Haringey
Session 1
Supporting fluency and
reasoning through rich
tasks
8 October 2014
Lynne McClure
Director, NRICH project
© University of Cambridge
National Curriculum
Become fluent in the fundamentals of
mathematics, including through varied and
frequent practice with increasingly complex
problems over time, so that pupils develop
conceptual understanding and the ability to
recall and apply knowledge rapidly and
accurately.
© University of Cambridge
National Curriculum
Reason mathematically by following a line of
enquiry, conjecturing relationships and
generalisations, and developing an
argument, justification or proof using
mathematical language
© University of Cambridge
Reach 100
Choose four different digits from 1−9 and
put one in each box. For example:
This gives four two-digit
numbers: 52,19, 51, 29
In this case their sum is 151.
Can you find four different digits that give
four two-digit numbers which add to a total
of 100?
© University of Cambridge
• What is the mathematical
knowledge that is needed?
• Who would this be for?
• What is the ‘value added’ for higher
attaining children/struggling children.
5
© University of Cambridge
Strike It Out
6 + 4 = 10
10 take away 9 makes 1
1 add 17 is 18
18……
Competitive aim – stop your partner
from going
Collaborative aim – cross off as many
as possible
6
© University of Cambridge
• What is the mathematical
knowledge that is needed to play?
• Who would this game be for?
• What is the ‘value added’ for able
children/struggling children of
playing the game?
• How could you adapt this game to
use it in your classroom?
7
© University of Cambridge
How do these rich tasks contribute to
fluency?
reasoning?
© University of Cambridge
Efficiency
An efficient strategy is one that the student
can carry out easily, keeping track of subproblems and making use of intermediate
results to solve the problem.
© University of Cambridge
Efficiency
Accuracy
depends on careful recording, the
knowledge of basic number combinations
and other important number relationships,
and checking results.
© University of Cambridge
Efficiency
Accuracy
Flexibility
requires the knowledge of more than one
approach and being able to choose
appropriately between them
(Russell, 2000
http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm)
© University of Cambridge
Procedural &
conceptual fluency
Automaticity
Automaticity with recall
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Fluency
Procedural without conceptual
Conceptual without procedural
Computation without meaning
Computation which is slow, effortful
and frustrating
Inability to adapt skills to unfamiliar
contexts
Inability to focus on the bigger picture
when solving problems
Difficulty reconstructing forgotten
knowledge or skills
Difficulty progressing to new or more
complex ideas
© University of Cambridge
Using the same rules is it possible to cross all
the numbers off?
How do you know?
14
© University of Cambridge
Two types of reasoning
Inductive reasoning
• Can be incorrect
• Can’t be used to ‘prove’
Deductive reasoning
• Follows rules of logic
• Can be used to prove
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In a problem:
• Reasoning is necessary when:
• The route through the problem is not clear
• There are some conflicts in what you are
given or know
• There are some things you don’t know
• Theres no structure to what you’re given
• There are different possible solutions
• All of which require mental work….
© University of Cambridge
Reasoning is…
• A critical skill to knowing and doing maths
• Enabling – it allows children to make use of all
the other mathematical skills – it’s the glue that
helps maths to make sense.
© University of Cambridge
Structuring
children’s reasoning
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Questioning: closed v open
Listening
Acknowledging
Improving
Modelling KS1: good 'because'
statements, short chains
• KS2: logic, convincing
© University of Cambridge
Session 2
Problem solving
© University of Cambridge
National Curriculum
Can solve problems by applying their
mathematics to a variety of routine and nonroutine problems with increasing
sophistication, including breaking down
problems into a series of simpler steps and
persevering in seeking solutions
© University of Cambridge
Historically
• learning the content v problem solving
• theory versus practice, reason versus
experience, acquiring knowledge versus
applying knowledge.
• problems seen as vehicles for practicing
applications ie computation procedures are
acquired first and then applied
• problem-based learning
© University of Cambridge
Dominoes
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Dominoes – have a play….
Have you got a full set?
How do you know?
Can you arrange them in some way to
convince yourself/others that you have/
haven’t got full set?
© University of Cambridge
• What number knowledge/skills did you
use?
• What mathematical processes did you
use?
• What ‘soft skills’ did you use?
© University of Cambridge
Amy has a box containing ordinary
domino pieces but she does not think it is
a complete set. She has 24 dominoes in
her box and there are 125 spots on them
altogether. Which of her domino pieces
are missing?
© University of Cambridge
• What number knowledge/skills did you
use?
• What mathematical processes did you
use?
• What ‘soft skills’ did you use?
© University of Cambridge
Order of events
• Free play –Montessori ‘work’
• Closed activity: structure of the
apparatus
• Task which uses that knowledge
• Multistep
• With or without apparatus
Ruthven’s
Exploration
Codification
Consolidation
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Dominoes v houses
Sort – have you got them all?
How do you know?
Tasks using that knowledge
Guess the dominoes/ houses
© University of Cambridge
Rich tasks….
