document resume se 014 287 author marshall, j. laird ... · a large-scale, summative, comparative...
TRANSCRIPT
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DOCUMENT RESUME
24 SE 014 287
AUTHOR Marshall, J. Laird; Fischbach, Thomas J.TITLE Evaluation of Patterns in Arithmetic in Grades 1-4,
1970-71: Effects on Teachers.INSTITUTION Wisconsin Univ., Madison. Research and Development
Center for Cognitive Learning.SPONS AGENCY National Center for Educational Research and
Development (DHEW/OE), Washington, D.C.REPORT NO TR-225BUREAU NO BR-5-0216PUB DATE Mar 72CONTRACT OEC-5-10-154NOTE 93p.
EDRS PRICE MF-$0.65 HC-$3.29DESCRIPTORS Arithmetic; *Curriculum Development; *Elementary
School Mathematics; Inservice Programs; *Instruction;*Research; Teacher Attitudes; *Teacher Education;Televised Instruction
ABSTRACTA large-scale, summative, comparative evaluation of
"Patterns in Arithmetic (PIA)", a modern televised arithmeticcurriculum for grades one through six, was carried out in grades onethrough four in both rural and urban schools during the 1970-71school year. This report deals with the ways in which PIA affectsteachers: in basic mathematical knowledge, knowledge of PIA-specificcontent, and attitudes toward teaching arithmetic. Findings indicatethat PIA can be used effectively as in-service education,particularly for those teachers with relatively lower initialknowledge of the basic mathematics which underlies a contemporaryelementary school mathematics program. PIA does not seem to changeteachers' attitudes, however; nor is it beneficial in increasingknowledge of concepts not specifically related to PIA. These resultsseem to hold equally for both rural and urban schools. (Author/4M)
EVALUATION. OFPATTERNS IN ARITHMETIC
IN GRADE 1- 4, 1970 71:EFFECTS ON TEACHERS
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Technical Report No. 225
EVALUATION OF PATTERNS IN ARITHMETIC IN GRADES 1-4, 1970-71:
EFFECTS ON TEACHERS
by J. Laird Marshall and Thomas J. Fischbach
Report from theQuality Verification Component of Program 5
Mary R. Quilling, Program Coordinator
Wisconsin Research and DevelopmentCenter for Cognitive LearningThe University of Wisconsin
Madison, Wisconsin
-March, 1972
Published by the Wisconsin Research and Development Center for Cognitive Learning, supportedin part as a research and development center by funds from the United States Office of Education,Department of Health, Education, and Welfare. The opinions expressed herein do not necessarilyreflect the position or policy of the Office of Education and no official endorsement by the Officeof Education should be inferred.
Center No. C-03 / Contract OE 5-10-154
3
Statement of Focus
The Wisconsin Research and Development Center for CognitiveLearning focuses on contributing to a better understanding of cognitivelearning by children and youth and to the improvement of related educa-tional practices. The strategy for research and development is compre-hensive. It includes basic research to generate new knowledge aboutthe conditions and processes of learning and about the processes ofinstruction, and the subsequent development of research-based instruc-tional materials, many of which are designed for use by teachers andothers for use by students. These materials are tested and refined inschool settings. Throughout these operations behavioral scientists,curriculum experts, academic scholars, and school people interact,insuring that the results of Center activities are based soundly on know-ledge of subject matter and cognitive learning and that they are appliedto the improvement of educational practice.
This Technical Report is from the Quality Verification Program,whose principal function is to identify and invent research and devel-opment strategies taking into account current knowledge in the fieldof statistics, psychometrics and computer technology. The QualityVerification Program collaborates in applying such strategies in re-search and development. The translation of theory into practice andpresentations of exemplars of methodology are challenges which theQuality Verification Program strives to meet.
iii
Contents
Page
List of Tables vii
List of Figures vii
Abstract ix
I Introdu ction 1
Background 1Previous Evaluations 1
Reasons for Present Study 2Goals of the Study 2
II The Evaluation Plan 5Subjects 5Method 5Measuring Instruments 5
III Analysis of the Data 7Theory 7Preliminary Steps in the Analysis 7
Step 1: Grade 8Step 2: Site 8
Test of the Main Hypotheses 11Further Analyses 1 2
IV Summary and Discussion 21
AppendicesA. Overview of the Key Mathematical Ideas
in Patterns in Arithmetic, Grades 1-6 23B. Pre-QuestionnaireSeptember, 1970 29C. Post-QuestionnaireMay, 1 971 33D. PretestSeptember, 1970 39E. PosttestsMay, 1971 57
PIA-specific content (Questions 1-15)Grade 1Grade 2Grade 3Grade 4
Basic Mathematics (Questions 1 6-35)F. Additional Statistical Tables and Figure 93
References 99
List of Tables
Table Page
1 Approximate Number of Participants in the Study 6
2 Analysis of Variance: Knowledge 9
3 Analysis of Variance: Attitude 10
4 Determination of Covariate Set for Analysis of Covariance:Knowledge 13
5 Analysis of Covariance: Knowledge 14
6 Estimated Parameters for Target Population with and without PIA 16
7 Analysis of Covariance: Attitude 19
1-F Sample Mean Scores 94
2-F Weighted Mean Univariate Analysis of Variance 95
3-F Estimates of Parameters: Knowledge 96
4-F Estimates of Parameters: Attitude 97
List of FiguresFigure
1 Diagram of the experimental design.
2 Regression of PIA-specific posttest scores on basic mathematicspretest scores.
3 Regression of basic mathematics posttest scores on basic mathe-matics pretest scores.
4 Performance levels on knowledge and attitude variables for targetpopulation with and without PIA: Estimated means.
1-F Performance levels on knowledge and attitude variables for experi-mental and control teachers: Sample data.
6
Page
6
15
15
21
98
vii
Abstract
A large-scale, summative, comparative evaluation ofPatterns in Arithmetic, a modern televised arithmetic curric-ulum for Grades 1-6, was carried out in Grades 1-4 in bothiural and urban schools during the 1970-71 school year. Thisreport deals with the ways in which PIA affects teachers: inbasic mathematical knowledge, knowledge of PIA-specificcontent, and attitudes toward teaching arithmetic. Findingsindicate that PIA can be used effectively as in-service educa-tion, particularly for those teachers with relatively lowerinitial knowledge bf the basic mathematics which underliesa contemporary elementary school mathematics program. PIAdoes not seem to change teachers' attitudes, however; nor isit beneficial in increasing knowledge of concepts not specifi-cally related to PIA.
These results seem to hold equally for both rural andurban schools.
Introduction
Background
Patterns in Arithmetic (PIA), a comprehen-sive elementary mathematics program for Grades1-6, originated in 1959 and was developedover the next decade under the leadership ofProfessor Henry Van Engen with the supportof the Ford Foundation and, starting in 1964,the Wisconsin Research and Development Cen-ter for Cognitive Learning. The componentsof PIA are television lessons on videotape,teacher suggestion manuals, and pupil exer-cise books. When PIA is used as an instruc-tional program, a class watches a 15-minutevideotaped lesson once or twice a week (theschedule varies at different grade levels) andthe teacher supplements this by using the PIAexercise books and suggestions in the PIAmanual. The normal sequence of events wouldbe: a few minutes of rre-telecast warm-up,suggestions for which come from the teacher'smanual; the telecast, during which the pupilsmay be asked by the TV teacher to manipulatevarious objects or work a few problems; post-telecast activities , suggested in the manual,which enlarge upon or review the importantconcept presented in the telecast; and then,over the next few days, supplementary activi-ties from the manual and exercises in thepupil exercise books, some of which are re-lated to material in the telecast and some ofwhich are review of concepts presented inprevious telecasts. Short tests are also givena few times a semester so that teachers canassess pupil progress.
PIA was designed to be utilized in theperiod of transition to a modern mathematicsprogram at the elementary level. To quoteProfessor Van Engen,
Remember that the telecasts are not in-tended to replace the classroom teacher.They are intended to help introduce anddemonstrate new mathematical ideas.
r"' "
You, the teacher, are still the most impor-tant element in the development of a soundand meaningful mathematics program inyour school... .PIA is one answer to a newprogram in mathematics, but after havingused PIA for a year or two, you will beready to find other answers for your school.Whatever happens, remember that changeis inevitable. You must help determinethe direction of that change. l
The dual objectives of PIA are to providea sound program in mathemat1cs for the elemen-tary school child and to provide in-serviceeducation for the elementary school teacher.Television was chosen as the medium by whichto accomplish these aims, particularly for pu-pils and teachers in rural areas, where distanceand sparse population often make it difficult tocarry out in-service meetings or visits, and inthe centers of large cities, where high turnoverin both pupil and teacher populations and otherfamiliar ills frequently make educational stabil-ity and quality difficult to achieve.
For more complete information on the history,mathematical content, behavioral objectives, andpedagogical principles of PIA, see Braswell andRomberg (1969). For an overview of the keymathematical ideas in PIA, see Appendix A.
Previous Evaluations
As part of the research and developmenteffort, a summative, noncomparative evaluationof the PIA program in Grades 1 and 3 was car-ried out during the 1966-67 school year (Bras-well & Romberg, 1969). Several hundred first-
1 From the preface to the Patterns in Arith-metic Teacher's Manuals.
1
and third-grade classrooms in Alabama andWisconsin took part in the study, carried outin communities of four different sizes, the twolargest of which were further broken down intothree socioeconomic levels. Very briefly, theresults of that study indicated that most of theobjectives set forth by the PIA staff were beingaccomplished. In addition, further analysisrevealed no relation between performance andcommunity size or socioeconomic level. Theonly difference appeared to be between thetwo states, and then only on the standardizedcomputation test.
A field test for Grade 2 was carried outin the 1967-68 school year (Braswell, 1969a).Again the results indicated that PIA was accom-plishing most of its goals. At the end of theschool year, the children in the 30 classesin the study scored significantly higher thanthe norms on a standardized test of conceptsfor that grade level, despite having scoredlower than the norms group at the beginningof the year.
A formative evaluation of the PIA programfor Grade 5, with about 80 classes participat-ing, was also carried out in 1967-68 (Braswell,1969b). Results indicated that children learnedtraditional computation skills and importantconcepts in arithmetic commensurate with theirstage of development. Computation problemstended to be easier at the end of the schoolyear than they were shortly after the skillswere presented, which is good evidence thatskills learned early in the year are not forgot-ten, but are reinforced by the structure of thePIA pupil exercise books.
In the 1968-69 school year, the PIA staffat the Wisconsin Research and DevelopmentCenter for Cognitive Learning conducted aformative evaluation of the PIA program forGrade 6 (Braswell, 1970). The study sucoess-fully used the techniques of item sampling toevaluate the year's curriculum. This evalua-tion engendered several specific improvementsin the telecasts.
