A Study of the Case for Measurement in Elementary School Mathematics

A Study of the Case for Measurement inElementary School Mathematics

Lloyd ScottElementary School Science Project,

University of California,Berkeley, California.

Measurement has never been given a proper role in the elementaryschool mathematics program. As a topic of study, it has been treatedas a very distant relative of the theoretical program. Yet the reasonsfor its neglect are obscure. Even in times when there was a greaterregard for the applications of elementary mathematics than there istoday, measurement received little attention. Enlightened classroompractice which favors concrete, practical uses of number knowledgeconsistently has failed to focus upon measurement as a topic of im-portance. Likewise, the role of applied mathematics in modern tech-nology and the anticipated demand for applied mathematical skills inthe future appear to have but little impact upon contemporary pro-gram formulations.

Indications of the effect upon children of this inferior regard formeasurement have accumulated over the years. Typical of a numberof studies in this area is one reported by Wilson and Cassell in 1953[1]. This study was represented by the authors as a culmination ofmuch investigation dating back to 1918. They confirmed the fact thatthe portion of the school program devoted to measurement was in-adequate over this entire period. The children they studied in 1952 didnot have a clear understanding of the measurement concepts investi-gated. The authors concluded that, "Most of the teaching of weightsand measures has been ineffective. ... If the work on measures in theschools had been omitted entirely, the results apparently would havebeen little different." A similar and equally forthright observationgrew out of Dorothy Wilson’s study [2] in 1937. Gunderson [3] docu-mented the meager attention given measurement through an analysisof children^ arithmetic textbooks in the early 1940’s. She found thatonly eight percent of the program for grade three was devoted to mea-surement. Analysis of items included in the eight percent reveals thatthey treated marginal or trivial aspects of this important topic. With-out exception, the reported studies corroborate the view that skill inapplications involving measurement requires substantial direct ex-perience. That is, it cannot be assumed that the skills and concepts ofmeasurement will be mastered incidentally as the child proceedsthrough abstract arithmetic processes. Corle [4] gave further sub-stantiation to this need for experience in his study of children’sfamiliarity with measurements. He reported that the findings of this

714

Measurement in Elementary Mathematics 715

study "suggest a lack of pupil identification with the factual materialthat is commonly taught about measurements. "An editor’s noteappended to the Corle report contributes the subjective analysis that"the evidence is sufficiently clear to make a strong indictment againstour present elementary program in this quarter."

Despite such evidence of the neglect of measurement to the presenttime, there is little indication that the revolution in elementary schoolmathematics has provided for its correction. The tendency for mostmodern programs to emphasize pure rather than applied mathematicshas served to maintain the disregard for measurement as a study topic.Fortunately, there are indications of a growing dissatisfaction withthis neglect, and certain recent efforts have been directed toward itscorrection. Of considerable importance is a position paper signed bysixty-five prominent mathematicians and mathematics educatorswhich condemns the barren theoretical program and supports a pro-gram which gives increased attention to applications [5]. If this ex-pression of opinion is awarded the attention which it merits, appliedmathematics eventually will be given its proper place in the programand the imprudence with respect to measurement will cease. Whetheror not the aforementioned position paper has been directly influential,it is interesting to note that such curriculum development projects asthe School Mathematics Study Group, the University of IlliniosCommittee on School Mathematics and the Madison Project havebegun to inject some applied mathematics into their theory-biasedformulations [6]. At the same time such groups as the MinnesotaMathematics and Science Teaching Project, the University of Cali-fornia Elementary School Science Project, the Elementary SchoolProject of the American Association for the Advancement of Science,the University of Illinois Elementary School Science Project, and theStudy of a Quantitative Approach in Elementary School Science ofthe State University of New York continue to stress applied mathe-matics as they have done since their inceptions.