• combine fluency, problem solving and
mathematical reasoning
• are accessible
• promote success through supporting
thinking at different levels of challenge (low
threshold - high ceiling tasks)
• encourage collaboration and discussion
• use intriguing contexts or intriguing maths
© University of Cambridge
• allow for:
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learners to pose their own problems,
different methods and different responses
identification of elegant or efficient solutions,
creativity and imaginative application of
knowledge.
• have the potential for revealing patterns or
lead to generalisations or unexpected
results,
• have the potential to reveal underlying
principles or make connections between
areas of mathematics
(adapted from Jenny Piggott, NRICH)
© University of Cambridge
Tasks
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Non-routine
Accessible
Challenging
Curriculum linked
Rich tasks/LTHC tasks
Implications for your teaching?
© University of Cambridge
Valuing mathematical
thinking
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Process as well as end product
Talk as well as recording
Questioning as well as answering
…………
© University of Cambridge
Purposeful activity
Give the pupils something to do,
not something to learn;
and if the doing is of such a nature
as to demand thinking;
learning naturally results.
John Dewey
© University of Cambridge
Session 4
Games are more than fillers
© University of Cambridge
3
4
Dotty
6
2
1
5
3
5
1
2
Green wins!
© University of Cambridge
• What is the mathematical knowledge that
is needed to play?
• Who would this game be for?
• What is the value added of playing the
game?
• Could you adapt it to use it in your
classroom?
• Contribute to F, R, PS?
© University of Cambridge
Board Block
© University of Cambridge
• What is the mathematical knowledge that
is needed to play?
• Who would this game be for?
• What is the value added of playing the
game?
• Could you adapt it to use it in your
classroom?
• Contribute to F, R, PS?
© University of Cambridge
Four Go
© University of Cambridge
• What is the mathematical knowledge that
is needed to play?
• Who would this game be for?
• What is the value added of playing the
game?
• Could you adapt it to use it in your
classroom?
• Contribute to F, R, PS?
© University of Cambridge
Nice and nasty
© University of Cambridge
• What is the mathematical knowledge that
is needed to play?
• Who would this game be for?
• What is the value added of playing the
game?
• Could you adapt it to use it in your
classroom?
• Contribute to F, R, PS?
© University of Cambridge
© University of Cambridge
“If I ran a school, I’d give all the average
grades to the ones who gave me all the right
answers, for being good parrots. I’d give the
top grades to those who made lots of
mistakes and told me about them and then
told me what they had learned from them.”
Buckminster Fuller, Inventor
© University of Cambridge
• What were these children’s views of
maths?
• Would you get the same answers?
© University of Cambridge
Session 3
Maths Working Group
© University of Cambridge
Purpose of study
Mathematics is a creative and highly interconnected discipline that has been developed over
centuries, providing the solution to some of
history’s most intriguing problems. It is essential to
everyday life, critical to science, technology and
engineering, and necessary for financial literacy
and most forms of employment. A high-quality
mathematics education therefore provides a
foundation for understanding the world, the ability
to reason mathematically, an appreciation of the
beauty and power of mathematics, and a sense of
enjoyment and curiosity about the subject.
© University of Cambridge
Purpose of study
Mathematics is a creative and highly interconnected discipline that has been developed over
centuries, providing the solution to some of
history’s most intriguing problems. It is essential to
everyday life, critical to science, technology and
engineering, and necessary for financial literacy
and most forms of employment. A high-quality
mathematics education therefore provides a
foundation for understanding the world, the ability
to reason mathematically, an appreciation of the
beauty and power of mathematics, and a sense of
enjoyment and curiosity about the subject.
© University of Cambridge
• interconnected subject in which pupils
need to be able to move fluently between
representations of mathematical ideas.
• make rich connections across
mathematical ideas to develop fluency,
mathematical reasoning and competence
in solving increasingly sophisticated
problems
• apply their mathematical knowledge to
science and other subjects.
© University of Cambridge
• interconnected subject in which pupils
need to be able to move fluently between
representations of mathematical ideas.
• make rich connections across
mathematical ideas to develop fluency,
mathematical reasoning and competence
in solving increasingly sophisticated
problems
• apply their mathematical knowledge to
science and other subjects.
© University of Cambridge
The new National Curriculum
What’s important to teachers?
© University of Cambridge
Aims
• All equally important
• First two feed into third
© University of Cambridge
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Big ideas
Fluency
Reasoning
Problem solving
Arithmetic/calculation (fractions)
Multiplicative/proportional reasoning
Pre-algebra/algebra
Connections within and without
No probability at KS1/2
Reduced data handling at 1/2/3
© University of Cambridge
Year 6
Pupils should be taught to:
•use simple formulae
•generate and describe
linear number sequences
•express missing number
problems algebraically
•find pairs of numbers that
satisfy an equation with two
unknowns
•enumerate all possibilities of
combinations of two
variables.
