A STUDY OF THE COGNITIVE DEVELOPMENT OF LENGTH AND AREA MEASUREMENT Thesis for the Degree of Ph‘ D. MICHIGAN STATE UNIVERSITY ROBERT J. KOSANOVICH 1 9 7 1 Date 0-7539 Michigan State This is to certify that the thesis entitled A STUDY OF THE COGNITIVE .—DEVELOPMENT OF . LENGTH AND AREA MEASUREMENT presented by ROBERT J. KOSANOVICH has been accepted towards fulfillment of the requirements for PIT . D. degree in EDUQATIQN Octqber 26: 1971 LIBRARY BINDERS II SPIIIIMIIY “INI- - ”“2— same? «an m INC if . ABSTRACT A STUDY OF THE COGNITIVE DEVELOPMENT OF LENGTH AND AREA MEASUREMENT By Robert J. Kosanovich It was the stated purpose of this study to investigate the cognitive development of length and area measurement relative to four common component properties (congruence, conservation, additivity, and unit measure) and chronological age. The intent was to conduct a comparative investigation to lend support to one of the two con- trasting points of view identified to be: (1) There is no difference between the ages at which a child attains corresponding levels of understanding relative to length and area measurement and that both of these concepts are finally attained at approximately the same age; (2) There is a difference between the ages at which a child attains corresponding levels of understanding relative to length and area measurement and that a child finally attains length measurement prior to area measurement. The need for this study was based on the conflicting con- clusions made from two separate investigations: (1) Piaget, lnhelder, and Szeminska concluded that there is a simultaneous development of and final attainment of the cognition of length and area measurement; (2) Beilin and Franklin concluded that the component properties of length measurement are understood prior to correSponding properties of area measurement and that the cognition of length and area measure- ment are finally attained in that order. Robert J. Kosanovich Pilot Study_and Sample The population for the study was the student body of a public elementary school in a northern Michigan city serving a middle class neighborhood. Prior to the actual study, a pilot study was conducted to determine the age groups to be used and to refine the tasks. As a result of the pilot study, twenty children in each of the five age groups (age seven through eleven) were randomly selected to be included in the sample. Collection of the Data Each subject was given nine tasks. The first was a vocabulary task of measurement terms used to determine inclusion in the final sample. Four length measurement tasks (concerning the properties of congruence, conservation, additivity, and unit measure) corresponding to four area measurement tasks were given to each child to determine their level of cognitive development. Analysis of Data Two research hypotheses were develOped for study. Operational hypotheses derived from the research hypotheses were submitted to test. A seven step inference process was employed to determine whether the operational hypotheses should be accepted or rejected. The Chi-square test for independence and the Phi-coefficient were used as test statistics. The Phi-coefficient was then used as an indicator of the association between scores on the length measurement tasks and scores on corresponding area measurement tasks. Robert J. Kosanovich Research Hypothesis I I. The cognitive development of length measurement is simul- taneous to the cognitive development of area measurement relative to the properties of congruence, conservation, additivity, and unit measure. This research hypothesis was transformed into twenty opera- tional hypotheses relating a measurement property to chronological age. Seventeen of the twenty hypotheses were accepted. Four summariz- ing operational hypotheses across all ages were formed for each meas- urement property. Each hypothesis was submitted to test and accepted. Thus, there is evidence to indicate that there is a simultaneous cognitive development of length and area measurement. Research Hypothesis II II. The understanding of length and area measurement are attained simultaneously. This hypothesis was tested for each of the five age groups and accepted for all but the nine-year old group, although 75 per cent of the nine-year olds were scored the same regarding the final attainment of length and area measurement. A summarizing hypothesis across all ages relating the final attainment of these two measure- ment concepts was statistically tested and accepted. Thus, there is evidence to indicate that there is a simultaneous final attainment of the understanding of length and area measurement. A STUDY OF THE COGNITIVE DEVELOPMENT OF LENGTH AND AREA MEASUREMENT By j .L- L"; Robert JI‘Kosanovich A THESIS Submitted to Michigan State University in parital fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Education 1971 DEDICATION To my wife Mary Louise and children Tad, Keith, Marcia ii ACKNOWLEDGMENTS Space does not permit an adequate expression of appreciation to all who have given so generously of their time and have provided continuous encouragement and leadership. Appreciation is extended for the efforts and consideration of many people, of whom I am recognizing only a few. The author expresses sincere appreciation to Dr. Lauren Woodby for his personal interest and guidance of this research, as well as to Drs. Geoffrey Moore, Dale Alam, and George Myers for their advice and encouragement at various stages of the doctoral program. My gratitude goes to Dr. Robert Houston for an introduction to the work of Piaget and to Dr. Andrew Porter's research consultation staff who assisted in the design of this study. My gratitude also goes to the principal of the school, Larry Musser, and to the teachers and students who were most cooperative and willing to partake in this research project. Appreciation is also expressed to Brenda Brouwer for her technical assistance and typing of this thesis. Finally, my deepest gratitude ones to my wife, Mary Louise, and my children, Tad, Keith, and Marcia, whose understanding and patience were constant sources of inSpiration. TABLE OF CONTENTS ACKNOWLEDGMENTS ...................... LIST OF TABLES ...................... LIST OF FIGURES ...................... CHAPTER I THE PROBLEM AND ITS SETTING ........... Introduction to the Problem .......... Need for the Investigation .......... Purpose of the Investigation ......... The Research Problem and the Hypotheses . . . . Research Hypothesis .......... Mathematical Considerations and Definition of Terms . . . ............... Congruence Preperty . . . . ....... Conservation Property .......... Additivity Property ........... Unit of Measure Property ........ Organization of the Study ........... Summary ..... . . ........... II REVIEW OF RESEARCH AND RELATED LITERATURE Theoretical Background ............ Piaget's Description of the Cognitive Development of Length Measurement ..... Related Research - Length Measurement ..... Piaget's Description of the Cognitive DevelOpment of Area Measurement ...... Related Research - Area Measurement ...... Comparative Study of the Cognitive Development of Length and Area Measurement . . . . . ........... Summary . ................... III THE RESEARCH METHODOLOGY ............. Pilot Study .................. Sample .................... Page iii vii ix 38 43 46 47 CHAPTER IV V General Procedures ............... Criterion for Inclusion in the Sample ..... Vocabulary Task ............. Measurement Tasks ............... Congruence Task - Length ......... Congruence Task - Area .......... Conservation of Length Task ....... Conservation of Area Task ........ Additivity of Length Task ........ Additivity of Area Task ......... Unit Length Task . . . .......... Unit Area Task .............. Research Design ................ Analysis ................. Research Hypothesis I .......... Research Hypothesis II .......... Collection of Data ............... Summary .................... ANALYSIS OF DATA AND FINDINGS ........... Introduction .................. Evaluation of the Subjects ........... Results for Criterion for Inclusion in Sample ........... . ....... Research Hypothesis I ............. Congruence Task ............. Conservation Task ............ Additivity Task ............. Unit of Measure Task ........... Research Hypothesis II ............. Summary .................... SUMMARY OF MAJOR FINDINGS, CONCLUSIONS AND IMPLICATIONS .................. Introduction .................. Conclusions Concerning the Four Measurement Properties ................. Congruence Property ........... Conservation Property .......... Additivity Property ........... Unit Measure Property .......... Conclusions Concerning the Research Hypotheses . . . . . . . . . . ....... Implications .................. BIBLIOGRAPHY . . . . . ................... APPENDIX A DETAILED DESCRIPTION OF TASKS ......... V 131 136 136 138 139 139 140 140 I41 142 145 152 CHAPTER APPENDIX B APPENDIX C RECORDING SHEET . . . . RESULTS FOR EACH SUBJECT vi TABLE 3-1 3-3 4-1 4-3 4-4 4-6 4-7 4-8 4-9 4-10 4-11 LIST OF TABLES Demography of Sample ............... Reading Score Distribution ............ Arithmetic Concept Score Distribution ...... Congruence Task for the Seven-year Old Group . . . . Congruence task for the Eight-year Old Group . . . . Congruence Task for the Nine-year Old Group Congruence Task for the Ten-year Old Group . . . . Congruence Task for the Eleven-year Old Group . . . . . .............. Congruence Task for All Subjects in the Sample ....... . ........... Conservation Task for the Seven-year Old Group . . . Conservation Task for the Eight-year Old Group . . Conservation Task for the Nine-year Old Group Conservation Task for the Ten-year Old Group . . Conservation Task for the Eleven-year Old Group ................... Conservation Task for All Subjects in the Sample .................... Additivity Task for the Seven-year Old Group . . . . Additivity Task for the Eight-year Old Group . . . . Additivity Task for the Nine-year Old Group Additivity Task for the Ten-year Old Group . . . . vii Page 50 51 52 111 113 114 114 115 115 116 117 117 118 113 119 120 120 121 121 TABLE 4-17 4-18 4-19 4-20 4-21 4-22 4-23 4-24 4-25 4-26 4-30 4-31 4-32 Additivity Task for the Eleven-year Old Group .................... Additivity Task for All Subjects in the Sample Unit of Measure Task for the Seven-year Old Group .................... Unit of Measure Task for the Eight-year Old Group .................... Unit of Measure Task for the Nine-year Old Group .................... Unit of Measure Task for the Ten-year Old Group .................... Unit of Measure Task for the Eleven-year Old Group .................... Unit of Measure Task for All of the Subjects . . . . Final Attainment of Length and Area Measurement for the Seven-year Old Group .......... Final Attainment of Length and Area Measurement for the Eight-year Old Group .......... Final Attainment of Length and Area Measurement for the Nine-year Old Group ........... Final Attainment of Length and Area Measurement for the Ten-year Old Group ........... Final Attainment of Length and Area Measurement for the Eleven-year Old Group .......... Final Attainment of Length and Area Measurement for All of the Subjects ............. Summary of the Operational Hypotheses Tested for Research Hypothesis I ............ Summary of the Operational Hypotheses Tested for Research Hypothesis II viii Page 122 123 124 124 125 126 126 129 I30 131 133 LIST OF FIGURES FIGURE Page 1-1 Conservation Property ............... 15 1-2 Additivity Property - Area ............ 17 1-3 Unit Measure ................... 20 2-1 Area and Length Measurement ............ 40 3-1 Vocabulary Task - Area .............. 58 3-2 Conservation of Length .............. 65 3-3 Conservation of Area ............... 72 3-4 Additivity of Length ............... 78 3-5 Additivity of Area ................ 84 3-6 Unit of Length Measure .............. 88 3-7 Unit of Area Measure ............... 94 ix CHAPTER I THE PROBLEM AND ITS SETTING Introduction to the Problem One of the most significant devel0pments that has occurred recently in the school mathematics curriculum has been the inclu- sion of a considerable amount of geometric material throughout the program. Only a few years ago practically all of the geometry being taught was concentrated at the tenth grade. Now, in the more updated curricula, it is being taught at all levels. The Cambridge Conference on School Mathematics has suggested that geometry be studied with arithmetic and algebra from kinder- garten on with the aims of devel0ping the planar and Spatial intuition of the pupil, affording a source of visualization for arithmetic and algebra, and to serve as a model for that branch of natural science 1 One of which investigates physical space by mathematical models. the beneficial results of introducing geometry at an early stage is that it provides for a more fundamental develOpment of the nature of 1Cambridge Conference on School Mathematics, Goals for School Mathematics (Boston: Houghton Mifflin Company, 1963), p. 33. 2 measurement and the measuring process.1 It is the measuring process, specifically length and area measurement, that will be considered in this paper. Almy, Chittenden, and Miller contend "that the success of the various new programs in mathematics . . . is largely dependent on their appropriateness for the conceptual abilities of the chil- drenreceivinginstruction."2 This statement indicates the importance to the design of mathematics curricula of data regarding the cogni- tive development of mathematical concepts in children. Gibney and Houle indicate that emphasis on the cognitive development of mathe- matical concepts in children is lacking: "Geometry is an area of mathematics that has received much attention and space in contem- porary mathematics textbooks, but geometgy readiness 1§___t0pic that appears tg.gg!g.bggg_slighted [underline mine]."3 They believe that readiness for learning, as it relates to geometry, is vital, and questions such as those below need to be given consideration: 1. Has adequate attention been given to the factors of readiness in planning for geometric concepts in a course of study? 2. Are current textbooks being designed to accommodate factors of readiness at the respective grade levels for which the content is prepared? 1H. Stewart Moredock, “Geometry and Measurement," Mathematics Education, National Society for the Study of Education, Sixty-ninth Yearbook, 1970, p. 167. zMillie Amy, Edward Chittenden, and Paul Miller, Youn Children's Thinking (New York: Teachers College Press, 1 , p. 126. 3Thomas G. Gibney and Hilliam N. Houle, ”Geometry Readiness in the Primary Grades," The Arithmetic Teacher, October 1967, p. 570. 3 3. Is the preparation of teachers adequate for them to understand geometric concepts well enough to teach readiness for this content? 4. How can teachers interpret readiness factors and present geometric concepts in accordance with the need of the class? 5. What can be done to improve teacher effectiveness in establishing readiness for geometric concepts? 6. Hill neglect of readiness in the presentation of geometry destroy the possible benefits that might have been gained at previous or subsequent grade levels? 7. Are geometric concepts placed at appropriate levels in courses of study? Certainly geometric concepts have a place in the primary grades. But regardle§ of how worthy the content may be, the endeavor to help children develop intellectually will be unsuccessful if they are not ready to understand the concepts. Lack of readiness can render the best instructional situation ineffective. Hence, educators involved in the development of the mathematics curriculum must con- sider factors regarding readiness to understand geometric concepts. Need for the Investigation Jean Piaget, a Swiss psychologist, and his associates2 per- formed a series of experiments concerned with the development of an awareness in children of various pr0perties of length and area libid. zJean Piaget and his associates are sometimes referred to as the Geneva group. 4 measurement (Piaget's work will be detailed in Chapter 11).1 Follow- ing his study of the intellectual development of length and area measurement, Piaget stated that "The develOpment of conservation and measurement runs exactly parallel whether the objects are lengths or whether they are areas and the level at which they are finally graSped is the same for both."2 Copeland states that: There is a readiness stage that the child must reach before logical concepts such a§ those involved in measurement can be . . . learned . The necessary concepts . . . to measurement do not appear for many children until age seven to eight or until sometime during the second or third grade of school Yet many teachers attempt teaching measure- ment before this time. This study [Piaget's work in measurement] indicates then that if systematic measurement is to be "taught" it should not be presented before the latter part of what is usually the third grade. Even then, for' , most children it will have to be an experimental or trial-and-error readiness-type experience . . . The necessary concepts for measurement] will develOp (1) When the child is o d enough (Eight to eight and a half, according to Piaget) . . . . 1Jean Piaget, Barbe] lnhelder, and Alina Szeminska, The Child's Conception of Geometry, trans. E.A. Lunzer (New York: Harper and Row, 1970) . 21bid., p. 300. 3Richard N. Copeland, How Children Learn Mathematics-Teaching Implications of Piaget's ResearCh TNew Ybrk: The Macmillan company, 1970), p.’23. 41bid., p. 193. 5Ibid., p. 209. 