P)COMENT RESUME ED 051 185 SP 007 249 · P)COMENT RESUME. ED 051 185 SP 007 249. AUTHOR. Chandler,...

P)COMENT RESUME

ED 051 185 SP 007 249

AUTHOR Chandler, Arnold I.; And OthersTITLE Guidelines to Mathematics, K-6. Key Content

Objectives, Student Behavioral Objectives, and OtherTopics Related to Elementary School Mathematics.

INSTITUTION Wisconsin State Dept. of Public Instruction, Madison.NOTE 58p.; Bulletin No. 141

EDRS PRICEDESCRIPTORS

EDRS Price MF-$0.65 HC-$3.29*Curriculum Guides, *Elementary Grades, *ElementarySchool Mathematics, Grade 1, Grade 2, Grade 3, Grade4, Grade 5, Grade 6, *Kindergarten, *MathematicsCurriculum

ABSTRACTGRADES OR AGES: K-6. SUBJECT MATTER: Mathematics.

ORGANIZATION AND PHYSICAL APPEARANCE: The guide is divided intoseveral chapters. The central and longest chapter, which outlinescourse content, is further subdivided jut° 15 snits, one for each of15 content objectives. This chapter is in list Lfirm. The guide isoffset printed and staple-bound with a paper cover. OBJECTIVES ANDACTIVITIES: The central chapter lists 15 mathematical concepts suchas numeration systems, ratio and proportion, size and shape, ormeasurement. A list of related behavioral objectives for each conceptat each grade level is then presented. No specific activities arementioned, although one short chapter gives general guidelines ondeveloping problem-solving situations. INSTRUCTIONAL MATERIALS: A

short bibliography lists several references for each chapter. STUDENTASSESSMENT: A chapter on evaluation and testing gives guidelines fordeveloping tests and choosing appropriate methods for different gradelevels. (RT)

t41COr4e-4

C3C3 Guidelines to Mathematics

K-6

Key Content Objectives, Student Behavioral

Objectives, and Other Topics Related to

Elementary School Mathematics

issued by

William C. Kahl, State SuporintendontWisconsin Department of Public Instruction

Bulletin No. 141

6000.S8. 84000 27

I

U.S. DEPARTMENT OF HEALTH,EDUCATION &WELFAREOFFICE OF EDUCATION

THIS DOCUMENT HAS SEEN REPRO-DUCED EXACTLY AS RECEIVED FR':$MTHE PERSON OR ORGANIZATION ORIGINATING LT. POINTS CF VIEW OR OPIN-IONS STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE OF EDU-CATION POSITION OR POLICY.

This project was sponsored cooperatively

Wisconsin tr atment of Public InstructionTitle V of the Elementary and Secuadary Edocation Act

Title Ill of the National Defense Education Act

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Published by:

Th; NVisconsin Department of Public InstructionWi/liam C. Kahl, State Superintendent

Committee Members:

Arnold M. Chandler, CoordinatorWisconsin Department of Public Instruction

M. Vere DeVaultThe University of Wisconsin

Rondd P. FisherManitowoc Public Schools

Adeline HartungMilwaukee Public Schools

George L. HendersonWisconsin Department of Public Instruction

Thomas E. Kriewall, Graduate AssistantThe University of Wisconsin

John F. LeBlancRacine Public Schools

Walter W. LcffinMonona Grove Public Schools

Ted E. LosbyMadison Public Schools

Larry MillerRipon College, Ripon

Henry Van Et gcn, ChairmanThe University of Wisconsin

Raphael I). WagnerThe University of Wisconsin

Marshall E. WickWisconsin State University, Eau Claire

Margaret WoodsPlatteville Public Schools

Technical Assistance*.

Emily Spielman, Publications EditorWisconsin Department of Public Instruction

JoAnn E. Stein, Graphic Artist1Visconsin Department of Public Instniction

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Foreword

In a dynamic society such as ours, change is constant. People accept this factinure readily in many fields than in the area of mathematics particularly at theelementary school level. And yet, in recent years new projects, new courses, andnew materials remind us that a great potential exists for improved changes in theelementary mathematics program.

The primary purpose of Guidelines to MatIrmatics, K-6 is to help those whohave responsibility in local school districts for providing a well conceived elemen-tary mathematics program. Each of the sections of this guide has been designedto meet specific objectives. Although no section represents the final word on thetopic being discussed, it is intended that all sections represent definitive statementswhich can serve as I sound foundation for furth.:r study and investigation.

The main section of this guide has been devoted to a careful development ofthe major concepts of elementary school mathematics and associated behavioralohjectives. In addition, several other sections have been devoted to issues con-cerning the effective teaching of mathematics at the elementary school level.

The 'Wisconsin State Department of Public Instruction trusts that the effortsof all who helped make this pnblieation possible Nv II result in improved mathe-matical experiences for all Wisconsin youth.

William C. KahlState Superintendent

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Table of Contents

Mathematics Instruction: A Point of View 1

Key Mathematical Content Objectives and RelatedStudent Behavioral Objectives 5

Adjusting Nlathematics Programs to Student Abilities 3.5

Problem Solving 39

Evaluation and Testing 44

Inservice Education 49

Criteria for Program Development 52

Bibliography 57

Mathematics Instruction: A Point of View

'file variety of mathematics programs now avail-aide for the elementary grades promises to benefitboth teachers and pupils alike. :facing been freedfrom the fixed form that charactcriled the tradi-tional program, teachers now have a greater oppor-tunity than ever to select the niaterials and teachingtechniques best suited to a child's needs. The basisused for selecting methods and materials is impor-tant am l is dependent upon the viewpoint of theteacher in regard to his instructional task. Just asmathematics programs have changed, so might cer-tain aspects of the traditional view of instructionhe changed.

It has been COMII30/1 practice heretofore to con-sider certain mathematics skills and concepts as hie-ing "all-or-nothing" attainments. That is mastery ofa certain body of facts or skills has been tradition-ally rc garde(' as the proper province of the teacherat a particular grade level. The failure of !winechildren to achieve the expected level of proficiencyhas forced the teacher of the next grade to -reteaerthese areas, a task that has long been a source ofirritation to many teachers. This type of compart-mentalization of a course of study should he mini-mized, if not eliminated. InFt;.-ad of expecting allchildren at a given grade level to master in a ratherfinal sense a certain portion of the mathematicaltraining, educators should develop an approachwhich recognizes the existence of individual growthrates and stresses the continuity of the instructionalprocess.Teaching for Growth

Pciliaps the one factor most essential to the suc-cess of the mathematics curriculum is an emphasison understanding, that is, understanding of mathe-matics. This emphasis represents a marked shift inthe focus of educators' attention from overt be-havioral skills and social applications to under-standings developed as a child organizes and codi-fies mPthematieal ideas. It is no longer deemed suffi-cient for a teacher to be satisfied with competentcomputational performance by a child, in spite ofthe obvious necessity for "uch skill, The need to finda more efficient, a more enjoyable, and a more

illuminating method of instruction has led to theconsensus that clear and penetrating understandingof certain essential mathematics must precede, butcertainly not supplant the traditional point of em-phasis, computation. flow is this to he achieved?Experiencing the Physical World

Certain principles of instruction deserve renewedemphasis. It is agreed that one should traverse luny'the known to the unknown, from the particular tothe general; from the concrete to the abstract.Clearly, children should he encouraged to developtheir first concepts of mathematics from their ex-periences with physical objects.

This approach implies, in inany cases, that priorto the introduction of a concept, time should heprovided for each child to experiment with physicalobjects appropriate to the objective. For instancc.every child should have the experience of actmdlycombining sets or groups of things as a prelude tothe introduction of addition; he should Si (Irk 1111 anabacus or place-value device prim to any contactwith sophisticated positional notation; he shouldhave "pre-number'. experiences, such as Lulling tocompare the numerousness of groups thtotigh theprocess of one-to-one matching, rather than count-ing in the traditional fashion.

For example, the classroom situation in whichchildren are seated at their desks provides a naturalone-to-one correspondence between children andoccupied chairs, The number of chairs can be com-pared to the number of children by simply notingwhether any chairs are einpty, rather than countingthe members of each group. Many other imilar situ-ations arise naturally in which the child's interestreadily leads to non-verbal awareness of such con-cepts as equal to, less than, and more than. Ily suchpurposeful "playing:. important concepts that pre-viously have been left large'y to chance develop-ment will he given appropriate attention and willbe established on a firm foundation,Useful Unifying Concepts

Because understanding is an ambiguous term andbecause a teacher must make as clear as possiblethe relationship between former objectives and

present points of emphasis, it May be well to con-sider just a few of the many concepts that arcesp.cially important to clarify and develop fromthe earliest stages.

For the foreseeable future, the study of numbersystems will continue to he the essence of themathematics program. Involved in this study arethe collections of various kinds of number ideas,such as natural numbers, together with the opera-tions defined on these number ideas and the proper-ties of these operations (for example, commutative,associative, distributive properties). Obviously, thedevelopment of an understanding of such ideasmust take place over an extended period of timeand at considerably different rates for differentchildrem .There i- no such thing as complete understanding of any particular number system by achild. Rattler, teachers at each level should as-ur-tai that the instruction is directed toward deepen-ing and extending the broad mathematical streamscited in later sections of this report. llow eaki thisbe done?

As a child's grasp of mathematics grows, he musthe guided toward the acquisition of the broad andes..ential concepts which tic together seemingly dis-connected particulars into a coherent general struc-ture. The same operations are encountered in sev-eral peculiar and different contexts in the grades.Each particular use really represents a differentoperation.

For instance, the multiplication of two naturalnumbers such as 3 X 5 may be properly regardedas the process of starting with nothing and adding5 objects to a set 3 times. But can 1/2 X 1/3 besimilarly interpreted as the process of repeatedlyadding 1/3 objects for 1/2 times? Or xs:tat meaningcan be ascribed to (- 3)( -5)? Certainly not 5dings repeatedly added for 3 times! Such state.meats require different interpretations because theoperation called muhiplication has several meanings and definitions depending upon the kinds ofnumbers or set elements to which it is applied.

Natural numbers, integers, and rational numberseach combine somewhat differently under a givenoperation. But rather than invent new symbols torepresent the hanging definitions of an operationas it applies to the various number systems, oneuses the same symbol throughout. In this situation,as in many similar ones, children must be taught tointerpret symbols in context. Only in this way willmathematics shorthand serve the important purpose

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of clarifying and improving communication. Fur-thermore, the child should be led to understandthat although the specific interpretation of an oper-ation differs from one . ystera to the next, a struc-tural unity still remains which is founded on thecommutative, assoc.ative, and distributive propai-ties as they pertain to the operations. Because ofthis structural unity, one can rely on the context inwhich a symbol is used to determine its meaning,rather than create new symbols for each particularvariation of a concept.

Again, the need for techniques which lead thechild to discover ideas through his experiences withphysical objects must be stressed. Furthermore,attention must be given to rather subtle ideas thatmay have been overlooked or thought too obviousto mention in the past. For instance, when a childcombines two sets of objects into a single collection,he must come to the rcalizatioc that the abstractproperty of numerousness is conserved; that thereare exactly as many objects after as there were be-fore the two groups were joined. As fundamentalas such an assumption is to understanding the addi-tion operation, it is by 110 means obvious to all be-ginners. This fact is substantiated by the work ofPiaget and others.

Another matter often treated too lightly involvesthe freedoms or restrictions MIMI may or may notexist for a given operation. When one matches theelements of a set with dm word names of the ordinalnumbers, he must know that the order in which theobjects are matched to the numerals will not affectthe count. On the other hand, cases exist in whichthe order of presentation of information is impor-tant. One cannot mix up numerator and denomina-tor with impunity or name the coordinates of apoint by whim. The basis for free choke, where itexists, and the reasons for restrictions, as they occur,must be made c'ear if understanding is to beachieved. Such matters are rarely as obvious tochildren as they seem to adults.

To sum up, children should come to realize thatmathematics is a logical system which exists is crhuman invention, formulated, enriched, extended,and revised in response to the twin weds to perfectit as a logical structure and to use it as a convenientmethod of describing certain aspects of nature asseen by man.Undue Emp'iasIs on Particulars

Educators should avoid giving undue emphasisto any single aspect of the total mathematics pro.

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gram. The idea of sets, for instance, should not beglorified beyond its usefulness in contributing tothe attainment of the broad goals of the program.Similarly, it is unwise to go to the extreme of down-grading the importance of computational pro-ficiency to the point where long range goals arcplaced in jeopardy. Furthermore, big words or im-pressive terms and symbols must never interferewith a child's understanding of the concepts whichthe words or symbols represent. Clarity of commani-cation is the chief purpose of mathematical lan-guage and symbolism. If a particular t:rm or symboldoes not serve the cause of ch. 'ty, then it shouldnot be useti 1, of course, must eventuallyIcian how to use the language effectively and under-stand it in context. llowevor, teachers should exer-cise great care in working toward such a goal andshould avoid any proliferation of symbols or pre-mature verbalizations that will hinder goal attain-ment.Problem Solving

One of the greatest challenges for the elementaryteacher is the devel_pment of appropriate, real-lifeproblems which meaningfully involve learners.Classroom teachers recognize the limitations of text -hooks in providing such verbal problems. Manyproblems fail to challenge children; they fail to represent a world which is real to children; and toooften they fail to cortribute to the child's skill inproblem solving situations outside the textbook.Notwithstanding these limitations, textbook verbalproblems will continue to be prominent in mostelementary classrooms until individual teachersidentify other ways of fleeting the problem solvingobligation.

There is increasing evidence that through experi-mental programs and through the efforts of indi-vidual classroom teachers improved ways of meet-ing this obligation are being developed. Much isbeing heard about the importance of relating mathe-matics to science, of units of study in miAhematics,of enrichment activities in mathematics, of the studyof the history of mathematics, and of children cre-ating their own mathematics problems. Instead ofstressing the social development of the child. theseefforts emphasize the structure of mathematicsthrough problem solving experiences. These activi-ties are hopeful signs pointing to a greater stressbeing placed, in the near future, on the importance

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of meaningfully-structured problem solving experi-ences as a part of the elementary mathematics pro-gram.

