Transcript Count Me In Too

Count Me In Too
2009 Curriculum project
South Western Sydney Region
Count Me In Too
 The story
 Rationale
 CMIT in your classroom and school
 Resources for implementing CMIT
 CMIT and the syllabus
 The 2008 CMIT curriculum project
The story
 Count Me In was trialled in 1996 with 4 District
Mathematics Consultants and 13 schools
 Based on Assoc Prof Bob Wright’s Learning
Framework in Number
 Bob had developed a mathematics recovery
program – individual children with a tutor
 Count Me In was a whole-class program
The story
 The Count Me In trial in 1996 was successful
– in terms of student learning and teacher
learning
 Commencing in 1997, the basic ideas of the
trial were implemented, over and over, in
each district across the state as Count Me In
Too.
 The Learning framework has slowly
developed by including the work of other
researchers.
The story
CMIT is based on:
 Teacher knowledge of the Learning framework
 An initial assessment of individual students
 Teachers trialing the framework in their own
classrooms
 Teachers planning and designing activities which
are appropriate for students’ current knowledge
 School-based teams
The rationale
 The strategies and understandings that students
use to solve number problems can be identified
and placed in an hierarchical order
 Students need to develop and practise basic
mathematical concepts before they can move
onto more sophisticated concepts
The rationale
 Students need to construct their own
understanding of the number system and
operations on number. Mathematical concepts
cannot be learnt, remembered and applied
successfully, through rote teaching and learning
 As students learn, they modify or reconstruct
their current strategies
The rationale
Teachers who work together in a team will have
the support and common interest to:
 persist with an innovation
 cater for the needs of all students in the grade
 ensure that implementation of the teaching focus
continues from one year to the next.
CMIT in your classroom and school
 Teachers become familiar with the Learning
framework in number
 They administer the SENA to students and
analyse the responses
 They determine the strategies used to find
answers (not just right or wrong answers)
 Teachers use the results to plan number lessons
CMIT in your classroom and school
 As students develop and practise more
sophisticated strategies, teachers refer back to
the LFIN to guide their programs
 Teachers enhance their understanding of the
LFIN by using the stages and levels to describe
what their students are doing
 Teachers find that the shared use of the LFIN
terminology assists in discussing student
progress with colleagues
Resources for implementing CMIT
CMIT professional development kit
 Implementation guide
 Annotated list of readings
 The Learning Framework in Number
 SENA 1 and SENA 2
Developing Efficient Numeracy Strategies
(DENS) 1 and 2
Mathematic K-6 Syllabus and Sample Units of
Work
CMIT and the syllabus
 The success of the CMIT teaching strategies
and the documented results of student learning
were reflected in the outcomes of the 2002
syllabus
 The syllabus support document has numerous
examples of CMIT activities
 The philosophies of both CMIT and the syllabus
are drawn from the same research base
CMIT and the syllabus
 The CMIT Learning framework provides finer
detail of how to assist students to acquire more
sophisticated strategies
 CMIT is not a collection of fun activities – it is the
teacher’s approach to teaching and learning
mathematics
 When teachers implement CMIT they are
implementing the syllabus
Table 1: Building addition and
subtraction through counting by ones
Stage 1: Emergent
Stage 2: Perceptual
Stage 3: Figurative
Stage 4: Counting on and back
Table 1:
Stage 0, Emergent counting
 The student cannot count visible items. The
student either does not know the number
works or cannot coordinate the number words
with items.
Students at the emergent stage are working
towards:
Counting collections
Identifying numerals
Labelling collections
Table 1:
Stage 1, Perceptual counting
The student is able to count perceived
items but cannot determine the total
without some form of contact.
This might involve seeing, hearing or
feeling items.
Students may use a “three count”.
Table 1:
Stage 1, Perceptual counting
Students at the perceptual stage are working
towards:
 Adding two collections of items
 Counting without relying on concrete
representations of numbers
 Visually recognising standard patterns for a
collection of up to 10 items without counting
them
 Consistently saying the forward and backward
number word sequence correctly
Table 1:
Stage 2, Figurative counting
 The student is able to count concealed items but
counting typically includes what adults might
regard as a redundant activity.
 When asked to find the total of two groups, the
student will count from “one” instead of counting
on.
