Preparing K-5 Students for the Focus,
Coherence and Rigor of the Common Core
State Standards for Mathematics
1
Shelbi K. C...
"The world is small now, and we're not just competing
with students in our county or across the state. We
are competing wi...
Common Core State Standards
• Define the
knowledge and
skills students need
for college and
career
• Developed
voluntarily...
What questions will we try to answer today?
• How can teachers plan lessons that take
into account the shifts in the CCSS-...
The Three Shifts:
What do they really mean?
5
The Background of the Common Core
Initiated by the National Governors Association
(NGA) and Council of Chief State School ...
The CCSSM Requires Three Shifts in Mathematics
• Focus strongly where the
standards focus
• Coherence: Think across
grades...
Mathematics
topics
intended at
each grade by
at least two-
thirds of A+
countries
Mathematics
topics
intended at
each grad...
9
Shift #1: Focus
Traditional U.S. Approach
K 12
Number and
Operations
Measurement
and
Geometry
Algebra and
Functions
Stat...
10
Grade
Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual
Understanding
K–2
Addition ...
Shift #1: Focus
Content Emphases by Cluster
11
The Smarter Balanced Content Specifications help support
focus by identifyi...
12
Shift #2: Coherence
Think Across Grades, and Link to Major Topics Within
Grades
• Carefully connect the learning within...
Shift #2: Coherence
Think Across Grades
Example: Fractions
“The coherence and sequential nature of mathematics
dictate the...
14
Shift #2: Coherence
Think Across Grades
4.NF.4. Apply and extend previous understandings of
multiplication to multiply ...
15
Shift #2: Coherence
Link to major work within grade
Example: Data
Representation
Standard 3.MD.3
Shift #3: Rigor
In Major Topics, Pursue Conceptual Understanding,
Procedural Skill and Fluency, and Application
16
• The C...
17
Grade Standard Required Fluency
K K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2
2.OA.2
2.NBT.5
Add/sub...
• Students can use appropriate concepts and
procedures for application even when not prompted
to do so.
• Teachers provide...
Resources
• www.achievethecore.org
• http://corestandards.org/assets/Math_Publishers_Criteria_K-
8_Summer%202012_FINAL.pdf...
Five Basic Characteristics to Support
Quality Mathematics Teaching
• Precision: Mathematical statements are clear and
unam...
“Just as technicians in any kind of
engineering must have a „feel‟ for their
profession in order to avert disasters in the...
22
Topic 1: Fractions
The Progression Across Grades 3-5
Definition of “fraction”
23
3.NF.A Develop understanding of
fractions as numbers.
• On a piece of paper, draw a line segment that is 6
inches long.
• ...
3.NF.A Develop understanding of
fractions as numbers.
• Next, label all fractions with denominator
3 along the part of the...
Which standards were addressed in the
activity so far?
• 3.NF.A Develop understanding of fractions as numbers.
• 3.NF.A.1....
Which aspects of this standard have
been addressed so far?
• 3.NF.A.3. Explain equivalence of fractions in special cases, ...
28
A Brief Detour
The Parallel Curriculum Model
What is the Parallel Curriculum Model?
The Parallel Curriculum Model is a set
of four interrelated designs that can be
use...
The Parallel Curriculum Model
CURRICULUM
OF CONNECTIONS
CURRICULUM OF
PRACTICE
CURRICULUM
OF
IDENTITY
CORE
CURRICULUM
Why Four Parallels?
• Qualitatively differentiated
curriculum isn‟t achieved by
doing only one thing or one
kind of thing....
Four Facets of Qualitatively Differentiated
Curriculum
• Core: The essential nature of a discipline
• Connections: The rel...
Concepts, Principles and Skills
• Concepts: A general idea or
understanding, especially a generalized
idea of a thing or c...
Building a CCSS-Aligned Lesson
• Discuss the number line activity.
– How would you implement a similar activity
with stude...
Learning from the Group
DON‟T start filling in the lesson template
before hearing the GOOD IDEAS of other
educators workin...
Work Time
• Begin to populate the template with some
of the ideas we‟ve heard that represent
important learning opportunit...
Increasing Engagement, Thinking Across
Grades
• Grade 1 – 1.OA.A (standards 5&6)
• Grade 2 – 2.MD.B (standards 5&6)
• Grad...
• And the number line remains relevant
even after grade 5…
38
Grade 3
3.NF.A.3b Express whole numbers as fractions, and recognize fractions that are
equivalent to whole numbers. Exampl...
Grade 7
Grade 8
Beyond the Number Line: Other
Representations that Support Student
Understanding of Fractions
• What fraction is represent...
Notice the use of “represented.” The
shaded area is not “equal” to a fraction.
• The fraction represented by the shaded ar...
Linking Operations with Fractions to
Operations with Whole Numbers
“Children must adopt new rules for
fractions that often...
Fractions Example
The shaded area represents 3/2. Drag the figures below to
make a model that represents 3 x 3/2.
BA C D
Slide 46
Student A drags three of shape B, which is equal in area to the shaded region. This
student probably has good und...
Slide 47
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...
Slide 48
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 s...
How can K-2 work on operations with whole numbers
and work on fractions in grade 3 support students’
thinking about these ...
At what grade should students be able to
solve these problems?
1
3
1
3
1
?
1
3
2
5
2
3
1
10
4
6
?
?
Explain the flaw in the chain of reasoning.
In the first group, 3/5 of the cats have spots. In the second group,
1/3 of th...
Closing Task – Fractions
• You have 4 strips of paper each 1 foot in
length.
– Partition one into thirds.
– Partition one ...
53
Topic 2: Whole Numbers
0
3/2
1 + 3 = ?
Operations with fractions should be a natural
extension of operations with whole numbers
54
2 4 5
0
1 3
1
...
55
Future impact
1.OA.B Understand and apply properties of
operations and the relationship between
addition and subtraction.
...
