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Curriculum Focal Points Curriculum Focal Points

Mathematics & Science PartnershipsMathematics & Science PartnershipsAnnual State Coordinators MeetingAnnual State Coordinators Meeting

June 15, 2006June 15, 2006

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Curriculum Focal PointsCurriculum Focal Points

Charge from the NCTM Charge from the NCTM BoardBoard• To develop a document to guide To develop a document to guide

curriculum focus at the district and curriculum focus at the district and state levels state levels

• To describe a set of 3-5 curriculum To describe a set of 3-5 curriculum focal points per grade that serve as a focal points per grade that serve as a foundation for a preK-8 curriculum foundation for a preK-8 curriculum framework framework

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Curriculum Focal PointsCurriculum Focal Points

Three Lenses – Three Lenses – the selection the selection processprocess• Is it mathematically important?Is it mathematically important?

– Preparation for further studyPreparation for further study– Use in applications inside and outside of Use in applications inside and outside of

schoolschool• Does it fit with what is known about Does it fit with what is known about

learning mathematics?learning mathematics?• Does it logically connect with the Does it logically connect with the

mathematics in earlier and later mathematics in earlier and later grade levels?grade levels?

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Curriculum Focal PointsCurriculum Focal Points

Members of the Writing Members of the Writing TeamTeam

• Janie Schielack, Janie Schielack, ChairChair

• Doug ClementsDoug Clements• Sharon LewandowskiSharon Lewandowski• Randy CharlesRandy Charles• Paula DuckettPaula Duckett• Sybilla BeckmanSybilla Beckman• Emma TrevinoEmma Trevino

• Rose Zbiek, Rose Zbiek, Chair, Chair, NCTM Essential NCTM Essential Understandings Writing Understandings Writing TeamTeam

• Francis (Skip) Francis (Skip) Fennell, Fennell, NCTM President NCTM President Elect, Board LiaisonElect, Board Liaison

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ReviewersReviewersDavid BressoudDavid BressoudBill BushBill BushJoan Ferrini-MundyJoan Ferrini-MundyAnne CollinsAnne CollinsLinda GojakLinda GojakJeremy KilpatrickJeremy KilpatrickDenise MewbornDenise Mewborn

Ann MikesellAnn MikesellJim Milgram Jim Milgram Barbara ReysBarbara ReysNorm WebbNorm WebbMike ShaugnessyMike ShaugnessyNorma Torres-Norma Torres-

MartinezMartinez

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Additional Reviewers - Additional Reviewers - MathematiciansMathematicians

• Susan AddingtonSusan Addington Yorum SagherYorum Sagher• Richard AskeyRichard Askey Marjorie SenechalMarjorie Senechal• Tom BanchoffTom Banchoff Maria TerrellMaria Terrell• Hyman BassHyman Bass Virginia WarfieldVirginia Warfield• Jerome DancisJerome Dancis Steve WilsonSteve Wilson• William DwyerWilliam Dwyer• David HendersonDavid Henderson• Rodger HoweRodger Howe

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Additional Reviewers – Math Additional Reviewers – Math EducatorsEducators

• Michael BattistaMichael Battista Joe RosensteinJoe Rosenstein• Gail BurrillGail Burrill Susan Jo RussellSusan Jo Russell• Cliff KonoldCliff Konold Jan ScheerJan Scheer• Karen FusonKaren Fuson Grayson WheatleyGrayson Wheatley• Glenda LappanGlenda Lappan• Mary LindquistMary Lindquist• Johnny LottJohnny Lott• Gerald RisingGerald Rising

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Additional Reviewers – Policy Additional Reviewers – Policy MakersMakers

• William SchmidtWilliam Schmidt• Cathy Dillender Cathy Dillender • Steve LeinwandSteve Leinwand• Debbie NixDebbie Nix• Patsy Wang-IversonPatsy Wang-Iverson• Iris WeissIris Weiss

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Additional Reviewers - Additional Reviewers - SupervisorsSupervisors

