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**Implementing Mathematics K-6 **

Using the syllabus for consistency of assessment and moderation The focus for this workshop will be to engage in a moderation process that will look at work samples from the Area sub strand of Mathematics. Preparation: have copies of the work samples available for teachers to work together in across stage groups. © 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 Discuss

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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. Read and discuss. The message is that assessment practices should model what we believe is important for our students to learn. It relates to the quality teaching question: Why does the learning matter?

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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) Read and discuss. The message is that assessment practices should model what we believe is important for our students to learn. It relates to the quality teaching question: Why does the learning matter?

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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. Discuss

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**What do we mean by consistency?**

Consistency of teacher professional judgement refers to the degree to which judgements about a student’s performance are independent of which teacher is assessing the student. Discuss: What do you understand from this statement?

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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. Discuss

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**What is consistency of moderation?**

Consistency of teacher professional judgement refers to the degree to which judgements about a student’s performance are independent of which teacher is assessing the student. Discuss: What do you understand from this statement?

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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. Discuss

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Some challenges In studying children’s 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 Discuss: Point-in-time decisions, collection of a wide range of data, the relevance of observational assessment, the importance of teacher professional judgement. Observation is not always informal assessment as informal assessment is that which is not used to make judgements on achievement.

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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. Discuss the importance of collecting a range of information, formal and informal, about student achievement.

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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 Read and discuss. Teaching experience provides information of what students are expected to demonstrate at certain points in time. Professional discussion with colleagues clarifies understanding.

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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? Working Mathematically Students learn to Knowledge and skills Students learn about Quality teaching question #1: What do you want the students to learn? The syllabus provides a clear description of content and processes to be developed at each stage. The following workshop activity is based on this sub strand of the syllabus.

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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. Quality teaching question #2: Why does that learning matter? Explain the rationale for teaching area. Discuss the importance of making connections with measuring length and volume.

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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 cm2. Students will demonstrate working mathematically processes through applying strategies and communicating their reasons for being able to find all possible solutions using whole numbers. Quality teaching question #3: What are you going to get the students to do (or to produce)?

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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. Quality teaching question #4: How well do you expect them to do it? Explain that, as an introduction to the task, we will review the continuum of learning for the sub strand of Area.

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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) Early Stage one teachers explain what is taught and what students are expected to demonstrate in Kindergarten.

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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) Stage one teachers explain what is taught and what students are expected to demonstrate in Stage 1.

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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) Stage 2 teachers explain what is taught and what students are expected to demonstrate in Stage 2.

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**Strand based decisions: Area Stage 3**

Outcome MS3.2: Selects and uses the appropriate unit to calculate area, including the area of squares, rectangles and triangles Foundation Statements (Years 5 & 6) 2005 Students select and use the appropriate unit to estimate, measure and calculate length, area, volume, capacity and mass. 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 Stage 3 teachers explain what is taught and what students are expected to demonstrate in Stage 3.

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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) This measurement task can be found as part of a unit of work from the DET resource, 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 Explain the moderation exercise. Provide each across stage group with a set of the eight work samples. The next slide is an example of how the information is to be recorded. Information on the expected standard is on the following slides 21 and 22. 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?**

Below Sound Above (Name of student) This work sample is a C because… An example of how teachers can record their decisions.

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**Key Idea: Estimate, measure, compare and record areas in square **

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. This slide can be shown as a reference whilst teachers are looking at the work samples.

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**Descriptions of achievement**

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 This slide can be used by teachers to reflect on their decisions re the student work samples.

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**What is the basis for your decision?**

Below Sound Above 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. Show this slide after teachers have made and explained their decisions.

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**What is the basis for your decision?**

Below Sound Above Claire: made and labelled all possible rectangles using whole numbers; understands factors and use of the abbreviation cm2 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 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. She needs to learn how to record area using square centimetres and to discuss and compare areas using some mathematical terms.

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Work sample: Claire Claire: made and labelled all possible rectangles using whole numbers; she understands factors and use of the abbreviation cm2. She needs to justify how she knew that she had found all possible solutions using whole numbers.

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Work sample: Edwin 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. He may be able to extend his understanding of area to using fractions but the task, as described, does not ask for this. A follow up question for students like Edwin could be: Can you find more solutions? Edwin can extend his understanding of area to using metres in practical situations such as covering a floor area.

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Work sample: Jai Jai did not provide an explanation for his work. He found most possible solutions using whole numbers; incorrect use of symbolic representation; provided extra information to show his understanding of perimeter. Jai needs to learn how to use the abbreviation for square centimetres correctly and to discuss and compare areas using some mathematical terms.

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Work sample: Jessica Jessica: found and explained factors for 24; she 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. Jessica needs to learn the appropriate use of the abbreviation for square centimetres.

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Work sample: Justin Justin: recorded all possible examples that use whole numbers; error in drawing 2 x 12; demonstrated knowledge and understanding of factors and was able to compare areas using some mathematical terms. Justin needs to learn how to use the abbreviation for square centimetres.

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Work sample: Lachlan Lachlan: no correct examples or explanation of what he has done; no understanding of the task although he has made different sized rectangles. Lachlan requires extra support to use a grid overlay to find the area of simple shapes. He may need support with counting by multiples to record area of rectangles.

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Work sample: Lincoln Lincoln’s explanation, possibly, is: I did math with the boxes and that is how I did it and I knew the boxes were one centimetre. Lincoln: two incorrect examples and two correct examples which are the same; he possibly understands the concept of area as the measure of a surface. He needs more hands on work in estimating and measuring a variety of surfaces using a centimetre grid overlay.

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**Teachers work in stage/year groups and follow**

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. Ask for suggestions on what the next strategy will be. A-E decisions are based on a teacher’s professional judgement after many forms of assessment information have been gathered. A-E decisions need to be made at the four reporting points within a stage.

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The Mathematics section of the curriculum support website provides practical strategies for supporting student learning in Mathematics.

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See this section of the Mathematics K-6 web site for sample assessment tasks and moderated work samples. Additional material will be added to this site during 2007.

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