Year 6 Improving Outcomes for Mathematics October 2018

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1

Year 6

Improving Outcomes for

Mathematics

October 2018

OverviewSession 1:

• Introductory game

• Key messages/reminders

• KS2 SATS 2018: 15 Things you need to know

• FDPs – Using and applying

• Developing vocabulary and children’s talk for maths

Session 2:

• A CPA Approach: Using concretes in year 6 to teach an

aspect of algebra

• Craig Barton: Takeaways – ‘Goal free problems’ and

‘SSDD problems’

• Reflection/actions

© Focus Education UK Ltd. 2

Factors and Multiples


Key messages/reminders

1) Setting, pre-assessments and ‘no labels’


➢ David Hargreaves argues that ability labeling

leads to ‘destruction of dignity so massive and

pervasive that few subsequently recover from it.’

➢ Rather than creating ‘low ability’ or ‘high ability’

groups, we can create flexible groupings based

on pupils’ current depth of understanding of the

relevant concept or skill.



2) ‘Two Ambitions’ – raising achievement for everyone

➢ What is standing in the way of ‘low achievers’ succeeding

in maths?

➢ What do the high achieving mathematicians in your class

need?

“People often say: ‘I teach them but they don’t learn.’ Well, if you know that, stop teaching. Not resign from your job: stop teaching in the way that doesn’t reach people, and try to understand what there is to do…”

Caleb Gattegno


3) High expectations… transform achievement.


✓ It’s not ‘ok’ to be ‘bad at mathematics.’

✓ We have a responsibility to provide learning experiences

that give every child the opportunity to succeed.

✓ There is sufficient evidence to show that most pupil

underachievement is due to deficiencies in the teaching

and learning environments rather than to the pupils’

genetic make-up.


4) Concretes and pictorial images (CPA approach)


“Children whose mathematical learning is

firmly grounded in manipulative experiences

are more likely to bridge the gap between the

world in which they live and the abstract world

of mathematics.”


5) Reasoning and problem-solving…



❖What is problem-solving?

❖What is reasoning?

❖ Is there a difference?

The application of knowledge, skills

and understanding to a real life

situation or a mathematical problem

in order to find a solution.

OR

Problem Solving is...

“Using what you know,

to find what you don’t!”


Reason mathematically by following a line of enquiry,

conjecturing relationships and generalisations, and

developing an argument, justification or proof using

mathematical language (NC Aim)

Mathematical reasoning – thinking through maths

problems logically in order to arrive at solutions. It

involves being able to identify what is important and

unimportant in solving a problem and to explain or justify a

solution. (NCETM)

Line of enquiry – is there a link between the number of sides

of a 2d shape and the number of lines of symmetry it has?

What is Reasoning?

Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language (NC Aim)

Conjecturing relationships and generalisations - I think the number of lines of symmetry is the same as the number of sides………. I think it is only for regular shapes.

Developing an argument, justification or proof – range of examples with different types of shapes (specialising), noticing when the statement is true, seeking to understanding why (generalising)

What is Reasoning? (cont.)

• Describe – say what you see, hear or do

• Explain – offer some reasons for above

• Convince – yourself or a friend that you have

a solution or case

• Justify – say why you are convinced

• Prove – to others with a ‘watertight’

argument including when challenged!

Progression in reasoning (Nrich)

1. What Do You Notice?

2. Draw Me, Make Me, Tell Me Why!

3. Tell Me a Story

4. What’s the Same, What’s Different?

5. Odd One Out

6. Zoning In / Guess My Number or Shape

7. Here’s the Answer, What’s the Question?

8. Because I Know…

9. Hard and Easy / POG (Peculiar, Obvious, General)

10. Spot the Mistake / Good Mistake / Silly Answer

11. What’s Missing?

12. Sometimes / Always / Never (Correct Answer / T or F)

Which one(s) could you use tomorrow?

What If?

The All-New, Top 12 Adaptable Reasoning Structures

12 21 32 36 Which of these numbers is the odd one out and why?

How many different answers can you find?

Odd One Out

This is ¼ of a shape.

What could the whole shape look like?

What if this was 1/5 of

the whole shape?

What if?


Here’s the Answer, What’s the Question?

Find two numbers that multiply together to make 60.

Find an obvious, a general and a peculiar example…

Obvious – 10 x 6

General – 15 x 4

Peculiar – 240 x 0.25

How many different answers can you find?

Extending ThinkingPOG - peculiar, obvious, general


6) Use of vocabulary…


✓Precise e.g. ‘equation/calculation not sum’

‘ones not units’ ‘equation not number

sentence’ ‘square not diamond’

✓We also need to build in frequent use - you

need to practise a word at least 50 times

before it is acquired and retained!


7) White Rose – new materials


This release of our schemes includes New overviews, with subtle changes being made to

the timings and the order of topics. New small steps progression. These show our blocks

broken down into smaller steps. Small steps guidance. For each small step we provide

some brief guidance to help teachers understand the key discussion and teaching points. This guidance has been written for teachers, by teachers.

