DOCUMENT RESUME - ERICDOCUMENT RESUME ED 287 719 SE 048 679 TITLE Mathematics: Comprehensive...

DOCUMENT RESUME

ED 287 719 SE 048 679

TITLE Mathematics: Comprehensive Curriculum Goals. A Modelfor Local Curriculum Development.

INSTITUTION Oregon State Dept. of Education, Salem.PUB DATE Oct 87NOTE 225p.; For a related document, see SE 048 678.PUB TYPE Guides Non-Classroom Use (055)

EDRS PRICE MF01/PC09 Plus Postage.DESCRIPTORS Academic Achievement; Curriculum Development;

Educational Objectives; *Elementary SchoolMathematics; Elementary Secondary Education;*Mathematics Curriculum; *Mathematics Instruction;*Secondary School Mathematics; Staff Development;State Curriculum Guides; *State Standards; StudentEducational Objectives; Student Evaluation

IDENTIFIERS *Oregon

ABSTRACTIn June 1984, the State Board of Education adopted

the Oregon Action Plan for Exctalence which established the directionfor school improvement in the state for the next decade. A centralconcept of the Action Plan is that while the sate will determinewhat must be taught in public schools, the schools will determine howit will be taught. This document is intended to provide the essentialinformation which local districts need to merge state curriculumexpectations with their own local determinations for mathematicsK-12. The report contains the following sections: (1) introduction;(2) executive summary of content strands; (3) grade level teachingemphasis; (4) instructional themes; (5) staff development; (6)organizational patterns and teaching considerations; (7)comprehensive goals and outcomes (comprising more than half thedocument and charting expectancies for grades K through 8 as well asgrade 11); and (8) bibliography. (RH)

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Reproductions supplied by EDRS are the best that can be madefrom the original document.

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MATHEMATICSComprehensive Curriculum GoalsA Model for Local Curriculum Development

October 1987

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/ g9

Oregon Department of Education700 Pringle Parkway SE

Salem, Oregon 97310-0290

Verne A. DuncanState Superintendent of Public Instruction

OREGON SCHOOLS--A TRADITION OF EXCELLENCE

BEST COPY AVAILABLE

U S DEPARTMENT OF EDUCATIONOffice of Educational Research and Improvement

EDUCATIONAL RESOURCES INFORMATIONXCENTER (ERICA

This document has been reproduced asreceived from the person or organizationoriginating 1

[. Minor changes have been made to improvereproduction Qv WO

Points Of JI er or opinions stated in this docu

mem do not necessarily represent offictalOEM position or policy

"PERMI TO

MATE AS B GRANTED BYTHIS

TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)"

rla

FOREWORD

In June 1984 the State Board of Education adopted the OregonAction Plan for Excellence which established the direction forschool improvement in the state over the next decade. The ActionPlan drew upon the insights of teachers, administrators, schoolboard members and community and business leaders.

A central concept of the Action Plan is that while the state willdetermine WHAT must be taught in public schools, the schools willdetermine HOW it will be taught. This document is intended toprovide the essential information which local districts need tomerge state curriculum expectations with their own localdeterminations for Mathematics.

All who have joined in the spirit of the Action Plan forExcellence have shared a commitment to high-quality performance.We are continuing to learn about how to provide children with thevery best in public education, and we welcome your comments andquestions. For further information about this guide, contact theSpecialist for Mathematics Education, 378-3566.

Verne A. DuncanState Superintendentof Public Instruction

4

It is the policy of the State Board of Education and a priorityof the Oregon Department of Education that there will be nodiscrimination or harassment on the grounds of race, color, sex,marital status, religion, national origin, age or handicap in anyeducational programs, activities, or employment. Persons havingquestions abcut equal opportunity and nondiscrimination shouldcontact the State Superintendent of Public Instruction at theOregon Department of Education.

3206619873000

ii 5

ACKNOWLEDGMENTS

Grateful acknowledgment is made to the followingpeople for the many hours they have contributedto developing ideas and writing materials forthis guide. They studied the literature, thenshared their thinking and experiences asclassroom teachers and as teachers of teachersat numerous inservice and preservice teachereducation programs. School districts, colleges

and math projects released these people for workon this publication. Special thanks goes tocEief writer Judy Johnson, a middle schooltez.cher and math specialist, for putting it all

together. Fond appreciation is acknowledged toMeredith and Oscar Schaaf for their career ofteamed help, advice and consulting wit'..1 the

mathematics education community.

COMMON CURRICULUM GOALS FOR MATHEMATICS OREGONDEPARTMENT OF EDUCATION STEERING/WRITING TEAM

Phil Bartsch, Carus School District 29Jack Hopper, Portland School District 1JJudy Johnson, Eugene School District 4JDonna McBride, Portland SO:L*1 District 1JDiane (Price) Stone, Philomath SD 17JOscar Schaaf, University of OregonTom Stone, Eugene School District 4JRon Zaraza, Portland School District 1J

CHIEF WRITERJudy Johnson, Eugene School District 4J

HELPER/ADVISOR/CONSULTANTMeredith Schaaf, Lane ESDOscar Schaaf, University of Oregon

iii

Two major curriculum documents served as theprimary basis for the development of this

publication: (1) Essential Learning Skills and

(2) The Oregon Mathematics Concept Papers. The

concept papers were developed as an OregonMathematics Education Council (OMEC) curriculum

project. Nearly 100 volunteers worked on their

own time for about two years in study, debate,

presentations, and writing. Their conclusions

present "forwardlooking" ideas for school

mathematics. Their names and schools at the

time the OMEC curriculum project was launched

are listed below.

OREGON MATHEMATICS CONCEPT PAPERSOMEC Writing Teams

OMEC PRESIDENTS/PROJECT FACILITATORSMike Morgan, LinnBenton Community CollegeRon Morgali, Western Oregon State CollegeGary Musser, Oregon State UniversityRon Smit, University of Portland

OREGON MATHEMATICS CONCEPT PAPER NO. 1:ELEMENTARY MATHEMATICS CURRICULUM, K-5

Jill Board, Oakridge Elementary SchoolDouglas Cruikshank, Linfield CollegeBarbara Fuhrer, Clear Lake Elementary School

Jan Heaton, Oak Elementary SchoolDennis Jones, Lincoln County School DistrictSusan Knapp, Chapman Elementary SchoolDiane (Price) Stone, Philomath Elem SchoolKathryn Warrior, Yaquina View Elem School

OREGON MATHEMATICS CONCEPT PAPER NO. 2: MIDDLESCHOOL MATHEMATICS, 6-8

Fred Board, Westridge Junior High SchoolDianne Erickson, Talent Junior High SchoolArlene Foss, Ainsworth Elementary SchoolJudy Johnson, Kennedy Middle SchoolScott McFadden, Henry D. Sheldon High SchoolDon McNair, McLoughlin Junior High SchoolJulie (Nakata) Keener, Beaumont Middle SchoolOscar Schaaf, University of OregonLarry Slceman, Philomath High SchoolJim Specht, Neil Armstrong Middle SchoolMarcia Swanson, Calapooia Middle SchoolKathryn Warrior, Yaquina View Elem SchoolJoanne Wilkie, Tubman Middle School

OREGON MATHEMATICS CONCEPT PAPER NO. 3:GEOMETRY, K-12

William F. Burger, Oregon State UniversityBarbara Culpepper, Franklin High SchoolDennis Dedrick, Medford Mid High SchoolJean Eversole, McKay High SchoolTony Gerlicz, Catlin Gabel SchoolVirginia O'Donnell, Binnsmead Middle SchoolDiane (Price) Stone, Philomath Elem SchoolLarry Sleeman, Philomath Middle School

OREGON MATHEMATICS CONCEPT PAPER NO. 4:STATISTICS AND PROBABILITY, K-12

Mildred Bennett, Portland State UniversityDavid Bohlmann, Hood River Valley High SchoolCindy Brown, Elmira High SchoolTim Hahn, Creston Elementary SchoolKaren Higgins, Oaklea Middle SchoolRon Morgali, Western Oregon State CollegeLeona Pittenger, Oaklea Middle SchoolFred Rectanus, Portland School District 1JMarcia Swanson, Calapooia Middle SchoolJoanne Wilkie, Tubman Middle School

8iv

OREGON MATHEMATICS CONCEPT PAPER NO. 5: AN

INTEGRATED MATHEMATICS CURRICULUM FOR HIGH SCHOOLCharles Barnhart, Obsidian Junior High SchoolDavid Erickson, Mt. View Senior High SchoolMike Gardner, Redmond High SchoolDaniel Gossack, North Salem High SchoolWendell Hall, North Eugene High SchoolFloyd Halvorsen, Churchill High School

Seymour Hanfling, Graduate Student at U of 0Karen Higgins, Oaklea Middle SchoolKathie Hledik, Willamette High SchoolDon Leslie, North Eugene High School

Judi Mathis, Graduate Student at U of 0Carolyn Miller, South Salem High SchoolKathy Pfaendler, Mt. View Junior High SchoolRick Purn, Pilot Butte Junior High SchoolOscar Schaaf, University of OregonRon Steffani, Southern Oregon State College"ick Thomas, South Eugene High SchoolLynne Tracy, Henry D. Sheldon High SchoolJohn Vogt, Springfield High School

OREGON MATHEMATICS CONCEPT PAPER NO. 6: OREGONGRADUATE REQUIREMENTS: MATHEMATICS

Mike Bolduan, Catlin Gabel School

Jack Hopper, Portland School District 1JSue McGraw, Lake Oswego High SchoolKatie Miller, Columbia High SchoolRichard L. Moon, Crescent Valley High SchoolHoward Wilson, Oregon State UniversityRon Zaraza, Wilson High School

OREGON MATHEMATICS CONCEPT PAPER NO. 7:VOCATIONAL AND TECHNICAL MATHEMATICS

Bob Holly, Oakridge High School

Richard L. Moon, Crescent Valley High SchoolJudy Paulus, Central High SchoolDoug Stensrud, Wilson High SchoolRon Zaraza, Wilson High School

OMEC PAPER: ENRICHMENT/ACCELERATION (Unpublished)Jim Stewart, Oregon Institute of Technology

9

TABLE OF CONTENTS

Foreword i

Acknowledgments iii

Introduction 1

Executive Summary of Content Strands 5

Grade Level Teaching Emphases 8

Instructional Themes 23

Staff Development 32

Organizational Patterns and Teaching Considerations . . 34

Comprehensive Goals and Outcomes 42

Bibliography 103

vi0

INTRODUCTION

THE OREGON ACTION PLAN FOR EXCELLENCE

The Action Plan identified seven areas ofimprovement, one of which called for a state-

wide definition what students should learn:

The Oregon Department of Education,working with local school districts

and higher education institutions,shall define the required commoncurriculum goals for elementary andsecondary schools in terms of thelearning skills and knowledge stu-dents are expected to possess as aresult of their schooling experience.

Local school districts, with assis-tance from the Oregon Department ofEducation, shall be responsible fororganizing the curriculum and deliv-ering instruction to aclJeve the

common curriculum goals.

Common Curriculum Goals

The first stage in defining the Common Curricu-lum Goals was to develop the Essential LearningSkills--the basic skill and performance expecta-tions for all students in the areas of reading,

writing, speaking, listening, mathematics,reasoning and study skills. The second stage

was to develop Common Knowledge and Skills in

individual subject areas. Together with theEssential _Learning Skills, they form the Common

Curriculum Goals for all students.

1

A. Essential Learning Skills

The Essential Learning Skills are consideredbasic to all students' learning, and allteachers are expected to provide instruction

in these skills. Only to the degree thatstudents develop these skills and form the

habit of using them, can instruction insubject matter areas be successful. The

skills are not specific to any one disci-pline but provide a link across all disci-

plines. Furthermore, the skills do not grow

in isolation from content; they arestrengthened through practice and use in all

subject areas.

B. Common Krowledqg and Skills

Looking beyond the Essential Lglrning

Skills, MATHEMATICS: Common Curriculo1.,mGo0als defines more fully the essentials in a

strong mathematics program for all Oregon

public schools.

C. Comprehensive Mathematics Curriculum

Each district will want to extend andelaborate upon the skills outlined inEssential Learning Skills and MATHEMATICS:Common Curriculum Goals in order to create

its own unique, comprehensive mathematic,

curriculum. Students should have the

opportunity to demonstrate their achievement

in a variety of ways. Equal opportunity to

learn and the special needs of students areprimary considerations in determiningacceptable performance levels. Mathe-

matics: Comprehensive Model Curriculumoffers specific suggestions for districtconsideration in meeting such student needs.

Is 4

In anticipating the three-stage curriculumdevelopment by the Department, and inresponse to the national awakening of thegrowing need to update school mathematics,the Oregon mathematics education communitybegan work in the spring of 1984. Aftermore than two years of development workfacilitated by the Oregon Mathematics Educa-tion Council The Oregon Mathematics ConceptPapers were published in the summer of 1986.The concept papers summc-ize the nationalliterature and translate it into recommenda-tions for an updated mathematics curriculum.

Both Common Curriculum Goals and Comprehen-sive Model Curriculum include many ideasfrom the concept papers.

State Standards

The Common Curriculum Goals receive theirauthority from the Oregon State Standards forPublic Schools, OAR 581-22-420 and 581-22-425.These rules were amended by the State Board ofEducation in January 1986. The Essential Learn-ing Skills deemed appropriate for mathematicsinstructional programs are included in theCommon Curriculum Goals. This document onComprehensive Curriculum Goals builds upon thefirst two by offering suggestions to districtsfor developing their own comprehensive programwhich meets Standards 581-22-402, 420 and 425in the program area of mathematics.

PHILOSOPHY/RATIONALE UNDERLYING THIS CURRICULUM

The 1980s have been a time of educational reformin the nation at large. There has been a strongand pervasive quest for excellence and equity ineducation in general, and mathematics educationin specific. The Oregon Action Plan for Excel-lence established the direction for school

13 2

improvement in the state and the EssentialLearning Skills document outlined the commonskills across all program areas for elementaryand secondary education. MATHEMATICS: CommonCurriculum Goals, was written in relationship tothe preceding documents specifically, as well asto the reform effort generally. MATHEMATICS:Comprehensive Model Curriculum, the third stageof curriculum development driven by the OregonAction Plan for Excellence, builds on MATHEMA-TICS: Common Curriculum Goals and offers manyspecific suggestions for districts to considerin developing their comprehensive programs.Together the three documents present acurriculum plan for an updated mathematicsprogram to meet the needs of a technology-basedsociety. Thoughtful readers will note thatthese three curriculum publications describemajor changes in school mathematics.

Althoub the underlying principles of mathe-matics are constant, the optimum structure forthe presentation and use of mathematics has beenshifting in response to the rapidly expandingimportance of technology in solving problems.Today's world demands the ability to think anduse mathematical ideas to solve problems and tomake decisions. The time our pupils spendlearning mathematics can no longer be limited topracticing long, repetitive or tedious proce-dures which are more efficiently done withcalculators. The impact of technology and itsimplications for mathematics education arereflected in this document.

The increasingly common uses of calculatorsunderlines the need for mental computations andestimations. Although the development of theseskills has always been implicit in mathematicsinstruction, they have not always been -aughtsystematically and fully. Deliberate and

14

thorough development of the ability to estimateand do mental arithmetic is a regular part ofthe computational strand at all school levels inthe curriculum outlined in this document.

Even though the calculator is becoming anincreasingly available tool for problem solving,

its use is of little value if the user hasinadequate conceptual understanding of number

and operation. To help students understand"which buttons to push," the curriculum mustplace significant emphasis on the conceptswhich underlie basic mathematical skills.Increasingly, research supports the use ofconcrete materials as models for number concepts

and for operations. The curriculum outlined in

this document provides that a broad range ofmanipulatives be used to introduce new concepts

at all levels. In addition to going from

concrete to the abstract, opportunities areprovided for finding concrete representationsfor abstract concepts and their symbolic

representations. The use of manipulatives as

tools for increasing understanding extendsnaturally to their use as tools to assist in

learning problem-solving skills. The use of

concrete materials as problem-solving tools isincorporated into a variety of curriculumcontent areas as outlined in this document.

As hardware becomes both more sophisticated andless expensive and as software developmentcontinues to expand and become more "userfriendly," computers will inevitably become

major tools for mathematics education. The

value of computers in creating geometric dis-plays, organizing and graphing data, simulatingreal-life situations, and generating numericalsequences is recognized in this document.

5 3

Mathematics is about making sense of the world.The mathematics outlined in this document is

consistent with the nature of the subject. This

means that pupils are learning mathematics in

ways that allow them to explore relationships

and to develop understandings. The fundamental

premise on which this document is based is thatevery aspect of school mathematics that pupils

encounter should enhance their understanding of

mathematical ideas and promote the growth of

thinking.

Organization

In order to provide a curriculum consistent with

the philosophy outlined above, the common curri-culum goals for math have been organized into

nine content strands. They are:

1.0 Number and Numeration: Students demon-

strate an understanding of number andnumeration concepts and use these under-standings to interpret and solve problems.

2.0 Appropriate Computational Skills: Students

select and use the most appropriate form ofcomputation - manipulative, mental, paper/pencil, estimation or calculator usage tosolve problems and check all computations

for reasonability.

3.0 Problem Solving: Students use problem-solving skills and strategies to solveroutine and nonroutine problems.

4.0 Geometry and Visualization Skills: Stu-

dents recognize geometric patterns andrelationships and apply them in solvingproblems and making predictions.

5.0 Measurement: Students measure quantitiesand use measurements to keep :.ecords,problems and make predictions.

6.0 Statistics and Probability: Studentscollect, organize, record and interpretdata, and be able to predict probableoutcomes based on collected data.

7.0 Mathematical Relationships: Studentsrecognize and use number patterns, relationships and logical thinking skills tomake predictions and to solve problems.

8.0 Oral and Written Communication Skills:

Students use vocabulary, speech, numeralsand other symbol systems essential foreffective individual and group problemsolving, and for effective oral and writtencommunication of mathematical concepts,problemsolving processes and results.

9.0 Appropriate Study Skills: Students selectand use appropriate study skills in orderto accomplish mathematical learning tasks.

The content outlined in each content strandincludes both the essential learning skillsdeemed appropriate for mathematics instructionand the common curriculum outcomes unique tcmathematics.

17

4

EXECUTIVE SUMMARY OF CONTENT STRANDS

The content of mathematics programs is organized

under nine content strands. This section

presents a brief summary and rationale for each

strand.

1. Number and Numeration: Basic Concepts.

Principles and Meanings

The mastery of a particular bit or area ofknowledge at a level that makes it genuinelyfunctional in one's life requires that it beunderstood thoroughly, that it be connectedto related bodies of thought, and that it beintegrated with other knowledge/attitudes/perceptions. Applied to mathematicsspecifically, this means that students needto apply in a conscious way the basics ofnumber and numeration if they are to bemasterful users of the algorithms they are

learning. To assist in this objective themathematics content provides opportunitiesto build concepts and demonstrateunderstandings through the use of concretemodels for whole number, fraction and

decimal numerations.

2. tip_pr giridecopytaLom n I Skills

Mastery of appropriate computational skillsis a necessary outcome of a mathematics

program. Computational skills need to berelated to realworld situations and seen as

a means of enhancing a person's ability to

use mathematics in daily living. Memorization of all onedigit basic facts at thequick recall level is imperative.Appropriate computational skills include

5

mental arithmetic, estimation, andcalculator use as well as paper/pencil

computation. Instruction should includeopportunities to select the more appropriatemode of computation and to determine whether

or not answers are reasonable. Furthermore,

the instructional content provides opportunities to demonstrate conceptual understandings and reasonability of answersthrough the use of concrete models and

materials.

3 Problem Solving

Problem solving has been designated as thecentral goal for mathematics. A problem is

a perplexing situation in which an

individual or group accepts the challenge offinding ways to clarify or resolve the

difficulties involved. Frequently, the

problem can be approached in many ways.Occasionally, some of the resulting

investigations are nonproductive. Sometimes

they are so productive as to lead to manydifferent solutions or suggest more problems

to solve.

Problemsolving skills and strate-'es shouldbe explicitly emphasized and problem solvingshould be incorporated frequently into theapproaches used in teaching the requiredtopics throughout the grade levels.

4. Geometry and Visualization Skills

We live on a sphere called Earth and work in

a threedimensional world. Citizens, con

sumers, and workers require some knowledge

of geometry. Much more knowledge is needed

in certain occupations such as plumbing,

I

carpentry, forestry, interior decorating,architecture and engineering. In additionthen to acquiring knowledge of certaingeometric concepts and properties and theirapplications, students need to develop theirspatial and visualization skills.

The instructional content emphasizes

exploration activities, informal reasoning,and the use of problemsolving skills. Muchuse is made of tools which aid in geometric

explorations, including the compass,protractor, the straightedge, squared (grid)paper, and the computer.

5. Measurement

Measurement concepts surround us. We use anunderstanding of measures to quantify andinterpret the world. Modern technology istotally dependent upon measurement. Hiddenin most of humanity's spectacularaccomplishments are innumerable

measurements, each related to or dependentupon a myriad of other measurements.

Measurement can be taught more successfullyif estimation is a content objective. Toteach measurement we should be concernedwith more than a system of measures. Wemust teach a "doing" kind of mathematics.Activity gives meaning to measuring skills,makes the resultant learning personallysatisfying to the pupil and begins thedevelopment of a process that will be usedthroughout life.

206

6. Statistics and Probability

We live in a society in which we are confronted daily with quantitative informationor data. Statistics is the science or studyof data. Quantitative literacy is arequirement for all educated individuals whowant to be informed citizens and hold jobsin technical businesses and industries.Throughout life, decisions about health,citizenship, parenthood, employment, financial concerns, and sports are based uponquantitative information. Statistics ismost often concerned with using informationin the face of uncertainty. Probabilitygives us a way to measure uncertainty.

In addition, studying statistics and probability can help in the development ofstudents' critical thinking skills. In

carrying out experiments, students developways to cope with uncertainty as they aresearching for truth in a situation andlearning to report it faithfully.Approaching situations statistically canhelp make students face up to their ownprejudices, think more consistently aboutarguments, and justify their thinking withnumerical information.

n ^,. 1

7. Kltkinatiml_RelAtignshipl

Topics in this content strand are used toshow how one thing changes as anotherchanges. Formulas, tables, and graphs oftenhave close relationships ;:o the "real world"and frequently can be used in making usefulpredictions. Ratic, proportion, and percentare also used extensively in prooemsol4ingsituations. Program content emphasizespatterning, using logical thinking skills,modeling of concepts and real worldapplications.

8. Oral and Written Communication Skills

Two emerging trends are reflected in "CommonCurriculum Goals" and "Comprehensive ModelCurriculum" for math; integration oflearning across content areas, and thedevelopment and use of enabling skills whichhelp students learn mathematics and otherdisciplines. The strand, "Oral and WrittenCommunication Skills" offers the basiccommunication skills needed to effectivelyread, discuss and share mathematical ideasand problems.

9. Appropriate Study Skills

It seems fitting that the ninth contentstrand be "Appropriate Study Skills." A

full repertoire of study skills enablesstudents to learn school mathematicsefficiently while becoming independentlearners. Only by reaching this level canstudents become lifelong consumers andusers of mathematics.

7

oGRADE LEVEL TEACHING EMPHASES

Kindergarten

1. Number and Numeration

Initial number and numeration activitiesinclude numerous counting experiences usingactual objects. Pupils demonstratecomprehension of whole numbers up to 10 byreading, writing, ordering, modeling andcomparing. They also use concrete models toshow simple place value and to demonstratehalves and wholes. Other number experiencesinclude skip counting by 5 and 10; usingordinals (1st to 5th); and estimating thenumber of actual objects and checkingreasonableness of answers by counting.