• DistrictDistrict– Nancy AcconciamessaNancy Acconciamessa– John CarterJohn Carter– Eric HartEric Hart– Lisa KasmerLisa Kasmer– Jana PalmerJana Palmer– Caroline PiangerelliCaroline Piangerelli– Kay SammonsKay Sammons– Dorothy StrongDorothy Strong

• StateState– Frank Marburger – PennsylvaniaFrank Marburger – Pennsylvania– Donna Watts - MarylandDonna Watts - Maryland

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Additional Reviewers - GroupsAdditional Reviewers - Groups

• College Board – via John DosseyCollege Board – via John Dossey• Dallas area – via Dinah ChancellorDallas area – via Dinah Chancellor• EDC – via Paul GoldenbergEDC – via Paul Goldenberg• Howard County (MD) – via Sharon Howard County (MD) – via Sharon

LewandowskiLewandowski• LTCAC Committee – via Gregg McMannLTCAC Committee – via Gregg McMann• D.C. Teachers – via Paula DuckettD.C. Teachers – via Paula Duckett• AMTE – via Rose ZbiekAMTE – via Rose Zbiek• MSDE – via Donna WattsMSDE – via Donna Watts

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Additional Reviewers – NCTM Additional Reviewers – NCTM BoardBoard

• Jennie BennettJennie Bennett• Cindy BryantCindy Bryant• David DeCosteDavid DeCoste• Shelley FergusonShelley Ferguson• Bonnie HagelbergerBonnie Hagelberger• Kathy HeidKathy Heid• Mari MuriMari Muri• Richard SeitzRichard Seitz

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PreK (4 year olds) K

Meaning of Whole NumbersAttributesShapes & Spatial Relationships

Representations of NumberSorting & ClassifyingGeometric Shapes & Space

1st 2nd

Number RelationshipsMeaning of Addition & SubtractionShapes

Place valueAddition & Subtraction (basic number combinations & strategies)Linear Measurement

3rd 4th

Addition & Subtraction ComputationMultiplication & Division (meaning, basic number combinations & strategies)Properties of Polygons

Multiplication ComputationMeaning of Fractions & Decimals Perimeter & Area

5th 6th

Division ComputationDecimal ComputationProperties of SolidsRepresentations of Data

Fraction Computation Meaning of IntegersMeaning of PercentVolume/Capacity & Surface Area

7th 8th

Ratio, Proportionality & PercentInteger ComputationCongruence, Similarity &SymmetryComparisons of Data Sets

Expressions & Equations ProbabilityGeometrical Reasoning & Proof

FIRST DRAFT – April 2005

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Grade Level Curriculum Focal PointsGrade 3 Number and Operations: Understand

Multiplication & Division Meanings & Develop Strategies for the Basic Multiplication Facts and the Related Division Facts

Number & Operations: Understand the Meaning of Fraction and Fraction Equivalence

Geometry: Describe and Analyze Properties of Two-Dimensional Shapes

Grade 4 Number & Operations: Develop Immediate Recall with Multiplication Facts and the Related Division Facts and Understanding of and Fluency with a Standard Procedure for Multiplying Whole NumbersNumber & Operations: Understand the Meaning of Decimals and Connections Between Fractions and DecimalsMeasurement: Understand and Find the Area of Two-Dimensional Shapes

FEBRUARY 2006 - DRAFTNote: title page

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Grade 4 Curriculum Focal Points Connections to the Focal Points

Number & Operations: Develop Immediate Recall with Multiplication Facts and the Related Division Facts and Understanding of and Fluency with a Standard Procedure for Multiplying Whole Numbers Students use understandings of multiplication to develop immediate recall with basic multiplication facts and the related division facts. They apply their understanding of models for multiplication (i.e., equal-sized groups, array and area, number line), place value, and properties of operations (e.g., distributive property) as they develop, discuss, and use efficient, and accurate methods to multiply multi-digit whole numbers. They select and accurately apply appropriate methods to estimate products and mentally calculate products depending upon the context and the numbers involved. They develop fluency with standard procedures for multiplication of whole numbers through three-digit by two-digit numbers.