Future impact
1.OA.C Work with addition and subtraction
equations.
1.OA.C.7. Understand the meaning of the equal sign,
and...
Within grade impact
• Look at the grade 2 standards in the
domains NBT and OA. Discuss how the
standards in NBT might be i...
Smarter Balanced Assessment Consortium
An Assessment System to Support Teaching and Learning
The Assessment Challenge
How do we get from here... ...to here?
All students
leave high school
college and
career ready
Co...
Concerns with Today's Statewide Assessments
• Each state bears the burden of test development;
no economies of scale
Each ...
The Purpose of the Consortium
• To develop a comprehensive and innovative
assessment system for grades 3-8 and high school...
• 26 member
states and
territories
representing
39% of K-12
students
• 22 Governing
States, 3
Advisory States,
1 Affiliate...
Theory of Action Built on Seven Key
Principles
1. An integrated system
2. Evidence-based approach
3. Teacher involvement
4...
A Balanced Assessment System
Common
Core State
Standards
specify
K-12
expectations
for college
and career
readiness
All st...
A Balanced Assessment System
School Year Last 12 weeks of the year*
DIGITAL LIBRARY of formative tools, processes and exem...
Using Computer Adaptive Technology for
Summative and Interim Assessments
• Provides accurate measurements of student growt...
What is CAT?
Administered by computer, a
Computerized Adaptive Test (CAT)
dynamically adjusts to the trait level of
each e...
How CAT Works (Binet’s Test)
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) 8 x □ = 4 x □
Give two differen...
0
3/2
Put a point at the number 1.
Some Important “Constraints” in the
Smarter Balanced Summative
71
2 4
0 3
0
0
3/5
Grade 1
Add and subtract within 20.
6. Add and subtract within 20,
demonstrating fluency for addition and
subtraction with...
K-12 Teacher Involvement
• Support for implementation of the
Common Core State Standards
(2011-12)
• Write and review item...
Higher Education Collaboration
• Involved 175 public and 13 private
systems/institutions of higher education
in applicatio...
Assessment System Components
Summative Assessment (Computer Adaptive)
• Assesses the full range of Common Core in English
...
Assessment System Components
Interim Assessment (Computer Adaptive)
• Optional comprehensive and content-cluster assessmen...
Assessment System Components
• Extended projects demonstrate
real-world writing and analytical
skills
• May include online...
Assessment System Components
Few initiatives are
backed by evidence
that they raise
achievement.
Formative assessment
is o...
Assessment System Components
Data are only useful if
people are able to
access, understand and
use them… For
information t...
Support for Special Populations
• Accurate measures of
progress for students
with disabilities and
English Language
Learne...
“Students can demonstrate progress toward college and
career readiness in English Language arts and literacy.”
“Students c...
“Students can demonstrate progress toward college and career
readiness in mathematics.”
“Students can demonstrate college ...
April 2012 Item/task specifications and guidelines v1.0 completed
May 2012 Training materials for writing & reviewing item...
Viewing Sample Items & Tasks
• Preliminary Sample Items
– http://sampleitems.smarterbalanced.org/itemp
review/sbac/
• Full...
85
Usability, Accessibility,
Accommodations
Overall New Document
• Introduction
– Purpose and intended audience
– Briefly addresses Framework and ISAAP, includes
link...
Speech-to-Text
• Available as a documented accommodation on
all math and ELA items:
– Available for students providing a r...
American Sign Language
• Available for students who are hard of
hearing or deaf as a documented
accommodation
• Available ...
Closed Captioning
• Available for students who are hard of
hearing or deaf as a documented
accommodation
• Available on EL...
Calculator
• Universal embedded calculator: available on
items as per Smarter Balanced item
specifications and targets
• N...
Survey Feedback—Participation
91
Universal Tools, Designated Supports,
and Accommodations
• Universal tools are access features of the assessment that
are ...
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...
Guidelines & Frameworks
• Smarter Balanced Usability, Accessibility
and Accommodations Guidelines
http://www.smarterbalanc...
97
Making Items More Accessible
Drag and Drop
Equal to 5.42 NOT Equal to
5.42
2.36 + 3.06
1.80 x 3
2.16 + 3.36
9.53 – 4.11
2.71 x 2
8.01 – 2.69
Table
Translation
Translation
Translation
Translation
Embedded
Embedded
106
Important Notes on Item
Development
107
http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/For an essay on similar themes, see
Features of CC...
What are progressions?
Many or most of the content standards in K-8 represent
steps or stages along a progression of learn...
109
Features of CCSSM and Implications for Assessment
Assessing at All Levels of the Content Hierarchy
Standards in number...
110
Features of CCSSM and Implications for Assessment
Assessing at All Levels of the Content Hierarchy
Number and Operatio...
111
Features of CCSSM and Implications for Assessment
Assessing at All Levels of the Content Hierarchy
5.NBT.1 5.NBT.2 5.N...
• There are a finite set of key opportunities to assess at the
cluster, domain, and grade levels. Smarter Balanced is
expl...
Features of CCSSM and Implications for Assessment
Assessing at All Levels of the Content Hierarchy
http://tinyurl.com/math...
114
Features of CCSSM and Implications for Assessment
Minimizing pitfalls of traditional multiple choice
questions
• High-...
Some considerations: Risks of technology in
assessing of the Standards.
• Technology for technology‟s sake - The construct...
Grade 3
Grade 3
Grade 5
119
Fluency as a special case of assessing individual
content standards.
• Fluent in the standards means fast and accurate...
Topics for Discussion
1. Assessing individual content standards or parts of
standards
2. Assessing at all levels of the co...
“Designers of curricula, assessments, and
professional development should all attend to the
need to connect the mathematic...
Features of CCSSM and Implications for Assessment
Smarter Balanced’s Claims Embody Specific Mathematical
Practices in the ...