Individual DifferencesIf mathematics instruction is viewed as a procesi

of initiating understanding and of carefully nurtur-ing this understanding as the child matures, it willthen be necessary to discover effective techniquesfor accommodating the widely differing rates atwhich children develop. Implicit in this statementis the need for schools and teachers to recognizethat some children may not be able to grow sub-stantially or may seem to terminate their potentialfor growth at some point along the way. In suchcases, provision should be made for experiencesmost appropriate to the child's welfare.

To date, no satisfactory method has been devel-oped to cope with the vast range of individualdifferences. Flexible grouping procedures, non-graded classrooms, individualized instruction, team.-teaching, and television teaching are receiving ex-tensive testing and evaluation. Perhaps the experi-ments currently underway vial yield effective tech-niques once more is learre3 about the problem. Inthe meantime, each teacher must exerciso profes-sional judgment and common sense in adopting anoptimal arrangement that is compatible with hisown abilities, with the characteristics of pupils inhis class, and with the physical facilities and admin-istrative policies of the school.

In summary, mathematics instt action should heviewed as a continuous effort to develop in thechild a knowledge of mathematics that is charac-terized by its depth and connectedness. To the ex-tent possible, the child should be encouraged toexperiment with the objects of his environment.Thus prepared, he may be led to the invention ordiscovery of those ideas which provide both a broadbasis for further exploration and a sense of delightin a well - founded mastery of the subject. The taskis a challenging one and deserves much effort. Tothis end, the teacher should do everything possibl,to see that instruction is well-planned and is pro-videi on a regular basis. As an additional, but essen-tial measure, cooperative action such as inscrvicceducation should be taken by school personnel todevelop in the leaching staff a view of instructionappropriate to present day needs and opportunPies.

Key Mathematical Co.ltent Obiecti'Jes

and

Related Student Behavioral Obis,( i1/:;F

in the fo1lowinit, (nitline, an attempt has beenmade to point out where kt y content objectives ofmathematics might be introduced and developedin grades K-6. The corresponding behavioral objec-tives suggest a sequence of development of thesemathematical ideas which provides for their -

inforeement and continuity throughout the K-6mathematic: program. By means of a sequence ofdraveloprnmt, key ideas can be (Mewled to conformwith the maturity and background Nix Hence ofthe student.

The suggested content objectives have been or-ganized uncle fifteen main topics: Sets am! Num-bers; Numerate.., Systems; Order; Number Systems,Opera:ions and Their Properties; Ratio and Propor-tion; Computation; Size and Shape; Sets of Points;Symmetry; Congruence Similarity; Coordinate Sys-tems ane. Graphs; Constructions; Measurement; andMathematical Sentences.

It is not intended that the placement of topics inthis outline be considered as the only correctarrangement or that all of the topics necessarily betaught at every grade level as presented here. Theideas listed for each grade level should be regardedas a suggested guide for introducing various topics:the outline is not intended to be all-inclusive. The!artier will find it necessary to alter the order of

topics to meet the needs of the children or theneeds of a particular group in a given classroom.

Furthermore, no fixed amount of Hine or emphasishas been suggested for any objective in this outline.A disproportionate amount of space has been de-voted to the geometric objectives due to their recentii.troduction in elementary mathematics courses.Many of the geometric objectives can be attained in

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a relati aly short tinea c.)n ',tied with the tineand empl.asis necessal a to ,etaiii those objectives ofelementary schnol arithmetic which are central to

understandir g and us- of the rational muralsystem.

This outline should praysT useful to the is 101administration, the mathematics Olrriculum'Mace, and the teachers of a school district in oneor more of the fidloing ways:

an orientation to the key content and be-havioral objectives of the eleir.fntary sc'toot.mathematics curriciihmaAs a source of knowledge of the related objec-tives of elementary school mathematics, ;heirdevelopment and reinforcement throughout theentire K-6 mathematics programAs a means of evaluating the content and be-havioral olijctives of the present mathematicsprogram of the school on the basis of the needsof the school community.As a guide for the preparation of a mathe-matics curriculum guide, K-6, for use by theschool district.As an aid in the evaluation and scleeiien oftextbooks.As 0 guide in determining the concepts thatneed to he highlighted in inservice activitiesfor teachers.As an aid ill choosing standardized tests.

Throughout this outline, au asterisk (*) appearingaorder any behavioral objective indicates that theunderstanding and skills listed for that objectiv( inpro, kills grades arc to be expanded arid developed.

Mathematical ConceptsArithmetic

Sets and NumbersA sat is a collection of things. Sets .,ray have many

properties, but a property peculiar to a collectionor set is that of number. When an orderly arrange-ment of these numbers has been made, one has thetools needed for counting. Numbers can be used inthe cardinal sense, that is, to indicate numerousness.For example, a triangle has three vertices. They canalso be used in he ordinal sense, that is, to indicatethe arrangement of numbers by order. For example,

Concepts

a student in fifth grade is studying page WO (thehundredth page) of his textbook. Sometimes a mun-ber is used in the nominal sense, that is, as a name.A social security number is used this way.

In this guide. the concept of number is extendedto include positive rational numbers and then posi-tive and negative integers. These number conceptswill be more fully de./eloped in the junior andsenior high school mathematics programs.

Students should be able to:

Numeration SystemsNames are given to the cardinal numbe,s. These

names are nu '2rals and are used to convey theidea of number. Numerals are symbolizeL cormsused for communication. A given number hc.s manydifferent names which can be used depending on

the idea one wishes to express. Organized methodsof arranging the numerals represent the numerationsystems of the past and present. Study of differentnumeration systems will give the student insightand appreciation of the more efficient ones.


OrderThe concept of order is a generalization made as

a lesult of experience in comparing unmatched setsof objects to determine which has more members.Order is expressed in sentences by means of the

symbols > (is greater than) and < (is less than).The concepts of equality and of inequality (num-bers that are not equal) are considcred here forconvenience.


Number Systems, Operations and TheirProperties

A number system consists of three parts: a set ofnumbers, two operations defined on the numbers inthe set, and the properties (laws or rules) of theoperations.

Two number systems, the whole numbers and thepositive rational numbers, are studied in detail inthe elementary school. A third system, the integers,is introduced in the elementary school when thephysical situation demands thaL the number systembe extended to include negative numbers. The inte-gers and a fourth number system, the real numbers,are studied in more detail in junior and senior highschool. An understanding of the integer and realnumber systems and their properties deper,:scareful and complete development of the wholenumber and (positive) rational number systems inthe elementary school.

The operations on numbers studied in the ele-menta:y school are addition and natiplicat ionandtheir respective inverses, subtraction and dia n.

The important and useful properties of the 1,7-ations on all number sy:,:ems studied in ,::-!thrneticare given here. They are:

AdditionCommutative: a --f-- b + aAssociative: (a + b) c = a + (h c)Identity element: a + 0 = aInverse element: a + ( a) = 0

MultiplicationCommutative: a X b=bx aAssociative: (a X b) X c=a X (bx c)Identity element: a X 1 = aInverse elerm nt a X 1/a -= 1, a ri- 0Property of zcre: a X 0 = 0Distributive: a X (la c) = (a X b) + (a X


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Behavioral ObjectivesKindergarten

Sets and NumbersI. Identify two equivalent sets by placing the mem-

bers of the set in one-to-one correspondence.2. Select the set of objects associated with a given

number.

3.

4.

5.

Count orally by matching numerals with setshaving a given number of objects.Identify, without counting, the number of setswith two, three, or four objects.Use such terms as more than, as many as, fewerthan when comparing sets of objects,

Numeration Systems1. Identify the numeral, 0 through 9.

Order1. Determine whether two sets are equivalent (can

be matched or placed in a one-to-one correspond-ence).

2. Compare two non matching sets of fewer than10 objects and decide which set has more men,-

hers and which set has fewer meml..-±rs.3. Determine that 3 is greater than 2 and that 2 is

less than 3 by comparing appropriate sets of ob-jects and do this for any two numbers less than 6.

4. Utilize the idea "one more than" in organizingsets in the natural order.

Number Systems, Operations end TheirProperties1. Rearrange sets of objects to demon strate the

joining and separating of sets, and thereby de-velop a readoess for addition and subtraction.

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2. Tell a story about a problem represented by theabove activities.

3. Use objects to represent the "action" or condi-tions of a problem.

WOC11,41,,,,Mftnewm.

Behavioral ObjectivesFirst Crade

Sets and Numbers 4. Use "0" as the symbol for the number of de-l. Count the members of a set containing one bun- ments in the empty set.

dyed or fewer members. .5. Insert missing sets, such as a set of 3 elements2. Demonstrate one -half, one-fourth of a physical between a set of 2 elements and a set of 4 ele-

unit. ments.3. Use the ordinal numbers through fifth.

Numeration Systems1. Read and write any numeral from 0 thro,igh 100.2. Give different numerals for a given number such

as 6 4- 2, 10 2, and 8 for eight.

3. Read number words through ten.4. Interpret the place -value concept for writing

whole numbers to one hundred; such as, 69 isthe same as 8 tens, 9 ones.

Order1. Compare two non matching sets of less than 1(X)

objects to decide which set has fewer (more)"members.

2. Determine that 8 is greater than 5 and that 5is less than 8 by comparing appropriate sets ofobjects and do this fur any two numbers les.;than 10.

Number Systems, Operations and Their :3.

Properties1. Identify the process of addition through experi- 4.

ince with joining two sets of objects.2. Identify the process of subtraction through ex-

perience with separating a subset from a set of 5.objects.

129

Recognize examples of the commutative prop-erty for addition in the set of whole numbers.Demonstrate with sets of objects the relation-ship between such sentences as 4 4- 2 = 6,6 2 4, and 6 4 2.Use the symbols +,--, and to form sentencessuch as 3 + 6 = D.

Behavioral ObjectivesSecond Grade

Sets and Numbers1. Determine the cardinality of a set to one thou-

sand when the objects are already grouped. Forexample, 4 bundles of 100 sticks, 9 bundles of10 sticks, and 4 sticks are the same as 494 sticks.

2. Use ordinal numbers through tenth.3. Identify 1/2, 1/4, 1/3 of a whole by using physi-

cal objects.

Numeration Systems1. Head and write any numeral through 999.2. Write many symbols for the same number; for

example, 6 + 3, 5 + 4, 17 8, and 9 fornine.

3. Count by 5's, 10's, and 100's.4. Use bundles of sticks to demonstrate place value

through 999; for example, 2.34 = 200 + 30 + 4.

5. Write three-digit numerals in expanded notation;for example, 765 = 700 + 60 + 5.

6. Use physical obj-Tts to demonstrate regrouping;for example, 2 bundles of 10 sticks and 16 stickshave the same number of sticks as 3 bundles of10 sticks and 6 sticks. Also 4 bundles of 10 sticksand 8 sticks have the same number cf sticks as3 bundles of 10 sticks and 18 sticks.

Order1. Use the terms greater than and less than, and

equals in sentences.2. Use symbols >, <, and = in mathematical sen-

tences.

3. Insert missing numbers, such as 3 between 2and 4.

4. Name successors and predecess: :s of each num-ber through 99.

5. Determine the order of numbers through 99.

Number Systems, Operations and TheirProperties1. Recognize that there is no largest whole number.2. Use the associative properly of addition in the

set of whole numbers, for example, (3 + 4) + 53 + (4 -i-- 5).

3. Recognize that subtraction is not commutative4. Identify the process of multiplication through

experience with joining several equivalent setsof objects.

5. Discover from the addition table number pat-terns through the sum 18.

6. Recognize zero as the identity element for addi-tion in the set of whole numbers and its specialrole in subtraction.

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Behavioral ObjectivesThird Grade

Sets arid Numbers1. Determine the cardinaIity of a set to 10,000

through appropriate experkuees.2. Use ordinal numbers beyond tenth.

3. Identify 2/3 an 1 5/4 of a whole.4. Show 2/4 = 1/2, etc., by the use of physical

objects or pictures.

Numeration Systems1. Read and write any numeral to 10,000.2. Write many symbols for the same number, such

as 7 + 5, 4 X 3, 10 + 2, and 12 for twelve.3. Interpret place value to 10,000.

4. Write four-digit riamerals in expanded notation;for example, 4567 = 4000 -;- 500 + 60 + 7.

5. Recognize that numerals such as 57 can be ex-pressed as 40 + 17.

6. Read and write roman numerals through X (10).

Order1. Determine betweenness, greater than, or less

than for numbers through 999.

2. Recognize greater than or less than for the frac-tions 1/4, 1/3, 1/2 with physical objects. (Notethat fractions is being used here for rationalnumbers.)

Number Systems, Operations and theirProperties 4.

1. Use the commutative and associative propertiesof multiplication in the set of whole numbers; 5.

for example, 4 X 3= 3 X 4 and (4 X 3) X 2=-. 4 (3 X 2).

2. Recognize the role of 1 as the identity elem....,for multiplication in the set of whole numbers. 6.

3. Recognize the distributive property of multipli-cation over addition in the set of whole numbers;

1'

for exampl^, 3 X (2 4- 4) = (3 X 2) + (3 X 4).Discover number patterns from the addition andmultiplication tables.Demonstrate the concept of division by partition-ing a set into several equivalent subsets; for ex-arrmle, separate a set of 12 objects into sets of3 objects.Demonstrate with sets of objects the relationshipbetween such sentences as 4 X 7 =---= 28, 28 ÷4= 7, and 28= 7 4.

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Behavioral ObjectivesFourth Grade

Sets and Numbers1. Determine the factors of a counting number (a

whole number other than zero).

2. Determine common factors of two counting num-bers.

Numeration Systems1. Read and write numerals as needed.

2. Interpret place. value for large numbers.3. Use roman numerals through XXV.

Order*

Number Systems, Operations and TheirProperties1. Recognize the inverse relation between addition

sentences and two subtractions sentences, such as725 + 342 = 1067 and 1067 725 312

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and 1067 342 = 725.2. Recognize the special role of 1 as a divisor.3. Use parentheses to show order of operation; for

example, 2 -4- 4 X 3 = 2 + (4 X 3) = 14and (2 + 4) X 3 = 6 X 3 = 18.

