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Standards Toolkit—Mathematics - 25 -

HANDOUTS

Standards Toolkit—Mathematics

1

Handout 1

Handout 1 is the “Notes” format of the “Setting the Stage” General Overview PowerPoint slides. When printed, the handout will contain 7 pages with 2-3 slides per page.

Hawaii State DOE, OCISS, Instructional Services Branch

4/30/03

Rollout for Complex Areas 1

Introducing the Standards Toolkit

OCISS Instructional Services Branch

Setting the Stage, General Overview

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Desired Outcomes

• Develop a clear understanding of the content and use of the Standards Toolkit.

• Plan ways to help schools understand the content and use of the Standards Toolkit.

Setting the Stage, General Overview 3

Part I. System of Standards

§ CONTENT STANDARDS(Benchmarks)

§ PERFORMANCE STANDARDS(Performance Indicators, Student Work, Commentary)


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System of Standards

§ Content Standards–K-12 statements–WHAT: know, be able to do, care about–Essential content (concepts,

processes, facts, generalizations) of a content area


System of StandardsOrganization of Content

Content and process interact--process embedded in content, and content embedded in process.

Social Studies

Process (Domain I).

Content (Domain II)

Science

Primarily Content. Some process. Process embedded in content.

Math

Process. Content embedded in process

Language Arts


System of Standards

Organization: First level at a Glance

§5 DisciplinesSocial Studies

§2 Domains (Strands within each domain)

Science

§5 StrandsMathematics

§3 Areas§4 Strands

Language Arts


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System of Standards

§ Benchmarks– Describe more specifically what the content

standard is by the end of a grade level cluster.

– Identify WHEN students can reasonably be expected to know a given content.

– Are clustered to acknowledge different rates of learning, and to provide flexibility for curricular choice.


System of Standards

§ Performance Standards describe evidence of student learning required to meet standards. They include:–Student Work–Performance Indicators–Commentary


Performance Indicators

Performance indicators are one element of the Performance Standards.


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Performance Indicators

§ Are organized by grade levels and linked to a standard and benchmarks.

§ Are descriptors of student learning toward a standard.

§ Describe what should be in student work.

§ Reflect growing sophistication required in student work and performance.


System of Standards

§ Six General Learner Outcomes–Essential overarching goals for all

grade levels, regardless of content area

–Are supported by the content and performance standards


System of Standards

§ Six General Learner Outcomes– Self-directed Learner– Community Contributor– Complex Thinker– Quality Producer– Effective Communicator– Effective and Ethical User of Technology


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Part II. Introducing the Toolkit

• Components– Instructional Guide

– Curriculum Framework° Grade Level Performance Indicators° Scope and Sequence


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Introducing the Toolkit• Uses

– Assessment Planning• Instructional Guide and Curriculum Framework

– Instructional Planning• Instructional Guide and Curriculum Framework

– Curriculum Planning• Instructional Guide and Curriculum Framework


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Caveat Emptor

• Standards and Benchmarks are not a curriculum.

• The Instructional Guide contains SUGGESTIONS for instruction and assessment.


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Implementing Standards

Using the Toolkit to Understand the Content Area Standards

Breakout Sessions: 25 Per Session


Remainder of Today (Day #1)

• Implementing the Content Area Standards

• Break• Implementing, continued• Lunch



• Implementing, continued• Cross Sharing• Considerations for

Professional Development• Q and A• Evaluation


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• Implementing the Content Area Standards breakout

• Break• Implementing, continued• Lunch• Cross-sharing content• Complex Planning• Evaluation



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Handout 2

Handout 2 is the “Notes” format of the “Setting the Stage” In the Context of Mathematics PowerPoint slides. When printed, the handout will contain 7 pages with 2-3 slides per page.

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6/13/2003 Setting the Stage, In the Context of Mathematics

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INTRODUCING THE STANDARDS TOOLKIT for Mathematics

Curriculum Framework for Mathematics

° Grade Level Performance Indicators° Scope and Sequence

Instructional Guide


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Mission

To ensure that all public school graduates will achieve the General Learner Outcomes and Content and Performance Standards in order to realize their individual goals and aspirations and become responsible and productive members of society.


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Vision of a Hawaii High School Graduate

Our graduates will:v Realize their goals and aspirations.v Contribute positively and compete

in a global society.v Exercise their rights and

responsibilities of citizenship.v Pursue post-secondary education

and/or careers without need for remediation.

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General Learner Outcomes

b Self-directed learnerbCommunity ContributorbComplex thinkerbQuality producerb Effective communicatorb Effective and ethical user of technologyK-12 essential overarching goals for all

content areasSupported by HCPS II – all content areas


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Board of Education Policy 2015

b“To ensure high academic expectations for all students, the Department of Education shall implement the Content and Performance Standards.”

bAdopted in October, 1994


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Act 238: Educational Accountability

The accountability system shall involve:b A statewide student assessment program that provides annual data on student, school, and system performance…relative to HCPS II.b An annual assessment in core subjects for each grade level, conducted by each school.

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HCPS II Statewide Assessment

bGrades 3 – 8, 10 from SY 2004 – 2005b SAT 9 Abbreviated – Problem Solvingb Standards-based: MC, Short-

constructed response and Extended response items

b Proficiency Levels: Exceeds, Meets, Approaches, and Well Below


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No Child Left Behind Act

b Goal 1: Proficiency in reading / LA, & mathematics by SY 2013-2014.

b Goal 2: Proficiency in reading / LA, & mathematics for all LEP students.

b Goal 3: Instruction by qualified teachers & paraprofessionals by SY 2005-2006.

b Goal 4: Learning environments that are safe, drug free, & conducive to learning.

b Goal 5: All students will graduate from high school.