5 Often measurement in one dimension is taught before the child is at the operational or readiness level to understand it, and yet two-dimensional or area measure- ment is deferred several years past the age at which children can understand it. Children at age nine in general are ready for measurement in two dimensions using the method of superposition of a unit square. Children in the age range seven to nine should be tested first for an understanding of conservation of area Land length} . . . when they are at the con- servation or operat onal level, they are ready to begin measurement using a unit [of measure] . . . and countin the number of times i is contained in the [object . . . being measured. Thus, Piaget's research has promoted criticism by Copeland of the present manner in which length and area measurement are being taught in American elementary schools. It appears to be Capeland's belief (as indicated in the previous quotes) that we presently begin teaching length measurement approximately one to two years too soon and begin to teach area measurement several years later than it could be taught without any loss of effectiveness. Beilin and Franklin conducted a study regarding the intel- lectual develOpment of length and area measurement on a comparative basis (This study will be discussed in detail in Chapter 11).2 The subjects were New York City school children from the first and third grades. Contrary to the finding of Piaget, Beilin and Franklin's results indicate that the majority of the children studied achieved 1Ibid.. p. 238. 2The discussion of this study is based on Harry Beilin and Irene Franklin, “Logical Operations In Area and Length Measurement: Age and Training Effects," Child Development, 33, 1962, pp. 607-618. 6 length measurement prior to area measurement. This finding is not consistent with the aspect of Piaget's developmental sequence of mathematical concepts in which length and area measurement are said to be finally achieved at the same level, and, hence, at the same age. It suggests, rather, that "length and area measurement . . . are achieved in that order" and that "the component operations [con- gruence, conservation, additivity, unit measure, etc.] are applied most easily first to a single dimension then to two dimensions . . ."1 Thus, there are two contrasting points of view regarding the development of and final achievement of length and area measurement: (1) Piaget states that there is a parallel development of and simul- taneous achievement of length and area measurement. Based on this finding Copeland expresses criticism of the present manner in which these two measurement concepts are taught. (2) Beilin and Franklin state that length measurement is learned prior to area measurement and in fact the component operations of the two measurements are first learned in one dimension, then in two dimensions. This he- lief is consistent with the order of appearance of length and area measurement in elementary school textbooks. Piaget and his associates rarely described the samples used in their studies, except for the age factor. It is assumed that extensive work by this group involving American children is absent. This assumption is based upon Hunt's description of Piaget's work: 1Tbid., p. 617. 7 In these early studies, Piaget's empirical data came almost completely from the language behavior of pairs of children observed in preschool situations at the Rousseau Institute in Geneva. The second period began with his observations of the origins of intelligence and reality constructions in his own three infant children. Rosenbloom, commenting on the importance of Piaget's work and the lack of American experimentation,stated that: The implications of Piaget's theories for mathe- matics education have not yet been realized. Studies by competent researchers involving American children are badly needed. New curricular materials, based on sound psychological evidence should be written. And, in teacher education, more work involving Piaget's theories and their implications would serve as land- marks ig improving instruction in the elementary school. Lovell states that "although there are a number of points on which I find myself in disagreement with the Geneva school, I strongly urge readers to study the books written by Piaget and Inhelder and to repeat for themselves some of the experiemnts de- scribed."3 1J. Mcv. Hunt, “The Impact and Limitations of the Giant of Developmental Psychology," David Elkind and John Flavell, Studies In Cognitive Development (New York: Oxford University Press, 196D), pp. 4L5. 2Paul C. Rosenbloom, "Implications of Piaget for Mathematics Curriculum? Improving Mathematics Education, Conference Sponsored by The Science and Mathematics TeaEhing Center, Michigan State University and The National Science Foundation, 1967, ed. by Robert Houston, p. 49. 3Kenneth Lovell, The Growth of Basic Mathematical and Scientific Concepts in Chderen (LondOn: UnTversity of London Press LTD, 1961), p. T. 8 Purpose of the Investigation The purpose of this study is to investigate the cognitive development of four significant properties (congruence, conservation, additivitygand unit measure) of length and area measurement relative to the factor of age. The intent is to lend support to one of the two contrasting points of view identified earlier concerning the attainment of length and area measurement: (1) That there is no difference between the ages at which a child attains correSponding levels of understanding relative to length and area measurement and that both of these concepts are attained at approximately the same age. This view is shared by Jean Piaget as indicated in his state- ment, ”The development of conservation and measurement runs exactly parallel whether the objects are lengths or whether they are areas and the level at which they are finally grasped is the same for both.“1 (2) That there is a difference between the ages at which a child attains corresponding levels of understanding relative to length and area measurement and that the child attains length measurement prior to area measurement. This view is shared by Beilin and Franklin as indicated in their statement: "length and area measurement . . . are achieved in that order" and that "the component operations [con- gruence, conservation, additivity, unit measure, etc;] are applied most easily first to a single dimension then to two dimensions."2 lPiaget, Inhelder, and Szeminska, Conception of Geometry, p. 300. 2Beilin and Franklin, "Logical Operations In Area and Length Measurement." p. 617. 9 The Research Problem and the Hypotheses In a speech at New York University in March of 1967, Piaget comments as follows: A few years ago Jerome Bruner made a claim which has always astounded me; namely that you can teach any- thing in an intellectually honest way to any child at any age if you go about it in the right way. Hell, I don't know if he still believes that . . . it's prob- ably possible to accelerate but maximum acceleration is not desirable. There seems tp.be an optimum time. Hhat this optimum time is will surETy—depend on eaCh individual and on the subject matter.1 The question of optimum time to introduce a child to a mathematical concept is of utmost importance to anyone who is reSponsible for the intellectual development in children. As Gibney and Houle have indi- cated earlier in this paper (pp. 2-3) in the form of questions, the readiness level of the child and the grade placement of the concept are major factors involved in the success or failure of the concept to be learned. Piaget claims that a parallel development of conservation and of length and area measurement exists and that the level at which they are finally grasped is the same for both [noted earlier, p. 9]. Based on Piaget's research, Capeland claims that systematic measure- ment should not be presented before the latter part of what is usually the third grade and that often measurement in one dimension is taught before the child is at the readiness level to understand it, and yet two-dimensional or area measurement is deferred several years past the age at which children can understand it. lFrank Jennings, “Jean Piaget, Notes on Learning,” Saturday Review (May 20 , 1967). p.82. 10 If Piaget's theory regarding the parallel development of and the simultaneous achievement of conservation and measurement regard- less whether the objects are lengths or areas is accepted, then the implications described by Copeland become quite prominent since the ideas are contrary to prevailing modes of thought. That is, many mathematics textbooks present length measurement prior to area measurement. As an example, the textbook series used in the elemen- tary school from which the subjects of this study came introduces length measurement two years prior to area measurement.l Research Hypotheses This study is an attempt to investigate hypotheses regarding length and area measurement and their common component properties of congruence, conservation, additivity, and unit measure (terms to be defined in the next section). The specific procedure of study, the test instruments used, and the tests employed to analyze the data are explained in Chapter III. The operational hypotheses and the statistical alternatives tnat were develOped from the research hypo- theses listed below are described more completely in Chapter III. I. The cognitive devel0pment of length measurement is si- multaneous to the cognitive development of area measure- ment relative to the properties of congruence, conserva- tion, additivity, and unit measure. 1Joseph N. Payne, et 1,, Elementary‘Mathematics Concepts and Tppics from Readiness Ihrough Grade 6 (New’YoFk: Harcourt, Brace, and Morld, 1965). 11 This hypothesis asks the questions: 1. Does a child understand the congruence of length and the congruence of area at the same age? 2. Does a child understand conservation of length and conservation of area at the same age? 3. Does a child understand the additivity of length and the additivity of area at the same age? 4. Does a child understand the use of a unit of length measure and the use of a unit of area measure at the same age ? II. The understanding of length and area measurement are attained simultaneously. This hypothesis asks the questions: 1. Has a child who has attained (failed to attain) an understanding of length measurement also attained (failed to attain) an understanding of area measure- ment? 2. Has a child who has attained (failed to attain) an understanding of area measurement also attained (failed to attain) an understanding of length measure- ment? Each of the two hypotheses is suggested as a result of Piaget's investigation into the cognitive development of length and area measurement. The first hypothesis deals with Piaget's pr0posal 12 regarding a parallel development of the two concepts of length and area measurement. The second hypothesis deals with Piaget's pro- posal regarding a simultaneous attainment of length and area measure- ment. Mathematical Considerations and Definition of Terms Some of the newer high school geometry textbooks contain axioms concerning the measurement of the length of line segments and the measurement of the area of polygonal regions.1 The axioms focus on significant prOperties of measurement including the preperties of congruence, conservation, additivity, and unit measure. These four significant properties of measurement, relative to both length and area, are investigated in this study. In mathematics the word "measurement“ refers both to a pro- cess (the method or way measurements are performed) and to the end result of the process if the end result is reported using a numeral and a unit of measure such as an inch or a square inch.2 The posi- tive real number that is used to denote the measurement of an object is called the “measure" of the object.3 Considering only whole 1“A ol onal re ion is a plane figure which can be expressed as the union oi iinite numEEr of triangular regions, in such a way that if two of the triangular regions intersect, their intersection is an edge or a vertex of each of them." Edwin E. Moise, Elementary Geometry_from an Advanced Standpoint (Reading, Mass.: Addison- Hesléy, 1963). p. 153. 2James R. Smart and John L Marks, "Mathematics of Measure- ment,“ The Arithmetic Teacher, Aprilll966, p. 283. 3Ibid. 13 number measures, the measure of a line segment is the number of times the unit segment can be laid end to end along the segment being measured from one endpoint to the other. In the remainder of this text, the term'heasurement'refers to the process of finding the measure of an object. Congruence Ptpperty The property of congruence is the mathematical basis for the theory of measurement.1 In the sense of develOping Spatial percep- tions, it is clear that concepts relative to measurement begin well below the school level. The child will begin early to distinguish between such things as a round object and a square or triangular one. An individual who correctly selects the piece to fit into a given space in a jigsaw puzzle is exercising his perception of this extremely important geometric property called congruence. The impor- tance of this measurement preperty called congruence is illustrated by the devotion of an entire workbook regarding the pr0perty of con- gruence by the University of Illinois Committee on School Mathematics.2 In general, two geometric plane figures are congruent if they have the same size and shape, or, in other words, if one can be moved so as to coincide with the other.3 At the elementary level, 1Ibid.. p. 285. 2do McKeeby Phillips and Russell E. Zwoyer, Book 2: Congru- ence, University of Illinois Committee on School Mathematics TNew YorE: Harper and Row, 1969). 3Moise, Elementary Geometry, p. 58. 14 congruence is given an operational definition: two segments or plane figures are congruent if a copy of one may be made to fit exactly on the other. The tasks used to test for an understanding of the con- gruence property (i.e., does a child understand the operational definition of congruence) are presented in Chapter III. Conservation Property ”Underlying all measurement is the notion that an object remains constant in size throughout any change in position.“1 The property that the length of a line segment or the area of a plane region is unaltered under certain transformations is referred to as conservation. The measure axioms presuppose the concept of conser- vation of length and of area. For example, the measure axiom re- garding the addition of areas states that if a region is the union of two subregions (such that the subregions intersect only in edges or vertices), then the area of the region is the sum of the area of the two subregions.2 However, no restriction is placed on how the subregions are combined. Therefore, since they may be combined in more than one way by changing the positions of the two subregions, several regions of various Shapes may have the same area (see Figure 1-1, AREA: The area measure of region A is equal to the area measure of region B). In order to make a realistic attempt to solve 1Jean Piaget, Barbel lnhelder, and Alina Szeminska, The Child's Conception of Geometry, trans, E.A. Lunzer (New York: Harper andiRow, 1964), p. 90. 2Noise, Elementarnyeometgy, p. 154. 15 FIGURE l-l CONSERVATION PROPERTY AREA 1 2 1 Region A 1 2 1 | 2 l LENGTH Region B 1 2 1 2 Line A 2 , 1 Line 8 Scale: 1“ = 2“ 16 a task requiring the application of this axiom one must have achieved conservation of area. A parallel discussion could apply to lengths as well (see Figure 1-1 LENGTH: The length of line segment A is equal to the length of line segment B). The tasks used to test for conser- vation (i.e., is the child cognizant of the invariance of length and area under certain transformations) are presented in Chapter III. Additivity Property The measure axiom regarding the addition of areas states: "Suppose that the polygonal region R is the union of two polygonal regions R1 and R2 such that the inter- sections of R and R2 are contained in a union of a finite number of segments. Then relative to a given unit of area, the area of R is the sum of the areas of R1 and R " 2. Suppose we are given a five-inch by three-inch rectangular region denoted by R (see Figure 1-2 A) and are told that R is the union of R1, a two-inch by three-inch rectangular region, and R2, a three-inch square region. Let a one-inch square be the given unit area. Using the area axiom which states that "if R is any given polygonal region, there is a correspondence which associates to each polygonal region in Space a unique positive number such that the number assigned to the given polygonal region R is one"2, we have a correSpondence which assigns the positive number six to R1 and the positive number nine to 1School Mathematics Study Group, Geometr with Coordinates, Part II (New Haven: Yale University Press, IDBSI, p. 989. 