123
A Closer Look at Claim #3: Communicating Reasoning
Students can clearly and precisely construct viable
arguments to su...
124
Reasoning is a refrain in the content standards
Note generally such words as justify a conclusion, prove a statement,
...
125
Claim #3 tasks have fine “grain size.”
The Standards ask students not just to Reason, but to
“reason about X,” where X...
126
A Closer Look at Claims #2 and #4
Problem Solving
Students can solve a range of complex well-posed problems
in pure an...
127
Modeling is a mathematical practice (MP.4) – and in
high school, it is also a content category. Therefore
it is reason...
128
• Contextual word problems involving ideas that are currently at the
forefront of the student‟s developing mathematica...
Grade 5
Grade 5
Grade 5
Grade 3
Find Out More
Smarter Balanced
can be found
online at:
SmarterBalanced.org
Contact Information:
shelbi.cole@smarterbalance...
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Preparing K-5 Students for the Focus, Coherence and Rigor of the Common Core State Standards for Mathematics

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This is the powerpoint presentation presented by Shelbi Cole, Director of Mathematics at Smarter Balanced. Please contact Jill Bessette, April Schultz or Jackie Walsh if you would like to meet for an overview or a PLC session on the contents.

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  • Shelbi to present
  • Shelbi to hand off to BethBeth to present this section
  • {read slide}
  • Read the given definition of fractions. Use drawings and discussion to understand the whole definition. Why is it important to emphasize that no distinction is made between proper and improper fractions?
  • Explain which standards were addressed in the number line activity. Use this to discuss why treating the standards as a checklist results in an unconnected curriculum. Question to keep in mind: With standards based grading, which “level” of the hierarchy (domain, cluster standard) is appropriate for providing information about student learning?Which aspects of the standards might be further addressed using the same number line? What can we do to deepen student understanding at the cluster level (developing understanding of fractions as numbers)?
  • Describe how the first three are related to the shifts in the CCSS and the fourth (Identity) is built on understanding individual student characteristics to build engaging, relevant learning experiences.
  • Technology enhanced items allow for accurate and efficient scoring of items that would take much longer to score by hand. The degree of precision can be defined as an exact point or value or as a range. For example, this item might allow a student to place the point in the correct location with an error margin of 1/10 on either side.Also note that Smarter Balanced Items can be delivered across multiple grades to increase measurement precision, particularly high and low achievers.
  • Assign each table a grade that is 1, 2, 3, 4, etc. years later that is dependent on or related to the study of these standards in grade 1. Help participants understand what is intended by “coherence.”
  • Assessment system that balances summative, interim, and formative components for ELA and mathematics: Summative Assessment (Computer Adaptive)Mandatory comprehensive assessment in grades 3–8 and 11 (testing window within the last 12 weeks of the instructional year) that supports accountability and measures growthSelected response, short constructed response, extended constructed response, technology enhanced, and performance tasks Interim Assessment (Computer Adaptive)Optional comprehensive and content-cluster assessmentLearning progressionsAvailable for administration throughout the yearSelected response, short constructed response, extended constructed response, technology enhanced, and performance tasks Formative Processes and Tools Optional resources for improving instructional learningAssessment literacy
  • Technology enhanced items allow for accurate and efficient scoring of items that would take much longer to score by hand. The degree of precision can be defined as an exact point or value or as a range. For example, this item might allow a student to place the point in the correct location with an error margin of 1/10 on either side.Also note that Smarter Balanced Items can be delivered across multiple grades to increase measurement precision, particularly high and low achievers.
  • In earlier grades this decision is almost entirely based on if computation is required.Older grades, CCSS asks that we measure student understanding of the problem and a variety of execution issue, begins in grade 8. Most cases, target level. There are 2-3 targets where the target is divided for specific purposes.In the fall, a table will be available indicating the target that would allow or not allow use of a calculator
  • So on a holistic reading of this, assessment not only can but sometimes must include fractions like 6/3 that are equal to whole numbers, as well as fractions like 10/4 that are larger than 1.*********
  • The standards are not so much assembled out of topics as woven out of progressions.
  • Important connections are often not left to chance in CCSSM but are made the explicit target of individual content standards (e.g., 3.MD.7). Rich tasks in elementary should be “a little bit” rich as compared to middle and upper gradesSophistication increases in later grades.Examples of opportunities to assess at the higher levels of the content hierarchy:First two clusters of NBTOA, NBT – word problems
  • This allows some synthesis of the standards in a cluster or a domain.Rich tasks in elementary should be “a little bit” rich as compared to middle and upper gradesSophistication increases in later grades. Could be very important in high school.Examples of opportunities to assess at the higher levels of the content hierarchy:First two clusters of NBTOA, NBT – word problems
  • Important connections are often not left to chance in CCSSM but are made the explicit target of individual content standards (e.g., 3.MD.7). Rich tasks in elementary should be “a little bit” rich as compared to middle and upper gradesSophistication increases in later grades.Examples of opportunities to assess at the higher levels of the content hierarchy:First two clusters of NBTOA, NBT – word problems
  • Passes the test
  • ******Practices combine easily and a single student behavior could be thought of as exhibiting multiple practices at once.  not a checklistMathematical practices change through the grades as students grow in mathematical maturity and in the sophistication with which they apply mathematics.  Need to ensure grade-level appropriate expectations[From PARCC ITN Appendix D]The practices are also sometimes referred to as mathematical habits of mind. The word habit is important here. The nature of expertise is such that in general, it does not require prompting. A problem that asks a student to look for a pattern or structure does not summatively assess the practice of looking for patterns and structure. I would like to go back to the text and see what they say. Practices: One way to think about them is as varieties of expertise. Sometimes these are called ‘habits of mind’. As I say, we need to connect them. The content is sort of the subject matter and the practice is sort of the ways in which you engage with the subject matter as you grow in mathematical maturity and expertise. There is, as you read through the practices section, definitely a sense of growth across the grade bands in the sophistication or maturity of these behaviors. Fortunately or unfortunately, the Standards for Mathematic Practice are not actually grade banded and part of the work of assessment will be to figure out the reasonable specific performances that are grade band and grade level appropriate that translate these practices into action while connecting them with the content of the grade or, in cases, mastered content. In general, practices are things that one does without prompting, when valuable to do so. However, this is not to say that tasks that assess the practices can never be scaffolded, or can never directly ask the student to carry out a task such as constructing a viable argument (MP.3) or a mathematical model (MP.4) Some problems are “all or nothing” in that they effectively require the student to engage in a practices in order to solve the problem at all. In other problems, it is moderately advantageous to engage in a practice. Only the first type of problem summatively assesses the practice itself.Note that although practices can be discussed individually, they often intersect in various ways and can appear together. The word connect in this passage is important. Separating the practices from the content is not helpful and is not what the standards require. The practices do not exist in isolation; the vehicle for engaging in the practices is mathematical content.The Standards for Mathematical Practice should be embedded in classroom instruction, discussions and activities. They describe the kind of mathematics teaching and learning to be fostered in the classroom. To promote such an environment, students should have opportunities to work on carefully designed standards-based mathematical tasks that can vary in difficulty, context and type. Carefully designed standards-based mathematical tasks will reveal students’ content knowledge and elicit evidence of mathematical practices. Mathematical tasks are an important opportunity to connect content and practices.
