A Smarter Balanced System for Supporting Mathematics Teaching and Learning

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Shelbi K. Cole, Ph.D.Saint Michael’s College

March 7, 2014

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

Slide 3

What does his future look like?

What skill set do his future employers value?

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

Student populations are transient.

"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.

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: www.corestandards.org

Smarter Balanced Assessment ConsortiumAn Assessment System to Support Teaching and Learning

A National Consortium of States

The Assessment Challenge

How do we get from here... ...to here?

All studentsleave 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?

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 careerBased on state standards

• Inadequate measures of complex skills and deep understandingHeavy 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 securityMost administered on paper

Theory of Action Built on Seven Key Principles

1. An integrated system2. Evidence-based approach3. Teacher involvement4. State-led with transparent governance5. Focus: improving teaching and learning6. Actionable information – multiple

measures7. Established professional standards

A Balanced Assessment System

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

learningInterim assessments

Flexible, open, used for actionable

feedback

Summative assessments

Benchmarked to college and career

readiness

Teacher resources for formative

assessment practices

to improve instruction

Using Computer Adaptive Technology for Summative and Interim Assessments

• Provides accurate measurements of student growth over timeIncreased precision

• Item difficulty based on student responsesTailored for Each

Student

• Larger item banks mean that not all students receive the same questionsIncreased Security

• Fewer questions compared to fixed form testsShorter Test Length

• Turnaround time is significantly reducedFaster Results

• GMAT, GRE, COMPASS (ACT), Measures of Academic Progress (MAP)Mature Technology

How CAT Works (Binet’s Test)

Rating Item Difficulty

A) 9 x 4 = 2 x □ B) 9 x 4 = □ x 9C) 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?

K-12 Teacher Involvement

• Support for implementation of the Common Core State Standards (2011-12)

• Write and review items/tasks for the pilot test (2012-13) and field test (2013-14)

• Development of teacher leader teams in each state (2012-14)

• Evaluate formative assessment practices and curriculum tools for inclusion in digital library (2013-14)

• Score portions of the interim and summative assessments (2014-15 and beyond)

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

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

Example Grade 11

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Usability, Accessibility, Accommodations

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.

New Graphic

General Table, Appendix A

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

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Guidelines & Frameworks

• Smarter Balanced Usability, Accessibility and Accommodations Guidelines http://www.smarterbalanced.org/wordpress/wp-content/uploads/2013/09/SmarterBalanced_Guidelines_091113.pdf

• Smarter Balanced Translation Frameworkhttp://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/Translation-Accommodations-Framework-for-Testing-ELL-Math.pdf

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Focus, Coherence & Rigor in the Smarter Balanced Assessments

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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

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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

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 FocusThe shape of math in A+ countries

1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002). 32

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GradeFocus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding

K–2Addition and subtraction - concepts, skills, and problem solving and place value

3–5Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving

6Ratios and proportional reasoning; early expressions and equations

7Ratios and proportional reasoning; arithmetic of rational numbers

8 Linear algebra and linear functions

Shift #1: Focus Key Areas of Focus in Mathematics

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.

Shift #1: FocusContent Emphases by Cluster

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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.

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

1313

1?13

25

23

110

46

??

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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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Shift #2: CoherenceThink 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?

BAD BAD BAD

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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.

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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.

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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, 1989

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What is ?

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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.

BA C D

332

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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.

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.

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.

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.

Grade 7 – Number Line

Grade 8 – Number Line

Shift #3: Rigor In Major Topics, Pursue Conceptual

Understanding, Procedural Skill and Fluency, and Application

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• 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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Grade Standard Required FluencyK K.OA.5 Add/subtract within 5

1 1.OA.6 Add/subtract within 10

2 2.OA.22.NBT.5

Add/subtract within 20 (know single-digit sums from memory)Add/subtract within 100

3 3.OA.73.NBT.2

Multiply/divide within 100 (know single-digit products from memory)Add/subtract within 1000

4 4.NBT.4 Add/subtract within 1,000,000

5 5.NBT.5 Multi-digit multiplication

6 6.NS.2,3 Multi-digit divisionMulti-digit decimal operations

Shift #3: RigorRequired Fluencies for Grades K-6

• 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.

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Shift #3: RigorApplication

Fluency

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Application

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Conceptual Understanding

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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)

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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

Find Out More

Smarter Balanced can be found online at:

SmarterBalanced.org

Contact Information: sbac@wested.org 59