Romans for Present. Study
One of the original objectives in devel-oping PIA was to provide in-service educationfor elementary school teachers, but none ofthe evaluations mentioned heretofore hadinvestigated the effectiveness of PIA in thisrespect. Moreover, no evaluation of Grades1-3 had been carried out following revisionof the teacher's manuals for these three grades.The original manuals had been judged by teach-ers to be of limited use,, and the revised ver-
2
sions included, for each telecast; a statementof behavioral objectives, a description of themathematical background for the topics to becovered, an overview of the program itself,and specific descriptions of what to do priorto the telecast; the materials to be used bystudents during the telecast, what goes onin the television program itself, and what todo after the telecast is completed.
In addition, as a result of feedback re-garding utilization of PIA in various settings,a need was felt to ascertain the effectivenessof PIA among rural and urban children, andagain none of these studies had focused onthis particular question. Thus a large-scale,summative, comparative evaluation was car-ried out in the 1970-71 school year, focusingboth on teachers and on rural and urban pupilsin Guides 1-4. (The study would have beenextended longitudinally to Grades 5 and 6 wereit not for limited funds.)
The attribute of a rural setting which makestelevised education promising is the fact thatmany teachers have not had recent opportunityto update their skills. Thus, used properlywith the teachers viewing the program as theirpupils see itPIA becomes an in-service pro-gram in the concepts and pedagogy of modernmathematics. Both teachers and pupils learnthe new ideas through the medium of televisionfrom qualified teachers presenting contemporarymathematics content.
Urban settings are often characterized byhigh transiency of pupils, and eVen teachers,within the school district. A televised program,in addition to being a familiar and appealingmedium, provides for continuity in the basicskills program and is an educational touchstonefor the transient pupils. Moreover, sinceurban children are frequently deficient in thereading skills required for efficient learningfrom mathematics textbooks, television pro-grams in which manipulation of concrete mate-rials is a key feature may be a particularlyeffective means of presenting new concepts.
There are some factors, of course, whichare important in both types of demographicsituationstelevision's appeal to childrenand its efficiency and economy as an in-ser-vice cours .
Goals of the Study
In general, the data can be evaluated fromthree approaches: the effects of PIA on pupils,on teachers, and the interaction between thetwo. This report will concern itself only withthe second kind: in what ways, if any, are
teachers affected by using PIA in their class-rooms? Other reports in preparation will dealwith the other two aspects.
The objectives to be dealt with in thisreport are:
1. Do teachers who use PIA increasetheir knowledge of modern mathe-matics; i.e., is PIA an effective in-service course?
2. Do teachers who use PIA become morepositively disposed toward teachingmathematics, i.e., enjoy it more orbecome more confident?
3. Does the effect of PIA on teachers'knowledge vary with the
a. demographic characteristics ofthe school?
b. grade level taught?
c, level of teachers' initial generalknowledge of the mathematicswhich underlies contemporaryarithmetic?
4. Does the effect of PIA on teachers'attitudes vary with these same threefactors?
Ft 103
IIThe Evaluation Plan
Subl-
experiment was designed to be carriedout in urban and rural areas. Urban districtsselected were New York archdiocese schools,Chicago public schools, and the Portland,Oregon, school system. Rural sites were lo-cated in the state of Vermont and in three ruralcounties near Roanoke, Virginia. Within eachof the five sites, schools willing to partici-pate were identified and were randomly assignedto the experimental or the control group. In theexperimental schools, PIA was to be used asthe major or sole component in the mathematicscurriculum.
Approximately 10,000 children and 400teachers in about 85 schools took part in thestudy. Because of various unforeseen reasons,some of which are explained later, this samplewas reduced somewhat in the data analysis,but the numbers in the analysis are still ofthe same order of magnitude. Table 1 indicateswhich grades participated in which sites, andgives approximate numbers of children, teach-ers and schools in each site. It is sufficientto note that within each site there were approx-imately equal numbers of experimental and con-trol schools. In those sites where more thanone grade level participated, there were ap-proximately equal numbers of classes in eachgrade.
Method
Classes in the experimental schools usedthe Patterns in Arithmetic programs as theirsole or major component in the mathematicscurriculum. This included the television pro-grams, pupil exercise books, and the teacher'smanuals, all described earlier.
Classrooms in the control schools pro-ceeded normally through the school year with
11
whatever mathematics curriculum they choseto use. Although teachers in control schoolclassrooms were encouraged not to use thePIA telecasts, they were not prevented fromdoing so if they chose; in fact, some of theteachers did use the PIA telecasts, but gener-ally as a supplement to another curriculum.
Measuring Instruments
At the beginning of the 1970-71 schoolyear, teachers from both experimental andcontrol schools attended an informationalmeeting. At this time, data were gathered bymeans of a questionnaire and a test oi ele-mentary mathematics. Near the end of theyear, the participants were asked to fill outa second questionnaire and take a posttest.
The preliminary questionnaire, whichfollows as Appendix B, asked for biographicaldata, length and kind of teaching experience,and information about the recency and extentof training in mathematics and modern arith-metic concepts. Also included were questionsabout how much the teacher enjoyed teachingarithmetic compared with other subjects, andhow confident the teacher felt when teachingthe more difficult concepts of arithmetic. (Theterm "more difficult" was not defined, butrather left to the individual teacher's subjec-tive judgment.)
The post-questionnaire, Appendix C,asked for information about the extent to whichthe PIA telecasts and supplementary materialswere used and for teachers opinions as to theappropriateness and effectiveness of the PIApackage for their pupils, and their desire touse the program again next year. Two ques-tions from the pre-questionnaire about theteacher's enjoyment of and confidence inteaching arithmetic were also included, sothat attitudinal changes could be measured,
5
Table 1Approximate Number of Participants in the Study
Grade LevelSite 1 2 3 4 Schools Teachers Pupils
New York X X 40 150 4700
Chicago X X 10 90 2350
Portland* X X 10 40 850
Vermont X X X 10 70 1650
Virginia X 15 40 1050
TOTAL 85 390 10600
*Data not used in analysis.
The 50-item pretest, Appendix D, measuredhow familiar the teacher was with the basicmathematical concepts which underlie a con-temporary elementary mathematics program.There were items on sets, number, numerals,numeration systems, operations, properties,informal geometry, and a few other topics orcombinations of these topics. The intent ofthe test was not to measure those conceptsor vocabulary in the PIA telecasts, but ratherto get a broad measure of concepts basic toany modern approach to mathematics. Theitems were chosen from two sources: the SIOGFilmFilm Text Evaluation Studv MathematicsInventory (School Mathematics Study Group,1963) and the Sets and Systems AchievementTests I-V (Educational Testing Service, 1964).
The test taken in May, 1971 (Appendix
rade ember 197
E) consisted of two parts: 15 items speciallyconstructed to measure the specific contentfrom the year's PIA telecasts; and a 20-itemposttest, culled from the ;retest on the basesof item analysis statistics and PIA/grade levelcontent appropriateness. Although the 15-itemPIA tests were constructed to measure onlyPIA content, care was taken to include onlythose concepts or vocabulary which control(non-PM) teachers could reasonably be ex-pected to have encountered in any generalexposure to the basic concepts underlyingmodern arithmetic. These 15 items (PIA Tests1, 2, 3, and 4) were necessarily diffisrent forthe nost part for the four grade levels, 1-4, inthe experiment. However, the 20 items culledfrom the pretest were the same for all four gradelevels. A diagram of the design follows:
1
pre-question-naire
pre-test
During the school year
the experimental schools
used PIA in the classroom;
the control schools used
their regular math program.
post-questionnaire1
PIATest 1
post-test2(20 items;
2 PMTest 2
3 PIATest 3
4 PIATest 4
1Two attitudinal questions In the pre-questionnaire were repeated In the post-questionnaire.2Posttest items were imbedded in tha pretest.
6
Fig. 1. Diagram of the experimental design.
12GPO $17.794-2
HIAnalysis of the Data
Theory
We defined target population to be allteachers in the sites from which the data weresampled, as well as in sites to which the re-sults of the study could reasonably be gener-alized. As such, the ultimate questions ofresearch interest were stated in terms of twonull test hypotheses:
I. If teachers in the target populationuse PIA, there will be no effect ontheir knowledge either of PIA contentor of the basic general mathematicalconcepts and procedures (henceforthcalled "basic mathematics") whichunderlie contemporary mathematicsfor the elementary school,
2. If teachers in the target populationuse PIA, there will be no effect oneither confidence in or enjoyment ofteaching elementary school mathe-matics.
To determine the plausibility of the twohypotheses, two sources of data were avail-able. One was a randomly selected samplefrom the target population who had not usedPIA (the control group) but whose schools werewilling to participate in the evaluation. Thesample statistics from this source providedestimates of knowledge and attitudes exist-ing in the target population at the end of thestudy. A second sample was similarly select-ed from the same original target population,but the teachers therein were exposed to PIAduring the school year (the experimental group).The sample statistics from this zource providedestimates of knowledge and attitudes at the endof the study that would have obtained had allteachers in the target population used PIA inthe classroom.
If PIA had no effect, then these two datasamples come from the same theoretical pop-ulation and thus would yield the same perfor-mance levels and attitudes within the limitsof sampling and measurement errors. Becauseof the random assignment to experimental orcontrol groups of schools within each siteand because measurement and sampling errordistributions can be well-approximated byfairly simple statistical models, it was thuspossible to compute statistical tests of thehypotheses listed above.
Preliminary Steps in the Anolysis
The school was the unit of analysts sinceschools within a site were randomly assignedto either the experimental or control group.An advantage of us!ng the mean for a schoolas the statistic of analysis, besides its con-venient statistical properties, was the factthat if gain score means were used we wouldnot need to be concerned about teacher turn-over during the study: we could obtain un-biased estimates of unit (school) performanceand unit gains regardless of whether the per-sonnel changed (in some schools the personnelturnover was 50% in the grades studied).
Grade level was one type of grouping ofteachers within units, and is a within-unitsource of variation. A second type of group-ing, within the sample as a whole, was site,another source of variance. Performance levelsand/or attitudes might be expected to vary withboth of these factors. This would be of no con-sequence to subsequent analysis if neither fac-tor altered the effect of PIA on teachers' behav-ior, as memsured with the present instruments,but if such interactions existed, it was con-sidered to be important to discover and reportthem. Thus the first two steps in the analysiswere done to determine whether the effect of
137
PIA on teachers varies by grade level or site.If the findings were negative, the statisticalmodel for further analysis could be greatly
Step 1: Grade
Not all four grade levels were representedwithin any one sitethe number was eitherone, two, or three (see Table 1, p. 6). How-ever, sampling was done randomly withineach site and resulted in essentially equalnumbers of schools (and therefore all classeswithin a school) assigned to the experimentaland control treatments by grade. Thus, gradeand treatment were well balanced within eachsite.
lf, within a site, PIA' s effect did not seemto vary with grade level, grade could be ig-nored by averaging the grade means withineach school. This would result in but onestatistic (for each variable) for each school,and would simplify procedures still furtherand make hypothesis testing more powerful.
