A Smarter Balanced System for Supporting Mathematics Teaching
and Learning 1 Shelbi K. Cole, Ph.D. Saint Michaels 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 peoplewill now reap
societys 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: www.corestandards.org
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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 (Binets 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 (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)
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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
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 K2 Addition
and subtraction - concepts, skills, and problem solving and place
value 35 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 .
Based on this: Draw an area that represents . 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 . 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, 1989 42
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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 . 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 .
Since 9/2 is 9 pieces of size , 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 10 2 2.OA.2 2.NBT.5 Add/subtract within
20 (know single-digit sums from memory) Add/subtract within 100 3
3.OA.7 3.NBT.2 Multiply/divide within 100 (know single-digit
products from memory) Add/subtract within 1000 44.NBT.4Add/subtract
within 1,000,000 55.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 peoplewill now reap
societys 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: [email protected] 59