  • This is not a nice-to-have.
  • I think of the excellent New Standards tasks in this regard. Which is closer to 1, 4/5 or 5/4? And explain the answer.When the practices disconnect from the content, we end up with a deadening of content, when the goal is a vibrant classroom in which mathematics is a living subject of conversation.
  • This is not a nice-to-have.
  • In some previous state standards, modeling or using models meant using multiple mathematical representations of mathematical concepts, such as solving 2x2 systems of equations algebraically and graphically. But in CCSSM, modeling refers to applications of mathematics to real-world problems (CCSSM, pp. 7, 72, 73).
  • Preparing K-5 Students for the Focus, Coherence and Rigor of the Common Core State Standards for Mathematics

    1. 1. Preparing K-5 Students for the Focus, Coherence and Rigor of the Common Core State Standards for Mathematics 1 Shelbi K. Cole, Ph.D. Smarter Balanced Director of Mathematics September 24, 2013
    2. 2. "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.
    3. 3. 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
    4. 4. What questions will we try to answer today? • How can teachers plan lessons that take into account the shifts in the CCSS-M and meet the needs of all learners? • How will the new assessment system help educators understand what students have learned and how to support their future learning? 4
    5. 5. The Three Shifts: What do they really mean? 5
    6. 6. The Background of the Common Core Initiated by the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO) with the following design principles: • Result in college and career readiness • Based on solid research and practice evidence • Focused, coherent and rigorous 6
    7. 7. 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 7
    8. 8. 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). 8
    9. 9. 9 Shift #1: Focus Traditional U.S. Approach K 12 Number and Operations Measurement and Geometry Algebra and Functions Statistics and Probability
    10. 10. 10 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
    11. 11. Shift #1: Focus Content Emphases by Cluster 11 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.
    12. 12. 12 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.
    13. 13. Shift #2: Coherence Think Across Grades Example: Fractions “The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.” 13 Source: Final Report of the National Mathematics Advisory Panel (2008, p. 18)
    14. 14. 14 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
    15. 15. 15 Shift #2: Coherence Link to major work within grade Example: Data Representation Standard 3.MD.3
    16. 16. Shift #3: Rigor In Major Topics, Pursue Conceptual Understanding, Procedural Skill and Fluency, and Application 16 • 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.
    17. 17. 17 Grade Standard Required Fluency K K.OA.5 Add/subtract within 5 1 1.OA.6 Add/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 4 4.NBT.4 Add/subtract within 1,000,000 5 5.NBT.5 Multi-digit multiplication 6 6.NS.2,3 Multi-digit division Multi-digit decimal operations Shift #3: Rigor Required Fluencies for Grades K-6
    18. 18. • 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. 18 Shift #3: Rigor Application
    19. 19. Resources • www.achievethecore.org • http://corestandards.org/assets/Math_Publishers_Criteria_K- 8_Summer%202012_FINAL.pdf • www.illustrativemathematics.org • www.pta.org/4446.htm • commoncoretools.me • www.corestandards.org • http://math.arizona.edu/~ime/progressions/#products • http://www.smarterbalanced.org/k-12-education/common-core- state-standards-tools-resources/ 19
    20. 20. Five Basic Characteristics to Support Quality Mathematics Teaching • Precision: Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known. • Definitions: They are the bedrock of the mathematical structure. They are the platform that supports reasoning. No definitions, no mathematics. • Reasoning: The lifeblood of mathematics. The engine that drives problem solving. Its absence is the root cause of teaching and learning by rote. • Coherence: Mathematics is a tapestry in which all the concepts and skills are interwoven. • Purposefulness: Mathematics is goal-oriented, and every concept or skill is therefore a purpose. Mathematics is not just fun and games. (Wu, 2011) 20
    21. 21. “Just as technicians in any kind of engineering must have a „feel‟ for their profession in order to avert disasters in the myriad of unexpected situations they are thrust into, mathematics teachers need to know something about the essence of mathematics in order to successfully carry out their duties in the classroom.” (Wu, 2011) 21
    22. 22. 22 Topic 1: Fractions The Progression Across Grades 3-5
    23. 23. Definition of “fraction” 23
    24. 24. 3.NF.A Develop understanding of fractions as numbers. • On a piece of paper, draw a line segment that is 6 inches long. • Label the left endpoint 0 and the right endpoint 3. • Locate and label the numbers 1 and 2 on the number line with respect to locations of 0 and 3. • Locate and label the numbers 1/3, 5/3, and 9/3. If you were doing this with Grade 3 students, would you change any direction above? What additional support may be needed? Why? 24
    25. 25. 3.NF.A Develop understanding of fractions as numbers. • Next, label all fractions with denominator 3 along the part of the number line you have drawn. – How many fractions with denominator 3 do you have labeled? • Discuss with your table: What parallels are there between how whole numbers are introduced in kindergarten and this introduction to fractions? 25
    26. 26. Which standards were addressed in the activity so far? • 3.NF.A Develop understanding of fractions as numbers. • 3.NF.A.1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. • 3.NF.A.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 26
    27. 27. Which aspects of this standard have been addressed so far? • 3.NF.A.3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. 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. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 27
    28. 28. 28 A Brief Detour The Parallel Curriculum Model
    29. 29. What is the Parallel Curriculum Model? The Parallel Curriculum Model is a set of four interrelated designs that can be used singly, or in combination, to create or revise existing curriculum units, lessons, or tasks. Each of the four parallels offers a unique approach for organizing content, teaching, and learning that is closely aligned to the special purpose of each parallel.