Table 1:
Stage 2, Figurative counting
Students at the figurative stage are working
towards:
 Using counting on from one collection to solve
addition tasks
 Using counting down to and counting down from
to solve subtraction tasks
 Developing base ten knowledge
 Forming equal groups and finding their total
Table 1:
Stage 3, Counting-on-and-back
 The student counts-on rather than counting from
“one”, to solve addition or missing addend tasks.
 The student may use a count-down-from
strategy to solve removed items tasks e.g.17-3
 The student may use count-down-to strategies
to solve missing subtrahend tasks e.g. What did
I take away from 17 to get 14?
Table 1:
Stage 3, Counting-on-and-back
Students at the counting on and back stage are
working towards:
 Applying a variety of non-count-by-one
strategies to solve arithmetic tasks
 Forming equal groups and finding the total using
skip counting strategies
Table 2: Model for development
of part-whole knowledge
Combining and partitioning
Level 1 – to 10
 Students know 10+0, 9+1, 8+2 ….
 Know “how many more make 10”
Level 2 – to 20
 Students know 20+0, 19+1, 18+2 …
 Know 8 7
8 2 5
10
5
Table 3: Model for development
of subitising strategies
Level 0 – Emergent
 Students need to count by ones in a collection
greater then 2
Level 1 – Perceptual
 Students instantly recognise number of items
to about 6
Level 2 - Conceptual
 Students instantly state number of items in a
larger group by recognising parts of the whole
e.g. 5, 3 = 8
Table 4: Background notes
 Multiples of twos, fives and tens are usually
easier for counting and grouping than threes or
fours
 Students typically develop from:
 counting individual items,
 to skip counting,
 to being able to keep track of the process when the
items are not present,
 to using the “number of rows” as a number
 to produce “groups of groups” (three groups of four
makes twelve)
Table 4: Background notes
Students who understand how to
coordinate composite units are able to
make efficient use of known facts, e.g.
What is the answer to 8 x 4?
“8 x 4 is the same as 4 x 8,
If 5 x 8 = 40, 4 x 8 must equal 32”
(Year 2 student)
Table 4: Background notes
What is the answer to 9 x 3?
“Double 9 is 18,
18 + 2 is 20
20 + 7 is 27”
(Year 3 student)
Table 4: Background notes
An understanding of composite units is
important in place value and the
calculation of the area of rectangles and
the volume of rectangular prisms
Table 4: Calculating area by identifying
rows or columns as composite units and
adding, skip counting, or multiplying.
Table 4: Calculating volume by
identifying horizontal layers and adding,
skip counting, or multiplying.
36
24
12
Table 4: Calculating volume by
identifying vertical layers and adding,
skip counting, or multiplying the
number of layers
9
18
27
36
Table 4: Background notes
 Some students persist with counting by ones
and have difficulty in progressing to grouping
strategies
 By focusing on groups, rather than individual
units, students learn to treat the groups as
items
 Students need to develop understanding of
composite units and the coordination of
composite units
Table 4: Building multiplication and
division through equal grouping and
counting
Level 1
Level 2
Level 3
Level 4
Forming equal groups
Perceptual multiples
Figurative units
Repeating abstract composite
units
Level 5 Multiplication and division as
operations
Table 4:
Level 1, Forming equal groups
 Uses perceptual counting and sharing to form
groups of specified sizes. (Makes groups using
counters)
 Does not attend to the structure of the groups
when counting.
(Continuous count; doesn’t pause between
groups or stress final number in each group)
Table 4:
Level 2,Perceptual multiples
 Uses groups or multiples in perceptual counting
and sharing e.g. skip counting, one-to-many
dealing
(Voice or finger indicates that each group is
seen separately)
Table 4:
Level 3, Figurative units
 Equal grouping and counting without individual
items visible (Understands that each group will
have the same quantity or value)
 Relies on perceptual markers to represent each
group (Each group is symbolised before the final
count is commenced)
Table 4:
Level 4, Repeated abstract
composite units
 Can use composite units in repeated addition
and subtraction using the unit a specified
number of times (Groups can be imagined, but
are added or subtracted individually)
 May use skip counting
 May use fingers to keep track of the number of
groups while counting to determine the total
(Fingers are used to keep a progressive count)
Table 4:
Level 5, Multiplication and division
as operations
 The student can coordinate two composite units
as an operation e.g. “3 sixes”, “6 times 3 is 18”.
