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Counting Growth Point supporting resources�

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Counting: Contents

Growth Points Overview
Victorian Curriculum aligned to Counting Growth Points
Numeracy Learning Progression: Quantifying Number
Key Ideas and important concept knowledge
Counting GPs 0 - 1
Counting GPs 1 – 2
Counting GPs 3 – 4
Counting GPs 5 – 6
Additional teaching resources
Using Concrete Materials

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MOI - Counting Growth Points

The Counting Growth Points describe the development of children's counting by ones, as well as by other multiples. These growth points are concerned both with children's production of number name sequences and, with children being able to make the count-to-cardinal transition in word meaning, so that children are able to interpret number names as quantities.

High quality student assessment supports teachers to monitor student progress and inform next steps, determine the effectiveness of chosen teaching strategies – both for learning and engagement – and to measure student understanding. (Stronge 2002)

What works best: 2020 update, Centre for Education Statistics and Evaluation

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Counting Growth Points

It is useful to consider the possible course or trajectory that describes the pathway of learning to count when planning associated learning opportunities for children. Curriculum documents provide a guide for this, but it is also helpful to focus on the major growth points or big mathematical ideas associated with the learning pathway. The Counting Growth Points describe the key milestones long the pathway of children's construction of counting knowledge. These growth points are concerned both with the production of number name sequences, and with children being able to make the count-to-cardinal transition in word meaning (Fuson, 1992a) so that they are able to interpret number names as quantities. The Counting growth points do not describe children's use of counting strategies in addition, subtraction, and multiplication or division calculations. Such strategies are included in the Addition and Subtraction, and Subtraction, and Multiplication and Division domains.

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Counting Growth Points

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Victorian Numeracy Learning Progressions -

Quantifying Number - Part A and Part B

Part A - Foundation to Level 1

Producing number names
Counting items
Numeral recognition and identification

Part B - Level 2 to Level 6

Producing number names
Numeral recognition and identification of place value
Understanding place value

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The Quantifying Number Part A Learning Progression begins at Foundation Level of the Victorian Curriculum and concludes at Level 1. Seven progressions are provided in this span. Please see Quantifying Numbers (Part B) for five additional progressions.

Description: This Learning Progression describes how a student becomes increasingly able to count, recognise, read and interpret numbers expressed in different ways. Although number is an abstract concept which can be represented by a word, a symbol (numeral) or an image, it is central to quantitative thinking. This Learning Progression outlines key understandings needed to process, communicate and interpret numerical information in a variety of contexts.

Within this Learning Progression, place value is taken to mean more than being able to read, write and state the positional value of a digit. Place value relies on understanding the relationship between digits in a numeral, which then enables the numeral to be renamed in multiple ways. In addition to the base-ten positional value property, the place value system has both additive and multiplicative properties. That is, the quantity represented by a numeral is the sum of the values represented by its individual digits (326 = 300 + 20 + 6) and the value of a digit is determined by multiplying its face value by the value assigned to its position in the numeral (326 = 3 x 100 + 2 x 10 + 6 x 1).

Details of progression provide nuanced and detailed descriptions of student learning – what students can say, do, make or write. Examples of student learning in each step are not hierarchical, nor are they to be used as a checklist.

https://docs.google.com/document/d/1QObxVZBFrxvtnGYdScqb2Q3kWYhPHFAPfv1pATdI0Ak/copy

The Quantifying Number Part B Learning Progression begins at Level 2 of the Victorian Curriculum and concludes at Level 6. Five progressions are provided in this span. Please see Quantifying Numbers (Part A) for seven previous progressions.

Description: This Learning Progression describes how a student becomes increasingly able to count, recognise, read and interpret numbers expressed in different ways. Although number is an abstract concept which can be represented by a word, a symbol (numeral) or an image, it is central to quantitative thinking. This Learning Progression outlines key understandings needed to process, communicate and interpret numerical information in a variety of contexts.

Within this Learning Progression, place value is taken to mean more than being able to read, write and state the positional value of a digit. Place value relies on understanding the relationship between digits in a numeral, which then enables the numeral to be renamed in multiple ways. In addition to the base-ten positional value property, the place value system has both additive and multiplicative properties. That is, the quantity represented by a numeral is the sum of the values represented by its individual digits (326 = 300 + 20 + 6) and the value of a digit is determined by multiplying its face value by the value assigned to its position in the numeral (326 = 3 x 100 + 2 x 10 + 6 x 1).