    30. 30. The Parallel Curriculum Model CURRICULUM OF CONNECTIONS CURRICULUM OF PRACTICE CURRICULUM OF IDENTITY CORE CURRICULUM
    31. 31. Why Four Parallels? • Qualitatively differentiated curriculum isn‟t achieved by doing only one thing or one kind of thing. • Students are different. • Students have different needs at different times in their lives. • Students‟ styles, talents, interests, environments and opportunities are different. • Students have different levels of expertise.
    32. 32. Four Facets of Qualitatively Differentiated Curriculum • Core: The essential nature of a discipline • Connections: The relationships among knowledge • Practice: The applications of facts, concepts, principles, skills, and methods as scholars, researchers, developers, or practitioners • Identity: Developing students‟ interests and expertise, strengths, values, and character 32
    33. 33. Concepts, Principles and Skills • Concepts: A general idea or understanding, especially a generalized idea of a thing or class of things; a category or classification • Principles: Fundamental truths, laws, doctrines, or rules, that explain the relationship between two or more concepts • Skills: Proficiency, ability, or technique, strategy, method or tool 33
    34. 34. Building a CCSS-Aligned Lesson • Discuss the number line activity. – How would you implement a similar activity with students in a grade 3 classroom? – How can you determine if students are successful on individual standards while still building important connections across standards? – Should there be something that precedes the number line activity in introducing fractions to grade 3 students? If so, what does it look like? 34
    35. 35. Learning from the Group DON‟T start filling in the lesson template before hearing the GOOD IDEAS of other educators working on the SAME THING. The Common Core State Standards offer new opportunity for collaboration so that TOGETHER we can build and gather the BEST RESOURCES for students. 35
    36. 36. Work Time • Begin to populate the template with some of the ideas we‟ve heard that represent important learning opportunities for students. • But first, let‟s take a look at the lesson plan template. 36
    37. 37. Increasing Engagement, Thinking Across Grades • Grade 1 – 1.OA.A (standards 5&6) • Grade 2 – 2.MD.B (standards 5&6) • Grade 3 – 3.NF.A (standard 3) • Grade 4 – 4.NF.A (standards 1&2) • Grade 4 – 4.NF.C (standards 5-7) • Grade 5 – 5.NF.B (standards 3&4) Create a task or learning opportunity that relates the content of the indicated standards to the number line. Materials allowed: Ruler, Sidewalk chalk 37
    38. 38. • And the number line remains relevant even after grade 5… 38
    39. 39. Grade 3 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.
    40. 40. Grade 7
    41. 41. Grade 8
    42. 42. Beyond the Number Line: Other Representations that Support Student Understanding of Fractions • What fraction is represented by the shaded area? 42
    43. 43. Notice the use of “represented.” The shaded area is not “equal” to a fraction. • The fraction represented by the shaded area is ¾. Based on this: – Draw an area that represents ¼. – Draw an area that represents 1. • The fraction represented by the shaded area is equal to 3/2. Based on this: – Draw an area that represents ½. – Draw an area that represents 1. 43
    44. 44. 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 44
    45. 45. Fractions Example The shaded area represents 3/2. Drag the figures below to make a model that represents 3 x 3/2. BA C D
    46. 46. Slide 46 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.
    47. 47. Slide 47 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.
    48. 48. Slide 48 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.
    49. 49. How can K-2 work on operations with whole numbers and work on fractions in grade 3 support students’ thinking about these problems in grades 4 & 5? 49 4 5 2 ? 4 5 2 ? Even before learning the exact sum, can students tell you between which two whole numbers the answer lies? Even before learning the exact product, what can students tell you about the value of the product?
    50. 50. At what grade should students be able to solve these problems? 1 3 1 3 1 ? 1 3 2 5 2 3 1 10 4 6 ? ?
    51. 51. Explain the flaw in the chain of reasoning. In the first group, 3/5 of the cats have spots. In the second group, 1/3 of the cats have spots. All together, 4/8 of the cats have spots. Therefore, 3/5 + 1/3 = 4/8.
    52. 52. Closing Task – Fractions • You have 4 strips of paper each 1 foot in length. – Partition one into thirds. – Partition one into fifths. – Partition one into eighths. – Use the ruler as needed. – How can the strips be used to compare fractions? – What is the length, in inches, of each partition? Assume that the strip is exactly one foot and each of your partitions is exactly equal. 52
    53. 53. 53 Topic 2: Whole Numbers
    54. 54. 0 3/2 1 + 3 = ? Operations with fractions should be a natural extension of operations with whole numbers 54 2 4 5 0 1 3 1 2 3 2 ? 1/2 2/2 4/2
    55. 55. 55
    56. 56. Future impact 1.OA.B Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.B.3. Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 1.OA.B.4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. .
    57. 57. Future impact 1.OA.C Work with addition and subtraction equations. 1.OA.C.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 1.OA.C.8. Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?