Algebra Readiness: Students continue identifying, describing and extending numeric patterns involving all operations, and non-numeric growing or repeating patterns. Properties of multiplication are revisited.

Geometry: Properties of two-dimensional shapes are revisited from Grade 3 as students find areas of polygons at Grade 4. Through the design and analysis of simple tilings and tessellations, students deepen their understanding of two-dimensional space including measuring and classifying angles. Students classify and measure angles. Data Analysis: Frequency tables, bar graphs, picture graphs, and line plots are continued from Grade 3 as problem contexts and tools for solving problems. Students apply their understanding of place value to develop stem and leaf plots and use them to solve problems.

Number and Operations: Students develop understanding of strategies for multi-digit division using models based on division as the inverse of multiplication, division as partitioning, and division as successive subtraction.

Continued Development in Number and Operations: Students extend their understanding of place value and ways of representing numbers with numbers to 100,000 in various contexts. Fraction equivalence is revisited in the work with decimals. Students’ work with multiplication and division facts is applied to techniques for simplifying fractions.

Number & Operations: Understand the Meaning of Decimals and Connections Between Fractions and Decimals Students understand that decimal notation is the use of the base-ten numeration system to represent a number, including numbers between 0 and 1, between 1 and 2, and so forth. Students extend their understandings of fractions to reading and writing decimals greater than or less than one, identifying equivalent decimals, comparing and ordering decimals, and estimating decimal/fractional amounts. They connect equivalent fractions and decimals by comparing models and symbols and locating them together on the number line.

Measurement: Understand and Find the Area of Two-Dimensional Shapes Students recognize area as an attribute of the region enclosed by a two-dimensional shape. They understand that area can be quantified by finding the total number of same-size units of area it takes to cover the shape without gaps or overlap. They understand that a 1-unit by 1-unit square is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for estimating and measuring area. Students connect area measure to the area model used to represent multiplication, and they use this connection to justify and find relationships among the formulas for the areas of different polygons and solve problems involving area. They measure necessary attributes of shapes in order to use an area formula.

FEBRUARY 2006 - DRAFT

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FINALGrade 4 Curriculum Focal Points

Connections to the Focal Points

Number and Operations and Algebra: Developing Quick Recall of Multiplication Facts and Related Division Facts and Fluency with Whole Number MultiplicationStudents use understandings of multiplication to develop quick recall of the basic multiplication facts and related division facts. They apply their understanding of models for multiplication (i.e., equal-sized groups, arrays, area models, equal intervals on the number line), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to multiply multi-digit whole numbers. They select and accurately apply appropriate methods to estimate products and mentally calculate products depending upon the context and the numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers; understand why the procedures work based on place value and properties of operations; and use them to solve problems.

Algebra: Students continue identifying, describing and extending numeric patterns involving all operations and non-numeric growing or repeating patterns in order to develop understanding of the use of a rule to describe a sequence of numbers or objects. Geometry: Students extend their understanding of properties of two-dimensional shapes as they find areas of polygons. They extend their work with symmetry and congruence in Grade 3 to include transformations, including those that produce line and rotational symmetry. Through the use of transformations to design and analyze simple tilings and tessellations, students deepen their understanding of two-dimensional space. Measurement: As part of understanding two-dimensional shapes, students measure and classify angles.Data Analysis: Frequency tables, bar graphs, picture graphs, and line plots are continued from Grade 3 as problem contexts and tools for solving problems. Students apply their understanding of place value to develop and use stem-and-leaf plots. Number and Operations: Based on Grade 3 work, students extend their understanding of place value and ways of representing numbers to 100,000 in various contexts. They use estimation in determining the relative size of an amount or distance. Students develop understandings of strategies for multi-digit division using models based on division as the inverse of multiplication, division as partitioning, and division as successive subtraction. Fraction equivalence is extended in the work with decimals. Students’ work with multiplication and division facts in Grade 3 is applied to techniques for simplifying fractions.

