ED 046 716 AUTHOR TITLE INSTITUTION PUB DATE NOTE AVAILABLE FROM EDRS PRICE DESCRIPTORS ABSTRACT DOCUMENT RESUME SE 010 212 Brydegaard, Marguerite; Tnskeep, James F., Jr. Readings in Geometry from the Arithmetic Teacher. National Council of Teachers of Mathematics, Inc., Washington, D.C. 70 126p. National Council of Teachers of Mathematics, 1201 16th St., N.W., Washington, D.C. 20036 EDRS Price MF-$0.65 HC Not Available from EDRS. Curriculum, *Elementary School Mathematics, *Geometry, *Instruction, *Mathematical Enrichment, Mathematical Models, Mathematics, *Secondary School Mathematics, Teaching Guides This is a book of readings from the "Arithmetic Teacher" on selected topics in geometry. The articles chosen are samples of material published in the journal from its beginning in February 1954 through February 1970. The articles are of three major types. The first is classified."involvement." These articles describe geometry units in which the students build geometrical models, play games, and draw geometrical objects. Another article in this classification focuses on a teacher preparation course in which the future teachers experience the learning activities of the students. The second group of articles is categorized "instruction-techniques." These articles focus on the techniques of teaching units in informal geometry using mirrors, models, toys, and Mobius bands. The third type of article is termed "instruction-rationale." This type of article gives reasons why geometry should be taught in the elementary trades and tells what parts of geometry should be taught. Included in the book is a bibliography of articles published in the "Arithmetic Teacher" pertinent to geometry. (Author/CT)
Transcript
ED 046 716
AUTHORTITLEINSTITUTION
PUB DATENOTEAVAILABLE FROM
EDRS PRICEDESCRIPTORS
ABSTRACT
DOCUMENT RESUME
SE 010 212
Brydegaard, Marguerite; Tnskeep, James F., Jr.Readings in Geometry from the Arithmetic Teacher.National Council of Teachers of Mathematics, Inc.,Washington, D.C.70126p.National Council of Teachers of Mathematics, 120116th St., N.W., Washington, D.C. 20036
EDRS Price MF-$0.65 HC Not Available from EDRS.Curriculum, *Elementary School Mathematics,*Geometry, *Instruction, *Mathematical Enrichment,Mathematical Models, Mathematics, *Secondary SchoolMathematics, Teaching Guides
This is a book of readings from the "ArithmeticTeacher" on selected topics in geometry. The articles chosen aresamples of material published in the journal from its beginning inFebruary 1954 through February 1970. The articles are of three majortypes. The first is classified."involvement." These articles describegeometry units in which the students build geometrical models, playgames, and draw geometrical objects. Another article in thisclassification focuses on a teacher preparation course in which thefuture teachers experience the learning activities of the students.The second group of articles is categorized "instruction-techniques."These articles focus on the techniques of teaching units in informalgeometry using mirrors, models, toys, and Mobius bands. The thirdtype of article is termed "instruction-rationale." This type ofarticle gives reasons why geometry should be taught in the elementarytrades and tells what parts of geometry should be taught. Included inthe book is a bibliography of articles published in the "ArithmeticTeacher" pertinent to geometry. (Author/CT)
from the Arithmetic Teacher
!t- 2-11-NJ
*,
NATIONAL COUNCIL OFTEACHERS OF MATHEMATICS
US DEPARTMENT OF HEALTH. EDUCATION& WELFARE
OFFICE OF EDUCATIONTHIS DOCUMENT HAS BEEN REPRODUCEDEXACTLY AS RECEIVED FROM THE PERSON ORORGANIZATION ORIGIN \TING IT POINTS OFVIEW OR OPINIONS STATED DO NOT NECES-SARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY
Readings in Geometry
from the
Arithmetic Teacher
edited by
MARGUERITE BRYDEGAARD
and
JAMES E. INSKEEP, JR.
National Council of Teachers of MathematicsWashington, D.C.
"PERMISSION TO REPRODUCE THIS COPYR IGHTEDMAMA M I GROF I CHE ONLY BAS BEEN GRANTED_By 110 Uouncil Teachers MathTO ERIC AND ORGANIZATIONS OPERATING UNDERAGREEMENTS WITH THE U.S. OFFICE OF EDUCATION.FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM
REQUIRES PERMISSION OF THE COPYR IGHT OWNER."
The special contents of this book areC copyright 1970
by the National Council of Teachers of Mathematics, Inc.,1201 Sixteenth Street, NW, Washington, D.C. 20036. Allrights reserved. Printed in the United States of America.
Library of Congress Catalog Card Number: 74-138809
Contents
Introduction
1. InvolvementEditorial Comment
With Sticks and Rubber Bands(Comment by Don Cohen)
February 1970
Geometry Alive in Primary ClassroomsFebruary 1967
The Quest for an Improved CurriculumFebruary 1967
Topics in Geometry for TeachersaNew Experience in MathematicsEducation
February 1970
2. Instruction TechniquesEditorial Comment
An Example of Informal Geometry:Mirror Cards
October 1966
Kaleidoscopic GeometryFebruary 1970
Informal Geometry through SymmetryOctober 1969
SymmetryFebruary 1970
Developing Generalizations withTopological Net Problems
February 1965
An Adventure in TopologyGrade 5November 1959
Creatamath, orGeometric IdeasInspire Young Writers
May 1967
1
3
Joseph Slott 5
Janet M. Black 9
J. R. MacLean 13
Carol H. Kipps 18
23
Marion Walter 25
Carol Ann Alspaugh 30
J. Richard Dennis 32
Sonia D. Forseth andPatricia A. Adams 36
Charles H. D'Augustine 39
Jean C. Clancy 43
Emma C. Carrol 45
A Second Example of InformalGeometry: Milk Cartons
May 1969
Volume of a Cone in X-RayApril 1959
Geoboard Geometry for PreschoolChildren
February 1970
Pegboard GeometryApril 1965
Tinkertoy GeometryOctober 1967
Congruence and MeasurementFebruary 1967
3. InstructionRationale
Marion Walter 48
Sister M. Vincent 51
Werner Liedtke andT. E. Kieren 52
Lewis B. Smith 56
Pauline L. Richards 60
Stanley B. Jackson 62
Editorial Comment 71
Geometry All around UsK-12 John C. Egsgard, C.S.B. 73October 1969
Primary-Grade Instruction in Geometry James E. lnskeep, Jr. 82May 1968
Why Circumvent Geometry in thePrimary Grades? Nicholas J. Vigilante 87
October 1965
The Role of Geometry in ElementarySchool Mathematics G. Edith Robinson 92
January 1966
Geometric Concepts in Grades 4-6 Dora Helen Skypek 100October 1965
Geometry in the Grades Irvin H. Brune 107May 1961
Bibliography
iv
117
The editors wish to express their sincere appreciationto the panel of reviewers who studied the .
Arithmetic Teacher files and made recommendations concerningarticles to be repented here. The following personsare given special thanks for their assistance:
Dr. E. Glenadine Gibb, The University of Texas,former editor of the Arithmetic Teacher
Dr. Charlotte W. Junge, Wayne State University,editor of the journal's "In the Classroom"
Dr. Len Pikaart, University of Georgia,coeditor of the journal's "Focus on Research"
Dr. C. Alan Riedesel, Georgia State University,coeditor of "Focus on Research"
Dr. Marilyn N. Suydam, Associate Director,Center for Cooperative Research with Schools,The Pennsylvania State University
IntroductionThis is a book of readings from the Arithmetic Teacher on selectedtopics in geometry. The articles reprinted here are samples of materialpublished in the journal from its beginning in February 1954 throughFebruary 1970.
The selection of readings to represent the contribution of the Arith-metic Teacher to geometry for grades K-8 was not easy. In some casesai article was chosen because it develops a topic or idea that is uniquend original or because it covers an area for which few articles have
been written. When one article paralleled or overlapped another, achoice wP , made on the basis of an evaluation by the panel of reviewers.Since the number of pages was limited in order to provide an inexpen-sive publication, articles on the topics of research and measurement arenot included in the readings but are represented in the Bibliography.A rather complete list of these and other pertinent articles is containedin the Bibliography, in which the latest citation is from the issue ofMay 1970. This listing includes the articles reprinted h 3re.
In recent years the Arithmetic Teacher has carried a variety ofarticles dealing with geometry in grades K-8. It is interesting to notethe growth of attention given to the topic. t3efore 1959 no articles inthe journal dealt specifically with geometry. Even after that time theflow of published articles was minuscule until the middle sixties, whenthe increase in publication was dramatic. As an indication of the in-crease, note that the following six issues emphasized the theme ofgeometry: October 1966, February 1967, October 1967, December1968, October 1969, and February 1970.
The increase in publication developed in response to a new conceptof geometry for the elementary school. Geometry has classically beenassociated with Euclid and an axiomatic approach to study, and fromthis viewpoint geometry was not considered appropriate for theelementary school program. The new approach is neither axiomaticnor narrowly defined but rather places emphasis upon such ideas asthe environment, informal approaches, and readiness.
The term "geometry," from its Greek derivation, has the meaning"earth measurement." Today our concern is with the exploration, study,and measurement of space; and to coin a word "spaceometry"would more aptly identify the subject. The broader concept of space
1
2 INTRODUCTION
relationships is a significant part of the program for grades K-8. It isrealized that each learner intuitively and naturally senseL and respondsto his spatial environment. Ideas such as position, direction, distance,size, and shape are sensed in the child's earliest experiences. It appearsto be true that by the time a child enters school, his knowledge andunderstanding of geometry may be greater than his development ofconcepts of number.
As published information about the teaching of geometry in gradesK-8 has increased, so has the need to disseminate it. With evidence ofdemand and need, this book of readings has been assembled to promotewider distribution of quality material dealing with this subject.
Fri
Involvement
The articles in this publication have been grouped according to twomajor categories involvement and instruction. Those falling underthe second category have again been grouped according to whetherthe emphasis is on technique of instruction or its rationale. While noarticle is strictly limited to any one of these categories, it is thought thatthe selected articles highlight either involvement or the techniques orrationale of inst action.
Involvement! Personal involvement and learning are inseparable.One of the basic approaches to the mathematical involvement of chil-dren is through the study of geometry. When studied intuitively, theenvironment of a child opens a wide vista of geometric ideas. Thisenvironmental approach to geometry lays a foundation of readiness forinformal and formal geometry at a later age.
Involvement of the learner! In the beginning article we see Joey, agesix, involved with geometric construction and problem solving that leadhim to significant mathematical ideas. His inventions were guided bythe "teacher" who provided the materials and the freedom to do prob-lem solving and also by his sister, Maureen, who assisted him.
Involvement of a classroom of learners and their teacher is describedin Black's article. As its title suggests, geometry is alive! The authorpresents many ideas that help teachers to identify geometric conceptsand to sense how to organize a laboratory for experimental involve-ment. MacLean, in the article that follows and complements Black'sarticle, describes the K-6 Experimental Project of the Ontario Mathe-matics Commission. The eight activities that are described illustrateideas a teacher can adapt for his own classroom.
Involvement of preservice teachers is discussed by Kipps in de-scribing her experimental course for university students. Using alaboratory approach, she challenged her students to experience someof the things they should practice in their classrooms with children.
These four articles illustrate ideas of involvement of children andteachers. The message is simple and exciting. He who pursues involve-ment as a basic procedure for learning will not only search for tech-niques that are productive but will seek effective rationale for theconcepts that he teaches.
3
t
With sticks and rubber bands
JOSEPH SCOTTNorth Valley Stream, New York
Ilaci a. g oo a
i me mo, v,r.
Things.
Th 5 i5 1.1)e fi r.stin 0 e ni- i on -f-ba +
cikph is vvas eaS
iOs0
,,,1
....AP. --......ro. A
ik \ , : J ..464..,^...-,'i. 0 ''' :. . -
tt X
'Trc&V.,::Tif0-ParrTZ:'
This neni 0 ilsi-ert from0 ter s Petce..
1)
6 INVOLVEMENT
th fh 0 ehcs-V cb dqç
ink 0 ne, Time
i. S+-ucK so tilebo8ybut- 1-ke.y g ot
Out a.n easy vt/cLy.
.:'72 7 ' . ''':'.E.'":11%'="IMMEZATFATIE2
Tkis is The b 1 ci g 11,4 i wctsudg My is ter filcuire eh vvqsho td n g one oP 1--he corn e,ris
o (shed *he- buici Ing but H-- w°ektiderti=~
With Sticks and Rubber Bands / Scott 7
st-md tip 1)-y f-f-s elfeQct4seclic:11)1f hany trio 09 le \\
ave
11
at,
This is the nose con e.
r
111 is is the boitorri-One vocK
5: 3
,-,V` 4.
.,?:, -I
, r '
;Zip
I Finished the Ncifel,.
0.11111111r11111111
Joey, who is six years old, spent aboutseven hours with this project, about twohours at a time.
The sticks are 1/4" dowel rods, three
9
8 INVOLVEMENT
feet long, and available at lumber yardsat about $.08 apiece. About 30 sticks aresufficient. Regular rubber bands were used.
What learning was involved?
Mathematics: two-dimensional shapes(triangle, square, pen-tagon, and others).three-dimensional shapes(tetrahedron, octahe-dron, cube, and others).symmetry of figures,counting edges, corners,faces.
Physics: rigidity of the triangle as
13
used in bridges, build-ings.
Art: positioning shapes, sym-metry (one could coverthese shapes with crepepaper).
Language Arts: Joey wrote about whathe did and read hisstory.
Problem Solving: problem solving that in-volved concentration andfreedom.
-DON COHEN, Madison Project ResidentCoordinator, New York, New York.
Geometry alive in primaryclassrooms
JANET M. BLACK Barrie, Ontario, Canada
Janet M. Black is a classroom teacherShe has experimented extensively intypical lesson that she describes wasof Teachers of Mathematics meeting
Planning an experimental course ingeometry for forty-two Grade 3 studentswas quite a challenge. Five phases ofgeometry were to be included:
1. A study of solids and their basic prop-erties to develop in children an aware-ness of shape in connection with their
environment
2. A study of plane figures through theexamination of the surfaces of solidsand of real objects around school and
home3. A study of lines as the edges of solids
and of points as fixed locations in spacethrough construction of models andexperimentation with as many con-crete objects as possible
4. A review of concepts through the"building" of solid shapes from flatsurfaces and the construction of modelsin the environment
5. Symmetry, tiling, and nesting involving
solids and flats
During geometry periods, work stationswere set up. The children were challengedorally by assignment cards, by the taperecorder, or by charts to examine certainmaterials to see what could be discovered.The following description of a typical les-son involving geometric concepts tookplace at about stage two of the five phasesof geometry. Children were in a transitionalstage proceeding from the study of solids
9
in Barrie.teaching geometry to children. Thereported at The National Councilin Calgary, August 1966.
to the study of plane shapes, and theactivities were designed with three definiteideas in mind:
1. To make the transition from the studyof solids to the study of plane shapes asnatural for the children as possible
2. To establish through actual experi-mentation with physical materials an in-tuitive understanding of both solid andplane shapes
3. To provide for primary children thetime to discover and the freedom to ex-periment within the classroom at eachindividual's own chosen speed
Seating arrangements within the class-room were changed frequently, dependingupon the number of work stations requiredfor each lesson. For this particular sessionsix stations which could accommodate ap-proximately eight children each were setup with the required materials. The sta-tions are numbered on the following floorplan but were not numbered during the
actual lesson, since children proceededfrom one station to another in any orderthey chose as long as there was a placefree at the chosen site.
Children at centre one are using half-sheets of foolscap and a large woodensolid each. Their challenge was givenorally. They were asked to examine thesolid on the desk, and on the paper listall the things that they could think of
1 A
10 INVOLVEMENT
which have the same shape as the solid.For instance, the boy in the picture isexamining the cube, and he may list suchthings as the following:
1. My sister's jack-in-the-box2. The blocks I have at home3. The plastic pencil holder4. Dice
5. The box on the green shelf
el'
1 Fi
When he is finished he is encouraged todiscuss his list with the person next tohim. He may then move to another deskin the same section and prepare a similarlist for a different Solid, or he may proceedto a new work centrewhichever hechooses.
CENTRE TWO
Children working at this section havea large chart to guide their examinationof solids. Since the transition from three-dimensional shape to two-dimensionalshape is a general aim for this lesson, thechart contains questions such as these:
1. List the shapes which look like boxes.2. Which solids have flat surfaces shaped
like circles?3. Find all the solids which have more than
five flat surfaces.4. Which rolls more easily?
a) a solid with two flat surfacesb) a solid with five flat surfaces
5. Spin the square pyramid. What othersolid does it look like while it is spinning?
Children are encouraged actually to handleand experiment with the solids as theyprepare their charts.
The third centre deals with the problemof rigidity. Children have examined alarge cube and have built its "skeleton"
Geometry Alive in Primary Classrooms / Black 11
At this centre nine individual work cardshave been prepared, all dealing with theshapes of faces (triangles, rectangles, andsquares). In this particular instance, thefellow in the picture is working throughan assignment dealing with the positioningof one face shape over anotheran in-tuitive experience with intersection of lines,but directly mentioning only the shape ofa three-sided flat surface. Since each as-signment card at this station is different,the boy may choose to continue to workin this section at a different desk witha different card or to proceed to any otherstation which intaests him upon comple-tion of his present assignment. In manycases it is possible to put the answers toassignment cards on the back of the card,and the student therefore is capable ofmarking his own work.
CENTRE THREE
from straws and pipe cleaners. Their chal-lenge was also given orally. They wereto find out whether or not the skeletonof the cube would stand upright and holdits shape. If it wouldn't, the children wereto discover a way to make the structurerigid by adding more straws.
Children were encouraged to work intwos or threes at this centre so that theycould discuss several ideas and try themout before deciding which one workedbest. This centre was, by the way, a realstudy in "citizenship" as well as geometry,since a discussion involving ideas is notalways a placid one, and children some-times need help to work effectively in asituation such as this.
The day before this lesson, childrenhad traced the faces of the large woodensolids onto sheets of paper and reproducedthe shapes of the faces on geo-boards.
CENTRE FOUR
CENTRE FIVE
At this centre children worked individu-ally with an assignment card as a guide.The instructions involved choosing a cer-tain number of straws in order to builda shape that would stand by itself. Theboy in the picture has experimented withsix straws, joining them at the verticeswith pipe cleaners. He was, of course,amazed at the resulting tetrahedron, whichhe called a "four-sided pyramid". Childrenare encouraged to invent their own namesfor many shapes, since in some cases, par-ticularly in Grade 1, the technical name
12 INVOLVEMENT
pav-- '7...71117-7%
CENTRE SIX
is much too difficult for many of thechildren.
There are many games available whichwill help children progress in their under-standing of spatial relationships. The gameincluded in this lesson was originallyknown as "Kahla", an Egyptian gamesometimes played on folded paper withpopcorn for counters. On large cardboardsheets with thick plastic discs for counters,our version of Kahia, known as "Cala-bash", encourages estimation visually andmentally. The children follow the rules ofthe game and attempt to win all or mostof the calabash counters from a partner.The players then record their scores onthe blackboard and move on to anotherdivision.
Following are other examples of gameswhich develop geometry concepts:
4. Parquetry blocks involved in games5. Chinese Checkers
CULMINATION
At the end of the lesson period, variousreports are given orally by children work-ing at stations 3, 4, and 5. The chart fromcentre two is developed co-operatively byindividual members of the class whileothers mark their work (anyone who didnot reach the centre picks up an extrachart, and works along orally with therest of the class, filling in answers if hewishes to do so). The lists from centreone are read aloud by individuals andpinned around the "Shapes Table" in acorner of the classroom. The blackboardcontaining the individual scores from Cala-bash is examined, and a discussion followsconcerning the person who won by thehighest score, combinations of differentscores, and the possibility of recordinga "score in the hole" for some people.
Focal point J. Fred Weaver
The quest for an improvedcurriculum*
J. R. MACLEANAssistant Superintendent, Curriculum DivisionOntario Department of Education
EDITOR'S NOTE.Mr. MacLean was oneof tr e. speakers at the Calgary Meeting ofthe National Council of Teachers of Mathe-matics in August 1966. His article com-plements that of Janet Black in this issueof the journal, "Geometry Alive in Pri-mary Classrooms."MARGUERITE BRYDE-GAARD,
For the past several years teachers havebeen almost overwhelmed by successivewaves of propaganda that has generallybeen critical of our "traditional" mathemat-ics programs. Unless set notation and sym-bolism, non-decimal numeration, modulararithmetic and strict attention to correctterminology (such as distinction betweennumber and numeral) were included inour courses of study, we were not up-to-date. Our students would be ill-preparedfor their responsibilities in the Space Ageand we were effectively sabotaging the fu-ture of our country.
It is unfortunate that this "new math"concept has confused as much as it haschanged elementary school mathematics.There are many reasons why this has oc-curred, but the most significant are these:the haste with which the new ideas were
*Reprinted by permission of the author from theOntario Mathetisatic. Gasette (special elementary schooled.), September 1968.
13
applied; a lack of understanding on thepart of some of the mathematicians design-ing the programs about how children learnand what they need to learn; the generalunpreparedness of teachers and supervisorswhich has fostered the idea that the "new"completely replaces the "old" rather thanclarifying, supplementing and providingmore meaningful approaches; and finally,the lack of unbiased evaluation of the new-er materiali and techniques.
The K-6 Kingston Study Group
Conscious of these weaknesses and yetalert to the necessity of applying newknowledge, both in mathematics and inpedagogy, to the development of new andbetter programs in elementary school math-ematics, the Ontario Mathematics Commis-sion, with the financial support of theOntario Curriculum Institute, undertook thetask of exploring and collating the variousexperiments and approaches that are beingcarried out in Ontario and elsewhere. TheCommission appointed ten teachers to acommitteeeight directly involved in ele-mentary schools, one from the high schoolsand one from the universitiesand askedthem to suggest possible revisions of thepresent curriculum and to make recom-mendations for the implementation of thesuggested changes.
The K-6 Mathematics Study Group met
14 INVOLVEMENT
during the summer of 1965 at the RoyalMilitary College in Kingston, and produceda report which was subsequently publishedby the Ontario Curriculum Institute. Thisreport has been described as one of themost significant documents to appear onthe educational scene in many years. Allteachers who are concerned with the teach-ing of mathematics are urged to read thisreport. Some of the recommendations ofthe committee have been included in theInterim Revision in Mathematics recentlypublished by the Ontario Department ofEducation.
The K-6 Experimental Project
The proposals in the report for im-plementing a new program have been con-sidered by the Commission, especially item7 "that a full time group be established. . . to prepare teachers' handbooks andother materials for selected topics in specificgrades so that extensive experimental workcan be carried out in the academic year1966-67." Once more the Curriculum In-stitute agreed to finance a project thatwould involve many teachers and studentsthroughout the province. It was decidedto prepare materials for the study of geom-etry in Grades 1, 2, and 3, and Graphs,Mappings and Relations in Grades 4, 5,and 6. Selected teachers from all sizes ofschool jurisdictions met in Toronto duringJuly for training sessions in presenting thematerials.
Perhaps the most vital aspect of this ex-perimental program is the change in ped-agogical techniques which in turn transformthe classroom from the usual array of stu-dents sitting in desks arranged in neatrows to a virtual "laboratory for learning."Pupils are actively engaged in solving prob-lems with physical materials that they canmanipulate themselves. This approach isillustrated by an ancient Chinese proverb
"I hear, and I forget;I see, and I remember;I do, and I understand."
Many of the ideas and techniques pre-
sented in the Nuffield Mathematics Teach-ing Project publications have been adaptedfor use in the experimenting classrooms.The writing team was greatly encouragedthrough the knowledge that other peoplein other countries were working with ideasnot too divergent from its own.
The following sample lesson or moreappropriately "experience description" hasbeen taken from the outline prepared bythe writing team and presented to the teach-ers attending the summer course. Runningthrough it is the central notion that chil-dren must be set free to make their owndiscoveries and think for themselves.
Lesson Topic: Examining Faces
This lesson was presented after threeprevious periods spent examining, discuss-ing and manipulating solids. Each studentpicks a partner to work with during thelesson period. The partners are given apiece of paper and crayons, and eachpupil chooses a solid from the set provided.Their assignment is to trace the fiat sur-face of each solid onto a sheet of paper,attempt to select shapes that are different,and then discuss why they are different.
The class might choose a circular faceto examine. One member is appointedsecretary-chairman. Each pupil should drawa ring around the circular faces they havetraced and suggest things that make thecircle different from the other shapes. Thesecretary records all suggestions, and theclass then evaluates the chart he produces.Other faces should be examined in a sim-ilar manner before students proceed to theactivity centres, indicated in Figure 1
(next page).(Activity centres are suggested as one
possible way of providing for experimenta-tion-thinking-communication, which seemsto be an obvious line of development fora child.)
The students proceed from one activityto another, working through assignmentcards, chart activities, logic games andmeasurement experiences. When they finishan activity they are allowed to move to
Quest for an Improved Curriculum / MacLean 15
1. GEO-BOARD
Geo-boards are square (in this case 12"by 12") boards with nails driven in at 1"intervals. Assignment cards could be similarto the following:
any other centre provided there is an emptyspace. The teacher moves about the room,guiding, encouraging and helping when re-quested. In this classroom atmosphere theteacher's role is not to stop the childrentalking, but rather to ensure that there issomething worthwhile for them to talkabout. There is a place for lively discus-sion, and the quality of the discussion willbe directly dependent upon the qualityof the class-teacher relationship.
Do not expect any miraculous changein the behaviour of children immediatelyafter introducing these experience-centredclassroom groupings. Initially it may bebetter to try fewer groups and allow thepupils time to adjust to the freer atmos-phere and to develop respect for the opin-ions of others. This latter attribute followssmoothly the respect the teacher showsfor the student's point of view. Remember,real discussion, wherever it appears, isprovoked by experience. The situation sup-plies the starting point; the discussion thatensues should widen the children's ho-rizons and open up many new avenues ofexploration.
Here are brief desci iptions of the eightactivities.
Assignment:
1. On your geo-board make all the facesof the triangular prism.
2. Two of the faces are alike. Whatshape do they have?
3. Three faces are alike. What shapedo they have?
4. Count the number of sides on onetriangle.
5. How many faces do you have on arectangle?
Check your answers on the back ofthe card.
0
0
0 0f3
0 o I o
o
geo-boards
1 assignment cards
0
Set of solids
FIGURE 2
Students may use elastic to recreate theshapes of faces, answer the questions eitherorally or on paper, then check their an-swers.
2. ENLARGING SHAPES
Again students work from assignmentcards. They create designs using knownshapes and enlarge the same design ongraph paper.
Assignment Card Samples
1. Use graph paper todesign a houseuse a triangle and arectangle.
16 INVOLVEMENT
2. By making the sideslonger draw anotherhouse. It will bethe same shape, butit will be largerin size.
3. How much bigger doyou think the new shapeis than the first shape?
Possible Results
FIGURE 3
3. LOGIC GAME
Four logic games are set out. Each setcontains a large bristol board sheet withseveral shapes drawn on it, four differentcolours of yarn and a set of cards in thesame four colours as the yarn. The cardsmay be classified according to colour (red,
black
green, yellow or black), size (large orsmall) as well as shape. Students developtheir own methods of placing the shapesand yarn on the bristol board (Fig. 4).
4. GEOMETRY LOTTO
This game is played much like Bingoexcept that colour and shape are used in-stead of numerals. As flash cards are held
6 0WEI
A!OA
0 A () A-
A Gm ffI A
Ili % A Ma
flash card
disc to cover corresponding* shape and colour
FIGURE 5
up by one student, other students coverthe corresponding colour of the same shapewith plastic discs.
III
black yam
red,green& yellow
c2
FIG. 4.All other shapes remain outside the set.
black
black
red, green& yellow
Quest for an Improved Curriculum / MacLean 17
5. FACE FINDING, VIA ASSIGNMENT CARDS
At this activity two people work fromeach assignment card. The cards are placedon the table with a set of solids.
amshapedlike the
flat surfaceson some of the
solids. Make a listof the solids that
have faces shapedlike me. Check your
answers CM the back.
FIGURE 6
6. IDENTIFYING SOLIDS BY FACE SHAPES
Students begin at point A and walkaround the table from Card A to Card Ilisting the solids from which the surfaces(both flat and curved) have been traced.Then they repeat the rotation, this timechecking the answers on the backs of thecards.
a
Card front
A
Of=3
4 V
Card back
triangularprism
b
FIGURE 7
7. PERIMETER OF FACES
The pupils use yarn to find the distancearound the faces of the solids, and fastenthe yarn to a graph with gummed mosaicshapes.
i.e.
Distance around the faces of a square pyramid.
o
A
FIGURE 8
8. CREATING NEW SHAPES
Pupils work with solids, rulers and as-signment cards similar to the following:
1. Trace a square face from one of your solids.B
2. Label it in this manner: D"C3. How many sides has it? How many corners?
4. Join point A to point C. Can you see 3 shapes nowList them.
5. Join point B to point D. Can you find 9 shapes now?List the shapes.
6. Check your answer on the back.
FIGURE 9
ConclusionIn the 1966-67 session, the teachers
trained in Toronto in July 1966 will carrythe project further by experimenting withthe teaching of the topics and conceptsmentioned earlier. There will be threephases to the experiment, each for tenweeks' duration. In the fall, the materialwill be taught for ten weeks in westernOntario. It will then be revised, and triedout in modified form in eastern Ontarioin the winter term, again for ten weeks.Finally, after a second revision, it will betaught for a further ten weeks in northernOntario in the summer term.
By means of these successive revisions,it is hoped that the feasibility of both con-tent and approach will be put strongly tothe test. All those involved look forwardto the experimentation with intense interestand a fair measure of confidence in the lab-oratory approach.
is
Forum on teacher preparation Francis T. Mueller
Topics in geometr7 for teachersanew experience in mathematicseducation
CAROL H. KIPPSUniversity of California, Los Angeles, California
Carol Kipps, besides teaching courses atthe University of California, Los Angeles, is highly active with in-serviceprojects, among thew the Madison Project, the California Conferencefor Teachers of Mathematics, and UCLA Extension.
Can teachers capture by themselves theexcited enthusiasm shown by children inclasses sponsored by such curriculumgroups as the Madison Project or NuffieldProject? Can a teacher reared on lecture-drill-homework classes feel and show thedrama inherent in "I do and I understand"activities, in peer-group discussions, and inconcepts such as the concrete-ikonic foun-dation of abstraction? A new course atUCLA is focusing on these dynamic factorsso that teachers will know their value fromtheir own personal experiences and feel-ings.
Geometry is the vehicle, and grades K-8is the level. Geometry was chosen becauseprospective teachers have little backgroundin geometry and very often fear having toteach it. More and more geometry is beingintroduced in the elementary grades.Teacher-training research and the recom-mendations of professional organizations
18
show that geometry is more troublesomethan arithmetic or algebra.'