A more integrated approach to fluency, reasoning and problem solving.

Answers to all the problems in our new scheme.This year there will also be updated assessments. We are also working with Diagnostic Questions to

provide questions for every single objective of the National Curriculum.



8) Year 5 and 6 need to develop a high level of

confidence in moving between FDP equivalence

through application based activities – this is your

mathematical phonics!


9) All children need (and deserve) rich, varied

and challenging maths lessons.


Do your children…Play lots of games, including dice gamesSolve puzzlesFollow lines of enquiry or tackle investigationsHave experience of using a wide variety of reasoning structuresHave regular opportunities to use/apply vocabularyEncounter challengePractice their skillsUse concretes and pictorial imagesMake connectionsLove maths?

Do you…Use a wide variety of sources to get your resources?

SATS 2018 - 15 things you need to

know

Third Space Learning


SATS 2018 - 15 things you need to know:

a summary (Taken from Third Space Learning)

• Levels of difficulty remained about the same.

• Themes were evident eg. mixed numbers (8 mks)

• ‘Using what you know’

• Mental v Written – high arithmetic score is crucial! Weekly arithmetic lessons should be happening.

• Important to revise previous content (see slide below)

• 3 mark question appeared again.

• Visualisation was tested in a number of questions.

• ‘SATs Land!’ – melons, croc noses and mars

• May appear in 2019? Order of operations, algebra, area and perimeter, re-testing of infrequent areas (2D shapes, analogue clocks, properties of circles)




Fractions, Decimals and Percentages

Fractions, Decimals and Percentages


FDPs

What is 60% of £35?



3 + 1 =5 2

What might children do?

Confident Convertors!!!


When children are confident at converting

between fractions, decimals and percentages,

and are taught to make these links as often

as possible, their mental arithmetic skills,

and ability to solve simple FDP problems,

will greatly improve!

In the ‘best’ lesson…

✓Children would already know many of the key

equivalences and would move between these

with relative ease.


✓ Where appropriate, visual resources would be

used to assist those children who don’t yet

know all of their equivalences.

✓ Children can change to decimals when

appropriate to make the calculation easier.

✓ Children choose and use efficient strategies

depending on the numbers

In ‘best’ practice …

✓Regular counting in decimal and fraction

steps built into mental starters (see slide below)

✓Regular speed testing for fraction, decimal,

percentage equivalences.

✓Collaborative working and talking maths built

in to routines.

✓Strategies for finding fractions / percentages

of amounts regularly discussed.



• Percentage Chain


£80

£80 is 100%

What Other Facts

Can You Find?

£80

£80 is 100%

What Other Facts

Can You Find?

Percentage Chain

Intelligent Practice

• 10% of 40 =

• 5% of 40 =

• 15% of 40 =

• 30% of 40 =

• 30% of 80 =

• 3% of 80 =

• 30% of 8 =

• 300% of 8 =

• 8% of 300 =


In designing [these] exercises, the teacher is advised to avoid mechanical repetition and to create an appropriate path for practising the thinking processwith increasing creativity.

Gu, 1991

How do you present percentage

questions to children?

10% of £8 =


10% of £80

of £80 = £8

10% of = £8

% of £ = £8

10% of = (5 ways)

of £80 = (5 ways)

%

%© Focus Education UK Ltd. 41

In ‘best’ practice …

✓There isn’t a ceiling on expectation for fraction,

decimal and percentage equivalences – fifths,

eighths, sixths, ninths and elevenths are used

regularly, with as much confidence as quarters

and halves.


✓ Expectations are generally high, and reasoning

is employed on a daily basis.

✓Supporting models, images and apparatus are

available and used.




Giant Foam Rods (TTS) – great for demonstrating…

If you would like some more ideas around using

the cuisenaire rods, or want to ask me anything

else, please e-mail me.

[email protected]


mailto:[email protected]

Make explicit links between fractions and the

language of division…

¼ ÷ 2 =


Shared between

Show divisions in different ways.

4 ÷ ½ = How many halves in…

Do children know that 1 is 1 5?

5

÷


What are we doing to

develop children’s

knowledge of vocabulary

and their ‘talk for maths’ ?



We always need…

•High levels of accurate, quality

mathematical vocabulary

•The language of reasoning

•Dialogic talk prompts

•Quality discussion

•Explanations, descriptions, definitions

•Justification and proof


Where do you see this word in

everyday life?

Mathematical Symbols

(if there are any):

Use your word in a phrase or

in a statement:

Picture or diagram:

Describe what your word or

phrase means:

What other mathematical

words is it related to?Mathematical word

or phrase

Negotiating Vocabulary

Where do you see this word in

everyday life?

Mathematical Symbols

(if there are any):

Use your word in a phrase or

in a statement:

Picture or diagram:

Describe what your word or

phrase means:

What other mathematical

words is it related to?Mathematical word

or phrase

Divide

Negotiating Vocabulary

How many of the following names can I correctly use

for this shape? Reason with your partner!

