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Assessing the Common Core: Mathematics Practices and Content Standards

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### Assessing the Common Core: Mathematics Practices and Content Standards

Ted Coe, Ph.D., Grand Canyon University

December 3, 2013

- Ways of doing
- Ways of thinking
- Habits of thinking

http://www.youtube.com/watch?v=ZffZvSH285c

Learning Progressions in the Common Core

Ways of Thinking?http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/

http://ime.math.arizona.edu/progressions/http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/

http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdfhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

Image from Learning http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdfTrajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. Pp.48-49

Image from Learning http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdfTrajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. Pp.48-49

Example of focus and coherence error: Excessively literal http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdfreading.

Image from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. Pp.50-51

Ways of Thinkinghttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

Multiplicative comparison?

Where does “copies of” work in the curriculum?

Measurementhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

What do we mean when we talk about “measurement”?

Measurementhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- “Technically, a measurement is a number that indicates a comparison between the attribute of an object being measured and the same attribute of a given unit of measure.”
- Van de Walle (2001)

- But what does he mean by “comparison”?

Measurementhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- Determine the attribute you want to measure
- Find something else with the same attribute. Use it as the measuring unit.
- Compare the two: multiplicatively.

Measurementhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

Image from Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

http://tedcoe.com/math/circumferencehttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

Geometryhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- Area:

Geometryhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- Similarity

Pythagorean Theoremhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- Pythagorean Theorem

http://tedcoe.com/math/geometry/pythagorean-and-similar-triangles

Constant Ratehttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

http://tedcoe.com/math/algebra/constant-rate

Trigonometryhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- Right Triangle Trigonometry

http://tedcoe.com/math/geometry/similar-triangles

Irrational?http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

- Irrational Numbers

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),who probably discovered them while identifying sides of the pentagram.Thethen-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.

http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

Hippasushttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea,

http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

Hippasushttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans

http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

Hippasushttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”

“Too much math never killed anyone”http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

…except Hippasus

Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword.

http://en.wikipedia.org/wiki/Archimedes. 11/2/2012

The last words attributed to Archimedes are "Do not disturb my circles"

http://en.wikipedia.org/wiki/Archimedes. 11/2/2012

- Ways of doing my circles"
- Ways of thinking
- Habits of thinking

- Make sense of problems and persevere in solving them. my circles"
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriately tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
- Mathematic Practices from the CCSS/ACCRS

OK, so math is bigger than just “doing” my circles"

How do we assess these things?

Next-Gen Assessment “Claims” my circles"

PARCC Mathematics Update my circles"

May 2013

Assessment Design my circles" Mathematics, Grades 3-8 and High School End-of-Course

2 Optional Assessments/Flexible Administration

- End-of-Year
- Assessment
- Innovative, computer-based items
- Required

- Performance-Based
- Assessment (PBA)
- Extended tasks
- Applications of concepts and skills
- Required

- Mid-Year Assessment
- Performance-based
- Emphasis on hard-to-measure standards
- Potentially summative

- Diagnostic Assessment
- Early indicator of student knowledge and skills to inform instruction, supports, and PD
- Non-summative

PBA

EOY

MYA

4

PARCC Model Content Frameworks my circles"

Approach of the Model Content Frameworks for Mathematics

- PARCC Model Content Frameworks provide a deep analysis of the CCSS, leading to more guidance on how focus, coherence, content and practices all work together.
- They focus on framing the critical advances in the standards:
- Focus and coherence
- Content knowledge, conceptual understanding, and expertise
- Content and mathematical practices

- Model Content Frameworks for grades 3-8, Algebra I, Geometry, Algebra II, Mathematics I, Mathematics II, Mathematics III

Model Content Frameworks my circles" Grade 3 Example

How PARCC has been presenting Evidence-Centered Design (ECD) my circles"

ECD is a deliberate and systematic approach to assessment development that will help to establish the validityof the assessments, increase the comparability of year-to year results, and increase efficiencies/reduce costs.

