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ABSTRACT

DOCUMENT RESUME

24

Austin, Gilbert P.A Longitudinal Pivaluation of MathematicalComputational Abilities of Mew Hampshire's PighthGraders: 1063-1967, Final Pe2ort.New Hampshire Univ., Durham.Office of. Education (DHEW) , Washington, D.C. Bureauof Research.BR-9-A-02315 Aug 690EG-1-9-090023-0106(010)110p.

PDPS Price 1F-$0.0 HC-`5.60*Evaluation, Grade A, Grade 10, *LongitudinalStudies, Mathematics, *Pesearch, ,Secondary SchoolMathematics, *Student Characteristics, TextbookContent

The purpose of this study was to evaluate theeffects of using different mathematics textbooks on the mathematicalcomputational ability of students as a method of assessing theeffectiveness of different mathematics instruction. This studyresulted from a 1963 report which discussed the results of the mewHampshire Statewide 7ighth Grade mesing Program and the observationthat a significant drop in the arithmetic computation scores occurredin 1964 and 1065. A study of the data collected in 1967 involvedthree different phases. The results of phase one indicated that theintroduction of modern mathematics is somewhat responsible for thedecline in computational ability. The second phase compared thearithmetic computational ability of 196E eighth graders with 1067tenth graders. The results suggested no significant differences incomputational ability in grade ten between traditional, transitional,and modern groups. The third phase involved a select group of tenthgraders and their abilities in algebra and geometry. The conclusionswere that students who had studied either a modern or a transitionaltextbook did markedly superior work to those who had used only, atraditional textbook. (FL)

DEPARTMENT OF HEALTH, EDUCATION 8 WELFARE

OFFICE OF EDUCATION

THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE

PERSON OR ORGANIZATION ORIGINATING IT. POINTS OF VIEW OR OPINIONS

STATED DO HOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION

POSITION OR POLICY.

re\ FINAL REPOArC.) Project No. 9-A-023

Grant No. 0EG-1-9-090023-0106(010)

L.1.1

A LONGITUDINAL EVALUATION OF MATHEMATICAL COMPUTATIONALABILITIES OF NEW HAMPSHIRE'S EIGHTH GRADERS: 1963-1967

Gilbert R. AustinUniversity of New HampshireDurham, New Hampshire 03824

August 15, 1969

U.S. Department of Health,Education and Welfare

Office of Education

Bureau of Research

0

U.S.

UI

SUMMARY

A LONGITUDINAL EVALUATION.OF MATHEMATICAL COMPUTATIONALABILITIES OF NEW HAMPSHIRE'S EIGHTH GRADERS 1963-67

The intent of this study was to assess the impact of the introductionof modern mathematics text books on the computational ability of stu-dents in New Hampshire. In 1963 a report discussing the results ofthe New Hampshire Statewide Eighth Grade Testing Program, writtenby Dr. Walter N. Durost, indicated the mean raw score in the :a tatsof New Hampshire in arithmetic computation was 34 raw score paints,and its grade equivalent was 8.8. This was consistent with a pat-tern that had been established for some years. New Hampshire hadalways done well in the area of arithmetic computation.

In 1964 the pattern changed markedly. It became apparent that therehad been a significant drop in the arithmetic computation scoressince 1963. This drop continued into 1965 when the first studyauthorized by the New Hampshire State Department of Education wasconducted by the Bureau of Educational Research and Testing Services.The results of that study clearly indicated, for the three groupsdesignated as modern, traditional, and transitional, that in intel-lectual skills as measured by the Otis Intelligence Test, the moderngroup was clearly superior, followed in order by the group classifiedas being transitional, and the group classified as being traditional.On mathematics computation the exact inverse was true. The traditionalgroup scored highest, followed by the transitional group, followed bythe modern group.

The first phase of this study of the 1967 data was a replication ofthat earlier 1965 study. Tha results clearly indicate that againthe group classified as modern was intellectually superior to thegroups classified as traditional or transitional, as measured by theSchool and College Ability Test. The computation sub-test of theStanford indicated the modern group performs no better than thetransitional group. It needs to be pointed out, however, that thescores obtained by each of these groups on the Stanford ArithmeticComputation sub-test were markedly lower than they were in 1965and still lower than they were in 1963.. In 1963, using the Metro-politan in a fall administration, the grade equivalent was 8.8.In 1967, using the Stanford in a fall administration, the gradeequivalent was 6.8. This means that in five years, 1963-1967, therehad been a two-year decline in mathematics computation abilities inNew Hampshire. It is not possible to conclude from this study thatthis is a direct result of the kind of text books used, but it isthe author's opinion that the introduction of modern mathematics isat least somewhat responsible for the decline in computational ability.

It is believed by the author that even in traditional and trans4-tional schools, so designated by their text books, there has beenincreasing emphasis on understanding with a corresponding decreasein the time allocated to practice of meaningful drills in arithmeticcomputation.

A second phase of this study was to compare the performance of com-munity (1965) eighth grade means with community (1967) tenth grademeans to make comparisons of arithmetic computational ability betweeneighth and tenth grade. The findings clearly indicate that althoughthere were differences depending upon the kind of text book used ateighth grade, this is no longer true"at tenth grade. There are nosignificant differences in mathematical computational ability atgrade ten between these three groups as measured by the StanfordNumerical Competency sub-test. There are, however, interesting andprovocative differences using the Stanford High School Mathematicssub-test, Part A, which measures Algebra and Geometry. We find veryreal and important differences in students' abilities dependent uponthe kind of text book thei have been exposed to.

The third phase of this particular study was to take a select sub-group of tenth graders and compare their abilities in the area ofAlgebra and Geometry, still controlling for the typb of text bookthat they had used in their previous years of school. We findclearly that students who have studied in either a modern or a trans-itional text book do markedly superior work to those who have studiedusing only a traditional text book.

Results of these three studies in mathematics computational abilitiesin the state of New Hampshire would seem to indicate the following:There has been a serious decline in mathematics computational abilityat grade eight in the state of New Hampshire. By grade ten thisdifference is no longer statistically significant. It also becomesapparent that modern mathematics begins to pay its greatest dividendswhen students are being expected and requested to master the conceptsassociated with Algebra and Geometry. This, it seems to the author,provides information on which Superintendents, Principals, and cur-riculum workers in the area of mathematics might make important cur-riculum decisions. The implications of these studies argue cogently,the author believes, for the use of differentiated text books inschool systems.

FINAL REPORTProject No. 9-A-023

Grant No. OEG-1-9-090023-0106(010)

A Longitudinal Evaluation of Mathematical ComputationalAbilities of New Hampshire's Eighth Graders: 1963-1967

Gilbert R. AustinUniversity of New HampshireDurham, New Hampshire 03824

August 15, 1969

The research reportod herein was performed pursuant to a grant with theOffice of Education, U.S. Department of Health, Education and Welfare.Contractors undertaking such projects under Government sponsorship areencouraged to express freely their professional judgment in the conductof the project. Points of view or opinions stated do not, therefore,necessarily represent official Office of Education position or policy.

U.S. Department of Health,Education and Welfare

Office of Education

Bureau of Research

Acknowledgement

The evaluation reported herein, under the direction of Gilbert R. Austin,could not have been completed without the cooperation of many other persons.

The author gratefully acknowledges the support and assistance of thefollowing individuals who contributed to the report and who gave manyhelpful suggestions:

Fernand PrevostDirector, Mathematics EducationNew Hampshire State Department of EducationConcord, New Hampshire

Donald RandallNew England Educational Assessment ProjectNew Hampshire State Department of EducationConcord, New Hampshire

George SpoonerDepartment of MathematicsCentral Connecticut State CollegeNew Britain, Connecticut

Walter N. DurostDepartment of EducationUniversity of New HampshireDurham, New Hampshire

Robert Hart

Bureau of Educational Research and Testing ServicesUniversity of New HampshireDurham, New Hampshire

Donald Bailey

Bureau of Educational Research and Testing ServicesUniversity of New HampshireDurham, New Hampshire

Grateful acknowledgement is also expressed to Mrs. June Hall for herassistance in typing the rough draft and final copy and for constructingthe tables.

Criticisms and constructive suggestions should be addressed to the projectdirector, as he assumes responsibility for the contents of the final report.

A LONGITUDINAL EVALUATION OF MATHEMATICAL COMPUTATIONAL ABILITIES

OF NEW HAMPSHIRE'S EIGHTH GRADERS (1963) AND EIGHTH GRADERS (1967)

Table

1

2

3

4

5

6

7

8

9

10

List of Tables

Title Page

A Comparison of Means For The Otis Quick-ScoringMental Abilities Test: Gamma 4

A Comparison of Means For The Metropolitan AchievementTest: Computation 5

A Comparison of Means For The Metropolitan AchievementTest: Mathematical Concepts

A Comparison of School and College Ability Test

A Comparison of Stanford Paragraph Meaning, Spelling andLanguage Tests 11

A Comparison of Stanford Arithrettr: Computation, Concepts

and Applications Tests -.....12

A Comparison of Stanford Social Studies and Science Tests..13

6

10

A Comparison of Selected Percentiles forCollege Ability Teot

A Comparison of Selected Percentiles forParagraph Meaning, Spelling, Language

School and

Stanford

A Comparison of Selected Percentiles for StanfordArithmetic: Computation, Concepts and Applications

Tests

11 A Comparison of Selected Percentiles for Stanford SocialStudies and Science Test

ii

15

16

17

18

List of Normal Percentile Charts

Chart Title Page

I SCAT Verbal 19

II SCAT Quantitative 20

III SCAT Total 21

IV Stanford Paragraph Meaning 22

V Stanford Spelling 23

VI Stanford Language 24

VII Stanford Computation 25

VIII Stanford Concepts 26

IX Stanford Applications 27

X Stanford Social Studies 28

XI Stanford Science 29

iii

In the fall of 1963, under the direction of Dr. Walter N. Durost, the

Test Service and Advisement Center conducted the sixth consecutive yearly

statewide testing program at grade eight for the New Hampshire State

Department of Education. The program consisted of a mental abilities

test, the Otis Quick-Scoring Test, Gamma, Form Fm and the Metropolitan

Advanced Battery, Form Bm.

