An outline of strategies underpinning the strategies used by students in 7 aspects of the numeracy continuum

2. 1st August 2012
9-9:15 Welcome, housekeeping
9:15 – 9:35 Why a continuum?
9:35 – 10:45 Aspect 1, 2, 3 analysis and familiarity
10:45 – 11:15 Morning Tea
11:15 – 12:45 Aspect 4, 5, 6, 7 analysis and familiarity
12:45- 1:15pm Lunch
1:15 – 2:45 pm Developing knowledge, where to now
2:45 - 3:00 pm Closing

3. DEFINITION
 Research-based continuum of learning of the conceptual levels of
mathematical thinking that children move through from K-10
 From authentic assessment, teachers get a ‘SNAPSHOT’ in time that can be
used to plan and program explicit teaching
 It provides a means for observing & tracking students’ strategies
 Developed from constructivist theory i.e. scaffolding learning in series of
steps
 Current and new syllabus both present a scope and sequence of activities
based on it
 A clear and accessible representation of development of critical aspects
across 7 years of primary education & then into high school.

4. FEATURES/PROPERTIES
 Each aspect is overlapping and interrelated
 It features seven aspects
 Teachers can use it to map students (visual wall mapping)
 Teachers determine not just the right and wrong answers, but the
strategies used to find answers.
 Teachers can explicitly plan for and teach more sophisticated ones
 As students develop and practise more sophisticated strategies,
teachers refer back to the Continuum to guide their program
 Each aspect is aligned with a syllabus outcome, up to Stage 4
(Dr Peter Gould)

5. Jann Farmer-Hailey
(former Leader K-4 Initiatives, Teaching Services,
NSW CLIC )

6. ASPECTS
 Aspect 1 - Counting Sequences- verbal and written labels
 Aspect 2 - Counting as a problem solving process-
Early Arithmetical Strategies [EAS] Emergent, Perceptual,
Figurative, Counting on and back, Facile
 Aspect 3 - Pattern and number structure – Subitising –
partitions to 20 in standard and non standard form
 Aspect 4 - Place value, PV 0-5
 Aspect 5 - Multiplication and division Levels 1-5
 Aspect 6 - Fraction units
 Aspect 7 - Unit structure of length, area and volume

7. NOT INCLUDED (in development)
 The Scope and Progression of Space and Geometry
 Mass
 Time
 Temperature
 Chance
 Data

8. ANALYSIS AND FAMILIARITY

9. Counting Sequences
•Forward number word sequence.
•Backward number word sequence.
•Numeral identification.
•Counting by 10’s and 100’s.
This aspect identifies a student’s ability to count

10. Counting as a problem solving process- Early Arithmetical
Strategies [EAS]
•EAS refers to the range of counting strategies that are used to solve addition
and subtraction problems.
Levels:
• Emergent
• Perceptual
• Figurative (view video)
• Counting on-and-back (view video)
• Facile (view video)

11. Pattern and Number Structure
•The identification of pattern associated with
the structure of numbers.

12. Overview of Patterns and Algebra
The knowledge and skills that students acquire in Patterns
and Algebra are outlined in the syllabus in terms of:
 Number patterns
 Number relationships

13. • Patterns are central to mathematics
teaching and learning.
• Learning that includes deep
knowledge about patterns can
develop strong conceptual
understandings.
• Describing and discussing patterns
can develop a capacity to reason
and generalise leading to algebra.
• Lessons focussed on reasoning
about patterns can develop deep
understanding and promote
substantive communication.
Why teach Patterns and Algebra?

14. Overview of Patterns and Algebra
Number patterns includes:
 creating number patterns
 describing number patterns
 finding terms in a number pattern.
Number relationships includes:
 describing number relationships
 generalising about number relationships
 finding unknown elements.

16. Subitising
This is the ability to immediately recognise the number of
objects in a small collection without having to count them
Dot Pattern Cards
This is part of a card matching
activity for subitising.
There are 20 cards in this game
for students to match focusing on
the numbers 1-10. The aim of the
game is to find matching pairs

17. We have two counting facilities –
subitizing and enumerating.
Subitizing is a fundamental skill in
the development of number sense,
supporting the development of
conservation, compensation,
unitizing, counting on, composing
and decomposing of numbers.
Which way
is the
easiest one
to see a
group of 5?

18. http://illuminations.nctm.org/ActivityDetail.aspx?
ID=75
Ten Frames and Five Frames
OFF COMPUTER ACTIVITY
Five frames and ten
frames are effective
models to help students
anchor to 5 and 10.

19. Brainstorming activity
Create an activity for each level in this aspect
Link to eBook/ pdf
Sample activities

20. Aspect 1:
Counting sequences Aspect 2:
Counting as a problem-
solving process
Aspect 3:
Pattern and number
structure

21. Let’s have a look now
at the
Numeracy Continuum

22. Place Value
Student should be at least at the counting on and back stage to be placed on
the place value framework

24. Level Characteristic Syllabus
0
Ten as a Count
Counting by ones NS 1.2
1
Ten as a Unit
Ten is a countable unit. Visual
materials
NS 1.2
2
Tens and Ones
Two digit mental addition and
subtraction
NS 1.2,
NS 2.2
3
Hundreds, Tens and
Ones
Three digit mental addition and
subtraction
NS 2.2
4
Decimal PV
Decimal place value NS 2.4
5
System PV
Understands place value NS 3.2
Place Value Framework Summary