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Comprehensive Needs Assessment

b Images of Successb Standards-based LearningbQuality Student Supportb Professionalism and Capacity of the Systemb Focused and Sustained ActionbCoordinated Team WorkbResponsiveness of the System

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Strategic Implementation Plan

b Provide a standards -based education for every child

b Sustain comprehensive support for all students

bDeliver coordinated, systemic support for staff and schools

bAchieve and sustain continuous improvement of all student performance, and professional, school, and system quality


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Program SupportEfforts will continue to be directed at

establishing:v A system of standards that puts the

focus on the quality of students’ work;v A system that aligns curriculum,

instruction, and assessment;v Professional development to support

implementation;v Capacity building and leadership to

sustain improvement.


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Reform ProgramsElementary:

• Everyday Mathematics (K-6) – McGraw Hill

• Investigations (K-5) – Scott Foresman

• Math Trailblazers (K-5) – Kendall HuntSecondary:

• Connected Mathematics (6-8) – Pearson

• Math in Context (5-8) – It’s About Time

• MathThematics (6-8) – McDougal Little

• Interactive Mathematics (9-12) – Key Curriculum

• Math Connections (9-12) – It’s About Time

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Content of the Standards

Five content strands:v Number and Operationsv Measurementv Geometry and Spatial Sensev Patterns, Function, and Algebrav Data Analysis, Statistics, and Probability

Five process standards:v Problem Solvingv Representation v Reasoning and Proofv Connectionsv Communication


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Content Standards

Define WHAT every student should know, be able to do and care about.

Identify the essential content for mathematics

Organize the content of mathematics


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BenchmarksDescribe what the content of the

standard “looks like” within a grade level cluster

Describe WHEN students can be reasonably expected to know a given concept.

Clustered to acknowledge different rates of learning and provides flexibility for curricular choice.

6


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Performance Standards

Describe the evidence of student learning required by the standard

Includes:- performance indicators- student work- written commentary


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Standards Toolkit

Components

v Curriculum Frameworkv* Performance Indicator Progression

v * Scope and Sequence

v Instructional Guide


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Remember…v Standards and benchmarks are NOT a

curriculum.v The Instructional Guide contains

suggestions for instruction and assessment.v The Instructional Guide is neither

comprehensive nor definitive…do not assume achievement of the standards with one activity.

v Tasks/activities – in reality – should and can address more than one standard/indicator.

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Process for implementation

v Identify relevant standardsv Develop/Identify a taskv Determine learning experiencesv Implement instructionv Administer the taskv Analyze student work

Students should be involved in all steps of the process.


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Thank you and you and you…


3

Handout 3 Handout 3 is the “Notes” format of the “Curriculum Framework for Mathematics” PowerPoint slides. When printed, the handout will contain 5 pages with 3 slides per page.

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Curriculum Framework for Mathematics 1

Curriculum Frameworkfor Mathematics

Hawaii State Dept. of EducationOCISS, Instructional Services Branch


Outcomesü Understand the content and

organization of the Mathematics standards

ü Understand the System of Standardsü Be familiar with the content and

organization of the Curriculum Framework for Mathematics


Curriculum Framework

I. General Description of the Mathematics Education Program

• Definition • Rationale • Goals • Vision & Mission

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Vision and MissionVision: Current, relevant, research-based instructional practices

are a part of each teacher’s classroom repertoire. Students are engaged in stimulating math talk about challenging concepts & processes of important math in rich, meaningful tasks. Math tools and technology are accessible and used in the context of doing mathematics. Teachers collect assessment data of student progress towards meeting the standards. Students take responsibility for their own learning; self -assessments are common occurrences. Professional Development (P.D.) is a key support.

Mission: Provide the necessary leadership and supports so all students can make informed choices for post-graduation paths without remediation.


Curriculum Framework

II. The Mathematics Standards

• The Need for Standards• The Setting of the Standards• The Organization of the Standards• The Relationship to the GLOs • The System of Standards


The Mathematics Standards

Strands:Number and Operations (3)Measurement (1)Geometry and Spatial Sense (4)Patterns, Functions, and Algebra (2)Data Analysis, Statistics, and Probability (4)

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The System of Standardsn CONTENT STANDARDSn K-12 statementsn WHAT: know, be able to do, care

aboutn Essential content (concepts,

processes, facts, generalizations) of a content area


The System of Standardsn Benchmarks

n Describe more specifically what the content standard is by the end of a grade level cluster.

n Identify WHEN students can reasonably be expected to know a given content.

n Are clustered to acknowledge different rates of learning, and to provide flexibility for curricular choice.


The System of Standardsn PERFORMANCE STANDARDS…describe evidence of student

learning required to meet standards. They include:n Student Workn Performance Indicatorsn Commentary

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The System of Standardsn Performance Indicators

- Describe what should be in student work.

- Are organized by grade levels and linked to a standard and benchmarks.

- Are descriptors of student learning toward a standard.

- Reflect growing sophistication required in student work and performance.