• Pupils should be introduced to
the use of symbols and letters to
represent variables and
unknowns in mathematical
situations that they already
understand, such as:
• missing numbers, lengths,
coordinates and angles,
• formulae in mathematics and
science
• equivalent expressions (for
example, a + b = b + a)
• generalisations of number
patterns
• number puzzles (for example,
what two numbers can add up
to).
© University of Cambridge
Year 6
Pupils should be taught to:
•use simple formulae
•generate and describe
linear number sequences
•express missing number
problems algebraically
•find pairs of numbers that
satisfy an equation with two
unknowns
•enumerate all possibilities of
combinations of two
variables.
• Pupils should be introduced to
the use of symbols and letters to
represent variables and
unknowns in mathematical
situations that they already
understand, such as:
• missing numbers, lengths,
coordinates and angles,
• formulae in mathematics and
science
• equivalent expressions (for
example, a + b = b + a)
• generalisations of number
patterns
• number puzzles (for example,
what two numbers can add up
to).
© University of Cambridge
© University of Cambridge
10 + 10 + 8 + 8
6+6+4+4
25 + 25 + 23 + 23
s + s + (s-2) +( s-2)
= 4s - 4
© University of Cambridge
10 + 9 + 8 + 9
6+5+4+5
25 + 24 + 23 + 24
s + s-1 + (s-2) +( s-1)
= 4s- 4
© University of Cambridge
9 + 9 + 9+ 9
5+5+5+5
24 + 24 + 24 + 24
(s-1) + (s-1) + (s-1) +(s-1)
= 4s- 4
© University of Cambridge
10 + 10 + 10 + 10 – 4
6+6+6+6-4
25 + 25 + 25 + 25 - 4
s+s+s+s-4
= 4s- 4
© University of Cambridge
s + s + (s-2) +( s-2)
= 4s - 4
s + s-1 + (s-2) +( s-1)
= 4s- 4
(s-1) + (s-1) +( s-1) + (s-1)
= 4s- 4
s+s+s+s-4
= 4s- 4
© University of Cambridge
102 - 82
62 - 42
182 - 162
s2 - (s-2)2
s2 - (s-2)2 = s2 - (s2 - 4s +4)
= s2 - s2 +4s – 4
= 4s - 4
© University of Cambridge
The expectation is that the majority of pupils will
move through the programmes of study at broadly
the same pace. However, decisions about when to
progress should always be based on the security
of pupils’ understanding and their readiness to
progress to the next stage. Pupils who grasp
concepts rapidly should be challenged through
being offered rich and sophisticated problems
before any acceleration through new content.
Those who are not sufficiently fluent with earlier
material should consolidate their understanding,
including through additional practice, before
moving on.
Opportunities?
© University of Cambridge
The programmes of study for mathematics
are set out year-by-year for Key Stages 1
and 2. Schools are, however, only required
to teach the relevant programme of study by
the end of the key stage. Within each key
stage, schools therefore have the flexibility
to introduce content earlier or later than set
out in the programme of study.
Opportunity?
© University of Cambridge
IWADWADWAGWAG
If we always do what we’ve always done
we’ll always get what we always got…..
© University of Cambridge
Session 3
National Collaborative Projects
a.Mastery pedagogy for primary mathematics 1 – ChinaEngland research and innovation project
b.Mastery pedagogy for primary mathematics 2 – Use of
high quality textbooks (linked to Singapore) to support
teacher professional development and deep conceptual
and procedural knowledge for pupils
© University of Cambridge
1. Increasing supply of specialist teachers of mathematics
(including primary, secondary convertors, Post-16) (SO1a)
2. Developing specialist subject knowledge of teachers of
mathematics (all phases and including particular areas)
(SO1b)
3. Developing pedagogical knowledge of teachers of
mathematics (especially understanding of mastery
pedagogy and Shanghai & Singapore pedagogy) (SO1c)
4. Improving quality of mathematics teaching practice
(including the move from good to outstanding) (SO1d)
5. Supporting teachers to address new curriculum and
qualifications
© University of Cambridge
6. Improving quality of curriculum resources and activities
(especially to support mastery teaching) (SO3b)
8. Improving supply and developing specialist leadership
knowledge of mathematics subject leaders (SO2a/b)
9. Improving quality of and access to mathematics
enrichment experiences (SO3c)
10. Increased progress and achievement in primary and
secondary (including sustained progress through transition
phases) (PO1a/b)
11. Reducing the gap in achievement between
disadvantaged pupils and others (PO4c)
14. Developing confidence (can-do attitude) and resilience
in learning mathematics (PO3a)
© University of Cambridge
Key Findings
Successful schools
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Hands on crucial in FS and KS1
‘Traditional’ methods need to be underpinned by place
value, mental methods fluency, facts
Inverse operations important
Confidence fluency and versatility nurtured through
problem solving and investigations
Clear coherent calculation policy
© University of Cambridge
Key findings
Made to Measure
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Inconsistency within schools
Need to increase emphasis on problem solving
Teachers to be enabled to choose approaches that foster
deeper understanding
Checking understanding and reacting immediately
Attention on most and least able
© University of Cambridge