
•
•
•
Who will study maths-related subjects at degree level?
Who will study A-level maths? How will they do?
How will students achieve at 16?
Consider your new intake of Year 7 students.
Collaborative Community
“One of the most exciting things about the programme is the idea
that we will be able to talk to colleagues in different schools about
what we are doing. We would hope to learn from others and to
share the ideas we have that we find work well. ”
Shared
curriculum
framework
Online
•
•
•
•
•
Task banks
Assessments
Training
Videos
Blogs
Lesson
observation
tools
Collaborative
cluster
workshops
Mathematics
Mastery
Launch
training
• Teachers
• Leaders
In-school
development
visits
Our approach
You say:
Conceptual
“The mathematics team is firmly
committed to a problem solving
understanding
approach which will equip our students for later life.”
Mathematical
problem
solving
Mathematical
thinking
Language and
communication
Our approach: problem solving
What does it mean to teach through problem solving?
What does it mean to teach for problem solving?
Problem solving – you say:
“Our evidence shows that students are performing markedly
worse on problem solving questions in GCSE exams in
comparison to procedural questions.”
“We promote problem solving within the mathematics
department through our use of starters and as part of the
differentiated extension tasks. However, deeper problem solving
occurs more in the higher sets. Mathematics Mastery will enable
us to bring the culture of problem solving in the mixed ability,
lower sets and intervention sets.”
Mastery for all
Represent
Mathematical
problem
solving
Generalise
Communicate
Potential barrier 1: language and communication
Represent
Mathematical
problem
solving
Generalise
Communicate
Mastering mathematical language
Mathematics Mastery lessons provide opportunities for pupils to
communicate and develop mathematical language through:
• Sharing essential vocabulary at the beginning of every lesson and insisting
on its use throughout
• Modelling clear sentence structures using mathematical language
• Insisting on correct use of language – “I know what you’re trying to say” as
start not end
• Talk Tasks
• Continuous questioning in all segments which give a further opportunity
to assess understanding through pupil explanations
Potential barrier 2: reasoning
Represent
Mathematical
problem
solving
Generalise
Communicate
Mastering mathematical thinking
“Mathematics can be terrific fun; knowing that you can
enjoy it is psychologically and intellectually empowering.”
(Watson, 2006)
We believe that pupils should:
• Explore, wonder, question and conjecture
• Compare, classify, sort
• Experiment, play with possibilities, modify an aspect and
see what happens
• Make theories and predictions and act purposefully to see
what happens, generalise
Mathematical thinking – you say
“By focusing on fewer topics whilst increasing their skills
as independent learners (which fits fantastically with our
whole school policy of collaborative learning) we will
increase the confidence of a large majority of our
students in their key mathematical skills.”
Potential barrier 3: conceptual understanding
Represent
Mathematical
problem
solving
Generalise
Communicate
What are manipulatives?
Bead strings
Bar models
Dienes blocks
Fraction towers
100 grids
Conceptual
understanding
Number lines
Cuisenaire rods
Mathematical
problem
solving
Multilink cubes
Mathematical
thinking
Shapes
Language and
communication
Let’s do some maths...
Problem solving using bar
models!
• Pupils draw a visual representation of a word
problem.
• Taught early on in the programme, using
concrete and pictorial representations, in the
context of the four operations.
• Pupils are then expected to use models for
fractions, decimals, percentages, algebra, pie
charts....
Solving problems with unknowns
John gives his brother three marbles.
Now his brother has three times as many marbles as John.
Altogether they now have sixteen marbles.
How many marbles did John have at the start?
?
3
John
16
John’s brother
Ben scored 1,866
Abe scored 2,177
Conceptual understanding – you say
“It is essential that all of our teachers aim for all our
students to clearly understand a mathematical concept
rather than simply learning the process.”
“Our aim is to teach for understanding, but realistically
this is not happening in all classes all the time.”
“I feel that the use of concrete manipulatives and a
constant focus on problem solving will mean that
students are much more able to understand
mathematical concepts.”
Lesson structure
New
learning
Do Now
Talk task
Independent
task
Develop
learning
Ofsted outstanding:
• Planning is astute
• Time is used very well
• Every opportunity is used to successfully develop crucial skills (inc. literacy and numeracy)
• Lessons proceed without interruption
• Appropriate independent learning tasks are set
• Pupils are resilient, confident and independent
• Well judged and often imaginative teaching strategies are used
Plenary
YOU DON’T ACHIEVE MASTERY BY CLIMBING...YOU
ACHIEVE MASTERY THROUGH DEPTH
Generalising
Modifying
Comparing
MATHEMATICAL THINKING
Curriculum
with problem solving at the heart
Maths learning in your school
What is consistent across the department?
What happens in every lesson?
What does ‘students’ work’ look like?
How are students supported to:
•
use language to reason and communicate with accuracy?
•
represent mathematical concepts and techniques?
•
make connections within mathematics?
•
make connections beyond mathematics?
•
think mathematically and solve problems?
Using data and evidence
Fine grain detailed data analysis on a question level and by national
curriculum sub-levels are essential to ensuring that every student is
successful
The big picture is what’s important – the focus should be on the best way to
teach the students, and the best way to teach the concept or technique, with
their long term success in mind
‘Big picture’ data can tell us…
1) What the essential concepts and techniques are for
students to succeed at A-level and beyond.
2) What the essential concepts and techniques are for
students who might otherwise fall behind.
3) That these are the same!