The System of Standardsn Performance Assessment

- Students use what they have learned and demonstrate proficiency of the standard(s)

- Measure of whether or not AND to what degree students have met the standard(s)


The System of Standardsn Criteria

- Scope or characteristics of standards for judging student work

- Not to be confused with product/performance expectations

n Rubrics- Scoring guides; bridge between standards and assessments

- Elements: Dimensions/Scale; Descriptors

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III. Assessment, Curriculum, & Instruction

STANDARDS

Instructio

n

Curriculum

Student

Work Assessment

Informal

Formal

Formative

Summative


IV. Bibliography, Resources, and Glossary

V. Appendix

Tab: Scope and Sequence

Tab: Grade Level Performance Indicators


ResourcesMath Puzzles is a collection of interactive games that give practice in basic logic and arithmetic.

www.mathispower.org/mathnav.htmlPerformance Assessments Links for Mathematics provides complete lesson plans and accompanying scoring guides for different grade-level clusters.

www.palm@ sri.comCyberchase is a new PBS animated, adventure TV series about problem solving and math. On the Web, kids play interactive games & teachers access lesson plans. www.pbskids.org/cyberchase/classroom

Handout 4


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Standards Terminology

Content Standards: • The WHAT • K-12 statements

Benchmarks: • The WHEN students can reasonably be expected to know a given content • “Translations” of the WHAT by the end of the grade-level cluster (e.g., This is what

the standard looks like by the end of grade 3.) • Grade level benchmarks for mathematics and language arts: K-1, 2-3, 4-5, 6-8, 9-12 • Grade level benchmarks for the rest of the content areas: K-3, 4-5, 6-8, 9-12

Grade Level Performance Indicators:

• Evidence that needs to be in the student work that teachers use to determine where students are with respect to Benchmarks and Standards

• Descriptors based on Benchmarks and Standards • Descriptors written to be specific, yet not task-specific nor tied to a particular

lesson; general enough to be relevant to multiple tasks Performance Standards:

• Samples of the quality of work expected of all students • Student work that contain evidence of grade level performance indicators • Commentary is provided about the indicators in the work

Scope and Sequence:

• Topics based on grade-level performance indicators Performance Indicator Progression:

• Grade-level performance indicators arrayed in a matrix that show how the indicators progress through the grades

• Mathematics and Science indicators are by grade levels for grades K-8, then by courses

• Social Studies indicators are by grade levels for K-3, then by courses • Language Arts indicators are by grade levels K-12

Instructional Guide:

• Grade Level, Strand, Standard, Benchmark, Performance Indicators • Sample Assessment Task(s) • Sample Instructional Strategy(ies)

Instructional Guide - Considerations:

• Elements need to be coherent/aligned with standards and benchmarks • Suggestions of what teachers could do in instruction that would result in student

work that contains performance indicator(s)

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Sample Assessment Task(s):

• Question(s) students are asked or activity(ies) students perform that will result in work that contain evidencegrade-level performance indicators

• Tasks may address multiple benchmarks/standards • Can be administered at any time in the instructional sequence (for various

purposes) • Resultant student work provides information to the teacher on what the student

knows and can do with respect to the “building blocks” of the learning sequence or “culminating project”

Sample Instructional Strategy(ies):

• Suggestions to teachers on what can be done in instruction that will enable students to do the assessment task and provide work containing evidence of the indicators that meets the expectation

• Bulleted or narrative suggestions that may be procedural, a compressed version of the lesson, or a description of the general strategy.

Performance objectives:

• Intent or purpose of instruction

Handout 5


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General Learner Outcomes

GLO 1: Self-directed Learner The ability to be responsible for one’s own learning.

GLO 2: Community Contributor The understanding that it is essential for human beings to work together.

GLO 3: Complex Thinker The ability to perform complex thinking and problem solving.

GLO4: Quality Producer The ability to recognize and produce quality performance and quality products.

GLO 5: Effective Communicator The ability to communicate effectively.

GLO 6: Effective and Ethical User of Technology The ability to use a variety of technologies effectively and ethically.

Handout 6


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Content Standards – At-A-Glance

NUMBER AND OPERATIONS

MEASUREMENT

GEOMETRY AND SPATIAL

SENSE

PATTERNS, FUNCTIONS,

AND ALGEBRA

DATA ANALYSIS, STATISTICS, AND

PROBABILI TY

1. Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

1. Understand attributes, units, and systems of units in measurement; and develop and use techniques, tools, and formulas for measuring.

1. Analyze properties of objects and relationships among the properties.

1. 1. Understand various types of patterns and functional relationships.

1. Pose questions and collect, organize, and represent data to answer those questions.

2. Understand the meaning of operations and how they relate to each other.

2. Use transformations and symmetry to analyze mathematical situations.

2. Use symbolic forms to represent, model, and analyze mathematical situations.

2. Interpret data using methods of exploratory data analysis.

3. Use computational tools and strategies fluently and when appro-priate, use estimation.

3. Use visualization and spatial reasoning to solve problems both within and outside of mathematics.

1.

3. Develop and evaluate inferences, predictions, and arguments that are based on data.

4. Select and use different repre-sentational systems, including coordinate geometry.

4. Understand and apply basic notions of chance and probability.

The process standards of Communication, Connections, Problem Solving, Representation, and Reasoning and Proof are incorporated throughout the above content strands.

Handout 8


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PROCESS STANDARDS

Principles and Standards for School Mathematics

National Council of Teachers of Mathematics, 2000

Problem Solving

Instructional programs from prekindergarten through grade 12 should enable all students to—

• build new mathematical knowledge through problem solving;

• solve problems that arise in mathematics and in other contexts;

• apply and adapt a variety of appropriate strategies to solve problems;

• monitor and reflect on the process of mathematical problem solving.

Problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of learning mathematics but also a major means of doing so. Students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and should then be encouraged to reflect on their thinking.