25. Covered Item Task for Place Value

26. Covered Item Task for Place Value
Level 0 – Ten as a count
Student counts the dots by ones as each section is uncovered.
Level 1 – Ten as a unit
Student can add the numbers using 10 as a countable unit while the
dots are visible.
Student can add the visible collections of 10 and 20 dots.
Student can add the visible collections of 14 and 25 dots.
Level 2 – Tens and ones
The student can mentally calculate how many more dots are needed to
make 100.
Level 3 – Hundreds, tens and ones - Not assessed in this task
Level 4 – Decimal place value - Not assessed in this task
Level 5 – Understands place value - Not assessed in this task

29. Multiplication and division
•Using equal groups in multiplication as well as two different types of division.

31. Level Characteristic Syllabus
1
Forming Equal Groups
Counts the visible items in each
group by ones
NES 1.3
2
Perceptual Multiples
Counts using groups with visible
items
NES 1.3
3
Figurative Units
Counts using markers for each group NS 1.3
4
Repeated Abstract
Composite Units
Counts without group markers NS 1.3
5
Multiplication and
Division as Operations
Uses multiplication and division as
inverse operations
NS 2.3
Multiplication and Division Framework Summary

32. Covered Item Task for Multiplication & Division

33. Covered Item Task for Multiplication & Division
Level 1 – Forming equal groups
Student needs to see the dots inside the circles.
Student counts the dots by ones in a continuous manner.
Level 2 – Perceptual multiples
Student needs to see the dots inside the circles.
Student counts the dots using a rhythmic or skip count or a combination of both.
Level 3 – Figurative units
Student needs to see the circles but not the dots.
Student counts the dots using a rhythmic or skip count or a combination of both.
Student uses perceptual markers to keep track of the groups.
Level 4 – Repeated abstract composite units
Student does not need to see the circles or dots.
Student uses a composite unit to determine the number of dots, maybe through repeated
addition or fingers.
Level 5 – Multiplication and division as operations
Student does not need to see the circles or dots.
Student can determine the number of dots using multiplication facts.

35. 35

36. Fractions
Developing a quantitative sense of fractions, relies on forming partitions,
relating the part to whole and recognising the need for equal wholes.

37. 10 out of 7 students have
difficulty with Fractions Understanding Fractions

38. Level Characteristic Syllabus
0
Emergent
Partitioning
Attempts to halve by splitting without attention to equality
of the parts.
1
Halving
Forms halves and quarters by repeated halving.
Can use distributive dealing to share.
NES1.4
NS1.4
2
Equal partitions
Verifies partitioning into thirds or fifths by iterating one
part to form the forming of the whole or checking the
equality and number of parts
NS 2.4
3
Re-forms the whole
When iterating a fraction part such as one-third beyond
the whole, reforms the whole. NS 3.4
4
Fractions as numbers
Identifies the need to have equal wholes to compare
fractional parts. Uses fractions as numbers. 1/3 > 1/4 NS 4.3
5
Multiplicative
reasoning
Coordinates composition of partitioning. i.e. can find one-
third of one-half to create one-sixth. Coordinates units at
three levels to move between equivalent fraction forms.
Creates equivalent fractions using equivalent equal wholes.
NS 4.3
Fractions Framework Summary

39. LEVEL 1
LEVEL 2
LEVEL 4
LEVEL 5

40. Measurement
•Knowledge of the structure of units in length, area and volume.

41. LEVEL Characteristic Syllabus
0
Attempts direct comparison without attending to alignment.
May attempt to measure indirectly without attending to gaps & overlaps
1 Directly compares the size of two objects. (alignment) MES 1.1
2
Directly compares the size of 3 or more objects. (transitivity).
Uses indirect comparison by copying the size of one of the objects.
MES 1.1
3
Uses multiple units of the same size to measure an object, (without gaps & overlap).
Chooses & uses a selection of the same size & types of units to measure an object.
MS 1.1
4
States the qualitative relationship between the size & number of units.
Chooses & uses a selection of the same size & type of units to measure by indirect comparison.
MS 1.1,
MS 1.2
5
Uses a single unit repeatedly to measure or construct length.
Make a multi-unit ruler by iterating a single unit & quantifying accumulated distance.
Identifies the quantitative relationship between length & number of units
MS 2.1
6
Creates the row-column structure of the iterated composite unit of area.
Uses the row-column structure to find the number of units to measure area.
MS 2.2,
MS 3.2
7
Creates the row-column-layer structure of the iterated layers when measuring volume.
Uses the row-column-layer structure to find the number of units to measure volume.
MS 2.3,
MS 3.3
Measurement Framework Summary

42. Take a break
Time to refresh ………

43. WHAT TO DO NEXT ......
 Know where the students are ...What strategies? Where are
students on the continuum?
 Know the concept you are wanting to teach (be explicit- from
Numeracy continuum, indicators syllabus driven by WM)
 Plan activity/lesson that addresses the concept
 Choose outcomes from both skills and content and WM
 Differentiate the activity/lesson to cater for diversity of
students along given framework or at least above and below
the level

44. WHERE TO GO FROM HERE!
 Develop tracking sheets for students
 Develop differentiated learning material
 Align to NAPLAN-style questioning
 Develop your toolbox for teaching . . . .

45. TOOLBOX OF TEACHING
• Whole class lessons to explicitly teach new concepts
• Activity Resource Centre (ARC)
• Mixed or paired activities
• Meaningful engagement via Rich tasks, and Connected
Outcomes Group (COGS) activities
• Curriculum Support site
• Counting On and Count Me In Too activities
• Problem Solving (Newman’s, red dragonfly maths)

46. Dr Linda Darling-Hammond
(Charles E. Ducommun professor of education at Stanford University )
Dr Linda Darling-Hammond

47. CLOSING!
 Please complete online evaluation

48. Course completed: Congratulations!