2Ibid. 17 FIGURE 1-2 ADDITIVITY PROPERTY - AREA A a: 17//////1 Scale: 1" = 2" 18 R2; that is,six copies of the given unit area are required to cover Rl exactly and nine copies to cover R2 exactly. The intersection of the subregions R1 and R2 is the single line segment which is a common side of the two regions. Therefore, using the additivity axiom, the area measure of the region R is the sum of the area measures of R1 (Six) and R2 (nine), or fifteen. Suppose a second polygonal region S which is irregular in Shape (see Figure 1-2 B) is the union of R1 and R2, such that the intersection of R1 and R2 is contained in a single line segment. Then the area measure of S is the sum of the area measures of R and 1 R2, or fifteen. Therefore, although regions R and S differ in shape, they have equal area measures relative to a common measuring unit. A three-inch by four-inch region T is also the union of R1 and R2 (see Figure 1-2 C). However, in this case the intersection of R1 and R2 is a one-inch by three-inch plane region (see shaded region) which cannot be covered by a finite number of line segments. Therefore, the additivity axiom cannot be used to calculate the area measure of region T. A Similar discussion involving lengths would illustrate the use of the additive property of lengths. The tasks used to test for an understanding of the additive property (i.e., is the child aware of the fact that the whole is equal to the sum of its nonoverlapping parts) are presented in Chapter III. Unit of Measure The understanding of the notion of a unit of measure and the importance of its Size is necessary for proper measurement to take place. For example, consider two congruent rectangles with 19 dimensions of two inches by four inches (see Figure 1-3, AREA). The one-inch square and the isosceles right triangle whose legs are one inch long are to be used as measuring units. The area measure of rectangle A is found using the unit square and the area measure of rectangle B is found using the isosceles right triangle as measuring units. A child who understands the notion of a unit of measure and the importance of its size would determine that the area measure of rectangle A is eight and the area measure of rectangle B is sixteen relative to their respective units of measure. In addition, the child would also state that the two rectangles, A and B, are congru- ent since the square measuring unit is exactly twice the size of the triangular measuring unit. Similar statements can be made regarding lengths (see Figure 1-3, LENGTH). The tasks used to test for an understanding of the use of a unit of measure (i.e., does a child consider the number and the size of the units used in the measuring process) are presented in Chapter III. Qgganization of the Stody This thesis consists of five chapters. CHAPTER I. INTRODUCTION Introduction to and statement of the problem, need and pur- pose for the study, statement and explanation of the hypotheses, mathematical consideration and definition of terms. CHAPTER 11. REVIEW OF RELATED RESEARCH AND LITERATURE An over-all view of Piaget's theory of intellectual develop- ment, Piaget's description of the development of length and area 20 FIGURE 1-3 UNIT MEASURE AREA. A B E Square Unit Triangular Unit 1.5.4911. __2 A 3 Unit 1 Unit 2 Scale: 1" = 2" 21 measurement, other research related to Piaget's investigation of length and area measurement. CHAPTER III. THE RESEARCH PROCEDURE Demographic information regarding sample, description of tasks and criteria used in evaluation, the research design, opera- tional hypotheses, and a description of the statistical instruments and the analysis process. CHAPTER IV. ANALYSIS OF DATA AND FINDINGS Presentation of results, restatement of hypotheses, conclu- sions regarding acceptance of hypotheses, correlation analysis, and statistical tests. CHAPTER V. SUMMARY, CONCLUSIONS AND IMPLICATIONS Major findings, synopsis of the problem, conclusions, and implications. Summary This study examines the development of length and area measurement on a comparative basis relative to the factor of age. It also investigates the final attainment of length and area measure- ment relative to age. Two questions are central to the study: 1. Is there a parallel development of the significant prop erties (congruence, conservation, additivity, and unit measure) of length and area measurement? 2. Are length and area measurement finally achieved at approximately the same age? CHAPTER 11 REVIEW OF RESEARCH AND RELATED LITERATURE Theoretical Background Perhaps more than any other single person, Jean Piaget [the Swiss psychologist] ranks as the giant of contemporary research into the way in which children think.1 His work is concerned with inves- tigating the sequential deveIOpment of intelligence. In effect Piaget has theorized an ordered sequence of stages of intellectual development, and he and his colleagues have amassed a vast amount of research in support of this developmental sequence. Piaget used the “clinical method“ as his experimental pro- cedure, which is similar to that used by psychiatrists as a means of diagnosis. This technique involves a single child and an experimenter who interacts with the child by posing questions or presenting the child with a task concerning a particular phenomenon. Piaget believes that this type of exchange between child and investigator is necessary in order that the child's beliefs may be realized. He has remarked about the pitfalls, methods, and advantages of the clincial method: The good experimenter must, in fact, unite two often inconpatible qualities; he must know how to observe, this is to say, to let the child talk freely, without ever checking or side-tracking his utterance, and at the same time he must constantly be alert for 1David Elkind, "The Continuing Influence of Jean Piaget," _§!ade Teacher (May/June, 1971), p. 7. 22 23 something definitive, at every moment he must have some working hypothesis, some theory, true or false, which he is seeking to check.1 Using the clinical method, Piaget and his colleagues have collected data that has led tO a formulation Of a theory Of human intellectual develOpment.2 According to this theory, the development Of the intellect can be outlined in four stages: sensori-motor, pre- Operational, concrete Operational, and formal Operational. The theory holds that the order Of the four stages as listed above is invariant and that each stage or Substage is a necessary prerequisite for the development of each subsequent stage or substage. Piaget has desig- nated approximate chronological ages for each Of the four major stages, but repeatedly emphasizes that these are approximate and are not to be construed as limits or bounds. The following brief discussion Of Piaget's four stages of intellectual development does not pretend to be a comprehensive examination Of these develOpmental stages. The theory upon which the stages are based is elaborate and somewhat complex. Each stage is composed Of substages which are interwoven into a highly detailed theoretical structure. For detailed discussions Of the four stages 3 Of intellectual development the reader is referred to Flavell, 1963, and Phillips, 19694. 1Jean Piaget, The Child's Conception Of the World (Patterson, N.J.: Littlefield, Adams, 1963), p. 9. 2John L. Phillips, Jr., The Origins of Intellect: Piaget's Theory (San Fransico, Calif.: WTH. Freeman and COmpany, 1969). 3John H. Flavell, The Developmental Psychology Of Jean Piaget, Princeton: D. Van Nostrand CO., Inc. 1963). 4Phillips, Origins Of Intellect. 24 The first of Piaget's stages Of intellectual development, the sensori-motor stage, begins at birth and lasts until approximately two years Of age.1 It is during this stage that the child learns to coordinate and organize perceptual and motor functions and develOps simple behavior patterns for dealing with the external world. He learns that Objects do not cease to exist when outside his perceptual field and becomes capable of elementary symbolic behavior. The second stage, the pre-operational stage, begins with the advent of organized symbolic behavior, language in particular, and lasts until around seven years Of age.2 The essential difference between a child in the sensori-motorstege and one in the pre-opera- tional stage is that the former is restricted to direct interactions with the environment, whereas the latter is capable Of representing the environment with symbols (language)3. In the pre-Operational stage the child is capable of representational thought, but in a limited sense. He is continually victimized by his perceptual field and thinks in terms Of beginning and final configurations when con- fronted with transformations. One Of the most significant indicators of the pre-Operational stage is the child's failure tO understand that certain physical prOperties such as length, number, area, weight, amount (mass), and volume are conserved under certain transformations in the Shape or 3Phillips, Origins Of Intellectual Development, p. 54. 25 configuration Of Objects. TO exhibit these concepts of conservation a child must hold invariant in his mind a given physical property throughout Observed changes of state. The child makes perceptual judgments based upon the appearance Of the Object following the transformation and disregards the invariant qualities Of the object. The third stage Of Piaget's develOpmental scheme, the con- crete Operational stage, begins about seven years of age and lasts until about eleven years Of age.1 The rules Of mathematics and logic are used by Piaget as models Of the mental functioning Of children in this stage. Piaget believes that the rules of logic have developed out of the interaction Of humans with the demands of living in a lawful universe.2 The actions that were origniallv overt, and then internalized, now begin to form tightly organized systems Of actions. Piaget refers to any internal act that forms an integral part Of one Of these systems (such as combining, separating, placing 3 The development of in order, or substituting) as an "Operation," the "Operations“ characterizes this stage of intellectual develOp- ment.4 Since birth, the dominant mental activities Of the child have changed from overt actions (in the sensori-motor stage) to perceptions (in the pre-operational stage) to the intellectual Operations (in 1Phinips, origins Of Intellect, p. 51. 2Ibid., p. 68. 3Ibid. 4Flavell, Developmental Psychology, p. 166. 26 the concrete Operations stage). These operations occur within a framework Of class relations that make possible what Piaget calls mobility of thinking - reversibility, decentering, taking the view Of the other, etc.1 As a result, the concrete Operations child con- serves quantity and number, constructs the time and space that he will live with as an adult, and establishes the foundations of the logical thinking that is the identifying feature Of the next and final stage Of his develOpment. Piaget's final stage of intellectual develOpment. formal Operations, begins about the age Of eleven.2 AS a child grows Older and gains more experience, his construction Of reality becomes more precise and extended and that makes him aware of gaps in his under- standing that had been masked by the vagueness Of his previous con- structions. He fills those gaps with hypotheses, and he is able to formulate, and Often even to test, hypotheses without actually manip- ulating concrete Objects. For the first time the child is able tO think in terms Of all possible combinations when confronted with a problematic situation. For convenience in identifying the four stages Of intellectual development proposed by Piaget, sensori-motor, pre-Operational, con- crete Operational, and formal Operational, the remainder Of this writing will refer to them as stages I, II, III, and IV, respectively. 1Phillips, Origins Of Intellect, p. 90. 21bid., p. 91. 27 Substages will be identified by capital letters, e.g. I A, I B, II A, II 8, III A, III 8, IV A, and IV 8. Piaget's Description Of the Cognitive Development of Length Measurement Piaget's view of the inception Of length measurement is described in his tower experiment:l He invited children to build with blocks a tower equal in height to a tower already built by the experi- menter. This tower, however, was on a table which stood higher than the table on which the subject was to build his tower and some distance from it. Sticks longer, shorter, and equal to the height Of the model were avail- able to the subjects. Children at stage I have an exaggerated confidence in visual comparison; their measuring may be summed up with the words: “I look and I see“. That faith is undermined when they come to notice a difference in base levels Of the towers. As a result, the two per- ceptual fields are brought together by manual transfer (substage II A). When the child is required to com- pare the towers without moving them, they gO through the motions of manual transfer. They accommodate their hand movements to the size Of the towers, imitating their height. Through body transfer (substage II 8) they reach the idea Of a common measure. Because body trans- fer is inaccurate, sooner or later they reject it. A third Object is sought as a measuring instrument. This instrument is a common measuring Object independent Of the subject's own body. Transitivity (A = B and B = C, implies A = C) at a qualitative level is now present (substage III A). When transitivity is extended to include relations between separate parts of an overall length, the evolution Of a metrical system consisting Of the gse Of a unit measure is completed (substage III B . 1Piaget, Inhelder, and Szeminska, Conception Of Geometry, pp. 30-66. ' 2Ibid., p. 65. 28 Piaget and his associates have conducted other single task investigations regarding the cognitive development Of length measure- ment. A description Of the intellectual development Of the child through the various developmental stages defined by Piaget (similar to the description Of the tower experiment) is noted with each inves- tigation. One of these investigations concerns conservation of length and the extremities of the Tines:1 The subject was presented with a straight wooden rod Of length 5 cm. and a longer undulating thread of plasticine shaped like a snake. The Objects were placed Side by Side a few millimeters apart, with their endpoints in exact alignment, and the child was asked tO compare the lengths of the two Objects. If he said that they were equal, he was made to run his finger along the two lines and the question was re- peated. Next, he was shown what happened when the plasticine was straightened, and the question was repeated. Finally, the plasticine was twisted back to its original shape and the original question was asked again. Of approximately a hundred children who were given these questions, only 15 per cent Of those aged four years, six months and younger correctly recognized the inequality Of the two lengths. Of those children over the age Of five ears, six months, 90 per cent gave correct replies. [This is one Of the few times in which any statistics are presented in the descriptions Of Piaget's investigations.] The children in stage I compare the lengths of the lines by focusing on the endpoints. Judgment is modi- fied by movement Of fingers for the children in sub- stage II A. The children in substage II B make correct judgments on this task which implies that they are aware Of the intervals that lie between the endpoints. lIbid., pp. 91-94.’ 21bid., p. 92. 29 Another Of Piaget's conservation Of length investigations relates a comparison of lengths and a change in position Of the line:1 The experiment consisted Of showing the subject two straight wOOd sticks identical in length and with their extremities facing each other; one of the sticks was then moved forward 1 or 2 cm. (the sticks being approximately 5 cm. long), and the subject was asked to say once again which Of the two was longer or whether they were the same length. At all levels, the sticks were judged equal before staggering. After that change of position, subjects at the first stage maintain that the stick which has been moved forward is longer, thinking only in terms Of the further extremities and ignoring the nearer extremities. This response lasts into substage II A. Between levels II A and II B we find a series Of transitional responses, beginning with perceptual regulations and passing from intutitive regulations tO Operations, when conservation of length iS assured (stage 111). An experiment used by Piaget to describe the intellectual development Of the child regarding length measurement is noted below:2 The subject is asked to judge between strips of paper in a variety Of linear arrangements, involving right-angles, acute angles, etc., but these are pasted on cardboard Sheets. When he has given his replies, saying they are equal or that one is longer than the other, he is shown a number Of movable strips and asked to verify his judgment. . . . he is given short strips Of card 3 cm., 6 cm., sometimes 9 cm. long (these lengths correSponding with those Of segments on the mounted strips). At levels I and II A, subjects had no notion Of conservation and consequently they failed tO understand the concept of a middle term and that of a unit. At substage II 8 conservation is dimly perceived, and children at this level also begin to understand tran- sitivity. At substage III A measurement is conducted with reliance on the transitive property but without a metric unit while at substage III B the child now 11bid., pp. 95-103. 2Ibid.. pp. 117-127. 30 uses a metric unit in the iterative process Of mea- surement. Piaget investigated subdividing a straight line with the following experiment:1 Two wires, AC and DF which are equal in length are placed parallel to one another with their ends in align- ment. The child was told that a bead on the wire was a train traveling along a railway line. The experi- menter moved his bead from A to B, and the child was asked to move his bead to do a journey Of the same length. Subjects were provided with a ruler, string, strips of card Of varying length which they were invited, but not shown how, tO use. The experimenter commenced by moving his bead from A, the child being invited tO move his bead from D so that the segment AB equaled the segment DE. This pro- cedure was repeated with the subject having to move his bead from the other end F SO that AB = FE. Next D was moved 4 inches to the left Of A, so that F was 4 inches to the left Of C. The subject was again asked tO move his bead to E on DF, starting from F, and making FE = A8. Keeping DF in the same position rela— tive to A8, the experimenter then moved his bead 15 inches from A -- a distance longer than any Of the mea- suring instruments provided. The child was again asked to locate E so that AB = FE. Finally the wire DF was replaced by a wire G1 which was shorter than AB. The wires were still parallel but G1 was displaced 4 inches to the right of AC. The exper- imenter moved his bead 6 inches from A and the subject was asked to move his bead 6 inches from 1. During stages I and II A, the length of travel is determined solely by the point Of arrival SO the pro- blem is solved only when the points Of departure are in alignment. In substage II B a given length can be reproduced with reasonable accuracy by visual estimate. Measurement is possible in substage III A if the mea- suring rod provided is equal to, or longer than,the distance to be measured. During substage III 8, subjects apply a short ruler by iterate stepwise movements, thus illustrating the use Of a unit Of length. 11bid.. pp. 129-149. 31 Related Research - Length Measurement The accuracy Of Piaget's account Of the cognitive develOp- ment Of length measurement has been investigated by Lovell, Healey, and Rowland.l This study contains four replications Of the Piagetian studies described in the previous section. The sample for the study consisted Of seventy Primary School children and fifty Educationally Subnormal Special School children. The following discussion will pertain only tO the seventy Primary School children. The general procedure and the criteria for the evaluation at the various stages were kept as close as possible to those aSpects of Piaget's inves- tigation. Only the results Of Lovell's study will be noted. Regarding the first investigation described in the previous section concerning the conservation Of length and the endpoints Of the lines: Kendall's tau coefficient (tau = .26, Significant at the .01 level) indicates a positive correlation between chronological age and measurement stage.2 As the age Of the subjects increased, so did the measurement level. This finding coincided with that Of Piagets. Approximately 65 per cent3 Of the subjects six years Old and Older were aware Of the intervals which lie between the endpoints . 