Numeracy Continuum course

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

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• Forward number word sequence-ExampleCannot count to 10, counts to 10, counts to 30, counts 100 and counts beyond 100.Backward number word sequence-ExampleCannot backwards from 10 to 1, counts backwards from 10-1, counts backwards from 30-1, count backwards from 100-1 and counts backwards from any number.Counting sequences are also part of all the other 6 aspects. Eg counting in multiples, counting in centimetres, counting in fractions etc
• EAS markers include emergent counting, perceptual counting, figurative, counting and back- and facile[flexible] If students do not have counting on and back strategies they can only be plotted as “working towards” in place value.Aspect 2 - This is the problem solving framework – needs to be differentiated in the K-2 classroom.
• Aspect 3 – Patterns &amp; Algebra is part of every aspect – cannot be done in isloation. Eg doing multiplication also do P&amp;A patterns of multiples. Place Value – patterns of 10 &amp; 100 etc.Pattern and number- Point out how each step links. Emergent, instant [subitising] repeated, multiple, part-whole to 10, part-whole to 20 and number properties.
• Have the participants work in pairs to briefly look at the P&amp;A strand in the syllabus to clarify the meaning of and difference between:Number patterns Number relationships.Direct participants to p.72 of the Syllabus and the glossary in the Teaching Patterns &amp; Algebra e-book for definitions of number relationships/number patterns.Have the participants share their thoughts.
• What is Patterns and Algebra? Allow time for participants to record their thoughts Have participants keep the record of their thoughts for evaluation at the end of the workshop. What isn’t patterns and algebra? Allow a few minutes for participants to discuss their thoughts with a partner, then share with the whole group.Possible responses:Designs Non-repeating elements.
• These two components of Patterns and Algebra will be elaborated later in the workshop.
• People have two counting faculties. We can &quot;see&quot; instantly a handful of things and without knowing how many there are - this is called subitizing. The other way of counting is enumerating - counting up individual numbers. We can subitize up to about four or five, then we resort to enumerating.
• Five frames and ten frames are one of the most important models to help studentsanchor to 5 and 10.Five frames are a 1x5 array and ten frames are a 2x5 array in which counters or dotscan be placed to illustrate numbers.The five frame helps students learn the combinations that make 5. The ten-framehelps students learn the combinations that make 10. Ten-frames immediately model all of the facts from 5+1 to 5+5 and the respective turnarounds. Even 5+6, 5+7 and 5+8 are quickly seen as two fives and some more when depicted with these powerful models.
• Point out how each step links. (view video #1 ... Understanding view video #2 ..... Calculating )Aspect 4 – abstract unit of 10 is difficult to teach – involves explicit teachingUse authentic assessment to see ability to group 10 &amp; use strategies in order to be able to place them on this framework.New Syllabus – moving decimals into stage 3 to give more time &amp; weighting to develop sophisticated strategies in stage 2 for whole number.
• Quick view video in each section ...... Aspect 5 – goes from Kinder to stage 2 &amp; is supposed to be completed in stage2. This involves fluency (working mathematically) &amp; drill (yes drill).Keep pushing the integrated 5 week planning block to deepen conceptual knowledge &amp; allow 25 days of repeated drilling.“Plates” activity Brainstorm activity - using table printoutCreate activity and identify level
• Point out how each step links.Remind that this is new &amp; the current syllabus does not have the correct scope &amp; sequence of activities. Is in the new syllabus &amp; in fractions pikelets &amp; lamingtons.
• Use syllabus to identify link between LFIN and syllabus. Red highlighted are new components. Participants can be guided through what each level means using strip
• Aspect 7 also new and does not include time, mass, temperature.The alignment in ES1 related to length area &amp; volume.Level 6 iterating a unit (moving one single unit) sits between informal &amp; formal measuring and often gets left out yet this is how children develop a sense of structure and this is what NAPLAN assesses.
• Use syllabus to identify link between LFIN and syllabus. Participants can be guided through what each level means using strip
• See cheat sheets for Newman’s and Red DragonflyClick hyperlink
• Numeracy Continuum course

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