Framework of Mathematical Learning

From Term 1 2017, Victorian government and Catholic schools will use the new Victorian Curriculum F-10. Curriculum related information is currently being reviewed and may be subject to change.

The Victorian Curriculum F–10 - VCAA

The impetus for the Early Numeracy Research Project was a desire to improve mathematics learning and so it was necessary to quantify such improvement. It was decided to create a framework of key "growth points" in numeracy learning. Students' movement through these growth points in trial schools could then be compared to that of students in the reference schools. It was intended that the framework would:

• reflect findings of relevant research in mathematics education
• emphasise important ideas in early mathematics understanding in a form and language readily understood and, in time, retained by teachers
• reflect, where possible, the structure of mathematics
• allow mathematical knowledge and understanding to be described
• form the basis of planning and teaching
• provide a basis for task construction for assessment via interview
• allow the identification and description of improvement in learning
• enable a consideration of those students who may benefit from additional assistance
• have sufficient "ceiling" to describe the knowledge and understanding of all children in the first three years of school.

Not all possible mathematical domains are included in this framework. The decision was taken to focus upon the strands of Number (incorporating the domains of Counting, Place Value, Addition and Subtraction, and Multiplication and Division Strategies), Measurement (incorporating the domains of Length, Mass and Time), and Space (incorporating the domains of Properties of Shape, and Visualisation and Orientation). We describe growth points as key "stepping stones" along paths to mathematical understanding. They provide a kind of conceptual landscape. However, we do not claim that all growth points are passed by every student along the way. "The order is more or less the order in which strategies are likely to emerge and be used by children. ... intuitive and incidental learning can influence these strategies in unexpected ways." (ENRP Final Report, p. 39)

Early Numeracy Research Project growth points

A. Counting

1. Not apparent.
Not yet able to state the sequence of number names to 20.
2. Rote counting
Rote counts the number sequence to at least 20, but is not yet able to reliably count a collection of that size.
3. Counting collections
Confidently counts a collection of around 20 objects.
4. Counting by 1s (forward/backward, including variable starting points; before/after)
Counts forwards and backwards from various starting points between 1 and 100; knows numbers before and after a given number.
5. Counting from 0 by 2s, 5s, and 10s
Can count from 0 by 2s, 5s, and 10s to a given target.
6. 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.
7. 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

B. Place value

1. Not apparent.
Not yet able to read, write, interpret and order single digit numbers.
2. Reading, writing, interpreting, and ordering single digit numbers
Can read, write, interpret and order single digit numbers.
3. Reading, writing, interpreting, and ordering two-digit numbers
Can read, write, interpret and order two-digit numbers.
4. Reading, writing, interpreting, and ordering three-digit numbers
Can read, write, interpret and order three-digit numbers.
5. Reading, writing, interpreting, and ordering numbers beyond 1000
Can read, write, interpret and order numbers beyond 1000.
6. Extending and applying place value knowledge
Can extend and apply knowledge of place value in solving problems

C. Strategies for addition and subtraction

1. Not apparent.
Not yet able to combine and count two collections of objects.
2. Count all (two collections)
Counts all to find the total of two collections.
3. Count on
Counts on from one number to find the total of two collections.
4. Count back/count down to/count up from
Given a subtraction situation, chooses appropriately from strategies including count back, count down to and count up from.
5. Basic strategies (doubles, commutativity, adding 10, tens facts, other known facts)
Given an addition or subtraction problem, strategies such as doubles, commutativity, adding 10, tens facts, and other known facts are evident.
6. Derived strategies (near doubles, adding 9, build to next ten, fact families, intuitive strategies)
Given an addition or subtraction problem, strategies such as near doubles, adding 9, build to next ten, fact families and intuitive strategies are evident.
7. Extending and applying addition and subtraction using basic, derived and intuitive strategies
Given a range of tasks (including multi-digit numbers), can solve them mentally, using the appropriate strategies and a clear understanding of key concepts