Multivariate repeated measures analysesof variance (with crade as the repeated mea-sure) were used to test the hypotheses thatthe Grade x Treatment interaction effects arezero for each variable (Table 2). The depen-dent variables here, as elsewhere, deal withknowledge of mathematics and attitudes towardteaching it. The results, a multivariate sig-nificance level of 0.68, showed the hypoth-esis concerning knowledge is not unreason-able; in fact, the evidence against the hypoth-esis was not even of moderate strength. Thesame results (significance level of 0.76seeTable 3, p. 10) obtained with the hypothesisusing attitude as the variate.2
2For this multivariate analysis step, anunweighted mean was used for arriving at arepresentative score for each school. Sinceit was thought that using a weighted mean(weighted according to the number of classesin the school) might alter the results, a moreexpensive second analysis, using the univari-ate repeated measures approach with weight-ed means, was carried out. The results werenearly identical (See Table 2-F, Appendix F).Hence, for further analysis, the mean scorefor each school is determined by the simpler,unweighted, method.
8
Thus there was no reason to reject thehypothesis that Grade x Treatment interactionis zero for either the knowledge or the attitudevariable; we can safely continue the analysison the assumption that the effect& of PIA onteachers' knowledge and attitudes does notvary with grade, and can proceed with theabove-mentioned simplifications in the analy-sis model.
Step 2: Site
We now proceed to determine whether theeffect of PIA varies according to site.3 If itcan be assumed that the Site x Treatment inter-action effect is zero, then the analysis modelcan again be simplified and certain inferencesmay be made with greater precision. For eachsite, a weighted mean of the estimates oftreatment effect (school meansl, where theweights are inversely proportional to the vari-ances of the estimates, was used as the statis-tic in the analysis; this statistic, under theassumption of zero Site x Treatment interaction,has the smallest variance of all unbiased linearestimators, and therefore enhances precision.(For example, the mean from New York, as thesite with the most cases (schools), would begiven most weight.)
Since the assumption of zero interactionis the key to the final statistical model, thereasonableness of this hypothesis must be in-vestigated. This was done in two ways:through the classical analysis of varianceapproach involving a test of this hypothesis,and by examination of "residuals," i.e., thedifferences between the values that wouldobtain if the model holds true and the actualvalues observed in the data samples, for eachcombination of site and treatment.
3The proportion of incomplete and missingdata in the Portland, Oregon, site was se largeas to make the supposed randomization processhighly suspect. Hence these data were dis-carded and not used in any of these analyses.
Table 2Analysis of Variance: Knowledge
Dependent Variables: (1) PIA-Specific Posttest Score(2) Basic Mathematics Posttest Score
Source
0.
:4
641
ww... 0g
4.1
.?.4-4 0= m2
Results
Univariate Resultsd
PIA-Specific Basic Math
Ma .2 MS 2
DirectionIf Relevant
Mean 1 5521.89 =MOD 12627.29 =HMI= =HMI=
Among Schoolsa 72 ---Among Sites 3 .0053* 2.036 0.65 .59 38.82 5.06 .0033*Treatment (PIA vs.
Control)b 1 .009* 26.39 8.39 .0052* 5.043 0.66 .42 PIA higherTreatment x Site 3 .23 6.189 1.97 .13 9.112 1.19 .32Among Schools
Within Celle 65 --- 3.146 --- --- 7.676 --- ---Within Schools =1=1=66 =HMI= =HMI= OD= =1 =1=1= =1=1=
Grades Specificto Sites 4 .55 7.007 1.56 .20 2.525 0.31 .87
Treatment x GradesSpecific to Sites 4 .68 2.452 0.54 .70 7.968 0.97 .43
Grade x SchoolsWithin Cells 58 4.501 --- 8.240 --- =1=1=
*Statistically significant with 0.05 level as criterion.aAnalysis performed using unweighted average of grade means in each school.
bAssumes Treatment x Site and Treatment x Grade specific to site effects are negligible but anypotential site effects have been eliminated.
alined as error for Among Schools Analysis.
dIf multivariate test is significant, univarlate tests are performed with modified Roy-Bose tech-nique: in this case, the requirement simplifies to use of ordinary F ratios shown.
9
Table 3Analysis of Variance: Attitude
Dependent Variables: (1) Enjoyment of Teaching Mathematics(2) Confidence in Teaching Mathematics
Source
Mean 1
Among Schools4 72Among Cells 7Among Sites 3
Between Treatmentsb 1
Treatment x Site 3
Within Cellsc 65
Within Cells 65
Within Schools 66Grades Specific to
Sites 4Treatment x Grades,
Specific to Sites 4
Grade x Schools (error) 58
CD441
4.4
2
CD
CDLig
CD
MS
Results
Univariate Results
Enjoyment Confidence
E MS E
DirectionIf Relevant
I= =I= I= =I= =I= I= I= =I= I= I= I=
.17 5.787 81.45 .24 5.650 1.67 .18
.54 1.468 0.37 .55 1 . 674 0.49 . 48
.35 1.598 70.40 .75 1.259 0.37 .77--- 3.980 --- --- 3.384 --- ---.0006* 7.611 2.89 .0001* 6.468 1.18 .26
.93 1 . 669 O. 63 . 64 1.103 O. 20 .94
.76 2.487 0:.94 .44 1.699 0.31 .87--- 2.633 --- --- 5 . 482 --- ---
*Statistically significant,with 0.05 level as criterion.
aAnalysis performed using unweighted average of grade means in each school.
bAfter any site effects are removed; assumes Treatment x Site effects are zero.
cUsed as error for Among Schools Analysis.
10
16
For knowledge,4 the hypothesis of noSite x Treatment interaction was not rejected(Table 2). The F test attained a multivariatesignificance level of .23, well above the .05level for clear rejection or the .05-.15 rangeindicative of moderate evidence against thehypothesis. Using the univariate approach,the significance level for the basic mathe-matics score was .32, well out of the ques-tionable range; for the PIA-specific score, itwas .13, which falls barely within the gener-ally accepted area of moderate negative evi-dence. Although this in itself was not enoughto reject the hypothesis, the residuals (sam-ple values minus predicted values) for thePIA-specific scores were inspected. Thelargest t ratio for the residuals was -2.04.The effect in this site was the opposite ofthe effect in the others, but the value wasstill below the rather liberal Bonferroni criti-cal value of approximately + 2.39, for 3 dfat alpha = .05, and hence is not statisticallysignificant. Thus, for the knowledge vari-ables, the hypothesis cannot be rejected andthe simple statistical model is adequate.
For the attitude variables, the resultsshowed even less reason for rejection of thesimple model. The multivariate test attainedthe significance level of .35 and the univari-ate levels were .75 and .77 for enjoymentand confidence, respectively. The t ratiosfor all residuals were small. Thus, there isnot sufficient evidence to compel us to rejectthe hypothesis of no Site x Treatment interac-tion for either knowledge or attitude variables;the simple statistical model is adequate forinvestigating the main hypotheses stated atthe beginning of this section.
Test of the Main Hypotheses
The testing in May, 1971 yieldedscores on PIA-specific content and on basic
4For shorthand convenience, "knowledge"stands for investigations of the hypothesesconcerning variables related to scores on pre-test and posttestin the latter case both thePIA-specific test and the basic mathematics.Similarly, "attitude" refers to investigationsof hypotheses concerning confidence in andenjoyment of teaching elementary school mathe-matics, data for which were gathered from thetvio questionnaires.
17
mathematics. Using the simple statisticalmodel with the assumption that the effect ofPIA does not vary among grades or sites, bothof these knowledge variables were investigated.The multivariate test attained a significancelevel of .009, well below the necessary .05and indicative of strong evidence against thenull hypothesis (see Table 2). This impliesthat using PIA in the classroom changed thelevel of performance on at least one and POSsibly both of the two variables. The univari-ate analysis was used to determine which ofthe variables was thus affected. For PIA-specific knowledge, the F ratio yielded asignificance level of .005, well below the .05required; there is clear-cut evidence againstthe assumption of zero effect. For basic mathe-matics, however, the F ratio yielded a signifi-cance level of .42, and thus the assumptionof no effect on that variable cannot be rejected.PIA changes the level of performance in thetarget population on PIA-specific knowledgebut not, apparently, on other basic mathe-matics (see Figure 4, p. 21).
However, this apparent lack of effect ofPIA on knowledge of basic mathematics wasdetermined using only the posttest scores. Abetter method of investigation is to use thepretest scores as well, recalling that all itemson the posttest were also included in the pre-test. Thus a univariate analysis of variancewas performed on the gain scores, or differ-ences in score from pretest to posttest. Thismethod is better in the sense that not only isa gain score an unbiased estimate, as is aposttest score alone, but also there is lessvariance connected with gain scores, and thusresults can be obtained with greater precision.In a sense, each teacher acts as his own con-trol when a gain score is used.
The F ratio resulting from this ANOVAreached a significance level of .003, highlyindicative that the null hypothesis must berejected (see Table 2, p. 9). When oneinvestigates gain scores, PIA does affect levelof knoWledge of basic mathematics. However,the effect is in a negative directionthe gainscores of the iarget population as a whole arelarger than for those who used PIA in the class-room.
When attitudes are analyzed, the resultsare less striking. Although the observed dif-ferences were negative for enjoyment and posi-tive for confidence, neither of these came closeto being significant (see Table 3). The attainedsignificance level for the multivariate test was.54 and the univariate levels were .55 and .48for these two variables.
11
Further Analyses
While the analysis could end with whathas been discussed so far, there remains thefact that much of the variance in the datacame from differences among teachers, andthus among schools, the units of analysis.The precision of various estimates should beincreased by eliminating differences associatedwith pretest scores (for the knowledge analysis)and beginning attitudes as measured on the pre-questionnaire (for the attitudes analysis).