2. Appropriate Computational Skills

Computational activities are broad-based andinclude many experiences using manipula-tives, estimation and mental arithmetic.The emphasis is on helping pupils relatereal experiences with the abstract symbolsthat represent those activities, rather thanon automatic answers to written exercises.

3. Problem Solving

Pupils are provided with opportunities touse concrete materials to solve problems.They work cooperatively toward the solutionof prcblems and discuss and comparestrategies and solutions. Role playing with"real world" objects is used to solve simpleoral problems and to model "put together"and "take away" situations. Guessing and

8

checking, making a model, generating a

pattern ant, drawing a picture are other

problem-solving skills to which pupils areintroduced.


Emp,asis is placed on explontion activities'4ith both plane and solid shapes. Pupilsuse a variety of manipulative materials,

recognize geometric patterns andapplications in the environment, and sharpenvisualization skills.

5. Measurement

Emphasis is placed on "hands on" activitieswhich help deyelop measurement concants andunderstandings of pupils have numerousinformal experiences estimating .:iddetermining length, weight and capacityusing nonstandard uniform .nits. Theyexplore money, time and temperature conceptsas they occur in daily situations.


Pupils have numerous informal experienceswith activities that have outcomes thatdepend on chance. They collect and recorddata using tally marks, lists and simplecharts; and make and draw conclusions fromreal graphs (an array of actual objects on arectangular grid), picture graphs and charts.

7. Mathematical Relationships

Pupils sort and classify actual objects andreproduce a pattern in a sequence of actu,1objects. They identify characteristics ol

(-1 t,- - *-y

simple objects that remain the same eventhough some change occurs; staterelationships using terms such as "greater

than" and "less than"; and describe insimple terms how a solution was reached.

8. Oral and Written Communication

Pupils use and apply the basic communicationskills needed to effectively read, discussand share mathematical ideas and problems.


Pupils use and apply study skills whichenable them to learn school mathematicseffectively and become independent learners.

Grade One


Pupils explore number and numerationconcepts through counting and one-to-onecorrespondence activities with actualobjects. They demonstrate comprehension ofwhole numbers to 20 by reading, writing,ordering, modeling, and comparing. Concrete

models are used to demonstrate an under-standing of place value grouping andexchanges and to show halves and wholes.Other number and numeration activitiesinclude rote counting to 20; skip countingby 2, 5 and 10; counting on and countingbackward; estimating the number of objectsand checking the reasonableness of answersby counting; using ordinals (to 10th) anddemonstrating addition and subtractionproperties.

9


Pupils have experiences learning tocalculate using manipulatives, estimation,mental arithmetic, calculators and paper/pencil methods. Considerable attention isgiven to manipulatirg concrete objects andto connecting "real world" experiences tothe abstract mathematical symbols whichrepresent them.

3. Problem Solving

Problem-solving skills such as guessing andchecking, using concrete objects, making amodel, generating a pattern and drawing apicture are emphasized. Pupils have manyopportunities to engage in cooperativeproblem solving and to discuss and comparestrategies and solutions. Role playing with

concrete objects is used to solve simpleword problems and pupils create their ownstory problems to match addition andsubtraction number sentences.


Emphasis is placed on "hands-on' activitieswith both plane and solid shapes. Pupils

use a variety of manipulative materials,copy and extend geometric patterns,recognize geometric applications in the

environment and sharpen visualization andestimation skills.

5. Measurement

Emphasis is placed on the act of measuringand on "hands-on" activities which help

develop measurement concepts and under-standings. Pupils have numerous informalexperiences estimating and determininglength, weight an0 capacity usingnonstandard units. They explore time andtemperature concepts as they occur in dailysituations and identify and order coins(penny, nickel and dime) by value.

Statistics and Probability

Pupils have numerous informal experienceswith activities that have outcomes thatdepend on chance and make predictions aboutsimple future outcomes. They collect andrecord data using tally marks, lists andsimple charts; and make and draw conclusionsfrom real graphs, picture graphs, and charts.


Pupils sort and classify actual objects andreproduce and extend patterns in a sequenceof objects or numbers. They state relation-ships using such terms as "greater than,"and "less than", and describe in simpleterms how a solution was reached.





27 10

Grade Two


Pupils demonstrate comprehension of wholenumbers to 100 by reading, writing,ordering, modeling, and comparing. They useconcrete models to demonstrate two-digitplace value understandings, grouping andexchanging. Place value experiences alsoinclude activities in which pupils estimatethe number of actual objects and checkanswers by grouping and counting. Othernumber and numeration activities includeusing concrete models to demonstrate halves,wholes, thirds, and fourths; rote countingto 100; skip counting by 2, 5 and 10; anddemonstrating addition and subtractionproperties.

2. Appropriate Computatio,11 Skills

Pupils have opportunities to use allappropriate forms of computation includingmanipulatives, estimation, mentalarithmetic, calculators, and paper/pencil.Instruction is provided in thinkingstrategies to assist in basic addition andsubtraction facts. Pupils deal withaddition and subtraction problems togetherso they can compare the two processes andmake sense of them. Early experiences intwo-digit addition/subtraction are modeledusing concrete materials and includesituations in which regrouping may or maynot be required so that pupils learn fromthe beginning to always check to determineif regrouping is neces!;ary.

3. Problem Solving

Problemsolving skills such as guessing andchecking, using concrete objects, making amodel, generating a pattern and drawing a

picture are emphasized. Pupils have many

opportunities to engage in cooperativeproblem solving and to discuss and comparestrategies and solutions. Role playing with

concrete objects is used to solve simpleword problems and pupils create their ownstory problems to match addition andsubtraction number sentences andalgorithms. Opportunities to applyproblemsolving skills also occur in drilland practice activities such as games and

puzzles.


Emphasis is placed on "handson" activitieswith both plane and solid shapes. Pupils

use a variety of manipulative materials todevelop an understanding of perimeter, area

and volume concepts. They copy and extendpatterns, recognize geometric applications

in the environment, and sharpenvisualization and estimation skills.

5. Measurement

Emphasis is placed on the act of measuringand on "handson" activities which helpdevelop measurement concepts and under

standings. Pupils estimate and measurelength, weight and capacity using nonstandard units and begin to use standard

11

metric and customary units to measure

length. They identify, order and use the

U.S coins and recognize and use time and

temperature concepts.


Pupils have numerous experiences Withactivities that have outcomes that depend onchance; make predictions using such terms as

"best chance," "fair," "not likely"; applyintuitive probability concepts in games and

activities and predict simple future

outcomes. They collect and record data and

make and draw conclusions from real charts;picture graphs, bar graphs, and charts.


Pupils sort and classify actual objects andrecognize, reproduce and extend objectpatterns and number patterns in charts and

tables. They state relationships using suchterms as "greater than," "less than," and

"equal to."





SC)

ftad_1. Number and Numeration

Students read, write, order, and comparewhole numbers to 1000, commonly usedfractions, and decimals in tenths andhundredths (using money models). They useconcrete materials to model, place valueexchanges, whole number place values to1000; commonly used fractions; and decimals.Place value concepts are used to estimateand sense reasonableness of answers.Properties of addition, subtraction, and

single-digit multiplication are modeled withactual objects, used and applied.


Students have opportunities to select anduse all appropriate computational skills

including manipulatives, estimation, mentalarithmetic, calculators, and paper/pencil.Mastery of basic addition and subtractionfacts is encouraged and pupils are providedwith thinking strategies to assist withmeaningful acquisition of multiplicationfacts. Concrete materials continue to beused to model addition and subtractionalgorithms and to demonstrate the variousmeanings of multiplication. Instruction isprovided in specific appropriate mental

arithmetic skills (such as adding andsubtracting multiples of tens or hundreds)and pupils are encouraged to use roundingand cther skills to make approximatecalculations and to check all answers forreasonability.

3i12

3. Problem Solving

Problem-solving skills such as guess andcheck, use concrete objects, make a model,look for a pattern, and draw a picture areemphasized. Students have many opportun-ities to engage in cooperative problemsolving and to discuss and comparestrategies and solutions. They solveonestep word problems, including thoseinvolving money and measurements, and createtheir own word problems to match addition,subtractioes, and multiplication algorithms.Opportun'ities to apply problem-solvingskills also occur in drill and practiceactivities such as games, puzzles,calculator activities, and challengeproblems.


Emphasis is placed on "hands-on" activitieswith both plane and solid shapes. Studentsuse a variety of manipulative materials todevelop an understanding of perimeter, area,and volume. They copy and extend geometricpatterns, :sake simple constructions,

recognize geometric applications in theenvironment, and sharpen visualization andestimation skills.

5. Measurement

Emphasis is placed on the act of measuringand on activities which help developmeasurement concepts and understandings.Students estimate and then determine lengthand weight (mass) using nonstandard, metric,and customary units of measure and select

appropriate instruments and units forspecific measurement tasks. They identify,order, and use all U.S. coins; reading timeusing standard and digital clocks; andestimate, read and record temperature in C°and F°.


Students have numerous experiences withactivities which have outcomes that dependon chance; make predictions about simplepossible future outcomes; and applyintuitive probability concepts in games andactivities. They collect and record dataand make and draw conclusions from bargraphs and picture graphs.


Students sort and classify actual objectsand identify number patterns in charts andtable:. They use number patterns andrelationships to make predictions, makesimple tables of values and state mathe-matical relationships using terms such as"less than" and "equal to."


Students use and apply the basic communica-tion skills needed to effectively read,discuss and share mathematical ideas andproblems.


Students use and apply study skills whichenable them to learn school mathematicseffectively and become independent learners.

13

Grade Four


Students demonstrate an understanding ofplace values to one million and commonlyused fractions and decimals to hundredths bymodeling with concrete and pictorialmaterials and by reading, writing, orderingand comparing. They demonstrate, use and

apply properties of addition, subtraction,multiplication and single-digit divisionwith whole numbers; recognize and us?concepts related to multiples and factors;and recognize and use appropriate mathe-matical terms such as product, sum,remainder, and quotient.


Students have opportunities to select anduse all appropriate computational skillsincluding manipulatives, estimation, mentalarithmetic, calculators, computers, andpaper/pencil. Immediate recall of all basicfacts is essential and is assisted by athorough review and application ofappropriate thinking strategies. Concrete

materials are used to model multiplicationand division of whole numbers, addition andsubtraction of decimals, and addition andsubtraction of fractions with like

denominators. Considerable attention isgiven to the development and use of mentalmath skills, including operations withpowers and multiples of 10. Pupils are

encouraged to use rounding and other skillto make approximate calculations and tocheck all answers for reasonability.

3. Problem Solving

Problem-solving skills such as guess andcheck; look for a pattern, make a systematiclist; and make a model or drawing areemphasized. Students have many oppor-tunities to engage in cooperative problemsolving and to discuss and compare problem-solving strategies and solutions to routineand nonroutine problems. They solve one-step word problems, including thoseinvolving money and measurements, and createtheir own story problems to match addition,subtraction, multiplication and divisionalgorithms. Opportunities to apply problem-solving skills also occur in drill andpractice activities such as games, puzzles,calculator activities and challenge problems.


Emphasis is placed on "hands-on" activitieswith both plane and solid shapes. Studentsuse a variety of manipulative materials toestimate and then determine perimeter, areaand volume. They copy and extend geometricpatterns, make simple geometric drz..:ings andconstructions, recognize geometric applica-tions in the environment and sharpenvisualization skills.

5. Measurement

Emphasis is placed on the act of measuringand on estimation and other activities whichhelp develop measurement concepts andunderstandings. Students recognize and usemeters, centimeters, feet, yards, inches,grams, kilograms, ounces and pounds, and

14

select the most appropriate instrument andunit for a specific measurement task. Theyidentify, order and use U.S. currency; readtime using standard and digital clocks; andestimate, read and record temperature in C°and F°.


Students participate in probabilityexperiments and game activities which haveoutcomes that depend on chance, recognizethe concept of fair or unfair, and makepredictions about simple possible futureoutcomes. They make and draw conclusionsfrom charts, picture graphs, bar graphs, andcomputer generated graphs and tables.


Students sort and classify actual objectsand simple geometric figures and identifynumber pat rns in charts and tables. Theyuse number patterns and relationships tomake predictions, make simple tables ofvalue, state mathematical relationships, anduse the symbols >, < and -.


Students use and apply the basic communica-tion skills needed to effectively read,discuss and share mathematical ideas andproblems.



(ru

Grade Five


Students demonstrate an understanding ofplace values to one million; commonly usedproper fractions, improper fractions, andmixed numbers; and decimals to thousandthsby modeling with concrete and/or pictorialmaterials and by reading, writing, ordering,and comparing. They demonstrate and useproperties of whole number operations;recognize and use appropriate mathematicalterms; and recognize and use conceptsrelated to multiples, factors, primes, andcomposites.


Students have opportunities to select anduse all appropriate computational skillsincluding estimation, mental arithmetic,calculators, computers, and pencil/paper.They apply acquired strategies to aid in thequick recall of all basic facts. Concretematerials are used to model the variousmeanings of whole number multiplication anddivision and to interpret remainders.Models such as money or metrics are used todemonstrate addition and subtraction ofdecimals and multiplication and division ofdecimals by whole numbers. Concrete mo.,cis

are also used to demonstrate an under-standing of the addition and subtraction ofcommonly used fractions and the multiplica-tion of a fraction and whole number usingthe "of" concept. Students solve mentallyappropriate whole number, decimal and

15

fraction problems and use a calculatorand/or computer to solve problems withlengthy calculations or large numbers. Theyuse rounding and other techniques useful n

mental computation to make approximate wholenumber, fraction and decimal computationsand to check reasonability of all answers.

3. Problem Solving

Problem- solving skills such as guess andcheck, look for a pattern, make a systematiclist, make a drawing or model, and solve asimpler relate] problem are emphasized.Students have many opportunities to engagein cooperative problem solving and todiscuss and compare problem-solvingstrategies and solutions to routine andnonroutine problems. They solve one- andtwo-step word problems and create their ownstory problems to match whole number,fraction and decimal algorithms. They alsohave opportunities to apply problem-solvingskills in drill and practice activities suchas games, puzzles, calculator and computeractivities, and challenge problems.


Emphasis is placed on "hands-on" activitieswith both plane and solid shapes. Students

use a variety of manipulatives to estimateand then determine perimeter, area andvolume. They identify properties of commongeometric figures, make geometric drawingsand constructions, recognize geometricapplications in the environment and sharpenvisualization skills.

:

05. Measurement

Emphasis is placed on the ac* of measuringand on estimation and other activities whichhelp develop measurement concepts andunderstandings. Students recognize and usemetric and customary units to estimate anddetermine length and weight mass. Theyselect the most appropriate unit for aspecific measurement task; convert amongunits of length within the same measurement:

system; use U.S. currency; read and recordtemperature; and determine indirectmeasurements by locating points and givingcoordinates of points on maps.


Students generate, record and interpret datafrom probability experiments, predictchances of an outcome, and recognize certainand impossible probabilities. They usecharts, tables and lists to organize allpossible outcomes of an experiment; andread, interpret, construct and drawconclusions from bar graphs, line graphs,tables, and charts.


Students sort and classify actual objectsand simple geometric figures and identifynumber patterns in charts and tables. Theyuse number patterns and relationships tomake predictions, make and interpret tablesof value, state mathematical relationships,and use the symbols >, < and -. Decimal andfraction understandings and 100-grids areused to illustrate the "for every hundred"model of percent.

16


Students use and apply the basiccommunication skills needed to effectivelyread, discuss and share mathematical ideasand problems.



Grade Six


Students demonstrate an understanding ofplace value to one million; commonly usedproper fractions, improper fractions, andmixed numbers; and decimals to thousandthsby modeling with concrete and/or pictorialmaterials and by reading, writing, ordering,and comparing. They apply and use proper-ties of whole number operations; recognizeand use appropriate mathematical terms,grouping symbols, and order of operationsand divisthility rules; and recognize anduse concepts related to multiples, factors,primes, and composites.


Students have opportunities to select anduse all appropn ate computational skillsincluding estimation, mental arithmetic,calculators, computers, and pencil/paper.Concrete and pictorial models such as moneyand grid paper are used to model decimal andfraction operations. Students solve

mentally appropriate whole number, decimaland fraction problems and convert mentallybetween commonly used fractions and decimals.They use a calculator and/or computer tosolve problems with lengthy calculations orlarge numbers. Instruction in skills usefulin making approximate whole number, decimal,and fraction calculations is provided andthese skills are applied in checking thereasonability of all computational results.

3. Problem Solving

Problem-solving skills such as guess andcheck, look for a pattern, make a systematiclist, make a drawing or model, and solve asimpler related problem are emphasized.Students engage in cooperative problemsolving and discuss and compare strategiesand solutions to routine and nonroutineproblems. They solve one- and two-step wordproblems and create their own story problemsto match whole number, fraction and decimalalgorithms. They also have opportunities toapply problem-solving skills in drill andpractice activities such as games, puzzles,calculator, and computer activities andchallenge problems.


Emphasis is placed on "hands-on" activitieswith both plane and solid shapes. Students

estimate and determine perimeter, area and

volume. They use a variety of manipulativesto cover the plane and to create designswith pattern and symmetry. They identify

properties of common geometric figures, makegeometric drawings and constructions,recognize geometric applications in theenvironment and sharpen visualization skills.

41 17

5. Measur_lent

Emphasis is placed on estimation and otheractivities which help develop measurement

concepts and understandings. Students

estimate and then directly measure distance,angles and other quantities to the nearestand mallest unit on the measuring scale.They convert among units of length andweight (mass) within the same measurementsystem; locate points and give coordinatesof points on maps; make scale drawingJ;determine highway distances from maps; andfind measurements, including inaccessibledistances, indirectly by using formulas for

distance and area.


Students predict probable outcomes based oncollected data. Thar construct, read andinterpret graphs, tables and charts and makepredictions based on them; identify mis-leading methods of displaying or inter-preting data; and determine mean, median andmode using data meaningful to the students.


Students sort and classify geometric figuresand sets of numbers and identify numberpatterns in charts and tables. They use

number patterns and relationships to makepredictions and solve problems, and make,interpret and explore relationships found ir

tables of value. Decimal and fractionunderstandings and 100-grids are used toillustrate percent concepts.




Students use and apply study skills whichenable them to learn school mathematics

effectively and become independent learners.

Grade Seven


Students demonstrate an understanding ofplace value to one million, commonly usedproper fractions, improper fractions, mixednumbers, decimals, and percents by modelingwith concrete and/or pictorial materials andby reading, writing, ordering and comparing.They apply operational properties andrecognize and use appropriate mathematicalterms, grouping symbols, order of operationand divisibility rules, and concepts relatedto factoring.


Students have opportunities to select anduse all appropriate computational skills

including estimation, mental arithmetic,calculators, computers, and pencil/paper.Concrete and pictorial models and "realworld" examples are used to demonstrate anunderstanding of operations with wholenumbers, decimals, fractions and percents.Students mentally add, subtract, multiply,

4318

and divide whole numbers and decimals bypowers of ten anC multiples of ten; computementally percents such as 25% and 50% andconvert mentally between commonly usedfractions and ccimals. They use a..alculator or computer to solve appropriateproblems and use rounding and other techniques useful in mental computation toestimate and make approximate whole number,fraction, decimal and percent computations.

3 Problem Solving

Problemsolving skills such as guess andcheck, look for a pattern, make a systematiclist, make a drawing or model, and solve asimpler related problem are emphasized.Students engage in cooperative problemsolving and discuss and compare strategiesand solutions to routine and nonroutineproblems. They pose new problems, solvemultiple step word problems and create storyproblems to match exercises involving wholenumbers, fractions, decimals, and percents.They have opportunities to apply problem-solving skills in drill and practiceactivities such as games, puzzles,

calculator and computer activities, andchallenge problems.


Emphasis is placed on activities and applications with both plane and solid shapes.Students estimate and determine perimeter,area and volume. They use a variety ofmaterials to cover the plane and to createdesigns with pattern and symmetry. Theyidentify distinguishing properties of commongeometric figures, make geometric drawings

and constructions, recognize geometricapplications in the environment and sharpenvisualization skills.

5. Measurement

Emphasis is placed on estimation and otheractivities which help develop measurementconcepts and understandings. Studentsselect the most appropriate instrument andunit, estimate and then directly measuredistance, angles and other quantities to thenearest and smallest unit on the measuringscale. They convert among commonly usedunits of measurement within the same system;make scale drawings and determine actualdistances from scale drawings, blueprints,maps and globes; and find measurements,including inaccessible distances, indirectlyby using formulas.


Students predict probable outcomes based oncollected data. They construct, read andinterpret graphs, tables and charts and makepredictions based on them; identifymisleading methods of displaying orinterpreting data; and determine mean,median and mode.

7. Mathematica Relationships

Students use number patterns and relationships to collect, organize, record, andinterpret data, to make predictioos, and tosolve problems. They use models such as100grids, number lines or meter sticks toillustrate how percent can be expr,..t:sed as a

fraction or decimal.

19


students use and apply the basiccommunication skills needed to effectivelyread, discuss and share mathematical ideas

and problems.



Grade Eight

1. Number and Numen tion

Students read, write, order, model andcompare commonly used fractions, decimals,percents, and signed numbers. They express

large numbers in expanded exponentialnotation and in scientific notation. They

recognize and use operational properties,appropriate mathematical terms, groupingsymbols, order of operation and divisibilityrules, and concepts related to factoring.

2. Appropri-49 Computational Skills

Students have opportunities to select anduse all appropriate computational skillsincluding estimation, mental arithmetic,calculators, computers, and pencil/paper.Concrete and pictorial models (such as gridpaper) and "real world" examples are used tomodel operations with signed numbers, fractions, decimals, and percent applications.Students mentally add, subtract, multiply,and divide whole numbers and decimals bypowers of ten and multiples of ten, and

compute with commonly used percents. Theyuse and apply estimation techniques andcheck the results of all computations forreasonability.

3. Problem Solving

Problem-solving skills such as guess andcheck, look for a pattern, make a systematiclist, make a drawing or model, and solve asimpler problem are emphasized. Studentsengage in cooperative problem solving anddiscuss and compare strategies and solutionsto routine and nonroutine problems. Theypose new problems, solve multiple step wordproblems and create story problems to matchexercises involving fractions, decimals,percents, and ratio and proportions. Theyalso apply problem-solving skills in drilland practice activities such as games,puzzles, calculator and computer activities,and challenge prob


Emphasis is placed on activities andapplications with both plane and solidshapes. Students estimate and calculateperimeter, area and volume. They use avariety of materials to cover the plane andto create designs with pattern andsymmetry. They identify distinguishingproperties of common geometric figures, makegeometric drawings and constructions,

recognize geometric applications in theenv'ronment and sharpen visualization skills.

4720

5. Measurement

Emphasis is placed on using and applyingmeasurement concepts. Students estimate anddirectly measure distances, angles and otherquantities and indicate the precision of themeasurement. They can demonstrate thatanswers resulting from calculations withmeasurements are no morn reliable than themeavements used. They make scale drawingsand determine actual distances from scaledrawings, blueprints, maps and globes; andfind and record measurements usingproportions and formulas.


Students predict probable outcomes based oncollected data. They construct, read andinterpret graphs, tables and charts and makepredictions based on them; identify mis-leaing methods of displaying or inter-preting data; and determine, interpret andcompare advantages and disadvantages ofmean, median and mode.


Students use number patterns and relation-ships to collect, organize, record, andinterpret data, to make predictions, and tosolve problems. They interpret and use theconcepts of ratio, percent, proportion, andcommonly occurring rates.

r C)





Grade_ El even

I. Number and Numeration

Students read, write, order, and comparefractions, decimals, percents, and signed

numbers. They express large and smallnumbers in exponential and scientificnotation. Operational properties and orderof operation rules are used and applied asare place value and numeration concepts.