Behavioral ObjectivesFifth Grade

Sets and NumbersI. Identify prime numbers such as 2, 3, 5, 7, II,

13, 17, . . . .

2, Find the prime factors of numbers through 100.3. Identify composite numbers.4. Determine the least common multiple of two

counting numbers.5. Determine the greatest common factor of two

counting numbers.6. Construct Nets of equivalent fractious through

working with sets of objects. An example ofsuch a set is 2/3, 4/6, 6/9, 8/12 . .

Numeration Systems1. Write many names for the same rational number.2. Work with bases, such as 3, 4, 5, 6, and 7, to

demonstrate an understanding of the base of anumeration system.

3. Use simple exponents such as 10P = 100, 32 = 9and express 300 as 3 X 102.

4. Read and write simple decimals,5. Read and v:rite roman numerals.

Order1. Determine greater than, less than, and between-

ness for rational numbers.

Number Systems, Operations and TheirProperties1. Recognize the set of positive rational numbers 4.

(fractions) as an extension of the set of wholenumbers.

2. Recognize that zero is the identity element for 5.addition in the set of positive rational numbersas well as in the set of whole numbers. 6.

3. Recognize that subtraction is not always pus-

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,16

sible in the set of positive rational numbers andin the set of whole numbers.Use the commutative and associative propertiesfor addition in the set of positive rational num-bers.Recognize that there is no smallest positive ra-tional nu 1mier.Add, subtract, and multiply, simple rational num-bers by use of physical objects, diagrams, etc.

Behavior& ObjectivesSixth Grade

Sets acid Numbers1. Use negative numbers in many different situa-

tions.

Numeration Systems1. Represent rational numl,ers by decimals and

fractions.2. Express large numbers by using scientific nota-

tion, such as the distance from earth to the sunas 9.3 X 107 miles.

:3. Use exponential notation in representing num-bers; for example, 2345 2 X 103 + 3 X 102+ 4 x 10 + 5.

4. Demonstrate an understanding of the relation-ship between decimals and common fractions.

Order1. Determine greater than less than, and be;-ween-

ness for (positive, negative, and zero) integers.

Number Systems, Operations and TheirProperties

1. Recognize that 1/1 or 1 is an identity elementfor multiplication in the set of rational numbers.

2. Recognize the multiplicative inverse (recipro-cal) for every positive rational number exceptzero and use it in the division of rational nuin-bers. For example, 1/2 ÷ 3/4 = 1/2 X 4/ 3.

3. Recognize that the operation of division is theinverse of multiplication in the set of positiverational numbers. For example, the sentences3/4 X 2/3 = 1/2, 1/2 -÷ 3/4 = 2/3, and1/2 2/3 = 3/4 have this relationship.

4. Recognize that there is no soallest or largestrational number between two p sitive integers.

5. Acct. oize that the integers (positive and nega-tive hole numbers and zero) are an extensionof the whole numbers.

8. Find the additive inverse (opposite) for eachinteger by using the number line.

7. eecognize that the rational numbers (positive:an negative whole numbers, pos tive and neg-ative fractions, and zero) are an extension of theinteger

8. Recogri.1 that finding an integn11 power of anumber in -elves repeated multiplication of thesame numbc. For example, (2/3)3 = 2/3 X2/3 x 2/3. \,

9. Use the commut tire and associative propertiesm iltiplication hr rational numbers.

10. Use Ihe distributive Property of multiplicationwith rcspe-t to addit, m of rational numbers.

11. Use the commutative p.operty of r ddition forintegers.

12. Recognize that the rationa, number system isdense; 11 at is, between each two different rationa' numbers, there is a ratie.1;i1 number.

14

17

Mathematical ConceptsArithmetic

Ratio and ProportionA ratio is a pair of numbers used to compare

quantities or to express r. rate.Symbols comroonly used for a ratio are (a,b),

a:b, and a/b. Note that the symbol (a,b) is also usedto denote the coordinates of a point in a plane andthat the symbol a/b is generally used to name a

Concepts

rational number, when a and b are integers andb is not zero.

A proportion is a statement that two ratios areequivalent (that two pairs of numbers express thesame rate). A proportion is written in the form a/b=e /d, when a x d c b.

Students should be able to

ComputationA certain amount of proficiency in the use of the

various algorithms of arithmetic is necessary. Thegrade placement used for these computational skillsin this guide is only an approximation. The expected

use and recall of certain addition, subtraction, mul-tiplication, and division facts for the various gradelevels are listed under the behavioral objectives.

Students should he able to

Geometric ConceptsSize and Shape

The classification of and distinguishing chant:-t,:,risties of simple two and three dimensional figuresare determined by their size and shape. Informal

exarniluition of the size and shape of geometricfigures develops an awareness of geometric patterns.


Sets of PointsSpace. can be considered as a set of points. Intui-

tively, a point represents and is represented by aposition or location in space. Lines and planes ..re

subsets of space. Each line is a set of points of spaceand each plane is a set of points of space. Thesesubsets and man7 of their properties should be in-tuitively presented.

Students should he able to:

SymmetryMany geometric figures and designs have a lOnd

of balance called symmetry. If a figure can befolded so :hat corresponding rails coincide, then it

hal a line of symmetry. A figure has a center ofsymmetry if for every point of the figure there is asecond point such that a line segment joining themis bisected by this center.

Stridents should be able to:

CongruenceIntuiiively two geometric figures are congruent

if they "fit" each other exactly, that is if thry havethe same size and shape. More precisely, two setsof points are congruent if there is a one-to-one cor-

17

respondence between them which preserves dis-tances; that is, if two points of one set are one inchapart, the corresponding points of a congruent setare also one inch apart.



Ratio and Proportion

Computation

Size and Shape1. Recognize squares, rectangles, circles, and tri-

angles.

2. Use the terms inside., outside, and on as relatedto these figures.

Sets of Points

Symmetry

Congruence

Is


Behavioral ObjectivesFirst Grade

Computation1. Use the addition fact:; through the sum of 10

and the corresponding subtraction facts.

Size and Shapo jects.1. Observe distinguishing features of spheres, rec- 2. Use the terms round, face, edge, corner, and sur-

tangular prisms (boxes), cylinders, and other oh- face.

Sets of PointsI. Recognize physical rerresentations of points,

line segments, and portions of a plane (flat sur-faces).

2. Recognize that squares, rectangles, triangles, imdcircies are closed curves and tell whether a 1, lintis inside, outside, or on such a ..tire.

Symmetry

Congruence

19

20



Computation1. Recall the addition facts through the sum of 18

and the corresponding subtraction facts.2. Use the vertical algorithm in the addition of

three addends with one-place n metals; for ex-ampld:

4

-i-5

3. Use the multiplication facts through the product18.

Size and Shape*

Sets of Points 3.

1. Recognize a point as a position and a line seg-ment or curve as a set of poirts. 4.

2. Recognize a straight line as a set of points with 5.

no beginning and no end.

Recognize a simple curve (in a plane) as onethat does riot cross itself.Recognize closed and open simple curves.Recognize the inside and outside of simple closedcurves.

Symmetry

Congruence 2. Recognize congruent segments as segments has1. Recognize congruent, plane figures as figures ing the same le,iglh.

which fit on one another.

20

Behavioral ObjectivesThird Grade

Ratio and ProportionI. Interpret simple ratio situations, such as 2 angles

for 15r, written 2 (apples)15 (cents).

2. Recognize that ratios such as 2 (pencils) and5 (cents)

4 (pencils) arc equivalent ratios (represent theJO (cents)

same rate).

Computation :3,

1. Use the multiplication facts with productsthrough 45 and the corresponding division facts.

2. Use the vertical algorithm in addition and sub- 4.traction with two- and three-place numerals .5.

when iegrouping may be necessary.

Use the ver"-.:al algorithm to carry out multipli-ci4;on with multipliers less than 10 when re-grouping 'nay be involved..Nlultiply mentally by 10 and by 100.Estimate the sum of two numbers. For example.287 + 520 is apprexiiiiately 300 'j- 500 or 800.

Size and Shape*

Sets cu Points1. Recognize rays and angles.2. Recognize that there is only one line through

two poin's and that tut' lines can intersect atonly one point.

3. noeogijize (hat many lines nay pass through apoint.

Symmetry1. Recognize symmetry with respect to a line by

lolling paper containing symmetrical fig glites

such as

axes of SylnInet

along their xcrtical

Congruence1. Recognize congruent angles.2. Recognize that a ,octangular sheet of paper can

21

224 ,

he divided into tN,VO or snore conl,i;r 'ent partsthrough folding.

BehavioralFourth

Ratio and Proportion1 . Make op sets of equivalent ratios for given physi-

cal situations, such as 1/2, 2/4, 3/6, 4/8, . . . .

2. Determine if two ratios are eipiivalent by usingthe property of proportions commonly calledcross mulliplicalion. For example, 3/4 =. 9/12because 3 X 12 = 4 X 9, whereas 6/7 7/8because 6 X 8 7 X 7.

ObjectivesGrade

3, Find the missing whole number in two equiva-lent ratios like 2/3 = /9 or 5/0 = 25/70.

4. Use equivalent ratios to convert units of meas-ure, such as (gallons) 1 (gallons)

(pints) 8 Li (pints)to find how many 'hilts there iue in 3 gallons.

Computation1. Recall the multiplication facts through 10 x 10.2. Do column addithm with several four-place or

five-place addends.:3. Subtract using three-place numerals and four-

place numerals.4. Multiply a number by multiples of 10, by multi-

ples of 100.

5, U: a the multiplication algorithm with two-placemultipliers.

6. Use the subtractive division algorithm with two-place divisors ending in 1, 2, 3, 4.

7. Estimate the product of two numbers and thequotient of two numbers. For example, 21 X 88is approximately 20 X 80 or 1800 and 795 --:- 2;3

is approximately 800 -÷ 20 or 40.

Size and Shape1. Recognize isosceles and equilateral triangles and

parallelograms.

Sets of Points1. Interpret space as the set of all points.2. Recognize a plane as flat .:urface which con-

tains lines and points.3. Describe lines as intersections of planes.

4. Interpret a circle as the set of all points in aplane that are at the same distance from a fixedpoint.

5. Recognize parallel lines as lines in a plane whichdo not intersect.

SymmetryI. Recognize that some figures have two or more

axes of symmetry through 'doper folding. Forexainple, two axes of yininc,ry ;Lie indicated forthe rectangle :hewn at right.

Congruence1. Recognize congruent augies.

22

23

axes of sYmtnetrY%,

r

Behavioral ObjectivesFilth Grade

Ratio and Proportion1. Use the ideas of ratio and equisalent ratio with

problems that include fractions as terms. Forexample, find the missing number in 2/3720. 5

2. Find the missing term in a proportion such as2/5 = D/9 by using the cress multiplicationproperty and find the solution of the mathe-matical sentence 18 = 5 X E.

:3. Use members of sets of equivalent ratios ssitlithe same first term or the same second term tocompare different ratios. For Pxamp1.2, to corn.pare 5/9 and 3/4, show 20/36 < 27/35, andto compare 5/11 and 3/7, show 15/33 > 15/35.Also, 5/9 < 3/4 because 5 X 4 < 9 X 3. Like-wise 5/1J > 3/7 because X 7 > 11 X 3.

Computation 4.1. Add and subtract rational numbers.2. Use the multiplication algorithm with numerals 5.

up to four places.3. Use the subtractive division ahr,orithin.

Express the quctient of integers as a mixed nu-meral; for example, 24 5 = 4 4/5.Find many -ays to express a rati.mal niunber,fur example, 2/3 .= 2 x 4 = 8/12 and

3 x. 48 48/12

12.= 2/3.

Size and Shape1. Recognize common polyhedra, such as a tetra-

hedron, a cube rectangular prism.2. Identify faces, edges, vertices, and diagom.!:-. of

ec mmon polyhedra.

3. Use Eulcr's formula, namely, V + F E + 2Nacre V is the number of vertices, F is the num-ber of faces, and E is the number of edges ofany polyhedron.

Sets of Points1. Recognize acute, right, and obtuse angles.2. Recognize parallel planes.3. liceogaizc perpendicular lines.

4. Recognize the radius and diameter of a circ12.5. Reco-gnize that a plane is determined by three

points not all on one line.

SymmetryI. Recognize symmetry "h respect to a point by

folding a paper along a line through the center ofsuch geometric figures as a circle and a square.

Congruence1. Recognize that triangles arc congruent if (1u--

responding sides are congruent and correspond-ing angles are congruent.

03

4

Behavioral ObjectivesSixth Grade

Ratio and Proportion1. Interpret percent as a ratio in which the second

number is always 100.2. Solve all three cases of percentage problems as

problems in which they find the missing term cftwo equivalent ratios. For example, 20% of 30 and:20/100 = 0/30; 30 is what per,cent of 55 and:0/100 =- 30/55; 25 is 40:0 of what number and:40/100 = 25/0.

3. Use equivalent ratios to convert fractions todecimals and conversely; for example, to write3/5 as hundredths, solve for n in 3/5 = n/100;to write 44 hundredths as 25ths, solve for n ino/25 = 44/100.

4. Solve ratio problems where some or all oc theterms of the ratios Pre written as decimal.;.

5. Use proportions in problems about the lengthsof sides of similar triangles.

Computation1. Multiply and divide rational numbers.2. Use the conve.itional division algorithm.3. Add in

Size and Shape'

Sets of PointsL Recognize the properties of isosceles triangles,

equilateral triangles, and scalene triangles, suetas the fact that the longest side of a triangle isopposite the angle of grcate5t measure. 3.

Recognize that a line (one-dimensional space)is a subset ui a plane (two-dimensional space)and that both are subsets of space (three- dimen-sional space).Recognize the relationship between the circum-ference and the diameter of a circle.

Syrnmatry1. Recognize the reflection of a prone figure in a

mirror and draw diagrams such as the figureat right.