Details of progression provide nuanced and detailed descriptions of student learning – what students can say, do, make or write. Examples of student learning in each step are not hierarchical, nor are they to be used as a checklist.

https://docs.google.com/document/d/1Cb9tSLV2uXEXae16xFadgsmJVguZuPMiuQGwumeQuEw/copy

Numeracy Learning Progressions: https://www.vcaa.vic.edu.au/curriculum/foundation-10/crosscurriculumresources/Pages/Numeracy.aspx

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Victorian Numeracy Learning Progressions -

Quantifying Number - Part A

Producing number names

Foundation

Level 1

Counting GP 0

C - GP 1

C - GP 2*

C - GP 3/4**

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* Note that Step 6 of the Producing number names progression extends the number range to at least 30 so students are able to hear the repeating pattern within the number system. Beyond 20, the sequence of number words is well structured in that the same cycle of number words (i.e., one, two, … nine) is appended to the term used for the relevant multiple of ten (e.g., twenty-one, twenty-two, … twenty-nine) P Gould: Hurdles in acquiring the number word sequence (https://files.eric.ed.gov/fulltext/ED572391.pdf)

** Note that Step 7 of the Producing number names progression extends the number range to 120 and beyond. This is to address the substantial challenge of students not knowing the number word following 109 in the counting sequence. Not knowing that 110 follows 109 might be influenced by this being the first time ten is re-used in the forward counting sequence. Indeed, students are actively discouraged from using ten earlier in the counting sequence: Thirty-ten does not follow thirty-nine. P Gould: Hurdles in acquiring the number word sequence (https://files.eric.ed.gov/fulltext/ED572391.pdf)

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Victorian Numeracy Learning Progressions -

Quantifying Number - Part B

Producing number names

Level 2

Level 3

Counting - GP 5

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Victorian Numeracy Learning Progressions -

Quantifying Number - Part A

Counting items

Foundation

Level 1

Counting - GP 0

Counting - GP 2

C - GP 3 & 4*

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Number word sequence and counting: �What’s the difference?

The number word sequence, forwards or backwards, is the fixed order of number names.

There is a difference between reciting and counting a number word sequence.

Reciting a sequence of number words is by rote, whereas counting is the allocation of each spoken number word with an item (quantifying).

Students begin to learn the number word sequence and then use this sequence to count collections.

Students can typically produce a number word sequence in a higher number range before they can accurately count a collection of that size. For example, a student may produce the number word sequence to 100, but is only able to count a collection of around 20 items.

Counting pg. 7 - 8

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Counting Growth Points 0 - 1

Counting Growth Points
GP 0	Not apparent Not yet able to state the sequence of number names to 20.
GP 1	Rote Counting to 20 Produce the forward number name sequence to at least 20, but are not yet able to reliably count a collection of that size.

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Working towards Growth Point 1: Rote Counting to 20

Students need:

Hear, repeat and recite number word sequences, to develop fluency and automaticity
Use concrete materials to model and imagine the quantities represented by increasing or decreasing number word sequences. Concrete materials can include beads, blocks and fingers. The models assist students to create meaning for number names and counting sequences.
Opportunities to notice the final sounds in number words, particularly in teen and ty number names. Some students have not heard or noticed the differences in these sounds Using rhythm, rhyme melody and movement to help students learn, imagine and repeat number word sequences.
Opportunities to notice and focus on the repeating sound patterns in number word sequences.
Opportunities to observe and notice people using counting sequences to count quantities

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10 Rest point…. it’s been the expectation

12 Hurdle…. the first identifiable challenge

20 Rest point... what comes after the next multiple of ten

Victorian Curriculum expectations: students name, represent and order numbers, including zero to at least 20, using physical and virtual materials and numerals.

Number word sequence milestones : Range to 20

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Alarm!

Any student unable to count past 10 mid Foundation (signifies lack of progress)
Any student unable to count past 20 mid Year 1 (signifies lack of progress)

Potholes. Omitted numbers.

They will persist unless identified and addressed through targeted teaching.

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MACS Ochre Resources �to support teaching

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Counting Growth Points 1 - 2

Counting Growth Points
GP 1	Rote Counting to 20 Produce the forward number name sequence to at least 20, but are not yet able to reliably count a collection of that size.
GP 2	Counting Collections Confidently counts collection of around 20 objects.

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Working towards Growth Point 2: �Quantifying and Partitioning Collections of 20

Students need:

Working out the total of items in a collections by subitising, structuring, imagining or counting
Recognising that miscounting objects leads to an incorrect total .
Producing or imagining a given number of items.
Matching or imagining groups with the same quantity .
Structuring or partitioning sets into subsets as a means of working out the total more quickly or efficiently.
Organising the items to be counted to ensure that no items are missed or counted more than once.
Asking the student to make or imagine one more or one less than the collection being considered or investigated. This emphasises the idea of quantity.

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Ochre Resources to support teaching

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MACS Ochre Resources �to support teaching

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Counting Growth Points 3 - 4

Counting Growth Points
GP 3	Counting by 1s (forward/backward, including variable starting points; more/less) Counts forwards and backwards from various starting points between 1 and 100; knows numbers before and after a given number.
GP 4	Counting from 0 by 2s, 5s, and 10s Can count from 0 by 2s, 5s, and 10s to a given target.