Students have opportunities to select anduse all appropriate computational skillsincluding estimation, mental arithmetic,calculators, computers, and pencil/paper.Concrete and pictorial models and "realworld" examples are used to model operationswith decimals, fractions, signed numbers,and percent applications. Students use andapply estimation techniques, computementally when appropriate, and convertmentally, manually and electronically among

43 21

decimals, percents and commonly usedfractions. They use calculators and/orcomputers to solve appropriate problems andrecognize the limitations of a calculatorwhen dealing with lams numbers.

3. Problem Solving

Students select and apply the mostappropriate tools, methodologies, processesand operations in solving a variety ofroutine and nonroutine problems. They

engage in cooperative problem solving anddiscuss and compare strategies andsolutions. They translate "real world"problems into matheutical statements andmathematical problems and answers back into"real world" context.


Emphasis is placed on activities andapplications with both plane and solid

shapes. Students sketch and constructgeometric figures, recognize geometricapplications in the environment and sharpen

visualization skills.

5. Measurement

Emphasis is placed on using and applyingmeasurement concepts. Students estimate z.11

directly measure length, area, volume, time,and weight (mass) with reasonable accuracyand/or round a measurement to a given unit.They apply ratio and proportion concepts inmaking and using scale drawings and modelsand in solving problems; and find and recordmeasurements using proportions and formulas.


Students interpret evecyay uses ofprobability and predict probable outcomesbased on collected data. They construct,read, and interpret graphs, tables andcharts and make predictions based on them.They collect, display and interpretstatistical data using mean, median, moderange, and percentile.


Students use number patterns andrelationships to collect, organize, record,and interpret data, to make predictions, andto solve problems. They interpret and usethe concepts of ratio, percent, proportion,and commonly occurring rates.




Students use ,nd apply study skills whichenable them to learn school mathematicseffectively and become independent learners.

5 1

22

INSTRUCTIONAL THEMES

Problem Solving

Problem solving is the process of using one'sknowledge and experience when encountering newand unexpected situations. It is not a gift offate, but rather, something that must be taught,modeled, experienced, and reinforced throughoutthe mathematics curriculum. Problem solving is

also an attitude, an approach to thinking aboutmathematics, and must be nurtured at every stageand in every topic presented in the mathematics

program.

Students must understand that problem solving isa process, with solutions coming from exploringsituations, brainstorming, and devising andtesting strategies over a period of time. They

heed to develop a positive attitude towardrisk-taking and must realize that beingtemporarily perplexed is a natural and necessary

state in problem solving.

Teachers at all levels need to create aclassroom atmosphere that iF conducive to

problem solving. They need to reward studentsfor devising and trying a likely approach even

it a solution is not achieved. Students need to

feel free to experiment and think originalthoughts withrut the fear of failure. Teachers

need to prepare students by teaching themspecific problem-solving skills which can beselected and sequenced to form a strategy ar

solution path. Some skills which can be used

include: (1) make a model or diagram, (2)

guess, check, and refine, (3) make a systematiclist, (4) look for a pattern, (5) workbackwards, and (6) solve a simpler but related

52 23

problem. Next, the teacher must providestudents with problem-solving opportunitieswhich are interesting and challenging and yetprovide a good chance of success on the part of

the students. S(He) must recognize that achallenging problem-solving situation for one

group may well be routine for another, and atotally frustrating experience in failure for

still another group. Finally, the teachershould assume a facilitator's role as studentsactively engage themselves in theproblem-solving process.

The problem-solving process often includesformulating problems, analyzing the problem,finding solutions, and verifying andinterpreting the results. Before one can solve

a problem, s(he) must understand the setting,screen relevant and irrelevant information, andidentify what is needed to solve the problem.During this first stage it is often nelpful forstudents to work in small group., and share ideasand interpretations of the problem. The teacher

should check at this point to make sure thatstudents' interpretations are correct. The

second stage, analyzing the problem, isprimarily concerned with strategy development.Again, cooperative work in small groups of two,three, or four students can be particularlyeffective. In groups, students can brainstormto select and sequence problem-solving skills toform plausible solution strategies. The teacher

should monitor this stage and ask leadingquestions such as "What if . . . ?," "Have you

considered . . . ?," "Is that the only way?,"but should refrain from giving direct help. In

the third stage, students use the data from theproblem and apply their strategies to an answeror answers to the problem. This stage is oftenan individual activity, but it too can beaccomplished, cooperatively, in small groups.

eThe final stage, verifying and communicatingresults, is a critical component of theproblemsolving process. Here students shouldverify that they have, in fact, solved theproblem. They might also explore thepossibility that their solution strategy couldbe extended to a more general problem.Communicating results is a very importantculminating or closing activity in whichteachers encourage students '-n share theirstrategies with the rest of the class. This isimportant for several reasons: (1) verbalizingthe process tends to improve one's problem

solving ability, (2) hearing others explain howthey solved a problem develops an understandingthat there are often several successful

strategies for a given problem, and (3) itprovides an opportunity for students to berecognized for originality and creative thought.

In summary, problem solving is a process thatmust be learned through active participationbeginning in kindergarten. It is an attitudewhich teaches: need to teach and model at everystage of a student's mathematical development.It calls for a wide variety of instructionalformats including direct instruction,facilitating large and small group discussions,and patient guidance while interjecting

appropriate hints and answering questions withquestions. Problem solving is not magic, but a

collection of skills and a willingness toexplore ways in which these skills can fittogether to form a strategy which leads to asolution of the problem.

24

Cooperative '_earning

In a cc ierative learning environment pupils arearranged in small groups, often groups of four,where they work cooperatively toward thesolution of a task. Pupils working in smallgroups ha'e more opportunities to contributeideas, make speculations, question one anotherand explain their own thinking. This type oforganization is particularly well suited forproblemsolving activities because it invitesthe active participation of each pupil andprovides pupils the opportunity to offer ideasand receive immediate feedback.

Cooperative learning improves how students feelabout themselves, school, and others. Therealso is evidence that more pupils achieve athigher levels in cooperativ? learning situationsand that they retain the information longer.Cooperative learning has also been shown tofacilitate

mastery of concepts and principlesapplication of information to othersettings

problemsolving skillscreativityverbal skills

ability to take another person'sperspective.

Cooperative learning groups can be usedsuccessfully at any grade level. Kindergarten,as well as secondary students benefit fromincreased opportunity to interact with materialsand with other pupils while learning.

c i 0

Symbolic Thinking*

We use words in dealing with ffilost of our

perplexities. However, it is not easy to usewords properly in solving problems. Since wordsare not identical with the things they stand forand since one word may stand for many differentthings, clear thinking requires the most carefulrelating of words to things. Mathematicians areespecially adept at using abbreviations andwritten signs. In addition to saving time, acareful selection of ..ymbols assists in thediscovery of relationships, e.g., theHindu-Arabic notation along with the inventionsof zero and the decimal point. Mathematicsillustrates the gaining of intellectual powerand material power over the "real" world throughthe control of symbols.

Mathematics teachers can help students todevelop their own powers of reflective thinkingthrough a mo-e conscious attention to symbols.They can show explicitly how symbols are used inmathematics and frequently make comparisons andcontracts with the use of symbols in ordinarylanguage. Three uses of symbols in mathematics

are.

1. Symbols such as numbers, units ofmeasure and geometric figures serve asintellectual tools in the conquest ofnature. They serve as instruments fordescribing physical things precisely interms of size, order, and number.

*Adapted from Progressive EducationAssociation Mathematics in General Education1938.

25

2. Symbols are used as a shorthand forother symbols, e.g., a bar graph, "pie"

chart, or a curve plotted on coordinatepaper are shorthand for tables of valuesand for relations between data.

3. Symbols are used as symbols for other

symbols. Such a use helps free the mindfrom attention to unnecessary details,e.g., "P" stands for all triangles; agraph i: a generalization of a table ofparticular symbols, the symbols ofarithmetic are restricted to illustra-tive examples but algebra deals withgeneral principles and rules of

operation.

Mathematical symbols when compared to non-mathematical symbols are relatively unambiguousand relatively independent of content. They can

be substituted with ease for things to whichthey refer, may be defined once and for all byfiat (zero exponent, i, negative numbers) andare relatively free from emotive variation.

Thinking and Doing Matnematics

Mathematical knowledge is something more than amere record of the subject matter or the end

product of mathematical inquiry. It also

implies the "doing" of mathematics. This doing

cannot be viewed as an activity individualsengage in by following predetermined rules. In

a sense mathematical activity can be seen moreas embodying the elements of an art or craft

than as a pure technical discipline. Thinking

mathematical then implies that learningmathematics is an active process and thereforeis not the same as an absorption of the record

of mathematical knowledge.

eRomberg* identifies four related activitiescommon to all mathematics: abstracting,inventing, proving, and applying. Theseactivities are the natural consequences ofmathematical investigations. Students shouldbe exposed to both inductive and deductivereasoning and their roles in problem solving andin the search for relationships. Matheihaticalinstruction also should reflect that conceptscan be general 'd.

Reading and Writing in Mathematics

In many mathematics classrooms, the only readingdone consists of the twosentence story problemsat the end of each chapter. Demands on studetcs'ability to read are not often made, so theability level is difficult to assess, and littledevelopment is done. Regrettably, the realworld use of mathematics often involvessubstantial reading in which information ispresented in a form different from that commonlyencountered by students in their reading inother classes and decidedly different from thesimplistic format of math textbook problems. Ifstudents are to meet the goal of becomingprepared to use mathematics, they must gain someexposure to reading mathematics in a practicalcontext.

Coupled with the lack of reading in mathematicscourses is the lack of writing in the courses.Answers to problems are "x . 7," or somesimilar simple Expression. Often students arenot called on to explain processes, or

*Adapted from Thomas Romberg A CommonCurriculum for Mathematics, p 121-159, 1983Yearbook "National Society for the S'.udy ofEducation."

58 26

challenged by situations which require more thana simple mathematical expression for theanswer. This is even more artificial than thelittle reading done in math courses. This factis pointed out in several of The Oregon

Mathematics Concept Papers, which recommendinclusion of extended problems in themathematics curriculum, including formal,written reports on these problems.

One example of such an extended assignment whichutilizes a variety of mathematical, reading, and

written skills could be done in an algebra orgeometry course. Students can be given four toeight floor plans for homes with certainsimilarities (number of rooms, floor space,etc.) but significantly different designs(geodesic dome, Cape Cod, ranch, split level,etc.). They can be assigned to determine whatthe advantages and disadvantages of each designare, and to prepare a written report in whichthe data, calculations, and conclusions arepresented in an orderly manner. Heatingequipment (gas, electric, solar, heat pump),window type, insulation, facing, and othervariables can be included. Students can beencouraged to look up the necessary informationin the library, 4n a file in the classroom, orcan be given a file folder with the necessaryinformation. I'i any event, they would doextensive reading in determining the informationthey need to solve the problem. In doing thisreading, they will be evaluating it for

applicability acid extracting useful information,the basic purpose of much of the reading done inthe world of work. Once the information isobtained, it will need to be organized andutilized. Upon solving the problem, the studentshould show, through a formal written report,the process and conclusions, and justify theconclusicns, including any assumptions made.

Ex4n1ded problems such as discussed above allowthe opportunity to integrate mathematics withother disciplines, and use the EssentialLearning Skills all together. Though most

practical in high schoo., extended problems canbe developed in lower grades as well. Reading,

writing, and mathematics do not exist in a

vacuum. Only by working with them as aninterrelated group can mastery of one, or all be

obtained. While extended problems can be very

time consuming, shorter "exercises" can bedesigned which will bring all three disciplines

into use. It is essential that such activities

be used. Even the most gifted of our currentmathematics students are frequently unable touse their mathematics in "real world'' situationsbecause of their inability to read and writemathematics and the information which leads toor flows from the mathematics. If we do not

demand reading and writing in the mathematicsclassroom, we will be preparing students to domathematics, but not to be mathematicallyliterate for the "real world" in whichmathematics is applied.

Appropriate Comoutational Skills

Computation is the execution of a chosenstrategy to solve a quantitative problem. The

mode of execution may include any or all of the

following: (1) mental calculation, (2) paperand pencil figuring, (3) calculator, and (4)

computer. Determining either precise orapproximate results is common when u,:ing thefirst two modes. The operations involved in

execution include addition, subtraction,multiplication and division of whole numbers,decimals, fractions, percents, and integers.

When we observe computation being used insociety, we see a variety of computationalskills being used to solve problems. Most

nonjob related computation is done mentally,

much of it approximate calculations. One study

found that nonjob related computation is about70 percent mental and 25 percent

calculator-assisted. On the job, most

computation (up to 90 percent) istechnologically assisted (calculator or

computer).

There appears to be a marked discrepancy betweenthe types of computation being taught in schooland the actual needs of pupils outside the

classroom. An examination of basal texts

reveals that most deal vimarily with paper and

pencil calculations. One study of classroompractices found that 90 percent of instructioninvolved paper and pencil algorithms. If

mathematics educators are to meet the zctualcomputational needs of students outside the

classroom we must:

1. Recognize the need to provideinstruction in the several modes ofcomputation and approach thatinstruction in such a way that studentsgain skill in making the decisionregarding which mode(s) is (are)appropriate for each problem in whichcomputation is needed.

2. Recognize the need to provideinstruction in both approximate andprecise calculations and in helpingstudents decide which is moreappropriate for each problem which

includes computation.

sIn light of the varied computational needs oftoday's students, it is time to recognize thatcomputation is not an end. Rather, it is justone part of the problemsolving process--oneproblemsolving tool. Hence, teaching theseveral modes of computation is an effort toplace computation into context--a means to anend.

Calculat-aAndCORadeli

Since less than 1 percent of college freshmenplan on majoring in mathematics*, it seems

obvious that the primary goal of math educationis to prepare students to cope with the manyproblems they may encounter on the job, infuture education, and in their daytodayactivities. Keeping this in mind will providesome measuring stick for determining theappropriate use of calculators and computers inthe mathematics classroom.

It is reasonable, when introducing new concepts,to use calculators to do problems as examplesthat are more complex than those already done.It Is obviously not appropriate, however, toallow students to rely on electronic devices tocarry nit simple computations before theyunderstand the processes. Once understandinghas been achieved, the advisability of usingpencil and paper techniques rather thancalculators for solving many problems isquestionable. Applying the criteria of futureuse, even a casual survey of those around us whoare routinely involved in "number crunching"will reveal that few instances arise where

*Astin, et al., The American Freshmc.A:

National Norms for Fall 1980, as reported in theConference Board of the Mathematical SciencesReport of the Survey Committee, Volume VI, 1981.

64. 28

pencil and paper operations are performed. Thenormal process will involve either calculator,or for extremely involved or repetitiveoperations, computers. Continued emphasis onpencil and paper computation is not consistentwith this reality and serves primarily to testthe students' tolerance for drudgery, not theirmastery of topics. These considerations shouldapply io most testing situations as well asduring instructional activities.

Characteristically, the American educationalsystem has emphasized computation more heavilythan the European nations and has devoted moretime to it. Both the National Council ofSupervisors of Mathematics* and the NationalCouncil of Teachers of Mathematics** have issuedposition papers in which the importance of othercontent strands has been emphasized and theimportance of computation diminished. The NCTMin particular has emphasized the role whichcalculators*** can play in shifting thisemphasis. If the mathematics curriculum in theprimary, middle, and secondary grades is toreflect the recommended increase in time devotedto other aspects of mathematics, particularly

problem solving, time must be gained by the useof calculators in place of pencil and papercomputation once concepts have been mastered.

*National Council of Supervisors ofMathematics, "Position Paper on BasicMathematical Skills," The Mathematics Teacher 71(February 1978: 147-152)

**National Council of Teachers ofMathematics, An Agenda for Action:

Recommendations for School Mathematics oc the1980s, 1980

**The Impact of Computing

Technology on School Mathematics***

A Position Statement onCalculators in the Classroom, September 1985.

Finally, some aspects of other content strandscan be explored conveniently only through the

use of calculators.

Calcula'.ors, therefore, become an essential part

of a forwardlooking mathematics curriculum.Students should be expected to obtaincalculators, or access to calculators should beprovided for them. Certainly, by the thirdgrade, calculators should be considered anessential tool for doing mathematics. In the

middle school and high school courses, studentsshould be allowed, if not encouraged, to usecalculators in math classes. The logic ofcontinuing to have Algebra I and Algebra IIstudents perform calculations involved lasolution of complex or involved eql...7.'ions is

questionable. The concepts being learned are

not computation concepts. Improved accuracy,

and thus a decrease in frustration, can beobtained by the use of calculators.

While calculators are now within virtuallyeveryone's reach, and are being usedincreasingly frequently, computers have made

less impact in the mathematics classroom. With

the exception of computer literacy andprogramming courses, and the use of relativelysimple programs designed for CAI, primarily inthe elementary grades, computers have not become"tools" used in the many mathematics classrooms.

As more sophisticated software is developed,computers will become more common. CAI programs

for most of the standard secondary mathematicscourses are now available, and steadilyimproving. Specific application of software,performing such tasks as solving first andsecond order equations, factoring, graphing,differentiation, and integration are also

29

appearing. Examining "real world" use of suchprograms, it seems likely that their use innonschooi professional environments willincrease. This suggests that there use in theclassroom should also increase. The same

criteria previously applied to calculators,namely, is the task central to the concept beinglearned or merely a timeconsumirg taskpreviously mastered, should be applied. Where

the task is not central, students should beencouraged to use a computer. Obviously, thissuggests that school districts will need toprovide increased budgets for the software andhardware needs of the mathematics curriculum.

Applications

According to Zalman Usiskin*, 'Skill is relatively useless without the ability to apply theskill; all courses should devote some attentionto 'real world' applications of the mathematicsbeing studied. In the collegebound stream,applications are often not included in the

curriculum. Tear.hers often skip applications

even when they are in the book." If we agree

with the idea that the primary purpose of themathematics curriculum is to prepare studentsfor its use later in life, application must beconsidered at least a vital as Usiskin maintains.

Applications should be included in all units inthe mathematics curriculum. The selected applicatirls should be consistent with students'level of development and perceived by them asreal and relevant to their world. This simplified diagram suggests the relationship that

*Usiskin, Zalman, A Proposal of ReFormingthe Secondary School Mathematics Curriculum,NCTM Annual Meeting, 1983.

should exist between mathematics instruction andthe "real world":

Real world:

problems, relationships,

t- inology

Abstractions

Translations

ApplicatiJns

Wcrla of mathematics:

symbolism, equations

prnblems, solutions

Notice that the "real world" influencesmathematics and mathematics influences the "realworld." Thus, mathematics instruction mustreflect not only aspects of the world andmathematics but of their interactions.*

Thus, in doing applications in mathematics weare concerned with developing relationalthinking. The concept of relation isfundamental to human thought. Knowledge ofrelationships allows humans to go beyond theimmediate and make reasonably reliablepredictions of future events.** Scientists andengineers use applications of mathematics toexplain physical phenomena and to control them."It is in precisely this fundamental problem ofsearching for relationships and giving accurateexpression to them that mathematics has been sociccessful."**

*Schaaf, Oscar et al., Oregon Mathematics

COLULEMIE111211LMiddle School mathematics.6-8, 1986.

"'National Council of Teachers ofMathematics, The_Plate of Mathematics inS2condiry Education, NCTM 15th Yearbook.

o

30

A knowledge of mathematical relationships isnecessary, but more important still is thedevelopment of the ability to search forrelationships both in mathematics and everydaysituations. Students must learn to "translate"between mathematics and practical situations.This process must go both ways. The ability ofrelational thinking is more likely to develop ifthe instructional focus is as much on the searchprocess as upon the relationships Clemselves.

The reality as seen in the mathematicsclassroom, however, is that application plays aminor role in instruction if it is present atall. The reasons for this are involved andcomplex, but can be broken down into three basicjustifications. The first, often quoted byfrustrated teachers, is that "Students simplydon't seem to be able to do story (application)problems." To avoid frustrating students, whichresults in low grades, large numbers of studentsdropping courses, and declining registration inthe courses, and to minimize !eir own

frustration, teachers often d, 't even attemptto teach applications. The senond is that inmost texts the "applications" appear in the formof "word" or "story" problems. The relevanceand r3asonableness of these problems, or ratherthe lack of these qualities is a great source ofamusement to students and teachers alike. Aparody of such problems (If a chicken and a halfcan lay an egg and a half in a day ana a half,how long will it take a grasshopper with onewooden leg to kick all the seeds out of a Dillpickle?) has even been turned into a greetingcard.

The third, and perhaps the most compellingjustification teachers give for not doing,rplications is lack of time. The huge volumeof material teachers feel they must cover

C

intimidates them into giving up the attempt toteach applications. This reason, however, isthe most obviously incorrect. The importance ofapplications is far greater than the importanceof completing all topics in our textbooks.Careful analysis of textbooks will allow exper-ienced teachers to eliminate topics, freeingtime for application, possibly the most impor-tant skill taught in the math curriculum.Appropriate use of calculators and computers toeliminate some computational diudgcv can freemore time.

The first reason, students' liability to doapplications, is also easily countered. If thisidea were extended, most students would beexcused from most of the math, science, andEnglish curriculum.

Lack of materials is an equally weak argument.In the area of algebra, Usiskin's two-volumealgebra text* provides an excellent sourcebookfor those using other texts, even if not used asa primary text. Problems for general mathe-matics, algebra, geometry, and other coursesare available from a number of differentpublishers** and are discussed in more

*Usiskin, Zalman, Algebra ThrouohApplications NCTM 1979.

**Delmar Publishers, Practical Problems inMathematics for eleven different volumes dealingwith different voc/tech areas.

National Council of Teachers of Mathematics,Applications in School Mathematics - 1981Yearbook.

Oregon Department -f Education, OregonV. -Tech Mathematics Problem Sets, 1974.

Wiltsie, David, Vocational Mathematicsagries, Motivation Development Inc., 1982.

31

detail in the bibliography of the OMEC paper,Vocational and Technical Mathematicl*. These

problems can give students practice in applyingthe various mathematics content taught, while atthe same time demonstrating the usefulness ofthat content and developing skills in theprocess of "translation." In particular, theylend themselves to developing extended problemswhich use a number of concepts and applications.

Even without the various sources alreadyreferred to, a number of applications can bedeveloped by using material readily available tomath teachers from their colleagues in theirscience departments. A simple series ofelectrical circuit opens up the possibility ofexploring linear and inverse relationships.Exploration of Ohm's Law through a demonstrationor experiment can illustrate the use of standardand log-Olog graph paper in deteroining therelationship between two variables. Doing a

Hooke's Law demonstration with a spring canallow students to explore situations in whichequations apply over one range of values, but donot apply over all values, due to physicalconsiderations. In this way they begin toappreciate the limitations of mathematics asapplied to real situations. A moreunconventional approach, but an equally goodone, would be to first consider a physicalproblem, then use it as motivation fordeveloping the appropriate mathematics.

*Zaraza t al., Oregon Mathematics ConceptPaper No. 7: Vocational and TechnicalMathematics, 1986.

*STAFF DEVELOPMENT

The previous section of this documentInstructional Themes suggested a variety ofstaff development needs. The followingchecklist can be used as a tool in assisting

as a classroom teacher, you

created a worksheet for your students?

really felt good about a math lesson?

used an interesting approach to skillbuilding?

shared one of your ideas with anotherteacher?

found a good idea when visitinganother teacher's classroom?

asked the librarian to order a mathbook for your students or for yourown professional growth?

had students solve a problem usingthings other than paper and pencil ortextbooks?

took or helped plan a workshop orinservice?

asked for released time to do any ofthe above?