The research reported in this paper was cGAucted with the sponsorship of

the New Hampshire State Department of Education and a grant from the United

States Office of Education, grant number OEG-1-9-090023-0106 (010). The

purpose of the research was to evaluate empirically the effects of using

different mathematics text books on the mathematical computational ability

of students as a method of assiassing the effectiveness of different mathe-

matics instruction, where it is assumed that the choice of text reflects

methods of teaching.

In his 1963 report, Dr. Durost indicates that the median raw score for the

state of New Hampshire in arithmetic computation was 34 raw score points.

The equivalent standard score was 55 and the grade equivalent was 8.8.

This was consistent with a pattern that had been established for some years.

In 1961, for instance, the median grade equivalent was 8.7 and in 1962 it

was 8.7, so I believe it is safe to say that for a period of years New

Hampshire, in its 8th grade testing program, which was conducted in November,

was rather markedly above the national norm in terms of its achievement in

the area of arithmetic computation.

In 1964 the pattern changed markedly. The median raw score for New Hampshire

dropped to 31 raw score points, providing a standard score of 52 and a grade

equivalent of 8.3. In his 1964 report, Dr. Durost makes the following state-

ment:"It is suspected that the adoption of the new curriculumin mathematics in New Hampshire may have resulted in the

drop in Arithmetic Computation. This influence has been

noted in other studies in communities where data areavailable over a period of years and where the new curriculum

has recently been introduced."1

In 1965 the 8th grade testing program was conducted by the Bureau of

Educational Research and Testing Services at the University of New Hampshire.

The state report of that year indicates that the median raw score for the

state of New Hampshire in the arithmetic computation subtest of the Metro-

politan was 30. This equalled a grade equivalent of 8.1 which indicated

a continuing drop in New Hampshire's arithmetic compv.tational ability. As

a result of that finding the State Department of Education authorized the

Bureau to conduct a study which would attempt to determine whether in fact

the introduction of modern mathematics was having a detrimental effect on

the computational ability of New Hampshire's 8th graders. The study was

conducted in the following manner: in 1965 the total number of students

tested in the state was 4,724. Of this number, 4,182 were included in the

study. The 8th grade classes which participated in this voluntary statewide

program were placed in one of four categories which were designated as follows:

modern, traditional, transitional and other. 542 students were eliminatedfrom the study by being placed in the category called other. Placementin these three groups was primarily done on the basis of the text the schoolsystem had been using for three years previous to the 1965 eighth gradeyear. In other words, the texts the student used in grades 5,6, and 7were identified as being either traditional, transitional or modern. Theassignment of the texts and the school systems into one of these four groupswas done by Mr. Fernand Prevost, Director of Mathematics Education, NewHampshire State Department of Education. This classification is, at best,a very subjetive one but the following have been used as working definitionsfor this study of modern, traditional and transitional mathematics:

Working Definitions for Classifying Schools Based on Texts

If the mathematics text used by the school showed no de-viation from methods of presentation common in the late1950's or early 1960's, and introduced a minimal amountof new math, it was judged to be traditional. Such textswere more frequently filled with long exercise sections;little structure or rationale in concept development wasemphasized.

Texts which tended to approximate the California stranddevelopment were judged to be modern. Such texts placedstress on the development of concepts and concrete manip-ulations. Texts emphasizing mathematical systems,properties, functions and graphing, for example, met thecriteria for modern.

Those texts which the publisher had admitted, or whichMr. Prevost judged, to have a middle of the road approachwere considered transitional. These texts were somewherealong the continuum of traditional to modern.

Where a school system did not fit into a category it waseliminated from the study.

Using the raw scores on the Otis-Ganma intelligence tests, a one-way analysisof variance was computed loolcing for differences among these three groups.(Ferguson, 1959). This analysis indicated there was a significant differencein mean raw scores among the three groups. The computed F was 14.81 whichis significant beyond the .01 level. Following the analysis of variance, Ttests were run among the three groups on their intelligence scores. Theresults of this analysis indicated there was a significant difference inintelligence beyond the .01 level between students in the modern mathematicsgroup and those in the traditional mathematics group, favoring the moderngroup. It was found that there was a significant difference at the .01 levelbetween those students studying mouern mathematics and those studyingtransitional mathematics, again favoring the modern group. It was found therewas a significant difference at the .05 level between those students studyingtraditional mathematics and those studying transitional mathematics, favoring

the transitional group. The means, as well as the computed F's and T's,are given in Table 1.

The same procedure was followed in looking for significant differences amongthe three groups in the area of mathematics, as measured by the MetropolitanAchievement Test, using the computation and concepts subtests of the battery.The analysis of the computation scores on the Metropolitan Achievement Testindicated there was a significant difference among the means for computa-tional abilities of the three groups. The computed F was 6.87 which issignificant beyond the .01 level.

Following the analysis of variance, T tests were run between the three groupson their computation scores. There was a significant difference between themodern mathematics group and the traditional group which was significant atthe .01 level. There was a significant difference between the modern math-ematics students and the transitional ctudents at the .05 level. There was asignificant difference between the Traditional students and the transitionalstudents which was significant at the .05 level. The means as well as thecomputed F's and T's are given in Table 2.

The data from the Metropolitan Achievement Test, subtest Mathematical Concepts,was also analyzed but no significant difference was found among the threegroups. The reported F is .99. The means as well as the computed F are givenin Table 3.

The results of these two analyses indicate that there was a significantdifference among the three groups based on their IQ. The difference favoredthe students studying modern mathematics, followed by those studying transi-tional mathematics, followed by those studying traditional mathematics. Whenone looks at the differences in computational ability one finds here, too,there is a significant difference, only in reverse. The students who havestudied traditional mathematics did significantly better than those who studiedtransitional mathematics and those who studied modern mathematics. Those whostudied transitional mathematics did better than those who studied modernmathematics.

Table 1

A COMPARISON OF MEANS FOR THE OTIS QUICK-SCORING MENTAL ABILITIES TEST:GAMMA

Fall 1965

MODERN TRADITION' TRANSITIONAL

Number of Students 1215 591 2376

OTIS Means 36.69 33.59 34.70

Analysis of Variance

F = 14.81

.01 level of significance 4.60

T Tests Modern : Modern :

Trad. Trans.

Traditional :

Transitional

4.96 4.27 2.15

Significant .01 .01 .05

Table 2

A COMPARISON OF MEANS FOR THE METROPOLITAN ACHIEVEMENT TEST: COMPUTATION

Fall 1965

MODERN TRADITIONAL TRMjSITIONAL


COMPUTATION Means 28.56 30.08 29.22


F = 6.87



Trad. Trans.

Traditional :

Transitional

3.65 2.26 2.26


Table 3

A COMPARISON OF MEANS FOR THE METROPOLITAN ACHIEVEMENT TEST:MATHEMATICAL CONCEPTS

Fall 1965

Number of Students

MODERN TRADITIONAL TRANSITLQUL

1215 591 2376

MATHEMATICAL CONCEPTS Means 29.17 29.34 28.87


F = .99

Not Significant

In the fall of 1967 the statewide eighth grade testing program was again

conducted. This is the tenth consecutive year of this program and the fifthyear of this particular study. The tests used in the 1967 testing at grade

eight were the following: The School and College Ability Test, Form 3B andthe Stanford Achievement Test, Advanced Form W. The tests used in theeighth grade testing program had been changed subsequent to the 1965 testingbecause it was felt that the Stanford Achievement Test might more adequatelymeasure students' ability in arithmetic computation. It was a much morerecently nonmed test than the Metropolitan and therefore hopefully wouldreflect more of the "modern Mathematics material" and thereby give a morehonest picture of what New Hampshire's students' abilities were, in the area

of mathematical computation.

It is noted in the 1967 report of the eighth grade testing program, editedby Dr. Gilbert Austin, that mathematics computation has continued to drop.In 1966, using the Stanford, the median grade equivalent had dropped to 7.8.In 1967, again using the Stanford, it had dropped to 6.8. Because of this

very significant drop, in terms of grade equivalents, from 1963 - 1967,

(the median grade equivalent, in 1963, using the Metropolitan, being 8.8;the median grade equivalent, in /967, using the Stanford, being 6.8--atwo-year drop in terms of grade equivalents--) it was decided by the authorthat a replication of the 1965 study would be appropriate to see whetherthere still were significant differences in the intellectual ability as wellas the computational ability of students in grade 8, again classified by the

kind of textbooks they were using. It should be noted here, in terms of thegrade equivalents just mentioned above, that it is not Truly possible tocompare grade equivalents based on the Metropolitan with grade equivalentsbased on the Stanford since there are serious norming problems in terms ofthe two different tests. The technical supplement provided by Harcourt,Brace and World, equating these two tests in terms of grade equivalents,provides the following data: the 1963 grade equivalent of 8.8 on the Met-ropolitan is equated with an 8.6 grade equivalent on the Stanford. This is

done using the 1963 data. If one uses the 1967 data and goes in the oppositedirection and takes t'7, tanford Achievement Test grade equivalent in arith-metic computation whiqi is 6.8, and looks up the Metropolitan grade equivalent,one finds it at 7.5. Therefore we have two possible ways of viewing thisdata. One may subtract the 1967 equated Metropolitan score from the 1963Metropolitan grade equivalent; this subtraction, which is 8.8 (1963),minus 7.5, (1967) leaves, in the five year period, a 1.3 year grade equivalentdrop. If one uses Stanford to do the subtraction, then one finds he mustsubtract the 1967 median grade equivalent of the Stanford of 6.8 from the 1963Stanford grade equivalent of 8.6. This subtraction yields a drop in grade

equivalent scores of 2 years.

The following is a discussion of how the replication of the study was conducted.The principal investigator and Mr. Fernand Prevost, Mathematics Consultantfor the State Department of Education (New Hampshire) who had worked togetheron the original study in 1965, spent a considerable amount of time working onthe question of the classifying of the schools in this eighth grade popula-tion into four groups: traditional, transitional, modern and other. It was

finally decided that the best way to do this would be to create a questionnaire

-7-

for the schools covering the 5th, 6th and 7th grades. This questionnairewould list all of the presently available commonly used mathematics text-books. Mr. Prevost agreed to develop this mathematics text list from whicha questionnaire was created. A sample of this questionnaire may be foundin the appendix of this report. The questionnaire for grades 'five and sixincluded 17 texts; the questionnaire fcr grade seven included 13 texts.A total of 107 questionnaires were mailed to the school systems participatingin the 1967 grade 8 testing program. Of this number, 91 questionnaires werereturned. This represents a percentage return of 85%. Having collected thedata from the schools, information was compiled and sent to Mr. Prevost forfurther refinement and for the categorization of the school systems into oneof four groups. In assjgning school systems into one of the four groups, Mr.Prevost used essentially the definitions of modern, traditional and transitionalwhich were used in 1965 and which were reported earlier in this study. Whenthat selection was completed the data was statistically analyzed in..thefollowing manner: a complete analysis of variance was done across each of thethree groups using the School and College Ability Test and the StanfordAchievement Test. The following are the results of that analysis.