If the interest in applications continues, projections of the "secondrevolution" in elementary school mathematics such as that of New-port [7] will become truly prophetic. They predict that the program ofthe future will be an applied program based upon a thorough integra-tion of mathematics and science. In such programs the stature of mea-surement as a study topic will be materially enhanced.Without question, some important contributions to the solution of

this problem have been made by curriculum development projects.For example, the School Mathematics Study Group includes a shortunit on measurement in its junior high school program; and someaspects of measurement are included, usually with goemetry, in itsprogram for grades four, five and six. While there is no available in-

716 School Science and Mathematics

formation on the effect of the elementary school treatment, it seemsprobable that children following such a program will gain some insightinto selected measurement concepts. Certainly, at the seventh gradelevel, Friebel [8] has provided convincing evidence that the SMSGunit on measurement produces a desirable result. Friebel studied 185seventh graders following "traditional" and SMSG programs. Hefound the SMSG students to be significantly superior to their ^tradi-tional" counterparts in their understanding of measurement conceptsand in their skill at handling situations involving measurement. Asthe School Mathematics Study Group and other mathematics proj-ects begin to offer materials with a greater applied bias, the contrastbetween the "measurement" groups and the "non-measurement"groups should be even more marked.

Yet, commercial publishers apparently have failed to see the virtuein measurement activities in the elementary school mathematics pro-gram. While they have adopted enthusiastically and elaborated uponmany of the "modern" topics treated by the curriculum projects, mea-surement has been given only perfunctory attention. An analysis ofavailable commercial textbooks in this field discloses either grossneglect of measurement or a treatment of only its most trivial aspects.Most of the concepts in the following list are completely ignored in allmaterials. The few that are given attention in one or two books aretreated so lightly that there can be no reasonable hope that a childwill appreciate or remember them.

(1) No measurement of a continuous variable is exact.(2) All measurements are comparisons.(3) Measurements have varying degrees of precision.(4) Objects may be compared with one another indirectly by comparing each

with a measurement unit.(5) The measurement unit may be arbitrarily chosen.(6) The measurement unit has the same dimension as the variable to be

measured.(7) Commonly accepted standard measurement units are desirable for com-

munication of measurements.(8) Measurements have varying degrees of accuracy, wliich is expressed as a

ratio between maximum potential variation and the total measurement.(9) Errors in measurement may be expressed as absolute errors or relative

errors.(10) Errors in measurement may be consistent errors or random errors.(11) The concept of significant figures is important in communicating precision

and accuracy of measurements.(12) Computations with denominate numbers must give consideration to the

error component.(13) Measurements may be made directly or indirectly.

Of course, not all of these concepts are susceptible to exhaustivetreatment at the elementary school level. However, it appears likelythat all the ideas can be understood to a limited extent by children ofelementary school age; and, more important, the ideas have value for


enriching many aspects of a utilitarian mathematics program. Ignor-ing the neglect of the fundamental ideas, even the superficial aspectsof measurement receive short shrift in textbook series. An analysis offour of the most widely used elementary school mathematics series [9]was undertaken to determine the proportion of problems’which in-volved measurement units in today’s conventional programs. The re-sults of this analysis are presented in Table I. With respect to thetabular data, any problem which involved a numeral with a measure-ment unit, excluding money, was included in the measurement classifi-cation. Money was excluded because it received separate treatment inthese programs, and because it was not treated as a continuous vari-able.

TABLE I. PERCENT OF PROBLEMS INVOLVING MEASUREMENT UNITS INFOUR WIDELY USED ELEMENTARY SCHOOL MATHEMATICS

TEXTBOOK SERIES

Percent Measurement�Excluding MoneyTextbook Grades

Series ��

123456

A 4.9 3.8 4.6 5.3 12.9 13.0B 8.9 6.8 4.5 5.1 12.1 16.8C 14.0 6.7 5.8 10.6 15.8 16.7D

.4.9 5.4 7.2 10.5 12.5 15.0

It may be seen that, as a general rule, the use of measurement unitsincreases with the progression through the grades (r==0.70). Yet theuse of measurement terms never exceeds 17% of the problems whichchildren encounter. In contrast, measurement units are involved inmost of the uses of mathematics. It is clear that children involved in aconventional arithmetic program are given relatively little opportun-ity to become familiar with measurement terms, much less with thenature of measurement.