32

administrators and teachers in identifying thoseneeds and suggesting a variety of staffdevelopment approaches. When was the last time:

as an administrator. you

attended a math lesson?

taught students a math lesson?

helped teachers plan inservicemeetings or workshops in math?

encouraged your teachers to visitanother math class?

utilized the skills of your teachershaving special math interests?

helped a teacher order math materialsother than textbooks?

recommended a re3c'Jrce center for oneof your teachers to visit?

helped teachers take advantage ofreleased time to develop theseinterests?

No textbook can have the impact on studentlearning that a knowledgeable well-trainedcaring teacher can. However, some teacherscontinue to teach mathematics in the same waythat they learned mathematics: memorizing a

tedious sequence of algorithms and definitions.

Mathematics must be viewed differently. The

teacher must exhibit -)n attitude of explorationand invention, conveying the idea that allstudents can learn, enjoy, and use mathematics.

School districts should provide professionalgrowth experiences for their teacher. Teachers

who have used only a textbook approach must haveinservice training to develop skills whichenable them to provide more versatile instruc-tion. Districts must provide ongoing financialsupport, encouragement, and time for teachers to

attend conferences and workshops.

Teachers and buildings should be encoui%ged tomaintain current membership in professionalorganizations to insure access to current trendsand research. Teachers should be encouraged toactively participate in professionalorganizations.

Given the new suggested curriculum and requiredlearning skills, districts which do not providean ongoing staff development program will beunable to fully reach the mathematics goalsdescribed in this guide.

Education

Service

Districts

33

GROWTH OPPORTUNITIES

Local

Districts

.a

STATE DEPARTMENT

. Specialist;, etc.

Professional

OrganizatiuAs

Universities i

and

Colleges

C.armercial

Enterprises

ej

ORGANIZATIONAL PAiicRNS ANDTEACHING CONSIDERATIONS

Role of the Textbook

Ideally, the mathematics textbook should beregarded as a guide to the specific subjectarea. In general, it provides far more materialthan can be reasonably covered in a singleyear. By doing so, it provides a number ofoptions to the teacher, presenting a variety ofsomewhat different courses depending on whichtopics are used and which are omitted. At thesame time major topic strands recommended by theOregon Council of Teachers of Mathematics(OCTM), the National Council of Teachers ofMathematics (NCTM), and the Oregon Mathematics'Education Council (OMEC) a'e asent

In reality, the textbook is more frequently usedas a "cookbook." Too often teachers begin onpage 1 and proceed as far as they can in theinstructional time allotted. In doing so, theyoften cover material that some experiencedteachers would regard as unnecessary. The firstchapter(s) is (are) often review, yet may betaught with the same depth and time commitmentas new material of equal length. Later, moreimportant chapters, may never be reached.

This problem occurs because teachers appcar toassume that the authors know exactly what shouldbe taught and have included all essentialtopics. Additionally, they mistakenly believeall the material must be presented in order,that no topics are of lesser importance andcould be deleted or deemphasized. They areunaware that an excess of material is presented

34

to "cover all the bases," to insure that

potential purchasers will find what they regardas important.

Current TSPC standards for mathematics

certification insure that math teachers have anadequate, broad exposure to and understanding ofmathematics. Such teachers are better preparedto pick and choose topics from texts. Throughconsultation with colleagues and considerationof journal articles, reports and positionpapers which discuss appropriate content forthe various courses such as Usiskin's*, theycan make reasonable choices regarding thecontent of their courses. They should not berigidly bound by the pace, arrangement, orcontent of a textbook. The textbook shouldmerely serve as a starting point, and perhaps asource of problems and exercises. Only if wellprepared teachers are encouraged to exercisetheir professional judgment can change andimprovement be implemented at a reasonable pace.

In choosing supplemental materials, usefulsources should include books, newspapers,magazines, project materials, maps, games, andcomputer software. Teachers should feel free tomake eeensive use of materials outside thestandard textbook. In this way they can besttailor courses to meet the specific needs oftheir students.

*Usiskin, Zalman. "What Should NOT Be inthe Algebra and Geometry Curricula of AverageCollege-Bound Students?" Mathematics TeacherSeptember 1980: 413-424.

Providing for Individual Differences/Remediation

A common problem which many Americans sufferfrom is "math anxiety." The implication of this

"ailment" is that there is something wrong withthe individual who suffers from it. S(He) is

somehow flawed because s(he) cannot learnmathematics as well or as quickly as the norm,

or the average. When this norm is examined, orwhen tone withdrawal and flunking rates inmathematics courses are reviewed, it becomesobvious that most of us do not achieve at the

"norm." The conventional mathematics curriculumhas been geared toward a single goal: calculus.

In such programs, from primary mathematicsthrough high school the pace and content of mathcourses are designed to prepare students forcalculus. Any student who fails to keep up thepace is viewed as "having a problem" withmathematics.

This is a misperception. The student does not

have the problem. The curriculum does. Our

conventional mathematics curriculum tends tr..%

lack the flexibility which allows students toreach their own maximum ability. Instead, it

serves to discourage those who cannot achieve ata standard pace. A recent survey indicated thatFewer than one percent of graduating high schoolstudents have taken calculus. Even in college,

fewer than 20 percent of students major infields which regularly require or use calculus.This means rust students "drop out" ofmathematics along the way.

Rather than designing the curriculum to fit the

needs of students, we design the curriculum,then force students to fit into it or leave

mathematics. More emphasis on individual needs

is necessary.

35

One level at which an awareness of individualdifferences and attempts to accommodate thesedifferences has been successful is the primarymathematics program. Primary teachers routinely

ability group students across the variouscontent areas. This grouping allows students to

progress at a more comfortable rate. The

difference in achievement that occurs at then,grade levels, however, helps generate problemsin later grades. During middle school, andcontinuing throughout high school, allowanceshould be made for these increasing individualdifferences.

If students are to be encouraged to continuein mathematics, if they are to become mathematically literate, appropriate learningopportunities must be provided. The twoyeargraduation requirement provides both anopportunity and an obligation to develop acurriculum which allows for differences inability and interest in mathematics. Courses

which are not explicitly preparation forcalculus, but rather oriented towardapplication, or toward general mathematicalliteracy (including the broad content strandsrecommended by the NCTM, NCSM and the OMECconcept papers) must be developed. The

curriculum must be broadened to allow moreoptions for students. This must take place not

only at the low end, where inclusion of all therecommended content strands 'an provide a moreuseful and interesting course, but at the upperend, as well, sn that students who have beensuccessful have an option other than calculus.

Organizing Systems for Secondary

Historically, the major system or vehicle bywhich mathematic instruction has been deliveredhas been through a sequential system of courseofferings. An alternative approach, one inwhich mathematics is taught as a "unified" dis-cipline rather than as neatly compartmen-talized series of discrete topic areas, is morecommon in the European school systems. The ideahas received substantial support from themathematics community in this country in thepast. the Mathematics Association of America(MAA), in 1923, recommended a program of"composite," "correlated," "unified," or"general" courses to be introduced intosecondary schools so that students might graspthe important interrelationships betweenarithmetic, algebra, and intuitive geometry inorder to gain insight into mathematicalmethods.* The 1940 report of the JointCommission of MAA and NCTM also advocated aunified approach as its first option.** TheOregon Mathematics Concept Papers, developed byOMEC, frequently refer to position papers ofboth the NCTM*** and the National Council of

*Mathematics Association of America, TheReorganization of Mathematics in SecondaryEducation, 1923.

**Joint Commission of MAA and NCTM, ThePlace of Mathematics in Secondary EducationFifteenth Yearbook, 1940.

***National Council of Teachers ofMathematics, An Attend for Action:Recommendations for School Mathematics of the1980s, 1980.

036

Supervisors of Mathematics (NCSM)* whichadvocate inclusion of a variety of strands inall mathematics courses. Two of these strandsare topic areas which are normally separatecourses in the traditional sequentialpresentation of mathematics.

In view of the historic support for the unifiedapproach to mathematics instruction and the 'morerecent advocacy of such ideas by the profes-sional mathematics teaching organizations, tneunified approach must be regarded as a preferredmethod of presentation.

In the past, suggesting that the unifiedapproach be adopted was somewhat impractical.Few materials were available for suchapproaches. Recently, however, there has been atrend in that direction which promises tomotivate text authors and publishers to developa greater array of materials, making this optionnot only viable, but desirable for forward-looking school districts. (At the time thispaper was written, three publishers alreadyoffered complete sets of textbooks utilizing aunified program.) New York state has unifiedprogram for those students planning to take theRegent's examination before graduation. The

Superintendent of Public Instruction for theState of-Washington has recommended** three

*National Council of Supervisors ofMathematics, "Position Paper on BasicMathematical SKIlls," The Mathematics Teacher,71, (February 1978), pp. 147-152.

**Guidelings for the 9-12 Mathematics Curri-culum from the Office of the Superintendent ofPublic Instruction from the State of Washington.

sequences of courses for the unified study ofmathematics, each designed for students ofdiffering ability, interests, and aspirations.The recently published Mathematics Framework forCalifornia Public Schools K-12 mandates thatconcepts from the various content strands beincorporated into mathematics courses :... all

grade levels.

Based on the positions taken by mathematicseducators, both nationally and within Oregon,the unified approach to the mathematicscurriculum must be regarded as the wave of the

future. As a means of adequately serving theneeds of all students, rather than merely thecollegeprep, such a program is far superior tothe more traditional or L..quential approach. It

allows ali students to progress beyond merecomputation, opening up a world of interrelated

mathematics. Additionally, it allows moreflexibility for different rates of studentdevelopment, diminishing the "selecting out"process common to sequential programs in which astudent, placed in a class such as formalgeometry before they have developed thenecessary reasoning ability, is frequently lost

to mathematics forever. School districts arestrongly encouraged to explore the possib.lityof implementing a unified approach :omatnematics as they review their mathematics

curriculum. The OMEC concept paper "AnPaesrated Mathematics Curriculum for HighLhool" presents additional arguments supporting

this option, as well as outlining a possible

approach in more detail.

Emphasis on delivery of instruction techniquesin the unified approach is often more heavilyemphasized than in the sequential curriculum.If the fully unified curriculum cannot beimplemented by school districts, serious

83

attentioo should at least be paid to some of

these key instructional methodologies:

1. Students should be involved in activelearning situations for at least half of

the average class period.

2. Individual seat work should not exceed

onethird of the class period.

3. Oral and written description ofproblem solving processes should be

required of students.

4. Cooperative learning should be anessential element of the learning process

in the mathematics classroom.

5. Teacher modeling of problem solving is

essential.

6. Mathematics should be presented as asequential discipline (not purely withinalgebra, for example, but in the context

of the whole of mathematics).

7. Questioning, by both students andinstructor, must be encouraged. Straight

lectures must be minimized.

8. Calculators and computers should beemployed to eliminate the tedium oflengthy calculations, to allT,-; topics to

be developed in greater detail and toallow more realistic problem solving.

9. Extended or indepth problems (of severaldays in length) should be included toallow students to develop problemsolving, analytical, and reporting skills

in a more realistic environment.

37 F:

Those school districts which choose to continueto utilize a traditional sequential curriculumare faced with the necessity for substantialmodification particularly with the two-yeargraduation requirement. The Algebra I -Geometry - Algebra II - Advanced Topics sequenceso commonly used is actually designed for thecollege- bound, math and science oriented

students, a small percentage of all thosestudents actually subjected to it. To trulyfill the needs of students, alternatives at aillevels must be considered. A pc;sible programto achieve this is shown below:

Remedial

Math

Math Informal

AlgebraConcepts

_I

Applied

Math

Any class in

this sequence

may feed into

Applied Math

Informal

Geometrl,

Algebra

Geometry

1

Algebra II

Advanced

Classes

38

All of these courses should emphasize more useof calculators than is presently the commonpractice. They follow the same content strandswhich a-e being recommended for the middleschool mathematics (which is a unified program).

Remedial Math. This course is designated forstudents ranging from special education students

tnrough students who are no more than two orthree years below grade level. It is assumedthat these students have not yet have masteredthe basic concepts and operations of arithmetic.

Math Concepts. For students no more than twoto three years below age/grade level, thiscourse should follow the curriculum included inThe Oregon Mathematics Concept Papers on themiddle school curriculum, or alternatively,Zalman Usiskin's Transition Mathematic) course.The emphasis here should not be oncomputation. Course content should follow thestrands referred to throughout the discussion ofthe K-8 curriculum.

Applied Math. This course may be a singlecourse or several different courses, dependingon the size and needs of the school. Two of TheOugon -4thematics Concept Papers, "OregonGraduation Requirements: Mathematics" and"Vocational and Technical Mathematics," providea number of possible curricula. Emphasis shouldbe on the use of calculators and on "real-world"problem solving.

Informal Algebra, This course and the courseon informal geometry should not follow the oftenrepeated pattern of the introductory algebracourse--slower presentation but the samematerial. This should not follow the "slowerbut louder" teaching apprtacn used with less

Et)

accelerated students. Certain traditionalconcepts in algebra should oe omittedcompletely, or done only through the use ofcomputer programs designed to carry out theoperations. These include quadratic equations,factoring, and Amultaneous solution ofequations by substitution. The course shouldbuild a conceptual base, minimizing analgorithmic approach. Equations should hesolved intuitively when possible, with modelsused as well. The orientation must bepractical. Problems should provide themotivation for the development of the algebraicconcepts. The various mathematical propertiesshould be examined and developed, but notemphasized by name. Drawing and graphicinterpretation 'particularly with the aid of thecomputer) should be utilized wherever practical.

Informal Geometry. This course mustde-emphasize proof. Primary focus should be onconstruction, measurement,, drawing,categorization and classification, propertyidentification, and graphic interpretation.

Algebra I - Geometry - Algebra II. Thesecourses should include more integration of thevarious content strands than is presently thecase.

Advanced Courses. Current practice generallyoffers senior math, pre-calculus, or calculuscourses in this slot. While these may still beoptions, serious consideration should be givento placing discrete mathematics, probability andstatistics, or problem solving courses at thispoint in the curriculum.

While this curriculum is not as desirable as thefully unified program, it does provide for a

F 439

wide variety of students. Advanced studentsmay, of course, enter at high levels, such asGeometry, Algebra II, or even the advanced

course.

Teaching Considerations

At the high school level, there is a wide rangeof student ability, skill, and background.Thus, the content varies widely. It is true,

however, that the delivery of instruction can bEvery similar for the majority of the high school

popu ation.

Instruction in the high school mathematics

program should:

Involve students in active learningsituations. Individual seatwork should beminimized.

Require that students describe in writingand or7.,ily the process used in solving agiven problem and the results obtained.

Involve questioning techniques that createstudent interaction, encourageexploration, diagnose processes thatstudents have used, enhance transfer oflearning, motivate, and challenge students

to think.

Stimulate curiosity on the part of the

students and set the stage so thatstudents ask questions and contributecomments in a discussion.

Include the model of problem-solvingactivities by the teacher as a regularpart of the ins'. :tional program.

F

Demonstrate the continuity of mathematicsby explaining the relationship of thecurrent work to concepts learned ii thepast and to be learned in the future.

Use a variety of instructional formats,e.g., small groups, interactive groups,teacherled discussions, steentpresentations, and laborator learningexperiences.

Involve high but appropriate expectationsfor all students, for example, studentsare expected to complete all assignmentsand to do well on examinations.

Involve the use of calculators andcomputers to: (1) enhance the study ofmathematical relationships, and (2) speedup the development of mathematicalconcepts by reducing the time spent onintermediate computations.

Encourage students to solve problems in avariety of ways and to accept solutions inmany different forms.

Provide opportunities for students tobecome involved with indepth problemsituations for periods of several days.As they work toward solutions, studentsshould investigate and experiment, developnew mathematical skills and concepts,apply previously acquired knowledge, andbe able to express the solution in writtenform.

Increase the student's ability to estimatesolutions and to always judge thereasonableness of calculated results.

Alternative Approaches to TwoYear GraduationEgquirement

With the adoption of the twoyear graduationrequirement in mathematics, the Oregon Board ofEducation has created an opportunity for thedevelopment of a relevant and effectivemathematics component for all students. A

survey of Oregon high schools taken by the OMECSubcommittee on Vocational and TechnicalMathematics* indicated that 20 percent of thegraduating class of 1984 took only one year ofmathematics. That same survey revealed that35.8 percent of r Auates took no math beyondAlgebra I and 51., Arcent did not go beyondGeometry. Thi- indicates that a large number ofOregon students have been choosing to not takecourses in the traditional mathematicscurriculum. They preseA a large "clientele"which has not always been well served Lilt nowmust be serred.

For many of the high school students who will betaking the minimum oatheratics requirement, thetraditional sequences are unsatisfactory. Atthe lowest achievement level, students havenormally been placed in classes which emphasizeco4Jutation, a content strand in which thesestudents have already demonstrated they havelittle facility. At a slightly higher level,students are placed in courses which areessentially part of the algebra sequence whichha; been designed as a "feeder" for Calculus.While algebra may be a suitable course for thecalculusbound student, it does not serve theneeds of students for whom it will be the lastcourse in mathematics.

*Zaraza et al , Report to the OregonMathematics Education Council on Vocational andTechnical Mathematics, 1985.

Either of the traditional sequences taken bystunts taking minimal mathematics emphasizecontent which is often not useful to them.Appropriate courses for these students shouldfocus on mental computation, application and useof simple algebraic formulate, interpretation ofgraphs and statistics, information geometry,measurement, and problem solving*. Mastery of

these concepts will e; 51e these students tocope more effectively with the adult world.

Accordingly, courses should be designed forthese students which address content other thansimple computation or algebra. A number ofoptions for such courses are outlined in theOMEC concept papers on the two-year requirement

and on voc/tech mathematics. Both of these

papers emphasize that courses designed tofulfill the two-year requirement should followthe NCTM and OCTM recommendations that allcontent strands of mathematics be developed.For these students, this is a radicallydifferent approach, but a more realistic one.

dc 3220g10/23/87

*Hopper et al., Graduation Require.

Committee Report, OMEC 1985.

is3

41

Another student population which must beconsidered is that group which completes Algebraor its equivalent in its first year of high

school, but historically has not takenadditional math. Relatively poor performance inalgebra, fear or dislike of geometry, or ageneral fear of mathematics or not recognizingits importance in our current technologicalsociety are the major reasons for not takingadditional mathematics. With the two-yearrequirement, these students are now placed inthe position of having to take additional mathcourses. Again, the option of offeringnontraditional mathematics courses such as thosesuggested in the OMFC concept papers seems to be

a viable approach. Such courses, when properlydesigned can provide interesting and usefulmathematics for a far broader ability range ofstudent than the more conventional courses.

3

1.0 Number and Numeration: Students demonstrate an understanding of number and numeration concepts and use these understandings to interpret and solveproblems.

1 1 READ. WRITE, ORDER. COMPARE AND USE NUMBERC (ELS 1.4)

KINDERGARTEN

4 Rote count to 20 anddemonstrate comprehention of whole numbersup to 10 by reading,writing, ordering,modeling, and comparing

b Use number patterns tofacilitate skip countingby 5 and 10

c Estimate number ofobjects and check reasonableness of answers bycounting

d Use ordinals (1st, 2nd,3rd, 4th, and 5th)

GRADE 1 GRADE 2 GRADE 3 GRADE 4

a Rote count to 20 and a Rote count to 100 and a Read and write whole a Read and write wholedemonstrate comprehen demonstrate comprehen numbers to 1.000, numbers to 100,000; modeltion of whole numbers tion of whole numbers commonly used fractions commonly used proper fractions;up to 20 by reading, up to 100 by reading, (1/2. 1/3, 5/10). and mixed numbers and improperwriting, ordering,modeling, and comparing

writing, ordering,modeling, and comparing

decimal in tenths and fractions (1 1/2, 2 1/4, etc.)hundredths using moneymodels

b Demonstrate the counting b Demonstrate the counting b Demonstrate the counting b Demonstrate skip countingskills of skip counting skills of skip counting skills of skip counting skillsby 2, 5, and 10; by 2, 5, and 10; by 2, 5, and 10;"counting on" and counting "counting on" and counting "counting on" and countingbackwards to and from 20 backwards to and from 20 backwards to and from 1.0

c Estimate number of c Estimate number of Use place value concepts c Use place value conceptsobjects and che-k reason objects and check reason to estimate and sense to estimate and senseableness of answers bycounting

ableness of answers bycounting

reasonableness of answers reasonableness of answers

d Use ordinals to 10th

* The outcomes for mathematics in the skills and grade level expectancies tha' are underlined reflect the requ,red outcomes from the Common CurriculumGoals and includes the Essential Learning Skills.

