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Strategy is about how children solve number problems, in particular the mental processes
they use.
Creates new knowledge through use
Knowledge considers the key items of
knowledge that children need to acquire.
Strategy Knowledge
Provides the foundation for strategies
What is the difference between strategy and knowledge? Give some examples
Knowledge and Strategy Examples
• Knowledge – What they need• Number Identification, Number sequence and
order, Grouping and place value, basic facts, written recording
• Strategy – What they do in• Addition and Subtraction, Multiplication and
Division, Fraction and Proportions
Numeracy Teaching Model
• Model and support children' understanding using a researched teaching model.
Using materialsImaging -Thinking about what would happen on the materials Properties -Working only on numbers
• Teach to achieve next learning steps.
Working only with numbers
ImagingMaterials
UsingMaterials
Stages of Development
Stage 0 Emergent
Stage 1 1 – 1 counting
Stage 2 Counting from 1 on Materials
Stage 3 Counting from 1 by Imaging
Stage 4 Advanced Counting
Stage 5 Early Additive
Stage 6 Advanced Additive/Early Multiplicative
Stage 7 Advanced Multiplicative/Early Proportional
Stage 8 Advanced Proportional
Cou
ntin
g
Part-W
hole
Level One
Level Two to Five
Emergent – Stage 0(Level One - Stages 1 to 3 After one year at school )
1,2,3,5,8...?
Can you get me 7 counters from the pile please?
What can’t emergent counters do?
Emergent
STRATEGY• This child is unable
to count a set of objects
KNOWLEDGE• Rote count to 5 at
least.
One to One Counting -Stage One
1,2,3,4,5,6,7,
8.
Can you get me 7 counters from the pile please?
What can this child do?
One To One Counting
STRATEGY• Count a set of
objects to 10 by one to one matching
KNOWLEDGE• Rote count to 10 at
least
Counting or Adding?
• One of the key progressions is when the students move from counting to adding.
• Counters usually do not have basic number facts and /or do not know how to use them.
• Counters usually have an incomplete understanding of place value.
Count From One on Materials – Stage Two
1,2,3,4,5,6,7.
There are 4 counters and another 3 counters. How many are there altogether?
The child solves the problem by using their fingers or other materials . What else do they do?
Counting from one on Materials
STRATEGY• Solve simple addition and
subtraction problems to 20 by counting all the objects.
KNOWLEDGE• Rote count to 20 at least• Instant recognition of
patterns to 5 including finger patterns
• Forward and backward number word sequence 0 – 20
• Order numbers to 20• Numbers before and after
in the range 1 - 20
Count From One By Imaging -Stage Three
Counts in
head 1,2,3,4,5,6,7,8.
There are 4 counters and another 3 counters. How many are there altogether?
The child counts all the objects from one by imaging visual patterns of the objects in their mind.
Counting From One By Imaging
STRATEGY• Can solve addition and
subtraction problems to 20 by counting all the objects and or imaging numbers in my head.
KNOWLEDGENeed …
• Instant recognition of patterns/add/sub facts to 10 including finger patterns
• Ordering numbers 0-20• Forward and backward word
sequence in the range 0 –20• Doubles to 10• Say the number before and after
a given number in the range 0-20• Record in pictures, diagrams,• 5 and 2 is 7, 5 minus 2 equals 7
or 7-2 =7
Advanced Counting – Stage Four(After two years at school)
Counts on
9, 10, 11, 12,
13.
There are 9 counters under there and another 4 counters under there. How many are there altogether? What if I remove 4 counters?
The child counts on from the largest number. Is it OK to use their fingers at this stage?
Advanced Counting
STRATEGY• Solve addition and subtraction
problems by counting on or back in my head from the largest number using supporting materials then moving to imagery.
• Solve addition and subtraction problems by counting on in 10’s and 1’s.
• Solve multiplication problems by skip counting in 2s, 5s 10s.
KNOWLEDGENeed …
• Recognising numbers 0 –100• Ordering numbers 0-100• Forward and backward word
sequence 0-100• Numbers before and after a
given number from 0-100• Skip count in 2s, 5,s 10s forwards
and backwards.• Teen numbers 10+• Doubles to 20• BF to 20• Compatable decade numbers to
100Arizona Monica
The Reality?
To become a Part-Whole thinker children needautomatic recall of …
• Facts to Ten• Doubles Facts• Ten and ….10 + 6 = 16
To Become a Multiplicative thinker children needto be able to recall the times tables
Early Additive Part-Whole - Stage 5 (Level Two After 3 years and/or End of Year 4)
“I know that If I take one off the 6 and put it on the 9 it =10. 10 + 5 = 15”
There are 9 counters under there and another 6 counters under there. How many are there altogether?
The child uses simple strategies to solve addition and subtraction problems mentally
Early Additive Part Whole
STRATEGY• Solve addition and subtraction
problems in their head by working out the answer from basic facts they know.
• Solve addition and subtraction problems with 2 or 3 numbers using groupings of 10 and 100.
• Use addition strategies to solve multiplication strategies
KNOWLEDGE• Recall doubles to 20 and
corresponding halves• Recall the names for 10 • Recall the teen numbers• Skip count in 2s,5s, 10s forwards
and backwards
Hannah Kate Louise
I think tidy numbers would be smartest.