Since there is the apparent tendency to incorporate a greater use ofmeasurement terms as the children grow older, designers of thesecurricula may well feel that there are inherent difficulties in the use ofcontinuous rather than discrete variables. The assumed effect of thesedifficulties upon children of elementary school age may dictate the ob-served avoidance of measurement in the school program for thesechildren. The concepts simply are reserved for educational levelswhere children are presumed to have the appropriate experience andsophistication to deal with them.

If such a position has merit, it should be revealed in children^problem-solving perfomance where both continuous and discrete vari-


ables are involved. Problems involving continuous variables shouldpresent much greater difficulty for children of elementary school age.This general theory was examined in the spring of 1964. A procedurewas developed to appraise children’s problem-solving performance onitems in t^o classes�non-measurement problems involving discretevariables and measurement problems involving continuous variables.Tests were devised for each of the grades three through six consistingof a series of "word problems" to be solved. The problems werecreated so as to be consistent in vocabulary and arithmetic complexitywith the conventional textbooks the children were using. They in-volved all of the basic operations and covered the range of conven-tional problem types. For example, subtraction problems called fortaking apart a collection as well as comparing two collections; divi-sion problems consisted of both comparison and partitioning types.The difficulty of the items was adjusted so as to be within the range ofcapability of average or slightly below average achievers at eachgrade level. Of the first sixteen items on the test, nine involved stan-dard measurement labels while seven did not. Using the four basictests (one for each grade) as models, a second form for each grade wasmade. Form II differed from Form I principally in the labels whichwere assigned to values. The nine items which were measurementproblems in Form I were converted into non-measurement problemsin Form II. The seven measurement items on Form II, were the non-measurement items from Form I. Care was taken to make no othersignificant changes. Exactly the same numerical values were used andvocabulary was carefully screened to insure that the problem re-mained reasonable and understandable to children. For example, anon-measurement item used in the third grade test was, "Dan had 7sea shells in a box. Mother gave him 3 more shells to put in it. ThenDan had how many shells in the box?" The measurement counterpartof this item was, "Ann took 7 minutes to write a note to her aunt and3 minutes to address the envelope. How long did it take her alto-gether?" Thus, this part of the testing program consisted of thirty-two items arranged into sixteen pairs. In each pair was a measure-ment and a non-measurement item. In order to assure that the changein wording had no constant effect upon item difficulty for the pairs,the two test forms were compared. The significance of differences be-tween means in the forms was determined through use of the / test forcorrelated means, except for the third grade tests. The distribution ofscores in grade three was sufficiently skewed so as to require analysisby the non-parametric Wilcoxon Test for Two Matched Samples.With order of presentation of the forms randomized, no significantdifferences were found between children’s mean scores on Form I andForm II at any grade level, and the forms were adjudged equivalent


with respect to difficulty. -It also is comforting to note that the effectof problem wording would be approximately random with respect tomeasurement and non-measurement if one assumed that the firstpreparation of the problem in each case was the more natural.Selected teachers from the University of California LaboratorySchools examined the tests for appropriateness and five children ineach of the grades three through six were given the tests in a pilottrial. They were modified in response to the suggestions of teachersand the performance of the children.The tests were then administered to 662 children in the three Uni-

versity of California Laboratory Schools in Berkeley, California, intwo sessions during May 1964 [10]. The three laboratory schools serveattendance areas which immediately surround them, and represent abroad range of social class, although none of the schools draws fromthe extremes of the socio-economic spectrum. Of the total number ofchildren tested, 159 were in the third grade, 156 were in the fourthgrade, 198 were in the fifth grade, and 149 were in the sixth grade. Atthe first testing a random selection of approximately half the childrenat each grade level received Form I of the test and the balance re-ceived Form II. At the second testing session, which followed the firstsession by three weeks, each child was given the appropriate alternatetest form. This distribution pattern was followed in order to equatethe effect of order of presentation for measurement and non-measure-ment problems. Credit was given for correct numerical solutions toproblems without regard for the manner of derivation, etc. Analysismerely involved the determination of the significance of the differencebetween the mean of measurement items and the mean of non-mea-surement items for the total group of children. The analysis was per-formed through use of the / test for correlated means. The results ofthe analysis are presented in Table II.