There was a significant difference among the three groups as measured by theSchool and College Ability Test on their verbal skills, F = 18.53; theirquantitative skills, 1' = 15.71; and their total score, F = 19.88. All of thesedifferences favored the modern group. Following this analysis a series of Ttests was conducted and it was found there was a significant difference betweenthe modern and traditional group and the modern and transitional group on allof the tests except the SCAT Quantitative, in which there was no differencebetween the modern and the traditional students. The differences betweenthe traditional and the transitional group were not significant. The means aswell as the computed F's and T's are given in Table 4.

There was a significant difference among the three groups as measured by thefirst three tests of the Stanford Achievement Test; these tests being Para-graph Meaning, Spelling and Language tests. The computed F's are: 20.50 forParagraph Meaning; 11.91 for Spelling; 13.98 for Language. The means forthese three subtests all favor the modern group. Following this analysis aseries of T tests was conducted and it was found that on Paragraph Meaning therewas a significant difference between the modera and traditional groups and themodern and transitional groups There was a non-significant difference betweenthe traditional and the transitional group.

On Stanford Spelling there was a non-significant difference between the modernand the traditional group, a significant difference between the modern andthe transitional group and a non-significant difference between the traditionaland the transitional group. On Stanford Language there was a significantdifference between the modern group and the traditioonal group and between themodern and the transitional group. There was a non-significant differencebetween the traditional and the transitional group. The means as well as thecomputed F's and T's are given in Table 5.

There was a significant difference among the three groups as measured by thethree Arithmetic subtests of the Stanford, these tests being Arithmetic

Computation, Arithmetic Concepts and Arithmetic Applications. The computedF for Arithmetic Computation is 11.78; for Concepts, it is 19.95 and forApplications it is 15.01. Following this analysis a series of T tests wasconducted and it was found that on Arithmetic Computation there was a non-significant difference between the modern and the traditional group, a sig-nificant difference between the modern and the transitional group and asignificant difference between the traditional and the transitional group.It should be noted hers: that this difference favored the modern group. Onthe Concepts subtest there was a significant difference between the modernand the traditional group and between the modern and the transitionalgroup. There was a non-significant difference between the traditional andthe transitional group. This difference favored the modern group. On theApplications subtest of the Stanford there was a non-significant differencebetween the modern and the traditional group, a significant differencebetween the modern and the transitional group and a non-significant differencebetween the traditional and the transitional group It should be noted thatthis difference favored the modern group. The means as well as the computedF's and T's are given in Table 6.

There was a significant difference among the three groups as measured by theStanford Social Studies Test and as measured by the Stanford Science Test.The F for Stanford Social Studies was 9.89 and for Stanford Science 12.81.Following this analysis a series of T tests was conducted and it was foundthat on Stanford Social Studies there was a significant difference betweenthe modern and traditional group and between the modern and the transitionalgroup. There was a non-significant difference between the traditional groupand the transitional group. On Stanford Science there was a significantdifference between the modern and the traditional group and the modern andthe transitional group. There was a non-significant difference between thetraditional and the transitional group. The difference in the means in allcases favored the modern group. The means as well as the computed F's and T'sare found in Table 7.

It is interesting to note that in the 1967 study of the eighth grade we findthat the pattern is totally consistent. The modern group is not only intel-lectually superior as it was in the 1965 group, but it is also academicallysuperior as measured by the 8 subtests of the Stanford Achievement Test. Ofparticular interest, it is academically superior in the area of mathematics,in all three cases, Computation, Concepts and Applications.

Table 4

A COMPARISON OF SCHOOL AND COLLEGE ABILITY TEST

GRADE 8: Fall 1967

elIeflaMMY

SCAT SCAT SCATI. Total

Modern MeanGroup 1 32.60 21.89 54.49

Traditional MeanGroup 2 30.06 21.21 51.27

Transitional MeanGroup 3 30.62 20.45 51.06

F's

Significant

18.528

.01

15.713

.01

19.875

.01

T's and Significance

1 : 2 = 4.358 .01 1.689 NS 3.599 .01

1 : 3 = 5.325 .01 5.607 .01 5.994 .01

2 : 3 = 0.940 NS 1.853 NS 0.222 NS

-10-

Table 5

A COMPARISON OF STANFORD PARAGRAPH MEANING, SPELLING AND LANGUAGE TESTS

GRADE 8: Fall 1967

StanfordSpelling

Stanford--------P.M.

StanfordLanguage




............

Significant

20.507

.01

11.914

.01

13.980

.01


1 2 = 4.151 .01 1.402 NS 2.114 .05

1 : 3 = 5.869 .01 4.881 .01 5.254 .01

2 . 3 = 0.396 MS 1.682 NS 1.219 NS

Table 6

A COMPARISON OF STANFORD ARCOMPUTATION, CONCEPTS AND APPL

ITHMETIC:ICATIONS TESTS

1967GRADE 8: F.4

StanfordComputation

StanfordConcepts

StanfordApplications


Traditional MeanGroup 2 1 8.75 18.80 14.65

WWISWMI


waMOVilr.M.I1.1.1,* VIM., reVearta

F's

Significant

11.775

.01

19.952

.01

5.012

.01

T's an Signifirince

1 : 2 =

1 : 3 =

2 : 3

0.633

4.737

2.376

NS

.01

.05

4.641

5.448

1.134

.01

.01

NS

1.103

3.160

0.902

ITS

.01

NS

....wwwwwww.Ilww..11b7n.I.I.41....

-12--

o

Table 7

A COMPARISON OF STANFORD SOCIAL STUDIES AND SCIENCE TESTS

GRADE TplA 1967,

0+0.,**,.00wromowswoormft.

Stanford

Soc$4441.-alai2A-

StanfordScience,W1.60.10011 ,.* ...,e

Modern Mean

. .Group 1 48.16 33.02

.. .. .W ......./..

Traditional Mean

.,

Group 2 46.22 31.72... ...1111, 1.0

Transitional MeanGroup 3 46.38 31.60

....geoworwow

F's 9.894 12.813

Significant .01 .01

11prollooll.........


1 : 2 = 2.846 .01 2.827 .01

1 4 3 4.097 .01 4.839 .01

2 3 = 0.227 VS 0.255 US

-13-

In the mathematical development of the analysis of variance, a numberof assumptions are made. One assumption is that the distribution ofvariables and the polul tion from which the samples are drawn are nor-mal, Since this study is not based upon the drawing of a sample froma population, but is, in fact, a population itself, the use of analysisof variance, can seriously be questioned. Because of the failure tomeet this requirement, the project director, in consultation with otherstatisticians, decided that to pursue the project as originally pro-posed, the analysis of co-variance for the 1965-67 study would beinappropriate since the assumptions for simple analysis are not met.The assumptions are certainly not met for the analysis of co-variance;therefore, it was decided thilt the 1967 grade 8 data would be subjectedto further analysis by computing selected percentile ranks as a basisfor determining differential effects for above and below average stu-dents.

Five selected percentiles were chosen for this study: they are the90th, 75th, 50th, 25th, and 10th. The comparisons that are providedin the following tables indicate the raw scores at each of these se-lected percentiles for the students involved in the study. In the en-tire state in 1967 there were 7,139 students tested. In this studythere were 4,658. Of that number, 2,269 were classified as modern;514 were classified as traditional; and 1,875 were classified astransitional. Tables 8 - 11 present the comparisons for these threegroups in terms of selected percentiles. We have also prepared normalpercentile charts which visually present the same information. Thesewill be found in the following charts, I - XI.

Table 8 and Chart I clearly indicate that we are dealing with threedistinct populations, at least as measured by the SCAT Verbal portionof this test. Only at the 75th percentile do the traditional and trans-itional groups attain the same scores. Other than that, the moderngroup performance is superior to the transitional and the transitionalsuperior to the traditional group. On the SCAT Quantitative test thepicture is not quite as clear. At the 10th, 25th, and 50th percentilesthe modern group separates and becomes the superior group. The transi-tional group starts off more poorly than either the modern or thetraditional group and only at the 90ch percentile crosses the tradi-tional group's line and has the superior score. On the SCAT Totaltest the modern group is again clearly, different from the other twogroups. The traditional and transitional group overlap at the 25thand 75th percentiles. The results of this analysis on the School andCollege Ability Test would seem to clearly indicate that at all levelsthe modern group is superior intellectually to the other two groups andthat in general the transitional group is superior to the traditionalgroup.

On the first three tests of the Stanford Achievement Test, (ParagraphMeaning, Spelling and Language), the same pattern is consistently fol-lowed. The modern and the traditional group at the 10th percentile areequal. At the 25th percentile the traditional group is superior. Atthe 50th percentile and the 75th percentile they are similar. At the90th percentile the modern group is clearly superior while the tradi-

-14- (a)

tional group has now dropped below the standing of the transitionalgroup. The transitional group at the 10th, 25th, 50th, and 75th per-centile was poorer than either the modern or the traditional group andonly at the 90th percentile does it pass the traditional group. On theStanford Arithmetic Applications subtest there are no differencesamong any of the three groups at the 10th, 25th or 50th percentile.At the 75th percentile there is no difference between the modern andthe traditional group. However, at the 90th percentile the moderngroup is superior. At the 75th percentile the transitional group isperforming more poorly than either of the other two groups; at the90th percentile it is scoring equally well with the traditional group.

On the Social Studies and the Science subtests of the Stanford thepattern once again re-emerges of the modern group being markedly super-ior to the other groups. On the Social Studies subtest of the Stanfordthe modern group is superior at all percentile levels to the other twogroups and only at the 25th percentile do the traditional and the mod-ern groups perform equally well on the Science subtest. At all otherlevels the modern group is superior to the other two. Again, as hcsbeen the case in earlier subtests, the traditional and the transitionalgroups tend to overlap each other at various percentiles.