Claims Driving Design: Mathematics my circles"

Students are on-track or ready for college and careers

MP: 3,6

MP: 4

How PARCC has been presenting Evidence-Centered Design (ECD) my circles"

ECD is a deliberate and systematic approach to assessment development that will help to establish the validityof the assessments, increase the comparability of year-to year results, and increase efficiencies/reduce costs.

Overview of Evidence Statements: my circles" Types of Evidence Statements

Several types of evidence statements are being used to describe what a task should be assessing, including:

- Those using exact standards language
- Those transparently derived from exact standards language, e.g., by splitting a content standard
- Integrative evidence statements that express plausible direct implications of the standards without going beyond the standards to create new requirements
- Sub-claim C and D evidence statements, which put MP.3, 4, 6 as primary with connections to content

Overview of Evidence Statements: Examples my circles"

Several types of evidence statements are being used to describe what a task should be assessing, including:

- Those using exact standards language

Overview of Evidence Statements: Examples my circles"

Several types of evidence statements are being used to describe what a task should be assessing, including:

- Those transparently derived from exact standards language, e.g., by splitting a content standard

Overview of Evidence Statements: Examples my circles" Several types of evidence statements are being used to describe what a task should be assessing, including:

- Integrative evidence statements that express plausible direct implications of the standards without going beyond the standards to create new requirements

Overview of Evidence Statements: Examples my circles" Several types of evidence statements are being used to describe what a task should be assessing, including:

- Sub-claim C & Sub-claim D Evidence Statements, which put MP.3, 4, 6 as primary with connections to content

How PARCC has been presenting Evidence-Centered Design (ECD) my circles"

ECD is a deliberate and systematic approach to assessment development that will help to establish the validityof the assessments, increase the comparability of year-to year results, and increase efficiencies/reduce costs.

Overview of Task Types my circles"

- The PARCC assessments for mathematics will involve three primary types of tasks: Type I, II, and III.
- Each task type is described on the basis of several factors, principally the purpose of the task in generating evidence for certain sub-claims.

Source: Appendix D of the PARCC Task Development ITN on page 17

Overview of PARCC Mathematics Task Types my circles"

For more information see PARCC Task Development ITN Appendix D.

Design of PARCC Math Summative Assessment my circles"

- Performance Based Assessment (PBA)
- Type I items (Machine-scorable)
- Type II items (Mathematical Reasoning/Hand-Scored – scoring rubrics are drafted but PLD development will inform final rubrics)
- Type III items (Mathematical Modeling/Hand-Scored and/or Machine-scored - scoring rubrics are drafted but PLD development will inform final rubrics)

- End-of-Year Assessment (EOY)
- Type I items only (All Machine-scorable)

Factors that determine the Cognitive Complexity of PARCC Mathematics Items

- Mathematical Content
- Mathematical Practices
- Stimulus Material
- Response Mode
- Processing Demand

For further reading on the PARCC Cognitive Complexity Framework see, “ Proposed Sources of Cognitive Complexity in PARCC Items and Tasks: Mathematics “ Aug. 31, 2012

1. Mathematical Mathematics ItemsContent

At each grade level, there is a range in the level of demand in the content standards--from low to moderate to high complexity. Within Mathematical Content, complexity is affected by:

- Numbers: Whole numbers vs. fractions
- Expressions and Equations: The types of numbers or operations in an expression or equation ( 3/7, √ )
- Diagrams, graphs, or other concrete representations: may contribute to greater overall complexity than simpler graphs such as scatterplots.
- Problem structures: Word problems with underlying algebraic structures vs. word problems with underlying arithmetic structures.

2. Mathematical Mathematics ItemsPractices

MPs involve what students are asked to do with mathematical content, such as engage in application and analysis of the content. The actions that students perform on mathematical objects also contribute to Mathematical Practices complexity.