Modem curricula aimed at optimal se-quencing capitalize upon the child's earlycuriosity about shapes, the relations be-tween shapes, and patterns. Informal ex-ploratory geometry provides the necessarybasis for later symbolization and abstrac-tion. Also, an active learning approachrequires a different kind of teacher be-havior. When small groups of students areinvolved, the role of the teacher is more
1. See for example: Goals for Mathematical Edu-cation of Elementary School Teachers: A Report ofthe Cambridge Conference on Teacher Training(Boston: Houghton Mifflin Co., 1967). CourseGuides for the Training of Teachers of ElementarySchool Mathematics, rev. ed. (Berkeley: Committeeon the Undergraduate Program in Mathematical As-sociation of America, 1968). Carol 1Cipps, "Elemen-tary Teachers' Ability to Understand Concepts Usedin New Mathematics Curricula," THE ARITHMETICTEACHER 15 (April 1968): 367-71. Marilyn Suydam,"Research on Mathematics Education, Grades K-8,for 1968," THE ARITHMETIC TEACHER 15 (October1968):53144.
Topics in Geometry for Teachers / Kipps 19
demandingand far more rewarding. Asthe teacher moves from group to group lis-tening to the dialogue, she must considerwhen to ask a question, when to be silent,and when to withdraw altogether.
Many people tend to teach the way theyhave been taught. This can be a virtue aswell as a hazard. In the experimental classtaught at UCLA during the winter quarterof 1969 and then repeated in the summer,the class was conducted in the same waythat corresponding classes ought to betaught in the elementary school. For notonly can the process be modeled, but theteacher can evaluate it from personal ex-perience, choosing appropriate learningactivities and peer groupings with greaterinsight and precision.
Method
The course began with a discussion ofthe goals pupils should achieve by the endof the eighth grade. These behavioral ob-jectives free the teachers from completereliance on the basic text and focus onindividualizing the learning opportunities toachieve at the 100-percent level. The fol-lowing objectives were suggested as basicand minimal.
SHAPES IN GEOMETRY
1. The child will name tat or space figureswhen shown a physical model or a pictorialrepresentation of the following: triangle, quadri-lateral (square, rhombus, trapezoid, parallel-ogram, rectangle), circle, ellipse, cube, cone,rectangular solid, sphere, prism, pyramid.
2. The child can show where he would mea-sure a flat or space figure to find the length of itsdiagonals and its altitude. Also, the child candraw a line on a pictorial representation to indi-cate what he would consider the altitude or adiagonal of the figure to be.
3. The child will state whether flat or spacefigures have point (turning) symmetry or line(folding) symmetry and define these ideas.
4. Given a physical model, a picture, a verbaldescription, or a description in set language, thechild will state whether the figure is open orclosed and whether it is convex.
5. The child will construct a physical modelor sketch and will describe essential properties oftriangles and tetrahedrons; squares, rectangles,cubes, and rectangular prisms; circles, cylinders,cones, and ellipses; polygons, regular plane, andspace figures.
6. The child can name, define, and represent
P.7
the foundation elements of geometry: points,lines, line segments, rays, planes, and angles.
RELATIONS BETWEEN. SHAPES
1. The child will pick out congruent shapesand verify his decision by fitting. He will definecongruent figures as those that can be made tofit together and use the notation = for congruent.
2. The child will classify from a set thosefigures that are similar.
3. The child can identify examples, list ex-amples, and sketch examples of the followingrelations between geometric shapes: covering(tessellate), separating, inside, outside, on, andtopologically equivalent.
4. The child will identify, construct, sketch, oruse set notation to describe the possible intersec-tion sets for lines; li.;es and plane regions: linesand solid figures; two plane regions; and a planesolid figure.
5. The child will locate a given point on thenumber line. Given a pair of coordinates (x, y)that belong to the set of rational numbers. thechild will locate the given point on a numberplane.
MEASUREMENT
I. Given practical problems involving measure-ment, the child will experiment (estimating, se-lecting appropriate units, gathering data fromobservations, constructing number scales, andcomputing) and will attempt to verify his solu-tion in some fashion.
2. Given a standard unit, the child can roughlyapproximate and measure the length, area, andcapacity of common objects in appropriateEnglish and metric units. The allowable marginof error will depend on instruments used, if any.Since all measurements are approximate, thechild can cite methods for reducing errors ofmeasurement.
3. The child will verbalize that measurementis the process of selecting an appropriate stan-dard unit and finding a number that comparesthe two The child will state generalizations thatfollow, based on measuremerts and the tables orgraphs in which the measurements are recorded.
LOGIC AND PATTERNS
1. Given one, two, or three criteria for selec-tion, for example, color (e.g., red), shape (e.g.,square), and size (e.g., large), the child canclassify elements of a set in a way that showswhich elements have none, one, two, or threeattributes.
2. The child can determine the pattern of asequence of figures or numbers, continue thesequence for at least three more elements, andverbalize the criteria that he is using.
The big change in method in these ex-perimental classes was that the universitystudents were asked to work together insmall groups on learning opportunitiesdesigned for specific behavioral objectives.What lecturing there was had to do mostly
20 INVOLVEMENT
with learning theoryPiaget, Bruner,Gagne. While solving the problems, thestudents were encouraged to use manykinds of resources, e.g., concrete embodi-ments of various mathematics ideas suchas Dienes Multibase Arithmetic Blocks,Cuisenaire rods, and geoboards. A curricu-lum library was available that includeddictionaries, textbooks, and teacher guidesfrom the experimental projects such asNuffield and Madison Projects as well asthose for standard texts. The instructoracted as a consultant, answering questionswith questions and suggesting references.As discussions waxed about the definitionsof words such as diagonal and altitude, theiradequacy for the present two-dimensionalproblems or similar ones in three dimen-sions were debated. Preservice teachersenjoyed "playing teacher" with each otherand using clues to draw out those morenaive mathematically. In an in-service situ-ation, the instructor would have many "as-sistants."
A second key difference from the usualuniversity course in education was the useof evaluation as one of the learning activi-ties. As part of the course, the students notonly answered questions, but proposedthem! Many educational taxonomies sug-gest that asking an insightful questionaimed at a particular learning behavior isa much higher cognitive skill than answer-ing questions. After solving a mathematicsproblem in their peer-group situation, stu-dents were asked to devise a suitable testitem. At this point each person was askedto make an individual contribution, but byall means to consult with the group aboutit. While the test item was to be based onthe stand objective, it might well includeprior skills or knowledge and need not bea paper-and-pencil type of test item.
To illustrate this classroom activity, thefollowing is a page taken from the classnotes. The students were required to maketheir own geoboards and to bring them toclass for this type of activity. First, as a
Topics in Geometry for Teachers / Kipps 21
teacher, each was to read the specific be-havior objective. Next, as a pupil, eachhad to solve the problem, discussing it withmembers of s small group. Last, againin the role of a teacher, each participantwas to read the sample test item and writeanother or his own based on the objective.
SAMPLE CLASS WORKSHEET
Specific behavioral objective.Identify the di-agonals of various plane figures and define theidea of a diagonal.
Learning opportunity.Make the followingshapes on your geoboard. Transfer each figureto dot paper and draw in all the diagonal linesusing a red pencil. In which shapes are thediagonals of equal length?
D
C
Test item.Define what you mean by the termdiagonal. Draw a shape that has no diagonal andtell why it doesn't by applying your definition.
Test item.Does a diagonal necessarily bisectthe angle at that vertex?
Comments.A follow-up discussion mightdevelop the idea of whether the definition given
would work for space figures, or for a line joiningtwo vertices which is not a side.
Evaluation
This methods course in mathematicseducation at UCLA is organized primarilyto develop
(1) skill in planning and evaluatinglearning opportunities in mathe-matics for pupils;
(2) skill in using the Socratic approach;
(3) knowledge and skill in applying ba-sic concepts of informal geometry.
At frequent intervals, test items writtenby the students on the worksheets werereviewed by the instructor and returnedwith suggestions or comments. Gradeswere not given for these worksheets, be-cause it was hoped that the content andexperiences would foster interest in thelearning activity rather than in some ex-trinsic reward. Relevance to the specificobjective and mathematical correctnesswere checked. Creative style and elegancewere noted with positive comments. It wasa matter of considerable delight and aston-ishment to the instructor when not onemember of the class of 44 asked about
22 INVOLVEMENT
getting a "grade." This attitude is mostappropriate to a study of teaching mathe-matics, for the "new math" was introducednot merely on the claim that it representedmore important content but equally on theargument that it would build a new spiritof inquiry and creativity.
Grades for the course were assigned onthe basis of an examination focusing onthe methodology. Here is a sample ques-tion:
Select one of the objectives from above anddescribe three learning experiences that youcould provide to enable children to achieve theobjective. Write one learning experience foreach of the following levels:Enactive (sensory-motor)Ikonic (representational)Symbolic (abstract)
Continual feedback of geometric contentwas possible during the class periods, sincethe instructor could spend time with a smallgroup or an individual student at no ex-pense to the rest of the class. It is the rarestudent who will display his ignorance ina conventional classroom; but in the small-group approach, important questions arereadily raised and discussed. When a stu-dent works in a small group, he finds itmuch easier to express his confusionsenough to enable others to help him.
A term paper on a mathematics topicselected by the student was required by thecourse. The basic text was broadly repre-sentative of arithmetic, algebra, and geom-etry. Prior to the use of the geometry notes
0")
and discussion "in fours," few students hadselected a topic from geometry for thispaper. It is of note, then, that 28 out of the44 felt comfortable with geometry and didtheir paper in this area.
Conclusion
Prospective teachers have little back-ground. in geometry and very often fearhaving to teach it. It is apparent that manypotential teachers exposed to this approachwill come away with a good feeling aboutgeometry, some of the confidence neededto teach geometry, some exploratory activ-ities they can use with children, and a muchbroader knowledge than that normallyobtained from a straight geometry ormathematics course.
Some very important questions cannotyet be answered. Does this approach toteacher training tend to make the contenteasier to retrieve and to reconstruct? Willteachers include geometry in their lessonsand use appropriate manipulative appa-ratus and resources beyond the pupil text?A follow-up is planned in the form of aquestionnaire at the end of the first teach-ing assignment.
Student evaluations of the format wereenthusiastic, and they expressed attitudestoward studying mathematics as well. Onestudent wrote, "I think you should keep thecourse as it is, because the workshop at-mosphere is what is needed in the class-room, and we as prospective teachersshould practice in the same atmosphere."
t.
InstructionTechniquesInstruction! This is the key word in any teacher-learner relationship.The involvement of the learner with the subject matter is directlyrelated to the teacher's instructional procedure. For genuine involve-ment, good instructional techniques and principles must be generated.
In this book the articles on instruction are divided into two groupsaccording to whether they deal with techniques or rationale. These arearbitrary categories, designed to facilitate reader interpretation. Tech-niques for instruction are the vital elements that lead to and are derivedfrom involvement. A rationale for techniques is often difficult toexpress. Some fine teaching may occur fortuitously, but it is generallytrue that good teaching grows out of good rationale and commendableplanning. Without a rationale, a technique is scarcely better than agimmick, time-occupier, or dramatic presentation.
The involvement of the learner is a function of instruction. Withouteffective instruction, few children discover the joy and excitement ofmathematics. The techniques and methods of teaching geometry lendthemselves to a creative, mathematical involvement of the learner.
Early instruction in geometry should be termed environmental ge-ometry because much of the learning is fostered by challenging thechild to discover his surroundings. Born into and enclosed in a three-dimensional world, the child has ready access to geometric models.Under the skillful hands of a good teacher, he can be moved to theexcitement of informal geometry and later led to the thrill of elegantgeometric abstraction.
Mirrors, models, toys, and Mobius bands are some of the tools forenvironmental geometry. The following articles present accounts ofteacher-proven techniques that use these and other readily availablematerials. Reporting the use of a mirror and some cleverly developedcards, Walter describes an approach to symmetry that can be used witholder students as well as very young children. Spatial perceptions andideas of geometric transformation are by-products of this approach.
Alspaugh pairs two mirrors to give a kaleidoscopic effect when linesand models are viewed in them. Interesting ideas of symmetry, geo-metric form, and pattern can be developed using this approach. Tilt.content discussion of symmetry by Dennis nicely balances the tech-
23
24 INSTRUCTION TECHNIQUES
niques and methods of Walter and Alspaugh. The teacher will need tostudy the ideas of symmetry to be effective in teaching them. Interest-ingly enough, Forseth and Adams reverse the traditional method ofusing art to teach geometry. They first introduce various transforma-tions and then use them as a means to develop art form and pattern.The fusing of art and geometry is clearly presented.
Topological ideas are somewhat new to the study of geometry. Piagethas shown that children develop intuitive ideas of topological geometrybefore those of traditional Euclidean space. D'Augustine illustrateshow a teacher may develop topological generalizations using commonexperiences of the child. His development is followed by that of Clancyin which the M8bius band is used as a means to discuss dimension.From the study of dimensions, it is but a small step to the one- andtwo-dimensional creatures of fanciful writing developed by Carrol'sclass. Carrot relates the imaginative expression of her children to theideas of dimension and geometric abstraction.
Milk cartons are reasonably easy to obtain. With these convenientmaterials, a teacher guided by Walter's second article can developexcellent lessons dealing with nets, patterns, and generalizations. Asthe reader inspects these ideas, questions of volume and volumetricrelationship may be raised. Vincent, in a very brief note, gives a finemethod for comparing the volumes of the cylinder and the cone. Hertechnique is easily modified for other volumetric as well as area com-parisons.
Any survey of instruction in geometry would be incomplete withoutsome mention of the useful geoboard. Smith, in one of the earliestarticles on techniques, discusses and illustrates the geoboard (peg-board). Liedtke and Kieren also discuss the use of the geoboard in thecontext of early childhood education. The results obtained suggestexperiences that could be used for preschool and kindergarten educa-tion. Another device that can be used for developing ideas of shape,congruence, vocabulary, and relationships is a Tinkertoy set. Richardsshows how Tinkertoys can be effectively used in geometry. As withother devices, the geoboard and Tinkertoy set can be best used by theteacher who knows the content of the geometry to be taught. Jacksonprovides some of this content in his discussion of congruence and therelationship of geometry to measurement. In the final selection in thisgrouping of articles, Jackson weds the technique to the subject-mattercontent.
Taken together, these readings indicate the direction a teacher maygo. The ideas expressed by the authors are creative and provide motiva-tion for the development of a teacher's own approaches. Those whodeal directly with children will find this selection of articles of particu-lar importance and practical usefulness.
9()
An example of informal geometry:Mirror Cards*
MARION WALTEREducational Services Incorporated, Watertown, Massachusetts
Marion Walter is a part-time mathematics instructor at the Harvard UniversityGraduate School of Education. She is on the staffs of EducationalServices Incorporated in the Elementary Science Study and the CambridgeConference on School Mathematics. She teaches mathematics to thestudents in elementary school education at the' arvard Graduate School of Education.
The need for informal geometry, es-pecially in the earlier grades, is beingrecognized by educators, psychologists, andmatnematicians. The Mirror Cards .verecreated by the author to provide a meansof obtaining, on an informal level, somegeometric experience that combines thepossibility of genuine spatial insight witha strong element of play.
The basic problem posed by the MirrorCards is one of matching, by means of amirror,' a pattern on one card with apattern shown on another card. For ex-ample, can one, by using a mirror on thecard shown in Figure 1, see the patternshown on the card in Figure la?
FIGURE 1 FIGURE la
This work was begun while the author was work.ing during the summer of 1963 with the ElementaryScience Study, a project supported by grants from theNational Science Foundation and administered by Edu-cational Services Incorporated, a nonprofit organizationengaged in educational research. She would like tothank the members of the group she worked with thatsummer and the group in optics of the previous sum-mer for their help and encouragement; she is especiallygrateful to Professor Philip Morrison, Mrs. PhylisSinger, and Mrs. Lore Rasmussen,
The reader should have a small rectangular pocketmirror handy before reading on.
25
The problems range from the simplest, suchas the one shown above, to more difficultones, such as the one shown in Figures 2and 2a.2 Some patterns are possible tomatch and others are not.3
=FIGURE 2
FIGURE 2b
FIGURE 2a
FIGURE 2c
Using the mirror on the card shown in Figure 2, whichof the patterns shown in Figures 2a, 2b, and 2c canyou make?
Each box of Mirror Cards contains, in addition tomirrors, 170 cards arranged in fourteen different sets.Although the instructions for the sets vary, the basicproblem is the same for all the sets and is the onedescribed above. A trial edition of Mirror Cards wasproduced and copyrighted by the Elementary ScienceStudy in June 1965. They are being used on a trialbasis in over 250 classrooms around the country. Theauthor would like to acknowledge the help receivedfrom Mrs. A. Neiman, Mrs. F. Ploye'r, and Mrs. J. Wil-liams in editing the guide and producing the cards.
The position of the pattern relative to the edge ofthe card is to be ignored.
26 INSTRUr LION TECHNIQUES
a.
ry
We have noticed that the children usuallyfind the colors and shapes pleasing andenjoy the challenge presented by the cards.They do not think of this work as "mathe-matics," and they often find the cardsstimulating over and above the actualgeometry involved. The cards may be ameans of reclaiming the children who al-ready dislike mathematics or are boredor frightened by it. The cards do not callfor verbal response from the children, andno mathematical notation is needed. Closerconnection with science and mathematicsclasses will be explored by the author inthe future, since the cards can give insightinto some mathematical and physicalprinciples.
One advantage that the cards have isthat the children can see for themselveswhether or not they have made a pattern.They don't need to resort to authority tocheck whether they have solved the prob.-lem correctly. In addition, while playingwith the cards they are, in effect, constantlymaking predictions and are immediatelyable to test these predictions and amendthem, if necessary; and it is fun to do so!Thus, while working with the cards theyshould gain confidence in their own powers
and learn through experience the natureof the scientific method.
While moving the mirror around on thecards, the children notice and experimentwith the position of object and image inrelation to the edge of the mirror. Theplayer can decide where to place the mir-ror; and he soon learns that he can controlits position, but that for any given positionof the mirror he cannot control the positionof the image!
The students also learn that a mirrordoes not carry out a translation. (See Figs.3, 3a, 3b.)
FIGURE 3 FIGURE 3a Flamm 3b
Can one by using mirror on Figure 3 make thepatterns shown in Moores 3a and 3b? alas, themirror does not carry out translation!
They learn by experience that congruenceof two parts is a necessary but not a suf-ficient condition for a pattern to be madeby use of a mirror. Most children do notknow the expression "symmetric with re-
Informal Geometry: Mirror Cards / Walter 27
spect to a line" or "reflection in a line."They may, nevertheless, by using the cards,gain experience that will enable them tounderstand the concepts that these expres-sions describe. This does not imply thatthey could give, or should be expected togive, a formal or verbal definition of theseexpressions. Eventually they do notice thatfor a pattern to be reproducible by use ofa mirror, it must have two parts that Heon either side of some line and that thesemust "match exactly." They soon learn,for example, that the pattern shown inFigure 4a cannot b6 made from the patternin Figure 4, and they probably have a goodfeeling for why this is so.
FIGURE 4 FIGURE 4a FIGURE 4b
Pattern 4a cannot be obtained from 4. What aboutthe pattern in 4b?
The cards provide opportunity to prac-tice recognizing congruent figures and se-lecting parts of figures congruent to another.
FIGURE 5 FIGURE 5a5a
When mud you place the mirror In Figure 5 to seethe pattern shown In Figure 3s?
The children must be observant, not onlyabout a shape and the position of thatshape, but also about its colors. Some ofthe patterns match in shape but not incolor.
They may also notice a variety of ge-ometric properties of figures. Consider, for
example, the circle. By putting the mirroron a diameter they can see the whole circle.More than that, any diameter will do andany chord not a diameter will not do. Thismay give young children their first feelingfor a diameter of a circle, long before theyknow the word "diameter."
With the diamonds (see Fig. 6) theynotice that there are two places where themirror may be placed to enable them tosee the whole diamond. On the other hand,
OFIGURE 6 FIGURE 7
the pattern shown in Figure 7 does nothave this propertyto the surprise ofmany!
Or, again, take the triangle (see Fig.8): the children may notice that the effectof putting the mirror along AB is in someway "the same" as that of putting it along
FIGURE 8
BC, but that it is quite different from thatof putting it along AC. What about BD?
Other patterns on the cards, such as theladybugs, arrows, etc., can b: explored insimilar ways.
For a few cards the children can obtainpatterns that look somewhat like the onerequired but are not congruent nor actuallysimilar in the mathematical sense. I intendto devise cards where congruent and similarpatterns are obtainable, and similar but notcongruent ones.
Unfortunately, none of the present cardshave circles with arrows on them to show,
90
28 INSTRUCTION TECHNIQUES
perhaps more clearly, that the orientationgets reversed under a mirror mapping orreflection. Thus Figure 9 becomes Figure9a.
FIGyRE 9 FIGURE 9a
The concept of orientation is, of course,brought out by the cards, although thearrows are not used for this purpose. Oftenpatterns with orientation reversed and notreversed are included to make the ideamore obvious. Examples taken from theladybug and the circle set are shown below.
FIGURE 10 FIGURE 10a FIGURE 10b
Can one by using the mirror on Figure 10 obtain thepatterns shown in Figures 10. and 10b respectively?
FIGURE 11 Rama Ila FIGURE 1lb
Can one by using the mirror on Figure 11 obtain Alepatterns shown in Figures 11r. ,.nd llb respectively?
The fact that a mirror does not carryout a rotation in the plane is often maskedby the symmetry of the figure. For ex-ample, one can make Figure 12a fromFigure 12, but not Figure 13a from Figure13.
04*FIGURE 12 Flame 12a
The imagined placement of points "A" and "B" illus-trates the fact that the mirror does not "rotate" thefigure. Actually the mirror "flips" the image. (Points"A" and "B" are not marked on the actual cards.)
i5,1=11
FIGURE 13 FIGURE 13a
The cards may be used at any age level.They have been used by children as youngas five and by sophisticated professionalscientists or mathematicians. It is interest-ing to note that some adults who "know
.M
Informal Geometry: Mirror Cards / Walter 29
all the rules" verbally (such as "There mustbe a line of symmetry" or "Image distance= object distance") often have more dif-ficulty. in working through the sets thanchildren who have not yet memorized suchphrases. The one barrier to the effectiveuse of the cards by adults appears to bean ingrained habit of respect for authority.Adults often do not want to rely on theirown ability to see whether they have madea pattern correctly.
When the children find the problemsbecoming too easy, they may want to addthe rule, "You may put the mirror downonly once for each pattern," so that all thetrial and error must go on in their heads.They may wish to make some of their owncards. When, as happens often, childrenare able to predict withnut using a mirror
at all whether a pattern can or cannot bemade, they will have a clear demonstrationof the power of reasoning based on ex-periencei.e., that it is possible to predictresults with confidence by thinking ratherthan doing! (And they are able to checktheir thinking if they wish.) In this waythey are savoring an essential part of thenature of rational thought.
There are many questions that still needto be answered. I meniion just a few. Willuse of the cards make children more ob-servant about other geometric pattern! IWill it enable them to see figures withinfigures more easily? Does it improve theirability to visualize? Will they be able todescribe patterns more clearly? Will ithelp or hinder cl-adren with reading dif-ficulties?
EDITORS' NOTE. Current information about Mirror Cards ( #18418)and Mirror Cards Teachers' Guide (#18417) can be obtained fromWebster Division, McGraw-Hill Book Co., New York.
is
tt
Kaleidoscopic geometry
CAROL ANN ALSPAUGH
Caro! Ann Alspaugh is doing research for her doctoral dissertationat the University of Missouri, Columbia, Missouri. She formerly taughtmathematics and computer programming at Southwest Missouri StateCollege and at other colleges and universities.
Kaleidoscopi.4aAeometry is an interestingtype of mirror getmetry that could be uti-lized to introduce tiometrical topics suchas regular polygons, 1A,:iordinates of pointsin a plane, reflections, symmetry. Mostchildren enjoy the pos4le explorationsoffered by this geometry; ;which wouldmake it useful to the teacheN desiring todevelop interest and motivatiot4when in-troducing new materials.
The physical materials needed fc-tk thisgeometry are simple and inexpensive. Nromirrors are hinged together either by gluirrz.them to one piece of cardboard or by using',44masking tape. In this way, the mirrors maybe set up at any desired angle on a table orfolded together to facilitate easy storage.Unbreakable mirrors are now available onthe market and could be used.
Students at the fifth- or sixth-grade levelwould be ready for this geometrical experi-ence. By this time, the student would havestudied various polygons, be familiar withthe use of a compass and protractor, andknow that there are 360° in a circle. Aninteresting point about this unusual geom-etry is the seemingly unlimited number ofmathematical concepts that can be illus-trated as the angle between the two mirrorsis allowed to change. When these mirrorsare placed in front of them, most studentsand teachers will find some fascinatingchallenges as they look into the new worldappearing before their eyes. A few of theconcepts that are possible to illustrate arepresented on the next page. However, formaximum learning and appreciation on the
, 5
part of the student, he should be allowedto discover these concepts himself by physi-cal manipulation of the materials in alaboratory situation.
1.
2.
3.
30
Using a wood cube (or a similar object)placed in he center of the table betweenthe positiouzd mirrors, it will be ob-served that athe angle 0 between themirrors increasetk, the number of cubes(images) decreasL to a minimum of2 when 0 = 180°, L:nd increases with-out bound as B 00.
The number of images, t'..07rhere I in.cludes the original object): is relatedto the angle 0 by the followine.iDrmula:I.0 = 360°.When 0 = 120°, three images are sC-?,n,and the following geometrical construct,tions are possible by drawing one linesegment on the table between the mir-rors (a ruler or extendable curtain rodcould be used as a line segment):
Kaleidoscopic Geometry / Alspaugh 31
a. equilateral triangles (any other kindsof triangles are impossible to con-struct)
b. circles (by making one arc betweenthe mirrors)
It is interesting to discover that squares,pentagons, etc., cannot be constructedwhen the angle between the mirrors is120°. When the angle is varied, it willbe noted that the constructions that canbe made will also vary. However, it isalways possible to construct a circle, andany polygons that can be constructedwill have equal sides.
The preceding table might be an exam-ple of a student's summary of the con-structions he discovered that were possi-
ble to make. Of course, the table couldbe extended.
5. With the mirrors at an angle of 90°,graphing in the coordinate plane canbe nicely illustrated. For this purpose,the teacher or student should cut a pieceof poster paper to lie on the table andfit between the mirrors. Then a coordi-nate system may be ruled off on thepaper, using the point where the mirrorsmeet as the origin. As a point is locatedon the paper, the reflections will simul-taneously locate the points in the otherthree quadrants.
Many types of poster paver overlayscwild be designed by the creative studentand teacher to illustrate mathematical andartistic ideas. Children who would be tooyoung to make some of the suggested tablesand generalizations would enjoy creatingor copying kaleidoscopic designs andpolygons using colored parquetry blocks,straws, sequins, etc.
It is hoped that this paper has suggestedthat the principles underlying the designof a child's toy kaleidoscope have manyteaching possibilities for elementary mathe-matics.
Informal geometry through symmetry
J. RICHARD DENNISUniversity of Illinois
I. Richard Dennis is coordinator of the Hawaii-UICSMCollege-School Cooperative Project in Honolulu, Hawaii,
The geometry component of the elemen-tary school program is th.:, basis of muchdiscussion today. Many of the efforts inthis area of mathematics have approachedelementary school geometry from the pointof view of Euclid's postulates. Another ap-proach to geometric topics is based on theidea of symmetry.* A unit using this ap-proach would involve five to six weeks ofclass time and would do much to augmentthe elementary school geometry program.
The first idea that we need to make clearis the meaning of the phrase "exactly alike"as it is used in geometry. Students alreadyhave opinions on the meaning of thisphrase, but their opinions frequently differ.Agreement is needed on an experimentaltest, the results of which will be acceptablein cases of differing opinions.
FIGURE
A pair of figures that appear to be ex-actly alike are shown in figure 1. How canwe tell for sure? The test that we agreeupon is to trace one figure 9..nd then try tomatch the tracing with the other figure.
* The ideas presented here grew out of my asso-ciation with the University of Illinois Committee onSchool Mathematics, I wish to express my gratitudeto Professor Max Beberman for the opportunity toparticipate in the UICSM activities and for his gen-erous counsel.
32
This match need not be achieved in anyparticular position or orientation. It maybe possible to achieve a matching in severalpositions. It must be possible to achieve amatching in at least one position (fig. 2).
FIGURE 2
When it is possible to exactly match atracing of one figure with another figurewe say that the figures are congruent, andthe various matchings are called congru-ences.
Teachers should create many opportuni-ties for children to experiment with the"trace and try to match" process describedabove. It is from such experiments thatbasic intuitions about congruence are de-rived. These intuitions will become a foun-dation for the discovery and exploration ofmore complicated properties of geometricfigures. For example, by experimentingwith a tracing, congruences can be immedi-ately separated into two types:
1. face-down congruences for which thetracing must be turned over to make itmatch (fig. 3)
FIGURE 3
Informal Geometry through Symmetry / Dennis 33
2. face-up congruences for which the trac-ing is not turned over, just movedaround to make it match (fig. 4)
FIGURE 4
Having distinguished between these twotypes of congruences, we concentrate onthe effects of each of them. We ask ques-tions like the following:
1. For the face-up congruence of these figures,with what part of triangle DEF does thetracing of segment AB match?
2. For the face-down matching, with what partof triangle DEF does the tracing of angleACH match?
3. Again for the face-down matching, withwhat part of triaagle DEF does the tracingof segment AC match?
Some important facts to observe arethese:
I. For both matchings,matches point E.
2. For both matchings,AC matches segmentindividual points ofmatch in the same way
the tracing of point B
the tracing of segmentDF but the tracings ofthe segments do not
Work of this nature gives an introductionto the notion of corresponding parts for acongruence, and, of course, since the trac-ing is, used to match these parts, corre-sponding parts of congruent figures arecongruent.
The next step in this development is toapply the notions of congruence and cor-responding parts to a single figure ratherthan to two figures. Specifically we study
self-congruences of a figure. Again theseare of two typesface up and face down.
Figures with face-down self-congruenceshave a very important property. There is aline each of whose points corresponds withitself for that face-down matching. For ex-ample, consider the triangle in figure 6.
AA C
FIGURE 6
For the face-down self-congruence of thistriangle, point B corresponds with itself.There is also a point of segment AC thatcorresponds with itself for this matching.In fact, if we draw a line through these twopoints, each point of that line correspoildswith itself for the face-down congruence.It is such lines that we shall call lines ofsymmetry.
Through this definition, each line of sym-metry is associated with a face-down con-gruence of a figure. So when given an ex-ercise such as to find all lines of symmetryfor a particular figure, the student needonly count the face-down matchings of atracing. After a little practice, students eas-ily move to the stage of just thinking aboutthe tracing. It is important to note, how-ever, that when all else fails, a tracing willmake answers to questions quite obvious.
We are now ready to begin our study oftriangles. It is assi.laed that many of thefigures used in the previous work have beentriangles. Students should see and experi-ment with triangles with no lines of sym-metry, triangles with one line of symmetry,and triangles with three lines of symmetry.An important exercise is to have studentstry to sketch a triangle with exactly twolines of symmetry. Another important ex-ercise is to try to sketch a triangle with asymmetry line that does not go through avertex.