    58. 58. Within grade impact • Look at the grade 2 standards in the domains NBT and OA. Discuss how the standards in NBT might be integrated with the standards for OA to create problems that challenge mathematically talented students. • Develop a problem that integrates concepts and skills from the two domains to increase the level of challenge. (You may use online resources such as Illustrative Math to find a base task and then further differentiate.) 58
    59. 59. Smarter Balanced Assessment Consortium An Assessment System to Support Teaching and Learning
    60. 60. 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?
    61. 61. 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
    62. 62. The Purpose of the Consortium • To develop a comprehensive and innovative assessment system for grades 3-8 and high school in English language arts and mathematics aligned to the Common Core State Standards, so that... • ...students leave high school prepared for postsecondary success in college or a career through increased student learning and improved teaching [The assessments shall be operational across Consortium states in the 2014-15 school year]
    63. 63. • 26 member states and territories representing 39% of K-12 students • 22 Governing States, 3 Advisory States, 1 Affiliate Member • Washington state is fiscal agent • WestEd provides project management services A National Consortium of States
    64. 64. 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
    65. 65. 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 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
    66. 66. A Balanced Assessment System School Year Last 12 weeks of the year* DIGITAL LIBRARY of formative tools, processes and exemplars; released items and tasks; model curriculum units; educator training; professional development tools and resources; scorer training modules; and teacher collaboration tools. ELA/Literacy and Mathematics, Grades 3-8 and High School Computer Adaptive Assessment and Performance Tasks Computer Adaptive Assessment and Performance Tasks Scope, sequence, number and timing of interim assessments locally determined *Time windows may be adjusted based on results from the research agenda and final implementation decisions. Performance Tasks • ELA/literacy • Mathematics Computer Adaptive Assessment • ELA/literacy • Mathematics Optional Interim Assessment Optional Interim Assessment Re-take option available Summative Assessment for Accountability
    67. 67. Using Computer Adaptive Technology for Summative and Interim Assessments • Provides accurate measurements of student growth over timeIncreased precision • Item difficulty based on student responses Tailored 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
    68. 68. What is CAT? Administered by computer, a Computerized Adaptive Test (CAT) dynamically adjusts to the trait level of each examinee as the test is being administered.
    69. 69. How CAT Works (Binet’s Test)
    70. 70. 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) 8 x □ = 4 x □ Give two different pairs of numbers that could fill the boxes to make a true equation. F) 8 x □ = 40
    71. 71. 0 3/2 Put a point at the number 1. Some Important “Constraints” in the Smarter Balanced Summative 71 2 4 0 3 0 0 3/5
    72. 72. Grade 1 Add and subtract within 20. 6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Work with addition and subtraction equations. 7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 8. Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?. Grade 4 Understand decimal notation for fractions, and compare decimal fractions. 5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
    73. 73. 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)
    74. 74. 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
    75. 75. Assessment System Components Summative Assessment (Computer Adaptive) • Assesses the full range of Common Core in English language arts and mathematics for students in grades 3–8 and 11 (interim assessments can be used in grades 9 and 10) • Measures current student achievement and growth across time, showing progress toward college and career readiness • Can be given once or twice a year (mandatory testing window within the last 12 weeks of the instructional year) • Includes a variety of question types: selected response, short constructed response, extended constructed response, technology enhanced, and performance tasks
    76. 76. Assessment System Components Interim Assessment (Computer Adaptive) • Optional comprehensive and content-cluster assessment to help identify specific needs of each student • Can be administered throughout the year • Provides clear examples of expected performance on Common Core standards • Includes a variety of question types: selected response, short constructed response, extended constructed response, technology enhanced, and performance tasks • Aligned to and reported on the same scale as the summative assessments • Fully accessible for instruction and professional development
    77. 77. Assessment System Components • Extended projects demonstrate real-world writing and analytical skills • May include online research, group projects, presentations • Require 1-2 class periods to complete • Included in both interim and summative assessments • Applicable in all grades being assessed • Evaluated by teachers using consistent scoring rubrics 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 “ ” Performance Tasks
    78. 78. Assessment System Components Few initiatives are backed by evidence that they raise achievement. Formative assessment is one of the few approaches proven to make a difference. - Stephanie Hirsh, Learning Forward Formative Assessment Practices • Research-based, on- demand tools and resources for teachers • Aligned to Common Core, focused on increasing student learning and enabling differentiation of instruction • Professional development materials include model units of instruction and publicly released assessment items, formative strategies “ ”
    79. 79. Assessment System Components Data are only useful if people are able to access, understand and use them… For information to be useful, it must be timely, readily available, and easy to understand. - Data Quality Campaign Online Reporting • Static and dynamic reports, secure and public views • Individual states retain jurisdiction over access and appearance of online reports • Dashboard gives parents, students, practitioners, and policymakers access to assessment information • Graphical display of learning progression status (interim assessment) • Feedback and evaluation mechanism provides surveys, open feedback, and vetting of materials “ ”
    80. 80. Support for Special Populations • Accurate measures of progress for students with disabilities and English Language Learners • Accessibility and Accommodations Work Group engaged throughout development • Outreach and collaboration with relevant associations Common- Core Tests to Have Built-in Accommodations - June 8, 2011 “ ”