1The discussion Of this study is based upon K. Lovell, D. Healy, and A.D. Rowland, “Growth Of Some Geometrical Concepts,“ in Logical Thinkipg In Children, ed. I.E. Sigel and F.H. Hooper (New York: THOlt, RTnehart, and Winston, Inc., 1968), pp. 140-157. 2Ibid., p. 144, 3 id. 32 (rated at substage II B) as compared to 90 per cent1 of Piaget's sub- jects who gave correct responses. The second replication of a Piagetian study concerns conser- vation Of length and a change Of position of the line: Kendall's tau coefficient (tau = .42, significant at the .01 level) indicates a positive correlation between chronological age and measurement stage.2 Increase in age implied a higher measurement level. Approx- imately 60 per cent Of the eight- and nine-year Old groups were rated at stage III while a considerably lesser percentage (15 per cent) Of those children in the younger groups were rated at this stage.3 This finding agrees with that Of Piaget : conservation Of length is achieved at a mean age of seven and one-half years.4 The third replication of one Of Piaget's investigations con- cerns length measurement with the use of independent Objects to be used as units: Kendall's tau coefficient (tau = .55, significant at the .01 level) indicates a positive correlation between chrono- logical age and measurement stage.5 The Older children were rated at a higher stage than the younger children. Lovell's results lPiaget, Inhelder, and Szeminska, Conception of Geometry, p. 92. 2Lovell, "Growth of Some Geometrical Concepts," p. 145. 3.1.21.9. 126 4Piaget, Inhelder, and Szeminska, Conception Of Geometry, p. . 5Lovell, “Growth of Some Geometrical Concepts," p. 146. 33 indicate that 70 per cent Of the eight- and nine-year old groups were rated at the highest level (substage III A) while only 30 per cent Of the younger subjects were rated at this stage.1 This finding is in agreement with that Of Piaget : length measurement is achieved at approximately eight or eight and one-half years.2 The fourth replication Of one Of Piaget's studies concerns subdividing a straight line: Kendall's tau coefficient (tau = .30, Significant at the .01 level) indicates a positive correlation between chronological age and measurement stage.3 The data indicates a slight increase in measurement understanding as the subjects increase in age. Only 13 per cent Of the seventy Primary School children studied were scored at measurement stage III regarding sub- division Of a line.4 There were no statistics presented in Piaget's study tO use for comparison purposes, but the subdivision task was noted as Piaget's most difficult length measurement task. Piaget's Description Of the Cognitive Development Of Area Measurement Piaget and his associates have conducted a sequence Of tasks to gain information regarding the cognitive development Of area mea- surement. As with length measurement, the stages of intellectual 1Ibid. 2Piaget, Inhelder, and Szeminska, Conception of Geometry, p. 126. 3Lovell, "Growth Of Some Geometrical Concepts," p. 147. 4Ibid. 34 development defined by Piaget are associated with the various levels Of attainment regarding the understanding of area measurement. One Of these investigations concerns subtracting smaller congruent areas from larger congruent areas:1 The child was shown two identical Sheets Of card- board painted green, each 20 cm. by 30 cm. These represent meadows. He is asked to compare the meadows and agree that there was the same amount Of grass on each. Following this, he was Shown a toy cow and asked if it had the same amount Of grass to eat in each of the fields. The experimenter then places small wooden houses (1 cm. by 2 cm.) one at a time in each Of the fields, In one field the houses were placed end to end, while in the other field the houses were Spread about. TO begin with, one house was placed in each field, then two, three, etc. After each increase in the number of houses the child was asked to compare the amounts Of grass left in each field for the cow to eat. The child in stage I had difficulty understanding what was being asked. At substage II A, equality of areas was recognized only when there was one house in each field. Children in substage II B determined equality Of remaining areas up to a certain number Of houses but this varied with the child. Conservation Of area was present at substage III A, i.e. equality was determined regardless Of the number of houses placed in the meadows. Another Of Piaget's area measurement investigations concerned unit iteration:2 The child was shown a number Of Shapes which are equal in area but which differ markedly in Shape. One is a square which can be composed out Of nine smaller squares. The others are irregular figures made up Of the same number Of small squares. The child was given a choice Of three counters to measure the figures. One is a square which is a quarter Of one of the lPiaget, Inhelder, and Szeminska, Conception Of Geometry, pp. 262-273. 2Ibid. .pp. 296-301. 35 figures to be measured. The second is a rectangle that can be composed of two unit squares. The third is a triangle equal to a square cut diagonally in half. The child is asked to compare the sizes of the regions of various shapes. Children at substage II A make judgments Of size by reference Of the perceptual appearance Of the fig- ure. At substage II 8 children make correct judgments if the regions being compared can be composed of all squares or all triangles. Squares and triangles are regarded as equivalent units. At substage III A com- parison of areas is made by transferring parts of one figure to vacant Sites Of another. Children at sub- stage III 8 measure the figures by unit iteration. The area of the unit square can be expressed in terms Of the area Of the unit triangle, etc. Piaget investigated subdividing areas with the following . 1 experiment: Each of the children (whose ages ranged from four to around seven) was shown a circular slab Of modelling clay. He is told that the clay is a cake. His first task was to cut the cake into two pieces so that each piece has the same amount. Next he is asked tO cut a Similar circular slab Of clay into three equal parts. Division into fourths, fifths, and sixths follow using the same procedure. After each request to cut the clay, the child was asked whether the sum Of the pieces equaled the whole- The children in stage I could not divide the clay equally. During substage II A, dividing into halves and quarters is possible but not trisection. Children in Substage II B begin to conserve the whole (whole is equal to the sum Of its parts) and trisection is accomplished by trial error. During substage III A, trisection is possible and the whole is conserved. 116id., pp. 302-325. 36 Related Research - Area Measurement The accuracy Of Piaget's account Of the cognitive develop- ment Of area measurement has been investigated by Lovell, Healey, and Rowland.l This study contains three replications Of Piagetian investigations described in the previous section. The sample con- sisted Of seventy Primary School children. General procedure and the criteria for evaluation at the various stages were similar to those aspects Of Piaget's investigation. Only the results of Lovell's study will be noted. Regarding the first investigation concerning subtracting small congruent areas from larger congruent areas: Kendall's tau coefficient (tau = .29, significant at the .01 level) indicates a positive correlation between chronological age and measurement stage.2 As the age of the subjects increased, so did the measurement level. Lovell's data indicates that 77 per cent3 Of the sample completed this task successfully, i.e. were rated at stage III. This is in agreement with Piaget's finding: At stage III (usually at seven and one-half but sometimes as early as six and one-half years) children recognize that remainders are always equal. Ibid., "Growth Of Some Geometrical Concepts, pp. 140-157. 1 2 bid., p. 152. 3Ibid. 4Piaget, Inhelder, and Szeminska, Conception Of Geometry, p. 264. 37 The second replication of a Piagetian study concerns unit iteration:1 Kendall's tau coefficient (tau = .47, significant at the .01 level) indicates a positive correlation between chronological age and measurement stage.2 Only 22 per cent3 Of those children whose ages are seven to nine years are rated as being in stage III. This finding is contrary to that of Piaget : stage III usually begins at the age Of seven. (It must be remembered that the ages assigned to the various stages of intellectual development are only approximations.) In agreement with Piaget's findings is the fact that only 8 per cent of those children seven years Old and younger are rated as being at stage III. The third replication of a Piagetian study concerns subdivi- sion of areas:4 Kendall's tau coefficient(tau = .59, significant at the .01 level) indicates a positive correlation between chronological age and measurement stage.5 Approximately 93 per cent6 of the six- and seven-year olds are rated as being in substage II B or higher. This finding is in agreement with that Of Piaget : in general, sub- stage II B occurs between six and seven years of age. NO comparison can be made using Lovell's eight- and nine-year olds since Piaget's sample for this task included children whose ages ranged from four to seven. 1Lovell, “Growth of Some Geometrical Concepts,“ p. 153. 2Ibid. 3Ibid. 4Ibid., p. 154. 5 bid. 6Ibid. 38 In summary, the main stages in the cognitive development of length and area measurement proposed by Piaget have been confirmed among English school children by Lovell and his associates.1 The protocols were classified into the stages enumerated by Piaget and a few intermediate substages Such as substage II B - III A. However, the number of children at the various stages were not always what one would expect from Piaget's results. For example, in Lovell's conservation of length task relative to a change Of position, only 27 per cent Of the seven year Old children were rated at stage 111.2 Piaget claims that "the third stage is reached about the age Of seven".3 Also, the data indicates that considerable variability in achievement Of an Operation may exist at a particular age level Thus, chronological age is not a very good guide to the stage Of cognitive development of some children. Comparative Study Of the Cognitive Development of Length and Area Measurement Beilin and Franklin conducted an investigation concerning length and area measurement. The study was conducted on a comparative basis to investigate whether the abilities to solve related problems of length and area measurement are acquired simultaneously, and whether there are age associated limits upon the acquisition of 11bid.. pp. 142-157. N Ibid., p. 145. w Piaget, Inhelder, and Szeminska, Conception of Geometry, 39 measurement Operations when a deliberate training effort is made.1 The discussion that follows will pertain to the first Of the two stated purposes. The subjects were New York City school children from the first and third grades of a public elementary school in a predominantly middle class area. The two groups were indicated to be above aver- age (by I.Q. scores). The mean age of the twenty-seven first graders is Six years, six months (range: six years, zero months to seven years, three months).2 The mean age Of the thirty-three third graders is eight years, eleven months (range: eight years, one month to nine years, four months).3 Piaget's unit measure tasks for length4 and area5 measurement were used for the tasks of this investigation. Figure 2-1 illustrates the length and area measurement testing materials? The area mate- vialswere made of white cardboard and the length materials consisted of strips of colored paper pasted on white cardboard. Lengths num- bered Six to ten were movable strips Of white cardboard. The 1The account of this experiment is taken from Harry Beilin and Irene C. Franklin, "Logical Operations in Area and Length Measure- ment: Age and Training Effects," Child Development, 33 (September, 1962), pp. 607-618. 2Ibid., p. 609. 3Ibid. 4Piaget, Inhelder, and Szeminska, Conception of Geometry, pp. 116-127. 5Ibid., pp. 296-301. 6Beilin and Franklin, "Logical Operations in Area and Length Measurement," p. 610. AREA AND LENGTH MEASUREMENT A1 40 FIGURE 2-1 AREA 5% A2 A5 L1 L4 L6 L7 4\/ A6 LENGTH L2 l:l L5 L8 A3 L3 1:] L9 I L10 Scale: 1. 1" = 4N 41 materials were SO devised that the measurement of both equalities and inequalities was tested. The shapes were so constructed that a conflict is generated between the perceptual properties of the Objects and their logical relations. Shapes equal in area were made to appear unequal. The procedures used in this study first required testing the subjects with the area materials. The intent was to determine whether the subject could measure the areas without aid from the experimenter. If the child did not answer correctly, the methods of superposition and unit iteration were demonstrated to him. Criteria used in evaluation is similar to the stage descriptions presented in Piaget's The Child's Conception of Geometry. Each child was classi- fied as to the level of measurement he achieved. The following is the Order of area measurement task presen- tations:l Step 1. The subject was given the three-inch square, A1, and the irregular Shaped figure, A2, of the same area measure (nine square inches). He was asked whether the Space in them was the same and to give a supportive reason for his response. He was permitted to manipulate the figures. Step 2. Figure A3 was substituted for figure A2 and a comparison asked for. These figures had the same area measure but not the same shape. Step 3. The Subject was given A1, A2, A3, and A4 together. He was told to verify whether his judgments were correct by using the one-inch square, A4. If necessary the experimenter demonstrated superposition and unit iteration processes. 1Ibid., pp. 611-12. 42 Step 4. The child was then given figures A5 and A6 which are unequal in area. The procedure of steps 1 to 3 were repeated. Step 5. Subject is given measuring instruments A7, A8, and A9 to verify his answers. A demonstration Of measurement was given if necessary. Regarding the order of length measurement task presentation:l Step 1. The subject was given L1, L2, L3, L4, and L5 and asked which of the lengths were equal and which were unequal. (L1 = L2, L3 = L4 # L5). Step 2. The subject was given three movable strips one, three, and five inches long (L6, L7, and L8) to be used as measuring units. Step 3. If measurement was not successfully achieved with L6 to L8, then L9 and L10 were given, which together provided the subject with measuring units that corresponded to all the strips mounted on the card. The experimenter demonstrated unit iteration if necessary. The results Of Beilin and Franklin's investigation support Lovell's Observation2 that considerable variability in achievement of an operation may exist at a particular age level. Also, the data indicates that first graders differ from third graders in their ability to utilize measuring concepts. The following tabie3 consists of the numbers of first and third graders who have achieved length and area measurement: 1Ibid., pp. 611-12. 2Kenneth Lovell, "A Follow Up Study of Some ASpects of the Work of Piaget and Inhelder on the Child's Conception of Space," British Journal of Education ngchology, 1959, p. 104. 3Beilin and Franklin, "Logical Operations in Area and Length Measurement". This is a portion of Table 1, p. 614. Per cents are in parentheses. 43 First Grade Third Grade Length Measurement 3 (11) 27 (82) Area Measurement 0 (O) 9 (27) As indicated by the data, a large proportion of third graders have achieved length measurement but not area measurement. A similar phenomenon exists with the first grade group, but to a lesser degree. In suimnarizing their investigation, Beilin and Franklin concluded that: On the basis of the data Of this study . . . , we would suggest that length and area . . . measurement are achieved in that order. Also the constituent Oper- ations to measurement (i.e. transitivity, subdivision, change Of position, etc.) are applied more easily first to a single dimension, then to two dimensions, . . . The order of achievement is a function of added dimen- sions . . . Although our data deny the Piaget view of the simultaneous achievement Of area and length mea- surement, we do not feel that this, of necessity, does violence to the unitary or structural interpretation of development . . . It seems likely that within the limits of a particular level (e.g. stage III) tasks which are ordered in difficulty because Of complexity (e.g. added dimensions) and which require no different Operations for their solution will be achieved in order of such complexity. Certainly more evidegce is needed before.this important issue is reSOlved’luhaETWTTRfif“ mine] .‘ Summary A synopsis of Piaget's theory of intellectual develOpment has identified four major stages: (1) sensori-motor, (2) pre-opera- tional, (3) concrete Operational, and (4) formal operational These stages were related to various levels of cognitive development regarding length and area measurement. Piaget concluded from his 1 bid., p. 617. 44 study concerning the cognitive develOpment Of length and area measure- ment that "The development of . . . measurement runs exactly parallel whether the objects are lengths or whether they are areas and the level at which they are finally grasped is the same for both."1 Lovell, Healey, and Rowland confirmed many of Piaget's find- ings through replications of his investigations. Seven such inves- tigations have been described. Beilin and Franklin conducted a comparative study concerning the ability to solve related problems of length and area measurement. On the basis of the data of this study, Beilin and Franklin concluded that "length and area measurement are achieved in that order" and that "constituent operations to measurement . . . are applied more easily first to a Single dimension, then to two dimensions."2 The present study will attempt to lend support to one Of the two stated contrasting viewpoints (i.e. that of Piaget and that of Beilin and Franklin). Measurement axioms found in modern geometry textbooks have been used to identify four common prOperties of length and area measurement: (1) congruence, (2) conservation, (3) addi- tivity, and (4) unit measure. This study will investigate the cogni- tive development of the four common properties to Obtain information regarding the two questions: 1Piaget, Inhelder, and Szeminska, Conception of Geometry, p. 300. 