 The student uses multiplication and division as
inverse operations
Table 5: Building fractions through
equal sharing
Level 1 Partitioning: halving
Level 2 Partitioning: sharing
Level 3 Re-unitising
Level 4 Multiplicative structure
Table 5:
Level 1, Partitioning: halving
 The student uses halving to create the 2partition and the 4-partition. Only one method to
create a 4-partition appears possible
Table 5:
Level 2, Partitioning: sharing
 The student can create a 3-partition (and
multiples) and a 5-partition and is able to identify
an image of the partition
Can you show me by folding, how much of this
piece of paper I would get if you gave me one
third of the strip?
Table 5:
Level 3, Re-unitising
 The student can describe the same “whole” by
recreating units in different but equivalent ways
e.g. What would we do if we had 9 pikelets to
share between 12 people? Can you draw your
answer?
Table 5:
Level 4, Multiplicative structure
 The student has a single number sense of
fractions and can order fractions by using the
multiplicative structure to create equivalences
and estimate location.
e.g. 2/4 is the same as 4/8 because 2 is half of 4
and 4 is half of 8
Table 6: Model for the development
of place value
Level 0 Ten as a Count
Level 1 Ten as a unit
Level 2 Tens and Ones
2a: Jump method
2b: Split method (SENA 2 only tests to the
end of Level 2)
Level 3 Hundreds, tens and ones
Level 4 Decimal place value
Level 5 System place value
Table 6:
Level 0, Ten as a count
 Ten is a numerical unit constructed out of ten
ones
 The student may know the sequence of
multiples of ten
 Ten is either “one ten” or “ten ones” but not
both at the same time
 The student must be able to count-on to be at
this level
Table 6:
Level 1, Ten as a unit
 Ten is treated as a single unit while still
recognising it contains ten ones
 Can count by tens and units from the middle of a
decade to find the total of two 2-digit numbersone must be visible
e.g. 4 tens and 2 units visible, 25 units hidden,
counts by tens and ones
Table 6:
Level 2, Tens and ones
 The student can solve two digit addition and
subtraction mentally
 Two methods are used:
the “jump” method
the “split” method
Table 6:
Level 2a, Jump method
 Ten is treated as an iterable unit. The student
can count by tens without visual representation
 The student can increment by tens off the
decade
 For the jump method the student holds on to one
number and builds on in tens and ones
Table 6:
Level 2a, Jump method
+10
28
+3
38
Jump method: 28 + 13
41
Table 6:
Level 2b, Split method
 Ten is treated as a unit that can be collected
from within numbers (abstract collectible unit)
 The student will partition both numbers, collect
the tens, collect the ones and then combine to
find the total.
Table 6:
Level 2b, Split method
Split method: 28 + 13
28 + 13
20 + 10
+
8+3
30
+
11
41
Table 6: Hundreds, tens and ones
Level 3a, Jump method
 The student can use hundreds, tens and units in
standard decomposition
 One hundred is treated as ten groups of ten
 The student can increment by hundreds and
tens to add mentally
 The student can determine the number of tens in
621 without counting by ten
Table 6: Hundreds, tens and ones
Level 3b, Split method
 The student can mentally add and subtract
reasonable combinations of numbers to 1 000
 The student has a “part-whole” knowledge of
numbers to 1 000
 Multiple answers can be provided to questions
such as: Can you tell me two three-digit
numbers that add up to 600?