    81. 81. “Students can demonstrate progress toward college and career readiness in English Language arts and literacy.” “Students can demonstrate college and career readiness in English language arts and literacy.” “Students can read closely and analytically to comprehend a range of increasingly complex literary and informational texts.” “Students can produce effective and well-grounded writing for a range of purposes and audiences.” “Students can employ effective speaking and listening skills for a range of purposes and audiences.” “Students can engage in research and inquiry to investigate topics, and to analyze, integrate, and present information.” Overall Claim for Grades 3-8 Overall Claim for Grade 11 Claim #1 - Reading Claim #2 - Writing Claim #3 - Speaking and Listening Claim #4 - Research/Inquiry Claims for the ELA/Literacy Summative Assessment
    82. 82. “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
    83. 83. April 2012 Item/task specifications and guidelines v1.0 completed May 2012 Training materials for writing & reviewing items/tasks completed May-June 2012 Automated scoring models developed June 2012 Engage students in “cognitive labs” for new and innovative items/tasks June 2012 Recruit classrooms for small scale trials Aug-Oct 2012 Conduct small scale trials May 2012 Recruit teachers to write and review first ~10,000 items/tasks (“Pilot” Phase) May-July 2012 Teachers trained and write pilot items/tasks Sept-Oct 2012 Teachers review pilot items/tasks Oct-Nov 2012 Solicit classrooms for pilot testing Feb-Mar 2013 Pilot testing of items/tasks May 2013 Recruit teachers to write/review next ~35,000 items/tasks (“Field Test” Phase) May-July 2013 Teachers trained and write field test items/tasks Sept-Oct 2013 Teachers review field test items/tasks Oct-Nov 2013 Solicit classrooms for field testing April-June 2014 Field testing of items/tasks Development of Items and Tasks
    84. 84. Viewing Sample Items & Tasks • Preliminary Sample Items – http://sampleitems.smarterbalanced.org/itemp review/sbac/ • Full Practice Tests Available – http://sbac.portal.airast.org/practice-test/
    85. 85. 85 Usability, Accessibility, Accommodations
    86. 86. Overall New Document • Introduction – Purpose and intended audience – Briefly addresses Framework and ISAAP, includes links • Three main sections on supports – Universal Accessibility Tools – Designated Accommodations – Documented Accommodations • Resources (References) • Appendices – Summary of supports – Research lessons
    87. 87. Speech-to-Text • Available as a documented accommodation on all math and ELA items: – Available for students providing a response, as a documented accommodation included in an IEP or 504. – Available to students when taking notes in preparation for a written response, included in an IEP or 504. • Students to use own Assistive Technology (AT) device, with AT device certification Note: Conventions not a challenge to the construct.
    88. 88. American Sign Language • Available for students who are hard of hearing or deaf as a documented accommodation • Available on math items • Available on ELA listening items
    89. 89. Closed Captioning • Available for students who are hard of hearing or deaf as a documented accommodation • Available on ELA listening items
    90. 90. Calculator • Universal embedded calculator: available on items as per Smarter Balanced item specifications and targets • Non-embedded calculator is a documented accommodation (i.e., Brailled calculator) – Student can use own device • Calculator is not allowed on non-calculator items – Decision based on intended construct to be measured by assessment targets.
    91. 91. Survey Feedback—Participation 91
    92. 92. 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.
    93. 93. New Graphic
    94. 94. General Table, Appendix A
    95. 95. 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 95
    96. 96. Guidelines & Frameworks • Smarter Balanced Usability, Accessibility and Accommodations Guidelines http://www.smarterbalanced.org/wordpres s/wp- content/uploads/2013/09/SmarterBalance d_Guidelines_091113.pdf • Smarter Balanced Translation Framework • http://www.smarterbalanced.org/wordpres s/wp- content/uploads/2012/09/Translation- Accommodations-Framework-for-Testing- ELL-Math.pdf 96
    97. 97. 97 Making Items More Accessible
    98. 98. Drag and Drop
    99. 99. Equal to 5.42 NOT Equal to 5.42 2.36 + 3.06 1.80 x 3 2.16 + 3.36 9.53 – 4.11 2.71 x 2 8.01 – 2.69 Table
    100. 100. Translation
    101. 101. Translation
    102. 102. Translation
    103. 103. Translation
    104. 104. Embedded
    105. 105. Embedded
    106. 106. 106 Important Notes on Item Development
    107. 107. 107 http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/For an essay on similar themes, see Features of CCSSM and Implications for Assessment Assessing Individual Content Standards or Parts of Standards Alignment in Context: Cluster, Domain, and Grade 3.c
    108. 108. What are progressions? Many or most of the content standards in K-8 represent steps or stages along a progression of learning and performance. Why are progressions important for item writers? They are context for alignment questions. Progression- sensitive tasks will help teachers implement the standards with fidelity. Where can I find more information? Progressions documents are narratives of the standards across grade levels, informed by research on children's cognitive development and by the logical structure of mathematics. 108 Features of CCSSM and Implications for Assessment Assessing Individual Content Standards or Parts of Standards Alignment in Context: Neighboring Grades and Progressions http://math.arizona.edu/~ime/progressions/#products
    109. 109. 109 Features of CCSSM and Implications for Assessment Assessing at All Levels of the Content Hierarchy Standards in number and operations that involve place value of decimals Items requiring procedural skill in expanded form, reading/writing, and comparing decimals, and rounding Q Q Q How Standards and Tests Used To Be Designed: Tossing Items Into Buckets Q
    110. 110. 110 Features of CCSSM and Implications for Assessment Assessing at All Levels of the Content Hierarchy Number and Operations in Base Ten—Understand the Place Value System 5.NBT.1 5.NBT.2 5.NBT.3a 5.NBT.3b 5.NBT.4 Q Q QQ Q Q Using the Design of CCSSM to Make Better Tests Step 1: Better Buckets! 1) Standards complement procedural skill with explicit expectations for conceptual understanding of specific content, and connections to practices such as constructing mathematical arguments (MP.3) 2) Individual standards make important connections explicit and inescapable— e.g., 3.MD.7 connects area to multiplication, division, and the distributive property. Buckets that connect buckets. 3) Standards are given a coherent context within and across grades through cluster headings
    111. 111. 111 Features of CCSSM and Implications for Assessment Assessing at All Levels of the Content Hierarchy 5.NBT.1 5.NBT.2 5.NBT.3a 5.NBT.3b 5.NBT.4 Q Q QQ Q Q Using the Design of CCSSM to Make Better Tests Step 2: Where Appropriate, Treat Headings As Buckets Where appropriate, aim some of the item development at the cluster level Where appropriate, aim some of the item development at the domain level Q Q 5.NBT.A 5.NBT