2Beilin and Franklin, "Logical Operations in Area and Length Measurement." p. 617. 45 1. Is there a Simultaneous cognitive development of length and area measurement? 2. Does the final attainment of length measurement occur at the same age as the final attainment of area measurement? Involved in this study will be four pairs of tasks (one regarding length measurement, the other area measurement) each testing one of the four common properties Of measurement. This procedure is unlike the investigations reviewed that consisted of single task studies. It is believed that with this procedure a more accurate assessment of the develOpment of measurement can be made. Criteria for evaluation will be Similar to that used in related studies so that a comparison Of results can be accomplished. Demographic data will be used to describe the sample. The data will be presented in tabular form and be subject to statistical analysis suggested by Bentler.l Similar procedures are absent from Piaget's work, a Situation that has produced a fair amount of criticism. 1Peter N. Bentler, "Monotonicity Analysis: An Alternative to Linear Factor and Test Analysis," Measurement and Piaget, ed. by Donald Green, et al., (New York: McGraw-Hill*BOOk Company, 1971), pp. 220-27. CHAPTER III THE RESEARCH METHODOLOGY Pilot Study Prior to the actual study, a pilot study was conducted to determine the age groups to be used and to refine the tasks based on the length and area axioms. Fourteen children from ages five to eleven comprised the subjects for the pilot study. Twelve of the subjects were students in the same public elementary school in which the actual study was conducted. The two five-year olds were children who would enroll in the kindergarten Of the same elementary school the following year. Chronological age was chosen as the population variable since many of the studies reviewed by the investigator, including those Of the Geneva group, relate the results to the ages of the subjects. Since this study relied heavily on the results Of the Geneva group for its theoretical basis, the ages of the children were used as a means of grouping the subjects. Almy, Chittenden, and Miller also state that the best predictor of ability to conserve is chronological age.1 The pilot study five- and Six-year olds had difficulty per- forming the operations required in the tasks. In addition, it could 1Almy, Chittenden, and Miller, Young Children's Thinkipg, p. 77. 46 47 be expected that a majority Of five- and Six-year olds would fail the length conservation task since Piaget and his associates have found that children attain conservation Of length at a mean age of seven and a half.1 Seven was taken as the study's base age. Regard- ing maximum age, the Geneva group has found that children aged eleven to twelve performed successfully in the doubling area task, their most difficult area task, hence, eleven was the pilot study maximum age.2 The eleven-year age group remained the maximum age group of the actual study. Specific modifications of the investigating procedure made as a result of the pilot study are discussed with the respective tasks. The responses obtained in the pilot study were used in con- nection with the Geneva group's results to determine the stages for each task. A recording sheet based upon this determination was developed and used in the actual study (see Appendix 8). Sample The population for the study was the student body Of a pub- lic elementary school in a northern Michigan city serving a middle class neighborhood. In the winter Of 1971 the names of nearly all the children in the school in the seven- through eleven-year age 1Piaget, Inhelder and Szeminska, Conception of Geometry, p. 126. 2Ibid., p. 337, 48 groups as of February were Obtained.1 There were 59 names in the seven-year Old group; 55 in the eight-year old group; 33 in the nine- year old group; 33 in the ten-year old group; and 34 in the eleven- year Old group. 2 the Following the procedure outlined by Walker and Lev investigator alphabetized and consecutively numbered each of the five sets of names. Then he reordered each group of names using a table3 of random numbers. The preliminary sample consisted Of chil- dren whose names were among the first twenty in each group. Children who did not pass the criterion for inclusion in the sample (the vocabulary task regarding measurement terms) were not included in the final sample of twenty children in each age group. The child whose name was next on the list was then added to the preliminary sample. In order to Obtain twenty children in each age group for the final sample,0ne nine-year Old and one ten-year old were replaced on the original preliminary sample.4 Demographic data for the final 1Seven years is Operationally defined as six years, seven months to seven years, six months; eight as seven years, seven months to eight years, Six months; nine as eight years, seven months to nine years, six months; ten as nine years, seven months to ten years, Six months; eleven as ten years, seven months to eleven years, six months. Children in the Special Education class were not included. 2Helen M. Walker and Joseph Lev, Elementar Statistical Methods (New York: Holt, Rinehart and Winston, 1958), pp. 202-212. 31bid.. pp. 280-281. 4The nine-year old was replaced due to her absence during the administration of the length and area vocabulary tasks. The ten-year old was replaced because of failure of the length and area vocabulary tasks. It was later learned that the ten-year Old Should have been placed in the Special Education class. 49 sample is given in Table 2-1. For notational purposes ages were recorded as years; months (e.g. eight-years, seven-months was recorded as B;7). The mean age for the seven-year old group is 7 years, 1.2 months; for the eight-year Old group is 8 years, 1.3 months; for the nine-year old group is 9 years, 1.0 months; for the ten-year old group is 10 years, 1.9 months; for the eleven-year old group is 11 years, 2.2 months. Table 2.2 and Table 2.3 give reading comprehension scores and arithmetic concept scores, respectively, in terms of grade level for the age groups of eight through eleven. These scores are the results Of the Stanford Achievement Tests published by Harcourt, Brace and World, Inc..administered in March Of 1971.1 Table 2.2 also gives the reading level of the children in the seven-year old group. The evaluation instrument used for the seven-year olds was the Basic Reading Test.2 The scores entered in Table 2.2 for the seven-year old group are percentile scores. The children in the sample came from two first-grade, two second-grade, one third-grade, one fourth-grade, and one fifth-grade classrooms. The mathematics textbooks used by grades are: 1. Grade one - One By One Elementary Mathematics, Joseph Payne, et al., *Harcourt, Brace and World,Inc.,1965. 1For those in Grade 2, Primary 2 Battery of Test W was used; for Grade 3, Primary 2 Battery of Test X was used; for Grade 4, Inter- mediate 1 of Test W was used; for Grade 5, Intermediate 2 of Test W was used. 2The children in the seven-year old group were tested for reading readiness. The Basic Reading Test, Sixties Edition, copy- right 1963 by Scott Foresman and Company was the evaluation instru- ment used. 50 NH m mvmgw mo m mumgo 0H m mange Ho m wumgw mo a mumew NH 8 ounce Ho N mvmgw mH H mumgw ON H mumgw monPDmHmhmHa mo HH mme> OH mkmw> m mme> w mme> N Hence: one smm> any mcorpanweumvo mm< “Dazem mm >zampm canoes» rucmmm as» com momeucm space .manHHc>c:: mew: naocm uHo Lemaicm>mm on» cm muumnnam Lao» com museum oppucmucwa one .mmcoum mHHpcmucma use aaocm uHo ccmxicm>om mg» com nmucumucm mmeoom use: N.m ccoz HcccHch N.m eccz HccoHch N.N ecoz HcccHch N.N ccom HecoHch m.m cmdz o.m cam: m.c cam: H.N can: N.mc cam: m.mH-N.N emcem m.m-m.N mmccm m.m-m.N dmccm m.m-m.H umcem mm-NH mmcmc H m.m H m.oH H m.m H N.m N m.m H e.m H N.N H m.N N N.m i. H m.m N N.N H m.m H m.N N mm c. m m.m H m.N H c.m H m.N H mm N N.N H N.m H N.m H m.N H Nm H N.m H N.m N m.m H N.N H mm N m.m H m.m H e.m H m.N N mm H N.m H m.m N N.m H m.N H mN H o.m N N.m H m.m H m.N H mN H N.m H m.c m N.N H m.N N mm H N.m N m.m H N.N H N.N H mm H N.N H m.m H H.N H N.N H mm H m.N H m.N N m.N N H.N H mm H m.N N m.N N m.N H m.N H mH H N.N H m.N H m.N N N.H H NH memm> HH mcco> oH mecm> m mcco> m mecm> m HHest ecccm amvs zmHmmmHmHmHm mammm mszcmm N-N mHmem we“ so» mHneHHe>ecz mew: megoum pamucou uHumszuHL HH meme» oH meme» m meow» m HHHm>mH mueew zmv onhzmHmkao uaoum hmmuzou uthzzhHm< mum m4m O. Hhere the symbol (Hi) denotes XY the statistical alternative hypothesis and form ¢xy indicates the population correlation coefficient. The statistical alternative hypothesis was employed to derive a corresponding null hypothesis (H0), in this case Ho: ¢xy §_O: this null hypothesis was submitted to test by means of the following seven step process: 1. Ho: oxy 5’0 2. Statistical tests employed: a) Chi-Square Test of Independence: (fojk - fejk)? 2 - fejk dof. ' l x = X E j k where fojk denotes observed frequency and fejk denotes expected frequency in a fourfold contingency table is used to test for independence of (x) length and (y) area relative to the four measurement properties studied: i.e. oxy = O.1 1Hays, Statistics For Psychologists, pp. 589—606. 104 b) Phi-coefficient: ¢ = bc - ad Ka+biTc+diTa+ch+Hi d-f- = 1 used to test for a negative correlation between length and area measurement: i.e. ¢xy < O.1 3. Level of significance is identified in a Chi-square (x2) Table. 4. A critical region is identified. 5. Values of "x2" and "o" are calculated. 6. Decision is made whether the Observed value of “x2" is greater than its critical value. If the value of "X2" is greater than its critical value or p > 0, then the null hypothesis (H0) is rejected and the statistical alternative hypothesis (Hi) is accepted indicating non- independence of the variables tested. 7. The value of "O" indicates the strength of association between variables.2 The seven step statistical process was applied to each of the twenty operational hypotheses developed from Research Hypothesis I. The statistical analysis procedure used in this study was suggested by the Research Consultation Center, Erickson Hall, Michigan State University. 1Ibid., pp. 604-06. 2 b d. 105 Research Hypothesis II II The understanding of length and area measurement are attained simultaneously. Operational Hypotheses of Research Hypothesis II 110 1 IIOZ 1103 110 110 The seven-year old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. The eight-year Old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. The nine-year old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. The ten-year old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. The eleven-year old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. 106 The same statistical procedure including the seven step test- ing procedure that was used on Research Hypothesis I was used on Research Hypothesis II. Collection of Data The following investigative procedures were used. Nine tasks were administered to each subject involving (1) vocabulary of measure- ment terms and (2) four properties common to both length and area measurement: (a) congruence (b) conservation (c) additivity and (d) unit measure. The tasks are discussed in Chapter III and detailed descriptims are contained in Appendix A. An interview recording sheet was used during the interview (see Appendix B). The criteria used in the evaluation process is defined with each task discussion in Chapter III and the recording of the evaluations for each subject is contained in Appendix C. In addition to evaluations on nine tasks, Appendix C also contains the following data: 1. Subject identification.1 . Age. Grade. Arithmetic concept level. 2 3 4. Reading level. 5 6. Conservation stage (attainment, transitional, none). 7 Measurement stage (II A, II 8, III A, III B). lSubject is identified by three letters and his age. 107 Summary In this chapter the pilot study was described and demographic data given to define the sample. General procedure of the study was followed by detailed descriptions of each task including: (1) a definition of the measurement property being tested, (2) a description of the materials and procedure, (3) criteria used for evaluation, and (4) sample interviews illustrating various levels of achievement. The research design involved a statement of research hypoth— eses :which were transformed into statistical hypotheses that will be tested using Pearson's Chi-square test for independence. The Phi-coefficient will be used as an indication of the strength of association between the length and area measurement properties studied. CHAPTER IV ANALYSIS OF DATA AND FINDINGS Introduction The statistical findings regarding the measurement properties of congrugence, conservation, additivity, and unit measure and the final attainment of length and area measurement are reported in this chapter. The first section concerns itself with sorting and classify- ing the data. Each of the twenty-five operational hypotheses are restated followed by: (1) supporting data in tabular fonn (fourfold contingency table), (2) values of the Chi-square and Phi-coefficient tests statistics, (3) the alpha level relative to the Chi-Square statistic, (4) a statement of rejection or acceptance of the null hypothesis, and (5) a restatement of the Phi-coefficient to indicate the strength of association between the compared variables. The summary of this chapter concerns a table which summarizes the tested hypotheses, the significance levels, and statements of rejection or acceptance. Evaluation of the Subjects All of the subjects in the sample were interviewed by a single investigator. The results for each subject were kept on a recording sheet (see Appendix B), and all of the interviews were tape recorded. The tapes were used where needed to clarify and substantiate 108 109 comments written on the recording sheets during the interviews. Fol- lowing the criteria for evaluation (noted with the discussion of each task in Chapter III) each subject was scored for each task using the information on the recording Sheet. This was done by the investi- gator who conducted the interviews. The interviews were numbered consecutively corresponding to the order in which they were conducted (grade one through grade five). Every fifth interview was selected and scored directly from the recording sheets by a second investigator who was familiar with the study. There was 93 per cent agreement between the scorings of the recording sheets by the two investigators.1 Results for Criterion for Inclusion in Sample The final sample for the study was composed of twenty sub- jects in each of the five age groups who passed the criterion for inclusion in the sample, the vocabulary task. In order to have this requisite number of subjects, the vocabulary task was administered to 101 children. One child in the ten-year old group had difficulty using terms regarding size ("more", "less", "larger", etc.) and thus 2 failed this task. The remaining 100 children all passed the 1There was 100 per cent agreement for the vocabulary and the congruence of length and area tasks, 95 per cent for the conservation of length and area tasks and the conservation stages relative to length and area, 90 per cent for the additivity of length and area tasks and the unit of measure tasks relative to length and area, and 85 per cent for the measurement stages relative to length and area. 2It was learned that this child should have been placed in the Special Education class. 110 vocabulary task. These results are consistent with the finding of Beilin referred to in Chapter III that post-kindergarten children reached near perfect levels of performance on a similar task.1 Research Hypothesis I I. The cognitive development of length measurement is si- mUkaneous to the cognitive development of area measurement relative to the properties of congruence, conservation, additivity, and unit measure. The following twenty operational hypotheses were developed to test Specifically this research hypothesis by converting them to Statistical alternative hypotheses (i.e. ny > 0). A null hypothesis (¢xy §_O) was formed from each statistical alternative hypothesis and was tested using the Chi-square and Phi-coefficient test statis- tics. Tabular data and a decision concerning the rejection or acceptance of the null hypotheses accompany each operational hypoth- esis. The operational hypotheses are grouped according to the measurement properties in the order of congruence, conservation, additivity, and unit measure. Each measurement prOperty is consid- ered relative to the factor of age (seven through eleven years). Then an operational hypothesis (HTi) is formed concerning each measurement prOperty and all subjects in the sample regardless of lBeilin, "Perceptual-Cognitive Conflict in the DevelOpment of an Invariant Area Concept," p. 217. 111 age. The seven step statistical procedure outlined earlier is used in detenni ni ng the acceptance or rejection of this summarizing hypothesis across all ages. All results are summarized in Table 4-31. Congruence TaS 101 TABLE 4-1. Co Length k The seven-year old subject's score on the Congruence of Length task will correlate positively with the subject's score on the Congruence of Area Task. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this operational hypothesis is accepted. The correlation between congruence of length and congruence of area is p = .63 ngruence task for the seven-year old group. £333; team; Pass 18 O X2 = 9.47 a = .05 Fail 1 l O = .63 112 The decisions of acceptance or rejection of the statistical alternative hypotheses for the Operational hypotheses 102, 103, 104, and 105 are not reported relative to an x2 value. Due to the fact that expected cell frequencies (fejk) of zero occur the Chi-square test for independence is not applicable (zero denominators are pres- ent.): 2 g (fojk - fejk)2 £2 jk 7e3k X The Phi-coefficient is not applicable to this type of situation (expected cell frequencies of zero) since it can be expressed in terms ¢ = fl 0 N There are two solutions to this problem: (1) make no decision 0f X2: regarding the acceptance of any of the operational hypotheses 102, 103, 104, or I05: or (2) if the observed cell frequencies indicate an extreme direction regarding either a pass-pass, fail-fail Situation or a pass-fail, fail-pass situation make a decision of acceptance or rejection, respectively, relative to the operational hypothesis based upon inspection of the fourfold contingency table. Based upon the observed cell frequencies of the contingency tables for the Opera- tional hypotheses 102,103, I04, and 105, the investigator chose the latter of the two solutions. 10 TABLE 4-2. Length 10 113 The eight-year old subject's score on the Congruence of Length Task will correlate positively with the subject's score on the Congruence of Area Task. The cell fre- quencies in the following contingency table indicate that 90 per cent of the eight-year Old subjects passed both the Congruence of Length and the Congruence of Area tasks. Based on the above statistic this operational hypothesis is accepted. Congruence task for the eight-year old group. Area Pass Fail Pass 18 2 Fail 0 O The nine-year old subject's score on the Congruence of Length Task will correlate positively with the subject's score on the Congruence of Area Task. The cell fre- quencies in the following contingency table indicate that 100 per cent of the nine-year old subjects passed both the Congruence of Length and the Congruence of Area Tasks. Based on the above statistic this opera- tional hypothesis is accepted. 114 TABLE 4-3. Congruence task for the nine-year old group. Area men Length Pass 20 0 Fail 0 0 ID The ten-year old subject's score on the Congruence of Length Task will correlate positively with the subjects score on the Congruence of Area Task. The cell fre- quencies in the following contingency table indicate that 100 per cent of the ten-year old subjects passed both the Congruence of Length and the Congruence of Area Tasks. Based on the above statistic this opera- tional hypothesis is accepted. TABLE 4-4. Congruence task for the ten-year old group. Area Pass Fail Length Pass 20 O Fail 0 0 105 The eleven-year old subject's score on the Congruence of Length Task will correlate positively with the sub- ject's score on the Congruence of Area Task. The cell frequencies in the following contingency table indicate that 100 per cent of the eleven—year old subjects 115 passed both the Congruence of Length and the Congruence of Area Tasks. Based on the above statistic this Operational hypothesis is accepted. TABLE 4-5. Congruence task for the eleven-year old group. foggy teem Length Pass 20 0 Fail 0 O In summarizing the property of congruence relative to length and area measurement, all of the subjects in the sample are consid- ered regardless of age. The following operational hypothesis is con- sidered for acceptance. HTl The subject's Score on the Congruence of Length Task will correlate positively with the subject's score on the Congruence of Area Task. Since the null hypothesis submitted to test was rejected, the statistical alterna- tive was accepted. Hence this operational hypothesis is accepted. The correlation between congruence of length and congruence of area is O = .88. TABLE 4-6. Congruence task for all subjects in the sample. Areg_ weal. Length Pass 96 2 x2 = 15.49 a = .001 Fail 1 1 o = .88 116 Conservation Task 106 The seven-year old subject's score on the Conservation of Length Task will correlate positively the subject's score on the Conservation of Area Task. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this opera- tional hypothesis is accepted. The correlation between conservation of length and conservation of area is o = .88. TABLE 4-7. Conservation task for the seven-year old group. Length 10 Area Pass 5 O x2 15.56 a = .001 Fall 1 14 ¢ .88 The eight-year old subject's score on the Conservation of Length Task will correlate positively with the sub- ject's score on the Conservation Of Area Task. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this Operational hypothesis is. accepted. The correlation between conservation of length and conservation of area is o = .50. 117 TABLE 4-8. Conservation task for the eight-year old group. 2 Length Pass 7 2 X = 5.05 a = .025 Fail 3 8 ¢ = .50 108 The nine-year old subject's score on the Conservation of Length Task will correlate positively with the sub- ject's score on the Conservation of Area Task. Since the null hypothesis submitted to test was not rejected, the statistical alternative could not be accepted. Hence this operational hypothesis cannot be accepted. The correlation between conservation of length and con- servation of area is O = .39. TABLE 4-9. Conservation task for the nine-year old group. Length 109 5559_ 229.2511]. Pass 6 1 x2 = 2.97 e = .05 Fail 6 7 ¢ = .39 The ten-year old subject's score on the Conservation of Length Task will correlate positively with the subject's score on the Conservation of Area Task. Since the null hypothesis submitted to test was rejected, the statis- tical alternative was accepted. Hence this Operational 118 hypothesis is accepted. The correlation between con- servation of length and conservation of area is O = .44. TABLE 4-10. Conservation task for the ten-year old group. Area Pass Fail 3.95 a .44 Length Pass 17 1 x2 .05 Fail 1 1 9- II 1010 The eleven-year old subject's score on the Conservation of Length Task will correlate positively with the sub- ject's score on the Conservation of Area Task. Since the null hypothesis submitted to test was not rejected, the statistical alternative could not be accepted. Hence this operational hypothesis cannot be accepted. The correlation between conservation of length and con- servation of area is o = .19. TABLE 4-11. Conservation task for the eleven-year old group. Area Pass Fail N ll \1 .h 8 II O 01 Length Pass 14 4 x Fail 1 1 ¢ = .19 In summarizing the property of conservation relative to length and area measurement, all of the subjects in the sample are 119 considered regardless of age. The following operational hypothesis is considered for acceptance. HT 2 TABLE 4-12 The subject's score on the Conservation of Length Task will correlate positively with the subject's score on the Conservation of Area Task. Since the null hypoth- eSis submitted to test was rejected, the statistical alternative was accepted. Hence this operational hypoth- esis is accepted. The correlation between conserva- tion of length and conservation of area is O = .59. Conservation task for all subjects in the sample. Area Pass Fail Length Pass 49 8 x2 34.73 a = .001 Fail 12 31 O .59 Additivity Task IO 11 The seven-year old subject's score on the Addtivity of Length Task will correlate positively with the subject's Score on the Additivity of Area Task. Since the null hypothesis submitted to test was rejected, the statis- tical alternative was accepted. Hence this Operational hypothesis is accepted. The correlation between addi- tivity of length and additivity of area is O = .73. 120 TABLE 4-13. Additivity task for the seven-year old group. Area Pass Fail Length Pass 3 2 X 10.5881 « = .005 Fail 0 15 O .73 1012 The eight-year old subject's score on the Addtivity of Length Task will correlate positively with the subject's score on the Additivity of Area Task.- Since the null hypothesis submitted to test was rejected, the statis- tical alternative was accepted. Hence this operational hypothesis is accepted. The correlation between addi- tivity of length and additivity of area is O = .80. TABLE 4-14. Additivity task for the eight-year old group. Ayeg_ eaten Length Pass 8 1 X2 = 12.74 a = .001 Fail 1 10 O = .80 I013 The nine-year old subject's score on the Additivity of Length Task will correlate positively with the subject's score on the Additivity of Area Task. Since the null hypothesis submitted to test was rejected, the statis- tical alternative was accepted. Hence this operational 121 hypothesis is accepted. The correlation between addi- tivity of length and additivity of area is O = .65. TABLE 4-15. Additivity task for the nine-year old group. Ayeg_ Pass Fail Length Pass 6 4 X2 = 8.57 a = .005 Fail 0 10 O = .65 1014 The ten-year old subject's score on the Additivity of Length Task will correlate positively with the subject's score on the Additivity of Area Task. Since the null hypothesis submitted to test was rejected, the statis- tical alternative was accepted. Hence this operational hypothesis is accepted. The correlation between addi- tivity of length and additivity of area is O .49. TABLE 4-16. Additivity task for the ten-year old group. Argo 89.55.5111 Length Pass 15 1 X2 = 4.80 a = .05 Fail 2 2 O = .49 1015 The eleven-year old subject's score on the Additivity of Length Task will correlate positively with the sub- jectS score on the Additivity of Area Task. Since the 122 null hypothesis submitted to test was rejected, the Statistical alternative was accepted. Hence this operational hypothesis is accepted. The correlation between additivity of length and additivity of area is O - .79. TABLE 4-17. Additivity task for the eleven-year old group. Areg_ 2922.69.11 Length Pass 17 1 x2 = 12.59 a = .001 Fail 0 2 O = .79 In summarizing the prOperty of additivity relative to length and area measurement, all of the subjects in the sample are considered regardless of age. The following operational hypothesis is considered for acceptance. HT 3 The subject's score on the Additivity of Length Task will correlate positively with the subject's score on the Additivity of Area Task. Since the null hypothesis submitted to test was rejected, the statistical alter- native was accepted. Hence this Operational hypothesis is accepted. The correlation between additivity of length and additivity of area is O = .76. 123 TABLE 4-18. Additivity task for all subjects in the sample. Length Area Pass Fail Pass 49 9 x2 58.38 « é .001 Fail 3 39 O .76 Unit of Measure Task 1016 TABLE 4-19. U Length 1017 The seven-year old subject's score on the Unit of Length Measure Task will correlate positively with the subject's score on the Unit of Area Measure Task. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this operational hypothesis is accepted. The correlation between the unit of length measure and the unit of area measure is O = .46. pit of measure task for the seven-year old group. Area Pass Fail 4.21 a .05 Pass 1 3 x Fail 0 16 .46 ‘9- II The eight-year old subject's score on the Unit of Length Measure Task will correlate positively with the subject's score on the Unit of Area Measure Task. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this 124 operational hypothesis is accepted. The correlation between the unit of length measure and the unit of area measure is O = .88. TABLE 4-20. Unit of measure task for the eight-year old group. Agog 1255.59.11 Length Pass 5 0 x2 = 15.55 a = .001 Fail 1 14 O = .88 I018 The nine-year old subject's Score on the Unit of Length Measure Task will correlate positively with the sub- ject's score on the Unit of Area Measure Task. Since the null hypothesis submitted to test was not rejected, the statistical alternative could not be accepted. Hence this operational hypothesis cannot be accepted. The correlation between the unit of length measure and the unit of area measure is 4' .38. TABLE 4-21. Unit of measure task for the nine-year old group. Agog, Pass Fail Length Pass 2 2 x2 = 2.81 c = .05 Fail 2 14 O = .38 125 1019 The ten-year old subject's score on the Unit of Length Measure Task will correlate positively with the subject's score on the Unit of Area Measure Task. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this opera— tional hypothesis is accepted. The correlation between the unit of length measure and the unit of area measure is O = .56. TABLE 4-22. Unit of measure task for the ten-year old group. Area Pass Fail Length Pass 11 2 x 6.28 a = .025 5 Fail 2 I020 The eleven-year old subject's score on the Unit of Length Measure task will correlate positively with the subject's score on the Unit of Area Measure Task. Since the null hypothesis submittted to test was rejected, the statistical alternative was accepted. Hence this Operational hypothesis is accepted. The correlation between the unit of length measure and the unit of area measure is O = .49. 126 TABLE 4-23. Unit of measure task for the eleven-year old group. Agog 59922.11. Length Pass 9 3 x2 = 4.85 a = .05 Fail 2 6 O = .49 In summarizing the prOperty of a unit of measure relative to length and area measurement, all of the subjects in the sample are considered regardless of age. The following Operational hypothesis is considered for acceptance. HT4 The subject's score on the Unit of Length Measure Task will correlate positively with the subject's score on the Unit of Area Measure Task. Since the null hypoth- 9515 submitted to test was rejected, the statistical alternative was accepted. Hence this operational hypoth- esis is accepted. The correlation between the unit of length measure and the unit of area measure is O = .64. TABLE 4-24. Unit of measure task for all of the subjects. Area Pass Fail Length Pass 28 10 X 40.59 a = .001 Fail 7 55 .64 '9 ll 127 Research Hypothesis II II. The understanding of length and area measurement are attained simultaneously. The following five operational hypotheses were developed to test Specifically this research hypothesis by converting them to statistical alternative hypotheses (i.e. O > 0). A null hypothesis xy (O < 0) was formed from each statistical alternative hypothesis and xy —- was tested using the Chi-square and Phi-coefficient test statistics. Tabular data and a decision concerning the rejection or acceptance of the null hypotheses accompany each operational hypothesis. The operational hypotheses are grouped according to the age of the subjects. Each of the hypotheses concerns the final attain- ment of both length and area measurement, i.e. is the Subject at sub- stage III B or is he at a substage lower than 111 B. An operational hypothesis is formed concerning the final attainment of length and area measurement and all of the subjects in the sample regardless of age. The seven step Statistical procedure outlined in Chapter III is used in determining the acceptance or rejection of this sunlnarizing hypothesis across all ages. All results are summarized in Table 4-32. 1101 The seven-year old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this 128 Operational hypothesis is accepted. The correlation between the final attainment of length measurement and the final attainment of area measurement is O = .46. TABLE 4-25. Final attainment of length and area measurement (sub- stage III B) for the seven-year old group. fflggg lll_§_ < III 8 Length III B 1 3 4.21 a .05 X ll < 111 B 0 16 O .46 I102 The eight-year Old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this opera- tional hypothesis is accepted. The correlation between the final attainment of length measurement and the final attainment of area measurement is O = .88. TABLE 4-26. Final attainment of length and area measurement (sub- stage III B) for the eight-year old group. O X N ll Length III B 5 15.56 a = .001 A H 0—. H W O... t—I .b '9 ll .88 110 129 The nine-year old subjects who have attained (failed to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. Since the null hypothesis submitted to test was not rejected, the statistical alternative could not be accepted. Hence this operational hypothesis cannot be accepted. The correlation between the final attainment of length measurement and the final attainment of area measure- ment is O = .14. TABLE 4-27. Final attainment of length and area measurement (sub- 5 Length 1104 tage III B) for the nine-year old group. Egggl Ill_§_ < III B 111 8 1 2 x2 .39 a .05 < 111 B 3 14 .14 '6- II The ten-year old subjects who have attained (failed to attain) an understanding of either length or area mea- surement have attained (failed to attain) an understand- ing of both length and area measurement. Since the null hypothesis submitted to test was rejected, the statis- tical alternative was accepted. Hence this Operational hypothesis is accepted. The correlation between the 130 final attainment of length measurement and the final attainment of area measurement is O = .56. TABLE 4-28. Final attainment of length and area measurement (sub- stage III B) for the ten-year old group. Area III 8 III 8 Length III B 11 2 x2 = 6.28 a = .025 < III B 2 5 O = .56 1105 The eleven-year old subjects who have attained (failed TABLE 4-29. Length to attain) an understanding of either length or area measurement have attained (failed to attain) an under- standing of both length and area measurement. Since the null hypothesis submitted to test was rejected, the statistical alternative was accepted. Hence this operational hypothesis is accepted. The correlation between the final attainment of length measurement and the final attainment of area measurement is O = .45. Final attainment of length and area measurement (sub- stage III B) for the eleven-year old group. Argo- LLLJ; III 8 III B 1 2 x 4.01 G .05 < 111 B 3 14 .45 9 II 131 In summarizing the consideration of the final attainment of length and area measurement, all of the subjects in the sample are considered regardless of age. The following operational hypothesis is considered for acceptance. HT5 The subjects who have attained(failed to attain) an understanding of either length or area.measurement have attained (failed to attain) an understanding of both length and area measurement. Since the null hypothesis submitted to test was rejected, the statistical alterna- tive was accepted. Hence this Operational hypothesis is accepted. The correlation between the final attain- ment of length measurement and the final attainment of area measurement is O = .61. TABLE 4-30. Final attainment of length and area measurement (sub- stage III B) for all of the subjects. foggy III B < 111 8 Length 111 B 17 10 x2 = 37.22 a = .001 <111 B 8 55 O = .61 Summary In this chapter, the statistical results concerning the measurement properties of congruence, conservation, additivity and unit measure have been stated. 