Thus analyses of covariance were alsodone. For knowledge, the choice of covari-ates was arrived at somewhat empirically: bydetermining the set of covariates which reducesthe mean square error the most for the leastproportionate reduction in degrees of freedomfor error (see Table 4). The final set of co-variates chosen in this manner was the scoreon those pretest items which were also on theposttest, dropping the assumption that theregression of posttest scores on pretest scoresis the same regardless of whether teachershave used PIA. (This was accomplished byusing as a variable the pretest subscore forthe experimental group, and zero for the con-trol group.) The use of several other possiblecovariates was examined, but did not provefruitful.
The results (see Table 5) agreed withthose from analyses of variance reported earlier.The multivariate significance level was .01;therefore the hypothesis that PIA has zero effecton knowledge must be rejected. Further uni-variate investigation revealed that this non-zero effect showed up on the PIA-specificpost-scores, but not on the basic mathematicspost-scores (LI test significance lei/els of.045 and .23, respectively). As before, theresults showed that using PIA in the classroomsignificantly increases knowledge related tothe PIA content.
Since dropping the assumption of homo-geneity of regression (of knowledge scoreson pretest basic mathematics score) proveduseful in the analysis of covariance, thehypothesis itself was investigated. The multi-variate significancelevel was .08, not belowthe .05 level for statistical significance, butnonetheless indicative of some evidenceagainst the hypothesis (see Table 6). Whenthis was further investigated, it appeared
12
that there was little difference between exper-imental and control groups for regression ofbasic mathematics posttest scores on pretestscores, but there was more apparent effecton the regression of PIA-specific posttestscores on basic mathematics pretest scores(see Figures 2 and 3). The regression coeffi-cient for the experimental group was estimatedat .27, while that for the control group was.49. This difference, although still only inthe marginal area statisticallythe significancelevel is .0-tends to indicate that amongthose with low to average scores on the basicmathematics pretest, using NA makes a markeddifference on the PIA-specific posttest whencompared with the target population as a whole,whereas among those with pretest scores morethan one standard deviation above the mean,the effect is just the opposite: using PIAproduces scores relatively lower than in thetarget population. Restated, the less theknowledge of basic mathematics at the startof the year, the more likely using PIA in theclassroom will be effective as a medium ofin-service training for PIA content. The samething is not indicated, however, for learningbasic mathematics not specifically in the PIAmaterial.
Although there were gains by both exper-imental and control groups from basic mathe-matics pretest to posttest, the gain of theexperimental group was estimated to be onlyabout 40% of that of the target population.79 as compared to 2.00 pointe, out of a totaltest score of 20 (see Table 6). Since the stan-dard error of the difference was .39, this dif-ference of 1.21 points is statistically signi-acant. This coincides, as it must, with thefinding from the ANOVA reported earlier.
The analysis of covariance of post-exper-imental attitude, using pre-experimental atti-tudes as covariates, produced the same resultsas did the analyses of variance; i.e., no ap-parent differences between experimental andcontrol groups (see Table 7). Additional analy-ses of covariance of enjoyment and confidence"gain" scores, using basic mathematics pre-test scores as covariates, indicated thatthere were gains in both enjoyment and con-fidence over the school year (in the latter,particularly for teachers whose initial basicmathematics scores were high), but that therewere no significant differences between exper-imental and control groups.
18
Tab
le 4
Det
erm
inat
ion
of C
ovar
iate
Set
for
Ana
lysi
s of
Cov
aria
nce:
Kno
wle
dge
(Est
imat
ed E
rror
Var
ianc
e w
ith V
ario
us C
ovar
iate
Gro
ups)
Res
ults
with
Cov
aria
tes
Cov
aria
tes
df f
orer
ror
Mea
n Sq
uare
Err
ors
PIA
-Spe
c.B
asic
Mat
h
Rel
ativ
e to
Sel
ecte
dC
over
iate
Grc
up, P
ropo
rtio
nIn
crea
se (
Dec
reas
e) in
df f
orM
Serr
orer
rcw
for
PIA
Non
e65
3.13
637.
663
1.0
320.
511a
Pret
est B
asic
Mat
h &
Pre
test
Bas
ic M
ath
x T
reat
men
t*63
2.07
542.
5582
---
Tot
al P
rete
st64
2.13
733.
0272
.016
0030
a
Ill-
Tot
al P
rete
st &
Tot
al P
rete
st x
Tre
atm
ent
632.
101
23.
0159
00.
012a
,-..
Pret
est B
asic
Mat
h O
nly
(Pre
test
I)
642.
1393
2.53
76.0
1600
31a
' Pre
test
s 1
& 2
632.
1423
2.57
550
0032
aPr
etes
ts 1
& 2
, Pre
test
1 x
Tre
atm
ent
622.
0687
2.59
79(.
016)
(000
3)b
La
Pret
ests
1 &
2, P
rete
sts
1 &
2 x
Tre
atm
ent
612.
0801
2.45
96(.
032)
0.00
2cC
O
*The
sel
ecte
d pa
ir o
f co
vari
ates
.R
atio
nale
: Cho
ice
of c
ovar
iate
s sh
ould
red
uce
MSs
for
err
or a
max
imum
am
ount
whi
le r
educ
tion
of d
f is
min
imiz
ed;
or a
t lea
st o
btai
n op
timal
com
bina
tion.
aInc
reas
e in
M.le
rror
> in
crea
se in
df.
bDec
reas
e in
Mer
ror
< d
ecre
ase
in d
f .
cdf
decr
ease
s an
d M
S in
crea
ses.
In a
ll ca
ses,
the
Mer
ror
for
PIA
-spe
cifi
c sc
ore.
is c
rite
rion
.
Tab
le 5
Ana
lysi
s of
Cov
aria
nce:
Kno
wle
dge
Dep
ende
nt V
aria
bles
:(1
) PI
A-S
peci
fic
Post
test
Sco
res
(2)
Bas
ic M
athe
mat
ics
Post
test
Sco
res
Sour
ce
lz I
DZ
o...
,= ol
ciD
irec
tion
> ..
..PI
A-S
peci
fic
Bas
ic M
ath
Ss-a
-a>
If R
elev
ant
trN
FA
-7M
S.E
.P.
ms.
r2
Res
ults
Uni
vari
ate
Res
ults
0
Scho
ols
Mea
n
Am
ong
Scho
ols
73 1 72--
---
- 1.,
Cov
aria
tes
(tot
ana
2.0
001*
36.5
478
17.6
1.0
001*
168.
4575
65.8
5.0
001*
Bas
ic M
ath
Pret
estb
1.0
001*
71.6
221
34.5
1.0
001*
322.
0518
125.
89.0
001*
Posi
tive
Bas
ic M
ath
x T
reat
men
tb1
.08
6.16
392.
97.0
91.
2535
0.49
.49
Dec
r. w
ith P
IA
Am
ong
Cel
ls7
---
Am
ong
Site
s3
.03*
0.82
970.
40.7
58.
1747
3. 2
0.0
3*B
etw
een
Tre
atm
ents
c1
.01*
8.73
484.
21.0
45*
3.71
411.
45.2
3PI
A h
ighe
rT
reat
men
t x S
ites
3.3
23.
8936
1.88
.14
1.15
970.
45.7
2
Res
idua
l for
Err
or63
2. 0
754
2.55
82
*Sta
tistic
ally
sig
nifi
cant
at .
05 le
vel.
aAft
er a
ll si
te a
nd tr
eatm
ent e
ffec
ts e
limin
ated
.bS
um o
f Sq
uare
s ar
e "e
xtra
" fo
r th
at p
artic
ular
cov
aria
te.
CA
fter
site
and
cov
aria
te e
ffec
ts e
limin
ated
but
ass
umin
g T
reat
men
t x S
ite in
tera
ctio
n ef
fect
is z
ero.
..11 7 8 9 10 11 12 13 14 15 16 17Basic Mathematics Protest 4100f0
Fig. 2. Regression of PIA-specific posttest scores on basicmathematics pretest scores.
16
(3
15
14
13
12
1
11
4.
Averse* Donsese0.113
1+ 2s9 10 1 12 13 14 15 16
Basic Mathematics Pretest SUM
Fig. 3. Regression of basic mathematics posttest scoreson basic mathematics pretest scores.
15
Tab
le 6
,
Est
imat
ed P
aram
eter
s fo
r T
arge
t PoN
latio
n w
ith a
nd w
ithou
t PIA
Lea
st S
quar
es E
stim
ates
Ass
umin
g Si
mpl
e St
atis
tical
Mod
el
Var
iabl
e
From
Ana
lyse
s of
Var
ianc
e
SIT
E
Urb
an S
ite 1
Mr.
-get
Pop
ulat
ion
With
PIA
With
oueP
IAE
st'd
S. E
.E
st'd
S. E
.M
ean
of E
st.
Mea
nof
Est
.
Urb
an S
ite 2
Tar
get P
opul
atio
nW
ith P
IAW
ithou
t PIA
Est
'dS.
E.
Est
'dS.
E.
Mea
nof
Est
.M
ean
of E
st.
Rur
al S
ite 1
Tar
get P
opul
atio
nW
ith P
IAW
ithou
t PIA
Est
'dS.
E.
Est
'dS.
E.
Mea
nof
Est
.M
ean
of E
st.
Kno
wle
dge
Pret
ests
Subt
est 1
(B
asic
Mat
hem
atic
s)-2
0 ite
ms
a1
2.19
.45
a9.
88.9
3a
10.3
6.8
8Su
btes
t 2 (
Oth
er P
rete
st it
ems)
-30
item
sa
11. 2
7.4
8a
9.62
.,99
a9.
66.9
4T
otal
-50
item
sa
23.4
6.8
9a
19.5
11.
82a
20.0
21.
73
Post
test
sPI
A-S
peci
fic-
15 it
ems
9.53
.35
8.32
.36
8.85
.62
7.64
.63
8.94
.60
7.73
.60
Bas
ic M
athe
mat
icsb
-20
item
s13
.28
.49
14.4
9.5
09.
64.9
510
.86
.96
11 .4
5.9
01
2. 6
7.9
0
Gai
n in
Bas
ic M
athe
mat
ics
1.09
.33
2.30
.34
-.24
.58
.98
.60
1.09
.56
2.31
.56
Atti
tude
cPr
e-qu
estio
nnai
reE
njoy
men
ta
8.76
.29
a7.
58.6
0a
7.93
.57
Con
fide
nce
a6.
46.3
6a
6.62
.73
a6.
17.6
9
Post
-que
stio
rnai
reE
njoy
men
t8.
79.3
99.
08.4
17.
36.7
07.
64.7
18.
15.6
78.
44.6
7C
onfi
denc
e8.
39.3
68.
09.3
87.
47.6
47.
16.6
68.
52.6
28.
22.6
2
From
Ana
lyse
s of
Cov
aria
nce
Kno
wle
dge
PIA
-Spe
cifi
cd8.
95.3
18.
