# Supporting Fluency and Reasoning through Rich Tasks Contributed by: Mathematics is a creative and highly inter-connected 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.
1. Haringey
Session 1
Supporting fluency and
reasoning through rich
8 October 2014
Lynne McClure
Director, NRICH project
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2. 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.
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3. National Curriculum
Reason mathematically by following a line of
enquiry, conjecturing relationships and
generalisations, and developing an
argument, justification or proof using
mathematical language
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4. 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?
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5. • 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
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6. Strike It Out
6 + 4 = 10
10 take away 9 makes 1
1 add 17 is 18
Competitive aim – stop your partner
from going
Collaborative aim – cross off as many
as possible
6
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7. • 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
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8. How do these rich tasks contribute to
fluency?
reasoning?
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9. An efficient strategy is one that the student
can carry out easily, keeping track of sub-
problems and making use of intermediate
results to solve the problem.
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10. depends on careful recording, the
knowledge of basic number combinations
and other important number relationships,
and checking results.
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11. requires the knowledge of more than one
approach and being able to choose
appropriately between them
(Russell, 2000
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12. Procedural &
conceptual fluency
Automaticity with recall
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13. Fluency
Procedural without conceptual Conceptual without procedural
Computation without meaning Computation which is slow, effortful
and frustrating
Inability to adapt skills to unfamiliar Inability to focus on the bigger picture
contexts when solving problems
Difficulty reconstructing forgotten Difficulty progressing to new or more
knowledge or skills complex ideas
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14. Using the same rules is it possible to cross all
the numbers off?
How do you know?
14
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15. 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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16. 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….
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17. 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.
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18. Structuring
children’s reasoning
• Questioning: closed v open
• Listening
• Acknowledging
• Improving
• Modelling KS1: good 'because' statements,
short chains
• KS2: logic, convincing
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19. Session 2
Problem solving
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20. National Curriculum
Can solve problems by applying their
mathematics to a variety of routine and non-
routine problems with increasing
sophistication, including breaking down
problems into a series of simpler steps and
persevering in seeking solutions
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21. 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
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22. Dominoes
• 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?
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23. • What number knowledge/skills did you
use?
• What mathematical processes did you
use?
• What ‘soft skills’ did you use?
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24. 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?
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25. • What number knowledge/skills did you
use?
• What mathematical processes did you
use?
• What ‘soft skills’ did you use?
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26. 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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27. Dominoes v houses
Sort – have you got them all?
How do you know?
Tasks using that knowledge
Guess the dominoes/ houses
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• 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
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29. • allow for:
• 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)
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• Non-routine
• Accessible
• Challenging
Implications for your teaching?
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31. Valuing mathematical
thinking
• Process as well as end product
• Talk as well as recording
• Questioning as well as answering
• …………
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32. 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
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33. Session 4
Games are more than fillers
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34. Dotty
4 6
Green wins!
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35. • 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?
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36. Board Block
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37. • 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?
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38. Four Go
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39. • 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?
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40. Nice and nasty
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41. • 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?
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42. © University of Cambridge
43. “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
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44. • What were these children’s views of
maths?
• Would you get the same answers?
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45. Session 3
Maths Working Group
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46. Purpose of study
Mathematics is a creative and highly inter-
connected 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
47. Purpose of study
Mathematics is a creative and highly inter-
connected 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
48. • 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.
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49. • 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.
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50. The new National Curriculum
What’s important to teachers?
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51. Aims
• All equally important
• First two feed into third
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52. 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
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53. Year 6
Pupils should be taught to: • Pupils should be introduced to
•use simple formulae the use of symbols and letters to
•generate and describe represent variables and
linear number sequences unknowns in mathematical
situations that they already
•express missing number understand, such as:
problems algebraically • missing numbers, lengths,
•find pairs of numbers that coordinates and angles,
satisfy an equation with two • formulae in mathematics and
unknowns science
•enumerate all possibilities of • equivalent expressions (for
combinations of two example, a + b = b + a)
variables. • generalisations of number
patterns
• number puzzles (for example,
what two numbers can add up
to). © University of Cambridge
54. Year 6
Pupils should be taught to: • Pupils should be introduced to
•use simple formulae the use of symbols and letters to
•generate and describe represent variables and
linear number sequences unknowns in mathematical
situations that they already
•express missing number
understand, such as:
problems algebraically • missing numbers, lengths,
•find pairs of numbers that coordinates and angles,
satisfy an equation with two • formulae in mathematics and
unknowns science
•enumerate all possibilities of • equivalent expressions (for
combinations of two example, a + b = b + a)
variables. • generalisations of number
patterns
• number puzzles (for example,
what two numbers can add up
to). © University of Cambridge
55. © University of Cambridge
56. 10 + 10 + 8 + 8
6+6+4+4
25 + 25 + 23 + 23
s + s + (s-2) +( s-2)
= 4s - 4
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57. 10 + 9 + 8 + 9
6+5+4+5
25 + 24 + 23 + 24
s + s-1 + (s-2) +( s-1)
= 4s- 4
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58. 9 + 9 + 9+ 9
5+5+5+5
24 + 24 + 24 + 24
(s-1) + (s-1) + (s-1) +(s-1)
= 4s- 4
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59. 10 + 10 + 10 + 10 – 4
6+6+6+6-4
25 + 25 + 25 + 25 - 4
= 4s- 4
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60. 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
= 4s- 4
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61. 102 - 82
62 - 4 2
182 - 162
s2 - (s-2)2
s2 - (s-2)2 = s2 - (s2 - 4s +4)
= s2 - s2 +4s – 4
= 4s - 4
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62. 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?
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63. 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?
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If we always do what we’ve always done
we’ll always get what we always got…..
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65. Session 3
National Collaborative Projects
a.Mastery pedagogy for primary mathematics 1 – China-
England 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
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66. 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)
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
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67. 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)
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68. Key Findings
Successful schools
• 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
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69. Key findings