-14- (b)

Table 8

A COMPARISON OF SELECTED PERCENTILES FOR SCHOOL AND COLLEGE ABILITY TEST

GRADE 8: Fall 1967

NAME OF TEST SELECTED PERCENTILES IN RAW SCORES

SCAT Verbal

10th 25th 50th 75th 90th

Modern Group 16 23 33 41 48Traditional Group 14 20 29 39 46Transitional Group 15 21 30 39 47

SCAT Quantitative

Modern Group 12 15 2.0 27 33Traditional Croup 12 15 20 26 31Transitional Group 10 14 19 25 32

SCAT Total


-15-

Table 9

A COMPARISON OF SELECTED PERCENTILES

FOR STANFORD PARAGRAPH

GRAD

MEANING, SPELLING, LANGUAGE

E 8: Fall 1967


STAITORDParagraph Meaning


Modern Group 18 25 34 43 49

Traditional Group 17 22 31 40 47

Transitional Group 16 23 31 40 47

STANFORDSpelling

Modern G roup 16 21 29 39 47

Traditi onal Group 15 21 28 37 46

Transi tional Group 14 19 27 37 45

STANFORDLanguage

Modern Group 68 81 97 111 120



-16-

Table 10

A COMPARISON OF SELECTED PERCENTILES FOR STANFORD ARITHMETIC:COMPUTATION, CONCEPTS AND APPLICATIONS TESTS

GRADE 8: Fall 1967


STANFORDComputation



STANFORDConcepts


STANFORDApplications


-17-

Table 11

A COMPARISON OF SELECTED PERCENTILESFOR STANFORD SOCIAL STUDIES AND SCIENCE TEST

GRADE 8: Fall 1967


STANFORDSocial Studies



STANFORDScience


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Conclusions

In 1964 it became apparent that there had been a significant drop in

the arithmetic computation scores since 1963. This drop continued

into 1965 when the first study authorized by the New Hampshire State

Department of Education was conducted by the Bureau of Educational

Research and Testing Services. The results of that study clearly

indicated that for the three groups designated as modern, traditional and

transitional, in intellectual skills, as measured by the Otis, the modern

group was clearly superior, followed in order by the group classified as

being transitional and the group classified as being traditional. On

Mathematics Computation the exact inverse was true. The traditional group

scored highest, followed by the transitional group, followed by the modern

group. In 1967 a replication of this study was undertaken by the Bureau

of Educational Rscarch and Testing Services. The results of that study

clearly indicate that again the group classified as modern is intellectually

superior to the groups classified as transitional and traditional as measured

by the School and College Ability Test. This finding is substantiated by

the same kinds of differences and in the same direction, based on the first

three tests of the Stanford Achievement battery, namely Paragraph Meaning,

Spelling and Language, This again clearly indicates that the modern group

is intellectually superior to the transitional or the traditional group.

On the Arithmetic subtests of the Stanford, (Computation) there seems to be

markedly less difference between the modern and the traditional group than

between either the modern and the transitional or the traditional and the

transitional group. The transitional group seems to be achieving much more

poorly than the other two groups. This is not true on the Arithmetic

Concepts subtest; there the modern group clearly does better work than

either the traditional or the transitional group. On the Stanford Arith-

metic Applications subtest there seems to be little difference among any of

the three groups at the lower selected percentiles. It is only at the

upper end of the continuum that there is any real difference and at that

point the modern group is clearly superior to the other two groups. The

Social Studies test and the Science test tend to confirm the fact that

the modern group is clearly superior to the traditional and transitional

groups as they were on the SCAT and the first three subtests of the Stanford.

What seems to be true, as a result of this study, is that we are dealing

with three significantly different populations, intellectually, and that

the group classified as modern is clearly superior to the other two.

In the area of Arithmetic Computation this difference does not hold up.

The modern and the traditional group both perform at superior levels to

the group classified as transitional. Perhaps this is caused by the fact

that students studying in a modern text or a traditional text are at least

being instructed in one systematic method while those students being

instructed with a transitional text are being somewhat confused by attempting

to understand modern mathematics and at the same time being taught

traditional mathematics. There does not seem to be any easy explanation for

this finding.

-30-

One should not lose sight, in this discussion, of the fact that scores

obtained by each of these groups on the Stanford Arithmetic Computation

subtest in 1967 are markedly lower than they were in 1965 and still

lower than in 1963. If we look up those median scores as grade equiv-

alents we find that the modern group and the traditional group have

median scores of 18 raw score points, which are equal to grade equiv-

alents of 7.2. The transitional group has a median score of 16 which

is equal to a grade equivalent of 6.6. This pattern is similar to one

we noted earlier in this paper. We can therefore safely conclude

that in five years the computational ability of the students at grade

8 in New Hampshire has markedly declined. We can also reasonably

safely conclude that this does nct seem to be a function of the kind

of textbook they are using in their classes, for this decline is al-

most equally great for all three groups involved in this study.

The data presented in this study clearly indicate that the type of

mathematics text book used does not differentially affect (in 1967,

at least) the ability of students to do computational arithmetic.

However, it is the author's opinion, based upon lengthy conversations

with Mr. Prevost of the State Department of Education as well as a

number of teachers in a variety of schools in New Hampshire, that most

teachers have in the last several years put more and more emphasis on

the understanding of arithmetic and less and less time on meaningful

drill and practice in the art of computation. The decline in mathe-

matics computational ability is probably due more to the broad and

generalized effect of the insistance on the part of mathematics spe-

cialists that understanding of mathematics be given a higher priority

than it has been given in the past. It is a general trend in mathe-

matics education, and the present preparation of teachers also con-

tributes in this particular direction. There, also, the general

understanding of mathematics has been stressed and so as new teachers

have moved out from the training institutions, they have gone out

with more concern to teach understandings and less concern with

teaching computation. As stated in the beginning of this paragraph,

these statements are not supported by hard data, neither are they

just ideal specualation. It is suggested that it would be an ap-

propriate kind of follow-up to this study to go and actually look at

the classroom behavior of a variety of teachers and attempt to assess

whether these statements are, in fact, true.

The original proposal for this study proposed that an item analysis

should be done on each of the three groups. Because data was avail-

able on students who were then in 10th grade, who had been tested in

1965 at eighth grade, it was decided by the project director that he

would forego the study of the item analysis and do, instead, two

other studies based on computational ability. They are reported un-

der the headings of A Longitudinal Study of Tenth Graders, 1965-1967

and A Study of Abilities and Achievements in Mathematics of Three

Groups of Tenth Graders in New Hampshire. The result of those two

studies seems to the project director to clearly indicate that this

was a wise choice.

-31-

APPENDIX

LIST FOR GRADE 5

SERIES NAME PUBLISHER

Elementary School Mathematics

Modern Mathematics Series

Discovering Mathematics

GCMP Math Program

Math Workshop for Children

Mathematics We Need

Growth in Arithmetic,Discovery Ed.

Elementary Mathematics

Modern School Mathematics

SRA Elementary Math Program

GCMP Math Program

Contemporary Mathematics

Seeing Through Arithmetic

Sets and Numbers

Modern Math Through Discovery

Elementary Mathematics:Concepts, Properties & Operations

SMSG Elementary Mathematics

Other (Please Specify)

Addison-Wesley

American Book Company

Charles Merrill Company

Ed. Res. Council of Greater *Cl.

Encyclopedia Britannica

Ginn and Company

Harcourt, Brace & World

Holt, Rinehart & Winston

Houghton, Mifflin Company

SRA

SRA

Sadlier

Scott, Foresman t. Company

Singer/Random House

Silver Burdett Company

Webster, McGraw-Hill

Yale University Press

This sheet prepared by

LIST FOR GRADE 6

SERIES NAME

Elementary School Mathematics

Modern Mathematics Series

Discovering Mathematics

GCMP Math Program

Math Workshop for Children

Mathematics We Need

Growth in Arithmetic,Discovery Ed.

Elementary Mathematics

Modern School Mathematics

SRA Elementary Math Program

GCMP Math Program

Contemporary Mathematics

Seeing Through Arithmetic

Sets and Numbers

Modern Math Through Discovery

Elementary Mathematics:Concepts, Properties & Operations

SMSG Elementary Mathematics


PUBLISHER

Addison-Wesley


Charles Merrill Company

Ed. Res. Council of GreaterC1.

Encyclopedia Britannica

Ginn and Company



Houghton, Mifflin Company

SRA

SRA

Sadlier

Scott, Foresman & Company

Singer/Random House


Webster, McGraw-Hill

Yale University Press


LIST FOR GRADE 7

SERIES NAME

Arithmetic Concepts and Skills

Basic Modern Mathematics

School Mathematics I

Structuring Mathematics

Mathematics We Need- J-1

Growth in Arithmetic Discovery,Ed. 7

Elementary Mathematics 7

Exploring Modern Math

Modern School Math -7

Math for Jr. High School, Vol. I

Contemporary Mathematics, 7

Seeing Through Mathematics I

Modern Math Through Discovery I


PUBLISHER

Addison-Wesley

Addison-Wesley

Addison-Wesley


Ginn & Company




Houghton Mifflin Company

SMSG-Yale Press

Sadlier

Scott, Foresman Company



Footnotes

1Durost, Walter N. Report and Summary. New EgazgAm StatewideGrade Eight Testing Program. Concord, New Hampshire: Test Serviceand Advisement Center, November, 1964, p.9.

References

BOOKS

Ferguson, George A. Statistical Analysis in Psychology and Education.

New York: McGraw-Hill, 1959.

REPORTS

Austin, Gilbert R. (ed.). New Hampshire Eighth Grade Testing Program.

Report for school year 1967-1968, sponsored by the New Hampshire

State Department of Education in cooperation with the Bureau of

Educational Research and Testing Services. University of New

Hampshire, 1967.

. (ed.). New Hampshire Eighth Grade Testing Program. Report for

school year 1966-1967, sponsored by the New Hampshire State

Department of Education in cooperation with the Bureau of Educational

Research and Testing Services. University of New Hampshire, Fall,

1966.

. (ed.). New Hampshire Eighth Grade TestingProgram. Report for



Research and Testing Services. University of New Hampshire, November,

1965.