30.2
99.
10.5
08.
46.5
38.
98.5
38.
32.5
4B
asic
Mat
hem
atic
s (P
ostte
st)d
12.4
9.3
413
.42
.33
10.7
4.5
711
.66
.60
12.1
8.6
013
.10
.60
Gai
n in
Bas
ic M
athe
mat
icse
1.44
.34
2.34
.33
-.32
.56
.57
.60
1.11
.54
2.01
.55
Atti
tude
(Po
st-q
uest
ionn
aire
)E
njoy
men
tf8.
47.3
28.
73.3
37.
70.5
67.
96.5
88.
35.5
38.
61.5
3C
onfi
denc
ef8.
32.3
07.
85.3
07.
57.5
27.
11.5
38.
74.4
98.
28.4
9
Tab
le 6
(C
ontin
ued)
SIT
E
Var
iabl
e
From
Ana
lyse
s of
Var
ianc
e
Rur
al S
ite 2
Tar
get P
opul
atio
nW
ith P
IAW
ithou
t PM
Est
'dS.
E.
Est
'dS.
E.
Mea
nof
Est
.M
ean
of E
st.
All
Site
s
Tar
get P
opul
atio
nW
ith P
IAW
ithou
t PIA
Est
'dS.
E.
Est
'dS.
E.
Mea
nof
Est
.M
ean
of E
st.
Kno
wle
dge
Pret
ests
Subt
est 1
(B
asic
Mat
hem
atic
s)-2
0 ite
ms
a11
.97
.70
a11
.10
.38
Subt
est 2
(O
ther
Pre
test
item
s)-3
0 ite
ms
a8.
48.7
4a
9.76
.41
Tot
al-5
0 ite
ms
a20
.45
1.37
a20
.86
.75
Post
test
sPI
A-S
peci
fic-
15 it
ems
9.08
.49
7.87
.49
9.10
.31
7.89
.32
Bas
ic M
athe
mat
icsb
-20
item
s13
.18
.72
14.4
0.7
211
.89
.43
13.1
0.4
3
Gai
n in
Bas
ic M
athe
mat
ics
1.21
.46
2.43
.46
.78
.30
2.00
.30
Atti
tude
cPr
e-qu
estio
nnai
reE
njoy
men
ta
8.56
.45
a8.
21.4
9C
onfi
denc
ea
6.22
.55
a6.
37.6
0
Post
-que
stio
nnai
reE
njoy
men
t8.
75.5
59.
04.5
58.
26.3
58.
55.3
7C
onfi
denc
e7.
38.5
17.
07.5
17.
94.3
37.
64.3
4
From
Ana
lyse
s of
Cov
aria
nce
Kno
wle
dge
PIA
-Spe
cifi
cd8.
66.10
8.00
.40
8.92
.19
8.27
. 29
Bas
ic M
athe
mat
ics
(Pos
ttest
)d12
.54
.11
13.47
.45
11.9
9.2
212
.94
.33
Gai
n in
Bas
ic M
athe
mat
icse
1.53
.46
2.43
.44
.94
.29
1.84
.30
Atti
tude
(Po
st-q
uest
ionn
aire
)E
njoy
men
tf8.
58.4
48.
84.4
48.
28. 2
88.
54.2
9C
onfi
denc
ef7.
44.4
16.
98.4
18.
02.2
67.
55.2
7
For
note
s, s
ee n
ext p
age.
,
Tab
le 6
(C
ontin
ued)
aPop
ulat
ions
"w
ith P
h,"
but b
efor
e PI
A w
as u
sed
In th
ecl
assr
oom
are
iden
tical
to p
opul
atio
ns "
with
out P
IA"
beca
use
ofra
ndom
ass
ignm
ent
to tr
eatm
ents
.bE
stim
ates
= e
stim
ated
targ
et p
opul
atio
n pr
etes
t mea
n +
est
imat
ed g
ain
(or
loss
).
cror
pur
pose
s of
ana
lysi
s an
d co
mpa
rabi
lity,
the
enjo
ymen
tan
d co
nfid
ence
var
iabl
es w
ere
tran
sfor
med
so
that
eac
hha
d a
poss
ible
ran
geof
1 =
low
to 1
3 =
hig
h.dC
ovar
iate
s ar
e B
asic
Mat
hem
atic
s Pr
etes
t Sco
re a
nd B
asic
Mat
hem
atic
s x
Tre
atm
ent I
ndic
ator
Sco
re.
el1.
10, t
he B
asic
Mat
hem
atic
s Pr
etes
t Sco
re m
ean
over
all
site
s, w
as u
sed
as th
e ba
sis
for
gain
sco
re in
eac
h si
te;
cova
riat
e Is
the
Bas
icM
athe
mat
ics
Pret
est S
core
.fb
ovar
iate
s ar
e E
njoy
men
t and
Con
fide
nce
"sco
res"
fro
m p
re-q
uest
ionn
aire
.
II
1
i
Tab
le 7
Ana
lysi
s of
Cov
aria
nce:
Atti
tude
Dep
ende
nt V
aria
bles
:(1
) E
njoy
men
t of
Tea
chin
g M
athe
mat
ics
(2)
Con
fide
nce
in T
each
ing
Mat
hem
atic
s
Sour
ce
0 0 I
am
o5
Enj
oym
ent
.... -
A -
4SI
o 4
1.)
m S
I'R
.M
SI'
o
Res
ults
Uni
vari
ate
Res
ults
Con
fide
nce
Dir
ectio
nIf
Rel
evan
t
Scho
ols
73
Mea
n1
Am
ong
Scho
ols
7 2
Cov
aria
tes
(tot
al)
2.0
001*
48.2
625
18.7
5.0
001*
43.4
351
20.5
6.0
001*
Enj
oym
ent (
pre-
ques
tionn
aire
)1
.000
1*51
.367
419
.95
.000
1*7.
3895
3.50
Con
fide
nce
(pre
-que
stio
nnai
re)
1.0
1*3.
1804
1.24
36.3
771
17.2
2.0
001*
Am
ong
Cel
ls7
Am
ong
Site
s3
.19
1. 6
080
O. 6
2. 6
05.
0761
2.40
.08
Bet
wee
n T
reat
men
ts1
.33
1. 2
061
0.47
.50
3.84
611.
82.1
8T
reat
men
t x S
ites
3.5
81.
0433
0.41
.75
2.54
681.
21.3
2R
esid
ual f
or E
rror
63--
-2.
5740
2.11
26
*Sta
tistic
ally
sig
nifi
cant
at .
05 le
vel.
IVSummary aid Discessios
Perhaps the best way to summarise is toanswer the questions posed as the goals ofthe study (see Figure 4):
1. Do teachers who use PIA increasetheir knowledge of modern mathematics; i.e.,is PIA an effective in-service course?
The answer is a clear Lei, provided werestrict ourselves to the general basicmathematics related to the PIA telecasts.However, if we look at content not speci-fically found in PIA, the answer is pa,for the gain scores for non-PIA-specificcontent, while definitely positive forboth groups, were generally higher forthe teachers who did not use PIA. Itseems that while PIA teachers were gain-ing in knowledge of PIA content, otherteachers made higher gains elsewhere thandid the PIA teachers.
2. Do teachers who use PIA become morepositively disposed tofiard teaching mathema-tics; i.e., enjoy it more or become more con-fident?
Both groups seemed to make slight gainsin enjoyment and larger gains in confidenceover the school year, but PIA teachers didnot show an advantage over control teach-ers in this respect.
3 a. Does the effect of PIA on teachers'knowledge vary with.the demographic charac-teristics.of the school?
Whereas there was a difference in levelof knowledge at the end of the study be-tween teachers in the two urban sites,this difference was not related to use ofPIA. PIA was equally effective in bothof these urban sites, as outlined in the
thliew INN IMOMOS Oft I Niel Taal
1 IA1114epsamieS. Tossedas10/0mislas(la
Ise POMO
KIMilloomafti peassre ammo Is NM roue*
IlleNsested pool mon h. Torr. Popidellen 01155.1 INASOUSAIMAS ow. let Taw. ralftlaNINt INA
Fig. 4. Performance levels on knowledgeand attitude variables for targetpopulation with and without PIA:Estimated means.
answer to question 1 above. isloi wasany difference found between urban andrural sites; PIA is equally effective asa medium of in-service education in bothsituations .
21
3 b. Does the effect of PIA on teachers'knowledge vary with the grade level taught?
The answer appears to be no.
3 c. Does the effect of PIA on teachers'knowledge vary with the level of teachers'initial general knowledge of basic mathema-tics?
Here, the answer is a tentative yet: PIAis very effective as an in-service toolfor those whose initial level of basicmathematics is low to average; fot thosewhose pretest scores fall more than astandard deviation above the mean, the
22
effect is lessened.
4. Does the effect of PIA on teachers'attitudes vary with these same three factors?
The demographic chracteristics of theschool seem to make no difference, nordoes the grade level taught. Whereasthere was a slight increase in teachers'enjoyment no matter what their initiallevel of knowledge, those who scoredhigh on the basic mathematics pretestseemed to have greater gains in confi-dence than those whose initial scoreswere lower; however, this occurred re-gardless of whether PIA was used.
GPO 1127-794-3
Appendix AOverview of the Key Mathematical Ideas in
Patterns in Arithmetic, Grades 1-6
The following is a brief description of the fundamental concepts which runthrough the Patterns in Arithmetic series.
1. at: The concept of a set is fundamental for developing and communicatingideas in mathematics. Since the idea of a set is a rather primitive one, asearly as Grade 1, pupils become familiar with the simple concepts whichinvolve sets, such as setting up a one-to-one correspondence (matching)between sets and comparing the numerousness of two sets. The set idea isfound to be particularly useful in the teaching of addition (set union) andsubtraction (set complementation). Set ideas are also applied in the latergrades to equivalent fractions, equivalent ratios, and geometric figures.
2. klumben One of the main features of the program is the logical develop-ment of the rational numbers, beginning with the positive integers. Nearthe beginning of Grade 3 the pupils will be able to determine the cardinalityof any set with less than 10,000 members and to use the ordinal numbers.During the First, and again the Second Grade, the pupil is taught the ideasof betweenness, lem, than, Greater tim, and eaual to.for numbers less than1,000. The number line is used to picture the order of integers and thepupil is taught how to use this line to perform addition and subtraction.These ideas are reinforced and extended so that near the end of Grade 6the pupil is totally familiar with the positive rational numbers. Other mis-cellaneous topics covered in the series are even and odd numbers, primenumbers, prime factorization of natural numbers, the least common multipleand greatest common factor of pairs of natural numbers, and the negativeintegers.