9042

GRADE 5

a Read and write wholenumbers to one million;commonly used properfractions. mingnumbers. and 'mp^oPerfractions: and decimalsto thousandths

41I,Demonstrate skipcounting skills

c Use place value conceptsto estimate and sensereasonableness of answers

GRADE 6 GRADE 7 GRADED GRADE 11

a Read and write whole a Read and write numbers, a Read ana write numbers a Read, write and ordernumbers to one million; including decimals, including decimals, numbers including decimals.commonly used proper commPnly used fractions commonly used fractions commonly used fractions.fractions, mixednumbers, and improper

and percents and percents Percents and numbers inlijentific notation

fractions; and decimalsto thousandths

b Write numbers in b Write numbers in b Express large numbers in b press 1010_aud smallexpanded notation expanded notation expanded exponential numbers in expanded(e.g., 623 = 600+20+3) (e.g., 623 - 600+20+3) notation exponential notation

c Order signed numbers and Order sigrtEimbers andcommonly used fractions. commonly used fractions,decimals and percents decimals and p rcents

43

d Express large whole num- d Express large ana smallbers in scientific notation numbers in scientific notation

1.2 USE CONCRETE ANO PICTORIAL MODELS TO UMONSTRATE NUMBER ANO NUMERATION CONCEPTS (ELS 1.4)

KINDERGARTEN GRADE 1

a Use concrete models toshow place value and todemonstrate halves andwholes

GRADE 2 GRADE 3 GRADE 4

a Use concrete models tztshow place value throuckgrouping and exchanging,and to demonstrate halvesand wholes

a Use concrete models todemonstrate twodigitplace value throughgrouping and exchanging,and to demonstrate halves,wholes, thirds, andfourths

b Identify the number ofones and tens in numbersless than 100 and useconcrete models todemonstrate understanding

44

a Order. compare and model(demonstrate comprehensionby use of obiects or Adrawing) place valligs to1000. commonly usedfractions and decimals(using money models) intenths and hundredths

b Identify the number ofones. tens and Kondred_s_innumbers less than 1000_4ndvse concrete models todemonstrate understanding

a Order, compare and modelplace values to one million;commonly used fractionsand decimals to hundredths,and use concrete and pictorialmodels to demonstrate an understanding of the above

b Identify the number of ones, III/tens, hundreds, thousands, andten thousands in numbers lessthan 100,000 and use concretemodels to demonstrateunderstanding

GRADE 5 GRADE 6 GRADE 7 GRADE 8 GRADE 11

a Order, compare and modelplace values to onemillion; coomonlv usedfractions and decimals tothousandths, and use concrete Pictorial modelsto demonstrate an understanding of the above

410) Identify the number o:Ines. tens. hundreds.lbausailds, tenthousands.and hundredthousandsin numbers less thanone million, and tenths.hundredths, thousandthsin numbers less than one

a Order, compare and modelplace values to onemillion; commonly usedfractions and decimals,and use concrete andpictorial models to demonstrate an understanding ofthe above

b Identify the number ofones, tens, hundreds,thousands, tenthousands,and hundredthousandsin numbers less thanone million, and tenths,hundredths, thousandths,etc , in numbers lessthan one

Describe the usefulnessof a place value nu'eration system in mentalcomputation givingexamples in common use

a

b

Order, compare and modelcemmonly used fractions,decimals and percents,and use concrete andpictorial models todemonstrate an understanding of the above

Identify the number ofones, tens, hundreds,thousands, tenthousandsand hundredthousandsin numb r: less thanone million, and tenths,:lundredths, thousandths,etc., in numbers lessthan one

Describe the usefulnessof a place value numeration system in mentalcomputation givingexamples in common use

d Compare the base tennumeration system withan ancient system

45

a Order. compare and modelcommonly used fractions.decimals. Percents. andsigned numbers. and giveexamples of Positive andnegative quantities (e.g..temperature, football.bank balances, altitude)

b Identify the number ofones. tens. hundreds.thousands. tenthousands.and hundredthous ndsin numbers less thanone million. and tenths,hundredths, andthousandths in numbersless than one

Describe the usefulnessof a place value numera -tion system in mentalcomputation givingexamples in common use

d Explain societal usesof modern, nondecimalsv:tems of numeration

e Illustrate the meaning ofsquare root by use of amodel (e.g., in a squarethe side represents thesquare root of the area)

a Order. compare and mndgicommonly used fractions.decimals. Percents, andsigned numbers. and giveexamples of positive andnegative quantities (e.g..temperature. Football.bank balances. altitude)

b Identify the number ofones. tens. hundreds.thousands. tenthousands.and hundredthousandsin numbers less thanone million. and tenths.hundredths, andthousandths in numbers

less than one

Describe the usefulnessof a place value numeration system in mentalcomputation givingexamples 'n common use

d Explain societal usesof rodern, nondecimalsystems of numeration,particularly binary

e Illustrate the meaning ofsquare root by use of amodel (e.g., in a squarethe side represents thesquare root of the area)

e1.3 RECOGNIZE AND USE NUMBER PROPERTY CONCEPTS

KINDERGARTEN GRADE 1 GRADE 2 GRADE 3 GRADE 4

a Demonstrate the proper a Demonstrate and use the a Demonstrate, use andties of addition and properties of addition and apply the properties ofsubtraction subtraction addition, subtraction

and single digitmultiplication

b Recognize and use mathematical terms (e.g.,equal, greater, less)

53

b Recognize and 'Ise mahemat'cal terms (e.g.,odd, even, equal,sum, and difference)

46

b Recognize and usemathematical terms(e.g., sum, total, difference, product, equal, odd,even)

a Demonstrate, use and applythe properties of addition,subtraction, multiplication,and single digit divisionwith whole numbers

b Recognize and usemathematical terms(e.g., product, factor,quotient, remainder, sum)

d Recognize and use conceptsrelated to multiples andfactors


a Demonstrate. use and a Use and apply the proApply the properties of perties of addition, subaddition. subtraction, traction, multiplication,multiplication. and and division with wholedivision with whole numbers numbers

b Recognize and usemathematical terms(e.g., product, factor,quotient, remainder, sum)

b Recognize and usemathematical terms

c Explain the reasons forthe rules for order ofoperations and use ofgrouping symbols

a Use and apply the properties of addition, subtraction, multiplication,and division with wholenumbers, positiverational numbers, anddemonstrate properties ofone and zero (includingnondivisicn by zero) andthe closure property


c Explain the reasons forthe rules for order ofoperations and use ofgrouping symbols

d Recognize and use con I Recognize and use con d Recognize and use concepts related to multiples, cepts related to multiples, cepts related to factoringfactors, primes and factors, primes And and determiningcomposites composites; and determine devisibility

divisibility by 2, 5, & 10

0"47

a Use and apply the properties of addition, subtraction, multiplication.and division with wholenumbers and positiverational numbers. anddemonstrate properties ofone and zero (includingnondivision by zero) andthe closure property


Explain the reasons forthe rules for order ofoperations and use ofgrouping wmbols

d Recognize and use concepts related to factoringand determiningdevisibility

a Use and apply operationalproperties


c Use and apply orderof operations rules asappropriate for mental.paper/pencil and calculatorusage

10

III/II2.0 A.)propr IPe Computational Skills: Students select and use the most appropriate form of computation manipulative. mental,_paper/pencil.

estimation or calculator usage to solve problems and check all computations for reasonability.

2.1 USE MENTAL. PAPER AND PENCIL. ESTIMATION AND CALCULATOR COMPUTATION:$ TO SOLVE APPROPRIATE PROBLEMS (ELS 1.4 and 1.7)


a Add and subtract with a Add and subtract with a Add and subtract actual a Use mental. manual oractual objects (up to 10) actual objects (up to 10) objects (up to 20) calculator processes to

pgrform qradelevelali_thrpetic operations

c Estimate and approximateanswers to problems

e Mentally add 10 to asingle number

102

Use estimating s%ills,such as rounding, tomake approximate wholenumber computations

a Use mental, manual orcalculator processes toperform gradelevelarithmetic operations

b Select the most appro b Select the most appropriate method of ccmputa

Ptro:te(mTreirglagverguell,girer(ZrrICItTecUlTetrioll paper/pencil, calculator)to use in a given situation to use in a given situation III

Use estimating skills,such as roundinc tomake approximate wholenumber computations

d Apply acquired strategies d Apply acquired strategiesincluding modeling patterns including modelinq Patterns(such as "counting on," (such as "counting on.""doubles," "neighbors," "doubles." "neighbors."etc.) and properties etc.) and properties(commutativit./ and (commutativity andassociativity), to aid associativity). to aidin quick recall of addi in quick recall of addition, subtraction, and tion, subtraction andmultiplication facts multiplication facts

e Mentally add or subtractmultiples of 10 to (from)a number

48

Solve mentally, appropri-ate addition and sutractionpr involvingplace value understanding,e.g.. add or subtract 10or 100 to (from) any a_digit number; add or subtract multiples of 10 or 100

C,

c Use rounding and othertechniques useful in mentalcomputation to estimate andmake approximate whole number,fraction and decimalcomputations

d Apply acquired strategiesto aid in quick recall ofall basic facts

e Use mental arithmetic skillsto solve appropriate problems(multiples of 10 and 100,addition of fractions withlike denominators, etc.)

GRADES GRADE 6 GRADE 7 GRADE 8 GRADE 11

4 Use mental, manual orcalculator processesto Perform grade levelarithmetic operation'

a Use mental, manual orcalculator processesto perform gradelevelarithmetic operations

Use mental, manual,calculator and computerprocesses to performmathematical operations

a Use mental, manual. a Use mental. manual.calculator and computer calculator and computerprocesses to Perform processes to perform

mathematical operations mathematical operations

b Select the most appropri b Select the most appropri b Select the most appro b Select the most appro b Select the most appro

ate method(s) of computa ate method(s) of computa priate method(s) of priate method(s) of priate method(s) of

tion (manipulative. meAtal. tion (manipulative, mental, computation (mental, computation (mental, computation (mental.

1110 paper/pencil. calculator) paper/pencil, calculator) paper/pencil, calculator) paper/pencil. calculator) paPerhencil, calculator)

to use in a given situation to use in a given situation to use in a given situation to use in a given situation to use in a given situation

C Use rounding and other c Use rounding and other c Use rounding and other c Use rounding and (Oar c Use and apply estimation

technicues useful in mental techniques useful in mental techniques useful in mental techniques useful ire mental techniques

computation to estimate computation to estimate computation to estimate computation to estimate

and make approximate whole and make approximate whole and make approximate whole and make approximate whole

number, fraction and number, fraction and number, fraction, decimal, numtJer, fraction, decimal,

decimal computations decimal computations and percent computations and percent computations

d Apply acquired strategiesto aid in quick recallof all basic facts

e Solve mentally. appropriate whole number, fracticaand decimal _Problems. e.g..10 x 64: 60 x 201_14.0047.000: 5.000 + 261; 3.000 x7: 1/4 + 3/4: 5/8 4/8:

3 0.5

e Mentally +, x,

whole numbers and decimalsby powers of ten andmultiples of ten, andcompute mentally amongcommonly used fractionsand decimals

G-x

e Mentally +, x

whole numbers and decimalsby powers of ten andmultiples of ten, percentssuch as 25% and 50%, andcompute mentally Imongcommonly used fractionsand decimals

49

e Mentally +, x.

whole numbers and decimalsby powers of ten andmultiples of ten andcommonly used percents

e Compute mentally when appropriate (e.g., 7 x 20; $1.50 +$.35; 4 1 1/4; $1.25 + $5,$7.29 + $3.82; $7.09 + $3.82is approximately $11or 17 15/16" 9 1/2" isapproximately 8 1/2")

(2.1 cont.)

KINDERGARTEN GRADE 1 GRADE 2

h Explore counting,addition and sub-traction wi h acalculator

k Recall informal exper-iences with multiplica-tion and division

h Explore counting,addition and sub-traction with acalculator

.1 Examine all answers forreasonableness

k Engage in informalexperiences with multi-plication and division

Perform addition and sub-traction algorithms withand without regroupiAgusing 1-2 digit wholenumbers

GRADE 3 GRADE 4

Perform addition and sub-traction algorithms withand without regroupingusing 1-3 digit wholenumbers

fUse paper/pencil to performaddition, subtraction,multiplication of wholenumbers, 1-digit division,addition and subtraction ofdecimals, addition and sub-traction of fractions withlike denominators

g Compute using measures oflength, weight (mass), timeand money

h Explore counting, addi- h Le a calculator to sollig h Use calculator and/ortion and subtraction with appropriate Problems_and computer to solve appropriatea calculator to check approximate cal- problems (e.g., real problems

culations (e.g., real with lengthy calculations orproblems with lengthvcal- large numbers)culations or large numbers)

.1 Examine all answers forreasonableness

k Soive addition/subtraction word problemsand create addition/subtraction word problemsto match a number sentence

50

J Use estimation and other j Examine all answers forskills tv check 4n1werS reasonablenessfor reasonableness

C, 7


f Use Paper/Pencil toperform addition, subtraction. multiplicationof whole numbers. 1 digitdivision, additionand subtraction ofdecimals. addition andsubtraction of fractions

1110with like denominators

g Compute using measuresof length, weightimass). time and money

h Use calculator and/orcomputer to solveappropriate problems(e.g., real problems withlengthy calculations orlarge numbers)

Use estimation and otherskills to check answersfor reasonableness

f Use paper/pencil toperform addition, subtraction, multiplicationof whole numbers, 1digitdivision, additionand -.ubtraction ofdec) als, addition andsubtraction of fractions

g Compute using measuresof length, weight(mass), time and money

h Use calculator and/orcomputer to solveappropriate problems

i Use estimation and otherskills to check answersreasonableness

'Jo

f Perform paper/pencilarithmetic computationon commonly used wholenumbers, decimals andfractions

g Compute using measures

h Use calculator and/orcoPuter to solveappr 7riate problems

i Use estimation and otherskills to check answersreasonableness

51

f fRecognize that the Recognize that thesame arithmetic algorithms same arithmetic algorithmsused with 1-3 digit numbers used with 1-3 digit numberscan be extended to multi can be extended to multidic,:t computations digit computations


h Use calculator and/orcomputer to solveappropriate problemsand recognize the limitations of a calculator whendealing with large numbers

Convert mentally,among decimals, percents and commonly usedfractions

j Use estimation and otherskills to check answersreasonableness


h Use calculator and/orcomputer to solveappropriate problemsand recognize the limitationsof a calculator when dealingwith large numbers

Convert mentally, manuallydecimals, percents andcommonly used fractions

1 Use estimation and otherskills to check answersreasonableness

2.2 DEMONSTRATE COMPUTATIONAL ALGORITHMS WITH CONCRETE MATERIALS OR REALWORLD EXAMPLES

KINDERGARTEN GRADE I GRADE 2 GRADE 3 Gmr,E 4

a Use concrete models toperform one digit addition/subtraction problems

a Use concrete models toperform one digit addition/subtraction problems

a Use concrete models toperform two digit addition/subtraction problems,and demonstrate placevalue exchanges (borrowingand carrying) up to 1000

-

52

a Use concrete models toperform whole numbercomputations, and demonstrate Place valueexchanges (borrowingand carrying) uP to 1000and to model the variousmeanings of multiplication

4

a Use concrete materials tomodel various meanings ofmultiplication and divisionand to interpret remainders

c Use models such as money ormetrics to demonstrate additionand subtraction of decimals andmultiplication of decimals bywhole numbers

d Use concrete models todemonstrate addition andsubtraction, of fractions withlike denominators

GRADE 5 GRADE 6 GRADE 7 6NADE 8 GRADE 11

a Use concrete materials tomodel the varioas meaningsof multiplication anddivision and to interpretremainders

c Use models such asmoney or metrics todemonstrate addition,subtraction and multiplication of decimalsby whole numbers

d Use concrete models todemonstrate addition andsubtraction of commonlyused fractions and tomodel the multiplicationof a fraction by a wholenumber and by a fractionusing the "of" concept

a Use concrete materials tomodel the various meaningsof multiplication anddivision and to interpretremainders

c Use models such asmoney or metrics todemonstrate addition,subtraction, multiplication and division ofdecimals by whole numbers

d Use rulers, grid paper,and/or other intuitivemethods for +, , and xwith commonly used fractions

1 1, 2

a Demonstrate an understanding of the variousmeanings of multiplicationand division (includingremainders) of whole numbers by drawings or byreferencing "real world"applications

b Model the various meanings of addition involvingsigned numbers in situations meaningful tostudents

c Use models such asmoney or metrics todemonstrate addition,subtraction, multiplication and division ofdecimals

d Use rulers, grid paper,and/or other intuitivemethods for +, x, and

with commonly usedfractions

53

a Demonstrate an understanding of the variousmeanings of multiplicationand division (includingremainders) of whole numbers by drawings or byreferencing "real world"applications

b Model the various mean b Use concrete (,r pictorialings of addition and sub models to demonstratetraction involving signed addition, subtraction andnumbers. and add, subtract. multiplication of signed

and multiply signed numbers nbers

in situations meaningfulto students

c Use concrete materials or"real world" examples todemonstrate operationswith decimals and percents

d Use concrete materialsor "real world" examplesto demonstrate operationswith commonly useJfractions

c Use concrete materials or"real world" examples todemonstrate operationswith decimals and percents

d Use concrete materialsor "real world" examplesto demonstrate operationswith commonly usedfractions

3.0 P bl use probl lving kill t lve routine and nonroutine roblems.

3.1 IDENTIFY PROBLEMS AND APPROACH THEIR SOLUTION IN AN ORGANIZED MANNER (ELS 6.3)

KINDERGARTEN GRADE 1 GRADE ? GRADE 3 GRADE 4

a Identify problems thatneed a solution

b Use simple questioning


b Use simple questioningto clarify problems to clarify problems

c Use data fromquestioning to developa problem-solving plan


b Use simple questioningto clarify problems

c Use data fromquestioning to developa problemsolving plan

a Identify problems that

nee4AJO.Ittigfi

b else simple question,to clarify problems

c Use data fromquestioning to developA problemsolving plan

d Solve problems using d Solve problems using d Solve problems using d Solve problems usingstrategies such as guessing strategies such as guessing strategies such as guessing strategies such asand checking, using con and checking, using con and checking, using con guessing and checking.crete objects, making a crete objects, making a crete objects, making a psing concrete_gbiects.model, generating a model, generating a model, generating a making a model.pattern or drawing a pattern or drawing a pattern or drawing a generating a _patternpicture picture picture or drawing a picture

e Identify alternativesolutions to a simpleproblem

f Choose and apply mental,manual and calculatorprocesses to problemsolving strategy(ies)



54


f Choose and aPPlm mental.manual and calculatorprocesses to Problemsolving strategy(ies)

a Identify problems, recognizeinformation necessary tosolve problems, and supplyadditional information ifneeded

b Use simple questioningstrategies to clarify problems

c Use data from the questioningprocess to develop a problemsolving plan

d Solve problems using avariety of strategies such asguessing and checking, makingpredictions based upon apattern, making a drawing ormodel

e Identify alternativesolutions to problems



a Identify problems.recognize informationnecessary to solveproblems, and supplyadditional informationif needed

b Use simple questioningstrategies to clarify

1111problems

c Use data from thequestioning processto develop a Problem-solving plan

d Solve problems usinga variety of strategiessuch as guessing andchecking, making pre-dictions based upona DattPrn, making adrawing or model

e Identify alternativeto Problems

f Choose and apply mental,manual and calculatorprocesses to Problemsolving strategv(ies)

a Define a problem, chooseinformation to solve theproblem and supplyadditional information,if needed

b Use a combination ofquestioning strategiesand observation to analyzeproblems

c Use data from severalsources to develop aproblem-solving plan

d Solve problems usingproblem-solving skillssuch as guessing andchecking, looking for apattern, making syst-maticlists, making a drawi.,or model, eliminatingpossible answers, orsolving a simpler relatedproblem


f Choose and apply menta'manual, calculator, andcomputer processes toproblem-solvingstrategy(ies)

g Select and apply appro-priate problem-solvingtools, includingcomputer software

4

a Define a problem, chooseinformation to solve theproblem and supplyadditional information,if needed

b Use a combination ofquestioning strategiesand observation to analyzeproblems


d Solve problems usingproblem solving skillssuch as guessing andchecking, looking for apattern, making systematiclists, making a drawingor model, eliminatingpossible answers, orsolving a simpler relatedproblem


f Choose and apply mental,manual, calculator, andcomputer processes toproblem-solvingstrategy(ies)


55

a Define a problem, chooseinformation to solve theproblem and supplyadditional information.if needed

b Use a combination ofQuestioning strategiesand observation to analyzeproblems


d Solve problems usingAppropriate strategiessuch as guessing andchecking, making asystematic list lookingfor patterns. making ordrawing e model.eliminating possibleanswers or solving asimpler problem

e Identify alternativesolutions to Problems

f Choose and apply mental,manual.calculator, andcomputer processes toproblem-solvingstrategv(ies)

g Select and apply appro-priate Problem-solvingtools, includingcomputer software

a Define a Problem, chooseinformation to solve theproblem and supplyadditional information.if needed

b Apply recognized resear1htechniques to analyzeproblems

c Design and f;arry out aplan for solving aproblem

d Solve Problems using themost appropriate tools.methodologies. processesand operations in solvinga variety of problems

e Identify alternativeSolutions to Problems

f Choose and apply menial.manual, calculator, andComputer processes toproblem-solvingstrategv(ies)


(3.1 cont.)


h Share successful andunsuccessful problemsolving strategies

i Engage in cooperativeproblem solving andcompare alternativesolution strategies

U



J Develop new suggestionsor approaches if problemis not solved

h Share successful andunsuccessful prob'emsolving strategies


J Develop new suggestionsor approaches if problemis not solved

56


Engage in cooperativeproblem solving andcompare alternativesolution strategies

J Develop new suggest'onsor approaches if problemis not solved



J Use fcrmative (in process)data to modify or ,;onfirmproblemsolving plan

GRADE 5 GRADE 6 GRADE 7 GRACE 8 WOE 11

h Share suLceisful andosuccessful Problemsolving strategies

i Engage in cooperativeproblem solving andcompare alternative

4111 solution strategies

Use formative (inprocess) data tomodify or confirmproblemsolving plan



J Use formative (inprocess) data tomodify or confirmproblemsolving plan

2k)

h Describe both successfnl and unsuccessfulsolution strategies


J Use summative (final)data to determine if theproblemsolving approachwas successful, and if cot,how it should be modified

57

h Describe both successful aAd wiSuccestflasolution _strategies

Engage in cooperativeproblem, solving anduswirs alternativesoIutipn strategies

J Use summative (final)data to determine if theprc5lemsolving approachwas successful, and if not.how it should be modified

h Evaluate problemsolvinqstrategies in terms oftools. methodologies.processes. operations

fngage in cooperativeproblem solving and

compare alternativesolution strategies

J Analyze formativeand summative data toconfirm or revise thsproposed solution

-) j,

3.2 CREATE AND SOLVE WORD PROBLEMS APPROPRIATE TO THE GRADE LEVEL (ELS 6.3)

KINDERGARTEN GRADE I GRADE 2 GRADE 3 GRADE 4

a Role play put togetherand take away situations

b Solve simple problemsby role playing

122

a Recognize appropriateoperation(s) (+, )for solutions ofword problems

b Solve simple word problems by role playing

c Create addition!subtraction word problemsto match number sentences

e Solve problems with morethan one possible solutionand recognize problemswhich cannot be solvedbecause they containtoo little information

a Recognize appropriateoperation(s) (+, )for solutions ofword problems

b Solve onestep problemspresented verbally byrole playing and use ofmanipulation

c Create addition/subtraction word problemsto match number sentences


58

a Recognize appropriateoperation(s) (+, x)

for solutions of wordproblems

b Solve onestep wordproblems including thoseinvolving money, measurement, and data presentedin graphs, tables andcharts

c Create word problems tomatch addition, subtraction and multiplicationalgorithms

e Solve problems with morethan one possible solutionand recognize problemswhich cannot :e solvedbecaus3 they containtoo little information

a Recognize appropriateoperation(s) (+, x, )for solutions of wordproblems

b Solve onestep wordproblems including thoseinvolving money, measurement, and data presentedin graphs, tables and charts

c Create word problems trmatch addition, subtr,xtion, multiplication, anddivision algorithms



a Recognize appropriateoperations) for solutions of word problems.and recognize informationnecessary to solve wordproblems. and supplyreasonable additionalinformation, if needed

Solve one and twostepword problems includingthose involving money.measurement, and datapresented in graPhs,tables and charts

Create word problemsto match whole number,fraction and decimalalgorithms


a Recognize appropriateoperation(s) for solutions of word problems,and recognize informationnecessary to solve wordproblems, and supplyreasonable additionalinformation, if needed

b Pose new problems andsolve multiple stepword problems

c Create word problemsto match whole number,fraction and decimalalgorithms


1 2 ,:rt

a Recognize appropriateoperation(s) for solutions of word problems,and recognize informationnecessary to solve wordproblems, and supplyreasonable additionalinformation, if needed

b Pose new problems andsolve multiple stepword problems

c Solve and create wordproblems to match exercises involving wholenumbers, fractions,decimals, and percent


f Translate "realworld"problems into mathematicalstatements, and mathematical problems and answersback into "realworld"context

59

a Recognize appropriateoperations) for solutions of word problems.and recognize informationnecessary to solve wordproblems. and suPPlvreasonable additionalinformation, if needed

b Pose and solvemultiplestep wordproblems

c Solve and create wordproblems to match exercises involving wholenumbers, fractlons,decimals, and percent

a Recognize appropriateoperations) for solutions of word problems.and recognize informationnecessary to solve wordproblems, and supplyreasonable additionalinformation, if needed

b Pose and solve wordproblems

c Solve and create Problemsto match exercisesinvolving ratios. proportions and formulas

d Identify, invent, or d Identify, invent, orcreate problems that can be create problems that can besolved by using ratio and solved by using ratio and

proportion: and use pro proportion: and use proportion to solve problems portion to solve problems


f Translate "realworld"problems into mathematicalstatements, and mathematical problems and answersback into "realworld"context

e Solve problems with morethan one Possible solutionand recognize problemswhich cannot be solvedbecause they containtoo little information

f Translate "realworld"problems into mathematicalstatements, and mathematical problems andanswers back into "realworld" context

nd Vipredictions.