Reasoning and Proof


• recognize reasoning and proof as fundamental aspects of mathematics;

• make and investigate mathematical conjectures;

• develop and evaluate mathematical arguments and proofs;

• select and use various types of reasoning and methods of proof. Mathematical reasoning and proof offer powerful ways of developing and expressing insights about a wide range of phenomena. People who reason and think analytically tend to note patterns, structure, or regularities in both real-world situations and symbolic objects; they ask if those patterns are accidental or if they occur for a reason; and they conjecture and prove. Ultimately, a mathematical proof is a formal way of expressing particular kinds of reasoning and justification.

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Communication


• organize and consolidate their mathematical thinking through communication;

• communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

• analyze and evaluate the mathematical thinking and strategies of others;

• use the language of mathematics to express mathematical ideas precisely.

Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public. When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others orally or in writing, they learn to be clear and convincing. Listening to others' explanations gives students opportunities to develop their own understandings. Conversations in which mathematical ideas are explored from multiple perspectives help the participants sharpen their thinking and make connections. Students who are involved in discussions in which they justify solutions—especially in the face of disagreement—will gain better mathematical understand ing as they work to convince their peers about differing points of view (Hatano and Inagaki 1991). Such activity also helps students develop a language for expressing mathematical ideas and an appreciation of the need for precision in that language. Students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically.

Connections


• recognize and use connections among mathematical ideas;

• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;

• recognize and apply mathematics in contexts outside of mathematics.

When students can connect mathematical ideas, their understanding is deeper and more lasting. They can see mathematical connections in the rich interplay among mathematical topics, in contexts that relate mathematics to other subjects, and in their own interests and experience.

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Through instruction that emphasizes the interrelatedness of mathematical ideas, students not only learn mathematics, they also learn about the utility of mathematics.

Representation


• create and use representations to organize, record, and communicate mathematical ideas;

• select, apply, and translate among mathematical representations to solve problems;

• use representations to model and interpret physical, social, and mathematical phenomena.

The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas. Consider how much more difficult multiplication is using Roman numerals (for those who have not worked extensively with them) than using Arabic base-ten notation. Many of the representations we now take for granted—such as numbers expressed in base-ten or binary form, fractions, algebraic expressions and equations, graphs, and spreadsheet displays—are the result of a process of cultural refinement that took place over many years. When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically.

The term representation refers both to process and to product—in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself.

Handout 8


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MATHEMATICS – Key Features

Standards Key Features Number and Operations: Students understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Number Sense • what numbers mean • how numbers relate to each other • relative size • how numbers can be thought about and represented in many ways • different number systems

Number and Operations: Students understand the meaning of operations and how they relate to each other.

Operation Sense • what operations mean • effects of operating with numbers

Number and Operations: Students use computational tools and strategies fluently and when appropriate, use estimation.

Computational tools and strategies; Estimation strategies • computational tools • computational strategies • estimation strategies

Measurement: Students understand attributes, units, and systems of units in measurement; and develop and use techniques, tools, and formulas for measuring.

Fluency with Measurement • measurable attributes of objects • measurement units and systems • measurement techniques (e.g., approximation) and tools (e.g.,

rulers)

Geometry and Spatial Sense: Students analyze properties of objects and relationships among the properties.

Properties and Relationships • components (e.g., sides) of figures and solids • properties of figures and solids • relationships among properties • logical reasoning, arguments

Geometry and Spatial Sense: Students use transformations and symmetry to analyze mathematical situations.

Transformations and Symmetry • transformations • symmetry

Geometry and Spatial Sense: Students use visualization and spatial reasoning to solve problems both within and outside of mathematics.

Visual and Spatial Sense • visualization and spatial reasoning

Geometry and Spatial Sense: Students select and use different representational systems, including coordinate geometry.

Representational Systems • different representational systems (e.g., number line, grids, arrays)

to demonstrate the meaning of operations and to represent location, direction, and distance

Patterns, Functions and Algebra: Students understand various types of patterns and functional relationships.

Patterns and Functional Relationships • different types of patterns and functional relationships

Patterns, Functions and Algebra: Students use symbolic forms to represent, model, and analyze mathematical situations.

Symbolic Representation and Notation • symbolic representation to model and analyze situations

Data Analysis, Statistics, and Probability: Students pose questions and collect, organize, and represent data to answer those questions.

Fluency with Data • useful, well-stated questions • data collection • organization and representation of data

Data Analysis, Statistics, and Probability: Students interpret data using methods of exploratory data analysis.

Data Analysis • methods of data analysis

Data Analysis, Statistics, and Probability: Students develop and evaluate inferences, predictions, and arguments that are based on data.

Statistics • Inferences, predictions, and arguments based on data •

Data Analysis, Statistics, and Probability: Students understand and apply basic notions of chance and probability.

Probability • probability notions (e.g., probably, and unlikely) • patterns of events through experiments

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STANDARDS IMPLEMENTATION PLANNING MODEL

�Identify relevant standards.

� Determine acceptable evidence. � Determine l earning experiences that will enable students to learn what they need to know and be able to do. � Teach and collect evidence of student learning. � Assess student work to inform instruction or evaluate student work and report on student learning results. Reteach or repeat process with the next step of relevant standards.

