Count Me In Too

Count Me In Too

2009 Curriculum projectSouth 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 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)


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)


What is the answer to 9 x 3?

“Double 9 is 18,

18 + 2 is 20

20 + 7 is 27”

(Year 3 student)


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


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 2a, Jump method

28 38

+10

41

+3

Jump method: 28 + 13

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


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


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


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”


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

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

Documents

Count Me In Too