Congrience*

24

25

Mathematical ConceptsGeometric; Concepts

SimilarityTwo geometric figures that have the sante sh ipe,

though not necessarily the same size, are said to besimilar. The measures of corresponding dimensionsof these figures will, thus, lw.ve the same ratio, Ideas

of similarity are needed for di? interpretation anddrawing of building plans, road maps, and scalemodels.

Stud:3'ms should he able to:

Coordinate Systems and GraphsA line is called a number line if a eve-to-one cor-

respondence exists betW2CP a given set of numbersand a subset of points on the'. line and if, by meansof this correspondence, the points are kept in thesame order as their cm-responding, numbers. The

number corresponding to a point is the coordinateof that poin:. The idea of assigning nunli;,,s topoint:: can be extended to points in a plane, that is,a one-to-one correspondence between ordered p;..irsof numbers and points in a plane.


ConstructionStudents can gain insight to geometry- by rising

a variety of materials and instruments to construct

models of geometric figures. Some useful materialsand instruments arc paper (for folding), NN ire, string,coins, riders, compasses, and protractors.


MeasurementThe process of rneastliMg associates a number

with a property of an object. Measuring an objectis done either directly or indirectly. In direct meas-urement the number assigned to an object is deter-mined by its direct comparison to a selected unitof measure of the same nature as the object beingmeasurec'. When a measuring instrument cannot beapplied directly to tha object to be measured, in-direct measurement is employed.

Measure of physical objects is approximate.The accuracy of the measure obtained is restri,:tedby the uaevenness of the object in asured, by the;imitations of the measuring instrument used andby human inabilities. The object to be measuredmust be measured by any unit wall the same char-acteristics: a tmit segment to measure segimmts; aunit angle to measure angles! a unit closed regionto measure closed regions; and a unit solid to meas-ure solids.


Malhematoal SentencesMathematical Sentences

Mathematical sentences and ordinary linguisticsentences have the following common characteris-tics:

1. Both types of sr ntcnees use words (symbols)to communicate ideas.

2. The construction of both sentences follows apredetermined set of rules.

3. Both sentences are true statements or falsestatements depending on the words or symbols

used and the contests in which they are used.M Ahematieal sentences like 2 4- 3 = 5 (an equation) and 3 + 4 < 6 can inequality) are eithert ue or false. Titus the first sentence is true whereasthe second one is false. Maihematjr-al sentences like11 3 7 and 12 N > 5 are tro.., for somereplacements of and N and are false for others.If the 5ct of posit iv e integers is used as replace-ments for and N in the above sentences, thenthe respective solution set_ are {4 and I. 2, 3, 4,5.

27

?'6

Similarity


Coordinate Systems and Graphs

Construction

Measurement of time (day, week, month, year).1. Recognize pennies, nickels, dimes. 3. Use appropriately such words as longer, shorter,2. Make comparisons in time and count whole units heavier, lighter, higher, lower, larger, smaller.

Mathematical Sentences

28

Behavioral ObjectivesFirst Grade

Similarity

Coordinate Systems and Graphs1. Use the number line to illustrate addition and

subtraction problems.

Construction

Measurement 3.1. Determine which of two line segments is the 4.

longer or the shorter, or whether they are thesame length.

2. Recognize the comparative value of coins (pen-nies, nickels, dimes) and use them in makingchange.

Tell time to the 1.,2arest half hour.Identify various instruments of measurement oftime, temperature, weight, and length, such asclocks, thermometers, scales, rulers.Use non-standard units of linear measure andliquid measure, such as a pencil or book forlength, and a paper cup for liquid measure.

lathematical Ser fences1. Write an appropriate mathematical sentence like

3 + 4 = or 5 2 =--_-- for a physical situ-ation or a story problem where the "action" ofthe problem suggests the operation of additionor subtraction.

29

28

2. Find th- "solution" for sentences like 3 -4- =Cl and 5 2

3. Find solutions for sentences like + i 7

in which many correct solutions are possible.4. Make up a problem situation to fit a given

mathematical sentence involving addition or sub-traction,


Similarity

Coordinate Systems and Graphs1. Use the number line to illustrate counting by

fives and by tens.

Construction

Measurement1. Make a ruler with divisions showing half units.2. Use standard units to the nearest whole unit for

linear measure (inches and feet), for weight(pounds), and for liquid measure (pints andquarts).

3. Identify proper instraments for measuring dif-ferent objects.

4. Tell time to the nearest quarter hour.5. Mahe change correctly for quantities up to 25c'.

Mathematical Sentences1. Use sentences like 5 + =-- 12, + 6 8,

12 = 8, and 5 = 6 to representphysical situations and find solutions for the sen-tences.

2. Make up story problems to fit sentences likethose shown above.

3. Find solutions for sentences like 3 + 2 == 8El, 4- 5 =-- 8 4- 7, 8 + (1) < 12, and4 + 9 > + 5, with the aid of sets of ob-

jects of the number line.4. Place the correct symbol (+ or ) in the place-

holder in sentences like 13 A 5 = 8 and 43 < 5.

5. Use se itences like 27 == 20 + 7 and 43 = 30+ 13 ;o indicate regrouping or the use of dii.fercnt iymbols for the same number.

8. Use equivalent sentences like 3 -4- == 7 and7 = 3 to show subtraction as the inverseof add lion.

Behavioral ObjectivesThlai Grade

Similarity1. Recognize that figures are similar if chey have

the same shape. For example, all squares aresimilar.

Coordinate Systems and Graphs1. Recognize that a point on a line can be described

by a number (coordinate).2. Recognize that different (uniform) scales can be

applied to the same line.3. Use the number line to illustrate multiplication

problems.

Construction

Measurement1. Use common standard units such a... inches, feet,

yards, in determining the measure of a distance.2. Use stk-ndard units of measure, such as cups,

gallons, ounces, in determining capacity andweight.

.3. Find the perimeter of a rectangle or parallelo-gram.

4. Make change for any purchase up to 81.60.

Mathematical Sentences1. Use sentences like 3 X 5 = . X 7 = 14,

and 4 : 12 to represent physical situa-tions and find solutions for the sentences.

2. Make r -) story problems to fit sentences likethose shown above.

.3. Place the correct symbol (<, >, ) in theplace hclder in sentences such as 3 X 5 117 7 + 8,25 + 87 28, and 39 0 5 X 7.

4. Demonstrate understanding of grouping and re-

31

30

grouping by completing sentences such as 458= + 50 + 8 and 394 = 3 hundreds +tens 4 ones, by means of tally boxes or otherdevices.

5. Find solutions for sentences like + 239 =239 + and 1987 + ( + 548) = (1987 +L1) -f- 5-13 to generalize the idea of the cumin li-tative and associative p,9perties for addition.

n. Use many different kinds of placeholders like. A. N, X in ma,hetnatical sentences.

Behavioral ObjectivesFourth Grade

Similarity1. Recognize that all congruent figures are similar,

but not all similar figures are congruent.

Coordinate Systems and Graphs1. Recognize that a line segment is a set of points.2. Recognize that points in a plane (the first quad-

rant) can be represented by (ordered) pairs of

numbers (coordinates).3. Use the number line to illustrate division prob-

lems.

Construction1. Demonstrate through paper folding an under-

standing of a line as an intersection of twoplanes.

2. Reproduce a line segment by using a compassand straightedge.

3. Bisect a line segment by using a compass andstraightedge.

Measurement1. Compare measures such as: 25 inches and 2 fret;

31 ounces and 2 pounds; 75 seconds and 1 min-ute; and 15 pints and 2 gallons.

2. Express different names for the same measure.

3. Measure perimeters of triangles and quadrilater-als.

4. Find areas of simple regions informally. Fur ex-ample, a rectangular region with dimensions 2"by 3" can be covered by six one-inch squares(regions).

Mathematical Sentences1. Use sentences like 36 4- 4 end X 3

= 12 to represent physical situations and findsolutions for the sentences.

2. Ma ke up story problems to fit sentences like 6.

those above. 7.

.3. Use sentences like X 5 = 45 and 45 .5

to show division as the inverse of multi- S.

Idication.4. Find solutions for sentences like 723 X []

0 X 723 and (C) X 176) X 19 =[] X(176 9.X 19) to generalize the idea of the commutativeand associative properties of multiplication.

5. Find solutions for sentences like 3 X 13 (32

X 10) + ( X 3), X A ==. (5 X 7) + (5X 8), and 0 X 16 = (r] x 10) + ( X 6)to generalize the distributive pn yerty of multi-plication over addition.Find solutions for sentences like X A 36.Recognize that 3 X = 7 has no whole 'Min-im solution.Find solutions for mathematical sentences in-volving more than one operation such as (2 X 51+ 4 = 0 and (3 X 2) + 0 = 10,\lake up problem situations to fit mathematicalsentences involving more than one operation.For example, make up a story to fit the sentence(3 X 4) -1- 2 =

Behavioral ObjectivesFifth Grade

Similarity1. Recognize the similarity of mops made with

different scales.

Coordinate Systems end GraphsI. Construct simple picture, bar, and line graphs2. Use the number line to represent positive ra-

tional integers.3. Use the number line to represent negative inte-

gers.

Construction1. Demonstrate an understanding of various poly-

hedra by making appropriate paper models.2. Bisect an angle. (Students may discover several

different constructions.):3. Reconstruct an angle and a triangle by using a

compass an,l a straightedge.

MeasurementI. Measure an angle by using a protractor.2. Demonstrate that the sum of the measures of the

angles of a triangle is ISO' by tearing off andmatching corners of a triangular piece of paper.

3. Find the area of a plane region, such as a rec-tangolar region.

4. Recognize informal concepts of volume; for es-ample, a box with dimensions 2" by :3" ay -I"contains 24 one inch cubes.

5. Recognize that a right angle has the measure90'.

6. Estimate distances to the nearest unit.7. Recognize that all measurement involves approx-

imation.

Mathematical SentencesI. Use all of the previously introduced sentence

forms with fractions.2. Write sentences using fractions to represent

physical situations.

33

3t

3. tire previously described sentence forms to gen-eralize the commutative property of addition,the associative property of addition, and the dis-tributive property of multiplication over addi-tion for rational numbers in fractional form.

Behavioral ObjectivesSixth Grade

Similarity*

CoordinP*1 Systems and Graphs*

Construction1. Construct a line perpendicular to a given line.2. Construct parallel lines,

3\ hike models of various prisms and find theirsurface areas.

Measurement1. Find the volume of a rectangular I rism.2. Estimate and compare perimeters of polygons,

such as rectangles, triangles, and parallelograms.:3. Estimate the area of an irregular plane region

by use of a grid where an approximation to thearea is the average of the inner and gutter areas.

4. Use formulas for the areas of rectangles, paral-lelograms, ann triangles.

5. Use the formula for the circumference of a circle.

6. Use the metric system of measure for length,7. Use formulas of volume for common solids,S. Work with approximate numbers. For example,

know that the area of a square whose sides meas-ure (3.5 and 3.6 inches to the nearest tenth of aninch has an area between 6.4 X 3.5 and 6.6X :3.7 square inches.

9. Solve problems involving the measurement ut in-accessible he'ghts and distances indirectly byusing the properties of similar triangles.

Mathematical Sentences1. Use all of the previously introduced sentence

forms with decimal numerals.2. Write sentences using decimal numerals to rep-

resent physic-al situations.

31

.3. Use previously described sentence forms to gen-craliie the commuhitive props rty of multiplica-tion, the associative property Of multiplication.;Ind the distributive propel Is of nnultidlicatiolluser ;Rldition for rational immilwrs in ;my form.

r

to

Adjusting Mathematics Programs to Student Abilities

Preliminary ConsiderationsFht fact that individual diflt.t.cnt.rs do e5ist among

stmlei.ts is not wily one of the most cooNistentldociimeited findings of echication.CI research. butalso one charactelistie of ;my .grolip of childrui,ok loos to evety teacher. cpericiiced 111'

this ( hosNfOt)t1i rcdlit fairlythat edncat ii, (lo something thotit it Teachers Icedgreat plessiire to adjust the program of instrocti'li,so it caresptimls to the of theirstorlents. This lid is especially true ni mallienhiticsinsirnction (\ here the range cncknintcrctiis e.traurdinarily large. l'nf -tnnalnly. the tecli-nique, fur gedrhig ti indiviclo.t1 differ-I !ices ;ire IPA as Oh% LIM.: or rt.,Rfil> .1, is

tI 11,,,k1 for it. It is `air to say th.it the prolilivi IIIiristro(tion is one of the 11111 or

Si IvcI l)ruloicrns in ("Itcnif)"htlY ec;nc,iliot, fig

or(lci to nuclerstaml this sitoalion. it is 11(1.4:Ns:Ir% ta

csairiirre }A. kinds of indkithral (I'lfcia.tices that ranrnuast talc int() account in :ittctripting h, solve II,

1); to (liffeletic k'N in 1(5(Cal achievemc nt. other kinds inch\ idnal dill,("ices. such as those of mental:lg.'. 11c1i1 s.

it's, talent, ancl voc.tlinni I

.ispirations. eSist "I hese (lilleren.i.li,ie .1 tic finite tilt, tifectiveness ol iiilrnctiun..1ltttong }t each of those I.ccters ina

iini)oll,not consider dim, I the construction of 1,.

tri;ction.t1 programs. the( .I4' III, 11 till] i,PV% 1111 h represent only une dimension It flit. pi 111,14 in

/)iffeti. 11(e, M 51'1101115, in sl l 4)141 Ne% stems. in 111111

111,1..!1111 in h.1.14cIs Inl,st irrt,r ralr,ith rah

in planning any serious program of individualizedinstruction.

Teachers, as a group, differ Ili probably as manyways as do their pnpils. From the standpoint ofmathematics instruction, the main differences inteachers that need to be considered are those ofsubject matter interest and knowledge. In mathe-matics instruction, the teacher's attitude toward thesubject may be as influential in determining hiseffectiveness as his subject matter knowledge. Inany case, both are important, and no effective pro-gram of mathematics instruction is possible withserious deficiencies of this type.