34 INSTRUCTION TECHNIQUES
You will recognize the triangles withone (or more) lines of symmetry as thoseusually called isosceles triangles. Thosewith three lines of symmetry are usuallycalled equilateral triangles. Having lookedat the possible line symmetries for triangles,students are in a position to find propertiesof each type of triangle. 1 example, con-sider a triangle with one line of symmetry,i.e., one face-down self-congruence (fig. 7).
FIGURE 7
1. There is a pair of congruent sides, becausefor the face-down self-congruence the trac-ing of one side of the triangle matches an-other side of the triangle.
2. There is a pair of congruent angles, becausefor the face-down self-congruence the trac-of one angle matches another angle of thetriangle.
3. The symmetry line goes through the middlepoint of one side, because for the face-downself-congruence the tracing of one part ofthis side matches the other part of this side.
4. The symmetry line bisects one angle (forsimilar reasons).
5. The symmetry line "divides" the triangleinto two congruent regions (for similar rea-sons).
At this stage some other important ques-tions should be considered:
1. Could a triangle have a pair of congruentsides without having a line of symmetry?
2. Could a triangle have a pair of congruentangles without having a line of symmetry?
For each of these questions, evidence iseasily gathered from an experimentalsketch and a piece of tracing paper. Theproperties of triangles with three lines ofsymmetry are presented in a like manner.
Before classifying quadrilaterals it is con-venient to introduce the notions of per-pendicular and parallel lines. For perpen-dicular lines we look at pairs of lines andask: In what cases is one line a line ofsymmetry for the other? For example, hereare a dashed line and a solid line (fig. 8).
FIGURE 8
The dashed line is a line of symmetry forthe solid line. We can also show this witha tracing.
Here it is important to have made clearthe idea that, at best, pictures of lines leavemuch to be desired. For figures like tri-angles, lines of symmetry appear cut thepicture in half. For lines, it is no longerpossible to judge lines of symmetry bychecking to see if the picture is cut in half.An important observation is that when oneline is a line of symmetry for another, thetwo lines make square corners with eachother. We say that two lines are perpen-dicular whenever one is a line of symmetryfor the other.
For parallel lines we again look at pairsof lines, but this time we ask if the lineshave a line of symmetry in common. Againwe can use a tracing (fig. 9). Parallel lines
FIGURE 9
are those lines which do have a line ofsymmetry in common.
Informal Geometry through Symmetry / Dennis 35
The face-down self-congruences gave uslines of symmetry. One type of face-up self-congruence is particularly important forthe study of quadrilaterals. Sometimes afigure has a face-up self-congruence for ahalf-turn of a tracing. When this hap-pens there is a point which correspondswith itself. Such a point is called a centeror point of symmetry (fig. 10). Again it is
FIGURE 10
important to examine several figures forpoints of symmetry, and to look for /.3or-responding parts under these half-turn self-congruences.
As we begin the study of quadrilaterals,we notice one important feature not foundamong triangt.s. Quadrilaterals may havelines of symmetry that do go through ver-tices or lines of symmetry that do not gothrough vertices (fig. 11). So we introduce
FIGURE I I
the phrase diagonal symmetry line for thosethat do go through vertices, and the phrasenondiagonal symmetry line for those thatdo not go through vertices.
When we classify quadrilaterals, we findthose with:
1. No lines of symmetry.
2. One line of symmetry.
3. Two lines of symmetry.
4. Four lines of symmetry.
5. A point of symmetry.
Notice that when quadrilaterals have twoor more lines of symmetry they also havea point of symmetry. There are no quadri-laterals with exactly three lines of sym-metry.
As was the case for triangles, the usualproperties about congruent and parallelsides, congruent angles, bisecting diagonals,perpendicular diagonals, etc., follow fromthe corresponding parts for the various self-congruences.
This has been a brief discussion of thetopics involved, an appropriate sequencefor these topics, and some sample questionsat the various points of the development.Probably the most important word of cau-tion to the teacher is to allow pl,mty oftime for making experimental sketches andfor conducting tracing experiments. Even-tually many students reach a point wherethey can answer questions by merely con-ducting a thought experiment, but very fewof them begin at this level.
Fl 0
Symmetry
SONIA D. FORSETH and PATR][CIA A. ADAMSUniversity of Minnesota, Minneapolis, Minnesota
Sonia Forseth is the art director at the Minnesota Mathematicsand Science Teaching Project (MINNEMAST).
Patricia Adams is an editor for the MINNEMAST Projectand is working on her master's degree in journalism.
Combining art projects with math andscience is an ideal way of promoting crea-tivity in children. They become aware ofhow to use math and science; they learnto make and solve creative problems; andthey have fun while doing it!
Symmetry is a basic geometric conceptand a common, pleasing design found innature, especially in leaves and flowers.Art activities using concepts of symmetrycan be used with almost any grade andwith every child, from the slowest to thebrightest. The children will have fun mak-ing fascinating designs, while at the sametime learning craftsmanship skills andachieving ideas for various types of sym-metry. With some modifications, theseactivities can be used with any given grade.
Begin by showing the children examplesof pictures of strip patterns in such thingsas architecture, pottery, tapestries, fabric,leaves, flowers, caterpillars, or centipedes.Older children may enjoy studying sym-metry in physics, chemistry, and biology.Molecular structure and X-ray crystallog-raphy provide examples of geometric sym-metry.
For materials, the children will needsheets of paper (such as construction,butcher, or bond), 3-by-5 inch index cards,color crayons or felt-tip pens, temperapaint, scissors, and common pins.
Here are seven basic strip patterns. Theprocess of making each pattern is calledan "operation" because you follow certain
36
rules. The patterns vary in difficulty, sochoose whichever one best suits your class.
1. Repeating Patterns, a very easy one,is made by simply moving the stencil afixed distance each time you trace it (fig.1). Have each child make a stencil bycutting an asymmetrical design in an indexcard. The children can save the cut outpart for other patterns.
LrFm. 1. Repeating Patterns
2. Translation Reflections is shown infigure 2. Trace the design, flip the stencilforward, and trace it again so that it lookslike a reflection of the first design. If youare teaching this pattern to young children,have them draw a line across their papers(fig. 2). Ask them what they notice whenthey fold the paper in half on the line andhold it up to the light.
Flo. 2. Translation Reflections
3. Two Reflections is shown in figure 3.Draw the pattern, flip the stencil over alongthe right edge, and draw it again. This isyour first reflection. Now flip the stencil
again along the right edge and draw thepattern. Now you have two reflections. Thechildren can test their reflections by draw-ing lines between each pattern (fig. 3) andfolding the two outside patterns into themiddle.
Fro. 3, Two Reflections
4. Three Reflections is more difficultand probably for older children only (fig.4). For this one the children will needtempera paint. Have them fold a sheet ofpaper, dividing it into eight parts (fig. 5).
Flo. 4. Three Reflections
A C 1 E G
B i D i F H
Line I
Fro. 5. Procedure for Three Reflections
Put a blob of tempera paint in the topleft box (labeled "A" in fig. 5). Fold thepaper in half on line 1 and press to repro-duce the paint blob on box B. Unfold thepaper and you have the first reflection.Now fold the paper on line 2, press, unfold,and you have the second reflection printedin boxes C and D. (If the paint is drying,apply more paint on boxes A, B, C, andD). Next fold on line 3 and press toproduce the third reflection in boxes E, F,G, and H. After the children understand
Symmetry / Forseth and Adams 37
the operation, they can make patterns withtheir stencils.
5. Half-Turns is an easy pattern (fig.6). Trace the stencil. Turn it 180° to theright and trace it again in this position.Move the stencil to the right while turningit 180° each time you trace it.
garz crozFIG. 6. Half-Turns
6. Half-Turns about a Point is a moredifficult pattern (fig. 7). Fold a sheet ofpaper to divide it into six parts (fig. 8).Mark points A, B, and C as shown. Labelthe paper "part 1, 2, and 3" along thebottom. Use a pin to hold the stencil atpoint A while you trace the design in thetop section of the paper. Without removingthe pin, turn the stencil 180° and trace thepattern in the bottom box. Remove thestencil and cut on line 1. Holding the pinat point B, turn part 1 of the paper 180°.Lay part 2 of the paper over part 1 andtrace these two patterns. Next cut online 2 and rotate part 2 of the paper 180°at point C. Lay part 3 of the paper overthis and trace these two designs.
Now tape the three parts of the paperback together as they were originally andyou have a design made following theoperation of half-turns about a point. Ask
Flo. 7. Half-Turns about a Point
'An. 2
A B
Pert 2 pert 3Pert I
Flo. 8. Procedure for Half-Turns about a Point
38 INSTRUCTION TECHNIQUES
the children how this design differs fromthe translation reflections design. At firstglance the two patterns may look verysimilar to the children.
7. Glide Reflections combines the rulesof moving a certain distance and reflection(flipping) the pattern along an axis (fig.9). Draw a line on your paper to use as anaxis. The pattern should be above the line(with part of it touching the line) as youtrace it. Move the pattern along the linea certain distance to the right. Reflect (flip)it below the line and trace it again. Movethe pattern along the line (below it) anequal distance to the right. Reflect it andtrace again above the axis. Continue mov-ing and reflecting the pattern alternatelyabove and below the axis.
Flo. 9. Glide Reflections
After the children understand some ofthese basic operations, they should makeup their own. There are dozens and dozensof operations that can be devised usingthese basic ones. This exercise will cer-tainly bring out the creativity in your chil-
dren. It also gives practice in inventingrules and carrying out procedures.
Now that the children know what to lookfor, they will enjoy finding examples ofstrip patterns in nature and art. Theyshould be encouraged to bring all theexamples they can find and to tell the classwhat operation was used to produce thepattern.
The children can think up games usingthese patterns. They might try to predictwhat a pattern will look like when usinga certain operation. Then they can carryout that operation and see how correct theywere. They can play games with partners.One child chooses the operation, and hispartner produces the first step. The firstchild does the next step, and so on.
These strip patterns suggest many waysof decorating the classroom. Encourage thechildren to make things using the patterns,such as mosaics, placemats, wall hangings,and paper chain designs. They can maketheir patterns using different techniques,such as block, potato, and sponge printing.Have them try rubbings (crayons rubbedover paper placed on object) with leavesand twigs. Just remind them that they mustchoose or invent an operation and followthe rules throughout the pattern. After thechildren make a few strip patterns on theirown, they can explain the operation to theclass or let the children guess the operation.
c.
Developing generalizationswith topological net problems
CHARLES H. D' AUGUSTINE Ohio University, Athens, OhioProfessor D' Augustine is a member of the college of education at Ohio University.
Topological net "games" are being in-cluded in contemporary junior high andelementary school mathematics texts. The"game" aspect is of questionable pedagog-ical soundness. Wouldn't these topics beof more value to the student if they wereintroduced systematically and in such away as to lead the student to make certainlogical generalizations? Wouldn't thesetopics be of more value to the student if he"possessed" certain generalizations whichwould enable him to create and solvehighly sophisticated net problems inde-pendently.
This article will be an attempt to showhow one type of net problem (that oftraversing nets) could be developed inboth a logical and an intuitive way to leadto generalizations which would enable astudent to create and solve "related" netproblems.
This article will presuppose that stu-dents have had previous experience withend points and cross points.
All examples will be intuitively treatedas follows:
Each situation will revolve around aperson attempting to deliver papers tohomes on a paper route. The followingrules will hold with respect to deliveringpapers:
1 In no instance may the person deliveringpapers get off his bicycle; he must con-tinue riding until he can ride no further.
2 In no instance may the person deliveringpapers leave the road.
3 In no instance may the person deliver-ing papers go over a portion of a roadwhich he has previously traveled.
With this basic set of rules we can nowstudy specific situations of paper delivery.
Example 1. Two end point problems(Fig. 1). Joe starts delivering his papers at
Figure 1
end point A. Can he deliver papers to thehouse at point M? (Yes) Where will hehave to stop his bicycle? (Point B)
Example 2. Three end point problems(Fig. 2). Joe starts delivering his papers at
39
Figure 2
end point U. Can he deliver papers to bothhouses? (No) What is the largest numberof houses that Joe could deliver papers to?(One)
Generalization: I am thinking of adelivery route with 5 end points. If there isa house at each end point, could Joe deliverpapers to each house? (No)
If he must start at one of the end points,what is the largest number of houses towhich he could deliver papers? (Two)Why? Because as soon as he reachesanother end point he must stop.
40 INSTRUCTION TECHNIQUES
What is the largest number of endpoints a problem can have in order to havea solution? (Two)
To make further generalizations it willbe necessary that we intuitively developthe idea of odd and even cross points.
Example 3. Joe is on one "road" (Fig.3). When he reaches the cross point he will
Figure 3
have a choice of two "roads" to take.Since 1+2 =3 and 3 is an odd number, wesay that cross point A is an odd crosspoint.
Example 4. Joe is on one "road" (Fig.4). When he reaches the cross point he will
Figure 4
haves choice of 5 different roads to take.Since 1+5 =6 and 6 is an even number, wesay that cross point C is an even crosspoint.
To focus our attention on various typesof cross points we will introduce a newrule. When Joe reaches or is in a dottedportion he may ride over this portion asmany times as he desires.
Example 5. Is point B in Figure 5 aneven or an odd cross point? (Odd) If Joestarts in the dotted area, can he deliverpapers to all three houses? (Yes) Wherewill his problem end? (At cross point B) IfJoe starts delivering papers at cross point
Figure 5
B, will his problem end at this point? (No!He will end up in the dotted area.)
Example 6. Is point C in Figure 6 an
Figvr1 6
even or an odd cross point? (Even) If Joestarts in the dotted area, can he deliver allthe papers to all the houses? (Yes) Wherewill his problem end? (In the dotted area)
1.$ this answer different from thesitt.eion with the odd cross point whereJoe started in the dotted area? (When hedidn't start at an odd cross point, he endedthere.)
If Joe starts at cross point C, can hedeliver papers to all the houses? (Yes)Where will his problem end? (At crosspoint C) How is this answer differentfrom the situation where Joe started at anodd cross point? (He ended in the dottedarea.)
Let us see what sort of generalizati -nswe can make about even and odd crosspoints.
Pretend that we begin a problem at across point whose number is N.
We will designate the road on which weleave the cross point as 1; the next road wecome back to the cross point on we willdesignate 2, and the next road we leave thecross point on we will designate 3, etc.
Would we be leaving or coming back on
II 5
Generalizations with Topological Net Problems / D'Augustine 41
a road designated 17? (Leaving) How dowe know this? (Because 17 is an odd num-ber, and we leave on the odd-numberedroads.)
Suppose that N=128. Would we beleaving or coming back on the road desig-nated 128? (Coming back) If we start ourproblem by leaving from an even crosspoint, how will our problem end (Condi-tion: assuming that we always get back tothis cross point)? Ti t. problem will end atthis cross point, because we won't have aroad to leave on.
If we start a problem at an odd crosspoint,, how will the problem end (Condi-tion: assuming that we always get back tothis cross point)? One cannot tell how theproblem will end, but one can say that theproblem will not end at this cross pointbecause one will always have a road toleave on since this is an odd cross point.
We are now in a position to make ageneralization concerning starting a prob-lem at an end point.
If we start at an cross point,(even, odd)
our problem will end at this cross point,providing we can always get back to thecross point. (even)
If we start at an cross point,(even, odd)
our problem will never end at this crosspoint. (odd)
The question now arises what happenswhen we come into even and odd crosspoints from some other portion of the net.
With the previous analysis of startingfrom a cross point as a background, see ifyou can now arrive at the generalizationthat one ca .. make with regards to cominginto odd and even cross points from someother artion of the net. Do not read fur-ther until you can.
As you have probably discovered, if youdo not start out on odd cross points thenyou must end your problem there, becauseas soon as you use one road to get to thecross point, then the problem becomes oneof leaving from an even cross point.
Example: Assume you ride your bicycleup to a "2N-I-1" cross point. Since youused 1 road in arriving at the cross point,you have 2N choices of leaving the crosspoint.
Similarly, if you do not start at an evencross point you do not have to end yourproblem there, because as soon as you ar-rive at the cross point the problem be-comes a "2N-1" cross point problem, andleaving from an odd cross point insuresthat one will not end there.
Now let us abstract the problem in thefollowing manner. Let "e" designate anend point, "0" an odd cross point, and "E"an even cross point. (Subscripts will de-note the different members with the sameclassification.)
Pretend that we have a single net com-posed of the following types of specializedpoints:
e1 E1 Es Os Eg
What point or points (based on previousgeneralizations) would be the most optimalplace to start? (Hint: If we don't start atone of these points we would certainly endat one of these points. But if we start atone of these points we don't have to endthe problem at the same point. Answer:Either the es or Os, because if we start atEl or Es or E3, we must end the problem atEl or Es or Ea respectively; but we alsohave to end the problem at both e1 and 01,and thus the problem could have no solu-tion.)
If we start our problem at e1, wherewould our problem end? (At 01)
If we start our problem at 01, wherewould our problem end? (At es)
Each problem can have only one begin-ning and one ending. In each of the follow-ing problems select one of the optimumbeginnings and then proceed to analyzewhether the problem has a solution.
1 E1 el Es Ea es 01 022 EL es Es E. ?:3 Es Es ES E4 Es4 01 02 03 E1 Es es
42 INSTRUCTION TECHNIQUES
(Problems 1 and 4 do not have solu-tions. In problem 3 you would end theproblem at the even cross point from whichyou started.)
We are now at a point at which thestudent can proceed creatively. We willneed to introduce the following notation:E14 will mean it is a "four" cross point(that is, four roads come together to makethis cross point), and the 1 will retain itsprevious meaning. 047 will mean it is a"seven" cross point (that is, seven roadscome together to make this cross point).The "e" will retain its previous designa-tions as these use only one type of endpoint.
Students are now in a position to createand attack problems, such as:
Given: el E18 E24 E38 013
1 Is this a solvable problem?2 How many different solutions does this
problem have?
3 How many different bicycles couldtraverse this net if each bicycle mustalways be on a different portion of theroad at the same time? (Hint: Considerhow at problem must end.)
4 If the problem is not solvable, what isthe greatest number of houses to whichpapers could be delivered?
These and many other types of problemscan be attacked through the formation ofgeneralization via logical analysis.
The placement of activities which repre-sent terminal learnings in mathematicaltextbooks should be examined closely.Students should always be provided withsufficient background in a topic so thatthey will be in a position to explore andsavor mathematics on their own. Onlyafter such a goal is reached may we hope tomake mathematics exciting and challeng-ing for students over an extended period oftime.
An Adventure in TopologyGrade 5JEAN C. CLANCY
Scarsdale Public Schools, Scarsdale, N. Y.
TO TAKE AN ADVENTURE implies the Un-
known, and that is just what a groupof fifth grade children did recently in mathe-matics. Before such a trip could be taken,the consent of all had to be obtained. In thiscase the pupils had a desire to learn and theteacher who acted as the guide, wished tostir the imaginations of her pupils.
In taking inventory she had to ask herselfsome questions. Was the present math pro-gram adequate for all pupils? Was compu-tation the most important phase of arith-metic to be taught? Were the pupils reallymotivated to think in mathematical terms?
To answer the first question she wouldhave to do some experimentation to dis-cover just how adequate the rogram was,for the realized that although many pupilscould handle advanced work easily, some-thing more was needed to broaden their ex-perience in mathematics.
As for computation, this was a skill worthdeveloping, but not to the exclusion ofothers. If mathematics was completelymanipulation of numbers, the electronicmachines could "take over." A child's arith-metic skill and such a machine have onething in commonboth are in need of re-pair from time to timebut unlike the ma-chine, a child can develop the ability toreason and use logic.
To motivate children to think in mathe-matical terms is a real challenge, for of allthe science mathematics breeds a motiva-tion that is different from that required inother academic areas. The "self-felt" needsarm': always seen in mathematics, and de-spite present day concern for greater compe-tence in math, this science has always beenthe most challenging group of ideas toteach students.
If the necessary skills beyond computingaren't apparent to all teacherswhateverlevelhow can "needs" be communicated
43
to students? To meet the challenge of moti-vation, she felt an obligation to approachmath in a way that was new to the pupilsand herself; to begin looking for patterns ofthought that have their basis in mathe-matics; to be able to generalize, work withtheory and then make some applicationsthat would be appropriate to the classroomsituation.
The ideas took form in the beginning ofthe year when the fifth grade class was dis-cussing length and width and then began toask questions about other dimensions, andthe meaning of our three dimensional world.When paper was represented as 2-D, theclass was asked if they had ever seen one-sided paper. After the initial laughter andquestions had subsided, they were asked tothink about this and share their thoughtswith classmates the next morning. Theirthinking ranged from: "paper stapled to thebulletin board so only one side showed,'to "a picture of a piece of paper becauseonly one side could be photographed." Cer-tainly there was thinking, and above all agenuine curiosity about one-sided paper.
When the teacher cut a long narrow stripof paper and pasted the ends together, inthe form of a headband, the class told herthat "it has two sides because we can see twosidesinside and outside." Then she tookanother strip of paperlonger than it waswideand gave the paper a twist beforejoining the two ends together (see Figures1 and 2). She told them this was called theMobius Band named for a German mathe-..:atician who studied it about 100 years ago.To help them discover why this was onesided paper, she had them run their fingersdown the middle of the length of the strip,following the twist carefully. After one triparound the band, they found that they hadreturned to the original point of departure,without having taken their fingers from the
C)
44 INSTRUCTION TECHNIQUES
A
B
C
D
no. I
no. 2
paper. No edges of paper had been crossed.When trying the same procedure with the
first named ring (in the form of a headband)they discovered that there was no way to getfrom the inside circle to the outside circlewithout crossing an edge. Therefore the con-clusion was that the Mobius surface was in-deed an example of one-sided paper whereasthe "ring" was the usual two-sided type.
Interesting to fifth graders? Yes, espe-cially when discussion led to the three di-mensional world we live in and applyingwhat they already knew about dimensions,to our world.
The motivation for more thinking was inthe new found surface before themtheMobius Band. Yet why learn about some-thing that had no apparent significance tothem in their three dimensional world?With more thinking and reasoning takingplace, some in the group ventured a "hunch"that it must have something do with outerspace. This proved to be a good idea for itwas pointed out that in four-dimensionalspace, closed surfaces such as Mobius cancAst. Mathematicians and scientists believethat it isn't at all impossible that astronomi-cal space is closed in on itself and twistedlike the Mobius Band. Amazing
Although this would be the closest theycould ever come to a Mobius Band, still itwould be fun to experiment with it to see ifit had any unusual properties. By cuttinglengthwise through the middle, everyonewas prepared to find that the band would
be cut into two pieces. But somehow theunexpected happened, for instead of havingtwo pieces, it still had one piece, buttwisted twice instead of once. And worst ofall it was now just an ordinary two-sidedband ! The fifth graders decided that thismagic was all right for demonstration, butwere just as glad that outer space in theform cr a giant Mobius surface was free fromprying scissors!
When someone recalled that you, theteacher, had mentioned four-dimensionalspace and "what did you mean by that be-cause here we only have three dimensions?"it demanded all your resources to explain toten year olds that time was the fourth di-mension.
It didn't seem too mysterious to under-stand when it was explained that events thathappen around us not only include distancein space but also the time that they happenedto be completely descriptive.
Now that the measurement of space wasin the thoughts of all, another word couldbe added to the growing ten's vocabularyalong with dimension, Mobius, surface,length, width, height. This word was Topol-ogy, the branch of geometry which meansthe study of locations (without reference tomeasurtments of lengths or angles in space).
"If this is a part of geometrythe subjectthat my brother is studying in high schoolcouldn't I learn something about it too, andhave sort of an adventure like we did withthe Mobius Band?"
"Yes, as a matter of fact you could"and all of us did have an adventure ingeometry that led us to the very nature ofthings around us.
Before taking that adventure however, afinal accounting was taken of what we hadlearned in mathematics and science as theIGY period came to its end. Of all the dis-coveries and experiments undertaken byscientists in all the world, our own encounterwith the Mobius Surface was the most dra-matic because it made us think about thepossibilities of unusual properties of spacestill waiting to be discovered by thinkingminds.
In the classroom Charlotte W. lunge
Creatamath, orGeometricideas inspire young writers
E M M A C. C A R R O L Carroll College, Waukesha, Wisconsin
It happened because of Flat Stanley, JeffBrown's delightful, two-dimensional boy,who became so when a falling bulletinboard flattened him outii My fourth grad-ers couldn't resist this fellow whose acci-dent cost him a dimension and gainedhim some adventures. Slithering underdoors was fun. Being flown like a kite waseven better! Fun ended, though, when the
Jeff Brown (Tomi lingerer, illus.), Flat Stanley(Evanston, Harper & Row, 1964).
boy-kite became entangled in the treetop!Brother Arthur and his bicycle pump ac-complished the feat of returning Stanley'slost dimension and making him a creatureof this world once more!
"Can you think about the points, lines,and planes in your math bookand thencreate characters from them as author JeffBrown did with Stanley?' an enchantedclass was asked.
Sly looks, chuckles, laughing eyes, andflying pencils answered my question in less
45
46 INSTRUCTION TECHNIQUES
than quarter of an hour. Authors sharedcharacters and adventuresand before theafternoon disappeared, delightful Stanleystood in the middle of the fourth-gradebulletin board in the company of twenty-four other geometric creations created byyoung mathematician authors!
Here are some for you to enjoy. Thecharacters are a little like Stanley, ofcoursebut their predicaments have origi-nal features and lost dimensions are re-gained in remarkable ways! Best of all,most of the chuckling authors deepenedand broadened their concepts of 1D, 2D,and 3D! Do have fun with the storieswhich follow!
The Flat Kid, Jim, by Pat
One day a boy was watching a steam-roller. The steamroller couldn't stop andran over him.
"Hey! Are you all right?" asked theoperator.
"I'm all right!" said Jim."You are flat!" said the operator. "It
is scientifically impossible!"Jim ran home as fast as he could go.
"Mom! Look at me! I'm Flat Jim!"Jim's mother was a fast thinker. She
said, "Get the gasoline. Drink it, Jim.""O.K.," said Jim, and he downed the
gasoline."Now, get a match!" said his mother."Boom!" There was Jim made 3D
again.Of course, Jim never went near a steam-
roller again!
Linda and the Printing Press, by Ann
Linda was a very curious girl. She wasmostly curious about her father's printingpress. One day, while watching her fatherat his press, she got too close and the bigpress squashed her. As soon as the presslifted, she ran away.
An inspector came by and saw thatsome of the paper had prints of a girl onit. He called Linda's father and asked,"What has happened?"
"That looks like my little girl! She mustbe 2D!"
Linda's father ran home and went rightto Linda's room. There he saw Linda as
a two-dimensional girl. Everyone tried toget her back to normal, but she alwaysstayed 2D!
A One-D Boy, by Bob
One day, Jim and his brother wereplaying with his tape recorder. Suddenly,Jim got caught and he came out 1D! Hecried and cried but he could find no wayto become 3D again.
It wasn't very good to become a jump
Connie Cross
rope for girls and that is exactly whathappened to Jim! He was pleased whensmite boys rescued hiM and decided touse him for a kitestring.
"How lucky can a boy-kitestring get?"said Jim, for as soon as he was high inthe air, he opened his mouth, the windsailed in and Jim became 3D again.
Steve and the Laundromat, by Diane
Steve and his mother went to the laun-dromat. Steve asked, "What would theworld be like if everyone were 1D?"
His mother told him, "I guess the worldwould be full of strings." While he wasthinking about this, he sat down on thewashing and soon was hidden in the dirtyclothes.
Steve's mother popped the washing into
the washer and, of course, Steve went rightin with the clothes. He screamed, but hismother had no idea where the scream was
aeatamath / Carrol 47
coming from. She shouted, "Where areyou, Steve?"
"I am in the washer!" answered Steve."I'll get you out!" said his mother. She
pulled and pL led and then saw Steve,looking like a string. She cried and cried,took him out, called the doctor and criedsome more.
"No bones are broken!" said the doctor."Can he eat? Can he talk?" Steve could,so the doctor left him as a 1D string witha crying mother.
A playmate came to visit. He asked forSteve and could scarcely believe that thestring who answered the door was hisfriend. "Yippy!" said the playmate. "We'lluse you as the tail on a kite!"
Outside the two boys went, and upwent Steve. He went higher and higheras the wind blew the kite into the sky.Then the friend let go of the string andno one ever saw or heard of Steve again!
EDITOR'S NOTE.Mrs. Carrol has shownone way children may relate arithmetic toother curricular areas, in this case creativewriting and literature. I suspect the childrenwho wrote these stories not only enjoyedwriting them, but also understood the mean-ing of one-dimensional, two-dimensional andthree-dimensional figures better because of it!CHARLOTTE W. JUNGE.
A second example ofinformal geometry:milk cartons*
MARION WALTERHarvard University, Cambridge, Mas,achusetts
Marion Walter is an assistant professor at the Harvard UniversityGraduate School of Education. Her concern is the education ofteachers in the mathematics program. She conducts manyin-service workshops and is particularly interested in thevisual aspects of learning mathematics.
This article describes some work that chil-dren can do with milk cartons. You willneed paper milk cartons and constructionpaper. Cut the top off the milk cartons sothat the height is equal to the width. Rulethe construction paper with two-inchsquares.
FIGURE 1
One can start the work in a variety ofways, depending on the age and interests ofthe children and the size of the group. Iand other teachers have worked withgroups as large as thirty-five and as smallas five. The description given here workswell with children in the third grade andabove. Appropriate modifications makethis an exciting unit for students from thefirst grade through college.
For a first example see "An example of informalgeometry: Mirror Cards," THE ARITHMETIC TEACHER,XIII (October 1966), 448-52.
48
Before reading on, try some of the workyourself. Please get a couple of the cut milkcartons, some construction paper, scissors,and a friend! Ask the friend to read thenext few questions and directions to you,because you won't be able to read them ifyou really do what I am going to ask youto do! I am going to start by asking you toclose your eyes!
Visuakinp the boxClose your eyes!
Visualize a box. Keep your eyes closed.How many sides does your box have?How big is the box you visualized?Can you think of a bigger (or smaller) box?Can you think of a longer (or shorter) box?Imagine a box all of whose sides are squares.Take the top off your box (are your eyes stillclosed?) so that you have a box without atop. Just to make sure that you have a goodfeeling for an open box, visualize filling itwith sand. Pour the sand out again.How many sides does your box have now?Now imagine that a box manufacturer wantsto ship these boxes flattened out. Can you im-agine how such a box looks flattened out?
Open your eyes!
Draw how a box with five square sides looksflattened out. What did you draw?
k.
