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- 1. 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
- 2. 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.
- 3. 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)
- 4. Jann Farmer-Hailey (former Leader K-4 Initiatives, Teaching Services, NSW CLIC )
- 5. 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
- 6. NOT INCLUDED (in development) The Scope and Progression of Space and Geometry Mass Time Temperature Chance Data
- 7. ANALYSIS AND FAMILIARITY
- 8. 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
- 9. 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)
- 10. Pattern and Number Structure •The identification of pattern associated with the structure of numbers.
- 11. 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
- 12. • 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?
- 13. 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.
- 14. 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
- 15. 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?
- 16. 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.
- 17. Brainstorming activity Create an activity for each level in this aspect Link to eBook/ pdf Sample activities
- 18. Aspect 1: Counting sequences Aspect 2: Counting as a problem- solving process Aspect 3: Pattern and number structure
- 19. Let’s have a look now at the Numeracy Continuum
- 20. Place Value Student should be at least at the counting on and back stage to be placed on the place value framework
- 21. 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
- 22. Covered Item Task for Place Value
- 23. 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
- 24. Multiplication and division •Using equal groups in multiplication as well as two different types of division.
- 25. 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
- 26. Covered Item Task for Multiplication & Division
- 27. 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.
- 28. 35
- 29. Fractions Developing a quantitative sense of fractions, relies on forming partitions, relating the part to whole and recognising the need for equal wholes.
- 30. 10 out of 7 students have difficulty with Fractions Understanding Fractions
- 31. 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
- 32. Measurement •Knowledge of the structure of units in length, area and volume.
- 33. 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
- 34. Take a break Time to refresh ………
- 35. 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
- 36. WHERE TO GO FROM HERE! Develop tracking sheets for students Develop differentiated learning material Align to NAPLAN-style questioning Develop your toolbox for teaching . . . .
- 37. 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)
- 38. Dr Linda Darling-Hammond (Charles E. Ducommun professor of education at Stanford University ) Dr Linda Darling-Hammond
- 39. CLOSING! Please complete online evaluation
- 40. Course completed: Congratulations!

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