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STANDARDS IMPLEMENTATION PLANNING MODEL

� Identify relevant standards. Revisit the vision in your SID of a literate student because your vision represents a compelling reason for doing what you do. It makes explicit what matters and gives direction and meaning to your actions. Connect the vision of a literate student to the standards. The standards are like your target. Stiggins says, “Think of a target with its bull’s-eye in the middle. That center circle defines the highest level of performance students can achieve…Each outside ring on the target defines a level of performance further from the highest level. As students improve, they need to understand that they are progressing toward the bull’s eye.” � Determine acceptable evidence. Examples of student work provide clear pictures of the ways students are expected to demonstrate what they know and can do. In the absence of this information, at best, the standards can do little more than inform teachers on the content of instruction. � Determine learning experiences. With a clear picture of the kind of work required to meet standards, instruction must be planned to help students produce standards-quality products and performances. � Teach and collect evidence. When collecting and analyzing samples of student work, consideration should be given to the type of evidence to be collected, the conditions for collecting the evidence, the degree of scaffolding, and whether students have had ample time and opportunities to learn. � Assess learning. It is important to understand the difference between assessment and evaluation. These are not the same. Anne Davies explains: “When we assess, we are gathering information about student learning that informs our teaching and helps students learn more. We may teach differently, based on what we find as we assess. When we evaluate, we decide whether or not students have larned what they needed to learn and how well they have learned it. Evaluation is a process of reviewing the evidence and determining its value.” (Making Classroom Assessment Work p.1) Make needed changes to improve students’ learning results. This is the most important part of the process and perhaps the most difficult. Assessment data must be used for instructional decisions where the focus is kept on improving student performance.

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Favorite Sports

Directions: Given the following clues, list some possible data sets. Then select one possible data set to display (bar graph, pictograph, or circle graph). Clues:

1. 25 students were asked what their favorite sport was. 2. More than half chose soccer. 3. Some chose baseball. 4. The amount of student who chose hockey and basketball were the same.

Use the table to list some possible data sets:

Soccer Baseball Hockey Basketball Total Select one possible data set (indicate your choice with a star to the left of the row) to display as a bar graph, pictograph, or circle graph.

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For Met Not yet met I noticed…

Conference requested o

Date(s) received:

Question(s):

Assessed by o teacher o self

o partner o other

Assignment:

Student:

Criteria

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Performance Grid For 3 2 1

Conference requested o

Date(s) received:

Question(s):

Assessed by o teacher o self

o partner o other

Assignment:

Student:

Criteria

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START

SUPPLEMENT CLASSROOM

INSTRUCTION • In-class assistance—tutors,

peer learning • Out-of-class assistance—

non-instructional time tutoring (e.g., after school, recesses, summer); learning labs; specialized classes; family support

MODIFY REGULAR

CLASSROOM INSTRUCTION

Differentiate instruction by modifying one or more of the following:

• Time • Program • Methods

A SIMPLIFIED FLOW CHART OF

INSTRUCTIONAL INTERVENTIONS

CONTINUE EFFECTIVE INSTRUCTION, WITH APPROPRIATE DIFFERENTIATION, EXTENSION, PRACTICE, ENRICHMENT

NO

ASSESSMENT:

Did student attain

standard(s)?

YES

ASSESSMENT:

Did student attain

standard(s)?

NO

REGULAR CLASSROOM

INSTRUCTION

YES

ASSESSMENT:

Did student attain

standard(s)?

504/SPED REFERRAL Level 4

INTENSIVE MULTI-AGENCY PROGRAM

Level 5

Level 1

Level 2

Level 3

YES

NO

ASSESSMENT:

Did student attain

standard(s)?

YES

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DIFFERENTIATING INSTRUCTION

Content Expectation: Student Need Action Plan Reflection



Handout 15


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Performance Task Blueprint Adapted from Jay McTighe, Maryland Assessment Consortium

Title of Lesson: Favorite Sports

Grade Level/Course Title: Grade 3

Developed by: Vickie Shiroma Date: September 2002

Content Standard(s): Data Analysis, Statistics, and Probability Standard 2: Students interpret data using methods of exploratory data analysis.

Benchmark(s): Given information about a data set, build possible representations (e.g., “In this package red is most and there are more blue than green. What could the graph look like?”)

Performance Indicator(s): The student: Determines what the possible representation could look like given information about the data set.

Assessment Task: Directions: Given the following clues, list some possible data sets. Then select one possible data set to display (bar graph, pictograph or circle graph).

Clues: 1. 25 students were asked what their favorite sport was. 2. More than half chose soccer 3. Some chose baseball 4. The amount of students who chose hockey and basketball were the same


Soccer Baseball Hockey Basketball Total + + + =

Select one possible data set to display as a bar graph, pictograph or circle graph. Assessment Tools (e.g., rubric, scoring guide, checklist, criteria, etc.): Criteria Checklist

Criteria MET NOT MET

Comments/Next Steps

Lists possible data sets accurately from given information about data set.

Represents one possible data set accurately.

*See also Rubric for Standard 2

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Overview/Prior Knowledge: Students need prior experiences with problem solving using deductive logic and some experiences with probability. The data analysis lesson for the previous benchmark for this standard introduces the concept of building possible data sets. Students need to have had experiences determining what the data could look like from information about the data.