Individual differences in schools and curriculaform yet a third dimension of the problem. Schoolsand school systems are as different as the peoplewho comprise them. There are inner city schoolsand suburban schools, each with distinct problem.,.The curriculum and philosophy of education ofthese schools also vary greatly. Sonic systems havemulti-tracked curricula; others have single-tracked.Although most schools establish a common pace andinstructional content for all pupils, some schoolsexperiment with individualized pac: ig and evencontent selection. Some schools tend to rely on theself-contained classroom taught by our teacher.Others use semi-departmentalized concepts of 0:-ganization. Non-graded and team teaching patternsof organization are becoming increasingly popular.Increased utilization of such devices as closed cir-cuit TV, edneatinoal network TV, pi ogrammed in-structing, and even computer-assisted instruction isevident. Although, the influence and effects of thesevariables are often subtle and difficult to analyze,they are real and, in fact, may quite possibly be-come the controlling factors in edorts to individual-ize mathematics instruction.Classroom Management and Grouping Devices

A primary obstacle to individualized instructionis related to problems of elassrooin management.When cliilchcn arc all lock-stepped in the curricu-linn, a condition of minimal administrative stress isestablished. It is easy to lecture the group en masse,to test thou, to sense who is competent and NV110 is

not, and so on. But as soon as the lock step isbroken, the teacher insist try to keep track of whateach' individual in the class is doing. With differentpupils doing different work, the conventional baseof comparison is lost, and evaluation of the pupils'progress becomes increasingly complicated. Finally.the demands of a classroom of individuals for indi.

vi 'mid attention can easily exceed the limits ofphysical en of any teacher.

To cope with these :na n a gemenf- problems, asari hy of organizational schemes has been utilized.One of the simplest is permanent intra-class groups.This technique customarily involves the use of anability criterion to separate tne class into, say, theA, B, and C sections. In theory, the idea is to con-struct a number of nearly homogeneous subgroupsin a heterogeneous class. This compromise pre ,ernesthe class structure within the subgroups. Theteacher can assign work on the basis of the abilityof the group, maintain contact with the sectionsmuch in the same mariner as he would with thewhole class, and continue to evaluate stir:amts. rog-ress in a conventional manner. Sonic problems occurin ranking :mid grading students, of course. Fe., ex-ample, should the best student is the least talentedgroup be given a higher grade theta say the leastable person in the most talented group? Someteachers determine the maximum grade to be givento any one group, e.g., A, B, or C, respectively.However, this grading policy tends to restrict stu-dents to certain performance levels. One apparentreined). is to make the intro -class groups flexible,thereby allowing a student to shift from one groupto another depending on his performance in a givengroup.

Another grouping technique is that of depart-mentalizing the school. In a departmentalizedschool, a single student could be placed in groupsof various ability levels depending on the subjectand li' level of performance in that subject. Thistype of rrangement, though meeting sonic of thedesirable democratic principles of classroom organ-ization, begins to produce considerable stress onthe teacher or the team of teachers. Keeping trackof the prolderns and progress of individuals, pre-servirq good articulation between groups, andsetting fair and consistent criteria for intro -classgroup transition are not easy tasks.

A somewhat Mare involved t eChnigne of grolipingis employed in non-graded schools and in schoolswith inter-class grouping. In these cases, the optionexists of fixing grouping arrangements for a give,'Period of time or allowing them to remain flexible.In a flexible arrangement, a child is placed in aparticular class if In meets certain criteria levels.The simplest criteria to use are those of perform-ance on achievement tests: however, a variety ofsra le and psychometric measures may be

used in addition to purely intellectual parametersor subjective judgment by the teacher. The particu-lar combination of criteria chosen depends on manyfactors; the school its organization and adminis-tration, the teacher or tcasn, the objectives of thecurriculum, the type of children, and so ore

There is no one grouping arrangement that is

hest for at childien, all teachers, all curricula, andall schools. The design used in meeting individualdifferences must be highly specific to each of theseconsiderations. It is a poor strategy to adopt totallyand uncritically someone else's scheme. The chancesof success in such a situation are small.The Instructional Variables

Once a grouping arrangement has been selected,decisions have or be made with respect to the kindof instructional differences one wishes to providefor groups of pupil;. There m'e three basic methodsof adjusting the mathematics curriculum; thesemethods, incidentally, are not mutually exclusive,but are mutually consistent.

The first and most obvious way is that of pacing.In general, one !night expect brighter groups ofstudents to learn faster. This assumption, however,does not always prove to be valid. The considera-tion basic to the notion of pacing is that the child'sbehavior be adaptive. In other words, ideal pacingfor each group of students should vary over timein such a way as to provide optimal progress towardeducational goals. Some children work effectively utrather high rates of speed, whereas othe:; benefitfrom more contemplative rates. In either case, theIlcal rate does vary with time and is not a simplefunction of intelligence or a Aiieveinent. Thus, onewould expect considerable variation among indi-viduals within gronfi.s.

Pacing may be externally controlled by . de-mands of the curriculum or by the teaco.. on theother hand, it may be self-determined by the pupil.The former method in.tintains teacher control andminimizes management problems. Iliivever, it runsthe risk of not producing optimmn learning effectsfur all concerned. Self-pacing, on the other hand,may be effective with students who have some in-sight into their own learning in«unisni and opti-mum pacing. Other students who, for reasons ofmotixation or maturity, are not so gifted may worktoo fast for their own good (or not at all). Recentexperiments with primary grade children have indiGavel that considerable acceleratien is possible forgifted children. Even in such a select group. enor-

:37

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mous differences in learning periods are encoun-tered. \Vhether acceleration is desirable or not isa decision that each school and its staff must make.If acceleration is encouraged, a school should takeinto consideration articulation problems from gradelevel to grade level, if these exist, or from school toschool. Care most be exercised in designing the longrange components of an individualized program.

A second instructional variable that can be ad-justed to the individual's needs is that of level ofdifficulty. In some mathematics textbooks, exerciseshave been arranged according to presumed levelsof difficulty for this purpose. A problem that oftenoccurs when this method is used by schools is thatthe factors determining difFeculty are poorly tinder-stood, In addition, students are often pressured intoworking toward "top-level" groups by their owndrives or by their parents for reasons not alwaysconsistent with efferti.e learning.

The third instructional variable, closely related tovariations in pacing and in level of difficulty, is

enrichment activities. if a child works effectively ata high speed and completes even difficult exerciseswith dispatch, he is often led to study other macesof mathematics not normally a part of the standardcurriculum. Enrichment activities have been usedmore frequently in recent years, not only for themathematically' talented, but also, to some degae,for students anywhere on the spectrum of individualdifferences.

Ail important infiiiene on enridiment activitiesin many elementary classes has been the "arithmeticcorner" or "arithmetic learning center." Many ele-mentary teadiers have found that individual differ-ent 's and needs arc' detected and best met if a richVariety of instructional materials is available to indi-vidual learners. Learning centers facilitate enrich-ment activities when they include an abundance ofsuch objects as counters, counting frames, measur-ing instruments, place value charts, geometricobjects, and number lines,

In A broad sense, the adivilment of these instrinRonal variables presents the possibility of anindividualized cuniculum for intra-class groups.inter-class groups, and, in the extreme, for single in-dividi.als. Little is known at this tine about hovesuch a program could be administered or, in fact,whether children could be expected to select topicsfor study as they are expected to pad' themselves orto select fitting levels of difficulty at which to work.However, trends in mathematics edhcation are

breaking down the uniformity, arid conformity whichcharacterized traditional programs. Hand-in-handwith this development has come the necessity of re-thinking the nature and objectives of the mathe-matics curriculum, Experiments with individualizedcurricula vill hopefully yield a theory of individual-ization which will subsume the special cases of en-richment and acceleration.Summary

The adjustment of mathematics programs to indi-vidual abilities is a challenging task. Success re-quires the effective cooperation of all concernedwith the instructional task. The school must meetits responsibilities with imaginative administration,especially in developing a well-integrated and artic-ulated curriculum, in providing assistance in class-ref-n management, in utilizing teachers effectively,and in devising grouping schemes. Teachers ofmathematics must meet their responsibilities by im-proving their suli'cc'. matter competence so as tobe more capable of evaluating pupil Progress, of

38

assigning students to groups, of advising studentsen enrichrneot materials, and of monitoring effec-tive working speeds of pupils.

School systems should do their best to provideOle waterhd assistance necessary for the program.Individualization may he fostered through the useof adequately supplied mathematics learning cen-ters in each classroom. Audio-visual aids are fre-quently useful in programs desigr ed to meet indi-vidual differences as is the use or instructional TVor closed circuit TV. Ii some cases programmed in-structional materials may be found beneficial. Goodlibrary resources arc necessary, both for enrichmentactivities and for accelerated students. In large-scale programs it may be necessary to turn to mod-ern technology for assistance. Efforts have becamade in the last few years to provide computerassistance for the classroom management aspects ofprograms that are adjusted to abilities of in lividualstudents.

is

r.

Problem Solving

A situation becomes a problem for a student onlyif the following conditions exist:

I. A question is posed (either implicitly by thematerials being studied or explicitly by thestudent) for which an answer is not immedi-ately available.

2. The student is sufficiently interested to feelsome intrinsic need to find a solution.

3. The student has enough confidence in himselfto believe a solution is possible.

4. The student is required to use more than im-mediate recall or previously established pat-terns of action to find a method for arrivingat a conclusion.

Not all pupils will consider a given situation aproblem. If a student can see a 'method of solution"immediately, then the situation is not a problem forhim, but simply an exercise or an application ofsome process which he has already mastered. Forexample, i,n exercise such as 3 X 14 is not a prob-lem for a r. rulent who has already mastered theidea of multiplication yid' two pI, e numerals.However, this example could very well be a prob-lem for a student who has studied only a few ele-mentary multiplication combinations. In order forthis student to obtain a correct solution, he wouldhave to think of some meth xl of renaming 14 suchas 10 -4- 4 and fies1 some method of reducing theproblem to si:npler stages with which he couldwork, such as 3 X 14 = 3 X (10 4) = (.3 X 10) +(.3 X 4). Heal problem solving is not recalling a nu-merical fact or fitting a situation into a memorizedpattern, but is discovering one's own method forextending previous !earnings to new situations.

It is not the intent of this guide to consider allof the general aspects of poblern solving. Onlysome of the problem situations that involve the useof known mathematical ideas by students will bediscussed. The verbal problems found in textbooksmake up a large portion of the problem situationsconsidered in mathematics programs. When suchproblems are presented. the mathematical ideas

necessary to solve them are generally known to thestudent, but the contexts in which they are Fe-scnted require the student to find the methods ofsolution or to decide which of the ideas lie alreadyknows will help him find the solution.

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The solution of a typical textbook verbal prob-lem involves a series of four interrelated steps.The complexity of each of the steps varies with theproblem, but the method of solution follows thisgeneral pattern.

Step 1 Recognizing the question being asked.Step 2 Translating the verbal problem into a

mathematical sentence.Step Finding a solution for the mathemati-

cal sentence.Step 4 Analyzing the solution for the sentence

to see if it provides a reasonable solu-tion for the original verbal problem.

Each step of the solution can involve many facets.Although all steps in the solution of a verbal prob-lem are important, Steps 1 and 2, the abstractionor translation phase, can be considered the ''heart"of the process.

The student may have to decide many things be-fore he is able to write the mathematical sentencewhich represents the problem. If the question is notexplicitly stated in the verbal problem, the studentmust formulate his own question or questions. Ilewill have to decide if all of the necessary data forthe solution is given in the statement, and he willhave to select the pertinent data from the giveninformation. Ile will have to decide which mathe-matical operation is suggested by the "action" ofthe problem. Ile must determine the order in whichthe data of the problem will appear in the corresponding sentence. The student must also decidewhat relationships, if any, are involved that willenable him to use previously solved problems as"models" and what approach is the most efficientor easiest for him to use in ;Attaining a correct solu-tion. Steps 1 and 2 could also involve some trialand error activity in which different sentences aretested as to which one best fits the given situation.These steps include problem solving techniquescommonly referred to in many textbooks andteacher guides as "Reading and understanding theProblem" and "Restating the problem in your ownwords."

Step 3 involves finding a solution to the mathe-matical sentence. This procedure can 1w very simplsor very complex depending on the given situation.In general this step involves the application ofknowledge that the student has already attainedthrough practice work with similar sentence forms.

Step 4 is the familiar "check" in which the stu-dent determines if the solution is acceptable for the

given problem situation, The student must decideif the solution is reasonable or "makes sense." If itis apparent that the solution is not correct, the stu-dent must then "rework" his problem, check hiscomputaions, examine his sentence to see if it

actually symbolizes the "story" of the problem, andlook for careless errors or possible misinterpretationsor misrepresentations of the data.

Some examples of typical verbal problems con-sidered It the elementar, school are given below.

In the first grade, simple "picture" or oral prob-lems arc presented which can be represented bysentences like 2 -=. where the sum doesnot exceed ten. The problem might he presentedto the stcdents in this form: "John has 5 toy cars.His mot' er gave him two more toy cars for hisbirthday. How many toy cars does he have now?"The "action" of the stnry is the joining of two setsof toy cars, and the thematical sentence whi:.11represents this problem. is 5 + 2 E. Throughsuch presentations, the student gill learn to assoei-ate the 1 otion of "joining" with the operation ofaddition. lie will thus have a better understandingof what addition means and will develop techniquesfor solving similar probrems.

In the second and third grades, problem situa-tions are extended to include verbal problems rep-resented by sentences like 3 + 8, fil] + 59,7 2 El, 8 5, and E 2 6,

A typical verbal problem at these grade levels i;:"Mary had 15 doll dresses. After site gave sonicthe dresses to her sister, she found that she had Sdresses left. !low many doll dresses did she gi)to her sister?" This problem is represented by thesentence = 8, The "action" of the problemis "taking away "; therefore, the corresponding oper-ation of the sentence must be subtraction. The 15represents the number of dresses Mary had in thebeginning, the represents the number of dressesshe gave to her sister, and the 8 represents thenumber of dresses she had left. Students shook'realize that although 15 -7 8, 8 + 15,

and 15 17] can be represented by differentphysical situations, the computation involved in

solving each of the sentences is the same.Similar "joining" and "taking away" problem situ-

ations can be represented by the other addition andsubtraction sentences mentioned above.