TABLE II. MEAN SCORES ON MEASUREMENT AND NON-MEASUREMENTPROBLEMS TOR VARIOUS GROUPS OF CHILDREN

MeasurementNon-MeasurementGrade-

Mean Score Mean Mean Score Mean(possible =16) Percent (possible= 16) Percent v

315911.4371.411.9774.81.10> 0.05415614.3589.714.5290.80.73> 0.05519813.0781.713.3883.61.01> 0.05614914.5891.114.7492.10.70> 0.05

Total 662 13.31 83.2 13.64 85.3 5.83 < 0.05Group


It may be noted that the subanalyses by grade level do not revealstatistically significant differences between the means of measure-ment and discrete items. On the basis of these subanalyses it is in-appropriate to reject the null hypothesis that children will show thesame performance on these problems regardless of the context of thenumbers. However, it is interesting that the mean differences are sys-tematic, all favoring the non-measurement problems. When the en-tire group is analyzed the effect of these systematic differences isnoticed. The hypothesis that the children perform equally well on thetwo types of problems must be rejected at the 0.05 confidence level.Yet the difference between the means is hardly of the order one

would expect if the problems involving measurement terms were in-herently more difficult. It is possible that the small difference isattributable to the documented neglect of measurement experiencerather than to intrinsic difficulty. In any event, the evidence derivedfrom this comparison does not offer much support to curriculumtheorists who w^uld advocate a preservation of the status quo on thebasis that problems involving measurement units cannot be managedby young children.

Because the children^ performance on the test was generally verygood, it may be possible that appropriate opportunity was not pro-vided for the impact of the measurement situation. That is, perhapsaccomplishment on measurement problems contrasts sharply withaccomplishment on non-measurement problems only when the requirednumerical task is near the limit of the child’s capacity. In order totest this notion, separate analysis was made of all children who re-ceived scores in the range 6-10 (37.5% to 62.5% correct) on the non-measurement items. For this group of children the problems offeredchallenge in their own right for reasons which are indeterminate. Anyadditional challenge contributed by the measurement situation, ifsuch situation is more difficult, should thus be clearly revealed. As amatter of fact, the mean score on measurement items was slightlyhigher for this group (8.01 vs. 7.81 out of the total of 16). Clearly, themeasurement situation does not materially effect this group of mid-range achievers.Both Form I and Form II of the test also included four items which

involved obsolete or contrived measurement units rather than themore familiar standard units. Such terms as cubits, sarfs, and mogswere included where feet, pounds, and quarts would be used normally.In addition, in a third testing session, the children were given fouritems involving metric units. The problems involving contrived ormetric units were designed to be of comparable numerical difficulty topreceding items, but did not involve matched numbers. It might be


expected that the introduction of these even less familiar terms wouldfurther handicap the problem solving performance, but as shown inTable III, such is not the case. The children’s proficiency (mean

TABLE III. CHILDREN^ PROBLEM-SOLVING PROFICIENCY ON ITEMSIN VARIOUS CATEGORIES

Grade

3456

Total

Non-measurement(16 items)

74.890.883.692.1

85.3

Percent

StandardMeasurement(16 items)

71.489.781.791.1

83.2

Correct

ContrivedMeasurement

(8 items)

70.889.182.090.6

83.0

MetricMeasurement

(4 items)