Durost, Walter N. (ed.). Report and Summary. New Hampshire Statewide

Grade Eight Testing Program. Report prepared by the Test Service

and Advisement Center, Concord, New Hampshire, November, 1964.

. (ed.). Report and Summary. New Hampshire Statewide Grade Eight

Testing Program. Report prepared by the Test Service and Advisement

Center, Concord, New Hampshire, November, 1963.

Dyer, Henry S., Linn, Robert L., and Patton, Michael J. Methods of

Measuring School System Performance. Report to the New York State

Education Department. Prepared by Ed\cational Testing Service,

Princeton, New Jersey, August, 1968.

Hogan, Thomas P. Some Notes .fin the Performance of Pupils in Modern and

Traditional Mathematics Programs on Standardized Arith :.etic Tests.

Report prepared by Test Department: Harcourt, Brace and World.

References

OTHER SOURCES

Cooperative School and College Ability Tests. Princeton: Educational

Testing Service, 1956-57.

Derrick, Clarence, Harris, David P. and Walker, Biron. Cooperative

English Tests. Princeton: Educational Testing Service, 1960.

Durost, Walter N. (General Editor). Metropolitan Achievement Tests.High School Battery. New York: Harcourt, Brace and World, 1962.

Gardner, Eric F. et al. Stanford Achievement Test. New York:Harcourt, Brace and World,1964)

. Stanford High School Mathematics Test. New York: Harcourt,

Brace and World, 1964.

. Stanford High School Numerical Competence Test. New York:

Harcourt, Brace and World, 1964.

Kelley, Truman L. at al. Stanford Achievement Test. TechnicalSuml.ement. New York: Harcourt, Brace and World, 1966.

Otis, Arthur. Otis uick-Scoring, Mental Ability Tests. New York:


A LONGITUDINAL EVALUATION OF MATHEMATICAL COMPUTATIONAL ABILITIES

OF NEW HAMPSHIRVS EIGHTH GRADERS (1965) AND TENTH GRADERS (1967)

Table

1 A Comparison of Means For The Otis Quick-ScoringMen:A.1 Abilities Test: Gamma 4

2 A Comparison Of Means For The Metropolitan AchievementTest: Computation 5

3 A Comparison of Means For The Metropolitan achievementTest: Mathematical Concepts 6

4 A Comparison of School and College Ability Test 9

5 A Comparison of Cooperative English Test 10

6 A Comparison of Stanford Numerical Competence andMathematics A Tests 11

List of Tables

Title Page

7 A Comparison of Selected Percentiles For School andCollege Ability Test 13

8 A Comparison of Selected Percentiles for Cooperative..English Test 14

9 A Comparison of Selected Percentiles For StanfordNumerical Competence And Mathematics A Tests .15

10 A Comparison of Selected Percentiles For 10th GradeTesting Program: Entire State 16

iii


Chart Title Page

I SCAT Verbal 17

II SCAT Quantitative 18

III SCAT Total 19

IV Reading Vocabulary 20

V Reading Level 21

VI Reading Speed 22

VII _Reading Total 23

VIII English Expression 24

IX Numerical Competence, 25

X Mathematics A 26

In the fall of 1965 the Bureau of Educational Research and TestingServices at the University of New Hampshire conducted the eighthconsecutive yearly statewide testing program at grade eight for the NewHampshire State Department of Education. The program consisted of amental abilities test, the Otis form Fm and the Metropolitan AchievementTest, battery form Am.

The research reported in this paper was conducted with the sponsorshipof the New Hampshire State Department of Education and a grant from theUnited States Office of EduLation, Grant No. 0EG-1-9-090023-0106(010).The purpose of the research was to evaluate empirically the effects ofusing different mathematics text books on the math:matical computationalability of students as a method of assessing the effectiveness of dif-ferent mathematics instruction, based primarily on a text.

For about a year previous to 1965 there had been a growing level ofconcern about dropping mathematics computation scores as measured by theMetropolitan Achievement Test. In his 1964 report, Dr. Walter Durostsaid:

"It is suspected that the adoption of the new curriculumin mathematics in New Hampshire may have resulted in thedrop in Arithmetic Computation. This influence has beennoted in other studies in communities where data areavailable over a period of years and where the new curriculumhas recently been introduced." (Durost, 1964)

In 1965 the total number of students tested in the state was 4,724. Ofthis number, 4,182 were included in the study. The eighth grade classeswhich participated in this voluntary statewide testing program were placedin one of four categories which were designated as follows: modern,traditional, transitional and other. 5 42 students were eliminated fromthe study by being placed in the category called other. Placement inthese groups was done primarily on the basis of the text the schoolsystem had been using for three years previous to the 1965 eighth gradiyear. In other words, the texts the student used in grades 5,6 and 7were identified as being either traditional, transitional or modern. Theassignment of the texts and the school systems into one of these fourgroups was done by Mr. Fernand Prevost, Director of Mathematics Education,New Hampshire State Department of Education. This classification is, atbest, a very subjective one but the following have been used as workingdefinitions for this study of modern, traditional and transitional math-ematics:

Working Definitions for Classifying Schools Based on Texts

If the mathematics text used by the school showed no de-viation from methods of presentation common in the late1950's or early 1960's, and introduced a minimal amountof new math, it was judged to be traditional. Such textswere more frequently filled with long exercise sections;little structure or rationale in concept development wasemphasized.

-1-


Those texts which the publisher had admitted, or whichMr. Prevost judged, to have a middle of the road approachwere considered transitional, These texts were somewherealong the continuum of traditional to modern.

Where a school system did not fit into a category it waseliminated from the study.

Using the raw scores on the Otis-Gamma intelligence test, a one-wayanalysis of variance was computed looking for differences among thesethree groups. (Ferguson, 1959). This analysis indicated there was asignificant difference among the mean raw scores for these three groups.The computed F was 14.81 which is significant beyond the .01 level. Fol-lowing the analysis of variance, T tests were run among the three groupson their intelligence scores. The results of this analysis indicatedthere was a significan difference in intelligence beyond the .01 levelbetween students in the modern mathematics group and those in the tradi-tional mathematics group, favoring the modern group. It was found thatthere was a significant difference at the .01 level between those studentsstudying modern mathematics and those studying transitional mathematics,again favoring the modern group. It was found there was a significantdifference at the .05 level between those students studying traditionalmathematics and those studying transitional mathematics, favoring thetransitional group. The means, as well as the computed F's and T's, aregiven in Table 1.

The same procedure was followed in looking for significant differencesamong the three groups in the area of mathematics, as measured by theMetropolitan Achievement Test, using the Computation and Concepts subtestsof the battery. The analysis of the Computation scores on the MetropolitanAchievement Test indicated there was a significant difference among themean computational abilities for the three groups. The computed F was 6.87which is significant beyond the .01 level. Following the analysis ofvariance, T tests were run between the three groups on their computationscores. There was a significant difference between the modern mathematicsgroup and the traditional group which was significant at the .01 level.There was a significant difference between the modern mathematics studentsand the transitional students at the .05 level. There was a significantdifference between the traditional students and the transitional studentswhich was significant at the .05 level. The means as well as the computedF's and T's are given in Table 2.

The data from the Metropolitan Achievement Test, subtest MathematicalConcepts, was also analyzed but no significant difference was found among

-2-

the three groups. The reported F is .99. The means as well as thecomputed F are given in Table 3.

The results of these two analyses indicate that there was a significantdifference among the three groups based on their IQ. The differencefavored the students studying modern mathematics, followed by thosestudying transitional mathematics, followed by those studying traditionalmathematics. When one looks at the differences in computational abilityone finds here, too, there is a significant difference, only in reverse.The students who have studied traditional mathematics did significantlybetter than those who studied transitional mathematics and those whostudied modern mathematics. Those who studied transitional mathematicsdid better than those who studied modern mathematics.

-3-

Table 1

A COMPARISON OF MEANS FOR THE OTIS QUICK-SCORING MENTAL ABILITIES TEST:

GAMMA

Fall 1965

Number of Students

NOMA TMILOITALTIAEZMONAL.

1215 591 2376

OTIS Means 36.69 33.59 34.70


F = 14.81



Trad. Trans.

Traditional :Transitional

4.96 4.27 2.15


-4-

Table 2

A COMPARISON OF MEANS FOR THE METROPOLITAN ACHIEVEMENT TEST: COMPUTATION

Fall 1965


COMPUTATION Means 28.56 30.08 29.22


F = 6.87


T Tests

Significant

Modern : Modern :Trad. Trans.

3.65 2.26

TraditionalTransitional

2.26

.01 .05 .05

Table 3

A COMPARISON OF MEANS FOR THE METROPOLITAN ACHIEVEMENT TEST:

MATHEMATICAL CONCEPTS

Fall 1965

INDEOL---TEAMIZIAL- TRANSIMITAL


MATHEMATICAL CONCEPTS Means 29.17


F = .99

Not Significant

-6-

010m.114,.../0/0/...4 powam../.0

29.34 28.87

,01.1.Mmammow

In the fall of 1967 when these same students were now in 10th gradethey were involved in another statewide testing program. In 1967,9,776 10th graders participated in the statewide testing program. Ofthis number, 3,439 students were involved in the follow-up study. Itshould be noted that traditionally many more school systems participatedin the 10th grade testing program than in the 8th and this accounts forthe large discrepancy. It should also be noted that due to populationloss and the difficulty of classifying high schools there was a loss ofstudents between the original 8th grade population and the 10th gradepopulation.

The 10th grade battery consisted of the School and College Ability Test,Form 2B; the Cooperative English Test, Form 2C; the Stanford High SchoolNumerical Competence Test, Form X; the Stanford High School MathematicsTest, Form X, Part A and B. It was decided to use the same proceduresas had been used two years earlier, when the students were in the 8thgrade, to conduct the study.

Having divided the children into three groups, again classified as modern,traditional and transitional, the following analyses were conducted: acomplete analysis of variance was done across the three groups using theSchool and College Ability Test, the Cooperative English Test, the StanfordNumerical Competence Test and the Stanford High School Mathematics Test,Part A alAd B. The following are the results cf those computations.