Informal Geometry: Milk Cartons / Walter 49
Before continuing, let me describe ex-amples of what takes place in a classroomup to this point. When one child, visualiz-ing a box, was asked how big his box was,he replied, "Big enough to put the wholeworld in." Another said, "I can hold it inmy h,:nd." Both statements could motivatea discussion of what the dimensions of suchboxes might have been. Occasionally chil-dren will give actual dimensions, and whenthey do they tend to give only two of thedimensions. For example, one boy said,"Mine is 5 by 7," but he didn't name theunit of measurement. After challenginghim, he said, "It's 5 feet by 7 feet." A dis-cussion that involved actual measurementof boxes brought out the need for knowingthree dimensions. The children also realizedthat one measurement was sufficient if oneknew ahead of time that the box had allsquare sides.
Not all the children were able to recog-nize squares, and you may need a discuss"squareness." Many children. who are quitesure that is a square are not certainwhen the pattern is turned.
Many children draw
when asked how their box looks flattenedout. Some have drawn , saying "All thepieces are on top of each other." Somehave tried to draw perspective drawingssuch as
Occasionally, a student has drawn
Drawing of five-square patterns
Do the two patterns pictured below fold intoa box without a top?
--[
Can you think of other patterns made of fivesquares, regardless of whether they fold intoboxes or not?Draw as many as you can find.
What takes place in the classroom? Oc-casionally, some children have had diffi-culty finding new five-square patterns. Theycan cut out squares from the constructionpaper and use them to push around to helpthem form new patterns. Sometimes chil-dren may want to draw and include pat-terns such as
L1
or
I then explain casually that for themoment we will make the rule "whole sidestouching." Sometimes children investigatelater how many patterns there are if therule is "corners only touching."
Eventually, the children find all twelvepatterns. The question of whether
and
or
are to be counted as "the same" or as "dif-ferent" is raised by the children. Theyusually decide to consider two patterns tobe "the same" if each of the patterns canbe covered by the same paper cut-out.
Folding and unfoldingWhich of the twelve patterns fold into boxeswithout tops?Can you predict just by looking at each pattern
50 INSTRUCTION TECHNIQUES
LI
FIGURE 2
in Figure 2 which are "box-makers"? Try!Check by cutting out all the patterns and fold-ing the paper.Can you predict which squ,,re(s) form thebottom?
(A)
(D)
(B) (C)
(E) (F)
(G) (n)
FIG. 3.The Box-makers
The children often disagree with oneanother when they predict which patternsare "box-makers" or which square formsthe bottom. They can always settle theirarguments by themselves by using thepaper patterns.
Now draw all the "box-makers" and labelthem. See Figure 3, for example.Choose one of the patterns [don't choose Al]and write its number on the bottom of a cutmilk carton. Try to cut the milk carton to ob-tain this pattern!
Children much enjoy this activity andlike to choose several of the patterns hisuccession. They are often surprised whenthey obtain one of the patterns but not theone they bargained for! Sometimes thecartons fall into two pieces.
Extension of the work
This work can be extended in manydirections.1 Many questions have been sug-gested by students themselves. Childrenhave worked with four or six squares. Theyhave explored enuilateral triangles andsolids made from them. Some investigatedrectangles. The children have often triedto cut patterns with minimum wastage ofconstruction paper.
Conclusion
This article describes only a small partof the work that can be done with patternsof squares. I chose to isolate this milkcarton section because it can be done byitself without the rest of the unit and be-cause milk cartons are being thrown awayevery day!
The children often do not realize thatthey are doing anything mathematical whileworking with these materials. What do youthink?
1"Polyominoes, Milk Cartons and Groups." Abrief description of the extended work written ferhigh school teachers appears in the English journalMathematics Teaching, No. 43 (Summer 1968),pp. 12-19.
r r-f
Volume of a Cone in X-RayManipulative materials: A cone made of
cellulose or sometransparent materialthat will retain itsshape, a cylindermade of the same ma-terial having, ofcourse, the sameheight and circumference and sawdust to beuscd to fill the figures.
Having learned how to find the volume ofa cylinder, pupils can through the followingperformance proceed to an understandingof how to find the volume of a cone. Let thepupils fill the cone to the brim with sawdustand then pour it into the cylinder. Uponmeasuring they'll lend accuracy to their ob-servation that only a of the cylinder is filled,and through further experience pupils willsee that the cylinder holds three times asmuch as the cone. Incidentally, it will helpconsiderally toward clarification if the pu-pils mark each third of the cylinder witha color. Through questioning as well as ac-tual performance, pupils will arrive at theconclusion that since the cone is a the sizeof the cylinder, it will logically follow thatthe volume of a cone will be a the volume ofa cylinder, and hence the formuln:V
The advantage to be found i.t the use ofthese manipulative materials is the directexperience the pupils have in seeing as wellas participating in actual solution of theirproblem.
Contributed bySister M. VincentCleveland, Ohio
51
Geoboard geometry forpreschool children
W. LIEDTKE and T. E. KIERENUniversity of Alberta, Edmonton, Alberta, Canada
Thomas E. Kieren is an assistant professor of
education (mathematics education) at The Pennsylvania State University,
and Werner Liedtke is a Laduate student in elementary
mathematics education at the University of Alberta.
There is increasing emphasis today onpreschool experiences in mathematics forchildren to capitalize on their eagernessto explore and interpret the world aroundthem. This paper explores a wide varietyof experiences based upon the use of ageoboard and rubber bands. The geoboardprovides many opportunities to acquaintchildren with geometric concepts. Learningfrom their own play and from imitationof adults and other children, it is not longbefore they can recognize, correctly label,and form for themselves many commongeometrical figures and instances of geo-metric properties, as well as such commonsha?es as letters of the alphabet. The illus-trations that follow were drawn from theauthors' observations of children aged 2-6as they worked individually and in groupswith the geoboard. Questions asked andpossible suggestions given are classified un-der three main headings: Familiar Shapes;Plane Figures; and Segments.
Free activity
An excellent way to begin is by provid-ing each youngster with a geoboard anda gubber band and letting him do whateverhe wishes. While he works, he could bemotivated to show and talk about whathe has made. Recent experiences and ob-jects from his immediate environment willbe represented by various ingenious con-
structions and configurations. Sum' sampleresponses from a "free activity" period arepresented on the following page (Our geo-boards were 5 by 5 inches, and the pegswere about 1 inch apart.) The commentsrecorded represent the first responses of thechildren. Often, slight modification or evenrotation of the geoboard led to the assign-ing of different names to very similarfigures.
One sequence may begin by giving thechildren one rubber band, later increasingthe number to two or even three, andchallenging them to make something thatwas not possible before.
52
One rubber band:
'
2 yearsBoat
21- yearsSandbox
21- yearsTent
. . .
K07
41- yearsHouse
Geoboard Geometry for Preschool Children / Liedtke and Kieren 53
5 yearsBed
4i- yearsRoad
Two rubber bands:
2 yearsStar
/ I/44- years
Train track and street
2-12- years
Garden and sandbox
44 yearsSteps
5yearsIce-cream cone
Three rubber bands:
2 yearsCorrals
. .\\/
. .4 years 5 years
Star Bird witli wings
5years"Fridge')
4-1- years
Sailboat
Familiar shapes:
SUGGESTIONS
On your geoboard, show how to make someshapes that look like something in this room.
. .Try to make something that can be found in
the kitchen.Try to make something that can be found in
thebasement;yard;grocery store;playground;garage.Show something your dad uses.Show something that is alive.
5 yearsCanada flag
4-1- years
Swimming pool
POSSIBLE QUESTIONS (INSTRUCTIONS)
Can you tell your friend what you have made?(Show him and explain.)
Look at something someone else has made and
54 INSTRUCTION TECHNIQUES
try to guess what it (Ask for a hint where itcan be found.)
Does your figure look the same if you turnthe geoboard around?
How many sides does your figure have?How many corners does your figure have?
(Are there more corners or more sides?)
Sample responses:
22 yearsTV
43- years
House and garden
Plane Figures:
22 yearssue,'
4-1. years
Hammer
SUGGESTIONS
Try to make figures with three sides that aresmall;large;"skinny";"fat."Try to make figures with four sides that are:long;short;long and wide;long and narrow;short and wide;short and narrow;"like a square";"not like a square."Try to make figures with "many sides."
POSSIBLE QUESTIONS (INSTRUCTIONS
What does the figure you have made remindyou of? Does it look like anything that is familiarto you? (Where did you see something like itbefore?)
Does the figure change if you turn your geo-board?
Make two figures that: (1) do not touch; (2)touch; (3) cut into each other. Look at thefigures you have made.
Can you make another one that looks justlike itbut smaller, (o bigger)?
Make a triangle and a square. How are theyalike? How are they different?
Sample responses:
/
years(Make anothertriangle like it.but bigger )
5 years(Two figures thatare different )
6 years 5 years(Two figures that (Figure with many
are alike) sides)
Segments:
SUGGESTIONS
Try, to make segments that areshort;long;straight;"crooked."Try to make segments thatdo not to4ch;touch;cross each other (intersect);will "never" touch (parallel);are exactly on top of each other.Try to make various segmentsleading to two (or more) points;various numbers of segments, i.e., two that areequal;two that are not equal;many different segments.
POSSIBLE QUESTIONS (INSTRUCTIONS)
How would you make a road?Can you make a very narrow r ;ad?Can you make one that is long and narrow?Make a railroad track. (If possible provide
rubber bands of different colors.)
Geoboard Geometry for Preschool Children / Liedtke and Kieren 55
Can you make a road and a train track thatcross? . . . do not cross? . . . will never moss?
Look at two pegs in different corners. Howmany different roadscrooked or straight; fewor many cornerscan you build between thesetwo pegs?
Which road would you like to travel on? Why?
Sample responses:
2-1-yeurs(Road)
4 years(Long and short road,
not crossing)
5 years(Many roads
between two towns)
Summary and additional suggestions
The previously outlined activities presentone possible way to begin a session witha geoboard. Since the children work withcreations on their own that differ in manyrespects, the activities are open-ended. Itwill soon become evident that any sessionwill be a combination of what has beensuggested. Depending on the age and back-ground of the children, they will interpretinstrucions and questions in various ways.They will give unique replies that often leadto some idea that was not intended at theoutset. While working, some children willrecognize configurations that suddenly re-mind them of something familiar. For ex-ample, while talking about roads and traintracks (segments), one girl looked at herconstructed figure and remarked, "A 'T' ona liner The question was raised, "Does itlook like a 'T' if you turn your geoboardaround?" "No, now it's an 'H'." The ques-tion "Can you make other letters on your
44- years
geoboard?" resulted in having two childrenmale the following:
f, 4.The last response led to an attempt to buildmore numerals. Thus the topic of segmentsled to letters, numerals, and sets, and itcould have also been used to discoversomething about angles (i.e., right angles).
.
2 years( Big squarelittle
square)
Similarly, thefigures can lead
2 years(The square that
grew rind grew)
topic of "big and little"to the discovery of some
of the properties for similar figures (corre-sponding sides and vertices). Having chil-dren attempt to copy a figure can leadto discovery of some properties of con-gruence. Some children will exhibit anawareness of symmetry (e.g., "birds withwings"). Most of them will easily pickup such terms as triangle, square, andrectangle and use these terms correctly.Some will talk of polygons and angles, anda few might even be led to discover suchpolygons as parallelograms, quadrilaterals,or trapezoids.
In allowing for these developments itshould be remembered that these childrendevelop ideas through both imitation andfree play. By imitating an adult or anotherchild, the child may get an idea he neverhad- before. But free play allows him toexpand on the ideas and capabilities thathe already possesses.
Pegboard geometry
LEWIS B. SMITH University of Wisconsin;ifr. Smith is a graduate student at the University of Wisconsin, where he ismajoring in elementary school mathematics. He has taught and servedas an administrator at the elementary school level.
Geometry has now won for itself a placein the elementary school mathematics cur-riculum. Although emphasis varies frompublisher to publisher, there is no questionthat geometry is included in each majorseries. The Cambridge Report urges amore distinguished place for it in the cur-riculum of the future.
Geometry finds its place because of thecontribution it makes to children's think-ing. Children are led to reason, to examinegeometric forms carefully, and to developwith a minimum of formal definition a skillin seeing and speaking about conditions invarious parts of a geometric construction.Clarity of thought, exactness of expressionand a greater feeling of self-dependency,trust in one's own ability to perceive, toexamine, to hypothesize, and to proveallassure geometry of its place in today's andtomorrow's mathematics curriculum forthe elementary school.
Professional mathematicians who sug-gest geometry topics for the elementaryschool call for giving children the oppor-tunity to use mathematical descriptions ofexternal reality; to sense that geometrictopics are intrinsically worthy of interestand respect; to interpret and composesatisfactory models involving spatial ideas;to be afforded a source of visualization forarithmetical and algebraic ideas; to dis-cover the meaning of symmetry, equality,inequality, congruence, and similarity;and to test the appropriateness, truth, andpertinence of idormation.1
Goal* for School Mathernages, the report of the CambridgeConference on School Mathematics (Boston: HoughtonMifflin Company, Ogg).
56
The experiences reported here are fullyin keeping with the spirit and vision ofthese proposals. Intermediate grade chil-dren responded to the following activitieswith enthusiasm, understanding, and in-terest.
Children were given 10' X 10' peg-boards. They were asked to form a rec-tangle 2 "X3" with four pegs and enclosethe pegs with rubber band. Then theywere asked to double the width of thefigure. The children set 2 new pegs and in-creased the stretch of the band to encom-pass the new figure. They were asked howthe perimeter and area were affected bythe change (see Fig. la). The children wereasked to return the band to the originalfour pegs and double the length by placingtwo new pegs and stretching the band toenclose the new figure. Again, they wereasked what effect this had on both thearea and perimeter (Fig. lb). Finally, thechildren were asked to double both thelength and width of the original figure. Inresponse to this request instances like thatshown in Figure lc appeared, and theseoffered opportunity for discussion. WhenFigure Id was agreed upon, questions ofperimeter and area were again raised.
I
a
Figure 1
The expanded rectangle in Figure ldwas used to extend experiences. Childrenwere asked to show ways of dividing the
Vj if
b
figure into two equal-sized areas. Earlyreplies were confined to halving by band-ing off equal, rectangular areas (Fig. 2).Illustrations here are limited to one phase,although in practice the children alsonoted the opposite or congruent phase.Equal areas with varyin perimeters werenoted.
Figure 2
Other replies were forthcoming in re-sponse to the idea that they halve theoriginal area, and much thought and dis-cussion were generated over the equiva-lence of areas described. Figures 3a and 3bwere quickly agreed to, but 3c and 3d werejudged correct only after construction andstudy of the complements.
a
Figure 3
b d
Opportunity arose for discussion of thearea of the triangles, their various shapes,and similarities of their areas. Childrenwere able to devise a fornwla for a tri-angle's area, based on the known dimen-sions of the rectangular figure. Experi-ences later in the lesson permitted them toverify its accuracy in many different situa-tions.
When the children understood triangu-lar regions, many smaller triangles ap-peared on their boards, although thedescriptions always represented half themarked-off area (Fig. 4).
Figure 4
Pegboard Geometry / Smith 57
Then came the more intricate figurescomposed by the fifth graders (Fig. 5).They occasionally composed the shadedareas by assembling smaller triangles withseveral bands. Other children stretched asingle band to encompass the largest areathe band could enclose.
C44
Finally, patterns developed whichcaused so much discussion and puzzlementthat manila graph paper was used to helpverify the similarity of areas (Fig. 6).Cutting the line-graph area proved helpfulin soh. a ;nstances.
Additional sophistication in communi-cating about diagrams can be encouragedby labeling the pegs with letter names andhaving the children describe their patternsin terms of the letter names. Childrengained accuracy in the description of areasenclosed by the rubber bands by namingthe pegs. Figure 7 was described as fourtriangles, AAEF, ABM, ABCF, andAnil, by children who had used fourrubber bands. However, other childrenpreferred to name the area as a quadri-lateral AEGF plus ABCF and AFHI.
Figure 7
Another experience involved the use offour pegs. The children were asked to
58 INSTRUCTION TECHNIQUES
form as many triangles as they could byusing three of the pegs.
Figure 8
We then moved peg P, as in Figure 8, tolocation P' in Figure 9 and tried again.
Figure 9
Children were then asked if they couldplace the band on the pegs in such a waythat they could make four-c;ded figures(quadrilaterals). They discovered thesefigures.
(Using Figure 8 model)
(Using Figure 9)
Figure 10
On a succeeding day the fifth gradersexamined the triangle with a base of 2 anda height of 8 peghole intervals (inches).They quickly found triangles of equal arealike the ones in Figure 11.
Figure 11
Questions arose when we consideredtriangles such as those depicted in Figures12 and 13. These triangles offered furtherpossibilities for discovery on the part ofthe children.
Figure 12
Figure 13
In Figure 12 children easily found thearea L.ABC and AA /J based on theexperiences described earlier. Since peg-hole gaps are one inch each, the remainingtriangular areas were computed by findingthe area of adjacent triangles and sub-tracting this from the enlarged triangle.For example:
Area /ACD=area /.A BD minus area L,ABC
Area AACD
=(1.8.4) minus (x.8-2)
Amp. AA CD
=8 square inches.
Similar treatment in Figure 13 yieldedaccurate results. The area of AABH
area A AZII minus area ABZH.(Twenty square inches minus 12 squareinnhcs ,-,crant. S square inches.)
We then explored with children whathappens Ivlien the AABH is extendedindefinitely to points I, J, and beyond.Several children were particularly fasci-nated with this discovery.
Another opportunity for analysis lies inthe examination of relationships of AB toAZ and AABH to AAZH as in
AABII ABAAZH AZ'
and with substitution
/A/111 2
20 5
5. AABH=40 AABH=8.
With, and sometimes without, en-couragement children will ask about theequality of areas in /ABC and ADEF inFigure 14.
Figure 14
Pegboard Geometry ,/ Smith 59
Figure 15
By this time the children's inquiry ledto Figure 15 (enlarged part of Figure 14)and analysis of equality of area for nu-merous triangles such as AXIT and,AX Y117 and AX Yr, and so forth.
Experiences described in Figures 11
through 15 served to extend children's in-sight. The formula for a triangle's areawould have sufficed to solve problemsraised in these situations. However, theauthor feels that many discoveries and'earnings await the child whose interpreta-tions are not limited too early by theformula.
Developing a sense of trust in one's ownability to perceive and an increased aware-ness of the varied choices or responsesavailable to the thinking person are a fewof the rewards awaiting the elementaryschool student of geometry. The methodsreported here amply provide these kindsof experiences. They also develop thepupils' ability to interpret physical situa-tions through the construction of simplemodels. The pegboard is a successful me-dium for portraying geometric idea::: andfor promoting exploration. Through itsuse boys and girls can be guided to dis-cover truths that reveal significant arith-metical and geometric understandings.
Tinkertoy geometry
PAULINE L. RICHARDSBethesda Elementary School, Bethesda, Maryland
Mrs. Richards' article grew out of a presentation thatshe made in a course at the University of Maryland. She is aclassroom teacher.
Finding suitable instructional aids formy fifth-grade geometry class was a prob-lem until I discovered the versatility ofTinkertoys.
The round "joints," or disks, thoughlarge, may be used as a representationof a point (Fig. 1).
FIGURE 1
A model of a line segment can beformed by joining short sticks and moredisks (points), as shown in Figure 2.
0-0-0-0-0FIGURE 2
The two end points and some of thepoints between are clearly represented, sothe property of "betweenness" can bepointed out by using this model.
If a cardboard arrow is attached, oneend point is eliminated and a ray is shown(Fig. 3A). If an arrow is used to repre-
FIGURE 3A
sent an extension in the opposite direction,a line is illustrated (Fig. 3B).
60
FIGURE 3s
These representations can be easilymanipulated. Geometric figures can beshown as being a part of different planesrather than as being just on the surfaceof a chalkboard or a textbook page.
One of the balls that are now includedin the kits can be used to illustrate apoint through which many lines pass (Fig.4).
4.Lines pass through the point in anendless number of directions.
Intersecting lines can also be shown,and the angles can be observed (Fig. 5).
FIGURE 5
Angles can be compared by matching onerepresentation with another and asking,"Are they congruent?" and, "Is the meas-ure of one greater than the measure of theother?"
As the study of angles is continued,two rays with a common end point canbe shown in a variety of ways. Namescan then be given to the different classesof angles observed, as shown in Figure 6.
OBTUSE
FIGURE 6
ACUTE
By the use of three intersecting linesegments, triangles can be introduced. Iencourage the students to make Tinkertoytriangles, to include angles of differentsizes, and to note the effect on the lengthof the sides of the triangles. The studentssoon want to know how to name triangles,and they quickly begin discussing scalene,isosceles, and equilateral triangles.
The study of polygons continues withmuch interest. By making representationsof polygons and using only four sticksbut a variety of lengths, students readilysee that quadrilaterals are not limited inshape to the familiar squares and rec-tangles. We are off again, naming quadri--laterals! (Actually, I had not planned togo into the naming of polygons so com-pletely with my fifth-grade class, but Ifound myself carried along by their in-quisitiveness.)
Illustrating a figure as the union ofdisjoint sets of points becomes a natural!
Tinkertoy Geometry / Richards 61
The representation of a line can beseparated into the three sets of pointsillustrated in Figure 7.
<-0-0- 0 --0-0-1>HALF-L1NE POINT HALFUNE
FIGURE 7
A plane may he separated into threedisjoint sets of points. For instance, byusing a model of a rectangle in a plane,it is possible to separate the plane intothree disjoint sets of points, as indicatedin Figure 8.
//,/,// /
FIG. 8.(1) indicates the points in the in-terior of the rectangle; (2), the points of therectangle; and (3), the points in the exterior ofthe rectangle.
By passing an object through the in-terior of the rectangle, a distinction canbe made between a rectangle and a rec-tangular region.
Geometric figures constructed from Tink-ertoys are light enough to be used for abulletin-board display. A small roll ofmasking tape on the back of each disk,and a pin through the hole, will hold anyfigure secure.
I am still discovering uses for my'Tinkertoysconcave and convex figuresand diagonals of polygons are but a fewpossibilities. Perhaps you can continueto discover, too!
Congruence and measurement
STANLEY B. JACKSONUniversity of Maryland, College Park, Maryland
Dr. Jackson is a priVessor of mathematicsat the University of Maryland.
The elementary ideas in mathen.atics areabstractions drawn from the experiencesan individual has with his physical en-vironment. These experiences are primarilyof two kinds. T',ere are, first, those ex-periences that are associated with count-ing. They arise naturally in dealing withcollections of discrete objects and leadultimately to the abstract notion of thecounting numbers and the operations onthe counting numbers. That is, the ex-periences of this type are those that induceus to invent the counting numbers andthe usual arithmetic of this system andits extensions. The second category ofexperiences includes those of a spatialnature. They are related to our perceptionsof size, shape, and form. The child dis-covers that this pencil is too long to fitin that box; that this peg will just fit inthat hole; that the faces of two of hisblocks fit exactly on earn other, and soon. These are the experiences that moti-vate us to identify and discuss variousfigures and to give names to the commonones like line s,r;ment, line, ray, plane,triangle, circle, and sphere. Thus the de-veloping space perception of the individualleads to the creation of the mathematicalideas normally associated with the word"geometry."
Both these strains of mathematical think-ing occur in the elementary school mathe-matics curriculum, with the larger por-tion of time devoted to the arithmeticwork for which it is nece.-ary to developnot only the concepts but also the appro-priate manipulative skills. The geometricphases of the curriculum are in large partrelated to the idea of measurement. This
6762
is wholly appropriate, but not infrequentlythe emphasis is placed so much on com-putational skills that the essential geo-meWc ideas involved are not fully per-ceived. It is the purpose here to explorebriefly the geometric ideas that underliethe measurement process.
It should be noted at the outset thatthe subject of geometry as it occurs inthe elementary school is quite differentfrom the corresponding subject in highschool. In the high school developmentthe emphasis is quite strongly on the de-ductive method, ,.vith discussion of axioms,theorems, how to give a proof, and the like.At the elementary school level, while theability to reason is to be encouraged inall possible ways, geometry is much morea development of perception, an exploringof spatial relationships. It might perhapsbe described as physical geometry, andis the intuitive basis against 'which thelater, more formal work can make sense.There will be much room for intuitionand discovery. A word of caution may bein order on this point, however. Care shouldbe taken to avoid giving the pupil theimpression that a few physical observationsconstitute a proof. They may form thebasis of a hunch or a conjecture, but topass this off as a proof is to precipitategreat confusion later when proofs are tobe discussed.
The idea of congruence
In the sense of developing spatial per-ceptions, it is clear that geometry beginswell below the school level. The childwill begin early to distinguish betweensuch things as a round object and a square
Congruence and Measurement / Jackson 63
or triangular one. An imaginative first-grade teacher of my acquaintance makesa deliberate effort to reinforce this dis-crimination by placing a number of objectsof different shapes in a bag. She then hasa pupil put his hand in the bag and de-scribe, on the basis of feeling alone, thecharacteristics of one of the objects. An-other activity in which children frequentlyengage is the assembling of puzzles. In-deed, the assembling of jigsaw puzzles as arecreational activity is by no means con-fined to children. The individual, of what-ever age, who correctly selects the pieceto fit into a given space in a puzzle isexercising his perception of an extremelyimportant geometric relation called con-gruence. That is, he is observing that aparticular piece of puzzle exactly fits ina particular hole. The author knew onestudent whose perception of shape was sokeen that he preferred to assemble a jig-saw puzzle with the pieces turned wrongside up so that he was not 'distracted"by the picture.
In general we speak of two figures asbeing congruent if one will fit exactlyon the other, or, more precisely, if amodel of one will exactly fit on the other.This concept of congruence proves to bea critical one for the understanding ofmeasurement. Once the idea has beenidentified, it can be noted again and again,both in common, everyday contexts andin guided experiences designed for thepurpose. For example, two sheets of paperfrom the same ream provide an excellentrepresentation of congruent rectangularregions, and two soda straws are a rea-sonably good representation of congruentline segments. Even in situations wherecongruence does not appear explicitly, itis often in the background. If we try tocompare two pencils by laying them sideby side on a desk with their erasers to-gether, it may happen, as in Figure 1,that the tips do not match. That is, the
t. LIU
FIGURE
pencils cannot be considered as represent-ing congruent segments. However, the veryprocess of laying them side by side sug-gests that one of them is congruent toa portion of the other. Thus the process ofobserving that two segments are not con-gruent amounts to identifying one of themas congruent to a part of the other.
Experiences with congruence
An observation concerning congruenceof angles may be obtained by using an ordi-nary sheet of paper. If one tears off twocorners from the sheet of paper, the tornpieces will almost certainly not he con-gruent, yet by placing one on the otherit appears (Fig. 2) that they can be soplaced as to "fit" near the vertices, This
FIGURE 2
B
leads in a natural way to the concept ofangles' being congruent even when theregions with which they are associated arenot. In particular, the four angles asso-ciated with the corners of an ordinarysheet of paper appear to be congruent.Moreover, if the four corners are tornoff, it is discovered that these four piecesappear to just fit together at a point, asindicated in Figure 3. This property is
FIC.URE 3
characteristic of what is called a rightangle. That is, an angle is a right angleif four congruent copies of the angleand its interior will just fit together to
64 INSTRUCTION TECHNIQUES
cover the space about a point in a plane.This leads to the possibility of making avery satisfactory model of a right angleby paper folding as follows. Let a sheetof paper (which is not assumed to haveany particular regular shape) be folded,creasing firmly. The crease will be anexcellent representation of a line segment,as indicated in Figure 4.
creaseFIGURE 4
A second fold is then made so that thefirst crease falls on itself (as shown inFigure 5), and is again creased firmly.
creaseFIGURE S
The two creases form a good representa-tion of a right angle. This becomes clear ifthe paper is unfolded again. The creasesare then seen (Fig. 6) to form four angles
FIGURE 6
that fit together at a point and yet arecongruent, since they lie on each otherwhen the paper is folded.
A second property of congruence whichlends itself to phr'zal verification con-cerns isosceles triangln, that is, triangleswhich have some pair of sides congruent.Consider the isosceles triangle ABC shownin Figure 7. We suppose that segmentsAB and BC are congruent. Imagine thatsuch a triangular region is cut from paper.If the paper is folded so that vertex Afalls on vertex C and if the paper iscreased, the situation is as shown in Figure8. That is, the angle with vertex A iscongruent to (will just fit on) the angle
B B
FIGURE 7
A,CFIGURE 8
with vertex C. This is an experimentaldiscovery of a basic property of isoscelestriangles which, at a later stage, the stu-dent will see as a theorem in geometry.The theoreri could be stated in some suchform as the following:
If two sides of a triangle are congruent,the ar,les whose vertices are oppositethese sides are also congruent.
Numerous other situations involving theconcept of congruence could be developed,but one more will suffice here. Imaginenarrow strips of paper which can be con-nected at their ends by fasteners. Thestrips are crude representations of linesegments. If three of these are fastenedtogether, as shown in Figure 9, they forma model of a triangle.
FIGURE 9
Congruence and Measurement / Jackson 65
Now suppose that several persons are eachgiven three such strips so that each twopeople have strips that are congruent(just alike), and suppose that each personproceeds to form a triangle as above. Itwill then be found that all the resultingtriangles appear congruent to each other.That is, in a certain sense there is onlyone way of forming a triangle whose sidesare congruent to three given segments.This is the physical indication for the
theorem that if three sides of one tri-angle are congruent to the three sides ofa second, then the triangles must be con-gruent.
The experiment above can be madeeven more striking by considering the cor-responding situation with four strips, whichcan be assembled to form a quadrilateral.Here experimentation readily shows thatit is not true that congruence of the sides
will insure congruence of the quadrilaterals.An illustration of this is shown in Figure
10.
FIGURE 10
The difference between the cases of thetriangle and quadrilateral is particularlyimpressive if the experiment is done withstrips of wood or metal rather than paper.The feeling of rigidity for the triangle isin sharp contrast to the easy distortionof the quadrilateral. Pupils often find it
interesting to look around for exampleswhere the rigidity of the triangle is usedin building as, for instance, a brace fora shelf or a diagonal reinforcement for agate in a picket fence.
Linear measurement
So far our discussion has concerneditself only with the intuitive meaning ofcongruence. What relation does this haveto measurement? What, indeed, do wemean by "measuring" something, say aline segment? The idea is to make use ofnumbers to describe in some sense "how
much" line segment is present. The wholenumbers (i.e., the counting numbers to-gether with the number zero), as we havenoticed, grow out of our experiences incounting and at first sight seem whollyinappropriate for working with an entitysnch as the line segment shown in Figure
FIGURE 11
11. What is there here to count? One isreminded of the frustrated golfer whobitterly described golf as a game in whichthe object is "to propel a little round ballfrom one place to another with instru-ments singularly ill adapted for the pur-pose." There is a sense in which the wholenumbers are "singularly ill adapted" forthe purposes of measurement. However, thetechniques by which numbers are attachedto geometric objects are among the mostfar-reaching ideas in all elementary mathe-matics. We meet here, so to speak, thewedding of arithmetic and geometry.
As has just been noted, when one looksat a segment, he sees nothing to count andhence at first sees no way of describing itby a number. In a sense his first task isto create something to count. This is done,as the reader is well aware, by selectingsome convenient segment as a unit. Thus,in Figure 12, to measure segment AB,using segment PQ as a unit. one asks howmany congruent copies of PQ are required
to cover segment AB. Using segment PQas a unit, it appears that AB is 8 units long.