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Working towards Growth Point 3: �Counting by 1s forwards and backwards from variable starting points between 1 and 120*; more/less

Students need:

Noticing that quantities between 10 and 99 are named in English by organising them into tens and ones and then noticing how many tens and ones are present, e.g. forty-five represents 4 tens and 5 ones
Emphasising the hundreds, tens and ones structure of collections and number names when working out the total of large collections (of at least 120 items)
Partitioning large collections of at least 120 items in various ways as a means of working out the total more efficiently than counting
Partitioning collections into subsets (including hundreds, tens and ones) so that the structure can be used for faster or abbreviated counting
Extending counting experiences beyond 100 so that students may form generalisations through identifying the sound and numeral patterns that recur beyond 100
Observing, discussing and describing sound and numeral patterns in the forward and backwards number name sequence (to at least 120)
Imagining and saying the total that is one more or one less than a given number, or imagining and saying three numbers more or less than a given number.

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Number word sequence milestones : range to 130

29 (&39)

Hurdle, a challenge to get into the

third multiple of ten

100 Rest point, expectation

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109 Significant hurdle

Victorian Curriculum expectations: Year 1 count to and from at least 120 from any starting point

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YR 1-Victorian Curriculum expectations: students connect number names, numerals and quantities, and order numbers to at least 120 and recognise, continue and create pattern sequences, with numbers by skip counting, initially by twos, fives, and tens.

YR 2-Victorian Curriculum expectations: recognise, describe and create additive patterns that increase or decrease by a constant amount.

Potholes. Omitted numbers. They can persist unless identified and addressed through targeted teaching. e.g. decuple numbers & double digits

Mid Year 1 Alarms!

Any student who stops at 11, 12, or 13 signifies lack of progress.
The continued existence of rest points at 49, 59, 69.
It’s usual to have trouble overcoming 20 and 30, not 60, 70, 80
Repeat: resting at 10, 20 or 100 is ok

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MACS Ochre Resources �to support teaching

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MACS Ochre Resources �to support teaching

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MACS Ochre Resources �to support teaching

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Working towards Growth Point 4: �Counting by 2s, 5s and 10s

Students need:

Using and imagining using objects such as bead strings to promote observing, discussing and describing increasing quantities in the forward and backwards number name sequence when counting by tens, fives and twos (to at least 120);
Exploring how addition and subtraction strategies can help to generate the next number in a twos, fives or tens counting sequence.
Imagining and saying the number name just before or just after a given number when counting by tens, fives or twos.

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MACS Ochre Resources �to support teaching

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Counting Growth Points 5 - 6

Counting Growth Points
GP 5	Counting from x (where x >0) by 2s, 5s, and 10s Given a non-zero starting point, can count by 2s, 5s, and 10s to a given target.
GP 6	Extending and applying counting skills Can count from a non-zero starting point by any single digit number, and can apply counting skills in practical task.

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Working towards Growth Point 5 & 6: �Counting by 10s, 5s and 2s (and other quantities) from various starting points�Can count from a non-zero starting point by any single digit number and can apply counting skills in practical tasks.�

Students need:

Using and imagining using bead strings to promote observing, discussing and Using bind increasing quantities in the forward and backwards number sequence when counting from X by tens, fives and twos (to at least 120).
Exploring how addition and subtraction strategies can help to generate the next number in a twos, fives or tens counting sequence (or other number sequences.
Imagining and saying the number name just before or just after a given number when counting from X by tens, fives or twos (and other quantities).

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MACS Ochre Resources �to support teaching

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Resources to support teaching: Growth Point Activity Companion

GPAC Appendix Link

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Counting Growth Point additional resources

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Using the Numeral Roll to teach number word sequences

Numeral roll

The numeral roll has a long sequence of numerals (e.g. 1 to 120). Display the roll and have the children say each number in sequence as the teacher points to it, either forwards or backwards.

Screen the sequence of numerals and display them one at a time.

Display a numeral in a window. Proceed along the sequence forwards or backwards, unscreening one numeral at a time: (a) first seeing then saying (See then say) and (b) first saying then seeing (say then see).

Variations

the numeral roll can be extended or shortened according to children’s levels of knowledge
use with numeral cards in a given range - turn over a card and count forward/backward from that number
use the numeral roll to keep track of the count.

Numeral roll printables

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Using the numeral roll video

- See then say (easier)

- Say then see(harder)

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Numeral Ladders

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Counting by 10s on and off the decuple

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Dynamic Counting

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Using concrete materials

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Using concrete materials to support student learning��Hands-on materials and manipulatives are important tools for supporting students to make sense of and develop their understanding of mathematical ideas.��The following slides are designed to support you in selecting and sequencing the use of appropriate concrete materials when exploring place value with students.��

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Counting Collections

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James Goold House, 228 Victoria Parade, East Melbourne VIC 3002

T: +61 3 9267 0228 E: execdirector@macs.vic.edu.au www.macs.vic.edu.au

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