4) The ‘habits of mind’ that students need to succeed
a) in maths
b) in applying their maths
Classroom data
1) In lessons: student responses, student engagement
2) Work scrutiny
3) Student voice
4) Student tracking
5) Case studies – progress of students in challenging
circumstances
6) Views of parents
7) Discussion with staff and senior leaders
Work scrutiny
Classroom data can tell us…
1) Which students are at risk of underachieving
2) Which concepts and techniques need re-teaching/teaching
differently next time round
3) Which students need additional challenge
4) Which pedagogic approaches were particularly effective
‘Exam’ data tells us…
1) Areas where the classroom evidence might be misleading
2) How students might perform in high stakes external exams
Assessment
Pre- and post-module assessments
Termly holistic assessments
Expectations
What does success look like?
Year 7 Term
Autumn
Spring
Summer
Evidence of success
Attainment at end of
term
Fewer topics, greater depth
“Our students have a brief knowledge of a large range of topics but their foundation
skills crumble as we approach more complex topics.”
“By focusing on fewer topics whilst increasing their skills as independent learners (which
fits fantastically with our whole school policy of collaborative learning) we will increase
the confidence of a large majority of our students in their key mathematical skills.”
“We want to close the achievement gap across all year groups so that it is not necessary
to put all of our resources and effort into just the Y11 groups each year and so that
maths can be truly understood and embraced by all year groups prior to their GCSE
year.”
Year 7
Knowledge, understanding and skills by week
Weeks
Integer place value
Round and estimate
Half terms
Multiple, LCM
Add and subtract
Word problems
Multiplication
Multiplication of decimals
Rectangle and triangle
area
Estimate measures
Read scales
Draw, measure and name
angles
Angle types
Fractions as numbers,
fractions as operators
Decimal numbers
Decimal place value
Factor, HCF
Division
Mean average
Triangles
Quadrilaterals
(Investigation)
Multiplicative
relationships with
fractions
Fraction of a quantity
Multiply and divide
fractions
Equivalent fractions
Compare and order fractions
Order of operations
Symbolic notation
Interpret pie charts
Convert fractions, decimals
and percentages
Find perimeter
(Investigation)
Substitute and simplify
Percentage of a
quantity
(Investigations)
Half term 1
Half term 2
Half term 3
Half term 4
Half term 5
Half term 6
Number sense
Multiplication & division
Angle and line properties
Fractions
Algebraic representation
Percentages & pie charts
Place value

Fractions, decimals and percentages
Addition and subtraction
Perimeter
Multiplication and division
Area
Using scales
Year 7
KEY
Half term topic
Big idea
Substantial new knowledge
mastered
Angle and line properties
Calculating with fractions
Algebraic notation
Support and challenge
Conceptual
understanding
Mathematical
problem
solving
Mathematical
thinking
Language and
communication
Low threshold, high ceiling
Pupils compare the area and perimeter of rectangles.
Which is bigger? Which is smaller?
8cm
2cm
4cm
4cm
Can/does this task involve modifying? How could the teacher prompt this?
Can/does this task involve generalising? How could the teacher prompt this?
Summary
What does Mastery mean?
• Assumption is that what has been learnt really has
been learnt – this doesn’t happen after one lesson
however!
• Achieving mastery through ‘less explicit’ repetition
• Mastery occurs as a result of repetition through
using skills continuously across genres
• Taking time to explore, clarify, practice and apply, not
memorise
• Valuing concrete and pictorial, not rushing to the
abstract
Supporting, monitoring and evaluating
What to look for in a lesson
• evidence of C-P-A
• use of language structures
• great questioning
• 100% involvement
• AfL
Looking at a lesson
What is 0.27 plus 1.009?
It’s one point two seven nine. That means it has nine thousandths.
Good. How many tenths are there in 0.27?
There are two tenths, in 0.27 and two tenths in the answer.
Actually there are twelve tenths aren’t there?
Yes. There are. What is 1.27 plus 1.009?
Supporting, monitoring and evaluating
Conceptual
understanding
Mathematical
problem
solving
Mathematical
thinking
Language and
communication
Looking at a lesson
I know you can already do this, but you need to show this using manipulatives.
But why, sir? We’re a top set.
It’s important. Show me which is bigger, 0.2 multiplied by £25 or 250% of £2.
How many different ways can you find of showing this.
Supporting, monitoring and evaluating
Conceptual
understanding
Mathematical
problem
solving
Mathematical
thinking
Language and
communication
Looking at a lesson
What is a fraction?
It’s a number less than one.
Good. Any examples?
It’s like three quarters, the top number is divided by the bottom number.
Brilliant. Division. Anyone else?
It’s part of a whole.
Excellent!
Supporting, monitoring and evaluating
Conceptual
understanding
Mathematical
problem
solving
Mathematical
thinking
Language and
communication
What are you looking for?
New
learning
Do Now
Talk task
Independent
task
Develop
learning
Plenary
Ofsted: Made to measure (2011)
Evidence from successful schools
•Pupil collaboration and discussion of work
•Mixture of group tasks, exploratory activities and
independent tasks
•Focus on concepts, not on teaching rules
•All pupils tackled a wide variety of problems
•Use of hands on resources and visual images
•Consistent approaches and use of visual images and models
•Importance of good teacher subject-knowledge and subjectspecific skills
•Collaborative discussion of tasks amongst teachers
Do the maths
I can't stress enough how vital it is for teachers to
complete the tasks before teaching them. By doing this I
have found it far easier to anticipate what my students
might do or where they may struggle (particularly with
the open problems) so I can plan scaffolding carefully. On
the occasions I haven't done this I have ended up having
very 'teacher led' lessons which were not as effective.
Emily Hudsmith, Head of Maths
Charter Academy