There was a significant difference among the three groups as measured bythe Verbal portion of the School and College Ability Test; F m 9.6. Therewas no significant difference among the three groups as measured by theSchool and College Ability Test, Quantitative, F = 1.5. There was nosignificant difference among the three groups as measured by the Schooland College Ability Test, Total: F = 2.39. Following this analysis a seriesof T tests was conducted and it was found there was a significant differenceon Verbal skills between modern and traditional groups and the modern andtransitional groups. There was a non-significant difference between thetraditional and the transitional group. The means as well as the computpdF's and T's are given in Table 4.

There was a significant difference among the three groups as measured bythe Vocabulary portion of the Cooperative English Test; F = 5.132. Therewas no significant difference in Reading Level; F = 2.94. There was nosignificant difference in Reading Speed; F = 2.10. There was a significantdifference among the three groups in terms of English Expression; F = 5.57.Following this analysis, a series of T tests was conducted and it was foundthat on Reading Vocabulary there was a significant difference between modernand traditional groups and the modern and transitional gidaps. There was nosignificant difference between the traditional group and the transitionalgroup. There was no significant difference among the three groups on ReadingLevel and Reading Speed.

On Reading Total there was a significant difference between the modernand traditional group and between the modern and transitional group.There was a non-significant difference between the traditional and thetransitional group. On English Expresbion, there was a significantdifference between the modern and the traditional group, and themodern and transitional group. There was a non-significant differencebetween the traditional and transitional group. The means as well asthe computed F's and T's are given in Table 5.

An analysis of variance based on Numerical Competence was computed andit was found there was a non-significant difference. There was a very

significant difference at the .,01 level with an F of 19.58 for Math-

ematics subtest A of the Stanford. We find here that there is a signi-ficant difference between the modern and traditional group, a non-sig-nificant difference between modern and transitional and again a signifi-cant difference between transitional and traditional. The means as wellas the computed F's and T's are given in Table 6.

Table 4


Fall 1967

SCATVe

SCATta

SCAT



Transitional MeanGroup 3 30.36 29.18 5948

F's

Significant

9.560

.01

1.488

NS

2.385

NS


1 : 2 =

1 : 3 =

2 : 3 =

3.888

3.427

1.771

.01

.01

NS

0.641

1.722

0.505

NS

NS

NS

2.110

1.393

1.282

NS

NS

NS

Table 5

A COMPARISON OF COOPERATIVE ENGLISH TEST

Fall 1957

ReadingVocab

ReadingLevel

ReadingSeed

ReadingTo a..

EnglishE r ss

Modern MeanGroup 1 35.54 18.56 29.44 64.79 45.08

Traditional MeanGroup 2 34.04 18.22 28.47 62.50 42.88

1..emaTransitional MeanGroup 3 34.51 18.07 28.69 63.13 44.04

411101

F's 5.132 2.937 2.102 3.240 5,571

Significant .01 NS NS .05 .01


1 : 2 = 2.614 1.093 1.569 2.011 3.160

.01 NS NS .05 .01

1 3 = 2.766 2.426 1.852 2.254 2.300

.01 NS NS .05 .05

2 3 = 0.870 0.514 0.388 0.582 1.770

NS NS NS US NS

-10-

Table 6

A COMPARISON OF STANFORD NUMERICAL COMPETENCE AND MATHEMATICS A TESTS

11111}111101..1=1111.=110,1171...144..11...11.......11,11.../1.=1(0..111.1.1.,

Fall 1967

Numerical

--CMONtence

27.04

Mathematics

26.09Modern MeanGroup 1

Traditional MeanGroup 2 26.51 19.41


F's 1.047 19.578

Significant NS .01


1 : 2 1.068 NS 6.157 .01

1 : 3 = 1.351 NS 0.783 NS

2 : 3 . 0.183 NS 5.934 .01

-11-

In the mathematical development of the analysis of variance, a numberof assumptions are made. One assumption is that the distribution ofvariables and the population from which the samples are drawn are nor-mal. Since this study is not based upon the draw_ng of a sample froma populatioft, but is, in fact, a population itself, the use of analysis

of variance can seriously be questioned. Because of the failure tomeet this requirement, the project director, in consultation with otherstatisticians, decided that to pursue the project as originally pro-posed, the analysis of co-variance for the 1967 study would be inappro-priate, since the assumptions for simple analysis are not met. The

assumptions are certainly not met for the analysis of co-variance;therefore, it was decided that the 1967 grade 10 data would be subjec-ted to further analysis by computing selected percentile ranks as abasis for determining differential effects for above and below averagestudents.

Five selected percentile ranks were chosen for this study: they arethe 90th, 75th, 50th, 25th, and 10th. The comparisons that are pro-vided indicate the raw scores at these selected percentiles for thestudents involved in the study, taking the test at grade 10 (3,439students). We have also prepared normal percentile charts which visu-ally present the same information. Shown are the raw scores for themodern group which numbered 1,107 students; for the traditional group,which numbered 404 students; and for the transitional group, which num-bered 1,928 students. These comparisons, as well as comparisons forthe entire state, may be studied in Tables 7-10 and normal percentilecharts I - X.

The modern mathematics group on SCAT Verbal seems to do markedly betterthan either of its two comparable groups at the 50th, 75th and 90thpercentiles. On the Quantitative subtest there seems to be little dif-ference at the upper percentile levels between the three groups butthere does seem to be some degree of difference at the 25th and 10thpercentiles favoring the traditional and transitional groups. A simi-lar pattern can be noted on SCAT total as well as on many of the Co-operative English Tests. A similar pattern can be noted also on theStanford Numerical Competence subtest. On the High School Mathematics

Test, Part A, we seam to find a very' real and important differencefavoring the students studying modern mathematics over those studyingtraditional mathematics. At the upper selected percentile ranks thesedifferences run between 7 and 8 raw score points, while at the lowerselected percentiles the difference is between S and 7 points. Thetransitional group again seems to fall between the two groups at theupper selected percentiles, but exceeds both at the lower selectedpercentiles.

-12-

Table 7


Fall 1967


10th 25th 50th 75th

SCAT Verbal


SCAT Quantitative


SCAT Total


-13-

Table 8

A COMPARISON OF SELECTED PERCENTILES FOR COOPERATIVE ENGLISH TEST

Fall 1967


Reading Vocabulary



Reading Level


1.,Reading Speed

Modern Group 15 20 29 37 44Traditional Group' 16 20 28 35 42Transitional Group 15 20 28 36 43

Reading Total


English Expression


Table 9

A COMPARISON OF SELECTED PERCENTILESFOR STANFORD NUMERICAL COMPETENCE AND MATHEMATICS A TESTS

Fall 1967


Numerical Competence

10th 25th 50th, 75th 90th




Mathematics A

"..101




-15-

Table 10

A COMPARISON OF SELECTED PERCENTILES FOR 10TH GRADE TESTING PROGRAM:ENTIRE STATE

Fall 1967


SCAT Verbal

SCAT Quantitative

SCAT Total

Reading Vocabulary

Reading Level

Reading Speed

Reading Total

English Expression


Mathematics A


16 21 29 39 47

16 21 29 35 47

34 45 58 72 84

22 28 35 41 47

10 14 18 22 25

15 20 28 36 43

38 49 64 77 89

28 35 43 52 59

14 20 26 32 37

16 21 26 31 34

-16-

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Conclusions

Using the statewide grade 8 test data it became apparent in 1964 thatthere had been a significant drop in arithmetic computation scores since1963. It was hypothesized that modern mathematics was contributing tothis decline in computational ability. As a result of that hypothesisa study was conducted which we have already discussed at length. Thisstudy clearly indicated that students in the modern mathematics groupwere intellectually superior to those in the traditional or transitionalgroup, while just the inverse was true in terms of their ability to doarithmetic computation. It should be noted here that one of the abviousweaknesses and problems associated with this study is, what is meant bythe terms modern, traditional and transitional. These terms are noteasily defined, but we have earlier presented Mr. Prevost's workingdefinition of these terms.

The same study was replicated when the former eighth grade studentswere in the tenth grade. It is noted, again as one of the problems inthe second study, that there were rather different degrees of student-loss between grade 8 and grade 10. There was a 9% loss for the moderngroup a 32% loss for the traditional group and a 19% loss for thestudents in the transitional group. A number of hypotheses seem toexplain this.

It is more likely that traditional students were classified or werefound in small eighth grade systems which sent their students to highschools in another town for their ugh school education and that some ofthese high schools had to be eliwirated from the study because the tenthgraders in them were not all from schools that had been classified formerlyas traditional, transitional or modern. The fact 4s possible for allthree groups but more likely to have happened, in the author's opinion,with the traditional and transitional students. Since the studentsclassified as modern tended to come from the more affluent communities,they tended to have their own high schools.

There is a second possible reason for the very significantly differentstudent loss and that is that the traditional group again may be comingfrom smaller or rural communities whose students tend to stay in school ashorter period of time. This is somewhat less true for the transitional

and the modern group.

A third possible reason is that the students who are classified astraditional found school, and its traditional orientation, not respon-sive to their particular needs and therefore were disappointed, andalienated from, the school at an earlier period since the school wasdoing little to sustain their interests in staying in school. This would

account for the higher student loss. Whether or not this student lossaffected the statistics reported for the tenth graders is not easilyanswered.

The results of the tenth grade study have already been reported in detail.It seems clear from those results that for the School and College AbilityTest, the Verbal section is probably the truest indicator

-27-

of mental abilities and that there still is a significant differencebetween these three populations at grade 10 in terms of their intel-

lectual prowess. This hypothesis .Ls supported by the fact that thereis a significant difference in English Vocabulary, Reading Total andEnglish Expression in the same direction favoring the modern group.At the very least one could say that the students in the modern groupare, in terms of their Verbal skills, significantly different from the

students in the traditional and transitional groups. The order in all

cases shows the modern group highest, transitional second and the trad-

itional, third. This is the pattern which has continued for the three

years of 'this study. In terms of arithmcAc abilities, particularlyarithmetic computation, it is no longer true at grade ten that thereare significant differences between these three groups. This is veri-

fied twice, once in the Quantitative portion of the SCAT and secondly,as part of the Stanford Numerical Competence Test. In either casethere is no significnat difference between the three groups. What this

seems to clearly say is that while the students classified as modern

have, at grade 8, less ability at arithmetic computation, apparently thetwo years of practice (grade 8, grade 9 and part of grade 10) causethem to increase their skills to the point where there are no longer

any significant differences between the three groups.