132 Concerning Research Hypothesis 1: I. The cognitive development of length measurement is simulta- neous to the cognitive development of area measurement relative to the properties of congruence, conservation, additivity, and unit measure. Twenty operational hypotheses relating each age level to each of the measurement properties were statistically tested. In addition four hypotheses were formed relating each of the four meas- urement properties to all the subjects in the sample. These sum- marizing hypotheses were also submitted to statistical test. State- ments of acceptance or rejection were made in addition to noting the correlation coefficient (O) for each operational hypothesis as a means of investigating Research Hypothesis I. These results are summarized in Table 4-31. Concerning Research Hypothesis II: II. The understanding of length and area measurement are attained simultaneously. Five operational hypotheses relating each age level to the final attainment of length and area measurement were statistically tested. In addition,ai hypothesis was formed relating the final attainment of length and area measurement to all the subjects in the sample. This summarizing hypothesis was also submitted to statistical test. Statements of acceptance or rejection were made in addition to noting the correlation coefficient (O) for each Operational hypoth- esis as a means of investigating Research Hypothesis 11. These results are summarized in Table 4-32. 133 Table 4-31 and Table 4-32 are the summarizations of all the operational hypotheses tested including: (1) the signifiance level “a” for the Chi-square test for independence, (2) a statement of rejection or acceptance, and (3) the value of the Phi-coefficient . . . . . 1 indicating the correlation between the variables tested. TABLE 4-31. Summary of the operational hypotheses tested for Research Hypothesis 1. Measurement Correlation Property Hypothesis Alpha level Accept Reject Coefficient Congruence 101 .05 X .63 102 - x - 103 - X - 104 - x - 105 - X - HT1 .001 X .88 Conservation 105 .001 X .88 107 .025 x .50 108 .05 x .39 109 .05 x .44 1010 .05 x .19 HT2 .001 x .59 1The variables tested in Research Hypothesis I are the attain- ment of each of the four measurement properties relative to length measurement and the attainment of each of the four measurement prOp- erties relative to area measurement. The variables tested in Research Hypothesis II are the final attainment Of length measurement (Substage III B) and the final attainment of area measurement (substage III 8) relative to the age levels studied. 134 Measurement Correlation Property Hypothesis Alpha level Aggept_ Rejegt_ Coefficient Additivity 1011 .005 X .73 I012 .001 x .80 1013 .005 X .65 1014 .05 X .49 I015 .001 X .79 HT3 .001 x .76 Unit Measure 1016 .05 x .46 ' 1017 .001 x .88 1018 .05 X .38 1019 .025 x .56 1020 .05 x .49 HT4 .001 X .64 Note that twenty-one of the twenty-four operational hypotheses tested concerning Research Hypothesis I are accepted. TABLE 4-32. Age Group 7 8 9 10 11 All Hypothesis II. 135 Hypothesis Alpha level Accepp 1101 .05 X 110 .001 X 2 1103 .05 1104 .025 X 1105 .05 X HT5 .001 X Summary of the Operational hypotheses tested for Research Correlation Reject Coefficient .48 .88 X .14 .56 .45 .61 Note that five of the six Operational hypotheses tested con- cerning Research Hypothesis IIare accepted. CHAPTER V SUMMARY OF MAJOR FINDINGS, CONCLUSIONS, AND IMPLICATIONS Introduction It was the stated purpose of this study to investigate the cognitive development of length and area measurement relative to four common component properties (congruence, conservation, additivity, and unit measure) and the factor of chronological age. The intent . was to conduct a comparative investigation to lend support to one of the two contrasting points of view identified to be: 1. There is no difference between the ages at which a child attains corresponding levels of understand- ing relative to length and area measurement and that both of these concepts are finally attained at approximately the same age. 2. There is a difference between the ages at which a child attains corresponding levels of understand- ing relative to length and area measurement and that a child finally attains length measurement prior to area measurement. The need for this study was based on the conflicting conclu- sions made from two separate investigations: (1) Piaget, Inhelder, and Szeminska have completed extensive research on the cognitive 136 137 develOpment of length and area measurement.1 Based upon the results of his work, Piaget stated that "The development of . . . measurement runs exactly parallel whether the Objects are lengths or whether they are areas and the level at which they are finally grasped is the same for both."2 (2) Beilin and Franklin conducted an investigation of length and area measurement on a comparative basis.3 One of the stated purposes of this study was to investigate whether the abilities to solve related problems of length and area measurement are acquired simultaneously. The intent was to test the validity of Piaget's conclusion regarding the simultaneous development of and final attain- ment of length and area measurement. Based on the results of Beilin and Franklin's study, they concluded that "He would suggest that length and area . . . measurement are achieved in that order. Also the constituent operations to measurement . . . are applied more easily first to a single dimension, then to two dimensions, . . ."4 Piaget's research has prompted Copeland to criticize the present manner in which length and area measurement are being taught in American elementary schools. COpeland stated that "Measurement in one dimension is taught before the child is at the Operational or readiness level to understand it, and yet two-dimensional or area lPiaget, Inhelder, and Szeminska, Conception of Geometry. 2Ibid., p. 300. 3Beilin and Franklin, "Logical Operations in Area and Length Measurement,“ pp. 607-618. 416i ., p. 617. 138 measurement is deferred several years past the age at which children can understant it.“1 If Piaget's research is to be an influence in the development of the mathematics curriculum of the elementary schools, verification of his results is necessary. Most of the related studies focused on only one of these two measurement concepts, either length or area, and had not investigated them on a comparative basis. In this study, four common component properties of length and area measurement have been identified using measurement axioms contained in modern geometry textbooks. These measurement properties were studied as a means of investigating the cognitive development of length and area measurement on a comparative basis. The study has collected and analyzed data regarding the four measurement properties in research of statistical evidence to test hypotheses central to the cognitive development of length and area measurement. Hithin the limitations of this study the following major findings and implications are presented. Conclusions Concerning the Four Measurement Properties The investigation of four common component properties of length and area measurement (i.e. congruence, conservation, additivity, and unit measure) has provided a more refined procedure for investi- gating the cognitive development of length and area measurement than has been previously available. The findings show that it is possible 1Copeland, How Children Learn Mathematics, p. 238. 139 to identify stages of cognitive development (as suggested by Piaget) relative to these two measurement concepts by the child's operational understanding of the four component properties common to both length and area measurement, although, in some instances,the number of sub- jects at certain stages were not what would be expected from Piaget's results. (e.g. The nine-year old group performed at a lower level than one would have expected: they are much nearer to the eight-year olds than the ten-year olds in ability to successfully complete the tasks concerning the conservation, additivity, and unit measure prOperties.) Copgguence Property The six Operational hypotheses concerning a positive correla- tion between the subject's score on the Congruence of Length Task and his score on the Congruence of Area Task within and across all ages were accepted. Thus, these results imply that a child under- stands what is meant by the property of congruence regardless whether the objects are lengths or areas. This conclusion needs to be veri- fied with younger children (e.g. four to seven years) due to the high rate of success of the subjects in this study on the congruence tasks (92 per cent and 97 per cent of the sample, respectively, passed the length and area measurement tasks concerning the congruence property). Conservation Property Three of the five Operational hypotheses concerning a positive correlation between the subject's score on the Conservation of Length 140 Task and his score on the Conservation of Area Task within each age group were accepted. The two hypotheses that were not accepted con- cerned the nine- and eleven-year old subjects, although 65 per cent and 75 per cent of the subjects in the nine- and eleven-year old groups, respectively, were rated the same on both conservation tasks (i.e. relative to length and area measurement). The operational hypothesis concerning a positive correlation between the subject's score on the Conservation of Length Task and his score on the Conser- vation of Area Task across all ages was accepted. Thus, there is evidence to support the claim that a child understands what is meant by the property of conservation regardless whether the objects are lengths or areas. Addi ti vi ty Property The six Operational hypotheses concerning a positive correla- tion between the subject's score on the Additivity of Length Task and his score on the Additivity of Area Task within and across all ages were accepted. Thus, these results imply that a child under- stands what is meant by the additivity prOperty regardless whether the objects are lengths or areas. Unit Measure Property Four of the five Operational hypotheses concerning a positive correlation between the subject's score on the Unit of Length Task and his score on the Unit of Area Task within each age group were accepted. The hypothesis regarding the nine-year old group was not accepted, although 80 per cent of the nine-year olds were scored the 141 same on the length and area measurement tasks concerning the unit measure property. The operational hypothesis concerning a positive correlation between the subject's Score on the Unit of Length Task and his score on the Unit of Area Task across all ages was accepted. Thus, there is evidence to support the claim that a child is cognizant of the size and number of the units used in the measuring process regardless whether the objects are lengths or areas. The following table indicates the percentage of subjects who have passed each of the measurement property tasks relative to length and area: Congruence Conservation Additivity, Unit Measure Length 92 56 58 38 Area 97 61 52 35 Consistent with the findings of Piaget, the findings of this study indicate the measurement properties in order of difficulty are: (1) congruence, (2) conservation, (3) additivity, and (4) unit measure. Conclusions Concerning the Research Hypotheses Two research hypotheses were established in this investiga- tion. The first research hypothesis states, in essence, that there is a simultaneous cognitive develOpment of length and area measure- ment. Seventeen of the twenty related operational hypotheses within age groups and all four related hypotheses across all ages were accepted in support of this research hypothesis. Based upon the analysis of the data collected in this study, Research Hypothesis 1 is Supported. 142 Piaget, Inhelder, and Szeminska also indicate the existence of a similar cognitive develOpment of length and area measurement: "The development of . . . measurement runs exactly parallel whether the objects are lengths or whether they are areas . . ."1 The second research hypothesis states that length and area measurements are finally understood at approximately the same age. Four of the five operational hypotheses concerning a simultaneous final attainment of length and area measurement within each age group were accepted. The hypothesis concerning the nine-year old group was not accepted, although 75 per cent of the nine-year olds were rated the same (substage III B for final attainment, < III B for lacking final attainment) regarding the final attainment of length and area measurement. The hypothesis concerning a simultaneous final attainment of length and area measurement across all ages was accepted. Based upon the analysis of the data collected in this study, Research Hypothesis II is supported. Piaget, Inhelder, and Szeminska have made a similar conclusion indicated by their statement: "The level at which they [length and area measurement] are finally grasped is the same for both".2 Implications Certain implications, over and beyond the study, warrant mentioning. 300 1Piaget, Inhelder, and Szeminska, Conception of Geometry, p. . 21 id. 143 The elementary school teacher would gain insight regarding the thought processes of a child through replicating some of the tasks presented in this Study. The stages of mental growth that would be observed should be taken into account when planning learning experiences. An effective individual interview technique would be a beneficial pedagogical skill for the elementary school teacher to acquire. Incorrect responses from the child could be pursued and misconceptions eliminated effectively through the use of this technique. Before introducing a new concept such as area measurement, the child should be tested with Piagetian type tasks to be sure that he has all the prerequisites for mastering the concept. If he is not yet ready for the concept, the child should be provided with experiences that will help him become ready. The English Nuffield Project is using this procedure to chart the cognitive growth of a child concerning mathematical concepts. Concrete materials Should be used wherever possible to provide children with certain experiences which will prepare them to learn a particular mathematical concept. The Mathematics Laboratory makes extensive use Of physical objects that can be manipulated by the children. The component properties of measurement (i.e. congruence, conservation, additivity, etc.) should be introduced using both lengths and areas simultaneously. As an 144 example, when the child is near the stage of understanding conservation (the age will differ with different children but may be found through the use of Piagetian type tasks) the physical objects used in the instructional process should be both lengths and areas. Implications for Future Research 1. If Beilin and Franklin's conclusion regarding one dimensional concepts being learned earlier than two dimensional concepts because of the complexity caused by the additional dimension is extended, one may assume that concepts of length, area, and volume measurements are developed cognitively in that order: this assumption is contrary to the results of Piaget. Thus, investigation is needed to determine the optimum order of placement regarding length, area, and volume measurements. Further investigation could involve weights, liquid volume, three dimensional surface area, etc. All investigations, including this one, need varification. 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Edited by Donald Green, Marqueritte Fora; and George Flamer. New York: McGraw-Hill Book Com- pany, 1971. pp. 1-11. ‘ .~‘~.-‘ - ‘07 - "a... Piaget, Jean and Inhelder, Barbel. The Child's Conception of Space. Translated by F.J. Langdon and'JiL. Lunzer. London: Routledge and Kegan Paul Ltd., 1956. Piaget, Jean, Inhelder, Barbel, and Szeminska, Alina. The Child's Conce tion of Geometry. 2nd ed. Translated by E.A. Lunzer. ew ork: Harper and Row, 1960. Ringenber, Lawrence A. Colle e Geometry. New York: John Wiley and Sons, Inc., 1 . Rosenbloom, Paul C. "Implications of Piaget for Mathematics Curriculum." Improving Mathematics Education for Elementary School Teachers. Edited by’Robert Houston. Conference sponsored—by_The Science and Mathematics Teaching Center, Michigan State University and The National Science Foundation. pp. -49. Rosskopf, Myron F., Sitomer, Harry, and Lenchner George. Modern Mathematics: Geometry. Morristown, N.J.: Silver Burdett Cb., 1966. Royden, H.L. Real Analysis. New York: The Macmillian Company, 1963. School Mathematics Study Group. Geometrnyart I. New Haven: Yale University Press, 1960. School Mathematics Study Group, Geometry_with Coordinates, Part II. New Haven: Yale University Press, 1963. 150 Shah, Sair Ali. ”Selected Geometric Concepts Taught to Children Ages Seven to Eleven." Edited by C. Alan Riedesel and Len Pikaart. The Arithmetic Teacher. February.1969, pp. 119-128. Smart, James R. and Marks, John L. "Mathematics of Measurement.“ The Arithmetic Teacher, April, 1966. pp. 283-87. Smedslund, Jan. Concrete Reasonin : A Study of Intellectual Develo ent. {"Monograph o the Society Tor‘Research in Child Bevelopment," 29, No. 2, Serial No. 93), pp. 389-405. Smedslund, Jan. ”Development of Concrete Transitivity of Length t in Children". Child Development, 34 (1963), pp. 389-405. ‘ Steffe, Leslie P. “Thinking About Measurement." Edited by C. { gggn Riedesel. The Arithmetic Teacher. May. 1971. PP- 332‘ , Stoll, Earline Lillian. Geometric Concept Formation In Kinder- 3; arten Children. UnpubTishedIDoctoral Dissertation, Stanford finiversity, 1962. Taloumis, Thalia. "The Understanding of Area Measurement." The_ Mathematics Education of the Elementar School Teacher. A Report at the lil Project in Science and Mathematics Tri- University Project in Elementary Education. Volume II, 1970. pp. 113-124. Towler, J.0. Training Effects and Concept Development: A Study of the Conservation of Continuous Quantityin Children. Wash- ington, D.C.: *United States Office of EducationiERIC. Department of Health Education, and Welfare. 1968. Turabian, Kate L. A Manual for Writers. 