© University of Cambridge Haringey Session 1 Supporting fluency and reasoning through rich tasks 8...

Haringey

Session 1Supporting fluency and reasoning through rich

tasks8 October 2014

Lynne McClure

Director, NRICH project

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.

National Curriculum

Reason mathematically by following a line of enquiry, conjecturing relationships and

generalisations, and developing an argument, justification or proof using

mathematical language

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?

• What is the mathematical knowledge that is needed?

• Who would this be for?• What is the ‘value added’ for higher

attaining children/struggling children.

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

Strike It Out

• 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?

How do these rich tasks contribute to

fluency?

reasoning?

Efficiency

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.

Efficiency

Accuracy

depends on careful recording, the knowledge of basic number combinations and other important number relationships, and checking results.

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)

Procedural & conceptual fluency

Automaticity

Automaticity with recall

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

Fluency

Using the same rules is it possible to cross all the numbers off?

How do you know?

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

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….

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.

Structuring children’s reasoning

• Questioning: closed v open• Listening• Acknowledging • Improving • Modelling KS1: good 'because' statements,

short chains • KS2: logic, convincing

Session 2

Problem solving

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

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

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?

• What number knowledge/skills did you use?

• What mathematical processes did you use?

• What ‘soft skills’ did you use?

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?

• What number knowledge/skills did you use?

• What mathematical processes did you use?

• What ‘soft skills’ did you use?

Order of events• Free play –Montessori ‘work’ • Closed activity: structure of the

apparatus• Task which uses that knowledge• Multistep• With or without apparatus

Ruthven’sExplorationCodificationConsolidation

Dominoes v houses

Sort – have you got them all?

How do you know?

Tasks using that knowledge

Guess the dominoes/ houses

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

• 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)

Tasks • Non-routine• Accessible• Challenging• Curriculum linked• Rich tasks/LTHC tasks

Implications for your teaching?

Valuing mathematical thinking

• Process as well as end product• Talk as well as recording• Questioning as well as answering• …………

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

Session 4

Games are more than fillers

Dotty 6

2 Green wins!

• 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?

Board Block

Four Go

Nice and nasty

“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

• What were these children’s views of maths?

• Would you get the same answers?

Session 3

Maths Working Group

Purpose of studyMathematics 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.

Purpose of studyMathematics 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.

• 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.

• 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.

The new National Curriculum

What’s important to teachers?

• All equally important• First two feed into third

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

Year 6Pupils 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 tothe 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).

Year 6Pupils 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 tothe 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).

10 + 10 + 8 + 8 6 + 6 + 4 + 4

25 + 25 + 23 + 23

s + s + (s-2) +( s-2) = 4s - 4

10 + 9 + 8 + 9 6 + 5 + 4 + 5

25 + 24 + 23 + 24

s + s-1 + (s-2) +( s-1)= 4s- 4

9 + 9 + 9+ 9

5 + 5 + 5 + 5

24 + 24 + 24 + 24

(s-1) + (s-1) + (s-1) +(s-1)= 4s- 4

10 + 10 + 10 + 10 – 4 6 + 6 + 6 + 6 - 4

25 + 25 + 25 + 25 - 4

s + s + s + s - 4 = 4s- 4

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

102 - 82

62 - 42

182 - 162

s2 - (s-2)2

s2 - (s-2)2 = s2 - (s2 - 4s +4) = s2 - s2 +4s – 4

= 4s - 4

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?

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?

IWADWADWAGWAG

If we always do what we’ve always done we’ll always get what we always got…..

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

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

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)

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

Key findings Made to Measure

• 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 Haringey Session 1 Supporting fluency and reasoning through rich tasks 8...

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