3. Numeration System: After some degree of understanding has been achievedin counting with both cardinal and ordinal numbers, the pupils are taughthow to write the numbers 0-9, and after 9, the concept of place value isintroduced. Although the pupils do not fully comprehend place value at thistime, extensive use of tally charts and regrouping is introduced in order toovercome some of the mystery of place value. Near the beginning of Grade 4the pupil should be able to write all numerals and interpret place value forlarger numbers. Although emphasis is on the decimal system (Base 10), nu-meration systems with other bases (Base 5, Base 2) are introduced to pro-
29 25
mote understanding of place value. Positive rational numbers and the sev-eral notations associated with them (common fraction, mixed number, deci-mal fraction) are introduced in Grades 4-6.
4. psrrations: A considerable portion of elementary school arithmetic is con-cecned with the four fundamental operations: addition, subtraction, multi-plication, and division; and how these operations act with various sets ofnumbers (natural numbers, positive rationals, integers). Of course, one ofthe major objectives of any elementary school arithmetic program is to de-velop accuracy and speed in computing. However, computing in itself isnot enough; therefore, we have placed considerable emphasis on ideasassociated with computing in an attempt to make the four operations morethan rote calculation. For example, in forming the sum 3 + 2, the pupil con-siders a stationary set with 3 objects and another set of 2 objects whichappear to be joining the given 3. Thus 3 + 2 is looked upon as a set of 3being joined by a set of 2 to form a set of 5. The concept of column addi-tion is approached through tally charts in an attempt to impart some under-standing of place value in our Base 10 numeration system. Rectangulararrays are used to analyze geometrically the operation of multiplication.Geometric arguments are made to lend substance to the operations appliedto the positive rationals. The notion of operation is extended in Grades 5and 6 by introduction of an operation with a simple geometric realization.
Beginning in the First Grade the commutative, associative, and dis-tributive properties of the operations receive considerable attention as aidsto computation. Not until Grade 5 are the properties formalized for thepupils. Upon completion of Grade 6 the pupil will be able to compute ef-ficiently with the positive rational numbers in their many notations.
Throughout PIA the pupils are presented verbal situations in which theycan apply their newly acquired computational skills. In these situationsspecial attention is devoted to developing the ability to formulate mathe-matical sentences in a clear and natural way.
5. Mathematical Sentence (Equation): Since language is the vehicle throughwhich we communicate our ideas, it is important that pupils begin to developthe ability to generate mathematical sentences as soon as possible. In theFirst Grade, the pupil encounters many experiences with pictures and objectswhich lead to basic sentence forms. In word problems pupils are requestedto "write a sentence which tells the story of the problem" before finding thesolution. (Problem: A box holds 6 apples. How many boxes are needed for30 apples? Sentence: N x 6 30.) Throughout the series the same im-portance is attached to the need to translate verbal problems into mathe-matical sentences.
26
6. Measurement: Owing to its importance in everyday life as a key link be-tween our physical and social environment, we begin a systematic study ofmeasurement in the First Grade. As an introduction, the First Grade pupilbecomes familiar with the concept of relative length (the desk is less thanfive pencil-lengths long) and a non-standard unit of measurement (the pen-cil above). Upon completing the first four grades, the pupil will ba able tocarry out approximate linear measurements in standard units (inches, feet)and he will be able to find the perimeter of some elementary geometricalforms (triangle, rectangle). In Grades 5 and 6 the concepts of area andvolume are discussed in terms of non-standard and standard units of mea-sure and approximation techniques are used to estimate areas and volumesof irregular regions. Formulas for areas or volumes of elementary regions,such as a rectangular region and right rectangular prisms, are introduced.
7. Geometry: One of the more unique features of the program is a systematicdevelopment of elementary geometrical concepts beginning in the FirstGrade. Aside from learning the names of the more common geometricalfigures, the pupil becomes familiar with open and closed curves, interiorand exterior of geometrical forms, points, lines and angles, intersection ofcurves, parallelism, and perpendicularity. The notions of congruence andsimilarity are introduced intuitively in the early grades and in the latergrades approached from transformations in the plane (reflections and dila-tions).
8. Practical Aspects: Althou.jh the practical aspects should not perhaps bementioned as a fundamental strand, these aspects are important enough inthe daily activities of pupils to warrant special attention. Beginning in theprimary grades the pupil is exposed to aspects of linear and cubic (cup,pint, etc.) measurement. By the end of Grade 4 the pupil will have someexperience in measuring the boundary and area of plane geometrical forms.Other practical aspects covered in the first three grades are money andmaking change and use of the thermometer.
27424
Apple& BPrONestiourireSspisabar, MO
29
-
30
TEACHER INFORMATION QUESTIONNAIRE
1. School: 2. City:
3. Your age: 4. Sax: M
(Do not mark inthis space)School Code
5. How many years have you taught?(Include this year but notpractice or intern teaching.)
6. What grade do you teach? (check one) First
Second
Third
Fourth
I do not teach one grade, but am an arithmetic/mathematics specialist in several grades.
7. How.many yearshave you taught this grade or in this capacity?
8. Prior to this school year, have you ever used a televisedarithmetic/mathematics program at lease once a week in yourclassroom? Yis
No
9. Number of semesters of mathematics courses taken in highschool:
10. Number of semesters of mathematics courses taken in college(undergraduate or graduate):
Mathematics for elementary school teachers.
College algebra
Other (e.g. .trigonometry, calculus, beyond calculus)
11. Number of semesters of mathematics education Courses taken'(undergraduate or graduate):
Methods of teaching arithmetic/mathematics
Seminars in mathematics education in elementary schools
12. When did you complete the following (give yearfor all which apply):.
High School 19
Bachelor's 19
Master's 19
Doctorate 19
13. Have you received college credit for course work taken Yessince your last degree?
NoIf "yes":
How many credits or points
In what year did you. last receive such credit? 19
14. Have you ever attended a National Science Foundation Yesor other institute in modern mathematics?
NoIf "yes":
How many.college credits did you receive for theinstitute?
In what year did you last attend such un institute? 19
15. Have you ever attended an inservice course in modernarithmetic or mathematics sponsored by your school orschool system?
Yes
No
If "yes":
In what year did you complete the inservice course(s)? 19
Approximanly how many hours were spent in theinaervice course(s) (check one):
1-4 hr 5-9 hr 10-19 hr 20-40 hr more than 40 hr
16. Compared to the other aubjects you teach, how.do you enjoy teachingarithmetic? (check one)
more than better moderately less than leastany other than most well, most of all
17. When teaching the more difficult concepts of arithmetic, how confidentdo you feel about your knowledge of those concepts?
completely quite somewhat not veryconfident confident confident confident
34
Appendix CPost.QuestionnaireMay, 1971
35
I. Name:
TEACHER QUESTIONNAIRE (Do not mark inthis space)Teacher Code
2. School: 3. City:
4. What grade do you teach? (check one) First
Second
Third
Fourth
I do not teach one grade, but am an arithmetic/mathematics specialist in several grades,
5. Have you taught the class you are now teaching for this entire school Yes No
year? If no, please explain:
6. Compared to the other subjects you teach, how do you enjoy teaching arithmetic?(check one)
more than better moderately less than least
any other than most well most of all
7. When teaching the more difficult concepts of arithmetic, how confident do you feelabout your knowledge of those concepts? (check one)
completely quite somewhat not very
confident confident confident confident
8. How many Patterns in Arithmetic telecasts did your class view this year? (check one)
ell (except for those shown MOst about half a few none
during vacation periods)
9. How many pages in the PIA student exercise book did your class complete? (check one)
34
all most about half a few none
PIA school teachers: go on to page 2, question #10.
Control school teachers: skip to page 5, pplanation of test.
10. Did you learn any new instructional methods from the Patterns in Axithmeticseries? many
11. Did you learn any new mathematics concepts from the PIA series?
S OSO
UOUS
many
S OSO
none
12. For how many lessons did you read the Mathematical Background section of yourPIA teacher's manual? . all
most
about half
a few
none
13. Now that you have had the experience of teaching with PIA for one year, how doyou think things would go if you used the program again next year? (Check the
response whtch comes closest to how you feel.)
This year went well, but next year Would be even better
This year went well, and next year would be about the same
This year dtd not go too well, but next year would definitely be better
This year did not go too well, and next year would be about the same
14. How do you feel about the effect PIA this year has had on your pupils' under-
standing of arithmetic?
Better than other classroom methods
Positive, but not necessarily better than other methods
Negative or no effect
(I have no basis to judge since this was my first year teaching)
15. Was it possible for you to individualize instruction using the PIA series?all of the time
much of the time
soms of the time
never or rarely
37 3 5
16. Vas Patterns in Arithmetic your major (basic) arithmeticprogram or a supplemental program this year? (check one)
major program
mupplemental program
I used PIA rarely or not at all
17. If PIA were available to you next year, how would youuse it? (check one)
major program
supplemental program
I would use it rarely or not at all
Directions: Please rate each aspect of PIA listed below byplacing a chick in one ma.
ZICAMPIR
Television reception
poor I 1 good
Value of the Patterns in Arithmetic teacher's manual
useless I I I I
Value of the pupil exercise books
useless I I I I
Value of the telecasts for students
useless I
36
I I I
38
Iuseful
Iuseful
Iuseful
Attitude of students toward telecasts
disinterested
Quality of the TV Teacher's presentations
interested
poor I I good
Pacing of the TV Teacher's presentation
too fast L I.
too short
aro 117.794-4
just right
Time interval between telecasts
just right
Level of difficulty of PIA forms class
too hard I
too slow
Jtoo long
just right
too easy
Your attitude toward PIA at the beginning of the school year
negative I I I I
4.Your attitude toward PIA now
negative
positive
positive
37/3?
Appendix DPretestSeptember, 1970
Note: Questions with an asterisk also appeared on the posttest.
40 3914D
1. Which of
to' the
the following number sentences are
Sentence whose model is shown on this
equivalent
Amber line?
5
II, 1'4 .
I I 4 5 6 7
(a) N * 7 + 5 (c) N + 5 7
(b) N 7 - 5 (d) 7 N 5
A. Only (a) and (b) D. Only (b),(c) and (d)
B. Only. (b) E. All of these.
C. Only. (b) anA (c)
2:k If x and y are positive even integers and if p and q
are positive odd integers, which of the.following is
FALSE?
A. x + y is a positive even intfter
B. p + q is a positivrodd integer .
C. xy is a positive even integer
D. pq is a positive odd integer
E. px is a positive even integer
3. Which of the following are true?
(a) # x
18(b) 75 is.the reciprocal of *(c)
(d) Pt x
A. Only (a) D. Only (q) and (d)
B. Only (b) E. All or these.
O. Only (a) and (b)
4141
4. In what base are the numerals written if 2 x'2 10?
A. Base four
B. Base three .
C. Base two
D. Base five
E. None of the above
is.correct,
"1° 0 011* 04'0
5. The equality of the arrays above illustrates which of
the following properties?
