count me in too 2009 curriculum project south western sydney region
TRANSCRIPT
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 2Developing Efficient Numeracy Strategies(DENS) 1 and 2Mathematic 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: EmergentStage 2: PerceptualStage 3: FigurativeStage 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 countingStudents 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 countingStudents 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 collectiongreater then 2Level 1 – Perceptual Students instantly recognise number of items to about 6Level 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.
12
24
36
Table 4: Calculating volume by identifying horizontal layers and adding, skip counting, or multiplying.
9 18 27 36
Table 4: Calculating volume by identifying vertical layers and adding, skip counting, or multiplying the number of layers
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 Forming equal groupsLevel 2 Perceptual multiplesLevel 3 Figurative unitsLevel 4 Repeating abstract composite
unitsLevel 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 2-partition 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 method2b: 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
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 numbers- one must be visible
e.g. 4 tens and 2 units visible, 25 units hidden, counts by tens and ones
Table 6: Level 1, Ten as a unit
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 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
28 + 13
20 + 10
8 + 3
+
30 11+
28 + 13
Split method: 28 + 13
41
28 + 13
Table 6: Hundreds, tens and onesLevel 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 onesLevel 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 seventy-five 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 100sLevel 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