Approximate Time Required: Five one-hour sessions

Brief Description of Lesson:

1. What’s in the bag?: Put 2 different colored cubes in a brown paper bag. Give students clues as to what might be in the bag (i.e., “There are more red cubes than blue cubes.”). Guide students to create possible representations of that data. (Note: A circle graph is easy to use for this, as the exact amount need not be represented, they would just have to color a circle with more than half of it red, the smaller part – blue). Elicit questions from students regarding what other information they need to determine a better data set (i.e. How many in all? Are there any other colors?). Provide a little more information about what’s in the bag (There are 10 cubes in all.) Guide students to list all the possibilities (it might be a good idea to teach students to use a table to organize their possibilities, see example below) and to revise their representation or make it more specific. (All possibilities could be represented on a chart or table, one of those possibilities could be represented using a bar graph or circle graph.) Repeat this experience numerous times using different colors, different amounts of colors, different total amounts of cubes, or other objects.

(Example of a chart to list possibilities) Red Cubes Blue Cubes Total 6 + 4 = 10 7 + 3 = 10

2. Problem Solving: Give students opportunities to solve problems that they would have to use

deductive logic to solve, given clues (similar to the more concrete, “What’s in the bag?” activity). Here’s an example: 12 people went to a movie. Most of them had popcorn and soda, a few had only soda and 2 had only popcorn. What are the possible data sets? (Again, it might be a good idea to teach students to use a chart to organize their thinking in lis ting possible data sets. See example below.) Get students to explain their reasoning, how they know it could be this or that and how they know it cannot be something else. Provide more opportunities using similar problems by changing the amounts, and the clues. Have students practice representing the possible data sets in charts, bar graphs or circle graphs.

(Example of table to list possible data sets: ) Popcorn and Soda Soda Only Popcorn Only Total

6 + 4 + 2 = 12

3. Assessment Task – Favorite Sports

Resource Materials: Paper bag or other container, colored cubes or other colored objects

Teacher Notes (context of lesson, special conditions): In groups, students could discuss possibilities to figure out what’s in the bag or to solve problems before sharing their ideas with the whole class. Students should definitely have opportunities to discuss and hear other’s ideas. This would provide students with new ideas or new ways of thinking that they might not have thought of on their own. This also gives students opportunities to be a model, as well as opportunities to learn from each other. Attention should be paid to students’ reasoning, get students to justify their thinking in words. They need many opportunities to practice this. They will learn from each other by justifying their thinking aloud. Encourage students’ questioning of each other’s thinking or justifications – let them prove to each other whether they’re right or wrong. This empowers them as learners and thinkers.

Handout 15


21

SAMPLE PERFORMANCE STANDARD FOR MATHEMATICS, GRADE 3 Data Analysis, Statistics, and Probability

Content Standard Benchmarks Performance Indicators 1. Students pose

questions and collect, organize, and represent data to answer those questions.

• Pose questions, and collect and organize small data sets.

• Represent data with bar graphs and with pictures.

The student:

1. Poses questions for data collection from everyday and mathematical situations.

2. Collects (e.g., observations, surveys, questionnaires) and records responses from a specified group.

3. Organizes responses by categorizing data (e.g., tally).

4. Represents data in more than one way (e.g., bar graphs, pictures, pictographs).

2. Students interpret data using methods of exploratory data analysis.

• Use a graph of organized data to build possible data sets.

• Given information about a data set, build possible representations (e.g., “In this package red is most and there are more blue than green. What could the graph look like?”).

The student:

1. Lists information obtained from a graph.

2. Categorizes information in different ways.

3. Determines what the representation of the possible data sets will look like.

4. Determines what the possible representation could look like given information about the data set.

3. Students develop and evaluate inferences, predictions, and arguments that are based on data.

• Answer questions and make predictions based on representations of data.

⇔

The student:

1. Uses information from a graph to answer questions and validate answers.

2. Uses information from a graph to make reasonable inferences or predictions and supports them with an explanation.

⇒

4. Students understand and apply basic notions of chance and probability.

• Explain concepts of certainty and fairness in real world situations.

• Predict the likelihood of random events and test predictions by experiment.

The student:

1. Explains what makes a situation an example of certainty.

2. Explains what makes a situation an example of fairness.

3. Uses logic to draw a conclusion about a real world problem situation.

4. Makes reasonable predictions regarding the likelihood of an event occurring.

5. Tests predictions by conducting an experiment and creates and uses representations to record the results.

Fluency with Data

* Useful, well-stated questions * Data collection * Organiza-tion and representa-tion of data

Data Analysis

* Methods of data analysis

Statistics

* Inferences based on data * Predictions based on data * Arguments based on data

Probability

* Probability notions (e.g., probably and unlikely) * Patterns of events through experiments

Handout 15


22

STUDENT WORK SAMPLE EVIDENCE OF THE

INDICATORS FAVORITE SPORTS

Directions: Given the following clues, list some possible data sets. Then select one possible data set to display (bar graph, pictograph or circle graph). Clues:

1. 25 students were asked what their favorite sport was 2. More than half chose soccer 3. Some chose baseball 4. The amount of students who chose hockey and basketball were the same.


Soccer Baseball Hockey Basketball Total 14 5 3 3 25 16 5 2 2 25 18 5 1 1 25 ¬ 13 6 3 3 25

15 4 3 3 25 20 5 0 0 25

Select one possible data set to display as a bar graph, pictograph or cicle graph. I chose 13 soccer, 6 baseball, 3 hockey, and 3 basketball to graph.

15 14 13 12 11 10

9 8 7 6 5 4 3 2 1

Soccer baseball hockey basketball

2.4 Using the clues given,

possible data sets are correctly stated.

1.4 The table is used to record the

possible data sets. 1.4 As a second representation, a

bar graph is used to record the possible data sets.

1.4

1.4

2.4

Handout 15


23

CRITERIA CHECKLIST:

CRITERIA

MET NOT YET

MET

COMMENTS / NEXT STEPS

Lists possible data sets accurately from the given information about a data set.

x

How many possible data sets do you think there are?