It is particularly interesting to look at the Stanford High School Math-

ematics Test, Part A, results, in which we find a very significant F of

19.57, and to look at the means for the three groups, which are: modern

group -- 26.09; transitional group--25.68; and traditional group--19.41.

There are no significant differences between the modern and the transi-

tional group but there are very significant differences between thetransitional group and the traditional group and the modern group and the

traditional group. I think it is worthy of note here that this difference

not only seems to be statistically significant, but perhaps more important,

educationally significant. It is beyond the scope of this present paper

to make an analysis of what is causing that particular very significant

difference but it is something that the author is presently engaged in

researching. But it is easily hypothesized, that the students in modern

mathematics, when asked to work in the area of algebra and geometry,which is what this test measures, do markedly better than do their tradi-

tional counterparts. I think it is safe to say that algebra and geometry

call for many more verbal skills than they do pure computational skills

and it is here, apparently, that the greatest pay-off comes for the students

who have studied modern mathematics.

One of the questions that needs to be raised relative to this piece of

research is: how representative has the performance in mathematics of

these eighth and tenth graders been to the entire eighth or tenth grade

population in the state of New Hampshire? If one makes the comparison

between Tables 7, 8 and 9, which present the raw scores obtained by the

three 10th grade groups, at five selected percentiles, with Table 10,

which is the same information for the entire state, one finds, particularly

at the medians, that there are not many serious discrepancies between any

two of the groups. The author believes it is safe to say that there is

-28-

at least as much variance within the three groups as there isvariance between any one of the groups and the entire state. It there-fore is the author's conclusion that these groups are in fact a represen-tative sample of the entire state and not a special or unique sub-set ofthat population. These conclusions can only be reached for grade 10because no such comparison was made on the eighth grade data in 1965.

Tables 7, 8 and 9 offer some additional interesting possibilities forstudy. It was noted in the body of the report that the students clas-sified as the modern group seemed to do quite well in computation ifthey were found in the upper end of the spectrum in terms of the selectedpercentiles, i.e. the 75th or 90th percentile. Students classified asmodern tended to do more poorly in computation if, in fact, they werefound at the bottom of that scale, i.e. the 10th and 25th percentiles.The reverse seemed to be true for students classified in the traditionalgroup. The students at the bottom of the scale, i.e. the 10th and 25thpercentile, seemed to do better than one would have expected and thestudents at the upper percentiles, i.e. the 75th or 90th. seemed to domore poorly than one would have expected. There are some interesting excep-tions to this in the paper so that it can not be presented as a flat set ofstatements. But it holds generally to be true. I think this findingargues very cogently for the concept of differentiated instruction,particularly at different ability levels.

APPENDIX

Footnotes

1Durost, Walter N. Report. and Summary. New itmoilt StatewideGrade Eight Testing Program. Concord, New Hampshire: Test Serviceand Advisement Center, November, 1964, p.9.

References

BOOKS

Ferguson, George A. Statistical Analysis in Psychology and Education.

New York: McGraw-Hill, 1959.

REPORTS

Austin, Gilbert R. (ed.). New Hampshire Eighth Grade Testing Program.

Report for school year 1967-1968, sponsored by the New Hampshire

State Department of Education in cooperation with the Bureau of

Educational Research and Testing Services. University of New

Hampshire, 1967.




Research and Testing Services. University of New Hampshire, Fall,

1966.




Research and Testing Services. University of New Hampshire, November,

1965.

Durost, Walter N. (ed.). Report and Summary. New Hampshire Statewide

Grade Eight Testing Program. Report prepared by the Test Service

and Advisement Center, Concord, New Hampshire, November, 1964.

. (ed.). Report and Summary. New Hampshire Statewide Grade Eight

Testing Program. Report prepared by the Test Service and Advisement

Center, Concord, New Hampshire, November, 1963.

Dyer, Henry S., Linn, Robert L., and Patton, Michael J. Methods of

Measuring School System Performance. Report to the New York State

Education Department. Prepared b: Educational Testing Service,

Princeton, New Jersey, August, 1968.

Hogan, Thomas P. Some Notes on the Performance of Pupils in Modern and

Traditional Mathematics Programs on Standardized Arithmetic Tests.

Report prepared by Test Department, Harcourt, Brace and World.

References

OTHER SOURCES

Cooperative School and College Ability Tests. Princeton: EducationalTesting Service, 1956-57.

Derrick, Clarence, Harris, David P. and Walker, Biron. CooperativeEnslih. Tests. Princeton: Educational Testing Service, 1960.

Durost, Walter N. (General Editor). Metropolitan Achievement Tests.High School Battery. New York: Harcourt, Brace and World, 1962.

Gardner, Eric F. et al. Stanford Achievement Test. New York:Harcourt, Brace and World, 1964.

. Stanford High School Mathematics Test. New York: Harcourt,Brace and World, 1964.

. Stanford High School Numerical Competence Test. New York:


Kelley, Truman L. et al. Stanford Achievement Test. TechnicalSupplement. New York: Harcourt, Brace and World, 1966.

Otis, Arthur. Otis Quick-Scoring Mental Ability Tests. New York:Harcourt, Brace and World, 1954.

A STUDY OF ABILITY AND ACHIEVEMENT IN MATHEMATICS

OF THREE GROUPS OF TENTH GRADERS IN NEW HAMPSHIRE

List of Tables

Table Title Page

1 A Comparison of School and College Ability Test 4

2 A Comparison of Cooperative English Test 5

3 A Comparison of Stanford Numerical Competence andMathematics A Tests 6

4 A Comparison of Selected Percentiles for School andCollege Ability Test 9

5 A Comparison of Selected Percentiles for CooperativeEnglish Test 10

6 A Comparison of Selected Percentiles for StanfordNumerical Competence and Mathematics A Tests 11

ii

Chart

I

II

III

IV

V

VI

VII

VIII

IX

X


Title Page

SCAT Verbal 12

SCAT Quantitative 13

SCAT Total 14

Reading Vocabulary 15

Reading Level 16

Reading Speed, 17

Reading Total 18

English Expression 19

Numerical Competence 20

Mathematics A 21

iii

In the fall of 1967 the Bureau of Educational Research anacting as a contractual agent for the State Department oconducted the annual 10th grade statewide testing progconcerns itself with the evaluation of the mathematicability of a small subset within that much larger prthere were 9,776 10th graders who participated in tprogram. Of this number, 3,439 students were invostudy comparing 8th grade computational ability wcomputational ability. Of the 3,439 students pastudy, 620 students had elected to take the StaPart A, an optional test in the 10th grade bain the area of algebra and geometry.

The 10th grade test battery consisted ofForm 2B; the Cooperative English Test, FoNumerical Competence Test, Form X; and tTest, Form X, Part A and B. The Stanfoommended by the New Hampshire State Deis appropriate to give all 10th gradStanford Mathematics Test, Part A aDepartment of Education as only suin advanced mathematics courses i

The original study in whichwith the question, does the tyclass make any difference inThe communities participatithree separate categoriesbased on the type of textthose classifications, a

Working De

t

t

d Testing Services,f Education,

am. This studyal computational

ogram. In 1967he statewide testinglved in a follow-up

ith present 10th graderticipating in the follow-up

nford Mathematics Test,tery which measures abilities

he School and College Ability Test,rm 2C; the Stanford High School

he Stanford High School. Mathematicsrd Numerical Competence Test: is rec-

partment of Education as a test whichers in the state of New Hampshire. The

nd Part B, is recommended by the Stateitable for those students who are involved

n their school systems.

hese 620 students were involved was concernedpe of text book that the teacher uses in theterms of the students' computational ability?

ng in that statewide program had been divided intoclassified as modern, traditional and transitional,

book they were using. The working definitions ofs used in the original studies, are as follows:

finitions for Classifying Schools Based on Texts

If the mathematics text used by the school showed no de-viation from methods of presentation common in the late1950's or early 1960's, and introduced a minimal amountof new math, it was judged to be traditional. Such textswere more freqmently filled with long exercise sections;little structure or rationale in concept development wasemphasized.


Those texts which the publisher had admitted, or whichMr. Prevost judged, to have a middle of the road approachwere considered transitional. These texts were somewhere

-1-

along the continuum of t

Where a school system deliminated from the st

In the original study of computait was noted that there was a vbetween those students that wemodern, on the Stanford High Sbeing as follows: the modernand the transitional group aor not these differences wetics Part A subtest or whetthroughout the rest of thefollowing numbers of stud226 students; in the trtransitional group we h

The following analysisT tests, was done onfollowing are the reAbility Test the F'significant and altransitional groutative the F is asignificant diffthe transitionagiven in Table

Following theanalysis wasare statiston which tthe patteTotal seitionalcompute

A simiand tthentheorCo

raditional to modern.

id not fit into a category it was

udy.

tional ability between 8th and 10th gradersery large raw score difference in the means

e classified as traditional, transitional or

chool Mathematics Test, Part A. The means

group, 26.20; the traditional group, 19.68;

s 25.93. This study is an attempt to see whether

re found only on the Stanford High School Mathema-

her or not these difference were consistently found

tests in this battery. In this present study the

ents were in each group: in.the modern group we had

aditional group we had 39 students; and in the

ad 355 students.

was conducted: a one-way analysis of variance, plus

10 of the 11 tests in the 10th grade battery. The

sults of those computations: on the School and Colleges for the Verbal and the Total scores are both statistically

1 the differences favor the modern group, secondly theand lastly, the traditional group. For the SCAT Quanti-

lso statistically significant. There is only a statistically

erence between the modern and the transitional group, favoring

1 group. The means as well as the computed F's and T's are

1.

analysis of the School and College Ability Test, a similardone on the Cooperative English Test. All of the computed F's

ically significant at the 1% or 5% level, except English Expression

here is a non-significant difference. All of the differences follow

n of results on the School and College Ability Test, Verbal andtions, with the modern group being the best, followed by the trans-

group, followed by the traditional group. The means as well as the

d F's and T's are given in Table 2.

lar analysis was conducted on the Stanford Numerical Competence Testhe Stanford Mathematics Test, Part A. On the Numerical Competence Test

e is a significant F at the .01 level but the pattern has 'changed. Here

transitional group is performing markedly better than either the modernthe traditional group, a pattern similar to that noted on the School and

liege Ability Test, Quantitative section.

