Transcript CMIT EAS wiki

The Learning Framework in
Number
Julie Rees
K-6 Quality Teaching Consultant
New England Regional Office
Noel Park House
A thought to begin
 Teacher:
‘Who can tell me what 7
times 6 is?’
 Student: ‘It’s 42!’
 Teacher: ‘Good, and who can tell
me what 6 times 7 is?’
 Student: ‘That’s easy. It’s 24!’
What research found
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Students arithmetical learning develops
through two complimentary processes:
counting and grouping.
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The transition from count-by-one
strategies to collection-based strategies
underpins the structure of the framework.
Underlying the framework is a
belief that it is important to
observe and take account of
children’s knowledge and
strategies.
3 components
Assessment – what can they do?
Framework - ‘where are they now’
Teaching tasks – ‘where to from here’
The details of the learning framework:
Early Arithmetic Strategies
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Where on the LFIN?
Some breakthrough moments
in this part.
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Emergent – can’t count or match name to count.
Perceptual – can count but needs to see, feel, hear
or touch.
Figurative – can count concealed collections but
needs to count from one.
Counting on and back – can add on to or take from
a group count.
Facile – confidently uses a variety of strategies to
count, add or subtract.
Addition and subtraction
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What key concepts are you trying to assess?
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To solve the task does the student:
 count perceptual items using 1:1 correctly?
 rely on perceptual items?
 count from one when visualising the groups?
 count on from the larger number?
 use groups to solve tasks?
 use knowledge of combinations of numbers to 10 or 20?
CMIT
Number/name confusion
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Sounding out the letters …b…a…t
Sound images for 1 to 10
Imagine a young child who learns how to
read 13, 15, 16, 17, etc correctly (i.e.
sound image starts from the right). How
might he or she read 43?
Confusion between the names of the
numerals 19 and 90
What is the next number after 19?
Order these!
33
12
13
28
30
20
32
Numerals
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The clothesline.
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Children can all be given the same challenge but this
can be differentiated to draw out more sophisticated
strategies.
The Clothesline
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Emphasis is on the order of numbers rather than
the exact position
Can start at any point. Not always zero or one
Models the concept of an empty number line
?
Just for fun
 How many times can you
subtract 7 from 83, and
what is left afterwards?
 As many times as you want,
and it leaves 76 every time.
Tape Diagrams
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Tape diagrams are developed in
the Japanese curriculum from
Grade 2 or Grade 3.
7
?
13
The empty number line
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Represents the linear model of numbers
Is used to record student thinking
Can be used flexibly by varying the
parameters
Using a diagram
John had 29 buttons. His brother gave
him some more buttons. Altogether
John had 73 buttons. How many
buttons did John get from his brother?
Beyond addition and subtraction
What is 14 x 10 = ?
Can you use your answer to work out
14 x 12?
What is 16 x 25 = ?
What is 16 x 24 = ?
How many groups of ten can be formed from
611?
Review
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Multiplication and Division relies on the
structure of a composite group.
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A composite group is a collection of individual
items that can be viewed as one thing.
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For example, the student must be able to see a
group of 5 items as one group of 5 and for it to
be one unit repeated.
Building multiplication and division
through equal grouping and counting
Equal groups
Perceptual multiples
Figurative units
Repeated abstract
composite units
Multiplication &
division as operations
Constructs composites
and coordinates the count
Uses multiplication and
division as binary operators
Eliciteffect
as many
as type
possible
The
ofstrategies
question
on strategy
NUMBERS
POSSIBLE STRATEGIES
40 + 20
"40, 50, 60" or "4 and 2 is 6, so
60"
47 + 20
"47, 57, 67" or "40 and 20 is 60. 7
more makes 67."
47 + 5
"45 and 5 is 50. 2 more is 52" or
"47 + 3 is 50 and 2 more is 52."
47 + 22
"40 and 20 is 60. 7 and 2 is 9. So
69."
47 + 25
"40 and 20 is 60. 7 more is 67
and 5 is 72." Or "40 and 20 is 60.
7 and 5 is 12, so 72."
Multiplication & division
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What key concepts are you trying to assess?
 To solve the task :
 Does the student form equal groups?
 Does the student need concrete material?
 Does the student use materials to represent
each item or each group?
 Does the student keep track of the groups while
counting?
 What counting strategies does the student use
to find the total?
CMIT
The why chart…
What does a maths lesson focus on?
What will the deep knowledge and
deep understandings
be?
What will a maths
lesson sound like?
This will be the
metalanguage and
substantive
communication.
What will a maths
lesson look like?
This will be
H.O.T.S.
background
knowledge,
problematic
Learning Plans