Represents one possible data set accurately.

x

What would your chosen data set look like as a circle graph? What would your chosen data set look like as a pictograph?

RUBRIC for Data Analysis, Statistics and Probability Standard 1: Students pose questions and collect, organize, and represent data to answer those questions.

Exceed At Below

Poses questions where answers to questions posed can lead to further investigations; or poses questions comparing two sets of data.

Poses questions where the answer is dependent on collection of data.

Asks questions that cannot be answered by data collected; or is unclear so that knowing what data to collect is difficult.

Is accurate and efficient while collecting data systematically; and can explain the process used to talk about issues of sampling.

Collects data accurately using appropriate method.

Collects incorrect or inaccurate data and is unable to keep track of data.

Organizes data in different ways to highlight different aspects of the data.

Organizes data by using tables, charts, tallies, etc.

Shows data in a disorganized way or in a way that is unclear.

Represents data fluently and in different ways depending on what needs to be shown regarding purposes and aspects of data.

Represents data with bar graphs, pictographs and other quick sketches of the data.

Represents data incorrectly, insufficiently or unclearly.

RUBRIC for Data Analysis, Statistics and Probability Standard 2: Students interpret data using methods of exploratory data analysis.

Exceed At Below

Obtains necessary information from graph(s) in order to make assumptions about the data.

Obtains information from a graph accurately.

Obtains some or partial data from a graph, or unable to obtain any information from a graph.

Categorizes information from a graph in different ways in order to make generalizations about the data or to compare the data.

Categorizes information from a graph in another way.

Categorizes data from a graph inaccurately, incorrectly or inappropriately.

Represents possible data sets (from a graph) fluently and in different ways depending on what needs to be shown regarding purposes or aspects of the data.

Represents a possible data set (from a graph) appropriately.

Represents a possible data set (from a graph) inappropriately or inaccurately.

Shows more than one or all possible representations of a data set given only information (no graph) about the data set in an organized manner.

Shows a possible representation of a data set given only information (no graph) about the data set.

Shows an incorrect or incomplete representation of a possible data set.

Handout 16


24

Handout 16, Sample Performance Standard for Mathematics, Grade 3 – Data Analysis, Statistics, and Probability (DASP) is an 81/2” x 14” version of the previous pages:

(1) Sample Performance Standard for Mathematics, Grade 3; (2) Favorite Sports – Student Work Sample with Evidence of the Indicators; (3) Criteria Checklist and Rubrics for Data Analysis, Statistics, and Probability Standard 1

and Standard 2.

Handout 17


25

RUBRIC for Data Analysis, Statistics and Probability Standard 1: Students pose questions and collect, organize, and represent data to answer those questions.

Exceed At Below Poses questions where answers to questions posed can lead to further investigations; or poses questions comparing two sets of data.

Poses questions where the answer is dependent on collection of data.

Asks questions that cannot be answered by data collected; or is unclear so that knowing what data to collect is difficult.

Is accurate and efficient while collecting data systematically; and can explain the process used to talk about issues of sampling.

Collects data accurately using appropriate method.

Collects incorrect or inaccurate data and is unable to keep track of data.

Organizes data in different ways to highlight different aspects of the data.

Organizes data by using tables, charts, tallies, etc.

Shows data in a disorganized way or in a way that is unclear.

Represents data fluently and in different ways depending on what needs to be shown regarding purposes and aspects of data.

Represents data with bar graphs, pictographs and other quick sketches of the data.

Represents data incorrectly, insufficiently or unclearly.

Handout 17


26

RUBRIC for Data Analysis, Statistics and Probability

Standard 2: Students interpret data using methods of exploratory data analysis.

Exceed At Below Obtains necessary information from graph(s) in order to make assumptions about the data.

Obtains information from a graph accurately.

Obtains some or partial data from a graph, or unable to obtain any information from a graph.

Categorizes information from a graph in different ways in order to make generalizations about the data or to compare the data.

Categorizes information from a graph in another way.

Categorizes data from a graph inaccurately, incorrectly or inappropriately.

Represents possible data sets (from a graph) fluently and in different ways depending on what needs to be shown regarding purposes or aspects of the data.

Represents a possible data set (from a graph) appropriately.

Represents a possible data set (from a graph) inappropriately or inaccurately.

Shows more than one or all possible representations of a data set given only information (no graph) about the data set in an organized manner.

Shows a possible representation of a data set given only information (no graph) about the data set.

Shows an incorrect or incomplete representation of a possible data set.

Handout 17


27


Standard 3: Students develop and evaluate inferences, predictions, and arguments that are based on data.

Exceed At Below Answers questions with valid inferences and conclusions drawn from the data.

Answers questions accurately and validates answers using information from the graph.

Answers questions incorrectly and/or is unable to validate answers with information from the graph.

Makes inferences and predictions using information from the graph that can be tested and supported with logical explanations.

Makes reasonable inferences or predictions using information from the graph and supports them with an explanation.

Makes inferences or predictions that cannot be supported with information from the graph.

Handout 17


28


Standard 4: Students understand and apply basic notions of chance and probability.

Exceed At Below Gives explanation of certainty that includes the range of probabilities from 0-1 and gives examples and counterexamples.

Explains and gives examples of situations that demonstrate the concept of certainty.

Shows incomplete or incorrect understanding of the concept of certainty.

Uses a list of possible outcomes to determine fairness of situations and shows understanding of what it means for situations to be fair.

