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A Smarter Balanced System for Supporting Mathematics Teaching and Learning 1 Shelbi K. Cole, Ph.D. Saint Michael’s College March 7, 2014

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The Big Picture “The future belongs to a very different kind of person with a very different kind of mind – creators and empathizers, pattern recognizers and meaning makers. These people…will now reap society’s richest rewards and share its greatest joys.” Daniel H. Pink, A Whole New Mind

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Slide 3 What does his future look like?

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What skill set do his future employers value?

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But are we modeling the collaboration that we need to be teaching kids?

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Student populations are transient.

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"The world is small now, and we're not just competing with students in our county or across the state. We are competing with the world," said Robert Kosicki, who graduated from a Georgia high school this year after transferring from Connecticut and having to repeat classes because the curriculum was so different. "This is a move away from the time when a student can be punished for the location of his home or the depth of his father's pockets." Excerpt from Fox News, Associated Press. (June 2, 2010) States join to establish 'Common Core' standards for high school graduation.

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Common Core State Standards Define the knowledge and skills students need for college and career Developed voluntarily and cooperatively by states; more than 40 states have adopted Provide clear, consistent standards in English language arts/literacy and mathematics Source:

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Smarter Balanced Assessment Consortium An Assessment System to Support Teaching and Learning

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A National Consortium of States

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The Assessment Challenge How do we get from here......to here? All students leave high school college and career ready Common Core State Standards specify K-12 expectations for college and career readiness...and what can an assessment system do to help?

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Concerns with Today's Statewide Assessments Each state bears the burden of test development; no economies of scale Each state pays for its own assessments Students in many states leave high school unprepared for college or career Based on state standards Inadequate measures of complex skills and deep understanding Heavy use of multiple choice Tests cannot be used to inform instruction or affect program decisions Results delivered long after tests are given Difficult to interpret meaning of scores; concerns about access and fairness Accommodations for special education and ELL students vary Costly, time consuming, and challenging to maintain security Most administered on paper

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Theory of Action Built on Seven Key Principles 1.An integrated system 2.Evidence-based approach 3.Teacher involvement 4.State-led with transparent governance 5.Focus: improving teaching and learning 6.Actionable information – multiple measures 7.Established professional standards

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A Balanced Assessment System Common Core State Standards specify K-12 expectations for college and career readiness Common Core State Standards specify K-12 expectations for college and career readiness All students leave high school college and career ready Teachers and schools have information and tools they need to improve teaching and learning Interim assessments Flexible, open, used for actionable feedback Summative assessments Benchmarked to college and career readiness Teacher resources for formative assessment practices to improve instruction

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Using Computer Adaptive Technology for Summative and Interim Assessments Provides accurate measurements of student growth over time Increased precision Item difficulty based on student responses Tailored for Each Student Larger item banks mean that not all students receive the same questions Increased Security Fewer questions compared to fixed form tests Shorter Test Length Turnaround time is significantly reduced Faster Results GMAT, GRE, COMPASS (ACT), Measures of Academic Progress (MAP) Mature Technology

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How CAT Works (Binet’s Test)

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Rating Item Difficulty A)9 x 4 = 2 x □ B) 9 x 4 = □ x 9 C) 4 x □ = 40 – 8 D) 8 x 5 = □ E)Put a different number in each box to make the equation true. 8 x □ = 4 x □ F) 8 x □ = 40

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Responding to Common Misconceptions about Adaptive Testing Can students return to previous questions if the test is adaptive? Will the test give students questions from higher and lower grades if they are performing very high or very low? –Do all adaptive tests work this way? 18

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K-12 Teacher Involvement Support for implementation of the Common Core State Standards ( ) Write and review items/tasks for the pilot test ( ) and field test ( ) Development of teacher leader teams in each state ( ) Evaluate formative assessment practices and curriculum tools for inclusion in digital library ( ) Score portions of the interim and summative assessments ( and beyond)

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Higher Education Collaboration Involved 175 public and 13 private systems/institutions of higher education in application Two higher education representatives on the Executive Committee Higher education lead in each state and higher education faculty participating in work groups Goal: The high school assessment qualifies students for entry-level, credit- bearing coursework in college or university

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Performance Tasks The use of performance measures has been found to increase the intellectual challenge in classrooms and to support higher-quality teaching. - Linda Darling-Hammond and Frank Adamson, Stanford University

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Example Grade 11 22

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23 Usability, Accessibility, Accommodations

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Universal Tools, Designated Supports, and Accommodations Universal tools are access features of the assessment that are either provided as digitally-delivered components of the test administration system or separate from it. Universal tools are available to all students based on student preference and selection. Designated supports for the Smarter Balanced assessments are those features that are available for use by any student for whom the need has been indicated by an educator (or team of educators with parent/guardian and student). Accommodations are changes in procedures or materials that increase equitable access during the Smarter Balanced assessments. They are available for students for whom there is documentation of the need for the accommodations on an Individualized Education Program (IEP) or 504 accommodation plan.