Comparative problem situations can also he de-scribed matheinatically. For example, the prohlcill;"Jack has 45 baseball cards and Tom has 62 cards.

10

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Ifow many more baseball cards does Torn havethan Jack?" can be represented by the scnter62 45 =

In the later elementary grades, problems are pre-sented that involve larger numbers as in sentencessuch as 48 + 327 or 230 76 andthat use the same principles as do the problemsfirst introduced in the primary grades.

In the third and fourth grades, problem situationsrepresented by sentences such as 3 X 7 ,5 X 20, and X 6 = 30 should be intro-duced. A sample problem for this grade level is:"Tom found that he needed sixteen small cartonsto cover the bottom of a packing case. If he neededfour layers of cartons to fill the case, how manysmall cartons were in the cas^?" The "action" of thisproblein is repeated addition or multiplication. Thusthe sentence used to represent the stop.: could be16 + 16 + 16 + 16 CI or 4 X 16 7-7- D Thestudent should be allowed to use either sentence torepresent the problem, but should be led to realizethat the sentence 4 X 16 is the shorter wayof writing a sentence for this type of problem. Inthe latter sentence, the 4 represents the number ofsets, the 16 represents the number of objects in eachset, and the represents the total number of ob-jects in all of the sets.

In the sentence 5 X = 30, the 5 representsthe number of sets, the represents the nomberof objects in each set, and the 30, the number ofobjects in all. Likewise the sentence X 6 = 30represents a problem situation in which the totalnumber of objects (30) is given, the number of ob-jects in each se', (6) is given, and the number ofsets, represented by the D, is the solution to theproblem. The sentences :z X , 30 andX 6 = 30 represent different physical situations,but both types of sentences can be solved by thesame computational method of repeated subtrac-tion if the multiplication fact that makes the sen-tence true is not known.

Problem situations introduced at the fourth andfifth grade levels are similar to the problem: IfJohn separates 12 marbles into small groups with4 marbles in each group, how many groups of mar-bles will he have?" This problem can be representedby the sentence X 1 12, for the questionof the problem can lie restated: "flow many setsof 4 are there in 12?" The problem can also heinterpreted as the division of a set of 12 objectsinto sets of 4 objects. With the latter interpretation,

41

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the problem can be represented by the division sen-tence 12 4 = fa Both sentences, x 4 == 12and 12 + 4 = , represent the same physical situa-tion and implicity ask the same question, "How-many sets of 4 are there in 12?" Both sentences aresolved by repeated subtraction if the number whichmakes the sentences true is not known.

As fractions, decimal numerals, and integers areintroduced, problem situations involving their usein sentence forms similar to those considered pre-viously should be presented.

The verbal "story" type problems are not the onlysource of problem situations for use in the elemen-tary school ma thema ties program. Many othermathematical problems arise from physical situa-tions, from social or mathematical applications ofMathematical ideas, or from situations made up bythe teacher or the pupils, These varied situationscan be real, imagined, or in the nature of a puzzle.

.Many of these problems fall into a category whichmight be regarded as a "higher order" of problemsolving than the typical textbook "story" problem.It is not possible to outline a regular progressionof steps to be followed in solving all such problemsituations as the process will vary with the natureof the problem presented and the ability and ex-perience of the pupil. The four steps outlined forthe solution of verbal problems do not necessarilyapply in general problem solving situations, for itis not always possible to translate such a probleminto a ssathernaticaI sentence.

Ilowever, it is possible to list a few of the activi-ties that are essential in the general problem solv-ing process. These activities include: trans' stingthe probleon into a simpler forin, critically examin-ing the given data, forming hypotheses or conjec-tures, reasoning on a trial and error basis, analyzingor evaluating results on the basis of past experience,and forming generalizations from similar problemsituations, Thes, activities are not the only onesthat may be involved, and not every problem situa-tions will require the student to become engaged inall of the activities listed above. It is important tonote that the order in which the student performsthese activities May vary for different problem situ-ations,

A few examples of "higher order" problems thatcan be included in the elementary mathematics pro-gram are given below. The types of problems pre-sented at various grade levels should vary only inthe complexity of the thinking required, not in the

method of thinking or the kinds of mental activitiesnecessary to solve the problems.Example 1 I have 80 cents in coins in my pocket.

I have the :aloe number of pennies,nickels, and dimes. How mmy coinsof each kind do I have in my pocket?

Answer -- Five coins of each kind.Example 2 Fill in the boxes so that the numbers

in each column, each row, and eachdiagonal have a sum of 18. (Do notuse any number more than once.)

711

3Answer 2 9 7

11 6 1 Other answers are possible.5310

Example 3 Is the point P inside or outside theclosed curve? Tell the class the meth-od you used to find your answer.

Answer The point P is inside the closed curve.If we place another point outside thecurve, we must cross the curve inorder to draw a continuous path be-tween the two points. Also note thatany straight lire connecting P to anoiside point cuts the figure an oddnumber of times, if you count a pointof tangency as two titres.

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4

Example -I Draw two straight lines through theclock face, dividing it :nto throeparts, so that the numbers represent-ed in each of the three rarts havethe rune sum.

Answer

Example 5 If 2, 3, 5, and 7 are examples ofPrime numbers, is 103 thee a primenumber?

Answer 103 is a prime number as it can bedivided evenly only by 1 aur: itself.(1 and 103 ore the only whole rutin-ber factors of 103.)

Example 6 Points A, B, and C all lie en :he samecircle Find the center of the circlethat contains points A, B, t.nd C.

A

.B

C

Answer Construct the perpendicular bisec-tors of *two of the segnr joiningthe three given points. The pointwhere the perpendicular bisectorsnuct is the center.

Example' 7 NNibat number can be divided by IIso that the remainder is zero, butwhen divided by 2, 3, -I, 5, or 6 has esa remainder of 1?

Answer 121

Example 8 -- Roy had four cards with 1/2 writtenon the first card and three other nu-merals written on the other threecards. Hoy claimed that he couldform all of the combinations from1/2 20, counting by 1/2 (that is,1/2, 1, 1 1/2, and so on), by usinga single card or by adding or sub-tracting the numbers represented ontwo or more of the cards. What werethe numerals Roy bad written on theother three cards?

Answer -- 1 1/2, ,4 1/2, 13 1/2.The teacher should provide some problem situa-

tions similar to the examples above in addition tothe typical l'stc,ry" type problems.

Teachers can do much to help students improvetheir problem solving abilities for all types of math-ematical problems. The teacher's function shouldbe ti create a questioning, challenging atmosphere;tG introduce poblem simations; and to guide andencourage students to develop their own problemsolving techniques.

A few suggestions that can help the teacher pro-mute good problem solving techniques are listedbelow.

Tie teacher should present problem situationsthat i.nolve many basic mathematical princi-ples. lie should be certain that these situationsare related to the kinds of experience the pu-pil has already had so that the pupil is ableto apply the principles.

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The teacher should let his students use theirown methods. Nlany problems have no single,best method of solution. Insistence on one"favorite" method of solution often destroysenthusiasm and original thinking.Whenever the problem permits, the teachershould emphasize the writing of a mathemati-cal sentence that shows the "action" of theproblem. If children are to acquire good prob-lem solving habits, they must be able to de-soribe the action of the problem in terms ofmathematical symbols whenever applicable.Writing the mathematical sentence for a prob-lem is as important mathematically as findingthe solution for the sentencP.The teacher show.' encourage his students touse diagrams, estimates, d,amatizations, orother techniques that help them understandthe problem.The teacher should si ggest that students tryvarious approaches to a problem when theyare not certain of the correct method. ,Andentscan evaluate the method used by ascertainingthat die solution is reasonable. Students shouldrealize that the trial and error method can bean effective approach to difficult problems.The teacher should confront students withsome situations in which they must formulatetheir own questions. This type of presentationis similar to the kinds of problems they areapt to face in their future vocations.

Evaluation and Testing

Evaluation is a continuous and integral part ofthe successful elementary school mathematics 'mi-gnon. It has many facets and necessarily involvesall staff niembers who are associated with the teach-int. of mathematics. Certain phases of evaluationmay he objective in nature, but the process must

ic..continodate the many vaine judgments that areboth necessary and desirable. The techniques ofevaluation must he both forma: id informal andmist include procedures for appraising the develop-ment of interests and attitudes. as well as skills andunderstandings.

Evaluating the Mathematics CurriculumThe purposes of evaluating the mathematics cur-

riculum are two-fold, but are very intimately re-lated. The first is to measure the extent to whichthe experience and belnivicq of the children rot!, ctthe content and helms oral objoctises of the math-matics The second is to use Ibis

illation in making adjustments in the instructionalprogram f' it is ill provide for even greater realiza-tion of the objectives. As curricula revised andthe objectives are perhaps modified, the evaluationprocedures must he adjusted so as to measure theinleruled outcomes of the revised eanrieula.

PI:Nisi-n(1 or standaidize-1 ubjccfive tests continue

4,3

to he an important source of illi0fIllati011 ;111011t theeffectiveness of the mathematics program. Testsmust be carefully chosen. They should measureonly those abilities that the curriculum is designedto develop, if the test scores are to have validity.As the objectives of the Mathematics curriculumare broadened, the task of selecting appropriatestandardized tests becomes even more importantand increasingly difficult. The once popular test withmajor emphasis on computational skills is of littlevalue when used alone to evaluate today's mathe-matics curriculum 'with its varied and extendedobjectives. Test scores take on significance only asthey are considered along with other measurementsof the effectis::.ncss of the curriculum.

The majority of standardized tests are of theobjective, short answer type, but the nature of thesetests does not necessarily limit them to measuringonly discrete, factual information. Properly designedobjective tests can measure mathematical insightand originality, as well as levels of learning ofdifficult concepts.

Regardless of how appropriate a standardizes;test may be in evaluating the mathematics corricii-loin, its use is justified only if the results can beemployed in improving instruction. System-wideanalysis of scores is certainly one legitimate wayof using test scores for evaluating and ultimatelyimproving the mathematics instruction. When testsare used for this purpose, the question always arisesas to what standards or norms should be used.Many school systems are less interested in coin, wr-ing their children with other groups than they arein studying the performance of the children withintheir own system over a period of years. In this case.school systems should be encouraged to developlocal norms. The use of a national or regional noneis not necessarily to he discouraged; however, itshould be used only in the light of complete infor-mation with regard to its development.

It is also important that standardized test resultsbe used to improve instruction within each teacher'sclassroom. Certainly these results embody signifi-cant information that will help increase the effec.tiveness of any teacher's instruction. The standard-ized test results may also be of some value in analyz-ing the instructional needs of the individual child.

Whatever use be made of standardized tests.schools mast be carelid that the material in the testdoes not dictate what is tanctlit in the classroom.The pitfall of teaching to the test must lie avoided

it all costs! Sonic of the motivation for L'aelain.; tothe test can be elincimited if everyone recognizesthat a given test, when administered, does notmeasure recent achievement alone. Tests nicaillrechicvement scud understandings developed by a

chi/di over several years. All teachers that a childhas had must share both in his success and in hislack of success as reflected in a given test re: alt.

Few questions associated with the evaluationprocess have definite "yes" or "no" answers, Con-templation of questions such as the following may,however, give sonic insight into the evaluation ofthe mathematics curriculum.

a) Does the curriculum differ markedly fromthat of most school systems?Does the curriculum develop and mainlaininterest in mathematics?

i Is the curriculum flexible enough to meet theneeds of each child?

d) Does the cu rricul Ill encourage creatisity,originality, and mathematical experimenta ionby the children?

c ) Does the curriculum develop adequate ftcil-ity with coniputational skills?

f) Does the curriculum give proper empliasii tothe unifying concepts and structure of mathe-matics?Does the curriculum prep the children to11SL' their mathematical skills and knowledgeoutside the mathem.lics class?

In) Does the curriculum present mathematics is

an enjoyable and fascinatingly alive sulij«t?The process of evaluating the curriculum must

not focus on simply whether the children are lei fil-ing what is being taught. Attenticn must be givento evaluating whether that which is bring taughtis that which should be taught. It must also lie rec-ognized that due to the varied backgrounds millabilities of the children, flexibility in the expect'dminimum levels of achievement is necessary.Evaluation of the Child's MathematicalDevelopment

Although evaluation of the effectiveness of theentire elementary school mathematics curriculum isof prime importance, the evaluation of the learningexperienees and achievement of each child as anindividual is of at least equal significance. in reality,such evaluation is the final test of the effectivenessof a mathematics curriculum in a society whichplaces IFittillate value on the worth of every indi-vidual.

g)

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The evaluation of a child's growth in developingmathematical concepts and abilities, like the evalu-ation of the entire mathematics curriculum, neces-sarily involves a variety of techniques and pro-cedures. This appraisal must include the manyaspects of learning beyond the acquisition of factsand skills. The expanded content and behavioralobjectives of today's mathematics curriculum aremaking the evaluation of each child's progress anincreasingly difficult task. Not only must educatorsknow whether the pupil is succeeding or failing,but they must also strive to ascertain the level ofsuccess he is experiencing with respect to each ofthe several objectives of the mathematics program.

The evaluation of a child's progress must be de-liberate and systematic, yet it must allow for themeasurement of growth (or lack of growth) at anytime when it can be observed. It matters littlewhether the evaluative procedure is the interview,a conversati a or exchange of ideas between teacherand child, a standardized or teacher-made test, orthe observation of the behavior of the child in amathematical situation. The relevance of an evalu-ation is determined by the degree to which it meas-ures the pupil's growth toward realization of specificobjectives of the mathematics curriculum.

With reasonable care and a small amount ofingenuity, the teacher can devise test questions oractivities to evaluate progress toward the realiza-tion of any of the content or behavioral objectivesof the mathematics program. The following areoffered as illustrations of possible evaluative activi-ties.