kill : ni nd 1 n hi 1 them in

4.1 RECOGNIZE AND USE GEOMETRIC PATTERNS. RELATIONSHIPS AND PRINCIPLES TO DESCRIBE AND CLASSIFY (ELS 1.5)

1111sol roblems and makin


a Identify the attributes a Identify the attributes a Identify similar and Identify, similar and a Identify attributes ofof rectangle, circle and of rectangle, circle and different attributes different attributes pclygons and common geometrictriangle triangle of two or more

geometric figuresof two or more solids

geometric figures

b Informally explore 2-D b Informally explore b Sketch, model and use b Identify. sketch, model b Sketch, model and manipulateshapes 2-D shapes 2-D shapes and manipulate squares. squares, rectangles, circles,

triangles, cubes, cones, and 4111cylinders

rectangles. circles.triangles. cubes

e Informally explore c Through informal exper- c Identify symmetry in c Identify symmetry and c Complete constructions andsymmetry (e.g., pattern iences identify symmetry environment, and create geometric forms in the designs which have one or moreblocks, in nature) in the environment and designs with symmetry environment (e.g.. con- lines of symmetry and identify

create designs with (e.g., construction with struction with colored parallel and intersecting linessymmetry (e.g., construc-tion with colored tilesor cubes)

colored tiles or cubes) tiles or cubes) and right angles in theenvironment

d Copy or extend patterns d Copy or P -end patterns d Copy or extend patterns d Corn or extend patterns d Copy or extend patternsusing concrete models using concrete model: and using concrete models and using concrete models and using concrete models and

drawing pictures drawing pictures drawing pictures drawing pictures

e Locate coordinate pointson graph paper

r.1

60


a Identify propertiesof common geometricfigures. includingquadrilaterals andgeometric solids

b Draw or model simple,common geometricalfigures with specificdimensions using ruler.tangrams. squared Paper,or other concrete materials

Identify, sketch or model c

intersecting lines. rightangles and lines ofsymmetry occurring in theenvironment

a

d Copy or extend patternsusing concrete models anddrawing pictures

e Locate Points on graphpaper, maps and globes.and graph coordinates(emphasize examples fromthe environment)

f Recognize the conceptsof diameter, radiusand circumference

Identify propertiesof common geometricfigures, includingquadrilaterals andgeometric solids

Draw or model simple,common geometricalfigures with specificdimensions using ruler,tangrams, squared 1.aper,or other concrete materials

Demonstrate parallel,intersecting, or skewlines, measure anglesand identify lines ofsymmetry

d Use drawings, models orcomputers to demonstrategeometric patterns andrelationships such ascongruence

e Locate points on graphpaper, maps and globes,and graph coordinates(emphasize examples fromthe environment)

f Recognize the conceptsof diameter, radiusand circumference

a Identify distinguishingproperties of commongeometric figures,including side orangle measurements

b Sketch or build commongeometric solids andtwo-dimensional figures

c Identify, sketch or modelparallel and intersectinglines, right angles andlines of symmetry occurringin the environment

d Use drawings, models orcomputers to demonstrategeometric patterns andrelationships such assimilarity and congruence

e Locate and give coor-dinates of points ongraph paper, maps, globes(emphasize examples fromthe environment andcomputer %pplications)

f Demonstrate by variousmeans the relationshipsamong radius, diameterand circumference of acircle

61

a Identify distinguishingproperties of commongeometric figures.including side or

angle measurements

b Sketch or build commongeometric solids andtwo-dimensional figures

c Identify. sketch or modelparallel and intersectinglines. right angles andlines of symmetry occurringin the environment

d Use drawings. models orcomputers to demonstrategeometric patterns andrelationships such assimilarity and congruence

a Identify and comparecommon two- and three-dimensional geometric shapesand solids according toattributes and properties

b Model or, make drawings of2- or 3-dimensional shapesand solids useful insolving problems

c Recognize and apply theconcepts of symmetry. con-gruency and similarity ofgeometrical figures as com-monly used in man-made objects

d Use drawings. models orcomputers to demonstrategeometric Patterns andrelationships such at

simLIAnity_tTisansatat

eLcsaiLirLdgi coor-dinates of points ongraph paper, mans. globesand other charts(emphasize examples fromthe environment andcomputer applications)

f Demonstrate by variousmeans the relationshipsamong radius. diameterand circumference of acircle, and the commonright triangle relationships

e Locate Points and lines anddetermine distance and areain a rectaagular coordinatesystem (emphasize examplesfrom the environment andcomputer applications)

f Aoplv and use cirryligand common right trianglerelationships in solvingoroblems

4.2 MAKE AND USE GEOMETRIC DRAWINGS AND MODELS. INCLUDING TESSELLATIONS


a Explore withmanipulatives (e.g.,pattern blocks, or tiles)

I 3



62

a Create simple constrsKtions with tills and copyon squared paper

j

a Investigate covering flatsurfaces with various congruent polygons

b Draw rectangultr prisms froma side view

c Use cubes to build structuressuggested by pictures, andmaterials such as patternblocks, and grid paper toinvestigate certain numberpatterns (e.g., squarenumbers, primes, area modelof multiplication facts)


a Find and use congruentpolygons which will covera surface without overlacoping

b Sketch to and side viewsOf rectangular solids

a Show how a flat surfacecan be completely coveredwithout overlap, by congruent rectangles, triangles or squares

b Make 2dimensionalsketches of geometricsolids such as pyramids,cones, spheres, cubes

C Use cubes to build struc c Construct a stack oftures suggested by pictures cubes given pictures ofand make 3D shapes from front, top and side viewspaper Patterns and make 3D shapes

from paper patterns

a Show how a flat surfacecan be completely covered,without overlap, by congruent rectangles, triangles or squares

b Make 2dimensionalsketches of geometricsolids such as pyramids,cones, spheres, cubes

c Construct a stack ofcubes given pictures offront, top and side views

d Show the importanceof using triangles inconstruction

63

a Explain why a flat surfacecan be completely coveredwithout overlap. by congruent triangles. andrectangles. or squares

b Use Protractor. compass.ruler. computer, andother instruments tomake common geometricconstructions

c Draw the net (2 dimentional pattern) for commongeometric solids. e.g.,cube. rectangularprism. cylinder

d Show the importance ofusing triangles in construction and surveyingby using a drawing ormodel

b Use Protractor. comoass,ruler. computer. andother instruments tomake common geometricconstructions

c Draw the net (2 dimentional Pattern) for commongeometric solids, makeorthographic and isometricdrawings of structuresbuilt with cubes, andvisualize and draw crosssections formed by slicinggeometric solids

d Use and apply trianglesin construction andsurveying

4.3 UNDERSTAND AND USE PERIMETEk. AREA AND VOLUME CONCEPTS

KINDERGARTEN GRADE 1 =DE 2 GRADE 3 GRADE 4

a Explore with a Explore with a Explore with a Develop an understanding a Develop an understandingmanipulatives to gain manipulatives to gain manipulatives to gain of perimeters area and of perimeter, area andexperiences with measure- experiences with measure- experiences with measure- volume using concrete volume using concretement of perimeter, area ment of perimeter, area ment of perimeter, area obiectl objectsand capacity and capacity and capacity

b Estimate and determine b Estimate and determine b Estimate and determine b Use common ob.nts to b Use common objects to"space covered" using "space covered" using "space covered" using estimate perimeter- area estimate perimeter, areasquare tiles or other square tiles or other square tiles or other and volume and volumeuniform objects uniform objects uniform objects

64

r,.:

GRADE 5 GRADE 6 GRADE 7 GRADES GRADE 11

a Develop an understandingof Perimeter. area andvolume using concreteobiects

b Estimate and determineperimeter and area ofrectanglca. and volumeof rectangulor solids. bymeans other than formula

c Use formula for findingperimeter and area ofrectangles

a Develop an understandingof perimeter, surface areaand volume using concreteobjects

b Estimate and determineperimeter and area ofrectangles and right triangles, and volume ofrectangular solids, bymeans other than formula

c Use formula for findingperimeter and area ofrectangles, and volumeof rectangular solids

a Demonstrate, other thanby using a formula, waysof finding perimeter andarea of right triangles,circles, rectangles

b Estimate and determinearea of right triangles,circles, rectangles, andvolume of rectangularsolids

c Use a formula for find-ing perimeter and area ofright triangles, circles,rectangles, and volume ofrectangular solids

d Calculate the surfaceareas of regularly shapedsolids (e.g., rectangularboxes and cubes)

65

a Demonstrate. other thanby using a formula. waysof finding perimeter andarea of general triangles,circles, parallelograms.and trapezoids

b Estimate and detemineperimeter. area andvolume of commongeometric figures

c Use a Formula for findingperimeter, area andvolume of common geometricfigures

d Calculate surface areasof regularly shaped solids(e.g.. cubes. cylinders..

r an 11C...tgLl rOLLQ) Lel( )

a Use and apply perimeter,area and volume concepts

b Estimate perimeter, areaand volume

c Calculate Perimeter. areaand volurle

Calculate surface areas0 regularly shaped solids(e.g., cubes, cylinders,rectangular boxes)

4

5.0 Measurillit: Students measure quantities and use measurements to keep ords_solve problems and make predictions.

5.1 POSE AND SOLVE PROBLEMS THAT INVOLVE TIME AND MONEY (ELS 1.7)


a Informal experiences a Identify and order by a Identify and order all a Identify and order b; a Identify and order bywith money value (penny, nickel and U.S. coins, role play value; make change using value, make change usingdime) and role play making making change and show U.S. coins U.S. currency, and use billschange up to a dime equivalent amounts up to

a quarterand coins to show equivalentamounts to $5

b Demonstrate an under b Demonstrate an under b Demonstrate an under b Read time using standard b Reao s'ime using standardstanding of time (day,week, month,tomorrow, yesterday)

standing of time (day,week, month, year,tomorrow, yesterday)

standing of time (day,week, month, year,tomorrow, yesterday) andorder days of the week

and digital clocks and and digital clocksorder months, seasons anddays of week

c Create and solve word c Create and solve word c Create and solve wordproblems which involve problems which involve problems which involvemoney money or time money or time

I 3

66


a Identify and order byvalue. and make changeusing U.S. currency

b Estimate elapsed timefgr given activities

Create and solve wordproblems which involvemoney or time

Create and sol,A problemswhich involve time ormoney

I El' 0

c Create and solve problems ' Create and solve problemswhich involv? time or which involve time ormoney money includiig consumer

and wage earner situationsof interest to students

67

Create and solve problemswhich involve time animoney. including conumgrand wage earner sttaationsof interest to students

5.2 SELECT AND USE APPROPRIATE INSTRUMENTS AND UNITS TO ESTIMATE AND MEASURE LENGTHWEIGHT; VOLUME AND CAPACITY: AND TEMPERATURE (ELS 1.7)


a Provide numerous informalexperiences, estimatingand measuring length,weight, and capacityusing nonstandarduniform units and line upmeasuring tool with itemto be measured

a Provide numerous informalexperiences, estimatingand measuring length,weight, and capacityusing nonstandarduniform units and line upmeasuring tool with itemto be measured

a Estimate and measureobjects using nonstandarduniform units, recognizethe need for standardunits of measurement andbegin using meters, Jritimeters, feet, yards tomeasure length; use abalance scale to determineheavier or lighter

68

a Estimate and determine a Recognize and use meters,length and weight (mass) centimeters, feet, yards, andusing nonstandard. metric inches to measure; and selector English U.S Customary the most appropriate instrumentunits of measure and select and unit for a measurement tasxand use appropriateinstrument and unit formeasurement task in

nonstandard metric.or English

3 z:

c Use squared paper,transparent grids, or othermaterials to estimate area/perimeter of irregularclosed figures


a Busy:min and use meters. a Determine the most appro- a Determine the most appro- a Detcrmile the most appro- a Determine the most appro-

ontimeters. feet. yards, priate unit and instrument priate unit and instrument priate unit and instrument priate unit and instrument

ind inches to measure and for a measurement task for a measurement task for a measurement task for a measurement task

select the most acormriateinstrument and unit for ameasurement task

c Use squared paper,transparent grids, orother wiaterial to esti-mate area/perimeter ofirregular closed figures

b Estimate and thendirectly measure dis-tances, angles, and otherquantities to the nearestand smallest unit on themeasuring scale

c Use squared paper,transparent grids, orother material to esti-mate area/perimeter ofirregular closed figures

d Convert among units oflength and weight (mass)within the same measure-ment system (English andmetric)

4 4

b Estimate and thendirectly measure dis-tances, angles, and otherquantities to the nearestand smallest unit on themeasuring scale

c Measure by some directmeans the area of apolygon or some 2 -U regionwith curves as boundaries

d Convert among commonlyused units of measurementwithin the same system

f Explain why allmeasurements areapproximations

69

b Estimate and directlymeasure distances. anglesand o,ner quantities.and indicate in some waythe precision of themeasurement. using metricand English ;f./.5. Customary)

c Measure by some directmeans the area of a polygonor some 2-D region withcurves as boundaries, andthe volume of 3 -D objects

e Give examples of theimportance of congruenceand precision in society

f Explain why allmeasurements arkapproximations and whyresults of all comPvtationFwith measurements areapproximations

Estimate and directlymeasure length. area.volume, time. weight (mass)etc.. with reasonable#ccuracv ane'or round ameasurement tv a. given unit

e Octernine the precisionof a measurirg tool,(e.g., 1° for a standardorrt:actor)

Explain why

approximations andresults of all computationswith measurements areapproximations

(5.2 cont.)


g Explore the concepts g Explcre the concepts g FNplore the concepts g Explore the concepts g Recognize and use grams,of weight (mass) using of weight (mass) using of weight (mass) using of weight (mass) using kilograms, ounces, and poundsa balance scale and a balance scale and a balance scale and a balance scale and to measure common classroomcommon classroom objects common classroom objects common classroom objects common classroom objects objects

hExplore temperature h Explore temperature h Expl--e temperature h

Estimate_._ read and record h Estimate, read anc: recordconcepts as they occur concepts as they occur concepts as they occur temperature in C° and F° temperature in C° and F°in daily situations in daily situations in daily situations

5.3 DETERMINE INDIRECT MEASUREMENTS (ELS 1.7)


is 6

70

. p,

a Estimate distances betweenPlaces represented by pointson maps

GRADE 5 GRADE 6 GRADE 7 GRADE 8 GRAuE 11

g Estimate and determine g Estimate and determine Estimate and determine g Estimate and determine

the weight (mass) of the weight (mass) of the weight (ma ;) of the weight (mass) of comM2nmagn classroom objects common classroom objects common classroom objects objects using metric and

in metric and English in metric and English in metric and English English (U.S. customary)(U.S. Customary) units (U.S. Customary) units (U.S. Customary) units ni

h Estimate. read and h Estimate, read and h Estimate, read and h atimate. read and

record temperature in record temperature in record temperature in record temperature inC° and F° C° and F° C° and F° in real and

givon situations


a Locate Points, givecoordinates Of Points

1111

on maps, and estimatejistances between placesrepresented by points onmacs

a Locate points and givecoordinates of points onmaps, make scale drawingsand determine highlydistances fror maps

b Find measurements,including inaccessibledistances, indirectlyby using formulas fordistance and area

a Make scale drawings anddetermi actual distancesfrom scale drawings,blueprints, maps, andglobes

b Find measurements,including inaccessibledistances, indirectlyby using formulas fordistance and area

71

a Make scale drawings anddetermine actual distancesfrom scale drawings,tlyeprints, maps. andglobes

b rind and record measureIts using proportions

and formulas

a Apply ratio and proportionconcepts in making and usingscale drawings and models,yld in solving problems

b Find and record measurements using proportionsAnd formulas

6.0 Statist csP and Probability: Students will be able to collect, organize. record and interpret data and be able to predict probable outcomes based

on collected data.

6.1 RECOGNIZE AND USE MATHEMATICAL PATTERNS, RELATIONSHIPS AND PRINCIPLES TO OUANTIFY PROBLEMS OR MAKE PREDICTIONS (ELS 1.6)


a Experience activitieswhich have outcomes thatdepend on chance

d Collect and recorddata (e.g., tally,lists, charts'

e Make and draw conclusions from realgraphs (an array ofactual objects on rectangular grin) picturegraphs, bar graphs andcharts

f Make and use real graphs(i.e., an array of actualobjects on rectangulargrid)


d Collect and recorddata (e.g., tally,lists, charts)

e Make and draw conclusions from realgraphs, picture graphs,bar graphs and charts

f Make and use real graphsand simple picture graphs


b Make predictions usingterms such as "bestchance," "most likely,""least likely," "fair,"etc.; apply intuitiveprobability concepts ingames and activities(dice, spinners, etc.)


e Make and draw conclusions frompicture graphs,bar graphs, and charts

f Make and use real graphsant simple picture graphs

72

a Apply intuitive probability concepts (e.g..make predictions in gamesby using terms such as"more likely." "lesslikely," "fair." etc.)

Make prediction usi%terms such as "bestchance," "most scikely,""least likely,' "fair,"etc.; apply intuitiveprobability concepts ingames and activities(dice, spinners, etc.)


e Collect and record datafrom picture graphs. bargraphs and charts todraw conclusions andmake predictions

f Make and use picturegraphs and bar graphs

a Collect data from probabilityexperiments

b Recognize the concept offair or unfair in gamevituations

d Gather and record datafrom a variety of sources

e Collect and record datafrom picture graphs, bargraphs and charts todraw conclusions andmake predictions

f Make bar graphs, andorganize information intotables/charts and diagramsgiven appropriate scale(e.g., box plots, boxandwhiskers, line plots.Venn diagrams)

g Collect random samples

GRADE 5 GRADE 6 GRADE 7 GRADE A GRADE 11

generate. record andinterpret data fromprobability experimentsand predict chances ofan outcome

b Recognize the concept of1111 fair or unfair in game

situations

c Recognize cert&in (1)and impossible (DIprobabilitiet

d Use charts tables andlists to organize allpossible outcomes of inexperiment

e Read. intypret. construct bar graphs. line'arias. tables andcharts and make Predictions based upon them

a Determine the number of a Determine the number of a Determine the number ofpossible events that could possible events that could pgiariTietvver(3sa:ndoutthc:me

occur in a probability occur in a probabilityexperiment (spinners, dice) experiment (spinners; dice) in a probability

experiment

b Identify and demonstratesituations in which probability or chance of anevent occurring is likely,unlikely, equally likely;and whether a game is"fair"

c Recognize certain (1)and impossible (0)probabilities

d Use charts, tables andlists to organize allpossible outcomes of anexperiment

e Read and interpretgraphs, tables and charts,and make predictionsbased upon them

I Organize information into f Make histograms andtables/charts and diagrams bar graphs from datagiven appropriate scale, meaningful to studentse.g.. box plots, boxandwhiskers. line plots,.Venn diagrams

g Collect random samples g Use data gathering procedures which will aid inanswering questions ofi terest to students(conducting polls, using.ables from almanacs)

b Identify and demonstratesituations in which probability or chance of anevent occurring is likely,unlikely, equally likely;and whether a game is"fair"

a Interpret everydavAses of

rroehtitliortIvs.s:tct:2wileather

forecasts or chances ofwinning a lottery

b Identify and demonstrate b Identify and demonstratesituations in which situations in which

probability or chance probability or chanceof an event occurring is of an event occurring islikely. unlikely. equally likely. unlikely. eoVaiLYlikely; and whether a g4mg likely; and whether a gameis "fair" is "fair"

c Understand the meaning c Understand the meaningof probabilities of 0 of probabilities of 0(impossible) and 1 (certain) (impossible) and 1 (certain)

d Use charts, tables andlists to organize allpossible outcomes of anexperiment

e Read and interpretgraphs, tables and charts,and make predictionsbased upon them

f Make histograms andbar and line graphsfrom data meaningfulto students

d Use charts. tables andlists to organize allpossible outcomes of anexperiment

e Read and interpretgraphs, tables and charts.and make predictionsbased upon them

Make line and circlegraphs from data meaningful to students

g Use data gathering pro g Use data gathering procecedures which will aid in dures which will aidanswering questions of in answering questionsinterest (conducting polls, of interest (conductingusing tables from almanacs) polls, sampling schemes)

73

e Read and interpretgraphs. tables and chartsand make predictions basedupon them

f Organize and displaydata using tables. charts.graphs and diagrams

g Use data gathering procedures which will aidin answering questions9f interest (conductingf:11s, samr inq schemes)

1 3

(6.1 cont.)


1 Predict simplepossible future outcomes or actions

1 Predict simplepossible future outcomes or actions

J 5 z;

j Read and interpretcomputer generated graphsand tables

1 Predict simple 1 Predict simple 1 Predict simplepossible future out possible future out possible future outcomes or actions comes or actions comes or actions

74

.4 - -i :- ,,

GRADE 5 GRADE 6

410 Read and interpretcomputer generatedgraphs and tables

GRADE 7 GRADE 8 GRADE 11

i Identify misleading orincorrect methods ofdisplaying or interpretingdata

j Read and interpretcomputer generatedgraphs and tables

1 Predict simple. possible 1 Predict simple, possiblefuture outcomes or future outcomes or

actions actions

m Determine mean, median,and mode from datameaningful to students

h Understand the relation h Understand the relation h Understand th7 relationship between size of sample ship between size of sample Ship between size of sampleand degree of certainty and degree of certainty and degree of certainty

i Identify misleading orincorrect methods ofdisplaying or interpretingdata

j Use computer softwareto generate graphsand tables

1 Predict probable futureoutcomes or actions

m Determine mean, median,and mode from datameaning1J1 to students

75

Identify misleading orincorrect methods ofdisplaying or interpreting data

j Use computer softwareto generate graph!;and tables

k Show the relationshipamong variables usingtables. graphs, formulas,and models

1 Predict probable futureoutcomes or actions

m Determine, interpretand compare advantagesand disadvantages ofmean, median and mode

Identify misleading orincorrect methods ofdisplaying or interpreting data

Use computer softwareto generate graphsand tables

k Show the relationshipamong variables usingtables, graphs, formulas,and models

1 Defend conclusionsfrom information given

m Collect. display, interpretstatistical data usingmean mode, median, rangeand percentile

6.2 GENERATE AND TES( INTERPRETATIONS* EXPLANATIONS. PREDICTIONS. AND HYPOTHESES (ELS 6.2)


a Identify facts thatsupport an explanationand a prediction

b Idtifv factors thatmay influence a behavioror a result

c Identify ways to determine whether a duplicateof an experiment willproduce the same results

76

a Identify facts thatsupport an explanationand a prediction

b Identify factors thatmay influence a behavioror a result

c Identify ways to determine whether a duplicateof an experiment willproduce the same results

GRADES

a Identify parts of anexplanation and aprediction notsupported by fact

b Predict what influencedifferent factors willhave on a behavior or

41,10 result

Follow directions toconduct an experimuland identify thqhypothesis used

dc 3115g10/23/87


a Identify parts of an a Interpret differences a Interpret differences a Critically analyze explana-

axplanation and aprediction not

between two explanations between two explanations tions and interpretations toconfirm or validate them

supported by fact

b Predict what influence b Develop a hypothesis b Develop a hypothesis b Dgyelop a hypothesis

different factors willhave on a behavior or

from observed data from observed data ing data from avariety of sources

result

c Follow directions to c Gather data that con- c gather data that con- c Design and conduct a test

conduct an experiment firms or negates a firms or negates a of a hypotheses and reportand identify the hypothesis hypothesis the results

hypothp-4 used

1 C77

III7.0 Mathema al Relationship; Students recognize and use number patternsIII, relationships, and logical thinking skills to make predictions and to

solve problems.