    112. 112. • There are a finite set of key opportunities to assess at the cluster, domain, and grade levels. Smarter Balanced is exploring this feature of the Standards.  Such opportunities should be identified carefully and must make obvious educational and mathematical sense within the framework of the standards. For example, the first two clusters in NBT are often important to link; and doing simple word problems to apply developing NBT skills requires some grade-level tasks that blend OA and NBT.  This is not a recommendation to make loose interpretations of the standards or go beyond what is written in the standards. Rather, it is an opportunity to measure plausible and immediate implications of what is written in the standard, without ever slipping into the imposition of additional requirements. . 112 Features of CCSSM and Implications for Assessment Assessing at All Levels of the Content Hierarchy
    113. 113. Features of CCSSM and Implications for Assessment Assessing at All Levels of the Content Hierarchy http://tinyurl.com/mathpubcrit Coherent Connections 113
    114. 114. 114 Features of CCSSM and Implications for Assessment Minimizing pitfalls of traditional multiple choice questions • High-quality machine-scoreable tasks are critical. • Some examples of how to do this in CBT that are machine scored:  Gridded response  True/False with multiple options  Drag and drop
    115. 115. Some considerations: Risks of technology in assessing of the Standards. • Technology for technology‟s sake - The construct should drive the task design, not functionality • Overuse of tech-friendly content - Technology can threaten focus; consider how technology can support focus and not dilute it • Innovative does not equal more difficult. Many tasks should give kids credit for learning the basics of the grade • Need cognitive labs or other tryouts to make sure the technology is adding value to the task 115 Features of CCSSM and Implications for Assessment
    116. 116. Grade 3
    117. 117. Grade 3
    118. 118. Grade 5
    119. 119. 119 Fluency as a special case of assessing individual content standards. • Fluent in the standards means fast and accurate. • The word fluency was used judiciously in the Standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity. • Assessing the full range of the standards means assessing fluency where it is called for in the standards.  Some of these fluency expectations are meant to be mental and others with “pencil and paper.” But for each of them, there should be no hesitation in getting the answer with accuracy. Features of CCSSM and Implications for Assessment
    120. 120. Topics for Discussion 1. Assessing individual content standards or parts of standards 2. Assessing at all levels of the content hierarchy 3. Identifying ways in which technology can promote quality assessment of the standards 4. Connecting mathematical practice standards and content standards 120 Features of CCSSM and Implications for Assessment
    121. 121. “Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.” (CCSSM, pg. 8) 121 Features of CCSSM and Implications for Assessment Assessing through authentic connections of content and practices
    122. 122. Features of CCSSM and Implications for Assessment Smarter Balanced’s Claims Embody Specific Mathematical Practices in the Presence of Content Standards Claim #2: Problem Solving. Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Claim #3: Communicating Reasoning. Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Claim #4: Modeling and Data Analysis. Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. 122
    123. 123. 123 A Closer Look at Claim #3: Communicating Reasoning Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. (Connection to MP.3: Construct viable arguments and critique the reasoning of others) Features of CCSSM and Implications for Assessment Connecting Content Standards and Mathematical Practices in Assessment
    124. 124. 124 Reasoning is a refrain in the content standards Note generally such words as justify a conclusion, prove a statement, explain the mathematics; also derive, assess, illustrate, and analyze. Features of CCSSM and Implications for Assessment Connecting Content Standards and Mathematical Practices in Assessment
    125. 125. 125 Claim #3 tasks have fine “grain size.” The Standards ask students not just to Reason, but to “reason about X,” where X is key grade-level mathematics such as properties of operations, relationships between addition and subtraction or between multiplication and division, fractions as numbers, variable expressions, linear/nonlinear functions, etc. Features of CCSSM and Implications for Assessment Connecting Content Standards and Mathematical Practices in Assessment
    126. 126. 126 A Closer Look at Claims #2 and #4 Problem Solving Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Modeling & Data Analysis Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Features of CCSSM and Implications for Assessment Connecting Content Standards and Mathematical Practices in Assessment
    127. 127. 127 Modeling is a mathematical practice (MP.4) – and in high school, it is also a content category. Therefore it is reasonable to say that modeling is enhanced in high school as compared to K-8, with more elements of the modeling cycle (CCSSM, p. 72, 73) – and equivalently, that modeling builds more slowly across K-8 and is less sophisticated there. Features of CCSSM and Implications for Assessment Connecting Content Standards and Mathematical Practices in Assessment
    128. 128. 128 • Contextual word problems involving ideas that are currently at the forefront of the student‟s developing mathematical knowledge. • Multi-step contextual word problem in which the problem isn‟t broken into steps or sub-parts • Micro models: These tasks define goals that can be met by autonomously apply a known technique from pure mathematics to a real-world situation in which the technique yields valuable results though it is obviously not applicable in a strict mathematical sense • Reasoned estimates: these tasks require students to use reasonable estimates of know quantities in a chain of reasoning that yield an estimate of unknown quantity. • Decisions from data: These tasks require students to select from a data source, analyze the data and draw reasonable conclusions from it. This will often result in an evaluation or recommendation. • Full models: These tasks require execution of some or all of the modeling cycle in high school. Source: Excerpted from Smarter Balanced Content Specifications, PARCC Item Development ITN, Appendix F Features of CCSSM and Implications for Assessment Modeling and Problem Solving tasks may come in a variety of types
    129. 129. Grade 5
    130. 130. Grade 5
    131. 131. Grade 5
    132. 132. Grade 3
    133. 133. Find Out More Smarter Balanced can be found online at: SmarterBalanced.org Contact Information: shelbi.cole@smarterbalanced.org 133

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