A similar analysis was conducted on the Stanford Mathematics Test, Part A. The

computed F is significant at well beyond the .01 level. The study of the means

of these three groups is worthy of our careful attention. The modern groupand the transitional group do very well on this test, while the traditionalgroup does very poorly. The means as well as the computed F's and T's are

given in Table 3.

-2-

No computations or analysis were conducted on the Stanford Mathematics

Part B test, because only students in the transitional and the modern

groups took this test. No student in the traditional group elected to

take Math B.

In the mathematical development of the analysis of variance, a number

of assumptions are made. One assumption is that the distribution, of

variables and the population from which the samples are drawn are nor-

mal. Since this study is not based upon the drawing of a sample from a

population, but is, in fact, a population itself, the use of analysis of

variance can seriously be questioned. Because of the failure to meet

this requirement, the project director, in consultation with other

statisticians, decided that to pursue the project as originally pro -

posed' the analysis of co-variance for the 1967 study would be inappro-

priate, since the assumptions for simple analysis are not met. The

assumptions are certainly not met for the analysis of co-variance;

therefore, it was decided that the 1967 special sub-group data would

be subjected to further analysis by computing selected percentile ranks

as a basis for determining differential effects for above and below

average students.

The test results of this special group were subjected to further analy-

sis by computing selected percentile ranks as a basis for comparing the

performance of above and below average students. Five selected per-

centile ranks were chosen for this study. They are: the 90th, 75th,

50th, 25th, and 10th. The comparisons provided indicate the raw scores

at each of these selected percentile ranks for the students involved in

the study. Table 4 present;; this information for the School and College

Ability Test; Table 5 for tLe Cooperative English Test; and Table 6 for

the Stanford Numerical Competence Test and the Stanford Mathematics Test,

Part A. The same' information is presented graphically on Normal Percen-

tile charts, I - X, which are found immediately following the selected

percentiles. These Normal Percentile charts say, in graphical form,

what the percentile tables say in tabular form.

-3-

Table 1


GRADE 10: Fall 1967

MATH A

SCAT

Verbal

SCAT

Quantitative

SCATTotal




Imartexmosoom.MillmO14100Mamew

F's

Significant

6.743

.01

6.464

.01

4.386

.05


1 : 2 =

1 3 -

2 : 3 =

3.666

0.893

3.319

.01

NS

.01

0.400

3.530

1.360

NS

.01

NS

2.357

1.010

2.928

.05

NS

.01

-4-

Table 2

A COMPARISON OF COOPERATIVE ENGLISH TEST

GRADE 10: Fall 1967

MATH A

Reading Reading Reading Reading English

voc)Leve3rotalEessiModern MeanGroup 1 40.26 20.77 34.13 74.39 50.23

Traditional MeanGroup 2 35.39 18.39 28.87 64.26 46.50

`......=101Transitional MeanGroup 3 39.08 20.44 32.79 71.86 50.11

F's 5.897 4.436 5.422 6.527 2.116

Significant .01 .05 .01 .01 NS

ONNE*.[T's and Significance

1 : 2 = 3.361 . 2.980 3.205 3.527 2.002

.01 .01 .01 .01 NS

1 : 3 = 1.675 0.852 1.678 1.804 0.134

NS NS NS NS NS

2 : 3 = 2.614 2.635 2.453 2.720 1.989

.01 .01 .05 .01 NS

-5-

Table 3

A COMPARISON OF STANFORD NUMERICAL COMPETENCE AND MATHEMATICS A TESTS

GRADE 10: Fall 1967

MATH A

NumericalCompetence

MathematicsA

Modern MeanGroup 1 29.45 26.20

Traditional MeanGroup 2 29.82 19.68


F's

Significant

7.527

.01

20.731

.01


1 : 2 =

1 : 3 =

2 : 3 =

0.301

3.782

1.585

NS

.01

NS

6.285

0.526

6.191.

.01

NS

.01

-6-

Conclusions

Table 1, 2, and 3 present the analysis of variance for these three groups.

It is clear from the results of that analysis that in those tests which measure

verbal skills the modern and the transitional groups are very similar and

that the traditional group is significantly different from them. There are

no significant differences on any of the 7 tests between the modern and the

transitional group on verbal measures. On the tests which measure mathematics

and quantitative skills the situation is quite different. On the SCAT

Quantitative and Numerical Competence tests, there are no significant dif-

ferences between the traditional and the modern group and there is no sig-

nificant difference between the transitional and the traditional group.

There is a significant difference between the modern and the transitional

group, favoring the transitional group. When we come to Mathematics A,

however, which is a test which measures knowledge of algebra and geometry, we

find the pattern re-establishing itself that was true with the tents which

measure verbal ability. There are no significant differences between the

modern and the transitional group but there is a very significant difference

betwee either the modern and the traditional group or the transitional and

the traditional group, in both cases, favoring the modern or the transitional

group.

The results of this analysis would seem to indicate the following is true

about computation ability: if the textbook used in the school classroom is

classified as either traditional or transitional, the students learn approximately

the same amount that students in a traditional textbook learn but they do not

learn as much as students in a transitional textbook do.

When, however, we look at the subject of algebra or geometry, it becomes very

apparent that students studying with either the aid of a modern text or a

transitional text apparently peiform markedly better than students who study

only with a traditional text.

Tables 4 - 6 and percentile charts I - X give us some additional information

about these three groups. These tables and charts confirm what we have already

said as the result of the analysis of variance, namely that we are dealing

with two separate populations here as measured by the testing of verbal skills.

The modern and transitional group seem to make up one group and the traditional

group seems to be a rather separate entity. The tables and charts seem to

clearly indicate that the modern and transitional group perform markedly

superior to the traditional group at all levels on tests which measure verbal

skills.

The separateness of these populations changes when we talk about quantitative

skills or computational skills. Here the transitional group stands by itself

while the modern and the traditional group stand together. On the Computation

subtest the charts and tables clearly indicate that at all levels the

transitional group is performing markedly superior to the modern and

traditional group.

On the Stanford High School Mathematics Test, Part A, the table and graphclearly indicate that the traditional group, at all selected percentiles, isperforming markedly poorer than the other groups, transitional and modern.The raw score differences found at these selected percentiles in this tableand chart can not, in the author's opinion, be attributed solely to the com-munity or genetic factors or environmental factors with which these childrencome to school. It seems clear to him that the great difference is alsomarkedly the result of the instruction, i.e. the textbook, and that thestudents who were taught using either a transitional or modern textbook hada marked advantage when it came to the subject of algebra and geometry, overthose students who had studied from a traditional text.

1

I

-8-

Table 4


GRADE 10: Fall 1967

MATH A


SCAT Verbal





SCAT Quantitative




SCAT Total




-9-

Table 5

A COMPARISON OF SELECTED PERCENTILES FOR COOPERATIVE ENGLISH TEST

GRADE 10: Fall 1967

MATH A


Reading Vocabulary

Modern G


29 34 40 45 51

Traditional GroupTransitional Group

25

27

30

33

35

39

39

45

4450

Reading Level




Reading Speed




Reading Total




English Expression




-10-

Table 6

A COMPARISON OF SELECTED PERCENTILESFOR STANFORD NUMERICAL COMPETENCE AND MATHEMATICS A TESTS

GRADE 10: Fall 1967

MATH A



i5th 50th 75th 90th


Mathematics A

Modern GrquR. 17 21 26 30 34Traditional Group 13 14 19 23 27Transitional Group 19 22 26 29 33

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APPENDIX

References

BOOKS

Ferguson, George A. Statistical Analysis in Psychology and Education.New York: McGraw-Hill, 1959.

REPORTS

Austin, Gilbert R. (ed.). New Hampshire Eighth Grade Testing, Program.Report for school year 1967-1968, sponsored by the New HampshireState Department of Education in cooperation with the Bureau ofEducational Research and Testing Services. University of NewHampshire, 1967.

-----. (ed.). New Hampshire Eighth Grade Testing Program. Report forschool year 1966-1967, sponsored by the New Hampshire StateDepartment of Education in cooperation with the Bureau of EducationalResearch and Testing Services. University of New Hampshire, Fall,1966.

. (ed.). New Hampshire Eighth Grade TesliltProgram. Report forschool year 1965-1966, sponsored by the New Hampshire StateDepartment of Education in cooperation with the Bureau of EducationalResearch and Testing Services. University of New Hampshire, November,1965.

Durost, Walter N. (ed.). Report and Summan. New Hampshire StatewideGrade Eight Testing Program. Report prepared by the Test Serviceand Advisement Center, Concord, New Hampshire, November, 1964.

. (ed.). Report and Summary. New Hampshire Statewide Grade EightTesting Program. Report prepared by the Test Service and AdvisementCenter, Concord, New Hampshire, November, 1963,

Dyer, Henry S., Linn, Robert L., and Patton, Michael J. Methods ofMeasuring School System Performance. Report to the New York StateEducation Department. Prepared by Educational Testing Service,Princeton, New Jersey, August, 1968.

Hogan, Thomas P. Some Notes Jn the Performance of Pupils in Modern andTraditional Mathematics Programs on Standardized Arithmetic Tests.Report prepared by Test Department, Harcourt, Brace and World.

References

OTHER SOUR

Cooperative School and CollegeTesting Service, 1956-57.

CES

la Tests. Princeton: EducationalAbility

Derrick, Clarence, Harris, DaviEnglish Tests. Princeton:

d P. and Walker, Biron. CooperativeEducational Testing Service, 1960.

Durost, Walter N. (General Editor). Metropolitan Achievement Tests.

High School Battery. New York: Harcourt, Brace and World, 1962.

Gardner, Eric F. et al.Harcourt, Brace and

Stanford Achievement Test New York:

World, 1964.

. Stanford High School Mathematics Test. New York: Harcourt,

Brace and World,

. Stanford Hi

Harcourt, Brac

Kelley, Truman L.Supplement.

Otis, Arthur.Harcourt

1964.

h School Numerical Competence Test. New York:

e and World, 1964.

at al. Stanford Achievement Test. TechnicalNew York: Harcourt, Brace and World, 1966.

Otis Quick-Scoring Mental Ability Tests. New York:

, Brace and World, 1954.

Documents

A Longitudinal Pivaluation of Mathematical Computational ... · STATED DO HOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. re\ FINAL REPOAr. C.) Project