Low Complexity

- Items at this level primarily involve recalling or recognizing concepts or procedures specified in the Standards.
High Complexity

- High complexity items make heavy demands on students, because students are expected to use reasoning, planning, synthesis, analysis, judgment, and creative thought. They may be expected to justify mathematical statements or construct a formal mathematical argument.

3. Stimulus Mathematics ItemsMaterial

This dimension of cognitive complexity accounts for the number of different pieces of stimulus material in an item, as well as the role of technology tools in the item.

Low Complexity

- Low complexity involves a single piece of (or no) stimulus material (e.g., table, graph, figure, etc.) OR single online tool (generally, incremental technology)
High Complexity

- High complexity involves two pieces of stimulus material with online tool(s) OR three pieces of stimulus material with or without online tools.

4. Response Mathematics ItemsMode

The way in which examinees are required to complete assessment activities influences an item’s cognitive complexity.

- Low cognitive complexity response modes in mathematics involve primarily selecting responses and producing short responses, rather than generating more extended responses.
- High Complexity response modes require students to construct extended written responses that may also incorporate the use of online tools such as an equation editor, graphing tool, or other online feature that is essential to responding.

5. Processing Mathematics ItemsDemand

Reading load and linguistic demands in item stems, instructions for responding to an item, and response options contribute to the cognitive complexity of items.

PARCC Content Specific Performance Level Descriptors (PLDs) Mathematics Items

- The PARCC PLD writing panels consisted of educators from across the PARCC States.
- The PARCC PLD writing panels were focused on staying true to the CCSS.
- The foundation of the PARCC PLDs are the PARCC Evidence Statements and the PARCC Cognitive Complexity Framework.

Capturing What Students Can Do Mathematics Items

- PARCC PLDs
- capture how all students perform
- show understandings and skill development across the spectrum of standards and complexity levels assessed

Looking at the PLDs Mathematics Items

Gives the Conceptual Concept the PLD is based on

Gives the PLD by performance level ranging from 2-5. Level 1 indicates a range from no work shown to Minimalcommand

Gives the Sub-Claim that the PLD is written for (A-Major Content)

High-Level Blueprints Mathematics Items

http://www.parcconline.org/sites/parcc/files/PARCC%20High%20Level%20Blueprints%20-%20Mathematics%20043013.pdfhttp://www.parcconline.org/sites/parcc/files/PARCC%20High%20Level%20Blueprints%20-%20Mathematics%20043013.pdf

- PBA and EOYhttp://www.parcconline.org/sites/parcc/files/PARCC%20High%20Level%20Blueprints%20-%20Mathematics%20043013.pdf
- Claim and Subclaims
- Task Types
- MCF
- Evidence Tables
- Complexity Framework
- PLDs
- High Level Blueprint

One (imperfect) Interpretation:http://www.parcconline.org/sites/parcc/files/PARCC%20High%20Level%20Blueprints%20-%20Mathematics%20043013.pdf

http://tedcoe.com/math/scc/making-sense-of-parcc

Grab a notebook and…http://www.parcconline.org/sites/parcc/files/PARCC%20High%20Level%20Blueprints%20-%20Mathematics%20043013.pdf

- Complete the task
- Determine accuracy of an alignment to an evidence statement.
- Complexity: Where might this task lie along the complexity of the PLD?
- What might a task look like to elicit a higher or lower level score?

http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdfhttp://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdf

http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdfhttp://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdf

- Grab an official PARCC prototype item. http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdf
- Determine:
- (a) Grade Level
- (b) Question Type
- (c) Bonus! Predict the Evidence Statement
- (d) Check your guesses.
- (e) Goodness of fit. Does the item truly show evidence of the statement?

- Grab a PARCC evidence statement http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdf
- Bonus: Pick a type “II” or “III” task.

- Develop a task that provides such evidence.
- Be ready to share and to have your ideas challenged.
- Bonus: Try to include a rubric.

- Implications for Practice?http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdf
- Challenges?
- Questions?

ted.coe@gcu.eduhttp://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMichelleConjecture_081913_Final_0.pdf

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