71.488.881.091.0

82.9

number correct/number possible) on these items closely approximatesthe proficiency on problems involving the standard measurementunits, and maintains the same relationship to the proficiency on thenon-measurement problems. Even without the more familiar mea-surement units, children show comparable adaptability to the rudi-mentary measurement situation and the non-measurement situation.In short, there are indications that the present program on measure-ment in elementary school mathematics should not continue to avoidmeasurement, if such avoidance is based on the assumption of unusualor extreme difficulty.With all the emphasis upon improving children’s understanding of

mathematics, it is surprising that greater concern has not been regis-tered for the documented neglect of measurement at the elementaryschool level. Piaget [11] has found the general concepts of measure-ment to be attainable for children beyond approximately eight yearsof age. When the child appreciates that a linear segment may be con-served and that subdivision of the segment is possible without de-struction of its totality, he is ready to learn measurement, accordingto Piaget. Yet substantial treatment of measurement continues toappear only at levels beyond theelementary school. Even though someof the experimental programs have made overtures in the direction ofstrengthening this important aspect of the curriculum, the conven-tional elementary school program has yet to reap the benefit. Further-more, the unpopularity of word problems in new curricula [12] mayserve to handicap the proper teaching of measurement for some timeto come. Since children can deal successfully with measurement


situations, there is little defense for a program which fails to treat themeaningful components of measurement or thwarts the emergence ofthis topic to prominence because of unrelated considerations. The im-portance of measurement to mathematics and to human affairs ingeneral is so well established that further neglect of measurement andrelated applied problems in the elementary school program may beattributed only to lack of perspective or indifference.

NOTES AND REFERENCES[I] WILSON, GUY M., AND CASSELL, MABEL. "A Research on Weights and

Measures," Journal of Educational Research, Vol. 46 (April 1953), pp. 575-85.[2] WILSON, DOROTHY M. "Teaching Denominate Numbers," Educational

Methods, Vol. 16 (January 1937), p. 177.[3] GUNDERSON, AGNES C. "Measurement in Arithmetic Textbooks for Grade

Three," The Mathematics Teacher, Vol. 35 (March 1942), pp. 117-21.[4] CORLE, CLYDE G. "A Study of the Quantitative Values of Fifth and Sixth

Grade Pupils," The Arithmetic Teacher, Vol. 7 (November 1960), pp. 333-40.[5] AHLFORS, L. V. AND SIXTY-FOUR OTHERS. "On the Mathematics Curriculum

of the High School," American Mathematical Monthly, Vol. 69 (1962), pp.189-93.

[6] American Association for the Advancement of Science and University ofMaryland Science Teaching Center. Third Report of the Information Clearing-house on New Science and Mathematics Curricula. College Park, Maryland,March, 1965.

[7] NEWPORT, JOHN F. "A Second Revolution," The Arithmetic Teacher, Vol. 12,(April 1965), pp. 253-5.

[8] FRIEBEL, A. C. "A Comparative Study of Achievement and Understandingof Measurement Among Students Enrolled in Traditional and ModernSchool Mathematics Programs," Dissertation Abstracts, Vol. 26 (August1965), pp. 903-4.

[9] Textbook series analyzed were those published by:Ginn and Company (1961)Holt, Rinehart and Winston, Inc. (1957)Scott, Foresman and Co. (1961)Silver Burdett Co. (1963)

[10] The teachers in the Hillside, Washington, and Whittier Laboratory Schoolsof the University of California contributed materially to the study reportedhere. The writer wishes to express his sincere appreciation to each of them.

[II] PIAGET, JEAN. The Child’s Conception of Number, (translated by C. Gat-tegno and F. M. Hodgson). London: Routledge and K. Paul, 1952. For asummary of Piaget’s observations with respect to children^ development ofmeasurement concepts see Coxford, Arthur F. "Piaget: Number and Mea-surement," The Arithmetic Teacher, Vol. 10 (November 1963), pp. 419-27.

[121 For a discussion of the role of word problems in the new curricula see VanEngen, Henry. "The Reform Movement in Arithmetic and the Verbal Prob-lem," The Arithmetic Teacher, Vol. 10 (January 1963), pp. 3-6.

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A Study of the Case for Measurement in Elementary School Mathematics