A gP-Q
FIGURE 12
Seen in this perspective, it becomesclear that congruence of segments is atthe very heart of measurement of seg-ments. Congruence is the concept that hasbeen used in obtaining the objects we count.To measure a segment, then, is to countthe number of (nonovcrlapping) congruentcopies of the unit segment necessary forcovering. Two segments on a line are called
(
66 INSTRUCTION TECHNIQUES
nonoverlapping if they have no interiorpoints in common, i.e., if they either haveno points in common or have a commonend point. As we shall note briefly below,the same essential idea of measurement ap-plies also to angles and to plane and solidregions.
While it would take too much space todiscuss in detail all the consequences of thisidea of linear measurement, it is possibleto mention some of the ideas that growout of it. First of all, any experience withlinear measure makes clear that the processis inherently approximate. Most of the time,when one seeks to cover one segment withcongruent copies of a second, it does not"come out even." For example, in Figure13 the length of segment CD using XYas a unit is more than 3 but less than 4units. Thus in terms of a given unit it isclearly not always possible to cover exactlywith any whole number of units, and thiswould still be true even if we could layoff the congruent copies of XY withoutany manipulative inaccuracies. The con-cept of the approximate nature of measure-ment is an important one.
C D
X Y
FIGURE 13
A second by-product of the idea of meas-urement of segments arises by observingthat, in laying off successive copies of theunit segment, it might be worthwhile tomark the end point of each copy to indi-cate how many times the unit segment hasbeen used. Thus, in Figure 14, point Bis marked with a 5 to indicate it is the
A
2 3 4 5 6
FIGURE 14
7 8
end of the fifth congruent copy startingat A. If we imagine the process to continueindefinitely, we have precisely the familiarnumber line. Presumably we would wish toassociate the starting point A with thenumber zero. A movable copy of a part ofsuch a number line is nothing but the
familiar ruler. It is to be noted that theunit may be any segment whatever. Thereis no requirement in any of the discussionthat a standard unit be used. The onlyreason for using a standard unit would beto facilitate communication with other peo-ple.
In considering the approximate natureof measurement, it inevitably occurs toone to abandon the requirement that themeasure be a whole number and allowfractions. For example, segment CD inFigure 13 may appear to have a length of
31 /s10units or units. This is certainly a3
legitimate idea and leads at once to theconcept of the number line with numericallabels for points at various fractional partsof units. A little reflection reveals, however,that the situation is not basically changed.
To say that CD has a length of 13 units
means simply that we have considered theoriginal unit segment divided into threecongruent parts. The statement then saysthat it takes 10 congruent copies of thissmaller segment to cover CD. Thus theuse of fractional measures, while frequent-ly useful and desirable, really amounts tonothing more than selecting a new andsmaller unit in which to measure.
Angle measurennitThe essential idea of the measure of a
segment as the number of congruent non-overlapping copies of a unit necessary forcovering extends readily to angles. Considerthe angle AOB in Figure 15, and let angleRS7' be selected as a unit angle. Then
FIGURE 15
by the measure of angle AOB with respectto this unit we mean the number of non-overlapping congruent copies of angle RST
(
Congruence and Measurement / Jackson 67
and its interior that are necessary to coverangle AOB and :is interior. The word"nonoverlapping" means that successivecopies have a ray in common, but thattheir interiors have no common points.Showing the successive congruent copiesby the dotted rays in Figure 16, it appearsthat angle AOB is between 2 and 3 units.
18 /
0it
FIGURE 16
That is, it is larger than an angle whosemeasure is 2 and smaller than an anglewhose measure is 3.
Just as the operation of measurement ofsegments leads to the invention of theruler as a measuring device, so the opera-tion of measuring angles leads to the in-vention of the protractor as indicated inFigure 17. Here angle LMN has a measure
unit angle
FIGURE 17
of 3 units. In this particular example, theunit angle is such that eight congruentcopies of the unit angle and its interiorjust cover a line and one of the half planesdetermined by it. A more common unit ofmeasure for angles, the degree, is suchthat it takes 180 congruent copies to coverthe same set of points.
Area measureLet us turn now to the question of meas-
ure for a region in a plane. By the word
"region" is meant a simple closed curve ina plane and its interior. The essential ideais the same as before. We select some con-venient region to serve as a unit and askhow many nonoverlapping congruent copiesof this unit region are necessary to coverthe given region. In this case, however,we find ourselves confronted with a be-wildering variety of possible shapes of re-gions and possible choices of unit. Threepossible examples are shown in Figure 18.
UNIT REGION REGION TO BEMEASURED
FIGURE 18
MEASURE OFREGION
12 units
4 units
12 units
In view of what is to be done with theunit region, it is clearly desirable to selecta unit region like those above such thatcongruent copies can be fitted together toprovide a paving of the plane. That is, wewant to cover as large a part of the planeas we wish with congruent copies of theunit region that do not overlap but thatde not leave spaces uncovered. (Circularregions, for example, will not fulfill thispurpose.)
The most common choice for a unit re-gion is, of course, a square region eachside of which is one linear unit in length;but this should not obscure the theoreticalpossibility of other choices and some of theinteresting geometric ideas connected withpaving the plane with differently shapedregions.
The measurement of a region, in the
1-'4 (-1
68 INSTRUCTION TECHNIQUES
sense just described, is called the area ofthe region. There are both similarities anddifferences between the measurements ofarea and of length. A notable similarity isthe fact that measurement is still approxi-mate. Indeed this is a characteristic of allmeasurement. In the examples of Figure18, the regions to be measured were chosenso that they could be exactly covered bysome integral number of congruent copiesof the unit region. This, however, is clearlythe exception; and in any case the processof fitting the congruent copies always in-volves manipulative error. Usually an at-tempt at covering some region with con-gruent copies of the unit region will showsome copies inside the given region andsome partly inside and partly outside. Forexample, using a square region as a unit,the rectangular region in Figure 19 is seento have an area between 8 and 15 units.iiiri
I
___I I
FIGURE 19
As in the case of linear measure, onemay introduce measures which are frac-tions but not whole numbers. As before,this is really a matter of choosing new unitregions that are fractional parts of theoriginal unit region. Whatever the unitused, however, most of the time the regionsto be measured are not exactly covered bya whole number of congruent copies of theunit region. Combining this with the factnoted above that in practice one cannotconstruct exactly congruent copies anyway,it is clear that though in theory we mayconceive a region to have an exact measure,in practice the answers we get are alwaysapproximate.
Possibly the most striking difference be-tween the measurements of length and ofarea is the lack of an instrument for meas-uring area that corresponds to the rulerfor length. Let us examine the ruler for a
moment. To measure the length of a seg-ment AB one may place the zero pointof the ruler at one end, say A, of the seg-ment and see where on the ruler the otherend B of the segment falls. If, as in Figure20, B falls at the point marked 5, thisindicates at once that the length of ABis 5 units, since exactly 5 congruent copies
A --3 3 4
FIGURE 20
of the unit are needed to cover AB. It isinteresting to ask why this simple instru-ment exists, and the answer is not difficult.We have noted that measurement of lengthmeans covering by congruent copies of aunit segment, and there appears to be onlyone reasonable procedure for doing this.We start at an end point A of the segmentto be measured and mark off a congruentcopy of the unit segment in the directiontoward B. If this does not yet cover thesegment AB, we start at the end of tkisfirst copy of the unit and lay off anothercopy in the same direction. This process iscontinued till AB is covered. At each stagethe next step is completely determined. Butthis is precisely the way in which the ruleris constructed. Thus, when we lay the rulerbeside the segment, the marks on it indi-cate exactly the steps we would take if wewent through the process of measuring forourselves. The number attached to the endpoint of the last segment needed is thenalways the number of congruent copies ofthe unit used in covering segment AB; i.e.,it is the measure of the length of AB. Thusa ruler is merely a set of congruent copiesof some unit segment laid end to end alonga line, each end point being associated witha number. In using the ruler, we can simplylook at the number associated with thelast point used, rather than having to goback and count the segments each time.
The discussion above is quite simple,and we can surely realize the saving ofwork in being able to read off lengths fromthe numbers indicated on the ruler instead
Congruence and Measurement / Jackson 69
of counting the number of congruent copiesof the unit segment. Surely it would beequally useful to have a similar instrumentfor measuring areas. What is to preventus from making one? For the ruler wecovered as large a section of a line asdesired with non overlapping congruentcopies of the unit segment. In the case ofthe plane, we can certainly cover as largea part of the plane as we wish with non-overlapping congruent copies of the unitregion. For example, using a square regionas a unit region, we can readily form anetwork, as shown in Figure 21.
onoREM
EMMENFIGURE 21
If we imagine Figure 21 drawn on asheet of transparent plastic, we shall havea device that could (at least in theory) befitted on any region to be measured. Tofind the area in terms of the given unitregion, it will then only be necessary tocount the number of unit square regionsneeded to cover the region being measured.This is indeed correct and on occasion canbe very useful. In the case of the ruler, how-ever, we were able to avoid the tediouscounting process by associating numberswith the points of the scale so that it wasnecessary only to look for the number asso-ciated with the end point of the last seg-ment used. Can something similar be donewith our network of square regions? Un-fortunately the answer is no, and for a veryinteresting reason. Essentially it is becauseof the variety of shapes of regions to bemeasured. Differently shaped regions mayrequire a different order for the placementof the covering unit-square regions. Thusthe order in which we would place the unit-square regions to cover the rectangular re-gion in Figure 22A is quite different fromthe order needed for the region in Figure22B. Hence if we number the square re-
FIGURE 22A FIGURE 22B
gions on our transparent sheet of plastic sothat the regions numbered 1, 2, 3, 4 willfit on Figure 22A, then no placement ofthe plastic will have these same squareregions covering the region of Figure 22B.Thus it is not possible to assign numbersto the squares on the plastic so that withproper placement an area can be read mere-ly by noting the largest number associatedwith a square in the covering.
It is this lack of a two-dimensional rulerthat makes the choice of the unit squareregion so desirable, since, in this way, onecan deduce the area of a rectangular regionin the usual way by using the linear meas-ures of the sides. Thus in Figure 23 if thelength of the rectangular region is 3 unitsand the width is 2 units, then the area ofthe region, i.e., the number of unit-squareregions needed to cover it, is 2 3, or 6.
I II
I
FIGURE 23
Paving the plane
As was remarked above, there are manypossible choices of region that allow usto pave the plane with congruent copies.While the paving with square regions isthe choice commonly made in discussingarea, there is interest in considering suchpavings in their own right. Pupils may findit interesting to discover as many suchpavings as possible. It may be of interestto observe that a paving can be givenusing a unit region bounded by any triangleor any quadrilateral and some hexagons.
In addition to their inherent interest,some of the pavings suggest geometric factsthat pupils will eventually see as geometrictheorems. Two examples of such resultswill be given as a conclusion to this dis-cussion.
70 INSTRUCTION - TECHNIQUES
Consider a paving with triangular re-gions. One such paving is shown in Figure24. The triangular regions are all con-gruent to the unit region shown. The small
FIGURE 24
numerals have been inserted in the draw-ing of the unit region to identify the threeangles and have also been shown in partof the pavin,;. An examination of the pav-ing about a point as shown in Figure 24indicates two angles congruent to angle 1,two congruent to angle 2, and two con-gruent to angle 3. This appears to suggestthat congruent copies of angles 1, 2, and 3(with their interiors) can be put togetherto fill up the space on one side of a line.Since angles 1, 2, and 3 are the angles ofa triangle, this is the geometric fact thatthe pupil will eventually see embodied inthe theorem that the sum of the degreemeasures of the angles of a triangle is 180.
As a second example, consider the pav-ing shown in Figure 25, with regions whoseboundaries are isosceles right triangles. Thisis actually the familiar paving with square
FIGURE 25
regions, except that each square tile hasbeen cut by a diagonal.
One of the triangular tiles in Figure 25has been shaded for convenience. For eachof the two shorter sides of this triangularregion note the square region, which hasbeen outlined with a heavier line. Eachof these square regions consists of twoof the triangular tiles. Hence if the tri-angular region is taken as the unit of area,each of these squares has an area of twounits. Next observe the heavily drawnsquare whose side is the hypotenuse of thetriangle. This is seen to consist of fourof the triangular tiles and so has an areaof 4 units. Thus the square region on thehypotenuse has an area that is the sumof the areas of the square regions on thetwo shorter sides. This is a special case ofthe famous theorem of Pythagoras thatclaims that this relation holds for all righttriangles. A legend suggests that it was bylooking at such a pattern that this theoremoccurred to Pythagoras.
Conclusion
The major purpose of this presentationhas been to exploit the idea that the con-cept of congruence is one of the mostfundamental and fruitful ideas that arisein our intuitive perceptions of geometricrelationships. In particular, it seems toprovide the appropriate means for describ-ing at the elementary school level theessential meanings for the measurement ofsegments, angles, and plane regions. Thesame ideas can readily be extended to thediscussion of the volume measures of solidregions.
InstructionRationaleTechniques and methods are of doubtful worth without good reasons
for their use. An effective teacher's rationale is more important thana bag of tricks. Many techniques are found in this book of readings,and nearly all of the articles have direct suggestions for teachingapproaches or special methods. The articles placed in this last groupingare no exception and contain many practical suggestions for theteacher. However, these articles particularly develop c'sect or impliedreasons for using techniques. Most of the authors in this section adhereto a similar philosophy, but each has a somewhat different point ofview to express.
The initial article of this section contains a comprehensive outlineof suggested content for the elementary school geometry program.Egsgard, in this paper, develops not only the scope of a program butalso illustrative methods and techniques. Clearly written and concise,it presents a very fine overview of a well-planned geometry courseof study.
Concern for the teacher is the focus of Inskeep's article. A briefrésumé of reasons why geometry should be taught is given, followed bysuggestions for the teacher who would implement geometry in histeaching. This discussion is directed to primary teachers, but it will beof value to any teacher who has not taught geometry before.
Vigilante's discussion helps to present some of the reasons, bothpsychological and pedagogical, why geometry should be taught. Thisdiscussion, also, is geared to the primary grades but develops a ration-ale that can be accepted for all grades. His essay will appeal to thereader for whom cursory mention of the need to teach geometry isnot enough.
Another aspect of mathematics instruction is the part geometryplays in contributing to other areas. Robinson presents the worth ofgeometry and geometric approaches to other topics in mathematics.For many mathematics educators this article will furnish a mostcompelling reason for teaching geometry.
Skypek has investigated the thinking of Piaget. Using some ofPiaget's conclusions and ideas related to the development of geometricconcepts in children, she emphasizes the impact upon the curriculum.There are good reasons from a psychological point of view to introduce
vje 71
72 INSTRUCTION - RATIONALE
and develop the study of geometry early in the grades. Skypek alsodeals with the content that would be appropriate for the curriculum,in keeping with the theories of Piaget.
The concluding article of this section is a nicely blended combina-tion of ideas for teaching and reasons for introducing geometry earlyin the experience of the child. Brune covers the scope of the curriculum,concurrently showing illustrative techniques. Moreover, he gives a log-ical basis for geometry in the early grades, noting the importance ofinformal geometric readiness to the formal axiomatic study to beundertaken in the secondary school.
Taken compositely, the articles in this section deal directly with thequestions of why geometry should be taught in the elementary schooland what geometry should be taught there. The rationale for the howof teaching is also included. The teacher will get a clear understandingof ways to involve his children in geometry. Involvement and instruc-tion! Instruction geared to involvement! These are the objectives forwhich this book of readings has been compiled.
Geometry all around usK-12
JOHN C. EGSGARD, C.S.B.
John Egsgard teaches at St. Michael's CollegeSchool, Toronto, Ontario, Canada. He served as amember of the Board of Directors of NCTM, 1965-68.
There is growing evidence among mathe-matics educators that geometry shouldbe experienced in each year of schoolingfrom kindergarten through grade 12. Ge-ometry is the study of spatial relationshipsof all kinds, relationships that can be foundin the 3- dimensional space we live in andon any 2-dimensional surfne in this 3-dimensional space. These relationships car:be discovered all around us. Observe themany different shapes in your environment.This is geometry. Listen to the descriptionof the path of the latest space rocket. Thisis geometry. Compare the photograph takenwith a polaroid camera to the object that itpictures. This is geometry. Notice the sym-metry to be found in a spherical or cubicalshape and the lack of symmetry in somemodern works of sculpture. This is ge-ometry. All of these involve spatial rela-tionships. Children are aware of spatialrelationships from their earliest days. In-troducing them to the idea of geometry asbeing concerned with shape and size in thematerial world will help them to rtalize andappreciate that mathematics is somethingthat plays an important role in the worldin which we live.
The geometry of the K-6 level shouldbegin in 3-dimensional space with the studyof solid shapes. From the earliest age thechild's experience is with solids, that is,things in 3-dimensional space. In the pre-operative stage, that is, K-2, the shape ofsolids should be emphasized. In kinder-garten the child should play with solids ofdifferent shapes such as cubes, cones,cylinders, spheres, rectangular boxes,prisms, and pyramids. Perhaps the names
'1873
of these should not be given in kinder-garten, but the children should be able tosort out solids that have a similar shapewhen the solids have been mixed together.In order to emphasize the idea of shape,similar solids of different sizes should beused, for example, cubes with edges of 1inch, 2 inches, 3 inches, etc. (It is interest-ing to note that some children do notrecognize a flat cylinder, such as a roundcandy box, as a cylinder.) The first opera-tion that the children should be able toperform in geometry is sorting accordingto shape. They should also be able torecognize things that have ::-:::se shapes.For example: rubber balls are like spheres;tents like pyramids; cans and some pencilslike cylinders, etc. Later they will discoverhow to order solids of the same and differ-ent shapes according to size and learn tomeasure their volume. Observe that theexamination of faces of such solids willlead the children from 3-dimensional shapesto 2-dimensional shapes. The concepts ofline segment and point will eventually growfrom the experiences the children have withthe edges and vertices of solids.
After the children have become familiarwith ,lifferent shapes they may use theseshapes to build walls. Assignments shouldbe given through which they may discoverwhich shapes fit together best without leav-ing a gap. In this way they will find someof the properties of these shapes. Hereare some examples of assignments. (Someof the assignments are similar to thosefound in the book "Shape and Size" of theNuffield Project.) In doing these assign-ments, four or five children may work
74 INSTRUCTION RATIONALE
together. ..7ach group will have 24 of eachshape of a given size, namely, cubes, rec-tangular boxes, spheres, prisms, cylinders,pyramids, etc.
Using all the bricks of one kind, try to build awall that has two thicknesses of brick.Repeat using all the different shapes.What shapes are most easily uscd for buildingwalls?Can you say why this is so?
In some groups all of the children willdecide to work on the same wall. In othergroups different children will work on dif-ferent walls at the same time. In anycase, the children will find that the cubeand rectangular-shaped brick fit togetherbest. These assignments will help them rec-ognize that this happens because thesebricks can occupy the same space in severaldifferent ways.
Use the cubes to build a wall 2 bricks thick.Take out a brick from the wall, turn it around,and replace it in its hole.How many different ways can you find to re-place a brick?Marking the faces of the brick in differentways will help.
Use the rectangular bricks to build a wall 2bricks thick.Take our a brick from the wall, turn it around,and replace it in its hole.How many different ways can you find to re-place a brick?Are rectangular bricks or bricks of cubes ordi-narily used to build walls?Examine a wall to see how these bricks arefitted together.Why do you think this type of brick gives astronger wall?
Observe that the turning around of thebricks is the beginning of the study ofsymmetry under rotation. It also has a moreimmediate purpose as the following assign-ment indicates.
Examine the wall you have built. Look at thecorners of the bricks.What type of corners do the cubes and rectan-gular bricks have?Why are these shapes used in building walls?
The ideas brought out will probably in-clude the facts that the bricks have squarecorners that fit together well, that thebricks can be replaced easily, that it is
easy to make the top of the wall level, etc.These ideas can be investigated further inthe classroom. The teacher will have tointroduce the children to the term "rightangle" for the "square corner" of rec-tangular faces and show them how to makea "right-angle tester" by folding a piece ofpaper twice (fig. 1).
FIGURE 1
First Fold
Seccnd cold(new edge is
folded overon itself)
Right Angle
The children can be asked to try to findother right angles in the classroom andabout the school and to test them by fittingtheir paper right angle onto the shapes.
Through these and similar assignmentsthe children learn how to distinguish thedifferent 3-dimensional shapes and discoversome of the simpler properties of theshapes. After these shapes have becomefamiliar, the children are ready to con-sider size and measure of volume. Chil-dren can be led to the notion of volumethrough the process of sorting similarshapes by size. Comparison of size can bemade by filling hollow shapes with thingssuch as water, peas, beads, pebbles, cubes,and sand. Assignments similar to the fol-lowing can be made.
fry 9
Geometry All around Us K-12 / Egsgard 75
2-dimensional shapes that they should beable to recognize are found as faces ofthe 3-dimensional solids they have beenusingnamely, the square, the rectangle,the triangle, the circle. An assignment suchas the following will help the children torecognize and sort different kinds of 2-dimensional shapes. An inked stamp pad,a set of solid shapes, paper, and scissorswill be needed.
You will need a large jar and a paper cup.Guess how many cupfuls of water you willneed to fill the jar.Pour water into the jar a cupful at a time untilit is full.How many cupfuls of water did you use?How close was your guess? Did you guess toomany or too few?
Other assignments can be given using ashoe box, a small match box, some sand,and also marbles and a glass jar. Studentswill realize that marbles are not very goodfor finding out how much space there is ina jar. Ultimately the children will come tounderstand that two cylindrical jars, forexample, are the same size (that is, havethe same volume) if the same number ofspoonfuls of sand are needed to fill each,and that a jar that needs a greater numberof spoonfuls of sand to fill it is larger thaneither of these. Once they can do this theycan sort any given set of hollow shapesaccording to size. Note that the sorting ofsizes in this way leads to the notion ofunits for measuring volume. The unit inthe last case is a spoonful of sand. The stan-dard units come at a much later stage,as does the formula £ x w x h.
So far we have seen that the study of3-dimensional space begins with the sort-ing of solid shapes, is followed by theordering of solids by size, which leads tothe measurement of volume. A similar pro-gression should be followed in the study of2-dimensional space as well: shape to sizeto measure. Nevertheless the examinationof 2-dimensional shapes should begin im-mediately after the children have becomefamiliar with different 3-dimensional shapesand before they are able to sort solidsaccording to size. In the handling of solidsthey will discover that some solids haveflat surfaces, some have curved surfaces,and some have both flat and curved sur-faces. When they encounter the cylinderas they sort solids according to the typeof their surfaces, flat or curved, they willgain one of their, first introductions to theidea of the intersection of sets. The first
Take one of your solid shapes that has a flatsurface.Press one of its flat surfaces on the stamp pad.Print a picture of this shape on a sheet ofpaper by pressing the inked face on the paper.Do the same for the other flat surfaces of yoursolid shape.Cut out the different pictures of faces from thepaper.How many pictures do you have?Are any of these pictures of the same shape?Which?Repeat with a different solid shape until allare done in this way.Write about your results.
Eventually the children will come torelate a specific set of 2-dimensional shapeswith each 3-dimensional shape and be ableto do the following assignment (fig. 2).
Use your set of solids to decide which of thethree solids in the top space has been usedto trace the set of faces in the bottom space.
OTI dile'0..z\v'
0 0al0 Dsi
FIGURE 2
Later on, assignments such as the fol-iming will help the children to realize that
oor I
76 INSTRUCTION - RATIONALE
2-dimensional shapes can be classified ac-cording to the number of sides. They shouldbe given a collection of polygons madefrom colored cardboard with three sides,four sides, five sides, etc.
Take the set of shapes and sort them into sub-sets like this: those with 3 sides, those with Asides, those with 5 sides, and so an. Make aloop of string around each of your subsets,or draw chalk rings around them on the floor.What is the name of the subset of shapes with3 sides?Do you know the name given to tho subset ofshapes with 4 sides?
The children will do something like. this.
oD o vc. A v 04Subset of Triangles
v 41 0o h<>0.of 4
Subset of Quadrilaterals
In this way the children can learn thenames such as triangle, quadrilateral,pentagon, hexagon, etc. Names such asequilateral triangle, regular pentagon, par-allelogram, and rhombus come at a laterstage after the children have learned some-thing about the measure of length andparallelism. The idea of congruence willalso be introduced later.
Once children, are able to sort 2-dimen-sional shapes according to the minter ofsides, they should be ready to order themaccording to size, i.e., according to area.This sorting is done through the processof covering these shapes with 3-dimensionalshapes or other 2-dimensional shapes. Thepatterns shown by the faces of the cubicaland rectangular bricks thai the childrenused in building walls can help to developthe idea of covering a 2-dimensional sur
face. Most of the tiles that children see inwalls, the floors they walk on, the side-walk; they jump on to and from schooluse rectangular shapes, so the coveringwork in the early stages will be largelyconcerned with these shapes. When childrenare looking at a wall to study the brickpattern, they can be asked such questionsas "What shape do you see?" (Rectangles.)"How could you make a pattern like thisusing bricks?" Some answers that will bereceived will be: "Draw around faces ofbricks"; "Use squared paper"; "Cut outshapes from colored paper and arrangethem in patterns," and so on. Groups ofchildren can be given square and rec-tangular tiles made from colored card-board. The following assignment can bemade.
Make tile patterns by fitting the square shapestogether.E,w many patterns can you make using therectangles?How many patterns can you make using thesquares and rectangles together?
Once the children have the feeling forcovering surfaces, they can begin to com-pare the size of surfaces using assignmentslike the one below. Shapes such as squares,equilateral triangles, regular pentagons,regular hexagons, circles should be avail-able with a sufficient number of each of thesame size to be able to cover the surfacesbeing used.
Take all of the squares. Guess how many youwill have to use to cover the front of yourworkbook.Now use the squares to cover your book.How many did you use? Was your guess toolarge, or too t..nall?Repeat with other shapes. Use only triangles,circles, hexagons, rectangles, and so on.With which of these shapes did you find youcould cover the surface?Which shapes were not very good for coveringthe surface?
The children are then given several booksor similar 2-dimensional shapes having dif-ferent areas of surface.
Geometry All around Us K-12 / Egsgard 77
can be added to introduce the notion ofsymmetry.
Order these books by finding the size of thesurface of the front cover.Which cover is largest? Which is smallest?Write about this in your own way.
These assignments should culminate in adiscussion of the various ways the shapesare used to cover the surfaces. This discus-sion should lead to the idea of using squares,triangles, regular hexagons for coveringsurfaces. The reason why pentagons andcircles are not useful should also arise. Byanalogy with the use of the cube for 3-dimensional space, the children will seethat the square is probably best. Later thearea of a surface will be measured by thenumber of standard squares required tocover the surface, so that assignmentscan be given in which the children usepaper marked in squares, say one inch orone-half inch.
Cover the front of your book with the squarepaper and find how big it is by countingsquares. Now do the same on the cover of adifferent book. Which cover was bigger? Howdo you know?
Draw around your right hand on the squaredpaper. When your group has done this, findout whose hand covers up the biggest surfaceon the paper.
Similar assignments can be given wherethe children measure the surface area ofa leaf, a cylindrical can, a box, and soon. Once again, it is important to point outthat no formulas have been used to deter-mine area. The formula t x w for a rec-tangle will be discovered later.
So far I have concentrated on that partof geometry for K-6 that can be calledthe study of shape, size, and measure.Occasionally I have made reference tosymmetry and the transformation of rota-tion. I shall now look further into thestudy of transformation in gr2des K-6.The tile patterns used in the study of areaere important aids to the understanding oftransformations. When studying the mak-ing of tile patterns and the covering of sur-faces, assignments such as the Following
Trace around the square shape. Trace aroundthe other rectangular shape.Cut out the shapes from the paper.How many ways can you fold these papershapes so that one half matches the other half?
The children should be asked to describewhat the shape on each side of the foldlooks like. Most will find this way of fold-ing:
Some will find this way of folding:
The same type of work can be done withother 2-dimensional shapes such as tri-angles and pentagons, but at a later stagein the development. The idea of symmetryor balance in shapes should be investigatedin other ways. One avenue for explorationis pattern work in arts and crafts (fig. 3).Another is in the study of blot patterns(fig. 4). In any case the children shouldtry to find the line or axis of symmetry.
VAVA TrA /AI11.
FIGURE 3
Observe that a line of symmetry is also amirror line for a reflection transformation.
78 INSTRUCTION RATIONALE
FIGURE 4
The following assignments help intro-duce the notion of translation. The childrenare given a cardboard :;quare and thesethree patterns in turn (fig. 5). When a
2
FIGURE 5
figure is moved along a straight line with-out turning the figure, then a translationtransformation has been performed on thefigure.
How could you show that the cardboard squareis the soma shape and size as each square inthe pattern?How could you use the cardboard square totrace out a pattern like the given one? Traceout the pattern.How many different paths can you find alongwhich you can slide the cardboard square fromposition 1 to position 2, so that the cardboardis always passing over a square in the pattern?You must not lift the square from the pagenor turn it around.
Similar questions can be asked for patternsof the same kind that contain rectangles,triangles, and rhombuses. The third pat-tern can be used to show that a singletranslation may be the "sum" of severalother translations.
After the ideas of translation, reflection,and rotation have been grasped, patternssuch as those in figure 6 can be used toemphasize the differences among the three.For each of the patterns I, II, and III, thechildren will be given a piece of coloredcardboard of the same shape and size of
each shape in the pattern and will be askedthese questions:
Is your cardboard shape the same shape andsize as any shape in the pattern?Which? How do you know?What type of transformation will take yourcardboard shape from position 1 to position 2?From position 1 to position 3? From position 1to position 4?
IWA.11AM
III
FIGURE 6
II
In pattern I, a reflection in the line com-mon to positions 1 and 2 carries the shapefrom 1 to 2, while a translation carries itfrom 1 into 3 or 4. In pattern II, a rota-tion about the midpoint of the segmentcommon to 1 and 2 brings the shape fromposition 1 to 2. A translation will take theshape from 1 to 3 or 4. In pattern III, arotation about the midpoint of the segmentcommon to 1 and 2 will carry the shapefrom 1 to 2. A translation carries it from1 to 3 and a rotation or reflection from1 to 4.
The idea of composition of transforma-tions can be introduced with these patternsby asking for the succession of transforma-tions needed to go from position 1 to posi-tion 5 in patterns I and II.
In pattern I a reflection and a transla-tion are most obvious; in pattern II a rota-tion and a translation or a reflection aresufficient; in pattern III a rotation, a trans-lation, and a reflection are usually selected.The idea of composition of transformationscan be introduced by asking for the trans-formation or transformations needed to gofrom position 1 to position 5 in patterns Iand II. In pattern I a. reflection to position
Geometry All around Us K-12 / Egsgard 79
2 followed by a translation to position 5
is usually suggested first. In pattern II arotation from position 1 to position 2 fol-lowed by a translation to position 5 willbe selected by most children. In pattern IIsome children will see that it is possibleto rotate from position 1 to position 5 usingonly one transformation.