3rd, edition. Chicago: The University of Chitago Press, 1967. Wagman, Harriet. A Study_of the Child's Concept of Area. Unpub- lished Doctoral Dissertation, columbia univerSity, 1968. Walker, Helen M. and Lev, Joseph. Elementgry Statistital Methods. New York: Holt, Rinehart, and’WinSton,i1958. Wallace, J.G. Concept Growth and the Education of the Child. New York: NewTYork university Press, 1967. Wertheimer, Max. Productive Thinking, Edited by Michael Wertheimer. New York: Harper and Biothers Publishers, 1959. Young, Beverly S. Inducing Conservation of Number, Weight, Volume, Area and Mass iniPre-School Chderen. Hashington, DlC.: Unitea States Uttice 0? Education ERIC. Department of Health, Education, and Welfare. February, 1969. Zimiles, 151 Herbert. "A Note on Piaget's Concept of Conservation." L ical Thinkin in Children - Research Based on Pia et's ifieor dgby IFving Sigel and Frank H. Hooper, 1963. O E. ”Edite pp. '381. APPENDIX A DETAILED DESCRIPTION OF TASKS Preparatory Remarks In preliminary conversation with each child, the following f. remarks were made: 1. "This is not a test; I would like to know what you think } about some of the materials that I have.“ 1i 2. "There aren't any right or wrong answers; just tell me 1;? what you think.“ 3. ”Don't guess on these tasks. Try to work things out." 4. "I'm going to write down some things and keep this tape recorder on during our meeting. In case I forget some- thing, I can listen to our conversation later.“ Each child was asked to state his name, grade, and teacher‘s name for identifying purposes. The child was allowed to listen to his voice on tape prior to the presentation of the tasks. Any variation in tenms used was indicated on the interview recording sheet (see Appendix B). 152 153 Vocabulary Task Purpose: To determine whether the subject is familiar with the vocabulary to be used in the measurement tasks. Only subjects who are will be included in the sample. Materials: 8 inch length of 1/8 inch diameter wire, colored black. 8 inch length of 1/8 inch diameter wire, colored red. 4 inch length of 1/4 inch diameter wire, colored green. 16 inch length of 1/8 inch diameter wire, colored white. 2 isosceles right triangles with 8 inch legs, one colored blue, the other green. 1 isosceles right triangle with 4 inch legs, colored red. 1 hexagon with area measure equal to 16 square inches. colored yellow. See Figure 3-1 for details of material. Procedure regarding lenggp: "I'd like you to look at these two pieces of wire, the black and the red. 00 they have the same amount of length or different amounts?" 4 Permit child to handle the wires and superimpose them if necessary. "Which one has more length? Which one has less length? Which one is longer? Which one is shorter?" Repeat the above with the black and white wires. Repeat the above with the black and green wires. I‘lllllllI‘lIlitI . 154 Variation: Child says, "What do you mean by 'length'?" or "I don't know what you mean." Investigator replies, "Is there the same amount of distance from one end to the other?“ If the child is still puzzled (rare), investigator says, "How are these two pieces of wire different? Are they the same in every way besides color? Is there more white or more black? Why?" Procedure regarding area: "I'd like you to look at these two pieces of paper, the blue and the green. 00 they have the same area? Do they have the same amount of space or different amounts?" Permit child to handle the triangles and superimpose them if necessary. "Which one has more Space? Which one has less Space? Which one is larger? Which one is smaller?" Repeat the above with the blue and red polygons. Repeat the above with the blue and yellow polygons. Variation: Child says, "What do you mean by 'Space'? or "I don't know what you mean." Investigator replies, "Do they have the same amount of room inside or different amounts?“ If child is still puzzled (rare), investigator says, "How are these two pieces of paper different? Are they the same in every way besides color? Is there more red or more blue, why?“ X , ~ ”“731 155 If the child uses “room" rather than "space“, when he indi- cates the polygon with more room, say, “So this one has more space." Similarly for the polygon with less room. Try using "space" in the questions when repeating the task with the blue and yellow polygons. If the child has trouble using “Space", then indicate same on record- ing sheet and substitute "room“ for "space" in the remaining tasks. q... V." ain‘tm x." err arr-u xxx-u 156 Congruence Task-Length Purpose: To determine whether the child can identify congruent lines (i.e. lines whose endpoints coincide when placed in a parallel man- ner). To determine whether the child can make correct judgements regarding equal and unequal lengths. Materials: 2 - 4 inch lengths of wire, 1/8 inch diameter, one made of black, the other of red wire. 1 - 8 inch length of wire, 1/8 inch diameter, made of white wire. The wires are identical in every way except the stated differences of color and length. Procedure: The three wires (black, red, and white) are placed on the desk in front of the child in no organized manner. The child is asked to respond to the following questions. Each question is fol- lowed by the child's response. "Are any of the wires the same length?" (response) "Which of the wires are the same length?" (response) "How can you tell that?" (response) "Are any wires of different length?" (response) "Which wires are of different length?“ (response) “How can you tell that?" (response) The child is allowed to manipulate the wires and superimpose them if desired. WW—Wfi'r-‘f—T'V ——--§ '7 w 157 Congruence Task-Area Purpose: To determine whether the child can identify congruent polygonal regions (i.e. polygonal regions that have the same size and shape). To determine whether the child can make correct judgments regarding equal and unequal areas. Materials: 2 isosceles right triangles with 4 inch legs.one made of blue, the other of green paper. 1 isosceles right triangle with 6 inch legs, made of white paper. Paper of the same composition and thickness was used in the construction of the triangles. Procedure: The three triangles (blue, green, and white) are placed on the desk in front of the child in no organized manner. The child is asked to respond to the following questions. Each question is followed by the child's response. "Do any of the pieces of colored paper have the same size and shape?" (reSponse) "Which pieces of paper have the same size and shape?" (reSponse) "How can you tell that?" (response) "Do any of the pieces of colored paper have different sizes or shapes?" (response) "How can you tell that?“ (response) The child is allowed to manipulate the triangles and superimpose them if he desires. 158 Conservation of Length Task Purpose: To determine whether the subject can conserve length, relative to both a change of position and subdivision. Materials: 2 - 3 inch long pieces of thin wire (same diameter), one colored black, the other red. See Figure 3-2. Procedure: This task is administered in two parts: Part A: Change of position "I'd like you to look at these two pieces of wire, the black and the red. Do they have the same length? How can you tell?" The subject is permitted to handle the wires and super- impose them if necessary. Eventually, with the assistance of the experimenter if necessary, the subject concludes that the wires are of equal length. The wires are placed in front of the subject in a par- 6119] manner so that their ends coincide. "Do the wires have equal length? Why?" The following arrangements as indicated in Figure 3-1, A-F are formed, each followed by the questions: "Do the wires have the same length? Why?" Part B: Subdivision "I'd like you to look at these two pieces of wire, the black and the red. Do they have the same length? 15 one 159 longer than the other? (Which one is longest?) How can you tell?" The subject is permitted to handle the wires and super- impose them if necessary. Eventually, with the assistance of the experimenter if necessary, the subject concludes that the wires are of equal length. The black wire is not altered, while the red wire is bent into the shape indicated in Figure 3-2 H. The child is asked: "Do the black and red wires have the same length? Is one longer than the other? (Which is longest?) How can you tell?“ 160 Conservation of Area Task Purpose: To determine whether the subject can conserve area, rela- tive to both a change of position and subdivision. Materials: 2 isosceles right triangles with 8 inch legs, one made of blue paper, the other made of green paper. See Figure 3-3. Procedure: This task is administered in two parts: Part A: Change of position “I'd like you to look at these two pieces of paper, the blue and the green. Do they have the same area? Do they have the same amount of Space? How can you tell?" The subject is permitted to handle the triangles and superimpose them if necessary. Eventually, with the assist- ance of the experimenter if necessary, the subject concludes that the triangles have the same area or same amount of space. The triangles are placed in front of the subject accord- ing to the arrangements indicated in Figure 3-3, A-D. After each arrangement, the subject is asked: "Are the areas the same? 00 they have the same amount of Space? (Which has more Space?) How do you know that?“ Part B: Subdivision “I'd like you to look at these two pieces of paper, the blue and the green. Do they have the same area? Do they 161 have the same amount of space or different amounts? (Which one has more Space?) How do you know that?" The subject is permitted to handle the triangles and superimpose them if necessary. Eventually, with the assist- ance of the experimenter if necessary, the subject concludes that the triangles have the same area or same amount of space. The blue triangle is not altered, while a segment is cut from the green triangle and placed in the positions indicated in Figure 3-3, E-F. The child is asked "Do the blue and green pieces of paper have the same area (same or different amount of space or room)? Which one has more space? How do you know that?“ 162 Additivity of Length Task Purpose: To determine whether the subject understands that the whole is equal to the sum of its nonoverlapping parts (except for possible common endpoints) regardless of the arrangement of the parts. The test for additivity of length involves a test for conservation and transitivity. Materials: The following lines were drawn on white paper (see Figure 3-4). 1 - straight blue line, 16 inches long. 1 - broken green line, with segments of 2, 8, 4, and 2 inches, respectively. 1 - oblique, straight, red line, 15 inches long. 2 - lengths of white wire, 4 inches long. 4 - lengths of white wire, 2 inches long. The wires were made of the same composition (solder) and were the same thickness (1/8 inch in diameter). Procedure: The blue, green, and red lines are drawn on a piece of paper. The white wires are the only movable pieces in this task. No indication of dimensions is given to the subject. The child is given all of the white wires and is asked to cover exactly each of the colored lines, one at a time, with the white wires. 163 The subject is asked, "Can you arrange the white wires so they cover up exactly the blue line by placing them end to end on the blue line?" (Assist the subject if necessary.) “Now, can you do the same with the green line?” "Let's think about the blue and green lines. Do they have the same length? Is one line longer than the other line? How do you know that? (Which line is longer? How do you know that?)" "Can you cover up exactly the red line with the white wires? Let's think about the blue, green, and red lines. Does the red line have the same length as the blue line? Is one line longer than the other line? How do you know that? (Which line is the longer line? How do you know that?)" 164 Additivity of Area Task Purpose: To determine whether the subject understands that the whole is equal to the sum of its nonoverlapping parts (except for possible common sides) regardless of the arrangement of the parts. The test for additivity of area involves a test for conservation and transi- tivity. Materials: The following polygonal regions were pasted on white paper (see Figure 3-5). 1 - 4-inch square made of blue paper. I - Z-inch by B-inch rectangle made of green paper. 1 - 2-inch by 7-inch rectangle made of red paper. 2 - 2-inch squares made of white paper. 2 - l-inch by 2-inch rectangles made of white paper. All the paper used was the same composition and same thickness. Procedure: The blue, green, and red pieces are glued to a large piece of paper. The white segments are the only movable pieces in this task. No indication of dimensions is given to the subject. The child is given all of the white pieces of paper and is asked to cover exactly each of the colored pieces of paper, one at a time, with the white paper. The subject is asked, "Can you arrange the white pieces of paper so they exactly cover the blue?" Assist the subject if 165 necessary. "Now, can you do the same with the green?" "Let's think about the blue and the green. Do they have the same amout of space or different amounts? How do you know that? (Which has more space? How do you know that?)" “Can you cover up exactly the red paper with the pieces of white paper? Let's think about the blue, green, and the red. Does the red have the same amount of space or different amounts as the blue or green? How do you know that? (Which has more space? How do you know that?)" 166 Unit Length Task Purppse: To determine if the child is aware of the use of a unit of length measure relative to its Size and number of units. Materials: The following lines were drawn on white paper (see Figure 3-6). 1 straight blue line, 16 inches long. 1 - broken green line, with segments of 2, 8, 4, and 2 inches, reSpectively. 1 - oblique, straight red line, 8 inches long. 6 - lengths of white wire, 4 inches long. 10 - lengths of white wire, 2 inches long. 10 - lengths of white wire, 1 inch long. The wires were made of the same composition (solder) and were the same thickness (1/8 inch in diameter). Procedure: The child is given the paper with the blue, green, and red lines drawn on it. No indication of dimensions are given. "Can you completely cover the blue line with these pieces of wire by placing them end to end on the blue line?" The child is given the six pieces of 4-inch long wires. If the child is not sure of what is being asked, the investigator gives assistance in the form of placing a few of the wires end to end on the blue line. "Now can you completely cover the green line with these pieces of wire?" The child is given the ten pieces of wire that are 2 inches long. The same type of assistance is given by the investigator if needed. 167 "Now, can you completely cover the red line with these pieces of wire?“ The child is given the ten pieces of l—inch wire. Again, similar assistance is given if needed. The child now has all three colored lines covered with the wires of different sizes. “Which line is longer, the blue or the green, or are they the same length?" (response) “How do you know that?" "Which line is longer, the green or the red, or are they the same length?" (reSponse) “How do you know that?“ Throughout this task, the child is permitted to manipulate only those pieces of wire that were not used to cover any of the colored lines (two pieces of 4-inch wire, two pieces of 2-inch wire, and two pieces of l-inch wire). "Without placing these small (l-inch) wires on the green line, can you tell me how many of the small wires it would take to completely cover the green line?“ (response) “How do you know that?“ 168 Unit Area Task Purpose: To determine if the child is aware of the use of a unit of area measure relative to its size and number of units. Materials: The following polygonal regions (blue, green, and red) were pasted on a piece of white paper (see Figure 3-7). 1.. 1- 10 - 10 - All square with 4-inch sides made of blue paper. rectangular region, 2 inches by 6 inches, with a 2- inch square adjoined (see Figure 3-6 B) made of green paper. rectangular region, 1 inch by 8 inches, made of red paper. squares, with 2-inch sides, made of white paper. rectangles, with l-and 2-inch Sides, made of white paper. squares, with l-inch sides, made of white paper. stated regions were made of paper that had the same composition and same thickness. Procedure: The child is given the white paper on which the colored polygonal regions are pasted. No dimensions of sides are indicated. "Can you completely cover the blue paper with these pieces of white paper?“ The child is given the six squares with Z-inch sides. If the child is not sure of what is being asked, the investigator gives assistance in the form of placing a few of the white squares on the blue paper. 169 "Now can you completely cover the green paper with these pieces of white paper?" The child is given the ten rectangular pieces of white paper. The same type of assistance is given by the investigator if needed. "Now can you completely cover the red paper wi th these pieces of white paper?" The child is given the ten squares of white paper with 1- inch sides. Again, similar assistance is given if needed. The child now has all three colored regions covered with the white pieces of paper of different areas. "Which has more Space, the blue or green paper, or are they the same?" (response) "How do you know that?" Which has more Space, the green or the red paper, or are they the same?" (reSponse) "How do you know that?" Throughout this task, the child is permitted to manipulate only those pieces of white paper that were not used to cover any of the colored regions (two squares with Z-inch sides, two rectangles with L-and 2— inch sides, and two squares with 1-inch sides). "Without placing these small (l-inch squares) pieces of white paper on the green paper, can you tell me how many of them it would take to completely cover the green region?" (response) "How do you know that?" APPENDIX B INTERVIEW RECORDING SHEET Name: Date: Age: Birth Date: Grade: Reading Score: 1. Summary of Performance:1 Task Length Area Comments (level) A. Vocabulary A. B. Congruence B. C. Conservation C. D. Additivity D. E. Unit Measure E. 11. Measurement level: Length Area Reason: lTasks are evaluated on pass (P) or fail (F) basis. Comments include statements regarding the child's responses and measurement level. 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