A. Commutative property of addition
B. Commutative property of multiplication
C. Associative property of addition
D. Associative property of multiplication
E. Distributive property of multiplication withrespect to addition
6. Given LRST as the unit angle
S30°
and t.he ZABC BT ,
A .
The sum of the measures of the angles.of AABC in
terms or the unit anale is:
A. 6 units
14. 7 units
C. 5 units
D. 180 units,
E. None of these.
7. A ball club won 4 of the 8 games already played.
If it wins the next two games, mhat percent of the
games will it then have won?
A. 80
B. .70.
C. 75
42 42
D. 50
E. 60
8. Which of the following represents the same number as
(6 x 104) + (6 x 10
2) + (6 x le)?
A. 6,066
B. 6,660
C. 60,066
D. 60,660
E. 66,600
QUestions 9 and 10 refir to the addition exercise below:
672 6 hundreds + 7 tens + 2 onesall. 8 hundreds + P tens + 4 ones
Q hundreds +30tens + 6 ones
R thousandc + S hundreds + T tens + U ones
= R, S T U (Final answer)
9.* P
A. 8
B. 3
C. 83
D. 34
B. 834
10. In the final answer, RI S T U represents:
A. The product of the numbers R,S, T and U.
B. The sum of the numbers R, S, T and U.
C. (11 x 1000) + (S x 100) + (T x 10) + (U x 1).
D. (R x 1) + (S x 10) + (T x 100) + (U x 1000).
E. Eleven thousand, flve hundred slx.
ll.* ln order to fInd Lhe product 6 x 9 we might:
A. Form the union of Lwo disjoint pets with six and
nine members.
D. Form a six by nine array,
C. Use the dIvIsion algorithm.
D. Notice that (6 x 9)base 10
3ten
x (6three
x 9three ).
E. Draw a triangle with sides 6" and 9" long.
4343
L
12. Which of the following sets of points beat represent
the inequality 2 < n < 7 where n is a whole num-
ber?
A.
B.
C.
D.
E.
<
0 1 2 3 4 5 6 7 8 9
<
7 6 5 3 2 1 0
<
15 16 17 18%19 20
u
%21 22
K23
2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9
5
420
13! Which of the following checks of ehe computation above
makes direct use of the commutative property?
4
A. 4 iiir B. 5 70- C. 5 D. 4 E. 4
20 20 5 4 ..2
0 0 5 4 20
....i4
20 420
4 x 376 = (4 x 300) + Cj + (4 x 6)
14.*
What number goes in place of Cj above?
A. 12
B. 24
C. 28
D. 240
E. 280
15.* which a the points A, Bp Cp Bp Or E On this num-
bet, line can be named by the fraction 6?
P0 A B E C D I
1 44 44 .0,fkol,'
.4
16. 612onine lo how many times as large as 812nine ?
A. twelve
B. ten
C. nine
D. five
E. None of the above iscorrect.
17. Indicate which English sentence or sentences corres-
pond to the open sentende, K < 13.
(a) The temperature is below thirteen degrees.
(b) Joan boughtmore than thirteen pairs of shoes.
(c) .There are fewer than 13 books overdue.
(d) Bill has thirteen dollars.
A. Only (a) and (c) D. Only (b), (c) and (d)
B. Only (c) and (d) E. All of them.
C. Only (a), (b), and (c)
18. Which of the following angles appear to be congruent
to LABC ?
A. Only LD and LFB. Only L E and z H
C. Only LD and La
D. Only LD, LE, and 4:0
E. None of these.
45 4. e 45
19. Which of the following are models for tne same rational
number as this one?
(c)
A. Only (a) /
B. Only (c) !
C. Only (a)/and (b)
(b)
(d )
D. Only (a), (b) and (d)
E. All of these
20. Whic of the following are true?
(4
(
(b)
(e)
( d )
If
(g. 3")
3 1
4 o5 !
1
70
5
112
3
A. Only (a)
B. Only (b)
C. Only (a) and (c)
D. Only (b) and (d)
E. None of these'.
21. The least common multiple of 18, 27, and 45 is:
46
A. 3 x3B. 2x 3 x5C. 2x3x3x3x5D. 2x3x3x3x3 x3x3x3x5E . None of the above.
46
Tom Mary Dick
7 )s1 7 598 7 59870 80 Is
22. The algorisms above by different pupils are based on
interpreting division as successive subtraction.
Which of the following statements can be made about
the step completed by the pupils?
A. All are wrong
B. Only Mary is right
C. Only Tom and Mary are right
D. All are right, but Dick's technique is the mostefficient
E. All are right but Mary' technique is the mostefficient
23. Four of the. following numerals are names for the same
number. Which one names a different number?
A. 0.025 D.724,
B. 2.5% E. 25 thousandths
C.
24. What is the, hundreds digit in the answer to
3375.49 7 861.1?
A.B.C.
524
. 1
25: What is the number of different prime factors of 60?
A. 0B. 1C. 2
D. 3E. 4
47
26. Choose the best way to complete this statement: Standard
units of measurement are used because
A. it is important for people to use the same unit in
dealing with each other.
B. standard units give more accurate measurements than
units which are not standard.
C. people have always used them.
D. they are all related to base 10 numeration.
E. the English Ostem is more natural than the metric
system.
2 . Which of the following are models for the same rational
number as this one?
ftc:
A. Only (a) and (b)
B. Only (a)
C. Only (d)
3 27
12 I 4
15
_10 k
9
2 30 9
(b) 1,1 1 vl
(d)
D. Only (a), (b) and (c)
E. All of thesC
28:k If the algorism above is based on interpreting division
as successive subtraction, then k
A. 2
B. 3
C. 5
D. 6
E, 9
48
29. Which of the following are empty sets?
(a) All odd numbers exactly divisible by 2.
(b) Women who have.been president of the U.S.A.
(c) All positive even numbers exactly divisible by 5.
(d) All whole numbers which do not have 1 as a factor.
A. Only (b)
B, Only (a) and (b)
C. Onli (b) and (c)'
D. Only (a), (b) and (d)
E. All of these.
30. Consider the set (p, 41, r, so 0 and the operation 0 between
two members of this set. The following is a table for 6
on this set.
ip rstPPPPPqpqrstr p r t I:I s
psqtt 'p t a r II
Which of the following statements is true?
I. rOss0rII. r 6 (t 6 11) (r 0 t) 0 9
III. p is the identity for 0
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
31! Suppose I is a binary operation with
1 J 1 0
2 _14_ 2 - 3
5 6
What is 6 _13A. 6
B. 3
C. 9
71 1.13'4 j k-15
9 _1_ 2 17
D. 17
E. 18
ir4949
SZ
x 2
460
864
32. The baste property used in the algorism above is
Illustrated by which of the following?
A. x(yz) (xy)z
B. X + (y + z) u (X y) + Z
C. x(y + z) xy + xz
D. x-1-y+z=z-1-y+xE. xyz zyx
33. The most Important common property of the sets (A, 2, )and (0, 1, 2) is suggested by
A. 2 D. empty
B. (,, ) E. 3
C. 6
3
3 1
96 2.
3
3 1
0 4
34* Which of the following number-line diagrams
. illustrates the subtraction operations depicted
in the division algorism above?
ir-Ne-"'Ne"N.O 1 2 3 4 5 6 7 8 9 10 11 12
B.
O 1 2 3 4 5 6 7 8 9 10 11 12
O 1 2, 3 4 5 6 7 8 9 .10 11 12
D
O 1 2 3 4 5 6 7 8 9 10 11 12
O 12345650
7 8 10 11 12
35! The inverse operation .for addition is:
A. addition.
B. Subtraction.
C. multiplication.
D. division
E. None of these.
- j(). What, la btic &tam or (3 4ind 3cight ?
A. 9eig1tD. lleight
B. 18 ei.ght E. None of these.:
C. 10eight
12 + 3 4
37. If the division above is interpreted as finding
how many groups of 3 each make up 12, Which of the
following .operations is most nearly analOgons.to
this?
A. Finding the size of a hand of cards if youknow the size of the deck and the number ofhands
B. Finding the area of a lot if a tract of landof.known area is subdivided into a given .
number of equal lots
C. Finding the number of feet in a 48-inch lengthwith a 12-inch ruler
D. Dividing a pie equally among a given numberof people
E. Finding what number can be subtracted fourtimes from 12
38. A point P lies 4.03 inches frivol the center of 4 circle,With a radius of 4.3 inches. ifhich of the followingis true?
A. P lies outside the circle
B. P lies at the center
C. P lies inside the circle, but not at the center
D. P lies somewhere on the circumference
E. It cannot be determined whether P lies inside,on, or outside the circle
.0;. f,011
39. Which of the following is the best example of somethingunique?
A. a symbol for the number of fingers 4fou have
B. the decond house'from a certain corner in a
residential distriCt
C. the number of eggs in a dozen
D. a point .on a ruler which is one inch from the
three-inch mark
E. a rectangle with one side five inches.long.
40.* If r < 0, s>.p, s< q, and 0 < po4hich of the
following numbeelines shows.the correct relatiwe
positions of p, q, re and i?
rB. 4.---116
C 4
D.
s q
I0
E.
41. If 13 . candy bars are to be given to 3 boys so that
each boy receLvea at least one candy bar, which of the
following statements must be true?
A. One boy receives 10 candy bars.
B. Each boy receives at.least 2 candy bars.
C. One boy receives at 'least 5 candy bars.
D. One boy receives at'least 7 candy bars.
E. The .boys all receive a different number of candy
bars.
52
42. If X = (0, a) and Y = (7,8), the most important
property Of the set .Xl.)Y is suggested by
D. the order b a, 7, 8
B. 2
43Y If x, ys z are numerals in base five notation, which
of the following' are correct?
(a)
(b)
(c)
x +y=y+ xx(y +z) = x.y + x.z
if x < y and y < z then x < z:
A. Only (a)
B. Only (b)
C. Only (a) and (b)
D. Only (a) and (c)
E. (a), (b), and (c)
44. Which of the. following are true?
(a)
(b) +
(a)
(d)
A. Only (a)
B. Only (a) and (b)
C. Only (c) and. (d)
D. Only (a), (b) and (c)
E. All of them.
45.*
If A and B are two sets which can be matched (put
in 1 to 1 Correspondence) then:
A. A and B must both be sets of numbers.
B. A and B. must have the same number of elements.
C. Neither A nor B can be the empty set.
D. A and B must both be sets of points.
B. A and B must each have one member.
.(5.3 53
46.* Which of the following is the Multiplicative inverse
of 4.0?