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New Graphic

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General Table, Appendix A

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Text-to-Speech On Items Designated support for math items Designated support for ELA items On ELA Reading Passages Grades 3-5, TTS for passages is not available Grades 6-HS: for passages available accommodation for students whose need is documented in an IEP or 504 plan 27

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Guidelines & Frameworks Smarter Balanced Usability, Accessibility and Accommodations Guidelines s/wp- content/uploads/2013/09/SmarterBalance d_Guidelines_ pdf s/wp- content/uploads/2013/09/SmarterBalance d_Guidelines_ pdf Smarter Balanced Translation Framework s/wp- content/uploads/2012/09/Translation- Accommodations-Framework-for-Testing- ELL-Math.pdf 28

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Focus, Coherence & Rigor in the Smarter Balanced Assessments 29

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The CCSSM Requires Three Shifts in Mathematics Focus strongly where the standards focus Coherence: Think across grades and link to major topics within grades Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application with equal intensity 30

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“Students can demonstrate progress toward college and career readiness in mathematics.” “Students can demonstrate college and career readiness in mathematics.” “Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.” “Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.” “Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.” “Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.” Overall Claim for Grades 3-8 Overall Claim for Grade 11 Claim #1 - Concepts & Procedures Claim #2 - Problem Solving Claim #3 - Communicating Reasoning Claim #4 - Modeling and Data Analysis Claims for the Mathematics Summative Assessment

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Mathematics topics intended at each grade by at least two- thirds of A+ countries Mathematics topics intended at each grade by at least two- thirds of 21 U.S. states Shift #1: Focus Strongly where the Standards Focus The shape of math in A+ countries 1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002). 32

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33 Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction - concepts, skills, and problem solving and place value 3–5 Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra and linear functions Shift #1: Focus Key Areas of Focus in Mathematics

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Grade 7 Example In grades 6 and 7, proportional relationships are a crucial pivot from multiplicative reasoning to functional thinking; sets stage for 8.F. Meanwhile, probability in grade 7 has potentially misleading grain size— uses almost twice as many words as for proportional relationships standards.

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Shift #1: Focus Content Emphases by Cluster 35 The Smarter Balanced Content Specifications help support focus by identifying the content emphasis by cluster. The notation [m] indicates content that is major and [a/s] indicates content that is additional or supporting.

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At what grade should students be able to solve these problems?

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37 Shift #2: Coherence Think Across Grades, and Link to Major Topics Within Grades Carefully connect the learning within and across grades so that students can build new understanding on foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

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38 Shift #2: Coherence Think Across Grades 4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 6.NS. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Grade 4 Grade 5 Grade 6

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Beyond the Number Line: Other Representations that Support Student Understanding of Fractions What fraction is represented by the shaded area? 39

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Do students really understand unit fractions and the concept of one “whole”? The fraction represented by the green shaded area is ¾. Based on this: –Draw an area that represents ¼. –Draw an area that represents 1. 40

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Do students really understand the concept of the unit fraction and of one “whole”? The fraction represented by the green shaded area is 3/2. Based on this: –Draw an area that represents ½. –Draw an area that represents 1. 41

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Linking Operations with Fractions to Operations with Whole Numbers “Children must adopt new rules for fractions that often conflict with well- established ideas about whole number” (p.156) Bezuk & Cramer,

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What is ? 43

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Fractions Example The shaded area represents. Which figures from below can you use to build a model that represents ? You may use the same figure more than once. B A C D

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Slide 45 Student A drags three of shape B, which is equal in area to the shaded region. This student probably has good understanding of cluster 5.NF.B he knows that 3 x 3/2 is equal to 3 iterations of 3/2. Calculation of the product is not necessary because of the sophisticated understanding of multiplication. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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Slide 46 Student B reasons that 3 x 3/2 = 9/2 = 4 ½. She correctly reasons that since the shaded area is equal to 3/2, the square is equal to one whole, and drags 4 wholes plus half of one whole to represent the mixed number. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Note that unlike the previous chain of reasoning, this requires that the student determines how much of the shaded area is equal to 1.

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Slide 47 Student C multiplies 3 x 3/2 = 9/2. She reasons that since the shaded area is 3/2, this is equal to 3 pieces of size ½. Since 9/2 is 9 pieces of size ½, she drags nine of the smallest figure to create her model. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. This chain of reasoning links nicely back to the initial development of 3/2 in 3.NF.1 “understand a fraction a/b as the quantity formed by a parts of size 1/b, illustrating the coherence in the standards across grades 3-5.

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Grade 3 – Number Line 3.NF.A.3b Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

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Grade 7 – Number Line

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Grade 8 – Number Line

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Shift #3: Rigor In Major Topics, Pursue Conceptual Understanding, Procedural Skill and Fluency, and Application 51 The CCSSM require a balance of: Solid conceptual understanding Procedural skill and fluency Application of skills in problem solving situations Pursuit of all threes requires equal intensity in time, activities, and resources.

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52 GradeStandardRequired Fluency KK.OA.5Add/subtract within 5 11.OA.6Add/subtract within OA.2 2.NBT.5 Add/subtract within 20 (know single-digit sums from memory) Add/subtract within OA.7 3.NBT.2 Multiply/divide within 100 (know single-digit products from memory) Add/subtract within NBT.4Add/subtract within 1,000, NBT.5Multi-digit multiplication 66.NS.2,3 Multi-digit division Multi-digit decimal operations Shift #3: Rigor Required Fluencies for Grades K-6

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Students can use appropriate concepts and procedures for application even when not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-5, 6-8, and HS. Teachers in content areas outside of math, particularly science, ensure that students are using grade-level-appropriate math to make meaning of and access science content. 53 Shift #3: Rigor Application

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Fluency 54

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Application 55

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Conceptual Understanding 56

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Smarter Balanced Collaborations CTB, American Institutes for Research, & DRC Illustrative Mathematics Khan Academy Desmos National Center for Research on Evaluation, Standards, & Student Testing at UCLA (CRESST) 57

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The Big Picture “The future belongs to a very different kind of person with a very different kind of mind – creators and empathizers, pattern recognizers and meaning makers. These people…will now reap society’s richest rewards and share its greatest joys.” Daniel H. Pink, A Whole New Mind

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Find Out More Smarter Balanced can be found online at: SmarterBalanced.org Contact Information: 59

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