D. Strategies for multiplication and division

1. Not apparent.
Not yet able to create and count the total of several small groups.
2. Counting group items as ones
To find the total in a multiple group situation, refers to individual items only.
3. Modelling multiplication and division (all objects perceived)
Models all objects to solve multiplicative and sharing situations.
4. Abstracting multiplication and division
Solves multiplication and division problems where objects are not all modelled or perceived.
5. Basic derived and intuitive strategies for multiplication
Can solve a range of multiplication problems using strategies such as commutativity, skip counting and building up from known facts.
6. Basic, derived and intuitive strategies for division
Can solve a range of division problems using strategies such as fact families and building up from known facts.
7. Extending and applying multiplication and division
Can solve a range of multiplication and division problems (including multi-digit numbers) in practical contexts

E. Time

1. Not apparent.
No apparent awareness of time, its descriptive language and features of clockfaces.
2. Awareness of time, its descriptive language, and some features of clockfaces
Can describe at least one feature and one purpose of clockfaces.
3. Knowing some clock times, some days of week and months of year, and relating key events (personal, community) to these
Knows some clock times, some days of week and months of year, and can relate key events to these.
4. Knowing clock times to half-hour, all days of week and months of year (including order)
Knows clock times to half-hour, all days of week and months of year (including order).
5. Facility with clocks and calendars
Can read analogue clock times to nearest five minutes and has good working facility with calendars.
6. Extending and applying knowledge, skills and concepts with time
Can solve a range of problems involving duration, and digital and analogue time to the nearest minute

F. Length measurement framework

1. Not apparent.
No apparent awareness of the attribute of length and its descriptive language.
2. Awareness of the attribute of length and use of descriptive language
Awareness of the attribute of length and its descriptive language.
3. Comparing, ordering, & matching with the attribute of length
Compares, orders, & matches objects by length.
4. Quantifying length accurately, using units and attending to measurement principles
Uses uniform units appropriately, assigning number and unit to the measure.
5. Choosing standard units for estimating and measuring length, with accuracy
Uses standard units for estimating and measuring length, with accuracy.
6. Applying knowledge, skills and concepts of length
Can solve a range of problems involving key concepts of length

G. Mass measurement framework

1. Not apparent.
No apparent awareness of the attribute of mass and its descriptive language.
2. Awareness of the attribute of mass and use of descriptive language
Awareness of the attribute of mass and its descriptive language.
3. Comparing, ordering, & matching with the attribute of mass
Compares, orders, & matches objects by mass.
4. Quantifying mass accurately, using units and attending to measurement principles
Uses uniform units appropriately, assigning number and unit to the measure.
5. Choosing standard units for estimating and measuring mass, with accuracy
Uses standard units for estimating and measuring mass, with accuracy.
6. Applying knowledge, skills and concepts of mass
Can solve a range of problems involving key concepts of mass

H. Properties of shape

1. Not apparent.
Not yet able to recognise and match simple shapes.
2. Holistic recognition of shape
Can recognise resemblances and match some simple shapes, using standard "prototypes".
3. Classification of shapes, attending to visual features
Can sort and compare shapes, using some geometrical language to describe features.
4. Identification of "classes of shapes" by some properties
Uses properties of shapes to classify shapes into classes, using appropriate language.
5. Definition of shapes using properties
States and understands conditions for defining key shapes

I. Visualisation and orientation

1. Not apparent.
Not yet able to visualise simple shapes.
2. Static, pictorial images formed in conjunction with models or manipulatives
Able to recognise static images in embedded situations.
3. Re-orientation of shapes mentally
Can visualise the effect of simple flipping, sliding and turning of shapes.
4. Dynamic imagery
Uses dynamic imagery to visualise manipulation of shapes by transforming and rearranging.
5. Extending and applying visualisation and orientation
6. Can combine a range of visualisation strategies in increasingly complex situations.