7.1 SORT AND CLASSIFY; USE LOGICAL THINKING (ELS 5.2)

KINDERGARTEN GRADE 1 GRADE 2 GRADE 3 G-(ADE 4

a Classify and sortobjects using one or



a igfiArid classify a Classify objects and simplegeometrical figures byobjects by attributes

more attributes by more attributes by more attributes by attributesobserving relationships :Jbserving relationships observing relationshipsand by making and by making and by makinggeneralizations generalizations generalizations

b Use evidence such as b Use evidence such as b Use evidence such as b Use evidence such as b Use evidence such asillustrations, examples, illustrations, examples, illustrations, examples, illustrations, examples, illustrations, examples,and verifiable sources and verifiable sources and verifiable sources and verifiable sources and verifiable sourcesto problem solve to problem solve to problem solve to problem solve to problem solve

c Select and organize c Select and organize c Se'ect and organize c Select and organize c Select and organizedetails to problem solve details to problem solve details to problem solve details to problem solve details to problem solve

d Organize information d Organize information d Organize information d Organize information d Organize informationor data using tallys or data using tallys, or data using tallys, or data using formats or data using formatsm!d real graphs tables, charts, and real tables, charts, and Nal such as outlining, making such as outlining, making

graphs graphs maps, tables, charts,and graphs

maps, tables, charts,and graphs

16278

r


a Classify objects andsimple geometricalfigures by attributes

b Use evidence such as4111illustrations. examples.

and verifiable sourcesto problem solve

c Select and organizedetails to problem solve

d Organize informationor data using formatssuch as outlining. makingmaps. tables. charts.and graphs

a Classify geometricalfigures and sets ofnumbers

b Use evidence such asii)ust;ations, examples,and verifiable sourcesto problem solve

c Select and organizedetails to problem solve

d Organize informationor data using formatssuch as outlining, makingmaps, tables, charts,and graphs

a Classify geometrical a Classify geometrical a Classify geometricalfigures and sets of figures and sets of figures and sets ofnumbers numbers numbers

b Use evidence from b Use evidence from b Use evidence fromverifiable sources tosupport own ideas andconcepts in problem solving

c Select and use detailsexamples, illustrations.evidence and logic toproblem solve

d Organize informationor data using formatssuch as outlining, makingmaps, tables, charts.graphs; and computerspread sheets

verifiable sources to verifiable sources tosupport own ideas and support own ideas andconcepts in problem solving concepts in problem solving

c Select and use details,examples, illustrations,evidence and logic toproblem solve

d Organize informationor data using format-such as outlining, makingmaps, tables, charts,graphs; and computerspread sheets

79

c Select and tne details.examples, illustrations.evidence and logic toproblem solve

d Organize informationc):- data using formats

such as outlining, makingmaps, tables. charts,graphs; and compute-spread sheets

7.2 COMPREHEND MEANINGS OF WRITTEN, oRAL AND VISUAL COMMUNICATIONS INVOLVING NUMBER PATTERNS AND RELATIONSHIPS (ELS 3.1)


a State relationshipsusing terms such as"greater than," "lessthan," and "equal to"

b Relate new informationto previous knowledge

c Draw logical conclusions from information presented

d Identify, verbalize,extend, and reproduce apattern in a sequence ofobjects and numbers, anduse to make predictions

a State relationshipsusing terms such as"greater than," "lessthan," and "equal to"



d Identify, verbalize,extend, and reproduce apattern in a sequence ofobjects and numbers, anduse to make predictions

a State relationshipsusing terms such as"greater than," "lessthan," and "equal to



d Identify patterns in asequence of objects ornumerical patterns incharts and tables, anduse to make predictions

80

a State relationshipsusing terms such as"greater than." "lessthan, " and "equal to"


logical conclusions from information

Find numerical patternsin charts and tables(e.g.. 100chart, additionand multiplication tables),and use number_pat,ernsand relationships to makepredictions

3+

a State relationships usingterms such as "greater than,""less than," and "equal to"and use the symbols >, <, =


c Draw logical conclusionsfrom information presented

d Find number patterns in 100chart, addition, and multiplication tables (e.g., primes,odd and even, square numbers),and use number patternsand relationships to makepredictions

GRADE 5 GRADES GRADE 7 GRADE 8 GRADE 11

4 State relationships usingterms such as "greaterthan." "less than." and"equal to" and use th,symbols >. <.

b Relate new informationto Previous knowledge

1111Draw logical conclusions c

from information presented

d

e

a Use equality andinequality conceptsand symbols

b

Find numerical patternsin 100charts and additionand multiplication tablesle.a.. odd/even. primes.square numbers). and usepatterns to completesimple charts and tablesand to make predictions

Explore relationshipsfound in tables of value

f Use decimal and fractionunderstandings and 100grids to illustrate the4111"for every hundrrd"

model of percent

e

a Use equali andinequality conceptsand symbols

Relate new information b Relate new informationto previous knowledge to previous knowledge

a Use equality andinequality conceptsand symbols

1 Use equality and inequalityconcepts and symbols

b Relate new information b Relate new informationto previous knowledge to previous knowledge

Draw logical conclusions c Draw logical conclusions c Draw logical conclusionsfrom information presented from information presented from information presented

Find numerical patterns d Find numerical patternsin 100charts and addition and use to completeand multiplication tables charts and table-(e.g., odd/even, primes,square numbers), and usepatterns to completesimple chart" and tablesand to make predictions

Recognize relationshipsfound in tables of value

f Use decimal and fractionunderstandings and 100grids to illustrate thepercent concept

f hs ow, using models suchas 100grids, number linesor a meter stick, howpercent can be expressedas a fraction or decimal

81

d Find numerical patternsand use to completecharts and tables

e Recognize direct andindirect cause andeffect relationships

c Synthesize informationand draw conclusions

d Find hJmerical patternsand use to completecharts and tables

e Infer direct andindirect cause andeffect relationships

f Show. using models such f Demonstrate fraction.as 100grids. number lines decimal. percentor a meter stick. 1w relationshipspercent can be expressedas a fraction or decimal.and conversely

g Interpret and use theconcepts of ratio. percent.proportion, and commonlyoccurring rates such asgrowth. speed and sportsapplications

Interpret and use the concepts of ratio, percent. Proportion. and commonly occurringrates such as growth, speed.interest and cost per unit

7.3 RECOGNIZE, CONSTRUCT AND DRAW INFERENCES CONCERNING RELATIONSHIPS AMONG THINGS AND IDEAS (ELS 6.1)

KINDERGARTEN

a Identify characteristicsof simple objects thatremain the same eventhough some change occurs,e.g., cutting objectsinto two pieces


a Identify characteristics a Identify characteristics a Identify characteristicsof simple objects that of simple objects that of simple objects thatremain the same even remain the same even remain the same eventhough some change occurs,e.g., cutting objects

though some change occurs,e.g., cutting objects

though some change occurs.e.g., cutting objects

into two pieces into two pieces into two pieces

b Make a simple table b Make a simple table b Make a simple table b Make a simp'e tableof values given a of values given a nf values given a o' values given aspecific rule and specific rule and specific rule and specific rule andmatch a table of values match a table of values match a table of valves match a table of valuesto its rule to its rule to its rule to its rule

7.4 REFLECT UPON AND IMPROVE OWN REASONING (ELS 6.6)

KINDERGARTEN GRADE 1 GRADE - GRADE 3 GRADE 4

a Describe in simple terms a Describe in simple terms a Describe in simple terms a Describe in simple terms a Describe the reasoninghow a solution was reachA how a solution was reached (low a solution was reached how a solution was reached process being used

b Act upon suggestions b Act upon suggestions b Act upon suggestions b Act upon suggestions b Act upon suggestionsfor improving reasoning for improving reasoning for improving reasoning for improving rslasoning for improving reasoningcapabilities capabilities capabilities capabilities capabilities

82


b Make a simple table b Make a simple table b Make a simple table b Evaluate or make a table b Evaluate or make a table

IllOof values given a

specific rule andof values given aspecific rule and match

of values given aspecific rule and match

:or twovariable formulas for twovariable formulaswhich have meaning to and match a graph or table

match a table of a graph or table of a graph or table of students and match a of values to its formula

values to its rule values to its formula values to its formula graph or table of valuesto its formula

c Explore relitionitips c Describe the nature of c Describe the nature of c Describe the nature of c Describe the nature of

illustrated in a table change of each variable change of each variable change of each variable change of each variable

of values as suggested by a table as suggested by a table as suggested by a table as suggested by a table

of values of values or graphs of values, graph, or of values. graph, orformula formula

GRADE 5 GRADE 6

Describet a Describe the reasoningprocess being used process being used

b Act upon suggestionsfor improving reasoningcapabilities

GRADE 7 GRADES GRADE 11

172

a Describe the reasoningprocess being used

83

a Describe the strengthsand weaknesses ofinductive and deductivereasoning

a Present arguments supportingthe use of deductive orinductive reasoning for aparticular purpose

III1111

III8.0 Oral a Writter Ovmmunioation Skills; The student uses vocabulary speech. numerals and other symbol _systems essential for effective individual

And group Problem solving and for effective oral and written communication of mathematical concepts, problemsolving processes and results.

8.1 RECOGNIZE ANO USE MATHEMATICS VOCABULARY COMMONLY USED IN GRADELEVEL MATERIALS (ELS 1.1)

KINDERGARTEN GRADE 1 GRADE 2 GRADE 3

a Recognize and identifynames and/or sounds ofletter symbols

b Use clues within theenvironment in orderto gain information fromprint

c Make oral distinctionsbetween compound andplural words

d Recognize mathematicalwords that are common tothe child's individualenvironment

I

a Use phonetic analysisskills

b Use illustrations andwords in a sentence toinfer correct word(s)

c Make oral distinctionsbetween compound andplural words

d Recognize mathematicalwords that are common tothe child's individualenvironment


b Use illustrations andwords in a paragraph toinfer correct word(s)

c Recognize zompound andplural words

d Use basic mathematicalterms (such as sum,difference, less than,equal, greater than,equal, greater than,circle, triangle) toconvey concepts ofquantity, order, operationand shape

84


b Use context clues inA_Paragranh to infercorrect wurd(s)

c Distinguish compoundand plural words

d Use basic mathematicalterms (such as sum. total.difference, product, lessWin. equal. greater than,rectangle) to convey concents of quantity, order.operation and shape

J

GRADE 4


b Use context clues ina paragraph to infercorrect wore(s)

c Distinguish compoundand plural words

d Use basic mathematical terms(such as sum, total, difference, product, quotient, lassthan, equal, greater than,square) to convey concepts ofquantity, order, operationand shape

GRADE 5 GRADE 6

b Use context clues ina Passage to infercorrect word(s)

Pistinguish affixesand root words

d U1Lmathematical term-to convey concepts ofmuo,t4tv. order, operation and shape (e.g..1:roduct. factor. Quotient,remeindet, sum,quadrilateral)

b Use context clues ina passage to infercorrect word(s)

c Recognize affixesand root words

d Use basic mathematicalterms and symbols toconvey concepts ofquantity, order,operation, and shape

e Recognize commonabbreviations(e.g., ft., in.) andsy...bols (e.g., cm, kg)

fr,

0

GRADE 7 GRADE 8

b Use context clues ina selection to infercorrect wcrd(c)

c Disting.iish affixesand root words

d Use basic mathematicalterms and symbols toconvey concepts ofquantity, order andoperation

e Recognize commonabbreviations(e.g., ft., in.) andsymbols (e.g., cm, kg)

85

GRADE 11

b Use context_clues ina selection to ;rife.-

correct word(s)

c Distinguish affixesand root words

d Use basic mathematicalterms and symbols toconvey concepts ofquantity. order,operation, and shape

e Recognize commonabbreviations(e.g., ft., in.) andsymbols (e.g., cm, kg)

b Use context clues ina selection to infercorrect wold(s)

c Distinguish affixand root wards

D Use basic mathematicalterms and symbols toconvey concepts ofquantity. order,pp @ration, and shape

e Recognize commonabbrevii-tions(e.g., ft., in.) andsymbols (e.g., cm, kg)

j

e8.2 DETERMINE MEANING OF UNKNOWN WORDS COMMONLY USED IN MATHEMATICAL MATERIALS (ELS 1.2)

KINDERGARTEN GRADE 1 GRADE 2 GRADc: 3 GRADE 4

a Use context clues from a Use illustrations and a Use illustrations and a Use adiacent words a Use adjacent words and/ororal presentation to adjacent words in a adjacent words in a to infer meaning of context clues in a passageinfer meaning of unknown paragraph to infer paragraph to infer unknown words to infer meaning ofword(s) meaning of unknown word(s) meaning of unknown word(s) unknown words

c Make oral distinctions c Make oral distinctions c Use knowledge of each c Use knowledr.o of each c Use knowledge of affixesbetween compound and between compound and part of a compound word part of a c.APound word and root words to determineplural words plural words to determine meaning to determine meaning word meanings

d Use glossary in d Use primary grade d Use dictionaries and d Use dictionary. glossary.curriculum materials dictionaries glossaries in grade and context to determine

level curriculum materials correct meaning of word

8.3 SPEAK WITH STANDARD PRONUNCIATION, APPROPRIATE VOLUME, RATE, GESTURES AND INFLECTIONS (ELS 1.3)


a Listen an,- practicebasic speech sounds

b Pronounce wards according to acceptable StandardEnglish

I

a Listen and practice a Listen and practice a Produce correct basicbasic speech sounds basic speech sounds s.eech_sounds

b Pronounce words accord b Pronounce words accord b Pronounce words according to acceptable Standard ing to acceptable Standard ing to acceptahle StandardEnglish English English

86

.111

J

a Produce correct basicspeech sounds

b Pronounce words ar, ,rding to acceptable StandardEnglish


a Use context cluesto infer meaning ofunknown wordi

b Re=gnize doublemeanings of words

4111. Use knowledge of affixesand root words to determine word meanings

d Use context to determine correct dictionarydefinition f word

a Use co0..ext clues,punctuation and syntax toinfer meaning of un'nownwords and concoptc

b Recognize doublemeanings of words

c Use knowledge of affixesand root words to dotermine word meanings

d Use dictionaries,glossaries and otherreference materials tofind word meanings

4 Use context clues,punctuation and syntax toinfer meaning of unknownwords and concepts

b Recognize doublemeanings of words

c Use knowledge of affixesand root words to determine word neanings

d Use dictionaries,glossaries and otherreference materials tofind word meanings

GRADE 11

a Use cortext clues,punctuation and syntax toinfer meaning of unknownwords and concepts

b Recognize doubl,meanings of words

c Use knowledge of affixesand root words to determine word meanings

d Use dictionaries.glossaries and otherreference materials tofind word meanings

a Use context clues.punctuation and syntax toinfer meaning of unknownwords and concepts

b Recognize doublemean.ogs of words

c Use knowledge of affixesand root words to determine word meanings

d Use dictionaries.glossaries and definitionsin footnotes to find wordmeanings


a Produce correct basicspenh sounds

b Pronounce words accordingto acceptable StandardEnglish

a Produce correct basicspeech sounds


ISO


87

b Pronounce words according b Pronounce words according

to accept-hie Standard to acceptable StandardEnglish English

c Make oral presentations c Make oral presentationsthat use verbal and non that use verbal and nonverbal communication skills verbal communication skillseffectively effectively

8.4 USE ORAL COMMUNICATION TO GIVE OR RECEIVE INFORMATION AND DIRECTIONS (ELS 2.3)


Paraphrase oral messages 4 Paraphrase oral messages a Paraphrase oral messages a Paraphrase oral messages a Paraphrase oral messages

b GiveGive accurate oral b Give accurate oral b Give accurate oral accurate oral b Give accurate oraldirections directic directions directions

c Ask questions design d c Ask questions designed c Ask questions designed c Ask questions designed c Ask questions designedto clarify, gain assis to clarify, gain assis to clarify, gain assis to clarify, gain assis to clarify, gain assistance or locate information tanze or locate information tance or locate information tance or locate information tance or locate information

d Share ideas and information orally with others

e Provide accuratedescriptive detail orally

9 Follow 2step oralinstructions



g Follow 2step oralinstructions




d Share ideas and ;nforltion orally with others

e .rovide accuratedescriptive detail orally

g Follow 2step oralirstructions



f Take notes based on oralpresentations and grout)discussions

g Follow 2-3step oralinstructions

8.5 DETERMINE THE SIGNIFICANCE AND ACCURACY OF INFORMATION AND IDEAS PRESENTED IN WRITTEN, ORAL, AURAL AND VISUAL COMMUNICATIONS (ELS 4.1)


88

C-

)


Paraphrase oral messages

b Give accurate oraldirections

Ask questions designedto clarify, r.pin assistance or local information

1111d Share ideas and infor

mation orally with others

e Provide accurate descriptive detail orally

f Take notes based on oralpresentations and groupdiscussions

g Follow 3 step oralinstructions

a Paraphrase oral messages


c Ask questions designedto clarify, gain assistance or local information


e Provide accurate descriptive detail orally

f Take notes based on oralpresentations and groupdiscussions



b Give accurate oraldirec'ions

c Ask ruestions designedto clarify, gain assistance or local information


e Prcvide accurate descriptive detail orally

f Take notes b ,d on oralpresentation, and groupdiscussions

g Follow multistep oralinstructions


b GiP accurate oraldi"ections

c !tck questions designedto clarify. gain assistance or local information


Provide accurate descriptive detail orally

f Take notes and preparesummaries based on oralpresentations and groupdiscussions

g Flow multistep ,-ralinstructions



c Ask questions designedto clarify, gain assi.,tance or local information


e Develop accurate detailbased on oral explanationsby others

f Take notes and Preparesummaries based on oralpresentations and groupdiscussions

g Follow multistep oralinstructions

GRADE 5 GRADE 6 GRAPE 7 /WADE 8 GRADE 11

3S4

a Separate between relevant a Separate between relevant a Distinguish between logical

and irrelevant information and irrelevant information and illoical conclusions

used to draw conclusions used to draw conclusions

Identify propaganda and b Identify Propaganda and b Identify propaganda and

other persuasion techniques other persuasion techniques other persuasion techniques

(e.g., use and misuse of (e.g.. use and misuse ofs. atistics) statistical, statistics)

(e.g., use and misuse of

89

o8.6 USE ORAL COMMUNICATION TO INFLUENCE OTHERS AND TO RESPOND TO PERSUASION (ELS 4.2)


8.7 LISTEN, READ, VIEW AND [VALUATE PRESENTATIONS OF MASS MEDIA (ELS 4.4)

KINDERGARTP!

a Ask questions and drawreasonable conclusionsfrom answers

a Ask questions and drawreasonable conclusionsfrom answers


... ...,

J ,

ci

90


a Provide logical answersbased upon factual data

b Use multiple sourcesto verify information






b Use multiple sourcesto verify informatiaa


b Use Primary and secondarysource materials to verifyinfornatioo

c Argue opposite sides ofissues

1 Recognize sources of persuasion and select appropriatepersuasive response

e Use verbal persuasiontechniques in a classpresentation


a Recognize persuasiontechniques found in visualcommunications (e.g., useand misuse of graphs)

91

a Recognize persuasiontechniques 'ound in visualcoma nications (e.g., useand misus,v of graphs)

a Recognize persuasiontechniques found in visualcommunications (e.g., usgand misuse vi graphs)

8.8 SELECT APPROPRIATE FORM OF WRITING (ELS 5.3)

KINDERGARTEN GRADE I GRADE 2 GRADE 3 GRADE 4

a Write in a variety of formssuch as reports, descriptions,or in problem posing or solving

b Use writing appropriate topurpose such as to inform,pose problems or solve problemill

8.9 PRESENT IDEAS IN UNDERSTANDABLE SEQUENCE ON THE TOPIC SELECTED (ELS 5.4)


8.10 ELE T AND E AE E T RE AND YMBLAPPR PRIATE Ti P R ' I E T1 PI' AND ETTIN WHEN MAKIN RAL PRESENTATIONS (ELS 5.5)


a Select words which make a Select words which makethe meaning clear the meaning clear

b Plan and make oral andvisual presentations


a Write in a variety of

fOTIELUgLitrePArtil.tescriptions. or inr^nblem posing or solving

b Use writing appropriateto Purpose such as to

1111inform. we problemsor solve problems

a Write in a variety of a Write in a variety of a Write in a variety of a Write in a variety of

forms such as personal forms such as personal forms such as personal forms such as personal

essays, journals, reports, essays, journals, reports, essays. Journals. reports, essays. journals. reports.

descriptions, or in problem descriptions, or in problem descriptions. or in problem descriptions, or in problem

posing or solving posing or solving posing or solving posing or solviqg

b Use writing appropriateto purpose such as toinform, pose problemsor solve problems

b Use writing appropriateto purpose such as toinfo.m, pose problemsor solve problems

b Use writing appropriateto purpose such as toinform, Pose problemsor solve problems

b Use writing appropriateto purpose such as toinform, pose problemsor solve problems


a Write complete sentences a Write complete sentences

b Write multiparagraphpersonal journals, reportsor problem solutionstrategies

a Write complete sentences

b Write multiparagraphpersonal journals reportsor problem solutionstrategies

a Write complete sentences

b Write multiparagraphPersonal Journals, reportsor problem solutionstrategies

a Write complete senter-11

b Write multiparagraphPersonal Journals. reportsor problem solutionstra_tegies

GRADE 5 GRADE 6 GRADE 7 GRADES GRADE 11

a Select words which make a Use a variety of techni- a Use a variety of techni- a Use a variety of techni- a Employ verbal, svmbolic,

the meaning clear ques and figurative ques and figurative ques and figurathe graphic and visual techniques

expressions to convey expressions to convey expressions to convey to convey information

meaning meaning meaning

b Plan and make oral and b Plan and make oral and b Plan and make oral and b Plan and make oral and b Plan and make oral and

visual presentations visual presentations visual presentations visual presentations visual presentations

192 93

(ELS 5.6)


a Revise own writing to a Revise own writing toenhance clarity and enhance clarity andmeaning meaning

b Use descriptive terms toemphasize facts andquantities

8.12 APPLY THE CONVENTIONS OF WRITING TO PRODUCE EFFECTIVE COMMUNICATION (EDITING AND PROOFREADING) (ELS 5.7)


94

.154t

a Edit for capitalization,end punctuation, andcomplete sentences

a Edit for complete andcorrect sentences, punctuationand usage

b Spell correctly

Produce legible final copy


a Revise own writing to a Revise own writing to a Revise own writing to a Revise own writing to a Revise own writing to

enhance clarity_and enhance clarity and enhance clarity and enhance clarity and correctiveness and

meaning meaning meaning meaning comprehensiveness

b Use descriptive terms b Use descriptive terms b Use descriptive andb Use descriptive and b Use descriptive and

to emphasize facts and to emphasize facts and connecting terms to connecting terms to connecting terms tc

quantities quantities enhance meaning, clarity enhance meaning, clarity enhance meaning, clarity

and precision and precision and precision

GRADE 5 G,ADE 6 GRADE 7 GRADE 8 GRADE 11

a Edit for complete and a Edit for complete and Edit for complete and a Edit for complete and a Edit to produce a correct

correct sentences, correct sentences,punctuation and usage

correct sentences,punctuation and usage

correct sentences, legible. effective piecepunctuation and usage punctuation and usage of writing

Spell correctly b Spell correctly b Spell correctly b Spell correctly b Spell correctly

c Produce legible final c Produce legible final c Produce legible final c Produce legible final c Produce legible final

COPY copy (mania' or copy (manual or copy (manual or copy (manual or

electronic processes) electronic processes) electronic processes) electronic processes)

1 S 6 95

9.0 Approariate_lAudy Skills: The student selects and uses appropriate study skills in order to accomplish mathematical learning tasks.