Assignments such as the preceding, to-gether with discussion should be means tothe clear understanding in grades K-6 ofthe concept of transformation. This knowl-edge will prove of considerable use in thegrade 7-10 level. For example, assignmentscan be made that will lead the children todiscover certain properties of the parallelo-gram and the rhombus. Strips of cardboardwith a hole punched about a quarter of aninch from each end will be needed. Thereshould be at least four strips of two differ-ent lengths for each child.
Fasten four strips of the some length, usingpaper fasteners, to make a rhombus.
Place this on a sheet of paper. Trace aroundthe inside of the framework. Cut out this
rhombus shape.How many axes of symmetry has this shape?Fold it along one of the axes of symmetry sothat one half matches the other.What can you discover about the lengths ofthe sides that fit on to each other?What can you discover about the angles?Now fold it along the other axis of symmetry.What does this tell you about the sides andangles this time?
The children should discover that the anglesat the opposite corners are congruent. Somemay even notice that the diagonals meet atright angles.
Rotation can be used to help the chil-dren discover a similar property for theparallelogram.
Fasten four strips together as before, two ofone length and two of another length, withstrips of the same length opposite to eachother to form a parallelogram.Use the strips to make different parallelogramframeworks.Draw some of these shapes on thin cardboardby drawing along the inside of the strips.
Cut out the parallelograms drawn on thecardboard.Now place the cutout shape on a sheet of un-lined paper and draw a frame around it.Discover how many ways you can fit the card-board shape into its frame without turningthe shape over.
Discussion should bring out that "halfa complete turn" or a complete turn will
do this. Immediately following the discus-sion the following assignment should becompleted.
Fit one of the cardboard shapes into its frame.Now make a half turn with the shape so thatit fits into the other half.Look at the angle colored red. Does it fit Intoits new position?Does the angle colored blue fit into its newposition?Repeat for several of your cardboard shapes.What can you discover about the angles of aparallelogram from this?
Children at the grades 7 and 8 level canalso use the composition of transforma-tions to get their first introduction to the
group properties without making use of anumber system. For rotations of a squareabout its center, closure is exemplified bythe fact that a rotation of 90° followed by
a rotation of 180° is equivalent to a rota-tion of 270° (fig. 7).
The lack of the commutative property
90°
1A
/1111 I-
VO°
FIGURE 7
r/
FIGURE 8
reflection
84
ICJ
reflection
90°
80 INSTRUCTION RATIONALE
under composition is evident from the factthat a rotation of 90° followed by a re-flection about a vertical axis of symmetrydoes not give the same result as a reflectionabout a vertical axis of symmetry fol-lowed by a rotation of 90° (fig. 8).
Children have great fun playing withsquare cards to test these and the othergroup properties.
Let us return to grades K-6. The fol-lowing chart summarizes what I thinkshould be taught in geometry at this level
and indicates some of the relationshipsamong topics (chart 1).
The next chart is a continuation of thefirst and is concerned with grades 7-10and 11-12. Vectors can be introduced assoon as the graph of a point is understood.The section on deductive proof in grades7-10 is not to be considered as a formalorganization of Euclidean geometry. Ratherthere should be short sequences of relatedtheorems based on the congruence and par-allel facts established in earlier grades.
CHART 1
SHAPE SIZE MEASURE
SHAPE & SIZEI first notions
of length
rSHAPElines & points
kfiZEclassification in 1 -D,lines, rays, segments,betweenness
SIZE & MEASURE"volumestandard unitsmodel construction
SIZE & MEASUREanglesamount of turning
IGRAPHICALREPRESENTATION
PROJECTIONenlargementsimilarityscale drawing
SHAPEclassification of2D shapes
MEASUREareastandard units
85
A
to2
TRANSFORMATIONSstudy of tile patternstranslation, rotation,reflection, discoveryof invariance
SHAPE & SIZE Icongruence
Geometry All around Us K-12 / Egsgard 81
Axiomatic deductive geometry has beendelayed until grade 11 or 12 and this forthe best. students. I am inclined to agreewith the British view that it is suitablefor the top 5 percent or less of the stu-dent body.
terpret the tangible world of spatial rela-tionships that exist in his environment.Discovering these relationships will helplearners interpret and appreciate mathe-matics. The simple ideas can lead to theabstraction of geometrical ideas of spaceand size.
SIMPLE DEDUCTIVE PROOFSproperties of lines andsimple polygonsEuclidean vectortransformations
AXIOMATIC DEDUCTIVEGEOMETRY
MATRICES &TRANSFORMATIONS
CONICS I
7to10
COORDINATE GEOMETRY IN 2Dvector geometryanalytic geometryreflections, translations
COORDINATE GEOMETRY IN 3Dvector geometry
Primary-grade instruction ingeometry
JAMES E. INSKEEP, JR.San Diego State College, San Diego, California
Dr. Inskeep is a professor at San Diego State College and isthe associate editor of THE ARITHMETIC TEACHER. He is well known to ourreaders through his editorial comment, "As We Read." This article grew outof a paper that he presented in November 1967 at the NationalCouncil of Teachers of Mathematics meeting at Richmond, Virginia.
ecommendations for teaching geometryto young children run as a thread throughmuch of the recent literature. In the Cam-bridge Conference Report of 1963, it wasrecommended that "geometry is to bestudied together with arithmetic and alge-bra from kindergarten on."1 The Cam-bridge Conference Report of 1967, dealingwith teacher education, suggested earlymerging of arithmetic and geometry, in-struction in spatial relationships, and thestudy of standard shapes.2 It was felt thatgeometry provided a rich area for an in-tegrated study of abstract mathematics andthe environment. In the preparation forstatewide adoption of mathematics texts,the State of California recently reaffirmedthe need for geometry as an integral part ofthe primary-grade curriculum.3 Other statesand groups have made similar recommen-dations.
It might be inferred that this desire forprimary-grade geometry is confined to
1 Edutational Services Incorporated, Goals for SchoolMathematics: The Report of The Cambridge Conferenceon School Mathematics (Boston: Houghton Mifflin Co.,1963), p. 33.
Educational Development Corporation, Goals for theMathematical Education of Elementary School Teachers:The Cambridge Conference (Boston: Houghton MifflinCo., 1967), pp. 8-9, 100-101.
'California State Department of Education, Mathe-matics Program, K-8: 1967-68 Strands Report, Part 1(Sacramento, Calif.: The Statewide Mathematics Ad-visory Committee, 1967). Also The Bulletin of the Cal-ifornia Mathematics Council, XXV (October 1967), 5-15.
82
postwar and "modern mathematics" em-phases. At the turn of the century, Wil-liam W. Speer wrote and published a smallbook designed for primary-grade arithme-tic instruction? Throughout this book geo-metric experimentation formed a kind ofunifying strand for much of the recom-mended arithmetic experiences. Solids be-came the media for initial experience withgeometry, and other geometric exercisesformed the basis for number and meas-urement concepts. Of earlier vintage,Thomas Hill's Preface to First Lessons inGeometry, 1854, included this commen-tary:
I have long been seeking a Geometry for be-ginners, suited to my taste, and to my convic-tions of what is a proper foundation for scientificeducation. . . . Two children, one of five, theother of seven and a half, were before my mind'seyes all the time of my writing; and it will befound that children of this age are quicker ofcomprehending first lessons in Geometry thanthose of fifteen.5
There is considerable agreement as to thedesirability of including geometry in theprimary grades.
William W. Speer, Primary Arithmetic: First Year.For the Use of Teachers (Boston: Ginn & Co., 1897),as reproduced in 1940.
'Thomas Hill, First Lessons in Geometry (1854),Preface, as quoted in The Teaching of Geometry, FifthYearbook of the National Council of Teachers of Math-ematics (Washington, D.C.: The Council, 1930), p. 10.
Primary-Grade Instruction in Geometry / Inskeep 83
Evidence in support of these recom-mendations for geometry in the primarygrades may be categorized into threeclasses: (1) primary-grade children's abil-ity to learn geometry, (2) the existence ofsuccessful ongoing projects in primary-grade geometry, and (3) the intrinsicworth of geometry. As discussed in thispaper, the primary grades are consideredto be the preschool (including kinder-garten) and the first three grades of theelementary school. The remainder of thediscussion is organized to include theabove three categories, implementation ofgeometry instruction in the primary grades,and teaching suggestions.
Children's ability to learn geometryThe most impressive evidence as to the
ability of primary children to learn geom-etry comes from the work of Jean PiagetsPiagetian results also indicate the form andcontent of these early experiences. Twogeneralizations from his work are worthnoting. The first of these deals with thepotential content of early instruction. Chil-dren understand topological ideas first, fol-lowed by those of projective geometry, andthen they finally grasp Euclidean concepts.The topological concepts include the ideasof proximity, order, enclosure, and con-tinuity. Children are able to determine theinterior and exterior of such closed curvesas a boy's marble ring or a girl's hopscotchpattern. Shapes and lines, including thenumber line and the idea of order, can begrasped by primary-grade children. Ideasof measurement associated with Euclideangeometry come later. For most children,the ideas of our formalized geometric sys-tems will be preceded by an intuitive de-velopment of simple topological conceptssingled out and fused into the less general-ized projective and Euclidean ideas.
Another important finding from thework of the Geneva school is the consistent
For a resume of these and other findings of Piaget,see John H. Flavell, The Developmental Psychology oflean Piaget (Princeton, N.1.: D. Van Nostrand Co.,1963), PP. 327 -41.
statement for developmental learning.Children will learn by manipulating theirenvironment, and geometry can providethe vehicle for this manipulation. Intuitiveideas follow and depend upon the en-counter of the child with his environment.For this reason, a primary-grade teacherwill probably give children solids to touchand feel. The primary-grade experiencewill also include much experimentationwith the visual presentation of shapes aswell as the manipulation of objects.
The results of other research tend tosupport the fact that children can and dolearn geometry at an early age. The em-phasis given here to the Geneva school isdue to the direction which the results givefor introducing and teaching geometry tochildren. In addition to the work of edu-cational psychologists, there is a growingbody of data from teachers to indicate thatchildren can handle geometry well andwithout harm to other emerging concepts.Some of these data will be dealt with in thesection that follows.
Projects in primary geometry
The British and Canadian educatorshave done considerable experimentationwith primary-grade geometry. There arealso schools and groups in this countrytaking active parts in the development ofgeometry for the primary grades. However,only two projects will be noted in this sec-tion, both of which reflect the influence ofthe Geneva school of educational psychol-ogy. These two projects are (1) the Eng-lish work as described in the publicationof The Schools Council, Mathematics inPrimary Schools,' and (2) the OntarioGeometry Projects
Ontario Institute for Studies in Education, Geometry:Kindergarten to Grade Thirteen (Toronto, Ontario: TheInstitute, February 1967). Also Ontario MathematicsGazette, Special Elementary School Edition, September1966, pp. 5-13, 42-48; October 1967, pp. 7, 8, 14-24,27-33; and THE ARITHMETIC TEACHER, XIV (February1967), 90-93, 136-40.
Pq
84 INSTRUCTION RATIONALE
than geometry and give a completely mod-ern flavor to the curricular offering. Someof the major topics covered in thy, projectare shapes, dimensions, symmetry, simi-larity, and some work with limits througha geometric approach. Emphasis is placedupon the learning of concepts by the ma-nipulation Of the environment and by di-rect experience with constructive and sen-sory activities. Some of these intriguingactivities include working with tiles andshapes to cover areas and the handling ofsolids with attendant description of faces,edges, and vertices. Among the older pri-mary-age groups, some intuitive grasp oflimits is developed by reference to thespiral and sets of contracting (or expand-ing) squares.
The Canadian experience was somewhatdifferent in that specific attention was de-voted to geometry as such. The strand ofgeometry was drawn from kindergartenthrough high school. The geometry de-veloped for the primary grades includedclass instruction in single topics, groupwork with "assignment cards," and flexi-ble grouping for exposure to many experi-ences. The emphasis was placed upon in-dividual experience with mathematicalmodels and personal experimentation bythe children. Classification of shapes bysize and other attributes, work with geo-boards, and applications to art were amongthe activities in which primary childrenparticipated. An interesting comment wasmade by one of the Canadian primaryteachers participating in the project: "Ge-ometry became the children's favouritesubject. Why? 'Because,' they said, wecan do it ourselves.' "0 The results of bothof these projects should challenge teach-ers to teach geometry in the primarygrades.
Intrinsic worth of geometry
There are two major intrinsic reasonsfor considering geometry as a primary-
9 Sandra Rivington, "A Primary Teacher's Impression,"Ontario Mathematics Gazette, October 1967, p. 7.
grade topic: (1) there is desirable mathe-matical content to be derived from a studyof geometry early in the child's experience;and (2) children are surrounded by obLjects which have geometric significance.
Geometry provides a sound mathemati-cal background for children. Many topicswhich are treated in the intermediategrades and secondary schools lend them-selves to geometric interpretation. Formany children, the only geometry they re-ceive is that offered in a secondary schoolformal course. By this time the potentialusefulness of initial experience in geometryhas been lost. In addition to readiness,every child should have some geometry tobe mathematically literate. Regardless ofhis plans for college or vocation, there aresome geometric ideas which should be inhis common knowledge. Many of theseideas can be introduced in the primarygrades. Geometry is desirable mathematicsfor young children.
Geometry is everywhere! This may bean exaggeration, but it is evident that welive in a sea of geometric shapes. We areinundated with terms and phrases whichhave geometric significance. The Pentagonin Washington is an example, as is the vil-lage square. Nature also weaves a beauti-ful, nonabstract array of geometric models.Parallel lines are seen in architecture andin the spacing on theme paper. Modernslang includes the term "cube," and thedelta wing suggests a particular shape. Thelist of applications in art, music, and scienceis nearly endless, and we live in homes orwork in offices where shapes of one sort oranother enclose us. Environmental geom-etry certainly has a place in the curricu-lum.
Implementation
Some readers may feel convinced thatgeometry should be taught but still hesi-tate to implement their convictions. Forthis reason, the following suggestions aregiven as helps in getting a geometry pro-gram started in the classroom. The fol-lowing four recommendations are not nec-
59
Primary-Grade Instruction in Geometry / Inskeep 85
essarily equal in substance or possibleresults.
Experiment with a modest lesson at first.A small dosage can be a kind of pilot studyfor the particular experiences the teacherfeels he can handle. Ideas from workshops,journals, student teachers, and othersources may give incentives. Then thesteps to implement the teaching can betaken deliberately and carefully. One well-executed lesson is a stimulus for others.
Set aside a special day specifically forgeometry. This suggestion is made forteachers who like to tackle one thing at atime. Geometry is probably best integratedinto the total mathematics program, butexperimentation in a concerted fashionmay give the teacher the confidence heneeds to weave it into the rest of his pro-gram.
Integrate geometry into your lessonplans. This suggestion requires more so-phistication and dedication than the twoprevious ones. A teacher will need to planhis program in such a manner that geom-etry is fitted into his regular schedule. Hemy decide to use a special day, or he mayhave "quickie" lessons at intervals. Theimportant point is that geometry shouldbecome a functional part of the entiremathematical offering.
Actively seek out information of the con-tent and pedagogy of geometry. Withoutideas and some knowledge of the subject,many teachers will hesitate to take any ofthe previous suggestions. For this reason,this last suggestion is crucial to the imple-mentation of any geometry in the class-room. Some sources include members ofthe local mathematics council, teachers inother grades, and interested administra-tors. THE ARITHMETIC TEACHER is a goodsource for content as well as pedagogy. 10Once a teacher gets a taste for the subject,he will find other sources as well. With
10A copy of an annotated bibliography of recentlypublished articles of THE ARITHMETIC TEACHER that per-tain to the teaching of primary-grade geometry may beobtained by request from the author, San Diego StateCollege, San Diego, California 92115.
background in content, a teacher will find thetopic a challenge to his ability to teach it.
In the final section of this paper somesuggestions for teaching geometry aregiven. The reader will wish to use his owncreativity in extending their usefulness.
Teaching suggestions and techniques
The most important suggestion forteaching geometry is neither specific norcomforting. Do some personal researchwith your own children and develop yourown techniques. This will seem obvious,but it will produce far more satisfying re-sults than all the "copied" or "adapted"ideas you may gain.
A more specific suggestion for the pri-mary-grade teacher is to start with simpleobjects. As has been pointed out, the en-vironment is rich in geometric models.Capitalize on these. Look for shapes inthe room. Compare shapes in dress pat-terns, room decorations, books, and otherpictorial material. Ask the children tobring shapes from home such as are foundin the leftover scraps of mother's newdress or the remainder ends of wallpaper.There are many examples of shapes suchas the circle, squLre, rectangle, and tri-angle to be obtained by simply searchingfor them. Simple solid objects can also beused in a creative manner. With commonbuilding blocks such as are found in manyprimary classrooms, faces and "points"may be counted or described. It is prob-ably wise to begin with solids for plane-figure identification. This type of instruc-tion differs from the formal approach togeometry via the point-, line-, shape-defini-tion route. Give children manipulative ex-perience with shapes and solids, and thedefinitions will come quite naturally.Proper mathematical names play an im-portant part in your instruction as theneed arises to talk about the children's dis-coveries.
Organize the classroom for small-groupexperimentation. Try arranging your chil-dren in small groups as was done by
86 INSTRUCTION RATIONALE
Black." (There were forty-two childrenin her classroom!) This suggestion will re-quire extra effort at first, but once someof the tasks and assignments are developed,the work will become less burdensome.The children in Black's room were sent tosmall-group centers, each of which had aseparate activity. Some centers includedsets of solids to examine and describe.Others had game-like activities and graph-ing. For each of the activities there wasadequate opportunity to experiment anddiscover ideas.
Another idea, which is hardly novel,deals with the room environment. Arrangeyour room to reflect geometry. Use the bul-letin boards for display and for the chil-dren's work. Such titles as "Watch for theMystery Shape!" and "What's My Line?"may serve to stimulate ideas for bulletinboard arrangements. Encourage the chil-dren to display and categorize shapes thathave been found in magazines and news-papers. Teachers may find pictures ofbridges, modern engineering feats, and in-teresting architecture to enhance the roomenvironment. Do not neglect the studycenter! On a table to one side of the roomthe teacher may wish to have games andassignment cards which the children mayuse when other work is finished.
Procure materials which can be used toteach geometry. Some of the materials anddevices which should become part of yourteaching aids are as follows: geoboards ofvarious sizes and shapes, pegboards, solids(homemade, "home-collected" or com-mercial), graph paper of various types andsizes, and assorted large pictures that havegeometric applications. Other materials willbe discovered as the teacher gives atten-tion to the problem of teaching geometry.If available, an overhead projector can
nIsmet M. Black, "Geometry Alive in Primary Class-rooms," THE ARITHMETIC TEACHER, XIV (February1967), 90-93.
provide the teacher with another means forvisual presentations.
As a final suggestion, combine the teach-ing of geometry with other areas of thecurriculum. The art period certainly lendsitself to the teaching of geometry. Eventhe youngest child can be given some tac-tile or sensory experience in art throughshapes and solids. Social studies and sci-ence lend themselves to geometry. Repre-sentations of various numerical ideas,graphs of simple designs, maps, and indi-cations of size occur frequently in theseareas. Look for geometry. The teacher whoemphasizes the aspect of "looking forsomething" will find his children "seeingit."
In conclusion, the following quotationfrom Thomas Hill's Preface expresses thephilosophy of this paper.
I have addressed the child's imagination ratherthan his reason, because I wish to teach him toconceive of forms. The child's powers of sensa-tion are developed before his powers of con-ception, and these before his reasoning powers.. . . I have, therefore, avoided reasoning andsimply given interesting geometric facts.12
Hill's developmental "stages" are very sim-ilar to those of certain psychologists today,even though his terminology may not bethat currently in use. His appeal to theintuitive experience, his approach to in-formal exercises, and his willingness topermit children to experiment with geom-etry are evident. However, no teacher willwork with primary-grade geometry unlesshe knows the subject himself and is madeaware of the possibility of teaching it. Onearticle can only raise interest. The test ofinterest is the teaching of some geometryin the classroom. Enter into this most ex-citing experience. Teaching geometry isfun!
op. clt.
Why circumvent geometryin the primary grades?
NICHOLAS J. VIGILANTEUniversity of Florida, Gainesville, Florida
Professor Vigilante has been teaching functional arithmetic coursesat Otterbein College, Ohio. His new duties at the University of Floridainclude teaching elementary mathematics coursesand supervising student teachers in the internship program.
Today the arithmetic program in the ele-mentary grades is being reevaluated.New ideas and approaches are being pro-posed, discussed, and tested. One of theseideas, the teaching of geometry in the ele-mentary school, is being introduced asearly as kindergarten. One of the most ex-tensive and well-known programs in ge-ometry for the primary grades has beendeveloped by Patrick Suppes and NewtonHawley at Stanford University. Their pro-gram is more formalized than the experi-ences some children have in their presentclassrooms. The presentation of geometrydiffers from one classroom or school sys-tem to another. Some teachers find it veryessential; others are still questioning itsvalue.
Is teaching geometry in the primarygrades really worthwhile? There is muchopportunity for tangible and visual ex-periences in this area which can makethings more interesting to the student andincrease his motivation. Nearly all the ex-periments done have shown that thechildren greatly enjoy working with thisaspect of mathematics. Facts can belearned through the number line andthrough geometrical shapes. The learningof the principles of our number systemthrough multisensory experiences withgeometry could bring about better results.For example, the teaching of fractions
87
could be greatly enhanced by the demon-stration of halves, quarters, etc., with geo-metric shapes. The child, acquainted withthe concepts related to these shapes before-hand, could be aided by learning themin this manner. Experiments by Hawleyand Suppes show that it is possible toteach concepts of geometry, and thatchildren can become efficient in construct-ing and talking about the line segment,perpendicular, bisector, angle, etc. Chil-dren have also applied themselves to andseemed extremely interested in learninggeometry.
It must be admitted, however, that noteveryone is certain about the value ofteaching geometry in the primary grades,or even in the intermediate grades. Lambfeels that this subject is not related to thelanguage arts program, and since thecurriculum is so concentrated with manysubjects already, geometry might not beworth the teachers' efforts [8].* Goldmarkfinds no practical application of geometryto any other subject studied in the pri-mary grades or to everyday life [4]. Sheasks, "If geometry is discontinued afterthe primary grades, has the learning beenmeaningful enough to be retained? If itcontinues through the grades, will the con-cepts become too difficult for children of
* Numbers ht brackets refer to the bibliographic entries atthe end of the article.
88 INSTRUCTION RATIONALE
nine, ten, or eleven years of age?" Bothof these writers have had experience withthe formal-type presentation of geometryto primary students. Perhaps the quota-tion by Goldmark might have some valid-ity in this case, although we must neverunderestimate the capabilities of children.One might ask, does a program need to beso formal in its presentation and concepts?
The objectives of any program woulddepend somewhat on the nature of theprogram, that is, whether the program wasmade up of formal or informal experiences.Either type of program would have as oneof its overall objectives the developmentof concepts regarding geometric figures.I. H. Brune suggests that this objectivecan be met by having the children studyand compare the form, shape, size, pat-tern, and design of geometric figures [2].They should also have experience in con-structing and measuring the figures. R.Briscoe points out that experiences in ge-ometry provide primary children withthe opportunity to develop scientificthinking [1]. Briscoe lists four specificcontributions of geometry:
1 Children can see space as somethingthey can understand, use, control, andmanipulate to their own advantage.
2 They learn the names and properties ofthe basic geometric shapes and their re-lation to space.
3 They learn how shapes can be useful tomankind. An example would be tri-angular supports used in constructionin building bridges and buildings.
4 Children learn to produce shapes.
Goldmark adds some other objectivesto the ones mentioned above. She pointsout that children learn to nianipulate theruler, compass, and other instrumentsused in construction. They derive theirsatisfaction from the mastery of theseskills while they actively utilize the con-cepts and terms which they are learning.Another very important aspect is thatstudents learn to follow step-by-stepdirections with precision. Many children
tend to be "skimmers"; that is, they learnthe general concepts of their various sub-jects, but they fail to be specific and be-come careless in their work. Followingdirections in the construction of variousgeometric shapes and patterns forces themto follow directions carefully and accu-rately. Teachers also must be very carefulto distinguish the marks on the paperwhich represent a point, line, circle, angle,etc., from the concept or idea which eachrepresents. Unless thill is done, childrenwill not have a proper understanding ofthe geometry which they are learning.
Children must be ready to study ge-ometry. Geometric readiness can be re-duced to two main factors: subject readi-ness and psychological readiness. Thereare several activities which children can doto become ready for experiences with ge-ometry. Handling objects develops theideas of square corners, straight edges, andcurved surfaces. Playing with blocks givesthem oppo;:tunity to arrange the blocksinto patterns. They can cut geometricfigures, or learn of line segments, mid-points, and congruency by working withdesigns.
When beginning instruction in ge-ometry, the teacher should relate the sub-ject to children's everyday experiences.The object of such a presentation is toshow the children meaningful relation-ships rather than make them learn ab-stractions from memory alone. Havingchildren notice geometric shapes in theworld about them can be meaningful.There are many geometric patterns to beobserved, including round clocks and tincans, rectangular chalk boards, perpen-dieular edges of a sheet of paper, window-panes, buildings, bridges, church windows,ice cream cones, wheels, sea shells, andtelephone dials. In presenting concepts ofrelationships, the teacher can also usefamiliar objects such as blocks, balls, andpatterns in cloth.
A good time to introduce geometry isduring Christmas or another holiday sea-son. Children can use many geometric
Why Circumvent Geometry . . . ? / Vigilante 89
shapes and figures to make decorations.They can cut out circular, triangular, andsquare regions, and make pictures bypasting them on another sheet of paper.They can also make three-dimensional ob-jects. Semi-circular regions are easilymade into cones, which in turn can beconverted into bells, or even Christmastrees. Spirals cut from circular regions aregood decorations for the tree. Easter liliescan be made from such shapes. There isno limit to the possibilities. While en-gaged in these activities, children experi-ence the changes necessary to convert oneshape to another--the cut necessary tomake a "circle" or a "semicircle," the twistnecessary to change a "semicircle" to acone, the diagonal line necessary to form atriangular region from a square region. AsSweet and DeWitt said, we should showthat mathematics, as well as art andmusic, has aesthetic appeal 1101. I believethese activities will also cultivate the in-terest of the children.
Children also may enjoy working withsubject-free geometric patterns. Theycould cut out various shapes and make adesign with them, or perhaps they wouldlike to make designs by dividing one basicshape into others. For example, Figure 1shows equilateral triangles divided into
Figure 1
other equilateral triangles, and Figure 2shows other patterns which children canconstruct from an equilateral triangle.
Not only can children create patterns atrandom, but they can be given problemswhich call for manipulation to challengethe student. Here are some examples:
1 See how many ways you can divide asquare into two congruent (identical)parts (Fig. 3).
Figure 3
2 See how many ways you can divide asquare into four congruent parts (Fig.4).
Figure 4
3 How many ways can you assemble threecongruent squares? (Fig. 5).
Figure 2 Figure 5
94
90 INSTRUCTION RATIONALE
4 Assemble four congruent squares in fiveways (Fig. 6).
Figure 6
5 Assemble two congruent equilateral tri-angles together edgewise. How manyshapes can be formed? With three equi-lateral triangles? (Fig. 7).
Figure 7
6 How many shapes can be formed byjoining four equilateral triangles edge-wise? (Fig. 8).
Figure 8
7 Divide an equilateral triangle in tothree congruent parts in as many waysas you can; into two parts (Fig. 9).
Figure 9
8 Divide the equilateral triangle into sixcongruent parts; eight parts (Fig. 10).
Figure 10
Paper folding is also a good way to pre-sent challenging ideas. Some suggestusing waxed paper rather than regularpaper for teaching because the wax makesa white mark when creased. Some good ex-ercises with paper folding for primarypupils would be a straight line, squarecorner, bisection, parallel lines, angles, di-ameter and center of a circle. A pentagonmay be made by tying a strip of paper in aknot, tightening and creasing flat, thencutting off the long ends. Children canlearn much about symmetric patternsalso. They should fold paper to make twoperpendicular creases which divide thepaper into four equal parts. Then theyshould fold to bisect the right anglesformed, and cut notches out of the paper.All these activities will help them gain in-sight into concepts of geometry throughpersonal experiences.
Another idea for the presentation of ge-ometry is the use of the pegboard. Pegscan be placed at the apex of the angles of ageometric figure and then outlined withyarn to form geometric shapes. Variousfigures, such as a square, rectangle, tri-angle, parallelogram, rhombus, pentagon,and others can be made. The use ofbrightly colored yarn will also make it easyto demonstrate how one shape is con-tained in another. This pegboard can alsobe used to teach measurement, and is espe-cially useful to set up a scale "drawing"for problems such as determining the areaof or sectioning a plot of ground.
Objects, such as rectangular and tri-angular prisms and cylinders, are usefulin the second grade to review arithmeticfacts which the children learned in the
Why Circumvent Geometry . . . ? / Vigilante 91
first grade. They can review counting byenumerating the number of sides or edgesof a prism. Then, by counting the numberof lar er sides and the number of smallersides, the children can add to find thetotal number of sides. They can subtractthe number of edges on a triangular prismfrom the number of edges on a rectangularprism to see how many more edges thereare on the rectangular prism. Here thechildren can review the facts in a newway, so that they are not bored with thesame activity they used in the first grade.
Another activity associated with ge-ometry is work with an array of dots. Thechildren can determine the number sequences as more rows are added. Theymay enjoy comparing an array in theform of a triangle with an array in asquare, both having the same number ofrows.
Geometry has other facets besides com-putation of geometrical formulas. A ge-ometry program for the primary gradesshould be more exploratory and informalat least in Grades 1 and 2. All studentsshould be given a purpose for learningthese facts; geometry needs application.
Children gain reasoning and deductive experiences through work in the applicationof geometry.
Bibliography1 BRISCOE, R. "Geometry of Christmas Orna-
2 BRUNE, I. H. "Geometry in the Grades,"THE ARITHMETIC TEACHER, VIII (May,1961), 210-219.
3 Co NDRON, B. F. "Geometric NumberStories," THE ARITHMETIC TEACHER, XI(January, 1964), 41-42.
4. GOLDMAREp B. "Geometry in the PrimaryGrades," THE ARITHMETIC TEACHER, X(April, 1963), 191-192.
5 HARPER, E. H. "Pegboard Geometry,"Grade Teacher, LXXX (April, 1963), 16-176.
6 HEARD, I. M. "New Content in Mathe-matics; First Through the Fourth Grades,"THE ARITHMETIC TEACHER, IX (October,1962), 314-316.
7 JOHNSON, D. A. "Geometry for the Pri-mary," Grade Teacher, LXXIX (Apri1,1962),52-94.
8 LAMB, P. M. "Geometry for Third andFourth Graders," THE ARITHMETIC TEA CHER, X (April, 1963), 193-194.
9 RANuccr, E. R. "Shape of Mathematics,"2G6-ra2d7e. Teacher, LXXXI (September, 1963),
10 SWEET, R., and DEWITT, M. "GeometricChristmas Decorations," School Science andMathematics, LXIII (December, 1963),701-704.
s
The role of geometryin elementary school mathematics
G. EDITH ROBINSON The University of Georgia, Athens, GeorgiaDr. Robinson is a member of the department of mathematicsat The University of Georgia, where her major responsibility is teachingmathematics courses for both elementary and secondary teachers.