1. In the lower elementary grades (even at the ',re-number level), the child's appreciation for num-ber properties can be appraised by confrontinghim with two collections of objects and askinghim Nxliether there are as many objects in onecollection as in the other. If the collections ofobjects are so large. that he cannot mentallymatch the objects, his understanding of the roleof one-to-one correspondence between sets withthe same number property can be appraised.

2. The appreciation of the kindergarten child forthe number property of a set can be observedby asking him which of two sets of objects ha.sthe greater cumber of objects, when one set hasa large number of small objects and the otherhas a smaller number of much larger objects.

3. The ability of the second or third grade childto generalize relationships can be evaluated by

-16

reminding him that 5 H- 7 is 12, 15 -l- 7 is 22,and 25 4- 7 is 32, and by then asking himwhat he thinks 85 + 7 equals. If his reactionis immediately "92," as probably made thedesired general,/ . 7 response is slower,or if lie gives evidei. .; counting or actuallyattempts to add the two numbers, he has prob-

ly not been able to make the generalization.4. This example is one way of presenting the sec-

ond or third grade child with an opportunityof showing his understanding of the distribu-tive property. State, perhaps orally, that Johnhas seven marbles and Ted has three. marbles.Then ask how many marbles they would havealtogether if they each had two times as manymarbles. The child who understands that thecorrect answer may be found by computing (2X 7) + (2 x 3) or by computing 2 X (7 -1- 3)has demonstrated his understanding of the dis-tributive law. lie understands the property evenif he is not able to verbalize the relationshipexpressed by it.

5. In the middle grades, evidence of the child'sability to generalize may be obtained in thefollowing way. Show him ti, .t: (12)2 = 144 and(13)2 7= 144 4. 24 + 1 c 169; and that (20)2= 400 and (21)2 400 + 40 -I 1 = 441.Then tell him (40)2 1600 and ask him whatOW equals. A further generalization would 1required for him to know what (39)2 equals.

6. In the middle grades, the feeling fur theassociative property of multiplication can beappraised by observing how he computes (13 X5) X 2. The student who -tinders:ands the asso-ciative property will probably recognize that(13 X 5) X 2 7= 13 X (5 X 21 and will thenhe able to carry out the operation mentally byusing 10 X 13. Such problems when given asmental exercises can I used both to developand to test an understanding of this basic property of multiplication.

7. The ability to translate an English statementinto a mathematical .;cntenee is a vital part ofsolving sonic problems. This aspect of problemsolving can L valuated in the middle gradesby giving the child the following exercise.

Choose the mathematical phrase or sentencethat best expresses the English phrase or sen-tence and write the letter of your rhoice in theblank preceding the F.nglich plias(' or sentence.

five times the number--the number is fivefive less than the number--five more than the number

A. 5 X 5B. H- 5C. 5 X 513. p 5

E. 5 --F. 5 +C. 5 x EH. 5

8, The following i.s an example of the type ofsimple problem situation that can he presentedto children in grades 4 to 6. Provide the child%ith a piece of waxed paper on which a circle hasbeen marked. Then challenge him to form, byfolding the paper, a square that just fits insidethe circle as in the manner slims'. in (a) below.

(a) ( b )/

6

3

5

Tell him that he can fold the paper as manytimes as he desires and in any way. Sketch (b)indicates at least one way in which successivefolds can be made so as to carry out the assign-ment. Observing the child in tie process ofsolving such a problem, even if a simple oneprovides the teacher with real insight into achild's ability to bring a variety of knowledgeto bear on a new or strange situation.

9. The following type of performance exercisecould be used in grade five or six to assess thechild's appreciation for the r cla t ion ship be-tween similar figures. Give the child a triangularpiece of paper and ask him to cut it so as toform a triangular piece of paper similar insave to the original triangle, but with one-halfthe perimeter. The child who proceeds to cutthe original triangular shape from the midpointof one side to the midpoint of another side willbe displaying an umlerstanding of the relation-ship between similar triangles.

10. A multiple choice exercise such as the followiagmight be useful in appraising a sixth grader'sunderstanding of why division by zero is notpassible.

47

1,c

Which of the following is a correct statementabout the problem 6 4- 0?

A. The ansv, er to 6 4- 0 is 6.B. The answer to 6 4- 0 is 0.C. There is no answer tc 6 ± 0 because

there is no number when multipliedby 0 gives 6 as a result.

D. Either (1 or 6 is correct so there is no oneanswer to the problem 6 .÷ 0. It dependson the ans,,q- you want to use.

The above examples of evaluation procedures arenot intended to be taken as models, but only asillustrations of the wide variety of techniques thatcan be used, It is frequently possible to present,with slight modification, the same basic question ortest situation to the pupil in the form of an oral,a written, or a performance experience. It ml, henecessary to adjust the evaluation procedure to thepupil, for not all pupils can be expected to achievethe same level of understanding of mathematicalideas. In order to determine the level of under-standing a child has reached with respect to a givenconcept, one may have to devise a series of ques-tions, each intended to measure a different level ofunderstanding. One may also employ a more lengthyquestion that calls for several immediate responses,each requiring a different level of undeestanding.In any event, the child's background and potentialmust be considered, both when deciding whichevaluation procedures to use and when iPterpretingthe degree of success the pupil has experienced.

An increasing number of rating scales and el.ecklists are available to help the teacher in evaluatingthe pupil's attitudes toward mathematics as well ashis interest in the subject. For the most part, theseinstruments are of unproven reliability and validity.Consequently, the tea-In-Cs value judgment remainsa significant source of evaluation in this area. Thesefactors along with the pupil's enjoyment of mathe-matics may be appraisal by observing his behaviorwhen confronted with mathematical situations. Isthe child number conscious in his recreational read-Mg or in his reading of other school subjects? Is heinterested in recreational mathematics? Is he fasci-nated by mathematical puzzles? Does he display aninquiring attitude toward Mathematics by raisingquestions about numbers, operations, or their prop-ertes that indicate he is thinking about conceptsbeyond those which he has studied?

Children in the lower grades can be asked ques-tions alxnit what die; like lxst or least about mum-

hers, adding, circles, or counting. Jr the uppergrades, the child can he asked to write a short essayor paragraph on what he thinks about when some-one says mathematics or on what he likes best orleast about mathematics. At times the teacher shouldrequest that his pupils leave their essays or para-graphs unsigned so as to encourage a freer responseand thus to obtain a more accurate picture of theattitude of the class as a whole.

Today's elementary school mathematics programcontinues to have as an objective, the developmentof computational skills. Both commercial andteacher-made objective tests are effective instru-ments in evaluating the degree to which this objec-tive is achieved. Tests should be used which havea wide variety of items, each designed to test aparticular skill. In this respect, there is some ad-vantage in using commercial tests in that more careand perhaps greater and more varied test writingexperience are employed in designing a given test.

Care must be exercised in interpreting a test

score when it is being used to evaluate a child'sprogress. A single test score has mean:rig only whenit is viewed in relation to other measures of hisachievement. Patterns formed over a period of timeby test scores are also useful in assessing a child'sn ia thema Berl development.

Evaluation of the mathematics curriculum or themathematics achievement of an individual childmust not be separated from the teaching of mathe-Inatics. Evaluation is an integral part of teachingand must be continuously brought into play in theprocess of directing and re-directing learning ex-periences. It is a vital part of the ongoing programof curriculum improvement. Evaluation in elemen-tary school mathematics must not be vi2wed as aprocess of appraising a finished product or a spentprogram. Instead, it most be recognized as theprocess by which a continuous record is maintainedof the pulse rate of every aspect of both the mathe-matics curriculum and the experiences of the chil-dren under the curriculum.

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47

Inservice Education

NeedCurrent demands that elementary mathematics

instruction be based on both an understanding ofthe learning process and an understanding ofmathematical content present a challenge to allteachers. For many decades the importance ofmathematics in the elementary school was mini-mized; little attention was given to it in teachereducation. The. need for more thorough teachepreparation in this area is now being recognize,.Several professional groups have proposed raisingthe minimum requirements in the traicing of ele-meutar:, teachers; e.g., the Committee on Under-graduate Programs in Mathematics (CUPM) rec-ommends a minimum of twelve hoots of mathe-matics for pre-service teaches. Gradually teachertraining institutions arc increasing their require-ments for graduation and are offering courses spe-cifically designed for prospective elemen tar yteachers.

There is also a greater realization that profes-sional growth is a developmental process. Learningbegun in pre-service edneation must be continuedif a teacher is to keep pace with changes and theireducational implications. The proliferation ofknowledge resulting in an ever-changing curricu-lum makes the present-day problem more acutethan ever before. NVith the introduction of new con-tent and modern teaching strategies, even teacherswith strong mathematical backgrounds feel a needfor cominnous education.

Planned programs of inservice education are o,,oway of meeting the need for professional growth.However, these programs, essential as they are, donot automatically insure the implementation of newcurriculum in the classroom. In the final analysis theeffectiveness or inservice work can best be measuredby what happens as a teacher works with childrenin a teaching-learning situation. Administrators andteachers share a joint responsibility for creating alearning situation which reflects the best currentthinking in the field. Teachers with a commitmentto the profession constantly seek to update their

background of knowledge as well as their teachingtechniques. They are open to change and availthemselves of professional growth opportunities.Administrators facilitate change by furnishing thenecessary resources: time, instructional materials,professional books, consultant and supervisory serv-ices; and by offering support, encouragement, andunderstanding of problems encountered.Objectives

The major purpose of inservice education is theimprovement of professional competence throughthe development of attitudes, understandings, per-ceptions, and skills that enable teachers to providestudents with a good math-maties program. Sys-tematic, locally-sponsored teach( r education pro-grams should be designed to:

develop a greater understanding of the struc-ture of mathematics, and to insure a thoroughunderstanding of the mathematical conceptstaught.relate what is known about child grow,b anddevelopment and the psychology of learningto the teaching of mathematics.translate the results of research studies intoch,ssroom practices.familiarize teachers with many 115e fill teach-ing materials.acquaint teachers with mathematic:al litera-ture, current hooks, articles, and research re-ports.stimulate enthusiasm and interest in improv-ing mathematics instruction.create an awareness of the need for continuedstudy.

GuidelinesThe approaches used to achieve the above objec-

tives will vary from district to district. however,past experience indicates that inservice programsare most effective when:

a systematic program of instruction is concen-trated in a series of meetings or workshopsessions.the content of the program is related to the

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specific local program that is being developedand is, therefore, relevant to the teachers'immediate needs.time is spent not only on mathematical ideas,but also on the demonstration of appropriateteaching techniques for various grade levels.participants are actively involved in discus-sions, demonstrations, mid work sessions.participants have an opportunity to -discover-mathematical ideas for themselves.inservice programs and classroom e%perimen-tat;on with new ideas are carried on concur-rently.

Organization of inservice ProgramsNo matter what type of organization is used, an

inservice program needs to be developed in stages.The first step should be the identification of prob-lems. The problems may be those which arc obvi-ous to teachers or those which are of concern tosupervisors. After the problem has been agreedupon and studied, implementation of the results inclassroom instruction must follow. The end resultof a quality inservice program will be desirablebehavioral changes in pupils.

1. WorkshopThe most common approach to inservice

education has been the workshop. Workshopsmay be conducted solely by local districts orby a local district in conjunction with a uni-versity or college. In either case careful con-sideration should be given to the followingfactors.

a. PlanningConducting an cPctive, wellorganized workshop requires ca ref u1planning. The objectives of the workshopshould be clearly defined, so that the pro-gram can be designed to meet the par-ticular needs of the participants. Whenvisiting consultants are used, it is essen-tial that local supervisors or administra-tors plan with the consultants prior to theworkshop.

h. SchedulesVarious times may he chosenfor workshop activities:

a series of regular afternoon or eve-ning sessions. e.g., S to 12 two-hoursessions.a series of Saturday sessions.a series of half-day or full-day sessiondinirig the school year.Summer workshops of varying peri

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ods, e.g., one, two, or three-weeksessions.

As inservice activities are planned, con-sideration should be given to the possi-bility of allowing some released time forteachers.

c. Leadership Consultants whether localor visiting should he knowledgeable inmathematics and in teaching techniques.Both aspects of the program may be han-dled by the same person, or the responsi-bilities may be divided among a team ofconsultants, In sonic instances contentmay be presented by a university or col-lege instructor, and methodology by localqualified supervisors or teachers.

d, Format feelings may he divided intotwo parts: one of which is devoted to thediscussion of mathematical content andthe other, to classroom application.

Frequently the content aspect of theworkshop is presented to the total groupof participants. However, in some cases,it may be desirable to arrange to havethe total group divided into cwo sectionsfor discussion of mathematical ideas:one for those who have had previouscourses or workshops, and the other forthose who have had no such experience.

For work on classroom application, di-vi -;ion of the group in primary and in-termediate sections is desirable.

2. Curriculum CommitteeCommittee work provides an excellent op-

portunity for teachers to become involved inthe development of new programs and in thepreparation of curriculum materials easeful inthe iinnlementation of these programs. Nlem-liers of such committees may be volunteers orselected representatives from schools within adistrict, Supervisory personnel or teachers withspecial skill in mathematics may provide lead-ership, and consultants may be brought in asneeded.

The content of this guide may be used as asource in identifying problems that can bestudied in lig], of the needs of a particularschool or school system. Committees might beorganized to:

develop a scope and sequence plan fora K-6 program.

prepare a guide which suggests variedlearning experiences related to the be.-:lavieral objectives.prepare units of study for particulargrade levels on such topics as geometry,measurement, ratio, problem solving, etc.evaluate instructional materials: text-books, audio-visual aids, manipulativematerials.evaluate standardized tests.prepare tests related to the objectives ofthe local curriculum.

3. TelevisiollTelevision provides an effective means of

reaching many teachers. Where educationaltelevision facilities are available, local schoolsystems may either prepare their own seriesof telecasts or use series produced by profes-sional organizations, educational institutions,or commercial concerns. Generally speAing,inservice education through television has

war

proved most successful when provisions havebeen made for some type of "feedback." Feed -hack may be achieved through a combinationof television lessons and group meetings.