9.1 IDENTIFY KAIN IDEAS. SUPPORTING DETAILS. AND FACTS AND OPINIONS PRESENTED IN 'MITTEN. ORAL AND VISUAL FORMATS (ELS 2.1)


a Locate facts in a Locate facts inoral and visual formats oral and visual formats

a Locate facts in a Locate facts in gradegradelevel materials level materials

a Locate facts in gradelevel selections

b Recall facts and supportingevidence

c Identify main idea in c Identify main idea ina problem situation a problem situation

9.2 USE INSTRUCTIONAL MATERIAL:, AS BASIS FOR GAINING KNOWLEDGE AND IMPROVING COMPREHENSION (ELS 2.2)


a Use table of contents a Use table of contents a Use table of contents a Use table of contentsto locate general to locate general to locate general and and index to locate generalinformation information specific information and specific information

c Use guide words in adictionary or glossaryto locate words

96

b Use supportive illustrations, 4111d ?tail and summations to obtaiinformation (e.g., captions,footnotes, glossary entries,graphs, tables, charts, maps)

c Use guide words anddiacritical markings tolocate and pronounce words


4 Locate facts in gradelevel selections

b Recall facts andSupporting evidence

1111

c Identify main idea i 1 a

problem situation


b Identify necessary andextraneous facts andrelated supporting details

c Identify main idea ina problem situation


b Identify necessary andextraneous facts andrelated supporting details

c Identify main idea ina problem situation


b Identify necuSary andextraneous facts andrelated supporting details

c Identify main idea in..2problem situation


b Ioentify neces1ary_andextraneous facts andrelated supporting details

c Identify main idea in aproblem situation


a Use table of contents a Use table of contents,index, and summaries tolocate information needed

b Use summaries andheadings

c Use diacritical markingsor respellings topronounce words

a Use table of contents,index, summarieb, chartsand graphs to locateinformation needed

b Use summaries andheadings

c Use diacritical markingsor respellings topronounce words

a Use table of contents, a Use table of contents.index, summaries, charts,and index to locate index. summaries, charts.

general and specific graphs and illustrations to graphs and illustrations to

information locate information needed locate information needed

111115 Use suPoortive illus ° Use organization of b Use organization pf

trations, detail and materials (summartel, materials (summaries.

summations to obtain headings and revtg:e headings and review

information Questions) question

C Use diacritical markin-li c Use diacritical markings c Use diacritica markings

or resPellinos to or respellings to or rospellings to

pronounce words pronounce words pronounce words

2G 097

201

9.3 CLARIFY PURPOSES OF ASSIGNMENT (ELS 7.1)


Repeat oral instructions a Repeat oral instructions

9.4 USE RESOURCES BEYOND THE CLAc$RQOM (ELS 7.2)

a Repeat oral instructionsin proper sequence andask questions to clarify

a Wermine general purposeIf...assignment and

clarification (In questionsif necessary

b Determine ideas andconcepts addressed inthe assignment

a

b

Determine general purposeof assignment and askclarification on questionsif necessary

Determine ideas andconcepts addressed inthe assignment

KINDERGARTEN GRADE 1 GRADE 2

,/21'2

GRADE 3 GRADE 4

98

or

a Locate, checkout andreturn books and othercirculating media materials

b Locate and use noncirculatingreference materials

c Use library classificationsystem and services to locatespecialized resources requiredto complete assignments


a Determine general purpose

clarifiSation on questionsif necessary

b Determine ideas, concepts. and generalitiesaddressed in the

1111 assignment

a Determine general purposeof assignment and askclarification on questionsif necessary

b Determine ideas, concepts, and ge ralitiesaddressed in tneassignment

a Determine general purposeof assignment and askclarification on questicnsif necessary

b Determine ideas, concepts, and generalitiesaddressed in theassignment

a Determine general purpose

of assignment and askclarification on questionsif necessary

b Determine ideas.concepts. vneralities orprinciples_ .ncluded inassignmer..

GRADES GRADE 6 GRADE 7

a Determine general purposepf assignment and askclarification on questionsif necessary

b Determine ideas.concepts. generalities orprinciples included inassignment

GRADE 8 GRADE 11

a Locate. checkout and a Locate, checkout and a Locate, checkout and a Locate. checkout and a Locate. check-out andreturn books and other return books and other return books and other return books and other return books and other

circulating media materials circulating media materials circulating media materials circulating media materials circulating media materials

Locate and use non

e,circulating referencematerials

c Use library classification system and servicesto locate specializedresources reguireu tocomplete assignments

b Locate and use noncirculating referencematerials

c Use library classification system and servicesto locate specializedresources required tocomplete assignments



d Use computer (e.g.,data bases, spread sheets)

99



d Use computer (e.g.,data bases, spread sheets)



d Use computer (e.g.,data bases. spread sheets)

2r-7-

9.5 SELECT AND USE APPROPRIATE STUDY TECNNIOUES (ELS 7.3)

KINDERGARTEN

a Select activities anduse time effectively

b Begin and complete task

d Keep materialsorganized and accessible

e Complete a task withina given amount of time


a Select activities and a Follow a study plan a Follow a study plan a Follow a study planuse time effectively including: time manage including: time manage including: time manage

ment, appropriate study ment. appropriate study ment, appropriate studyenvironment, processing environment. processing environment, processingof information of information of information

b Begin and complete b Begin and complete b Accomplish learning task b Accomplish learning taskassignment and ask assignment and ask using appropriate study using appropriate studyquestions to clarify questions to clarify techniques (read and techniques (e.g., read and

reread text, ask clarifyingquestions, seek help whenneeded, use memory techniquesand study with classmates)

c Vary reading rateaccording to purpose forreading the selection

reread text, ask clarifying questions. seek helpwhen needed, use memorytechniques)

c Vary reading rateaccordinq to purpose forreading the selection

d Keep materials d Keep materials d Keep study materials d Keep study materialsorganized and accessible organized and accessible organized and accessible organized and accessible

e Turn in assignments e Turn in assignments e Turn in assignments e Turn in assignmentson time on time on time on time

f Use appropriate test f Use appropriate test f Use appropriate test f Use appropriate testtaking techniques taking techniques taking techniques. taking techniques

100

C.'11t


a Follow a study planincluding: time management. appropriate studyenvironment. processingof information

a Follow a study planincluding: goal setting,time management, appropriate study environment,processing of information

b Accomplish learning task b

using appropriate studytechniques (preview andreview chapters. read andreread text. ask clarifvinqquestions, seek help whenneeded. use memory techniques. summarize, studywith classmates. useself questioning)

Vary reading rateaccording to purpose forreading the selection

d Keep study materialsorganized and accessible

e Turn in assignmentson time

Use appropriate.taking techniques

dc 3116g10/23/87

d

Accomplish 103rning taskusing appropriavt studytechniques (preview andreview chapters, read andreread text, clarifyingquestions, seek help whenneeded, use memory techniques, summarize, studywith classmates, useselfquestioning)


Keep study materialsorganized and accessible

Turn in assignmentson time

f Use appropriate testtaking techniques

C's

a Follow a study planincluding: goal setting,time management, appropriate study environment,processing of information

b

d

Accomplish learning taskusing appropriate studytechniques (preview andreview chapters, read andreread text, ask clarifyingquestions, seek help whenneeded, use memory techniques, summarize, studywith classmates, useselfquestioning)


Keep study materialsorganized and accessible


f Use appropriate testtaking techniques

ini

a Follow a study planincluding: time management. appropriate studyenvironment. processingof information

b Accomplish learning task b

using appropriate studytechniques (preview andreview chapters. read andreread text. ask clarifvinqquestions, seek help whenneeded, use memory techniques. summarize. studywith classmates, useselfquestioning)

c Vary reading rateaccording to purpose forreading the selection

d Keep study materials.log and related notesorganized and accessible

a

e Turn in assignments

on time

Use appropriate testtaking techniques

Follow a study planincluding: time management. appropriate studyenvironment. processingof information

Accomplish learning taskusing appropriate studytechniques (preview andreview chapters. read andreread text. ask clarifyingquestions, seek help whenneeded, use memory techniques. su- --rize. studywith classmal.4. useselfquestioning)

c Vary reading rateaccording to purpose tolreading the selection

d Keep study materials.log, related notes andfiling system organizedand accessible


Use appropriate testtaking techniques

2 r

BIBLIOGRAPHY

Ashlock, Robert B. Error Patterns in Compaa-tion. A Semi-Programmed Approach, FourthEdition. Columbus, OH: Charles E. Merrill

Puhlishing Company, 1986.

The reader is engaged in a careful analysisof systematic errors in computation; theability to identify the probable conceptualor procedural bases for errors is explicitlypromoted, as is the ability to decide whatdirections are appropriate for follow -upinstruction.

Baratta-Lorton, Mary. Mathematics Their Way.Menlo Park, CA: Addison-Wesley, 1976.

The goal of the activities in this book isto develop understanding and insight of thepatterns of mathematics through use ofconcrete materials. There are bothtotal-group and small-group experiences forprimary school children. Each activityincludes a list of the specific skillstaught and the materials needed, as well asa description of the activity itself.

Baratta-Lorton, Robert. Mathematics, A Way of

Thinking. Meno Park, CA: Addison-Wesley,1977.

The purpose of this book is to provide adetailed guide for teachers wishing toimplement an activity-centered approach tothe teacher of mathematics. The book may beused as a complete mathematics curriculumfor upper elementary or as a resource toteachers wishing to supplement textbooklessons with manipulative activities.

1 0103

Brown, Stephen and Marion Walter. The Art of

Problem Posing. Palo Alto, rA: Dale

Seymour Publications.

Attention is given to help the readerunderstand what problem solving consists ofand why it is important. Strategies for

engaging in and improving problem solvingare presented as a means of improvingproblem solving skills. Many examples fromsecondary school mathematics are used toillustrate the points the authors have inmind.

Burns, Marilyn. Meeting the Math Challenge.

Palo Alto, CA: Learning Institute, 19C2.

A handbook for leaders of mathematicsworkshops for teachers interested inimplementing ...he NCTM's Agenda for Action.

A strong emphasis is placed upon individualand group problem solving, cooperativelearning, and the appropriate classroomenvironment. Many activities fromelementary mathematics are used asillustrations.

Burns, Marilyn. The Math Solution. Teaching

for Mastery Through Problem Solving.Sausalito, CA: Marilyn Burns Education

Association, 1984.

These K-8 supplementary materials aredesigned to help make problem solving thefocus of mathematics curriculum.Mathematics is taught as a way ofthinking--a tool for solving problems.

Copeland, Richard W. How Children Learn Mathematics: Teaching Implications of Piaget'sResearch. New York: The Macmillan Company,1970.

This book is intended to help teachersdiscern a child's stage of development as a

basis for determining type of mathematicsfor which he or she is ready. It summarizesthe ideas of Piaget in a form most teacherswill find readable.

Driscoll, Mark. Research Within Reach:Elementary School Mathematics. NationalInstitute of Education, U.S. Department ofEducation, Washington, DC 20208. (Alsoavailable through the National Council ofTeachers of Mathematics.)

A researchguided response to concerns ofeducators, including bridging the gapbetween the concrete and abstract throughthe use of manipulatives, models, drawings,patterns. Also presents research findingson questions relating to drill, teachingbasic facts and algorithms, calculators,diagnosis, evaluation, grouping, motivation.

Driscoll, Mark. Research Within Reach:Secondary School NationalInstitute of Education, U.S. Department ofEducation, Washington, DC 20208. (Alsoavailable through the National Council ofTeachers of Mathematics.)

A researchguided response to concerns ofeducators, including effective mathematicsteaching, communicating mathematics,

-;4 t 4104

breaking the vicious cycles, reudiationin secondary school mathematics, problemsolving, understanding fractions,estimation, iearning and teaching algebra,and learning and teaching geometry.

Driscoll, Mark and Jere Comfrey. TeachingMathematics: Strategies That Hork: TeacherWriting to Teacher Series. Chelmsford, MA:Northwest Regional Exchange, Inc.

Articles written by a group of talented andenthusiastic professionals to theircolleagues about pedagogy in teachingmathematics.

Fennema, Elizabeth. Mathematics EducationResearch: Implications for the 80$.Association for Supervision and CurriculumDevelopment, 1981.

This publication is a collection of articlesby authors who have synthesized currentresearch in mathematics education. Theirobservations include findings on such areasas problem solving, calculators, computers,factors related to gender, andteacher/student interaction characteristics.

Hirsch, Christian R. Activities for Implementina_Curricular Themes From the Agendafor Action. Washington, DC: NationalCouncil of Teachers of Mathematics, 1986.

Selections from the Mathematics Teacherwhich can be used to implement problemsolving, expanding basic skills, and usingcalculators, computers, and manipulatives.

-;

4

Hirsch, Christian R. The Secondary School Mathe-matics Curriculum (Yearbook; 1985).Washington, DC: National Council ofTeachers of Mathematics, 1985.

This yearbook is a collection offorward-looking articles addressing theissue of what the secondary mathematicscurriculum should be in order to prepare ourstudents for the life they face in anever-changing technological society.

Labinowicz, Ed. Learning From Children: New,ieginnings for Teaching Numerical Thinking.

Menlo Park, CA: Addison-Wesley, 1985.

This book attempts to build a bridge betweentheory, research, and classroom practicewith the aim of supporting informed change

in schools. The research reported ischaracterized by clinical interviews ofindividual children and observation of theteaching - earning process throughexperiments with small groups of children.

Labinowicz, Ed. Learning From Children. NewBeginnings for Teaching Numerical Thinhing.Menlo Park, CA: Addison-Wesley Publishing

Company, 1985.

Specific suggestions from the Piagetianpoint of view for teaching the concepts andoperations with numbers.

105

Labinowicz, Ed. The Piaget Primer--Thinking.

Learning, Te_ching. Menlo Perk, CA:

Addison-Wesley Publishing Company, 1980.

This book presents the essence of Piaget'sideas with a minimum of technical vocabu-

lary. It also bridges the gap between pistheory, his research methods, and classroompractice in presenting mathematics and

science concepts.

Lane County Mathematics Project. Problem

Solving in Mathematics. Eugene, OR: Lane

Education Service District, 1983.(Available from Dale Seymour Publications,PO Box 10888, Palo Alto, CA 94303-0879.)

The activity-oriented materials providesupplementary units on problem col vi ray for

grades four through eight and algebra. A

full range of problem-solving skills is

included.

Lappan, Glenda. Middle Grades Mathematics

Project. Michigan State Jniversity:Addison-Wesley Publishing Company, 1986.

Excellent two-three week units utilizingmanipulation to teach units, such asProbability, Factors and Multiples, SpacialVisualization, Simulation, Mouse and

Elephant.

2

Laycock, Mary and Gene Watson. The Fabric ofPAsithmi:AlfrT3gr(Revised Edition). Hayward, CA: ActivityResources Company, Inc., 1975.

This resource book contains mafiy suggestedclassroom activities for teachers interestedin using concrete materials to introduce andreinforce elementary mathematics concepts.

Laycock, Mary and Margaret A. Smart, Hands OnMath for Secondary Teachers. Hayward, CA:Activity Resources Company, Inc., 1984.

This book was written to help fill the needof the secondary teacher who wants alterna-tive strategies in teaching basic mathe-matics to junior and senior high students.This book will help you integrate manipu-latives into your math program and help yourstudents gain understanding of mathematicsat a concrete level.

Lester, Frank K. and Joe Ganofalo. MathematicalProblem Solving: Issues in Research.Philadelphia PA: The Franklin InstitutePress, 1982.

Papers of leading researchers of mathe-matical problem solving are presented,including a chapter on "Building BridgesBetween Psychological and MathematicsEducation Research on Problem Solving."

Marcy, Janice and Steve Marcy. Math. Imagina-tion. and Pizzazz Series. Palo Alto, CA:Creative Publications, 1978.

Presents puzzles at many skill levels.

106

Mason, John with Leone Burton and Kaye Stacey.Thinking Mathematically. London, Engllnd:Addison-Wesley Publishing Company, 198,..

The authors believe that mathematicalthinking can be improved. They presenttheir theory and make their ideas explicitwith the use of problems in which the readeris expected to seek appropriate solutions.

Math and the Mind's Eye. Available (beginningOctober 1987) from the Math Learning Center,PO Box 3226, Salem, OR 97302.

A collection of 38 activities for grades 5to 9 utilizing visual methods for learningmathematics. The activities are separatedinto five units: I Seeing MathematicalRelationships, II Visualizing NumberConcepts, III Modeling Whole Numbers, IVModeling Fractions and Decimals, V Lookingat Geometry.

Mathematics Resource Project. Geometry andVisualizatio. Eugene, OR: Department ofMathematics, University of Oregon, 1976.(Available from Creative Publications, POBox 10328, Palo Alto, CA 54303.)

This lengthy collection of classroommaterials for middle school studentscontains units on Lines, Planes and Angles;Polygons and Polyhedra, Curves and CurvedSurfaces; Similar Figures; and Area andVolume Teaching emphases of the materialsincludt visual perception, grapnic represen-tation, calculators, applications, estima-tion and approximation, and laboratoryapproaches.

4.

Mathematics Resource Project. Mathematics in

Science and Society. Eugene, OR:

Department of Mathematics, University ofOregon, 1976. (Available from CreativePublications, PO Box 10328, Palo Alto, CA94303.)

This collection of classroom materials for-4ddle school students is organized into sixunits: Mathematics and Astronomy, Mathe-matics and Biology, Mathematics and Environ-ment, Mathematics and Music, Mathematicsand Physics, and Mathematics and Sports.Problem solving, teaching for transfer, andlaboratory approaches are stressed.

Mathematics Resource Project. Number Sense andArithmetic Skills. Eugene, OR: Departmentof Mathematics, University of Oregon, 1975.(Available from Creative Publications,PO Box 10328, Palo Alto, CA 94303.)

This collection of classroom materials formiddle school students contains resourceideas on whole number operations, numerationsystems, fractions, and decimals. Teaching

emphases of the materials includecalculators, applications, problem solving,mental arithmetic, algorithms, estimationand approximation, and laboratory approaches.

Mathematics Resource Project. Ratio. Proportionand Scaling. Eugene, OR: Department ofMathematics, University of Oregon, 1975.(Available from Creative Publications,PO Box 10328, Palo Alto, CA 94303.)

28107

This collection of classroom materials formiddle school students contains resourceideas on the topics, ratio, proportion,scaling, and percent. Teaching emphases ofthe materials include calculators,applications, problem solving, mentalarithmetic, estimation and approximation,and laboratory approaches.

McFadden, Scott, Keith Anderson, and OscarSchaaf. Math Lab--Junior High. Eugene,

OR: Action Math Associates, Inc., 1975.

This book consists of pages which can bereproduced to make activity cards presentingnine topic strands, including perimeter andarea, volume, calculators, and applica-tions. The tasks presented are problemsolving in nature and many require hands-oninvestigations.

McFadden, Scott, Judy Johnson, Mary Ann Todd,and Oscar Schaaf. Math Labmetric.

Eugene, OR: Action Math Associates, Inc.,1976.

This series of three books (separate edi-tions for grades 1-2, 3-4, 5-6) containsnumerous activity cards foi use in elemen-tary math classrooms. Fifteen math topicsare covered at each grade level, and thetasks presented are activity based.

2 L.

McFadden, Scott. Algebra Warm-Ups. ShortExercises for Review and Exploration.Palo Alto, CA: Dale Seymour Publications,1987.

Activities for the first few minutes ofclass with emphasis on skill maintenance,concept development, and problem solving.Content includes topics from first yearalgebra.

McFadden, Scott. Math Warm-Ups for Junior High(7-5). Short Exercises for ReviewExploration. Palo Alto, CA: Dale SeymourPublications, 1983.

Activities for the first few minutes ofclass with emphasis on skill maintenance,concept development, mental computation, andproblem solving. Content areas includewhole numbers, fractions, decimals, money,percent, measurement, and geometry.

National Council of Teachers of Mathematics, Inc.The Arithmetic Teacher. Reston, VA:National Council of Teachers of Mathematics,Inc.

This magazine is one of two officialjournals of the National Council of Teachersof Mathematics. It is published nine timesyearly and is distributed to NCTM members.It contains a wealth of ideas that can beput to use in the elementary and middleschool mathematics classroom.

2,O108

National Council of Teachers of Mathematics, Inc.The Mathematics Teacher. Reston, VA:National Council of Teachers of Mathematics,Inc.

This magazine is one of two officialjournals of 'he National Council of Teachersof Mathematics. It is published nine timesyearly and distributed to NCTM members. It

contains information on recent happenings insecondary mathematics education and includesa special activities section each issue.

Oregon Council of Teachers of Mathematics. The

Oregon Mathematics Teacher. Portland, OR:Oregon Council of Teachers of Mathematics.

This magazine is the official journal of tteOregon Council of Teachers of Mathematics.In contains many classroom-tested ideas foruse in both elementary and secondaryschools. The journal is available throughmembership in OCTM; for information,contact, Specialist, Math Education, OregonDepartment of Education.

Oregon Depa-tment of Education. MATH in OregonSchools. Salem, OR: Department ofEducation, 1977.

Oregon Department of Education. The Oregon

Mathematics Concept Papers. Salem, OR:

Department of Education, 1986.

These position papers were written byteachers for teachers. The information andrecommendations presented are based on astudy of research, reports, and recommenda-tions appearing in the national literature.The eight papers address forward-lookingissues at each school level.

Polya, G. How to Solve It. Princeton, NJ:

Princeton University Press, 1973.

This book contains a comprehensive discus-sion of the nature of problem solving andsuggests many specific problem-solvingstrategies.

Reys, Robert E., Marilyn N. Suydam, and MaryM.Lindquist. Helping Children I carn

Mathematics. Englewood Cliffs, NJ:

Prentice-Hall, Inc., 1984.

This book is for teachers of mathematics inelementary school. It is written to helpteachers help children learn importantmathematical concepts, skills, andproblem-solving techniques. The first partof the book provides a base for under-standing the changing mathematics curriculumand how children learn. The second part

discusses teaching strategies, techniques,and learning activities.

"2 109

Reys, Robert E. Helping Children Learn Mathe-

matics. Englewood Cliffs, NJ: Prentice

Hall, Inc., 1984.

Presents a basic overview of backgroundusing technology.

Richardson, Kathy. Developing Number ConceptsUsing Unifix Cubes. Menlo Park, CA:

Addison-Wesley, 1984.

This book is for teachers who want to makemathematics understandable for theirstudents. The concepts dealt with rangefrom beginning counting to the introductionof multiplication and division. The book is

based on the premise that children aretaught mathematics so it can serve as auseful, problem-solving tool in their lives.

Romberg, Thomas A. "A Common Curriculum forMathematics." Individual Differences andthe Common Curriculum Yearbook, Universityof Chicago: National Society for the Study

I of Education, 1983.

A case is made for learning by doing mathe-matics. Four related activities common toall of mathematics are identified:abstracting, inventing, proving, andapplying. These activities are the naturalconsequences of mathematical investigationsand can be made an integral part of schoolmathematics instruction.

4 4-

Schminke, C. W. and Enosh Dumas. Math Activi-ties for Child Involvement, Third Edition.Boston MA: Allyn and Bacon, Inc., 1981.

Children learn best when they are stimu-lated, surprised, amused, and aresuccessful. The activities emphasizeconcept development, basic skill develop-ment, and extended understanding. Contextincludes prenumber topics, numeration andplace value, number operations with wholenumbers and fractions, geometry, andmeasurement.

Schoen, Harold L. Estimation and Mental Computation (Yearbook; 1986). National Council ofTeachers of Mathematics, 1986.

This yearbook is a response to the growingrecognition among mathematics educators ofthe importance of teaching and learningestimation. Its articles make a strong casefor the importance of estimation in variousoomains such as numerosity, computation, andmeasurement.

110

Sharron, Sidney. Applications in School Mathe-matics (Yearbook; 1979). National Councilof Teachers of Mathematics, 1979.

Much of this yearbook deals with forming andusing mathematical models to make mathe-matics more meaningful to students. It alsoaddresses the problem that many teachershave incorporating real-life applications intheir instruction since most of their train-ing has dealt with the abstract rather thanthe practical or serviceable areas ofmathematics.

Sobel, Max A. and Evan M. Maletsky. TeachingMathematics: A Sourcebook of Aids.Activities. and Strategies. EnglewoodCliffs, NJ: Prentice-Hall, Inc., 1975.

This book presents a collection ofexperiments and activities for secondarymathematics classrooms. Also included is abibliography of resources and references.

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