Number and Operations: Developing Understanding of Decimals, including Connections Between Fractions and DecimalsStudents understand that decimal notation is the use of the base-ten numeration system to represent a number, including numbers between 0 and 1, between 1 and 2, and so forth. Students extend their understanding of fractions to reading and writing decimals greater than or less than one, identifying equivalent decimals, comparing and ordering decimals, and estimating decimal/fractional amounts in problem-solving situations. They connect equivalent fractions and decimals by comparing models and symbols and locating them together on the number line.

Measurement: Developing Understanding of Area and Determining the Areas of Two-Dimensional ShapesStudents recognize area as an attribute of two-dimensional regions. They understand that area can be quantified by finding the total number of same-size units of area required to cover the shape without gaps or overlap. They understand that a 1-unit by 1-unit square is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating and measuring area. Students connect area measure to the area model used to represent multiplication, and they use this connection to justify the formula for finding the area of a rectangle.

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Consider – from the Consider – from the PrefacePreface

• Organizing curriculum around focal points, Organizing curriculum around focal points, with a clear emphasis on the processes…, with a clear emphasis on the processes…, and, particularly problem solving…and, particularly problem solving…

• Immediate uses…stimulate discussionImmediate uses…stimulate discussion• Long term opportunity…starting point for Long term opportunity…starting point for

continued dialogue and discussion…next continued dialogue and discussion…next generation of curriculum standards, tests, generation of curriculum standards, tests, and textbooks.and textbooks.

• Use them, discuss them, argue about Use them, discuss them, argue about them, refine them.them, refine them.

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Consider – Why Identify Focal Consider – Why Identify Focal Points?Points?

• What mathematics should be the What mathematics should be the focus of instruction and student focus of instruction and student learning at the PreK-8 level?learning at the PreK-8 level?

• A Quest for Coherence provides A Quest for Coherence provides oneone possible responsepossible response to the question to the question of how to organize curriculum…of how to organize curriculum…

• ……wide variety of mathematics wide variety of mathematics curriculum standards, little curriculum standards, little consensus (see Reys, et al).consensus (see Reys, et al).

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Number of 4th grade learning expectations per state by content strand

Number&

Operations

Geometry Measurement Algebra Data Analysis, Probability & Statistics Total Number of LEs

California 16 11 4 7 5 43

Texas 15 7 3 4 3 32

New York 27 8 10 5 6 56

Florida 31 11 17 10 20 89

Ohio 15 8 6 6 13 48

Michigan 37 5 11 0 3 56

New Jersey 21 10 8 6 11 56

North Carolina 14 3 2 3 4 26

Georgia 23 10 5 3 4 45

Virginia 17 8 11 2 3 41

Reys, et al, 2006

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Consider – What are Consider – What are CFPs?CFPs?

• ……represent a set of important represent a set of important mathematical topics for each level…mathematical topics for each level…

• ……organizing structures, key topics, organizing structures, key topics, foundations for further mathematics foundations for further mathematics learning…learning…

• ……situated within applications of the situated within applications of the processes one view…, which is processes one view…, which is different from a set of grade-level different from a set of grade-level mastery objectives..mastery objectives..

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Consider – How Should CFPs be Consider – How Should CFPs be used?used?

• ……directed to those who are responsible for directed to those who are responsible for development of mathematics curriculum…development of mathematics curriculum…

• This document is presented as This document is presented as an an exampleexample from which the next from which the next generation…might be built.generation…might be built.

• ……identifying for curriculum developers identifying for curriculum developers grade level targets…grade level targets…

• This document does not include This document does not include instructional approaches…instructional approaches…

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Consider – How related…to Consider – How related…to PSSM?PSSM?

• ……the curriculum focal points are the curriculum focal points are targeted mostly toward content. The targeted mostly toward content. The mathematical processes described mathematical processes described within PSSM are expected to be within PSSM are expected to be addressed in the implemented addressed in the implemented curriculum through…to discuss and curriculum through…to discuss and validate their mathematical validate their mathematical thinking…and apply the mathematics thinking…and apply the mathematics they are learning to solve problems…they are learning to solve problems…

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Consider – How related…to PSSM? Consider – How related…to PSSM? (cont.)(cont.)