Explains and gives examples of situations that demonstrate the concept of fairness.

Shows incomplete or incorrect understanding of the concept of fairness.

Uses reasoning to develop arguments to defend conclusions/ decisions made and shows why alternatives were not justified.

Explains how conclusions/decisions were logically determined based on data about a real world situation.

Makes conclusions/decisions that are not supported by data and is unable to give reasons to justify them.

Explains how predictions were made and makes predictions on the probability of occurrences.

Makes reasonable predictions about the likelihood of events based on prior experiences and information given.

Makes predictions that are not logically based on information available.

Conducts many tests to strengthen arguments used to validate or refute predictions.

Tests predictions about the likelihood of events occurring through experiments involving collecting, organizing, and representing data.

Tests are not connected to the predictions made or experiment is done incorrectly.


29

POSTER MASTERS


30

Poster 1

Mathematics Content Standards for Data Analysis, Statistics, and Probability Grade Cluster Benchmarks

CONTENT STANDARDS

K - 1

2 – 3

4 - 5

1. Students pose

questions and collect, organize, and represent data to answer those questions.

• Pose questions, and collect and organize small data sets (e.g., tally).

• Represent the collected data with objects, and with pictures.

• Pose questions, and collect and organize data.

• Represent data with bar graphs and with pictures.

• Design investigations requiring data collections, including measured data.

• Systematically collect and organize data (e.g., using tables, line graphs, or pie charts).

• Translate among different representations of the same data.

• Analyze parts of a graph, (e.g., axes, scale, legend).

2. Students interpret

data using methods of exploratory data analysis.

• Describe parts of the organized data (e.g., “How many more red than blue?”) and the data as a whole (e.g., “What is the total?”).

• Identify those parts of the data that have special characteristics (e.g., “Which color was most? Least?”)

• Use a graph of organized data to build possible data sets.

• Given information about a data set, build possible representations (e.g., “In this package red is most and there are more blue than green. What could the graph look like?”).

• Describe the shape and important features of a set of organized data (e.g., range, mean, mode, median, where appropriate).

• Classify and describe data in different ways; analyze information highlighted by different classifications (e.g., order categorical data alphabetically as opposed to order by popularity).

• Compare related data sets.

3. Students develop

and evaluate inferences, predictions, and arguments that are based on data.

• Use representation to answer posed and related questions.

• Answer questions and make predictions based on representations of data.

• Describe how data collections methods can impact the nature of the data set.

• Explain the concept of representativeness of a sample.

• Describe the population based on a given sample.

• Propose and justify conclusions based on data.

4. Students understand

and apply basic notions of chance and probability.

• Identify certainty and fairness in real world situations.

• Explain concepts of certainty and fairness in real world situations.

• Predict the likelihood of random events and test predictions by experiment.

• Formulate questions or hypotheses based on initial data collection and design further studies to explore them.

• Describe the degree of likelihood of random events with fractions between 0 and 1.

• Estimate and test by experiment the probabilities of outcomes.

• List all possible outcomes of a simple experiment.


31

Poster 2

Mathematics Content Standards - Data Analysis, Statistics, and Probability Grade Cluster Benchmarks

CONTENT STANDARDS

6-8

9-12

1. Students pose questions and collect, organize, and represent data to answer those questions.

• Design experiments and surveys with consideration for issues of sampling (e.g., size, bias).

• Describe different types of data and organize collections of data.

• Choose, create, and use various representations of data (e.g., histograms, stem and leaf plots, box and whisker plots).

• Design and carry out investigations or experiments with two variables.

• Select appropriate methods for collecting, recording, organizing, and representing data; and describe how a change in representation affects the likely interpretation of the information.

2. Students interpret data using methods of exploratory data analysis.

• Describe and interpret measures of the center of a data set and know which measure to use for particular situations.

• Describe and interpret the spread of a set of data (e.g., range, quartile).

• Analyze and interpret relationships between variables (e.g., scatterplots).

• Analyze different representations of the same data to determine effects of the representation.

• Compute, identify and interpret measures of center and spread (including standard deviation).

• Look for patterns in data and understand their use in interpretation of the data.

• Explain how sample size or transformations of data affect shape, center, and spread.

• Explain trends and use technology to determine how well different models fit data (e.g., line of best fit).

3. Students develop and evaluate inferences, predictions, and arguments that are based on data.

• Develop conclusions about a characteristic in the population.

• Explain that differences in data may indicate an actual difference in the populations from which the data were collected or that the differences may result from random variation in the samples.

• Use data to answer the questions that were posed, describe the limitations of those answers, and pose new questions that arise from the data.

• Identify good models for phenomena (e.g., exponential model for population growth).

• Apply models to predict unobserved outcomes.

• Evaluate conclusions based on data and support inferences with valid arguments.

4. Students understand and apply basic notions of chance and probability.

• Judge the likelihood of uncertain events and connect these judgments to percents or proportions.

• Understand what it means for events to be equally likely and a game or process to be fair.

• Calculate theoretical probabilities based on assumptions about sample space, and compare with experimental results.

• Identify relationships among events (e.g., inclusion, disjoint, complementary, independent, and dependent).

• Compute probabilities of two events under different relationships, unions, intersections.

• Use fundamental counting principle, permutations, and combinations as counting techniques to solve problems.

• Compute the theoretical probabilities of repeated experiments with replacement and repeated experiments without replacement.

• Recognize random variables in real situations (e.g., insurance, life expectancy) and estimate and compute expectations.

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