I. 0012r Common. Insertice ActivitiesThe following activities may be used to aug-

ment the above-mentioned programs:faculty discussion groups.teacher visitation between schools.action research.demonstration lessons.

5. Other Professional Growth OpportunitiesIn addition to the programs sponsored by

schools or school systems, teachers may im-prove their teaching competency by

enrolling in college or university courses.attending professional meetings and con-ferences.applying for grants for summer institutes.reading professional literature.

INLOC___Nr

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500

Criteria for Program Development

What are the characteristics of a good mathe-matics program? What teaching tools (books, films,visual aids, etc.) should be selected to assist theteaching staff in achieving the objectives of the pro-gram? These are two of the most important ques-tions that every school must answer pc.%.iodicallyThey are also among the more difficult questions forany school staff to answer. It is the aim of this guideto provide the schools of Wisconsin with meanswhereby they can arrive at more satisfactory an-swers to important curricular questions.

The foundation of any good program incorpo-rates the ideas which society has found to be ofvalue in the education of its youth. These ideas arethe content objectives of a program. For the mathe-matics program, they consist of the properties ofnumbers, similarity, ratios, rational numbers, etc.,as outlined in the previous sections of this bulletin.In addition to content objectives, bohavioral objec-tives for a program must be established. These, too,are outlined in this bulletin. If to these objectives,one adds the goal of teaching pupils to apply mathe-matics through problem solving, then one has thesound framework needed to build a mathematicscurriculum for the schools.

This bulletin suggests the type of objectives theschool should consider in developing its curriculum.However, the same criteria that are used to judgethe adequacy of a program can, in part, be used tojudge the adequacy of the teaching tools selectedfor use in the school. These judgments Pant bemade by the teachers or their representative coin--duces. To assist the schools in making these judg.ments, a sample checklist of basic criteria is givenin this section.

Note that the sample criteria checklist consists offive main sections. Section 1 focuses on general con-siderations. Is the mathematical content sound? Isthe development spiral in nature? Are the objectivesappropriate for the various grade levels?

Section 2 is a listing of the content and lohavimalobjectives. Section 3 consists of pedagogical consid

orations, Section 4 contains criteria on which tojudge supplementary aids, and Seaion 5 brieflytouches on the importance of format in makingjudgments as to the usefulness of a teaching tool.

341=4

Committees of teachers will want to modify thissample criteria checklist to fit the needs of theschool for which a program is being formulatsxl.Furthermore, they may wish to assign weights tothe various sections. In the opin:on of many people.a gLat deal of weight should be placed on thevarious subsections of Section 2 of the sample cri-teria checklist, It is in this section that one can findthe spirit of the "new" mathematics programs forthe schools of Wisconsin,

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Sample Criteria Checklist

Name of Program Publisher Grade Teacher

Rating

1 2 3 4 5

1. Philosophy of ProgramThe programa) Is consistent with the K-12 mathematics pro-

gram of the school.h) Contains good mathematical development.e) Presents a continuous development of con-

cepts and skills.d) Is teachable and appropriate for the particular

level.

e) Uses a spiral development.f) Develops concepts through numerous exam-

ples.g) Uses consistent and understandable mathe-

matical language.

Low High

2. Kay Mathematics' Content and Behavioral Objectives of Program

A. ARITHMETIC CONCEPTS

a) The program provides for an understandingof sets and numbers, including:Cardinal numbers.Ordinal numbers.\Vhc!c numbersFractionsNegative integers.

b) The program provides for an understandingof numeration systems through:A variety of grouping E.:tivities.An emphasis on place values (base 10).An introduction to other numeration systems.

LowRating

High2. 3 4 5

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c) The program provides for understandingof the concept of order through:

Low

2

Rating

-1

High5

Experiences in determining "greater than,""less than," and "equals."The use of the symbols <, >,The ordering of a s2t of numbers (from small-est to largest).

d) The program provides for an understandingof number systems, operations and their prop-erties through:Experiences with addition (and subtraction itsits inverse).Experiences with multiplication (and divisionas its inverse).Experiences with the structure of the systemof whole numbers.Experiences with the structure of the rationalnumber system.

e) The program provide3 for an understanding ofthe concepts of ratio and proportion throm,h:Separate treatment of the concept of ri,tiofrom that of rational number.Many experiences entailing the use of ratioand proportion in problem solving.

f) The program provides for an understandingof computation through:A developmental program in basic facts.Experiences developing proficiency in com-mon computation procedures (algorithms).The use of non-drill activities to build compu-tational skills,

B. MATHEMATICAL SENTENCESRating

Low I I igh

g) The program provides for an understandingof mathematical sentences through:Equations and inequalities developed as"translations" and "interpretations" of physi-cal situations.Intuitive methods suggested for the solutionof equations and inequalities.Practice in the solution of equations andinequalities.

1 .3 .1 5

Use of equations an.i inequalities to solveproblems.

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C. GEOMETRY CONCEPTS

11w program provides for an understandingof the concepts of:

11) Size and shape through:Experiences in identifying plane rod solidfigures.

i) Sets of points through:Nlanipulative and conceptual experiences withpoints, lines, and planes.

Low1 2

1

Elating

:3 -1

High5

j) Symmetry through:The development of symmetry concepts.

k) Congruence through:The development of congruence concepts.

1) Similarity through:The development of similarity concepts.

m) Coordinate systems and graphs through:The use of number line activities and the Oe-velopment of coordinate systems and graphing.

n) Constructions through:Construction problems.

o) Measurement through:Experiences with money and time.

The development of concepts of basic unitsfor length, area, volume, angles.

The development of approximate measures.

Experiences with equivalent measures (andthe reduction of measures).

The development of area formulas for squares,rectangles, parallelograms, and triangles.

The development of the linear metric system,

D. PROBLEM SOLVING

p) The program provides for an understandingof problem solving through:Continuous emphasis on problem solving.

LowI

Rating

3 4

Iligh5

Practical help for solving problems.A wide distribution and variety problemsolving practice.

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3. Methodology Used in and Suggested for ProgramThe program provides:a) Mathematical concepts correctly and con-

sistently used in explanations.Is) A variety of approaches in developing a topic.c) A basic skills development and maintenance

program.d) Adequate material for written assignments.e) Sufficient material for oral exercises.f) An emphasis on estimation and -trial-and

error" techniques in problem solving.g) "Mental arithmetic" activities.

Independent thinking activities.i) Exercises graduated in difficulty.j) Consideration of individual differences,k) Clarity of examples,1) Supplementary activities for high-achievers.m) Supplementary-remedial activities for low-

achievers.n) Independent work in geometry or other topics.

4. Supplementary Aids to Programa) The teacher's manual

Contains mathematical background for con-tent of program.Provides adequate suggestions.suggests numerous activities for enrichment(high and low),

b) A suggested testing program for use in pupilevaluation is available.

c) The publishing company provides consultants,films, or other means of inservice aids.

5. Physical Format of Booka) The general format of the book is attractive

to the pupil.

RatingLow High

1 2 .1 5

LowRating

high3 -1 5

Rating1,o

1 2 3 5

b) The binding of the book is of good quality.

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Bibliography

I. Mathematics Instruction: A Point of View1. Ansubel, David 1'. The Psychology of Meaningful Verbal Learning.

New York: Grime & Stratton, 1963.2. Bruner, Jerome S. The Process of Education. Cambridge, Massachu-

setts: Harvard University Press, 1963.3. Educational Services, Inc. Coals for School Mgthematies: The Report

of the Cambridge Conference on School Mathematics. Boston: l lough-ton-Mifflin Co., 1963.

4. Phenix, Philip II. Realms of Meaning. New York: McGraw-Hill BookCo., 1964. Chapters 6 and 27.

Key Mathematical Content Objectives and Related Student BehavioralObjectives

1. Briggs, et al. "Implementing Mathematics Programs in California A

Guide, K-8." Pacific Coast Publishers, 1964.2. Educational Services, Inc. Goals for School Mathematics: The Report of

the Cambridge Conference on School Mathematics. Boston: Hough-ton-Mifflin Co., 1963.

3. National Council of Teachers of Mathematics. "The Growth of Mathe-matical Ideas, K 12." Twenty-Fourth Yearbook. Wasbington, D.C.:The Council, 1959.

III. Adjusting Mathematics Programs to Student Abilities1. Association for Supervision and Curriculum Development. bulb:411ml-

izing Instruction. 1964 Yearbook. Washington, D,C.: the Association,1964.

2. Guodlad, John and Anderson, Robert 11. The Nongraded ElementarySchool. New York: Harcourt, Brace and World, 1959.

3. National Council of Teachers of Mathematics. Instruction in Arithmetic.25th Yearbook. Washington: The National Council, 1960. Refer tobibliography of Chapter 6, page 146.

4. National Society for the Study of Education. Instruction.61st Yearbook, Part I. Chicago: University of Chicago Press, 1962.

.5. Smith, Rolland B. 'Providing for Individual Differences.' The Learningof AIathematics: Its Theory and Practice. TwentyFirst Yearbook.Washington, D.C.: National Council of Teachers of Mathematics,1953. Pp. 271-302.

6. Weaver, J. Fred. "Differentiated Instruction and School-Class Organiza-tion for Mathematical Learning wits in the Elementary Grades." TheArithmetic Teacher, NIII (October 1966), 495.506.

IV. Problem Solving1. Banks, J. Houston. Learning and Teaching Arithmetic. Boston: Allyn

Bacon, 1961. Chapter 14.

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2. llenderson and Pingry. "Pndilcm-Solving in Mathematics." The Learn-ing of Mathematics. Twenty-First Yearbook. Washington, D.C.: Na-tional Council of Teachers, 19.53. Chapter 8.

3. Nlarks, Purdy, and Kinney. Teinhing Elementary School Mathematicsfor Understanding. New York: McGraw -hill, 1965. Chapter 13.

4. National Council of Teachers of Nlathematics. -Instruction in Arithme-tic.- Twenty-Fifth Yearbook. 'Washington, 1).C.: The Council, 196(1.Chapter 10,

5. Ward and flargrove. Modern Elementary Mathennities. Beading. NIassa-chnsetts: Addison-Wesley, 1964. Chapter 10.

V. Evaluatinn and Testing1. Dutton, Wilbur II. Eca luating IlMlerstanding of Arithmetic.

Englewood Cliffs, New Jersey: Prentice-Ilan, 196.1.2. Johnson, Donovan A. -Evaluation in Nlathemati rut 9-Si.011 Year-

book of the National Council of Teachers of Mathei;etics. Washing-ton, The National Council, 1961.

.3. Meschkowski, Herbert. Evaluation of Alatheinatical Thought. San Fran-cisco: Ilulden-Day, 1965.

3. Nlyers, Sheldon S. Mathematics Tests Acailahle in the United States.Washington, 1).C.; National Council of Teachers of Mathematics,April 1959.

5. Spitzer, Herbert F. "Testing Instruments and Practices in Relation toPresent Concepts of Teaching Aritimietic.- The Teaching of Arithme-tic: Fiftieth Yearbook. Part 1, National Society for the Study of Edu-cation. Chicago: University of Chicago Press, 1951. Pp. 186-202.

6. Suck", B. A., Boynton, 11 and Sauble, 1. "The Nteasurement of Under-standing in ElemenLtry Sclnyd Nlathematics." The Measurement ofOuierstanding. Forty-Fifth Yearbook. Part I, National Society for theStudy of Education. Chicago: University of Chicago Press, 1916. Pp.138-156.

VI. Inservice Education1. Bruner, Jerome 11. The Process of Education. Cambridge, Massachu-

setts: Harvard University Press, 1960.2. DeVault, et al. Improving Mathematics Programs. Columbus, Ohio:

Charles E. Merrill Books, 1961.3. Deans, Edwina. Elementary School Mathematics: New Directions, C. S.

Department of Health, Education mid Welfare, Office of Edit(Shingt1)11, D.C.: Government Printing Office, 1963. 116 pp.

I. Grossuickle, Foster E., and BrueeImer, Leo V. Discovering Meaniirg inEtc e Vary School Mathematics. 4tli ed. New YOIL MAL iI(mi artmid 'Winston, 1963.

5. lleddens, James NI. Today's 3fafbemattes: A Guide to Concepts midMethods in Elementary School Mathematics. Science Research Asso-ciates, Inc. 1964.

6. Nlay, Lola. Major Concepts of Elementary Modern Mathematics. JohnCollmrn Associates, Inc., 1122 Central Avenue, Wilmette, Illinois.1962.

7. Mat/lunatics Background for the Primary Teacher. Wil-mette, Illinois: John Colburn Ass(oeiates, Inc., 1966.

S. Mueller, Francis J. Irpclating 3lathematies. New Izauton. Ginned icut;Croft Educational Services, 1961.

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9. National Cooneil of Teachers of Mathematics. Instruction inv,enty-ifth. Yearbook. NN'ashiligton, 1).C: The Council, 1960.

10. Totes in 31(iiheinatics for Elementary School Teachers,..fwenty-Ninth Yearbook. NN'ashington, 1).C.: The Council, 1961. 381

PP.I 1. Page, 1) Lvicl A. Number Lbws, FitnetioNs, and Toith s,

New York: The NIacmillan Company, 196112. NVerthlietilicr, Alielmel. 1 'au/fitive Thinking. Ness York: ILL rper &

Brothers, 1959,

VII. Criteria for Program Deve!QpmentI. Briggs, Afettler, and 1)er.00ltu. Progra in

California. 'Menlo Pail, California: Pacific: Coast Publishers.State Dcpatti tent of Education. Summary of the liepori of

the AtIcisiny Craninitice on Alailic.aaties to the California Shift. Cur-riculum (.70inini.slion. Bulletin, Vol. XXXI II, No. 0, Deceml,er 1963.

:3, Peak, Philip, et al. "Aids for Evaluators of Mathematics 'lc \,Iritlinictic Teacher. `.lay 1965. 388-391.

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