• The integrated nature of the CFPs is The integrated nature of the CFPs is highlighted…many relate to more than one highlighted…many relate to more than one content strand.content strand.

• Connections to the focal points…serve two Connections to the focal points…serve two purposes:purposes:– Recognize the need for introductory and Recognize the need for introductory and

continuing experiences related to focal points continuing experiences related to focal points that may be identified at other grade levels.that may be identified at other grade levels.

– Identify ways in which a grade-level’s focal Identify ways in which a grade-level’s focal points can be used to support learning in points can be used to support learning in relation to strands that are not focal points at relation to strands that are not focal points at that grade level.that grade level.

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Why is this important? Why is this important?

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Why should NCTM do Why should NCTM do this?this?

• Curricular CoherenceCurricular Coherence• Need (see next two slides)Need (see next two slides)• OpportunityOpportunity

– National Mathematics PanelNational Mathematics Panel– ACIACI

• Math Now – next yearMath Now – next year– PACEPACE

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Recommendations:Recommendations:Specification of Learning Specification of Learning

ExpectationsExpectations• Identify major goals or Identify major goals or focal pointsfocal points at at

each grade level. At each grade we each grade level. At each grade we recommend a general statement of major recommend a general statement of major goals for the grade be stated. These goals goals for the grade be stated. These goals may emphasize a few strands of may emphasize a few strands of mathematics or a few topics within mathematics or a few topics within strands.strands.

• ……This may also help reduce superficial This may also help reduce superficial treatment of many mathematical topics, a treatment of many mathematical topics, a common criticism of the U.S. mathematics common criticism of the U.S. mathematics curriculum.curriculum.

Reys, et al, in press.

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Recommendations:Recommendations:Collaborate to Promote ConsensusCollaborate to Promote Consensus• ……states will need to work together to states will need to work together to

create some level of consensus about create some level of consensus about important curriculum goals at each grade.important curriculum goals at each grade.

• ……if states build their curriculum standards if states build their curriculum standards from a “core curriculum” offered by from a “core curriculum” offered by national groups such as the NCTM, College national groups such as the NCTM, College Board, and/or Achieve.”Board, and/or Achieve.”

Reys et al, in press

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And…And…• Provide guidance and resources for establishing Provide guidance and resources for establishing

and enacting mathematics curriculum that is and enacting mathematics curriculum that is coherent, focused, well-articulated and consistent coherent, focused, well-articulated and consistent with PSSM (curriculum).with PSSM (curriculum).

• Engage in political and public advocacy to focus Engage in political and public advocacy to focus decision makers on improving learning and decision makers on improving learning and teaching mathematics (advocacy)teaching mathematics (advocacy)

• Advance professional development by creating a Advance professional development by creating a coherent framework of audience-specific products coherent framework of audience-specific products and services (professional development)and services (professional development)

NCTM’s - STRATEGIC PLANNING OBJECTIVESNCTM’s - STRATEGIC PLANNING OBJECTIVES

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The ImportanceThe Importanceofof

Big Picture ThinkingBig Picture ThinkingBasic Mathematical Skills (NCSM, 1977)Basic Mathematical Skills (NCSM, 1977)

An Agenda for Action (NCTM, 1980)An Agenda for Action (NCTM, 1980)Curriculum & Evaluation Standards (NCTM, 1989)Curriculum & Evaluation Standards (NCTM, 1989)

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Questions?Questions?

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Motion to:Motion to:• Approve the PreK-8 Curriculum Focal Approve the PreK-8 Curriculum Focal

Points as a position of the National Points as a position of the National Council of Teachers of Mathematics.Council of Teachers of Mathematics.

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QuestionsQuestions

Next StepsNext Steps

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Francis (Skip) Fennellffennell@nctm.org OR ffennell@mcdaniel.edu