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Implementing Mathematics K-6 Using the syllabus for consistency of assessment and moderation © 2006 Curriculum K-12 Directorate, NSW Department of Education and Training

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What is Assessment? Assessment is the process of identifying, gathering and interpreting information about students' learning. The central purpose of assessment is to provide information on student achievement and progress and set the direction for ongoing teaching and learning. Schools are to undertake assessment to collect information about students learning. This will occur through both formal and informal activities. Assessment of student learning will be undertaken for all learners, including students with disabilities: –enrolled in regular classes; –enrolled in special classes or in special schools; Policy Standards for Curriculum Planning and Programming, Assessing and Reporting to Parents K-12

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Planning for assessment It is through our assessment that we communicate to our pupils those things that we most value. David Clarke, The Mathematics Curriculum and Teaching Program, 1988.

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Policy guidelines Schools plan assessment so that: -students can demonstrate achievement for the relevant stage of learning -valid and reliable assessment strategies are used -assessment processes are time efficient and manageable. (Curriculum Planning and Programming, Assessing and Reporting to Parents K-12)

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Assessment tasks In practice, effective mathematics assessment is characterised by tasks which: connect to prior learning provide adequate time for students to think before responding engage students, are relevant and are valued by them allow students to demonstrate their mathematics skills in context.

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What do we mean by consistency? Consistency of teacher professional judgement refers to the degree to which judgements about a students performance are independent of which teacher is assessing the student.

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What is moderation? Moderation is a process where teachers compare judgements to either confirm or adjust them. The process involves close collaboration to establish a shared understanding of what achievement of syllabus standards looks like and whether or not the student has demonstrated achievement of the syllabus standard. Teachers work towards making judgements that are consistent and comparable.

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What is consistency of moderation? Consistency of teacher professional judgement refers to the degree to which judgements about a students performance are independent of which teacher is assessing the student.

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Why do we moderate? We moderate to: develop shared or common interpretations of standards and expectations of what constitutes achievement of syllabus standards develop shared understandings of what students achievements look like develop accuracy and reliability in making judgements ensure judgements are equitable in terms of implications for student learning strengthen the value of teachers judgements inform well-targeted teaching programs.

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Some challenges In studying childrens thinking we usually find all sorts of in- between patterns of performance: children who succeed on some versions of the task but not on others, and who thus seem sometimes to have the concept and at other times to not have it. Cognitive Development (p. 321) Flavell, Miller & Miller, 1993

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Two components of teacher judgement All teachers judgements have two components: the decision and the level of confidence in the decision. Sometimes you need more information to make a confident decision.

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Teacher judgement How do you form your expectations of what your students should achieve? Sources of information: - syllabus expectations - shared professional expectations and discussion

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Informing judgement: What do you want the students to learn? What parts of the syllabus do you currently draw on to inform your planning, teaching and assessing? Knowledge and skills Students learn about Working Mathematically Students learn to

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Why do students need to learn about area? Why does the learning matter? Students need to be able to measure accurately and efficiently using the appropriate unit. An understanding of area builds on an understanding of measuring length and is a prerequisite skill for understanding and calculating volume. Students apply understanding of area concepts to solve problems relating to the painting, tiling or carpeting of a defined area, preparing a garden plot or purchasing a length of material. Students can apply understanding of area when interpreting maps in HSIE and completing design and make tasks in Science and Technology.

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What will the students do? In this task, students are asked to calculate and draw rectangles with an area of 24 cm 2. Students will demonstrate working mathematically processes through applying strategies and communicating their reasons for being able to find all possible solutions using whole numbers.

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How well do you expect students to complete the task? What is the expectation of student achievement at each stage? An understanding of the continuum of learning in Area will provide information on syllabus expectations for Stage 2 students.

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Strand based decisions: Area ES1 Outcome MES1.2: Describes area using everyday language and compares areas using direct comparison Foundation Statements (Kindergarten) 2005 Students identify length, area, volume, capacity and mass and arrange objects according to these attributes. Expectation: Students superimpose shapes and order two or more areas according to size. Students explain why they think that the area of one surface is larger or smaller than another. (WM)

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Strand-based decisions: Area Stage 1 Outcome MS1.2: Estimates, measures, compares and records areas using informal units Foundation Statements (Years 1 & 2) 2005 Students estimate, measure, compare and record using informal units for length, area, volume, capacity and mass. They recognise the need for formal units of length and use the metre and centimetre to measure length and distance. Expectation: Students estimate, measure, compare and order two or more areas by placing the same informal unit in rows or columns without gaps or overlaps. Students explain why tessellating shapes are best for measuring area. (WM)