Table 6:
Level 4, Decimal place value
 The student used tenths and hundredths to
represent fractional parts with an
understanding of the positional value of digits
e.g. 0.8 is greater than 0.75 (read as seventyfive hundredths)
 The student can interchange tenths and
hundredths e.g. 0.75 may be thought of as
seven tenths and five hundredths
Table 6:
Level 5, System place value
 The student understands the structure of the
place value system (as powers of 10) that can
be extended indefinitely in two directions – to
the left and to the right of the decimal point
 Understanding includes the effect of
multiplying or dividing by powers of ten
 The student appreciates the relationship
between values of adjacent places (units) in a
numeral
Table 7: Model for the construction of
forward number word sequences (FNWS)
Level 0 - Emergent FNWS
 The student cannot produce the FNWS from
“one” to “ten”
Level 1- Initial FNSW up to “ten”
 The student can produce the FNWS from “one”
to “ten”
 The student cannot produce the number word
just after a given number. Dropping back to
“one” does not appear at this level
Table 7: Model for the construction of
forward number word sequences (FNWS)
Level 2 - Intermediate FNWS
 The student can produce the FNWS from “one”
to “ten”
 The student can produce the number word just
after a given number but drops back to “one”
when doing so
Table 7: Model for the construction of
forward number word sequences (FNWS)
Level 3 - Facile with FNWSs up to “ten”
 The student can produce the FNWS from “one”
to “ten”
 The student can produce the number word just
after a given number word in the range of “one”
to “ten” without dropping back
 The student has difficulty producing the number
word just after a given number word, for
numbers beyond ten
Table 7: Model for the construction of
forward number word sequences (FNWS)
Level 4 - Facile with FNWSs up to “thirty”
 The student can produce the FNWS from “one”
to “thirty”
 The student can produce the number work just
after a given number word in the range “one” to
“thirty” without dropping back
 Students at this level may be able to produce
FNWSs beyond “thirty”
Table 7: Model for the construction of
forward number word sequences (FNWS)
Level 5 - Facile with FNWSs up to “one hundred”
 The student can produce FNWSs in the range
“one” to “one hundred”
 The student can produce the number word just
after a given number word in the range “one” to
“one hundred” without dropping back
 Students at this level may be able to produce
FNWSs beyond “one hundred”
Table 8: Model for the construction of
backward number word sequences (BNWS)
Level 0 - Emergent BNWS
 The student cannot produce the BNWS from
“ten” to “one”
Level 1 - Initial BNSW up to “ten”
 The student can produce the BNWS from “ten”
to “one”
 The student cannot produce the number word
just before a given number. Dropping back to
“one” does not appear at this level
Table 8: Model for the construction of
backward number word sequences (BNWS)
Level 2 - Intermediate BNWS
 The student can produce the BNWS from “ten”
to “one”
 The student can produce the number word just
before a given number but drops back to “one”
when doing so
Table 8: Model for the construction of
backward number word sequences (BNWS)
Level 3 - Facile with BNWSs up to “ten”
 The student can produce the BNWS from “ten”
to “one”
 The student can produce the number word just
before a given number word in the range of “ten”
to “one” without dropping back
 The student has difficulty producing the number
word just before a given number word, for
numbers beyond ten
Table 8: Model for the construction of
backward number word sequences (BNWS)
Level 4 - Facile with BNWSs up to “thirty”
 The student can produce the BNWS from “thirty”
to “one”
 The student can produce the number word just
before a given number word in the range “thirty”
to “one” without dropping back
 Students at this level may be able to produce
BNWSs beyond “thirty”
Table 8: Model for the construction of
backward number word sequences (BNWS)
Level 5 - Facile with BNWSs up to “one hundred”
 The student can produce BNWSs in the range
“one” to “one hundred”
 The student can produce the number word just
before a given number word in the range “one”
to “one hundred” without dropping back
 Students at this level may be able to produce
BNWSs beyond “one hundred”
Table 9: Model of development of
counting by 10s and 100s
Level 1 - Initial counting by 10s and 100s


Can count forwards and backwards by 10s to 100 (e.g. 10, 20,
…100).
Can count forwards and backwards by 100s to 1000 (e.g. 100,
200,…1000).
Level 2 - Off-decade counting by 10s

Can count forwards and backwards by 10s, off the decade to 90s
(e.g. 2, 12, 22, …92).
Level 3 - Off-hundred and Off-decade counting by 100s


Can count forwards and backwards by 100s, off the 100, and on or
off the decade to 900s (e.g. 24, 124, 224, …924).
Can count forwards and backwards by 10s, off the decade in the
range 1 to 1000 (e.g. 367, 377, 387, …).
Table 10: Model for the development
of numeral identification
Level 0 - Emergent numeral identification
 May recognise some, but not all numerals in the
range “1” to “10”
Level 1 - Numerals to “10”
 Can identify all numerals in the range “1” to “10”
Level 2 - Numerals to “20”
 Can identify all numerals in the range “1” to “20”
Level 3 - Numerals to “100”
 Can identify one- and two- digit numerals
Table 10: Model for the development
of numeral identification
Level 4 - Numerals to “1 000”
 Can identify one-, two- and three- digit numerals
Level 5 - Numerals to “10 000”
 Can identify one-, two-, three- and four-digit numerals