63 – 40 = 23 23 + 1 = 24
63 people are on the bus and 39 people get off the bus. How many people are left on the bus?
The child can select from a wide range of strategies to solve various addition and subtraction problems mentally. How many strategies do they need to be functioning at stage 6?
Advanced Additive Part-Whole -Stage 6(Level Three -End of year 5 and 6)
Advanced Additive Part Whole
STRATEGYChoose from: • Compensation• Place Value• Compatible numbers• Reversibility• Equal Additions for subtraction• Decomposition to solve + and - problems.Use pencil and paper or caluclator to work
out answers where the numbers are large or untidy
Carry out column + and – with whole numbers of up to 4 digits (algorithms)
Solve multiplication and division problems using known strategies eg doubling, rounding.
KNOWLEDGE• Identify numbers 0-1000• Forward and backward sequence by
1,10,100 to 1000• Order numbers from 0-1000• Recall + and - facts to 20• Recall multiplication facts for 2, 5, and
10 times tables.
Advanced Multiplicative - Stage Seven(Level Four- After Year 7 and 8)
Tidy Numbers would be a smart strategy. 30 x 6
= 180180 – (2 x 6) =
168
There are 28 fruit trees in each aisle of the orchard. There are 6 aisles. How many trees are there altogether?
The child can select from a wide range of strategies to solve various multiplication and division problems mentally. What other strategies could you use?
Advanced Multiplicative Part Whole
STRATEGY• Solve +, - , x and ÷ problems with
whole numbers (and decimals) using a range of strategies.
• Solve problems involving fractions, decimals, proportions and ratios using multiplication and division strategies
KNOWLEDGE• Identify, order and say
forward and backward number sequence from 0 –1000000
• Recall multiplication and division facts.
• Order fractions, including those greater than 1.
Advanced Proportional – Stage Eight(Start Level 5 - Year nine)
I can see that 9:15 are both
multiples of 3. I can simplify by ÷3 and get a
ratio of 3:5 ?:10= 6
You can make 9 mittens from 15 balls of wool. How many mittens can you make from 10 balls of wool?
The child can select from a wide range of strategies to solve challenging problems involving, decimals, fraction percentages and ratios.
The brainbox of the framework!
Advanced Proportional Part Whole
STRATEGY• Choose appropriately from
a broad range of strategies to +, -, x and ÷ fractions and decimals.
KNOWLEDGE• Know equivalent
proportions for unit fractions with numbers to 100 and 1000
• Know fraction, decimal, % conversion for unit fractions.
• Order decimals to 3 places.
What does this mean for you?• Assessment of all students in your class.• On going use of formative assessment methods.• Students grouped according to their numeracy
strategy stages.• Planning and sharing learning intentions with
students.• Use of equipment to reinforce teaching and learning. • Sharing learning intentions with students.• Encouraging students to talk about their learning.• Using modeling books with each group.• Students record in their own book• Sharing ideas and supporting colleagues
•Clip art and 3D counters•Fly flip cards•Bead frame•Bead strings•Tens frames•Animal strips
•Place value equipment - unifix cubes - bean cannisters - iceblock sticks•Number line•Empty numberline•Hundreds board•Money
EquipmentModel concepts with many physical representations
Assessing what children know.
• Assess - where each child is at through oral interviewing and questioning
• Group according to a Childs strategy stage using the New Zealand Number Framework
• A useful tool - I CAN Portfolio Sheets• Encourage children to self assess (reflect) know
and own their next learning steps.
Grouping
• Examine your Class Summary sheet and look at how you might group the students.
• Strategy Stage for addition and subtraction is main indicator.
• Transfer data to Class Grouping sheet.
• With a partner discuss each other’s groupings.
Classroom Management• The children need to be able to
work in groups.
• You need to be able to plan for groups.
• Children must be able to work independently.
• Spending time establishing routines, systems and expectations is crucial.
Classroom Implementation
• Long term planning
• Weekly plan
• Model for daily lesson
• Learning outcomes/intentions
• Modelling book • Taskboard
3-Way Rotation TPA
Teacher Practice
Practice Activity
Activity Teacher
Writes and Wrongs, Student Recording
Why?• Records the
process• Avoids mental
overload• Encourages
Imaging• Clarifies (and
may extend) thinking
How?• Quality not
quantity• Separate pages
for thinking and formal working
• Equipment sketched
• Modelled by teacher
How do you want your children to record their working?
Why is written recording important?We all need to learn and practise symbol and diagram literacy. They help to and to “park” information while you work on sub-tasks. Symbols and diagrams can ease the load on your working memory.
Draw a diagram to help you solve this problem. Think about how the diagram helps you.
Katy and Liam went shopping. At the start Liam had only three-quarters as much money as Katy. Liam spent $14 and Katy spent half her money. Then they both had the same amount of money. How much money did each person have left?
Planning
Materials
Resource Documents
Assessment Information
Learner Needs
Teaching Model
Learning Intentions
Modelling Book
Task Board
Acknowledgements...
www.nzmaths.co.nz
Photos: Gray Clapham
Acknowledgements...
www.nzmaths.co.nz
Photos: Gray Clapham