1. Polygon

2. Quadrilateral

3. Parallelogram

4. Rectangle

5. Trapezium

6. Kite

7. Pentagon

8. Rhombus

9. 2-D Shape

10. Square

11. Oblong

12. Cube

Let’s play …

Numiculate!

Numiculate 1 –

Individual Competition (Guessing)

• Choose one player to describe the words

and phrases.

• As each word is revealed, the player must

describe it as accurately as possible to the

rest of the group.

• The first person to shout out the correct word

/ phrase scores a point.

Numiculate 2 –

Individual Competition (Describing)

• Choose one player to describe the words and phrases to the whole class.

• As each word is guessed, the describer moves quickly onto the next word.

• After 1 or 2 minutes, the describer scores the number of guessed words.

• Over the week, allow different people to describe.

➢ This can also be done as a group activity round a table.

✓Quadrilateral

✓Denominator

✓Prime number

✓Factor

✓Protractor

✓Obtuse

✓Quotient


Examples of words to use…

✓Product

✓Equation

✓Multiple

✓Angle

✓Place value

✓Percentage

✓Remainder

Talk about … Word Links!

Speaker

➢Describes/ talks

about a

mathematical

topic, concept

or idea for as

long as they

can

Listener

➢Chooses six words connected to the chosen mathematical topic, concept or idea that they hope the speaker will use in their description

➢Cross off the words as the speaker uses them

➢ If the speaker gets stuck give them clues to a word you need.

➢ Shout Bingo when you have crossed off all 6.

Subtraction

Fractions

Place Value

‘Flippin heck!’ – a quick-fire maths

vocabulary game

• teachtothehilt.com/maths/maths-

vocabulary-game


teachtothehilt.com/maths/maths-vocabulary-game


It’s even

It’s a multiple of 2, 4 and 8

It’s the product of 2 and 8

When you divide 32 by 2 it is the quotient

It’s a square number

It’s double 8

What is the mystery

number?

We always need…

•High levels of accurate, quality mathematical

vocabulary

• The language of reasoning



• Explanations, descriptions, definitions

• Justification and proof


Language Functions

It’s greater than

ten

comparing

If you double it

then you get …

expressing cause

and effect

It has three sides.

An acute angle is an

angle which ….

defining

describing

All multiples of even

numbers are even

numbers

generalising

First I added them

together and then I

multiplied by …..

recounting

68

We always need…

•High levels of accurate, quality

mathematical vocabulary

•The language of reasoning



•Explanations, descriptions, definitions

•Justification and proof


Dialogic Talk Prompts

How do you know……

How can you be sure…..

Why?

How might you record that for

someone else?

What do you already know?

Why do we……

What is the same?

What is different?

What do you notice or see?

If you know ………………..how

…………………………………….

could you find out…………….

……………………………..

• Discuss, in pairs or threes, how you might

describe the set of numbers below:

{12}

• Feedback

© Focus Education UK Ltd. 70© Focus Education UK Ltd. 70

Talk for Learning

Talk for Learning

• Discuss, in pairs, how you might describe the

set of numbers below:

{12, 6}

• Feedback

Talk for Learning



{12, 6, 18}

• Feedback

Talk for Learning



{12, 6, 18, 9}

• Feedback

Talk for Learning



{12, 6, 18, 9, 3}

• Feedback

Talk for Learning



{12, 6, 18, 9, 3, 2,}

• Feedback

Talk for Learning



{12, 6, 18, 9, 3, 2, 4,}

• Feedback

Talk for Learning



{12, 6, 18, 9, 3, 2, 4, 1}

• Feedback

Talk for Learning



{12, 6, 18, 9, 3, 2, 4, 1}

• Missing Number – 36

Factors of 36!

Break…


A CPA Approach

➢As well as using concretes, many schools

have now shifted their classroom practice in

terms of using pictorial representations.


➢Manipulatives don’t always make maths

easier – if modelled correctly by the

teacher, and used correctly by the pupils,

they often make pupils think more deeply.

➢ The use of concrete objects allows pupils

to visualise, model, and internalise abstract

mathematical concepts.



Teaching an aspect of Algebra

through a CPA approach

5m + 4 = 3m + 22



Teaching an aspect of Algebra

through a CPA approach

You will need stacking counters,

cups and a notepad for this part!

Work in pairs or threes.


Craig Barton

(‘How I wish I’d taught maths’)

Goal free problems…

✓ reduce the number of steps needed in a

complex problem

✓ are not as cognitively draining

✓ enable the student to feel that each step is

within reach, rather than trying to aim for

some far away, daunting final goal.





What can you work out?



SSDD problems…

SSDD problems…

‘Same Surface, Different Deep Problems’

The following slides give examples. The

philosophy behind it and many more examples

can be found at: www.ssddproblems.com


http://www.ssddproblems.com




List all the factors of 24 Draw 2 rectangles with an

area of 24cm.

Draw an oblong with a

perimeter of 24cm.

Which is bigger - 1 of 24 or

1 of 24?

2

12

8

‘24’


Documents

Year 6 Improving Outcomes for Mathematics October 2018