Many reasons have been given for theinclusion of more geometry in the lowergrades. Certainly, geometry is an im-portant subject in its own right, and ascertainly, it is unrealistic to postpone thestudy of geometry until it can be ap-proached in a systematic and rigorous way.(Precision and rigor can best be appreci-ated when we understand what we are be-ing precise and rigorous about.) It is alsotrue that geometry is encountered ineveryday life, and that children do find thesubject interesting. What has not oftenbeen pointed out is that geometry can ex-tend and enrich the study of arithmetic.
Historically, geometry and arithmetic"grew up together." The ancients studiedboth, and Euclid, in his Elements, compilednot only the geometry, but also the num-ber theory known in his day. Thus theseparation of the two subjects appears tobe an artificial one; actually, they com-plement one another.
In this article, we will discuss ways inwhich geometry can add new dimensionsto some of the understandings we expectchildren to gain in arithmetic. We willexplore five areas: measurement, prop-erties of the natural numbers, the meaningof fraction, order properties for the naturalnumbers and rational numbers, and theconcept of operation.
MeasurementIt is sometimes said that there are two
kinds of measurementthat related todiscrete quantities (called counting) and
92
that related to continuous quantities. Inthe study of counting, we make frequentuse of physical models, presenting collec-tions of pencils, marbles, books, and thelike which the child can see and touch.The property of "discreteness" can bemade very obvious for the child if he setsaside each object as he counts it or pointsto each object in turn. But what can weuse as models for continuous quantities?One continuous quantity we can measureis time, yet we cannot display any "time"for the child to see and touch. We measure"capacity" but display only the emptycontainer. Even if we show a container fullof sand, it is not the sand which we aremeasuring. Some reflection on the mattersuggests that "capacity" is a rather ab-stract idea.
Geometry provides both models fromwhich a sense of continuity can developand situations in which "discreteness" canbe distinguished from "continuity," pro-vided, of course, that we emphasize theseproperties. Consider, for example, a pathjoining two points in a plane. Such a pathis said to be connected: we can draw apicture of it without lifting our pencil fromthe paperthe pencil moves continuouslyfrom one point toward the other. Figure1 illustrates two such curves.
Figure 1
Geometry in Elementary School Mathematics / Robinson 93
We think of each of these curves asbeing "all in one piece." We also think ofeach of them as being a set of points. Theserepresent two fundamentally differentinterpretations. We might say that oneinterpretation is dynamic, the other static.To emphasize the difference in theseinterpretations and the consequencesarising from this difference, we might raisethe following questions:
1 Suppose we remove one point fromCurve I; will the curve still be "all inone piece"?
We see, of course, that the answer de-pends on our choice of the point to re-move; if we choose either point A orpoint B, the connectedness will be un-disturbed; selection of some other pointof the curve will separate the curve intotwo pieces. Each of the two separatepieces, however, will retain its con-nectedness.
2 If we remove one point from Curve II,will the curve still be "all in one piece"?
Again we see that this depends uponour choice of the point to remove; as inthe first case, removal of either "end-point" (C or D) will not disturb the con-nectedness property. Unlike the firstcase, however, it is possible to remove apoint other than an endpoint withoutseparating the curve (try a point on oneof the loops).
3 Is a proper subset of Curve I ever con-nected?
Looking back at question 1, we seethat a proper subset of a connected setdoes not have to be connected. We cansee, however, that some proper subsetsmay be connected. Suppose we selectsome third point X on Curve I and lookat the subset of the curve consisting ofX, A, and all points of the curve "inbetween." Such a subset is clearly con-nected (as, of course, would be true ofthe subset consisting of X, B, and allpoints of the curve "in between"). Thuswe see that a proper subset of a con-nected set may or may not be connected.
Another example of a proper subset ofCurve I which is not connected is thesubset consisting of the points A, X,and B. This, of course, is a discrete sub-set, and we note that a connected setmay have proper subsets which arcdiscrete or connected (or neither).
The property of being "all in one piece"is easily recognizable, and we need not tryto make it more precise. As a matter offact, preschool children may sense such aproperty in an object even before theysense its distinctness (or "oneness"). How-ever, the school experiences accompanyingthe study of counting may have disruptedthe perception of connectedness, and itmay be pedagogically wise to introduce ex-periences in which the child can learn todiscriminate between "discrete" and "con-nected" sets. As we have just seen, geom-etry provides appropriate experiences ofthis sort.
Extending our discussion, we can con-sider closed curvespaths from one pointto another and back again. Again we con-sider the two examples in Figure 2.
III
Figure 2
G H
IV
Here also we have sets of points, andsets of points which have the property ofbeing "all in one piece." The type of curvewe have in mind is one whose picture canbe drawn without lifting the pencil fromthe paper. In relation to these curves wecan also raise some questions:
4 If we remove one point from Curve III,Is the curve still connected?
We see a difference between this"closed curve" and the curves in theprevious examplesit is impossible toseparate this curve by removing a singlepoint.
5 Can Curve IV be separated by removalof one point?
94 INSTRUCTION - RATIONALE
There is exactly one "separatingpoint" for this curve; unless we removethe point where the curve "crosses it-self," the resulting set will still be connec-ted. Is it possible to have a closed curvewith more than one separating point?
6 Is a proper subset of Curve III alwaysconnected?
Our answer to question 4 tells usthat a proper subset of Curve III maybe connected. To see that a subset neednot itself be connected, we need consideronly that subset of Curve III which re-sults from removing two points, or thediscrete set consisting of two points ofthe curve. Students might think that aset of two points "right beside eachother" would still be connected; thisgives us an opportunity to emphasizethat there is no such thing as "the verynext point." We do not draw the pic-ture by drawing a "line of dots" becausesuch a procedure does not conform toour idea of what a "path" is.
We have looked at some sets of pointswhich we have called "paths," and havediscussed the idea of connectedness. Nowwe will examine some other sets of points.For example, we might consider the line,the plane, and space itself. "Space" isdefined as the set of all points. Each ofthese sets of points is considered to beconnected.
Following the same procedure as above,we might examine subsets of each of these.Some subsets of the line are the segment,the half-line, and the ray. Are these con-nected sets? We could also pick a subset ofa line which consisted of three points (orfour, or two, etc.); would this subset beconnected?
We have already looked at some subsetsof the plane (Curves IIV and the line),but there are many others. For example,there is the subset of the plane interior toCurve III, and the subset of the planeexterior to Curve III. Then there is theangle, the exterior (and interior) of theangle, and the half-plane. Each of these is
a connected set. If we look for subsets ofthe plane which are not connected, we canthink of any discrete set of points, or asubset such as the plane with some closedcurve (a triangle, for example), removed.
Some subsets of space which we havenot already considered are the solids andthe portions of space bounded by solids.We might use the cube as an example. Ouridea is that the cube is a connected set,and we see that the subset of space in-terior to (or exterior to) the cube is alsoconnected. Some subsets of space whichwould not be connected would be anydiscrete collection of points, or a subsetsuch as space with a cube removed.
Our purpose so far has been to emphasizethe difference between "discrete" and"connected" sets, since geometry providesus with examples of both. In looking atthe various examples, however, we canscarcely avoid laying the foundation formeasurement in the geometric sense. Onefeels intuitively, for example, that CurveII is "longer than" Curve I, that the pieceof the plane enclosed by a loop of Curve IIis "smaller than" that enclosed by a loopof Curve IV, and that some cubes are"larger than" others. To make such intui-tive ideas precise, we measure the lengthsof curves, measure subsets of the plane in-terior to closed curves, measure solids, andmeasure subsets of space !nterior to solids.In other words, the terms "perimeter,""area," "surface area," and "volume" arenames applied to the measurement of con-nected sets. We note also that we choose aconnected set as a unit of measurement ineach case.
Properties of the natural numbersGeometrically, a natural number, N,
which is greater than one is a composite ifany collection of N objects can be arrangedin some rectangular form; that is, if the Nobjects can be put into rows and columns,with both the number of rows and thenumber of columns being greater than one.Twelve, for example, is composite, sincewe can arrange twelve objects in any of the
Geometry in Elementary School Mathematics / Robinson 95
.1
9
000000000000e e
Figure 3
ways shown in Figure 3.' On the otherhand, seven is a prime number, because nomatter how we try, we cannot arrangeseven objects into rows and columns (ex-cept one row or one column) without hav-ing "one left over."
If, in working with our N objects, wecan put them into a square array (equalnumbers of rows and columns), we saythat N is a "perfect square."
Many teachers illustrate the com-mutative property for multiplication byplacing objects into rectangular arrays intwo ways; what is suggested here is thatthe children experiment with collections ofvarious sizes, determining for themselveswhich numbers are "rectangular" andwhich are not. Depending on when such anactivity is initiated, it could serve eitherto introduce the multiplication facts or toreinforce the learning of these facts.
Another geometric activity which canaid in the understanding of the propertiesof the natural numbers begins with acollection of square-shaped regions. Sup-pose we cut some of these, all the samesize, from a piece of paper. We can ask thechildren if they can put several of thesetogether to make a larger square region.As they experiment, it becomes obvious tothem that they cannot do this by puttingtwo pieces together, or three. Four pieces,
Some may object to calling this discrete colleotion "rec-tangular." However, it may be argued that the viewer canimpose the "rectangular" interpretation on this array only ifhe understands the 000neotedneee property.
however, can be put together to make alarger square region (Fig. 4).
\
Figure 4
Next, we ask the natural question: Canmore pieces be put together to make a stilllarger square region? Experimentationshows that this cannot be done with 5, 6, 7,or 8 squares; the next larger square re-quires nine small ones. Continuing theactivity, we see that we require succes-sively 1, 4, 9, 16, 25, etc., small squares tomake a larger one having the same shape.
Next, suppose we take triangular shapes(Fig. 5) and raise the same question: Canwe put some of these together to make a
Figure 5
larger one of the same shape? Throughtrial, we find that two will not do, or three;again we find that a larger triangularregion requires at least four small ones(Fig. 6).
Figure 6
(In case a child might think that threewill do, we emphasize that the interior ofour figure must be connected. See Figure 7.)
Proceeding, we discover that it requiressuccessively 1, 4, 9, 16, etc., small regionsto build a larger region of the same shape.To extend geometric understanding, we
ti
96 INSTRUCTION RATIONALE
Figure 7
might ask if other shapes would give thesame results; for example, can we buildlarger rectangles from smaller rectangularregions? What about parallelograms, orcircles? And, if so, how many will be re-quired in each case? We might also extendthe activity to cubes: How many cubesmust be assembled to make a larger cube?
Concept of fractionThe "fraction pie" is a familiar device
for the introduction of fractions, and isvery convenient to use because of its sym-metry. Geometrically, of course, the frac-tion pie is not a pie at all, but a circularregion, a subset of the plane. In using it asour unit, we are appealing to its propertyof being connected. In other words, whenwe begin the study of fractions, we select aunit to partition, and we choose for ourunit something which is connected. To de-velop understanding of the meaning offraction, then, suppose we choose for ourunit some other connected set. We mightconsider any of the sets in Figure 8. Ques-
Figure 8
dons relevant to the meaning of fractionwhich we could raise for each of the unitsin Figure 8 are:
1 Can each of these be partitioned intohalves with certainty?
2 Is there more than one way in whichthese units can be partitioned intohalves with certainty?
3 If there is more than one way to parti-tion one of these units into halves, inhow many ways can this be done?
4 Can the above units be partitioned intothirds with certainty? Into fourths?
A question which we must anticipatefrom such a discussion is: Must the partsof the unit be the same shape to be thesame size? A case in point is illustrated inFigure 9, each piece shown being one-halfof a half.2
Figure 9
Since either a closed curve or the regionbounded by that curve might be selectedas a unit, we might examine the diagramsin Figure 10. After studying the diagrams,
Figure 10
we might consider these questions:
5 If a region is partitioned into halves, isthe curve bounding that region neces-sarily partitioned into halves?
6 If a closed curve is partitioned intohalves, must that part of the plane in-terior to the curve necessarily be parti-tioned into halves?
Also, while we are extending ideas, weneed not confine ourselves to plane figures.
a It has been determined that third-grade children can findas many as twelve different ways of partitioning a rectangularregion into fourths,
I
t
Geometry in Elementary School Mathematics / Robinson 97
Figure 11
Five solids are illustrated in Figure 11.Are there ways of partitioning the solidsother than those illustrated? In how manyways can each of these solids be parti-tioned into fourths with certainty? Canwe devise ways of partitioning thesesolids into thirds with certainty?
Properties of orderfor the natural numbersand rational numbersWhen we count, we say the word "five"
immediately before we say the word "six."That five "comes before" six is not merelyconventionnotice that it is not the sameas saying that "a comes before b in thealphabet." Rather, the statement "fivecomes before six" expresses the mathe-matical fact that five is less than six. Wecan show this to children with discretesets of objects; we can also show this tochildren geometrically with a numberline.
When we looked at curves before, westressed that the property of being con-nected was related to drawing a picture ofthe curve without lifting the pencil fromthe paper. As we draw, we get to certainpoints in the curve before we get to others.In Figure 12, if we draw the path from Ato B, we get to point X before we get topoint Y. This fact is se obvious that the
Figure 12
reader may wonder why we bother tomention it. We mention it because thevery fact that this is so obvious may meanthat it is easier for children to understandwhat is meant by "five is less than six"with the aid of the number line.
For the natural numbers, we actuallyuse a ray rather than a line; in drawingFigure 13 we get to point X before we getto point Y.
X Y
Figure 13
For the natural numbers, we have notonly order, but also order in a very par-ticular way: 2 is 1 greater than 1, 3 is 1greater than 2, etc. Thus on our numberline we do not select points at random, butmake use of a unit, and lay off this unitin a regular fashion, beginning with theendpoint of the ray (Fig. 14). Since point
I ABC I D E
Figure 14
A "comes before" point B, and since B isone unit to the right of point A, it isnatural to assign the natural number 1 topoint A, the natural number 2 to point B,and so forth (Fig. 15).
1 2 3 4 5 6
Figure 15
The number line gives a more "pano-ramic view" of the very regular orderingof the natural numbers. For some children,this may be easier to understand than con-sidering order in relation to discrete sets ofobjects; for other children, it may providea visual summary of the order propertyobtained from working with discrete sets.
Order among the rational numbers isalso important. Since we established aunit to construct our number line, we canthen assign "fraction names" to other
r Sl
Sd
98 INSTRUCTION RATIONALE
I 2
Figure 16
4 5 6e a74.
points on the ray (Fig. 16). Then wemight think about which of the followingis more helpful in understanding that"1 is less than 1":
(a) "I is less than s because the morepieces into which a unit is partitioned,the smaller those pieces must be."
(b)
(c)
I ,
/4
Is it possible that different readers mightmake different choices from these three? Isit not possible that different childrenmight find one of these more meaningfulthan the others? Is it not possible that oneof these "explanations" would be moremeaningful for the beginner (in the studyof fractions), whereas another wouldbe meaningful after fractions had beanstudied for a while?
Sometimes we make statements aboutorder which say more than simply "5 isless than 6"; we may say, for example, "5is less than 6 and greater than 4," oris greater than and less than i." Alge-braically, we write these "4 <5 <6" and"f <I." At other times we say "5 isbetween 4 and 6," and "i is between Iand I." These latter versions:really expresswhat we see (geo.rnetrically) on the num-ber linenamely, that the point desig-nated "i" :lies between the points named"i" and "1." if we discuss this with chil-dren, they may ask if every point on thenumber line has a fraction name. Thisgives the teacher the opportunity to ex-plain that such is not the case, and that
other number systems will later be studiedwhich do provide number names for allpoints on a line.
Extending the concept of operationWe can demonstrate to children that
2+3 =5 with discrete sets of objectsthat is, we can join a set of three pencils toa set of two pencils and determine thenumerousness of the union. We can alsodemonstrate that 2+3 = 5 with connectedsets on the number line. To do this, wetlnk of "taking a step of size 2" followedby "a step of size 3." To represent thispictorially, we let arrows represent thestepa step of size 2 is represented by anarrow 2 units long, a step of size 3 by anarrow 3 units long. "Followed by" is rep-resented pictorially as being laid end toend (Fig. 17). The result can be seen (lit-erally) to be equivalent to an arrow 5 unitslong.
1 2 3 4 5 60111111DFigure 17
Furthermore, this interpretation of addi-tion applies more readily to addition forthe fractions than the concept of the join-ing of sets. Although it may not be easy tosee what the sum of and I is through theuse of arrows, at least this interpretationmay be shown, whereas it is a little diffi-cult to show what we mean by joining s ofa pie to of a pie. Actually, in the longrun, this geometric interpretation appearsto be more efficient; when faced with addi-tion for the integers, for example, it is diffi-cult to interpret (+2) + ( 3) in terms ofthe union of two discrete sets!
Multiplication can also be interpretedgeometrically: 2X3 can be interpreted as"2 steps of size 3," and can be representedpictorially by two arrows, each three unitslong, placed end to end. Furthermore, it iseasy to interpret subtraction as the inverseof addition, and division as the inverse of
1
Geometry in Elementary School Mathematics / Robinson 99
multiplication within this geometric con-text. Such an interpretation, like that foraddition, is appropriate for these processesin the system of integers. This, of course,is what we mean when we speak of "ex-tending" the concept of operationpro-viding a model which has wider applica-tion than the first one studied.
ConclusionWe do not know with any degree of cer-
tainty that all children learn in the sameway. We know that all children do notlearn at the same speed, and strongly sus-pect that they do not learn in the sameway. Thus a meaningful explanation toone child may leave another "in the dark."Knowing this, experienced teachers use avariety of ways in which to present arith-metic ideas to children. We have sug-
gested here that geometry has a contribu-tion to make to the extension and enrich-ment of certain understandings of arith-metic. The very fact that we need picturesto demonstrate geometric ideas meansthat we have visual models for the associ-ated ideas of arithmetic; all we need to dois emphasize the connection between thetwo.
Nor should we overlook the unique con-tribution which geometry can make toarithmetic: certain sets of points have theproperty of connectedness. This propertyof being connected is one which is closelyrelated to the concept of fraction, and isbasic to the concept of measurement.Moreover, it can be used to impart newmeaning to concepts such as those ofoperation, and of order and densenessamong elements of a number system.
1
is
E.
f.
Geometric concepts in Grades 4_6*
DORA HELEN SKYPEK Emory University, Atlanta, GeorgiaMrs. Skypek is a member of the faculty in the division of leacher educationand in the department of mathematics at Emory University.
In trying to decide what to say about thissubject with only forty-five minutes inwhich to say it, I asked a colleague for sug-g-bdon.s. He said, "Refute, if you can, thecharge of cpseudosophistication' currentlybeing made against much of what's new inthe elementary school mathematics curric-ulum."
I looked carefully, then, at units on ge-ometry in some of the contemporary text-books and could easily imagine many ele-mentary school children who, anxious toplease their teachers, could and wouldmemorize all the new words, the collec-tions of pictures that go with the words,and the symbols and measurement for-mulas that go with the pictures, just aschildren before have memorized their waythrough arithmetic without really under-standing the ideas represented by the lan-guage and the mathematical symbols.
While I deplore the charge of pseudo-sophistication, I concede that it could betrue. It is possible to teach a mathemat-ically sophisticated language and symbol-ism without realizing the basic purpose ofall our efforts: that children come topossess mathematically sound and mathe-matically rich concepts. Before decidingwhat geometric concepts to teach andwhen to teach them, we need to considerwhat research has to say about how chil-dren learn geometric and topological con-cepts.
The major contemporary contributor toresearch in concept formation in the child
* Adapted from a epeeeli made at the Atlanta Area Meetingof NCTM, November 20, 1964.
is Jean Piaget, the Swiss psychologist,philosopher, and educator. Piaget has con-ducted and published an amazing numberof studies of intellectual development andhas been largely responsible for many re-lated studies by other psychologists andlearning theorists.
At the grave risk of oversimplifyingPiaget's many-faceted theory, I havechosen to examine the implications of twoof his hypotheses for the teaching of geo-metric concepts in the upper elementaryschool.
Much of his theory is dominated by thehypothesis that action-involvement is thekey to progress in concept development.With . rgard to spatial relationships,Piaget's hypothesis is even a bit startling.He emphasizes that action on objects inthe child's world, rather than perceptionof the objects, is of primary importance.John H. Flavell, who is perhaps the mostimportant American interpreter ofPiaget's developmental theory, points outthat it just seems natural to us to assumethat we see space as it is and to assumethat we have always seen it that way. Notso, says Piaget. This effortless seeing isreally the end product of long andarduous developmental construction, andthe construction is more dependent on ac-tions than perception per se. One key im-plication, then, is this: action on objectsprecedes perception and, of course, con-ception.
Piaget's research led him to the conclu-sion that concepts involving topologicalrelations precede those of projective and
100, ,5
s.
Geometric Concepts in Grades 4-6 / Skypek 101
Euclidean relations. Moreover, topologicalrelations constitute the foundation onwhich the others are constructed. The im-plication here is one of order in the devel-opment of geometric concepts.
Topological relations of an elementarynature included in all contemporary curric-ula are these: the distinction betweenopen and closed curves; the ideas ofboundary, of regions interior to a closedfigure and exterior to a closed figure; theidea of betweenness on open figures.Piaget found these relations to be under-stood by the early school-age child. Thatthis is so seems reasonable when one con-siders the following instances of action onobjects in the child's world.
Little boys scratch out rings in the sandor clay in schoolyards for their games ofmarbles (Fig. 1). Marbles lying whollywithin the ring represent points (A and Bin the figure) inside, or interior to, theclosed curve; marbles lying outside thering represent points (C and D in thefigure) in the exterior region; and marbleslying on the ring represent points (E in theillustration) on the curve, or boundary.
Do little girls still play hopscotch?Children who sketch with sticks in thesand or with crayons on the sidewalk theunion of rectangles, as shown in Figure 2,are also action-involved with elementarytopological notions. According to therules of the game, one tries to toss a stoneor token of some sort so that it falls in aregion interior to one of the closed figures(point A in Fig. 2). However, it some-times falls on the boundary of a closedfigure (point B), sometimes on the com-mon boundary of two closed figures(point C), and sometimes in the exteriorregion of the union of the rectangles (pointD).
The conceptualizations of simple closedfigures, regions interior to closed curves,regions exterior to closed curves, andboundaries of the regions are already"working" concepts for the child who hasengaged in these or similar games. Theyoung child can also discriminate between
Figure 1
Figure 2
simple closed and open curves, for hetravels many paths or pushes things, suchas marbles, miniature cars, or doll car-riages, along pathsactivities which pro-vide the action-involvement from whichhe conceptualizes the distinction betweenopen and closed curves, points betweenother points, endpoints, etc.
Consider the example represented bythe open curves in Figure 3. Let's imaginethat Tom occupies a desk on the oppositeside of the classroom from the pencilsharpener. Let point A represent Tom'sdesk and point B represent the pencilsharpener. Tom has several choices forthe path from his desk to the sharpener:(1) He can travel the most direct pathfrom A to B (Fig. 3a); (2) he can go alongthe aisle by his desk to the back of theroom, across the back of the room, and
444 The Arithmetic TeacherI
102 INSTRUCTION RATIONALE
Figure 3a
A
Figure 3b
A. .8
down another aisle to B (Fig. 3b), or (3)he can go to the front of the room andacross (Fig. 3c.). If Tom has completefreedom of chase he may even travel apath up one aisle and down the next (Fig.3d). In fact, he may deliberately choosethis path so that (by chance, of course!) hejostles his friend Jack's desk (at point C),which lies between points A and B.
Through discussion and questioning,children can be guided tc organize theirideas about topological relations by re-lating their experiences to imaginativerepresentations of them; it is after a childacquires working concepts that it is mean-ingful to provide him with the moreformal language and representations toassociate with the generalized concepts.
Another topological property includedin contemporary programs is that of acontinuous infinite set of points. Piaget'sexperiments indicate that a child's con-ceptualization of a continuous figure as aconnected series of points, infinitely smalland infinitely many in number, is again theresult of developmental change, and is notusually available to the child before earlyadolescence.
There is need for more definitive re-search on this and other topological prop-erties in.cluddd in elementary school pro-grams. In the meantime, let us be carefulto avoid pseudosophisticated glibnessabout "continuous lines" and "infinitesets of points" without some assurancethat the concepts with which this Ian-
Figure 3c
Figure 3d
guage is associated are suitable ones forthe children in our classes. Our task is torecognize patterns of developmentalchange, a kind of rhythm in intellectualgrowth, and to "fit" learning experiencesand instruction into this rhythm.
Piaget's experiments involving con-cepts in projective geometry includestudies of those properties which remainperceptually invariant under changes inthe point of view from which a figure islooked at, i.e., spatial perspective. He de-scribes a task in which children were askedto arrange sticks in a straight line. Theyoungest were unable to do so. The nextage group could do so, if the arrangementfollowed a course parallel to the straight-line edge of a table; otherwise, the ar-rangement tended to drift toward somenonlinear reference curve. However, theseven-year-olds generally tested forstraightness by sighting along the array ofsticks from an end -on. position.
Of course, the arrangement of objects ina straight line is not of particular rele-vance for us, but Piaget interprets these re-sults as the child's growing awareness ofthe existence of points of view and thechoice of perspectives for assessing theproblem at hand. This interpretation is ofsignificance to us in the development ofperspective with regard to two-dimen-sional and three-dimensional space.
Children in the upper grades need tohandle, or "act on," physical models ofsimple open and closed curves and simple
Geometric Concepts in Grades 4-6 / Skypek
open and closed surfaces which can belooked at from various points of view.For example, let them view a large squaretile at some distance from them so that allpoints on the boundary (the square) areapproximately the same distance fromtheir eyestheir "point of view." Letthem draw the boundary as it appears tothem. The drawing should be the same asthe one we usually see representing asquare (Fig. 4a). Now rotate the tile sothat it still lies in the same plane as be-fore, but the vertices of the square do notoccupy the same points; have the childrendraw the boundary as it now appears tothem (Fig. 4b). Now tilt the tile so that itis no longer in the same plane as before, ex-cept for one of the edges; agc,in ask thechildren to draw the boundary as it ap-pears to them (Fig. 4c). Perhaps they canfind pictures of a tiled floor in which the"squareness" of the tiles is distorted be-cause of the perspective from which thepicture is made (Fig. 4d). At this point Ican't resist a reference to the rich possibil-ities of discussions on "Geometry in Art"with special attention to perspective.
Provide worksheets with representa-tions of plane figures as they might lookfrom many different points of view andlet children match the representations ofsimilar or congruent figures.
Figure 4a Figure 4b Figure 4c
Figure 4d
103
Figure 5
Figure 6
In later experiences, display on a bul-letin board drawings of three-dimensionalmodels as they might appear from severaldifferent points of view, as illustrated fora pyramid in Figure 5. Let children handlethree-dimensional models (they can easilybe made from construction paper) and tryto hold their models in the positionscorresponding to the points of view fromwhich each of the drawings was made.Such an experience also provides an ex-cellent time for using the language of"vertex," "edge," "face," and for discuss-ing the ideas of a closed surface, of a re-gion interior to a closed surface, of a closedsurface as a boundary of the interior re-gion, and of a region exterior to a closedsurfs -'e.
Still later, let the children draw a cube,a pyramid, or another three-dimensionalmodel as it would look if it were "undone"and opened out flat. Or let them matchthe three-dimensional models with two-dimensional paper patterns for makingmodels of their own. Let them match thepatterns for the models with correspond-ing drawings of the completed models(Fig. 6). Such experiences are also impor-
4.
iS
104 INSTRUCTION RATIONALE
tant for many later and more formal learn-ing experiences in mathematics.
In Piaget's treatment of concepts ofEuclidean geometry, his studies are pri-marily concerned with conservation andmeasurement of length, area, and volume.In these experiments he also finds a de-velopmental trend zcording to age. Sincethe idea of conserva... .a is central to muchof Piaget's theory of mathematical con-cept development, let me recount one ex-periment on conservation.
The experimenter and the child havetwo balls of clay, both of the same shapeand size. The shape of one ball is thenchanged by rolling it into a sausagelikeshape, or flattening it into a cake, orbreaking it into pieces, while the other ballof clay retains its original shape for com-parison. The experimenter tries to find outwhether the child thinks the amount, theweight, and the volume of the clay havebeen changed by the transformations, orwhether he thinks they remain unchanged,i.e., whether the amount, the weight, andthe volume of the clay have been con-aerved.
Piaget's subjects indicated working con-cepts of conservation of matter regardlessof perceptual changes at ages 8 to 10, con-servation of weight at ages 10 to 12, andconservation of volume at 12 years orlater. The important implication here isnot that there is a "natural" maturationprocess going on with increasing age andthat all we need to do is wait until thechild ages sufficientlywe know betterthan that! Instead, the important implica-tion is this: there is a pattern of sequentialstages which mark concept formation.Learning experiences and teaching tech-niques need to be devised to provide ac-tive participation by the children, ex-change of ideas with other people throughclass discussions, and refinement of thcchild's own developing processes accord-ing to a developmental sequence.
For instance, in working with the mea-surement of area, before any formal gen-eralizations or computational rules for
October 1985
finding area ere taught, we might provideexperiences such as the following, in whichchildren "act on" the plane regions to bemeasured and discover for themselrin theidea of conservation of area.
First, we will assume that the child al-ready possesses the concept of a linearunit for measuring line segments or theunion of line segments. Let me digressstill further and summarize the funda-mental notions involved in the concept ofmeasurement :
1 The measurement process requires thechoice of a unit that is of the same na-ture as the thing one wants to measure.
2 The unit needs to be such that it can bemoved around or copied for comparisonwith the thing one wants to measure.
3 Measurement is a process of assigning anumber to a set or an entity of somekind and yields an approximate numberof units.
4 In measurement, one chooses an ap-propriate unit.Now I shall use representations of rec-
tangular disks; however, in ole classroomone should begin with actual models ofrectangular regions, such as table tops,bulletin boards, or rectangular pieces ofcardboard which the child can "act on."The figures in Figure 7 represent a rec-tangular region. To measure the surfaceone must choose (1) a unit of the same na-ture as is the thing to be measured, i.e., aunit possessing surface; (2) a unit that canbe moved around for comparison with thcsurface to be measured, and (3) an ap-propriate unit. One possible unit is acircular disk (Fig. 7a). It has surface, and
Figure 7a
1 Pi 9447
Geometric Concepts in Grades 4-6 / Skypek 105
so is of the same nature as the thing to bemeasured. It can be moved around orcopied for comparison, but it is not con-venient or appropriatei.e., six circularunits of surface compared with the surfaceof the rectangular region do not evenclosely approximate the surface of the re-gion. Another possible unit is a triangulardisk (Fig. 7b). It has the same nature as
Figure 7b
the thing to be measured, and it can becopied and moved around. Although it is abetter choice than the circular disk, it isnot so convenient as a square disk (Fig.7c). Measurement also involves the process
Figure 7c
I II I
L lirr -;I I
of assigning a number to this plane rec-tangular region. Thus, we count the num-ber of square disks it takes to cover the re-gion. We say the measure of the plane re-gion is 6; there are 6 units of surfacemeasurement in the region.
