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Focus Education
© Focus Education UK Ltd.
Handouts and course material is strictly for use within the purchasing organisation only.
Material produced by Focus Education UK Ltd that is shared with non-attending schools is in breach of copyright.
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
Key messages/reminders
1) Setting, pre-assessments and ‘no labels’
© Focus Education UK Ltd. 4
➢ 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.
Key messages/reminders
© Focus Education UK Ltd. 5
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
Key messages/reminders
3) High expectations… transform achievement.
© Focus Education UK Ltd. 6
✓ 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.
Key messages/reminders
4) Concretes and pictorial images (CPA approach)
© Focus Education UK Ltd. 7
“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.”
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!”
© Focus Education UK Ltd. 11
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?
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
Key messages/reminders
6) Use of vocabulary…
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✓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!
Key messages/reminders
7) White Rose – new materials
© Focus Education UK Ltd. 12
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.
Key messages/reminders
© Focus Education UK Ltd. 21
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!
Key messages/reminders
9) All children need (and deserve) rich, varied
and challenging maths lessons.
© Focus Education UK Ltd. 22
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:
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)
© Focus Education UK Ltd. 24
Confident Convertors!!!
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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.
© Focus Education UK Ltd. 33
✓ 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.
© Focus Education UK Ltd. 34
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 =
© Focus Education UK Ltd. 38
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
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.
© Focus Education UK Ltd. 42
✓ Expectations are generally high, and reasoning
is employed on a daily basis.
✓Supporting models, images and apparatus are
available and used.
If you would like some more ideas around using
the cuisenaire rods, or want to ask me anything
else, please e-mail me.
em.williams@focus-trust.co.uk
© Focus Education UK Ltd. 46
Make explicit links between fractions and the
language of division…
¼ ÷ 2 =
© Focus Education UK Ltd. 47
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’ ?
© Focus Education UK Ltd. 49
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
© Focus Education UK Ltd. 51
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
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
© Focus Education UK Ltd. 58
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.
‘Flippin heck!’ – a quick-fire maths
vocabulary game
• teachtothehilt.com/maths/maths-
vocabulary-game
© Focus Education UK Ltd. 63
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
•Dialogic talk prompts
•Quality discussion
• Explanations, descriptions, definitions
• Justification and proof
© Focus Education UK Ltd. 66
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
•Dialogic talk prompts
•Quality discussion
•Explanations, descriptions, definitions
•Justification and proof
© Focus Education UK Ltd. 68
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
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18}
• Feedback
Talk for Learning
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18, 9}
• Feedback
Talk for Learning
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18, 9, 3}
• Feedback
Talk for Learning
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18, 9, 3, 2,}
• Feedback
Talk for Learning
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18, 9, 3, 2, 4,}
• Feedback
Talk for Learning
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18, 9, 3, 2, 4, 1}
• Feedback
Talk for Learning
• Discuss, in pairs, how you might describe the
set of numbers below:
{12, 6, 18, 9, 3, 2, 4, 1}
• Missing Number – 36
Factors of 36!
A CPA Approach
➢As well as using concretes, many schools
have now shifted their classroom practice in
terms of using pictorial representations.
© Focus Education UK Ltd. 80
➢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
You will need stacking counters,
cups and a notepad for this part!
Work in pairs or threes.
© Focus Education UK Ltd. 85
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.
© Focus Education UK Ltd. 86
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
© Focus Education UK Ltd. 92
© Focus Education UK Ltd. 95
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’
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