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Strand based decisions: Area Stage 2 Outcome MS2.2: Estimates, measures, compares and records the areas of surfaces in square centimetres and square metres Foundation Statements (Years 3 & 4) 2005 Students estimate, measure, compare and record length, area, volume, capacity and mass using some formal units. Expectation: Students use square centimetres and square metres to estimate, measure, compare and record areas. Students explain the strategies used to find area in square Centimetres e.g. knowledge of factors. (WM)

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Strand based decisions: Area Stage 3 Foundation Statements (Years 5 & 6) 2005 Students select and use the appropriate unit to estimate, measure and calculate length, area, volume, capacity and mass. Outcome MS3.2: Selects and uses the appropriate unit to calculate area, including the area of squares, rectangles and triangles Expectation: Students develop formulae in words for finding area of squares, rectangles and triangles. Students explain the relationship between the length, breadth and area of rectangles. Students investigate the area of rectangles that have the same perimeter. WM

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A measurement task Make a rectangle with an area of 24 square centimetres. How many rectangles can you make that have an area of 24 square centimetres? Explain how you made different rectangles with the same area. (Teaching measurement Stage 2 and Stage 3 p. 62) Discuss: What is the expected standard for Stage 2?

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An initial sort Look through the student work samples and, as a group, put them into three piles: Sound Below Above Discuss: What questions do you need to answer to be able to do this?

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What is the basis for your decision? BelowSoundAbove (Name of student) This work sample is a C because…

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Stage 2 Area Key Idea: Estimate, measure, compare and record areas in square centimetres and square metres (MS2.2) Students learn about: recording area in square centimetre e.g. 24 square centimetres measuring a variety of surfaces using a square centimetre overlay Students learn to: use efficient strategies for counting large numbers of square centimetres e.g. strips of ten or squares of 100 Foundation Statements (Years 3 & 4) 2005 Students estimate, measure, compare and record length, area, volume, capacity and mass using some formal units. Students ask questions and use appropriate mental or written strategies, and technology, to solve problems. They use appropriate technology to describe and link mathematical ideas, check statements for accuracy and explain reasoning.

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Descriptions of achievement A Outstanding The student has an extensive knowledge and understanding of the content and can readily apply this knowledge. In addition, the student has achieved a very high level of competence in the processes and skills and can apply these skills to new situations. B High The student has a thorough knowledge and understanding of the content and a high level of competence in the processes and skills. In addition, the student is able to apply this knowledge and these skills to most situations. C Sound The student has a sound knowledge and understanding of the main areas of content and has achieved an adequate level of competence in the processes and skills. D Basic The student has a basic knowledge and understanding of the content and has achieved a limited level of competence in the processes and skills E Limited The student has an elementary knowledge and understanding in few areas of the content and has achieved very limited competence in some of the processes and skills

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What is the basis for your decision? BelowSoundAbove Lincoln: two incorrect examples and two correct examples which are the same; possibly understands the concept of area as the measure of a surface. Lachlan: no correct examples or explanation of what he has done; no understanding of the task although he has made different sized rectangles. Justin: recorded all possible examples that use whole numbers; error in drawing 2 x 12; demonstrated knowledge and understanding of factors. Annika: understands the concept of area; she explained how she found three solutions; she crossed out when she found she already had 4 x 6; she attempted 24 x 1 but ran out of space. Edwin: found all possible examples that use whole numbers; understands factors and square centimetres; correct labelling of length of sides of rectangles and use of symbolic representation.

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What is the basis for your decision? BelowSoundAbove Claire: made and labelled all possible rectangles using whole numbers; understands factors and use of the abbreviation cm 2 Jessica: found and explained factors for 24; knew that she could have made 24 x 1 if she had more room on the page; incorrect use of symbolic representation when recording the dimensions of the sides of the rectangles. Jai: found most possible solutions using whole numbers; incorrect use of symbolic representation; provided extra information to show his understanding of perimeter.

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Work sample: Anika

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Work sample: Claire

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Work sample: Edwin

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Work sample: Jai

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Work sample: Jessica

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Work sample: Justin

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Work sample: Lachlan

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Work sample: Lincoln

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Where to from here? Teachers work in stage/year groups and follow the same process with other sub strands of mathematics using work samples from their students.

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