Consider the rectangular region repre-sented in Figure 8a. The rectangle en-closing the region measures 3 linear unitsby 4 linear units. What is the measure ofthe interior region? In other words, whatis the area of the rectangular region?
Figure 8a
L71:Figure 8b
There are 3 rows with 4 square units ineach row, or 12 square units. Notice thatthe area of the rectangular region inFigure 8b is also 12 square units, or 2 rowswith 6 square units in each row. The areaof the two rectangular regions is the same,even though the rectangles are not con-gruent; furthermore, the perimeters of thetwo rectangles are not the same. The firstrectangle has a perimeter of 14 linearunits; the second rectangle has a peri-meter of 16 linear units.
In these first experiences, let the chil-dren handle the square disks; let them "acton" the thing to be measured by placingenough square disks on it to cover the re-gion, and then let them count the numberof square unite of surface in the region.Also$ let them count the number of linearunits along the boundary, or rectangle.The words "area" and "perimeter" shouldcome after the children grasp the idea ofsquare units for measuring plane regionsand linear units for measuring line seg-ments or the union of line segments (as inrectangles and other simple closed linearfigures).
Consider the rectangular region repre-sented by Figure 8c. Do you suppose thearea of this region is the same as that ofthe other two? An attempt to use the sameunit of measurement poses a problem.Shall we use a smaller unit? To comparethe areas, we need to use the same unit.
aFigure 8c
448 The Arithmetic Teacher
106 INSTRUCTION RATIONALE
Give your children an opportunity to dis-cover the solution to this problem. Theyhave few inhibitions about cutting some ofthe square disks into halves and making a"fit." And, again, although the perimetersare not the same, the areas are.
The foregoing examples are all ratherelementary. The point I seek to make isthat experiences in which the child "actson" the objects in his world rather thanpassively observing someone else, usuallythe teacher, showing and telling him aboutgeometric relations and properties is afirst step in the developmental formationof geometric concepts. The child who hasthese experiences is less likely to confusethe concepts of perimeter and area thanthose who, without action involvementand small group "discovery" experiences,simply memorize computational rules fordetermining perimeter and area.
Of course, in still later learning experi-ences, children must learn that society hasadopted certain standard units of measure-ment, both the English system and themetric system. Conversion ratios from,say, square inches to square feet, and viceversa, will have real meaning for childrenwho have "acted on" the problem of howmany square inches are in a square foot byplacing 12 rows of 12 cardboard square
disks (each disk representing a squareinch) on a larger cardboard square diskwhich measures one foot on each side, andthen counting the number needed to coverthe larger disk.
In closing, I encourage you to experi-ment, to conduct your own action-re-search in your classrooms to determinewhat geometric concepts should be in-cluded in the upper elementary schoolmathematics program. Read the researchreport in the October, 1964, issue ofTHE ARITHMETIC TEACHER by CharlesD'Augustine. He reports a study onteaching topics in geometry and topologyto an average sixth-grade class. In thefinal paragraph he calls for more researchto determine what topics are teachable,suitable, and efficiently learnable at thevarious levels of the elementary school.The criteria of teachability, suitability,and learnability are worth remembering.
References1 D'AUGUSTINE, CHARLES H. "Topics in Ge-
ometry and Point Set Topology: A PilotStudy," THE ARITHMETIC TEACHER, XI(October, 1964), 407-412.
2 FLAVELL, Jost.: H. The Developmental Psy-chology cf Jean Piaget. Princeton, NJ.:D. Van Nostrand Co. Inc., 1963.
3 PIAGET, J., INHILDER, B., and SZEMINSEA, A.The Child's Conception of Geometry. NewYork: Basic Books, Inc., 1960.
Geometry in the grades
IRVIN H. BRUNE Iowa State Teachers College, Cedar Falls, IowaDr. Brune is professor of mathematics at Iowa State Teachers College.
The proper study of mankind is geome-try. Tiny Timothy chose a nickel ratherthan a dime, but it was worldly wisdom,rather than geometric sense, that helacked. Timothy's grandfather, however,who was shrewd in money matters, be-lieved that three-foot cubes rather thanthree-foot spheres should adorn the newcourthouse. He surmised that the cubes,covered only on five faces, would requireless gold leaf to gild. His geometric guesswas almost as naive as Timothy's moneymuddle.
Myra in Grade 5 wanted to growflowers. Her parents gave her fifty feet ofaluminum edging and told her that shecould have all the ground that the edgingwould enclose. What shape of flower bedwould give her the most space?
At a sale Mr. Handyman bought apiece of linoleum that was nine feet wideand sixteen feet long. He knew that withonly one cut he could fit it on a floortwelve feet by twelve feet. This, as wellas the other cases cited, requires geometry.
Or consider the circles in Figure 1. Obvi-ously the black circle on the right islarger than the black circle on the left. Butis it?
Similarly for Figure 2. Which has thegreater area, the outer black annulus (orring) or the inner black circle?
Another instance is the well-knownFigure 3. Which segment, the horizontalor the vertical, is the longer?
Perhaps you think such examples aretrivial. At least they are homely, ordinary.But there are more important applicationstoo; you see them daily. When mathe-maticians use higher geometry to solvecomplicated problems in the production
107
Figure 1
Figure 2
Figure 3
and distribution of goods, no one con-cerned thinks the matter trifling. Con-sider, for example, the problem of selectingthe most economical combinations of sometwenty ingredients that fluctuate in pricedaily. This problem arises in sausage-making. Which quantities of which meatswill satisfy fixed standards of high quality
11
108 INSTRUCTION RATIONALE
and at the same time cost the least accord-ing to today's prices? Here the mathema-tician employs geometry in an untrivialway.
Ageless geometry
To enumerate and to describe man'suses of geometry would take a trip in timefrom prehistory to the present moment.The subject began in earth-measuring, itgrew in planet-observing, it led the way inpure mathematics, and it pioneered inmodern mathematics.
Man has always needed geometric prin-ciples, however dimly he may atfirst haveperceived them. Similarly, children's livescannot be devoid of geometry, howeverunaware they may be of its formal aspects.For, irrespective of its many applicationsand regardless of its value as a system ofreasoning (and both of these phases meritattention), geometry embodies numerousideas interesting, in themselves.
Geometry for all
We suggest, therefore, that geometrydeserves a lifetime of interest. To study ftin only the tenth grade hardly suffices. Atthat level pupils pretiumably study one ormore kinds of geometry as deduction.There and in subsequent courses they alsolearn about applications. But the com-puting with geometric formulas that fre-quently represents the only planned ex-perience that pupils have in geometryprior to Grade 10 seldom prepares themfor Grade 10.
Grade-school geometry
Informal geometry in the elementarygrades can, therefore, counteract a seriousdeficiency. In these grades geometry is thestudy of form. Shapes, sizes, patterns, de-signsthese are the stuff from whichchildren form concepts. From studyingforms children discover numerous geo-metric relations; from making construc-tions pupils Iearn about geometric facts;from measuring figures learners acquire abackground of geometric information. The
41
es 0
Figure 4
work teems with both classical conceptsand contemporary concepts.
Fun and future
We believe, therefore, that children ofall ages should get wimple opportunities tofind out things about geometry. The goalis satisfaction, here and now, with thingsmathematical, and geometry abounds insuch ideas. An accompanying benefit willdoubtless be a preparation for a more for-malized geometry of proofs. Just as theancient Egyptians' surveying and theearly Chaldeans' star-studying opened apath for the deductions of the Greeks, sothe block-arranging of the curious kinder-gartners and the design-drawing of the en-thusiastic upper graders provide under-standings for the problems of the olderpupils. The pleasures of the momentoutweigh the preparations for the future.
Therein lies the heart of the matter.Teachers cherish in their pupils suchtraits as alertness, preparedness, andwillingness. And possibly the greatest ofthese is willingness. Seldom, though, dothese traits develop overnight; rather,they seem to stem from many things thatpupils do. Through the situations thatteachers encourage them to explore, pupilsdiscover relations, achieve insight, andgain satisfactions for the moment as wellas for later studies. Mathematics, youknow, is a cumulative subject. For exam-ple, clusters of dots, such as those inFigure 4, provide numerous helpful experi-ences. For infants the dots in Figure 4 aremany, whereas those in Figure 5 are few.
Figure 5
Figure 6a Figure 6b
0
Figure 8a
00 GOOFigure 6e.
Figure 8b
0
Figure 8d
Figure 9
Geometry in the Grades / Brune 109
Figure 6c Figure 6d
Later the dots represent nine. Then theyhelp with the ancient idea that somenumbers are squares: that nine corre-sponds to three threes. Furthermore, in
Figure 7
Figure 8c
**
0Figure 8e
Figure 10
Figures 6ae, the square arrays, one, four,nine, sixteen, twenty-five, etc., when or-dered and compared via differences, encour-age pupils to think about the odd numbers.
110 INSTRUCTION RATIONALE
Square numbers:
-- 1 4 9 16 25 36 49 .. .
DIfferences between successive square num-bers:
3 5 7 9 11 13 ...Does zero belong in the blank before 1
in the top row?For upper graders, finding a continuous
path containing four line segments andconnecting all the nine points will prob-ably be a fascinating challenge. It mightalso be preparation for later work in sim-ple topology, where, among other things,the study of pathsclosed paths and not-closed pathsreceives attention.
Geometric readiness
Examples such as the foregoing abound.Suppose that we look briefly at the pupils'readiness before we consider further in-stances.
We have hinted that readiness in ge-ometry implies at least two other "nesses":1. Preparedness, or adequate mathemati-
cal maturity to go on.2. Willingness, or enough emotional se-
curity to go on.Geometric preparedness begins at an
early age. Tots in kindergarten enjoyplopping the cutout figures into theirproper places (Fig. 7). Children quicklydiscriminate between right triangles andequilateral triangles, between squares andoblongs, and between trapezoids and par-allelograms. Already these youngsters areshape conscious.
Success in this sort of activity leadsyoung children on. Their handling ofsquares, cubes, disks, triangles, spheres,and so on, prepares them for further workwith forms.
All too frequently, however, such activi-ties terminate abruptly. This occurs be-cause courses of study encourage thepupils to put away "childish things" andsettle down to the stern business of memo-rizing facts and practicing operations withnumbers. Since perfection in these worthy
matters eludes most learners, the study offacts and operations flourishes while thestudy of forms !anguishes. Of course, les-sons in the upper grades deal with areasand volumes, but computing with num-bers and distinguishing between area andperimeter and between volume and sur-face have been known to monopolize theact.
Fortunately, the trend today points togeometry for the sake of geometry, ratherthan to geometry as further practice incalculating. In the elementary grades in-formal, or intuitive, studies get the empha-sis. Drawing, counting, and measuringlead pupils to observing, inferring, andgeneralizing. Consciousness of forms con-tinues to grow, and readiness for proofsin geometry also continues to grow.
Let us return momentarily to the tots.By handling wooden, paper, or plasticrepresentations of geometric figures, chil-Ctren appreciate numerous ideas; amongthese are notions of square corners,straight edges, round edges like faces,roundness of disks, and roundness of balls.These children gain a degree of under-standing to go on; they grow in geometric
adiness.But children gain in willingness too.
The shapes, the fitting of objects into pat-terns, the matching, the comparing, andthe counting all make children receptiveto further activities. One quite ordinaryfirst grader happened onto triangular-number arrangements, as in Figures 8a-e.Pupils do things, learn, and crave to learnsome more.
So, as they progress in mathematicalmaturity (preparedness), pupils tend toseek new mathematical worlds to P...snquer(willingness). Thus, willingness and alsopreparedness stem from activitiesthingsdone successfully. Junior is likely to won-der what the next step will lead to. If, forexample, thirty-six lines can be drawnthrough nine points such that no threepoints lie on aline (Fig. 9), then how manylines can be drawn if exactly three of thenine are on one line? (Fig. 10.) In the
11^._7
Geometry in the Grades / Brune 111
figures the joins of one point with each ofthe others constitute a hint; this is one wayto begin the problems.
In sum, teachers seek to challenge pu-pils, not to frustrate. Sometimes a teamapproach (or working as a class on a per-plexing problem) will prevent defeat.Whether pupils suffer more from frustra-tion than from boredom, however, is moot.
Some s;mple arrangements
Besides patterns of points previouslymentioned, we might look at a few otherconfigurations. In Figure 11 a side of thesquare ABCD measures 3, AE measures 2,and angle DEF measures 90°. In squareKLMN, a side measures 2, and angleLKP has the same measure as angle BEF.The sections thus cut form two separatesquares or one combined squarea kindof readiness for the Theorem of Pythag-oras.
In Figure 12 ABCD is a square with aside that measures 6, and BEFG is asquare with side 3. AL= BH =CJ =DK=11, and BN measures 3. M is the inter-section of HK and JL. Square MNOP hasthe same measure in square units assquares ABCD and BEM combined.
Now suppose that we begin with othersegments, half-squares and half-oblongs,for example. These, plus a few rectangles,semicircles, and quAdr.ants, form a varietyof designs. Figur( 4 13-17 ;llustrate someof them.
Abundant triangles
If the pupils start with three equal seg-ments and the question, "What can we dowith the segments?", they can soon comevp with an equilateral triangle (Fig. 18).By joining the mid-points of the sides ofthe triangle, they can produce four equi-lateral triangles (Fig. 19). By repeatingthe process successively for unshaded tri-angles, the pupils obtain Figures 20and 21.
Or, the pupils may choose not to shadeany of the component triangles and pro-ceed to successive quartering of all the
new triangles. One triangle yields four tri-angles, four yield sixteen, and sixteenyield sixty-four. Some pupils will moveahead and forge a fifth stage or even fur-ther proliferations of triangles. Theoret-ically, the fast workers need not growweary of waiting for their slower class-mates to finish a step. Endless steps awaitthose who wish them, and the steps getharder.
Some pathological curves
If the pupils begin again with threeequal segments, they can form anotherequilateral triangle (Fig. 22). By trisect-ing the sides, erecting equilateral triangleson the middle sections, and erasing the in-tersections of these four triangles, thepupils get Figure 23. Then in Figure 24further trisections and outward points ap-pear. Some pupils may wish to carry thisprocedure still further. Although the areaof this snowflake-like curve clearly cannotexceed the surface of the page, its perim-eter becomes infinitely large.
Similarly for other pathologic curves(Figs. 25-27), the pupils proceed from anequilateral triangle again. Here, 'owever,the open mid-sections are spanned byequal segments that intersect inside in-stead of outside the original triangle. Thisgives an inverted snowflake pattern. Heretoo, the perimeter can be made infinitelylarge, even though the area will not exceedthat of the drawing paper.
Figures 28,29, and 30 show what resultswhen pupils begin with a circle, divide itinto six equal parts, and invert alternatearcs. This procedure, repeated, leads toanother figure, the aesthetics of which maybe doubtful. It is known as an inside-out-side curve. It troubles almost everyonewho seeks to determine its curvature.
Tiling patterns
Ali pupils soon learn'wlien they begin towork with measures of angles, one fullturn 360°. A further interestinginvestigation results when pupils face thequestion, "What flat figures will fit around
112 INSTRUCTION RATIONALE
A E B
F
C
Figure 11
Figure 12
Figure 14
Figure 16
Figure 18
Figure 13
Figure 15
Figure 17
Figure 19i:
:.1r 117I-....,,
. .....4'1
Figure 20
Figure 23
Geometry in the Grades / Brune 113
Figure 21
Figure 24
Figure 26
Figure 27
Figure 30
Figure 22
Figure 25
Figure 28
AAVA
AYAATAT A
A A
Figure 31
114 INSTRUCTION RATIONALE
Figure 32
Figure 34
a point and fill in the flat surface?" Regu-lar polygons seem to be needed, althoughall rectangles will suffice, but not all reg-ular polygons will meet the requirement.
Considerable acquaintance with thesepolygons can result from such experiment-ing. How can we draw them? Later thepupils will study the straightedge-com-pass constructions for regular polygons,and still later they will study criteria ofconstructibility.
But strictly informal experimenting willreveal some combinations of polygonsthat, so to speak, cover the floor. Indeed,among sophisticates the whole subject offloor coverings; filings, and mosaics bearsthe impressive name of "tessellation."
Six equilateral triangles (Fig. 31), fourrectangles (Fig. 32), and three hexagons(Fig. 33) exhaust the possibilities of howmany flat figures will fit around a pointand fill in a flat surface. If, however, the
Figure 33
Figure 35
AA AMEMAYAWAYA
1011Figure 36
pupils do not limit the problem to poly-gons of one single sort, then the followingserve: two hexagons and two triangles(Fig. 34); two octagons and one square(Fig. 35); three triangles and two squares(Fig. 36); one hexagon, two squares, andone triangle (Fig. 37). Still other possi-bilities, not illustrated here, exist: onehexagon and four triangles; one dodeca-gon, one hexagon, and one square; two
Geometry in the Grades / Brune 115
dodecagons and one triangle. Your pupilsmay want to try them.
It will occur to the pupils that these arepossibilities when they experiment andconstruct the following table, referring toregular polygons:
Number of sides:
3 4 5 6 7 8 9 10 11 12
Measure of angles:
60 90 108 120 128+135 140 144 147- 150
From the increasing sizes of the angles, itappears that regular polygons having a
Figure 37
Figure 39
t. 12 1-11
still greater number of sides are not likelycandidates,
Centroids
Locating a centroid, or a center of massof a body, can become a thorny problem.Quite young children, on the other hand,can cut geometric forms from cardboardand readily locate lines and points of bal-ance.
Regular figures balance rather easily ona knife-edge. The intersection of two suchlines of balance determines a center ofbalance, or a center of mass. Equilateraltriangles, squares, regular hexagons, reg-
Figure 38
Figure 41
116 INSTRUCTION RATIONALE
ular octagons, and circles illustrate theseideas.
When the figures depart from regular-ity, the knife-edge procedure may becomemore difficult, as in the case of a convexpolygon. There the centroid lies outsidethe figure (Fig. 38). This might becomethe germ of the idea of a centroid for asystem of bodies, wIlich may interest thefuture physicists and astronomers in yourclasses.
Simple and nonsimple
Finding the centroid of an involved, yettechnically simple, curve could provide adifficult problem. A cutout again providesan intuitive approach (Fig. 39). Inci-dentally, when is a curve simple? If theleft circle were removed from Figure 39,the curve would remain simple. But if theright circle were removed, the figure wouldbecome nonsimple.
Figure 40 is another puzzler. It has twooutsides, one of which might seem at firstto be inside. When is an outside inside?
In case you wondered, here is one solu-tion to the path problem, referred to ear-lier, for nine points arranged as threethrees (Fig. 41).
Yin-yang
Rooted in antiquity, especially in ven-erable Chinese philosophy, is the symbolrepresented in Figure 42. From yang, lit-erally the south o: sunny side of a hill, theunshaded portion of the design representsthe bright, good, positive, male principlein Chinese dualism. Yin, on the contrary,
Figure 42
symbolizes the dark, evil, negative, and fe-male. The shaded part stands for yin.
Regardless of how pupils and theirteachers look on such mystical matters,the emblem appears to be easy to con-struct and to describe geometrically.
Summary
Suppose, now, that we summarize.From the kindergarten on the concepts,rules, and operations of arithmetic and al-gebra dominate pupils' experiences inmathematics. Few deny it, and if the teach-ing has been good, still fewer bemoan it.
New occasions, however, teach newduties. Mathematics grows apace; mathe-matics education accelerates its search.With reason we urge teachers to learn newmathematics and teach new courses. Wesee merit in helping young children togain acquaintance with good mathematicsearly. We note with pleasure the improve-ments in textbooks of elementary-schoolmathematics.
In our zeal, however, we run the risk ofletting abstractions get out of hand. Thiswe disapprove. The motto "be abstract"should be left to the habitants of Green-wich Village. The race, including those ofus who urge mathematics reforms, learnedmathematics through practical needs andreal problems. The abstractions, the gen-eralizations, and the deductions followedthe investigations, the approximations,and the corrections.
Surely, therefore, we should not denyyoung children the opportunity to exploreand learn.
A proper study for all children is geom-etrythe geometry of form. Here pupilsperceive, compare, measure, and general-ize. Here they sharpen intuition withoutplunging too far into abstractions. Aboveall, children see values in what they do. Ifwe can encourage pupils to discover forthemselves some principles in the scienceof space, then they will bring into theirgeometry classes a usable store of informa-tion about the Euclidean plane. They ),might also have a good start on three-space.
1 91
Bibliography1954-1956
Jenkins, Jen. "A Plan for Teaching Arithmetic Shorthand." 3 (November1956) :207-9.
1957-1959
Buck land, G. T. "The Prismoidal Formula." 6 (February 1959) :44-45.Forrest, Genevieve. "R. Ratio." 6 (February 1959) :49.Ragland, Elizabeth. "Art and Arithmetic." 6 (March 1959) :112.Vincent, Sister M. "Volume of a Cone in X-Ray." 6 (Ain it 1959) :132.Stone, Marshall H. "Fundamental Issues in the Teaching of Elementary School
Mathematics." 6 (October 1959) :177-79.Clancy, Jean C. "An Adventure in TopologyGrade 5." 6 (November 1959) ;
278-79, 256.
1960-1962
Forrest, Genevieve. "A Recipe for Angle, Circle, Construction Surprise." 7(May 1960) :266.
Mire, Paul A. "The Volume of a Sphere." 7 (May 1960) :268.Brune, Irvin H. "Geometry in the Grades." 8 (May 1961) :210-19.Rutland, Leon, and Max Hosier. "Some Basic Geometric Ideas for the Ele-
mentary Teacher." 8 (November 1961) :357-62.Hawley, Newton S. "Geometry for Primary Grades." 8 (November 1961):
374-76.Powers, Richard. "A Dream House Project." 9 (May 1962) :280-81.Clarkson, David M. "Taxicab Geometry, Rabbits, and Pascal's TriangleDis-
coveries in a Sixth-Grade Classroom." 9 (October 1962) :308-13.Heard, Ida Mae. "New Content in MathematicsFirst through Fourth Grades."
Goidmark, Bernice. "G:ometry in the; Primary Grades." 10 (April 1963):191-92.
Lamb, Pose M. "Geomety for Third and Fourth Graders." 10 (April 1963):193-94.
Condron, Bernadine F. "Geometric Number Stories." 11 (January 1964):41-42.
117 12?.4%
118 BIBLIOGRAPHY
Denmark, Thomas, and Robert Ka lin. "Suitability of Teaching Geometric Con-struction in Upper Elementary Gradesa Pilot Study." 11 (February 1964):73-80.
Miller, G. H. "Geometry in the Elementary Grades: A Comparative Study ofGreek Mathematics Education." 11 ( February 1964) :85-88.
Whitman, Nancy C., and Sadie Okita. "Constructing an Inexpensive Sphere."11 (April 1964) :261-62.
Todd, Robert M. "An Illustration of the Unrecognized Assumption." 11 (May1964) :317-18.
D'Augustine, Charles H. "Topics in G ometry and Point Set Topologya PilotStudy." 11 (October 1964) :407-12.
Josephina, Sister. "A Study of Spatial Abilities of Preschool Children." 11
(December 1964) :557-60.Sand ling, Dolores C. "Plane Polygons." 11 (December 1964) :569-70.D'Augustine, Charles H. "Developing Generalizations with Topological Net
Problems." 12 (February 1965) :109-12.Smith, Lewis B. "Pegboard Geometry." 12 (April 1965) :271-74.Felder, Virginia. "Geometry Concepts in Grades K-3." 12 (May 1965):
356-58.Knowles, Evelyn. "Fun with One-to-One Correspondence." 12 (May 1965):
370-72.Skypek, Dora Helen. "Geometric Concepts in Grades 4-6." 12 (May 1965) :
443-49.Vigilante, Nicholas J. "Why Circumvent Geometry in the Primary Grades?" 12
(October 1965) :450-54.Wahl, M. Stoessel. "Easy-to-Paste Solids." 12 (October 1965) :468-71.Powick, E. "Unexpected Discoveries." 12 (November 1965) :574, 578.Giddings, Marie. "Being Creative with Shapes." 12 (December 1965) :645-46.
1966
Robinson, G. Edith. "The Role of Geometry in Elementary School Mathe-matics." 13 (January 1966):3-10.
Woods, Dale, and William E. Hoff. "Introducing Models for N-DimensionalGeometry in the Elementary School." 13 (January 1966) :11-13'.
D'Augustine, Charles H. "Factors Relating to Achievement with SelectedTopics in Geometry and Topology." 13 (ICirch 1966) :192-97.
Hewitt, Frances. "Visual Aid for Geometry." 13 (March 1966) :237-38.Smart, James R., and John L. Marks. "Mathematics of Measurement." 13
(April 1966) :283-87.Weaver, J. Fred. "Levels of Geometric Understanding: An Exploratory
Investigation of Limited Scope." 13 (April 1966) :322-32.Walter, Marion. "An Example of Informal Geometry: Mirror Cards." 13
(October 1966) :448-52.Guraii, Peter K. "Discovering Precision." 13 (October 1966) :453-56.Major, James E. "Rings and Strings." 13 (October 1966) :457-60.Sganga, Francis T. "A Bee on a Point, a Line, and a Plane." 13 (November
1966) :549-52.
123
BIBLIOGRAPHY 119
Elliott, Richard W. "Head-Shrinking--an Introduction to Scale." 13 (December1966) :685.
Weaver, J. Fred. "Levels of Geometric Understanding among Pupils in Grades4, 5, and 6." 13 (December 1966) :686-90.
1967Paschal, Billy J. "Geometry for the Disadvantaged." 14 (January 1967) :4-6.Smith, Lewis B. "Geometry, Yesbut How?" 14 (February 1967) :84-89.Black, Janet M. "Geometry Alive in Primary Classrooms." 14 (February
1967):90-93.Jackson, Stanley B. "Congruence and Measurement." 14 (February 1967) :
94-102.O'Brien, Thomas C. "A Look at Triangle Congruence." 14 (February 1967):
103-6.Sullivan, John J. "Some Problems in Geometry." 14 (February 1967):107-9.Walter, Marion. "Some Mathematical Ideas Involved in the Mirror Cards." 14
(February 1967):115-25.MacLean, J. R. "The Quest for an Improved Curriculum." 14 (February
1967) :136-40.Tassone, Sister Ann Dominic, C.S.J. "A Pair of Rabbits and a Mathematician."
14 (April 1967):285 -88.Skidell, Akiva. "Polyominoes mid Symmetry." 14 (May 1967) :353, 382.Carrol, Emma C. "Creatamath, orGeometric Ideas Inspire Young Writers."
14 (May 1967) :391-93.Brune, Irvin H. "Some K-6 Geometry." 14 (October 1967) :441-47.Hillman, Thomas P. "Colors, Geometric Forms, Art, and Mathematics." 14
(October 1967) :448-52.Vigil mte, Nicholas J. "Geometry for Primary Children: Considerations." 14
(October 1967) :453-59.Buck, Charles. "Geometry for the Elementary School." 14 (October 1967):
460-67.Richards, Pauline L. "Tinkertoy Geometry." 14 (October 1967) :468-69.Gibney, Thomas C., and William W. Houle. "Geometry Readiness in the Pri-
mary Grades." 14 (October 1967) :470-72.Froelich, Effie. "An Investigation Leading to the Pythagorean Property." 14
(October 1967) :500-504.Overholt, Elbert D. "Manipulating Points and Figures in Space." 14 (November
1967) :560-62.Needleman, Joan R. "Discovery ApproachPolar Coordinates in Grade
Sullivan, John J. "Problem Solving Using the Sphere." 16 (January 1969):29-32.
Shah, Sair Ali. "Selected Geometric Concepts Taught to Children Ages Sevento Eleven." 16 (February 1969) :119-28.
Heard, Ida Mae. "Developing Geometric Concepts in the Kindergarten." 16(March 1969) :229-30.
Gogan, Daisy. "A Game with Shapes." 16 (April 1969) :283-84.Walter, Marion. "A Second Example of Informal Geometry: Milk Cartons." 16
(May 1969) :368-70.Higgins, Jon L. "Sugar-Cube Mathematics." 16 (October 1969) :427-31.Dennis, J. Richard. "Informal Geometry through Symmetry." 16 (October
1969) :433-36.
Egsgard, John C., C.S.B. "Geometry All around UsK-12." 16 (October1969) :437-45.
Farrell, Margaret A. "Patterns of Geometry." 16 (October 1969) :447-50.Ogletree, Earl. "Geometry: An Artistic Approach." 16 (October 1969):
457-61.Teegarden, Donald 0. "Geometry via T-Board." 16 (October 1969) :485-87.Liedtke, Werner. "What Can You Do with a Geoboard?" 16 (October 1969):
491-93.
1 95
BIBLIOGRAPHY 121
Overholser, Jean S. "Hide-a-RegionN ?. 2 Can Play." 16 (October 1969):496-97.
Lulli, Henry. "Editorial Feedback." 16 (November 1969):579-80.Ranucci, Ernest R. "Thumb-Tacktics." 16 (December 1969) :605, 630,664.
1970
Schaefer, Anne W., and Albert H. Mauthe. "Problem Solving with Enthusiasmthe Mathematics Laboratory." 17 (January 1970) :7 -14.
McCiintic, Joan. "Capacity Comparisons by Children." 17 (January 1970):19-25.
Alspaugh, Carol Ann. "Kaleidoscopic Geometry." 17 (February 1970) :116-17.Forseth, Sonia D., and Patricia A. !_dams. "Symmetry." 17 (February 1970):
119-21.Liedtke, Werner, and T. E. Kieren. "Geoboard Geometry for Preschool Chil-
dren." 17 (February 1970): 123-26.Wardrop, R. F. "A Look at Nets of Cubes." 17 (February 1970) :127-28.Buchman, Aaron L. "An Experimental Approach to the Pythagorean Theorem."
17 (February 1970) :129-32.Scott, Joseph. "With Sticks and Rubber Bands." Comment by Don Cohen. 17
(February 1970):147-50,Kipps, Carol H. "Topics in Geometry for Teachers--a New Experience in
Mathematics." 17 (February 1970) :163-67.Walter, Marion. "A Common Misconception about Area" 17 (April 1970):
286-89.Mann, Nathaniel, III, and Dale Philippi. "Volume and Surface Area of Rec-
tangular Prisms: A Maximum-Minimum Problem for the Grades." 17 (April1970):291-92.
Woodby, Lauren G. "The Angle Mirror Outdoors." 17 (April 1970):298-300.Wells, Peter. "Creating Mathematics with a Geoboard." 17 (April 1970):
Century Fashion." 17 (May 1970) :372-86.Fawcett, Harold P. "The Geometric Continuum." 17 (May 1970) :403-12.Hart, Alice G. "The Angle Mirror Indoors." 17 (May 1970) :419-23.