A. 0.04
B. 0.25
C. 0.4
D. 0.5
E. 1.25
47.* Which one of the.following division.exnmples wili result
in the quotient, 25.2 ?
A. 5T-7176
B. .5T.126c . .05 T.12S
D. .0357-77205
B. .0000-771215
48! A football team starts the following series of plays
from its own 20-yard line: gains 8 yards, loses 2
yards, loses 1 yard, gains 6 yards. On what yard
line is the ball now?
A. 11
B. 26
C. 31
D. 34
E. 37
49. Which of the following intersection sets is.impossible
for a straight ling and a trtangle?
A. The empty.set
13. A set' of exactly one point
C.. A.set of exactly two points
D. A set of exactly three poimte
E. A set of an Infinite number'of points.
54
54G PO 127.794-3
50. The equality 2 x (3 + 4) 2 + (3 x 4) is an
illustration of which of the following
principles?
A. The associative law of addition
B. The commutative law of addition
C. The distributive law of multiplicationwith respect to,addition
D. The commutative law of multiplication
E. None of these
Appendix EPosttestsMay, 1971
Note: Questions 1-15 were, for the most part, different foreach grade level. These appear in this appendix as PIA-specificcontent, Grades 1, 2, 3, and 4. Questions 16-35 were the samefor all grade levels, as they were culled from the common pretest.In the actual tests, there was no break between questions 15 and16. However, to avoid duplication, questions 16-35 appear onlyonce in this appendix, under the label Basic Mathematics.
57
EXPLANATION OF THE TEST
This is a test on some of the basic mathematical concepts which underlie
a modern arithmetic program. Many of these concepts have not, in the past, been
stressed in the elementary school curticulum; consequently, some of the ideas
in this test may be.unfamiliar to many teachers.
Results from this test will be compared with those from a similar test given
last fall. If you did not take a test similar to this last fall, check here:
I did mt take a
test similar to
this one last fall.
DIRECTIONS
All of the questions in this test are multiple choice. You are to select
the one best answer from the five choices given and circle the letter of that
answer choice.
Here are two sample questions:
EXAMPLE Oi
:Which of the following:is a multiple of 10?
A. 1 D. 8460
B. 5 E. 77777,
C. 101
The correct answer.is .since 8460 846 x 10. Therefore, the letter D
would be circled:
A. 1
B. 5
C. 101
58
.0 8460
S. 77777.
5'7
EXAMPLE 00.
Which of these is another name for 27?
(1) '2 x 10 + 7
(II) 3 x 3 x 3
(III) 72
A. ,Only I
B. Only III
C. Only I and II
D. Only I and III
E. I, II and III
Both 2 x 10 + 7 and 3 x 3 x 3 are equal to 27. Therefore the correct
answer is "C. Only I and II," and the letter C is circled.
You may use any extra space for scratch work. Go ahead and begin.
Selected items are reproduced by permission of Educational Testing Service, thecopyright owner. Others are copyrighted by the Board of Trustees of the LelandStanford Junior University, and are reproduced with author's permission.
59/60
PIA-specific content: Grade 1
59
1. Which of the points are on the same side of the curve?
A. P and Q
B. P and R
C. R and Q
D. p, Q and R
E. none of them
2. In which of the following pairs of numerals does the '8' have the sameplace value?
(/) 81 and 812 (II) 4802 and 876 (III) 8 .and
A. only I. D. only I and III
B.
C.
only II
only I .and II
E. .only II and III
3. Set A has members 2, '3, 4, 5, and 6. 'Which of the following numbers comprise'a subset of A?
A. 0, 1, 7, 8, 4
B. 1, 2, 3, 4, 5
C. 1. 2'
D. 2, 5
E. 0, 1
4. Which is an example of changing something from a closed curve to an opencurve?
A. breaking a rubber band 0, cutting a hole in a fish net
B.
C.
peeling a tangerine
bending a hula hoop .
E. snapping a rut% in a 6-.footladder
6062
5. If set A 11, 2, 31 and set B 3, 2, 1 then A and B are
A. equal D. associative
B. equivalent but not equal E. commutative
C. disjoint
6. Which point is inside the curve?
A. A
B. B
C. C
D. D
E. none of these
7. Which of these shows the meaning of inverse operations?
D. (3 + 5) - 5 3
E. (3 + 5) + 0 3 + (5 + 0)
A. 3 - 3 - 0
B. 3 + 0 3; 3 - 0 = 3
C. 3 + 5 5 + 3
8. Which of these is an example of the commutative property of addition?
A. 6 + 0 6
B. 2 + 2 mg 2 x 2
C. 2 + 6 6 + 2
D. 2 + 6 = 3 + 5
E. 6 + (2 + 3) = (6+ 2) + 3
9. If a, b, c, d and e are whole numbers, and if a > d, b < c, e > c
and a < b, which is the largest number?
D. d
E. e
63
L00000 I
0.0 0 0 0
10. Which of the following is a simple closed curve:
A. c51 D. CPC)
B. C.09 E.
11. Which of the following is exact? Finding out the number of
A. pounds a boy weighs D. miles between two cities
B. inches in a foot E. square inches in the areaof this page
C. cups of water in a bowl
12.
X 7 9
13.
On the number line above, X im
A. 0
B. 1
C. 4
lst'row
2nd row
D. 5
E. not necessarily any of these
A dhild says that there are more circles when they are spread out (as in thesecond row above) than when.they.are close together. This most probablymeans he is not yet able to
A. conserve numerousness
B. count from one to five
C. memorize simple addition facts
64 62
L recognize when one stick isshorter than another
E. say "five" when presented withthe numeral 5
14. If set A is equivalent to set B and set B is equivalent to set C, then
(I) A and C are equivalent
(II) A and C have the same members
(III) A and C can be put into one-to-one correspondence
A. I only
B. II only
C. I and III only
D. II and III only
E. I, II and III
15. Which number is associated with the empty set?
A. -1 D. 10
B. 0 E.
C. 1
6
CO
65/0,
PIA-specific content: .Grade 2
-6467
1.
X 7 9
On the number line above, X =
A. 0 D. 5
B. 1 E. not necessarily any of these
C. 4.
2. When performing the indicated subtraction, 34
34 should be considered as - 1816
A. 20 tens, 4 ones D. 2 tens, 14 ones
B. 20 tens, 14 ones E. 3 tens, 14 ones
C. 30 tens, 4 ones
3. Which of the following is exact? Finding out the number of
A. pounds a boy weighs
B. inches in a foot
C. cups of water in a bowl
D. miles between two cities
E. square inches in the areaof this page
4. Which is an example of changing something from a closed curve to an open
curve?
A. breaking a rubber band
B. peeling a tangerine
C. bending a hula hoop
68
D. cutting a hole in a fish net
E. snapping a rung in a 6-footladder
S 0
0 1 2 3 4 5 6 7 8 9 10 11This number line diagram shows which of the following situations?
A. 4 + 5 = 9 D. 9 - 4 - 5 = 0
B. 9 - 4 - 5 E. 0 + 4 + 5 = 9
C. 9 - 5 - 4
6. Which point is inside the curve?
A. A
B. B
C. C
D. D
E. none of these
7. How many members are in the solution set for n < 6 if the solutionset is draWn from the set of positive odd numbers?
8. The shaded area of the polyhedronis called a
A. prism
B. pyramid
C. edge
D. face
It. vertex
D. an infinite number
E. none of these
6669
.
90 Which of these would be solved using subtraction?
A. n + 7 = 12
B. n - 7 = 12
C. 12 + 7 = n
D. n - 12 = 7
E. none of these
10. In order for a pupil to decide which is the larger of 248 and 263, it
is sufficient for him to compare the
A. tens D. hundreds and ones
B. hundreds E. hundreds arid tens
C. tens and ones
11. If a, b, c, d and e are whole numbers, and if a > d, b < c, e > c
and a < b, which is the largest number?
A. a D. d
B. b E. e
C. c
12. Which of these shows the meaning of inverse operations?
A. 3 - 3 = 0 D. (3 + 5) - 5 = 3
B. 3 + 0 = 3; 3 - 0 = 3 E. (3 + 5) + 0 = 3 + (5 + 0)
C. 3 + 5 = 5 + 3
13. 419 can be rewritten
A. 2 hundreds + 10 tens + 9 ones
B. 2 hundreds + 20 tens + 19 ones
C. 3 hundreds + 9 tens + 19 ones
D. 3 hundreds + 10 tens + 9 ones
E. 3 hundreds + 20 tens + 19 ones
6770
GPO 827494-6
14. Geometric figures which are the,same size and shape are
A. equal
B. equivalent
C. proportional
15. The column.which is shaded inthe table shows that
A. n+Omn
B. nxlm n
C. n x 0 m 0
D. n n m 1
E. n+rmr+ n
D. synszetric
E. congruent
0 1 2 3 4.
40 1 2. 3
1 2 3 4 5
2 3 4 5 6A.
3 4 5 6 7
4 5 6 7 8
71/12
PIA-apecifIcContent: Grade 3
69
73
The equality of the arrays above illustrates which of the followingproperties?
A. Commutative property of addition
B. Commutative property of multiplication
C. Associative property of addition
D. Associative property of multiplication
E. Distributive property of multiplication with respect to addition
2. Which of these number pairs (ratios) would NOT be used when convertingbetween feet and yards?
2A.
9 17D.
51
12B.
4
3. Which point is inside the curve?
A. A
B. B
C. C
D. D
E. none of these
E. 227
4. When performing the indicated Subtraction,34 should be considered as
A. 20 tens, 4 ones
B. 20 tens, 14 ones
C. 30 tens, 4 ones
qw,1074
,
34-1816
D. 2 tens, 14 ones
E. 3 tens, 14 ones
5. Which of these figures contains the largest angle?
A
6. What is the name of the number property that links addition with multiplication?
A. Associative
B. Commutative
C. Distributive
D. Identity
E. Inverse
7. Which of the followlng is exact? Finding out the number of
A. pounds a boy weighs
B. inches in g foot
C. cups of water in a bowl
D. miles between two cities
E. square inches in the areaof this page
B. Geometric figures which are the same size and shape are
A. equal D. symmetric
B. equivalent E. congruent
C. proportional
9. Which of these shows the meaning of inverse operations?
A. 3 - 3 is 0 D. (3 + 5) -.3 3
B. 3 + 0 3, 3 - 0 - 3 E. .(3 + 5) + 0 3 + (5 + 0)
C. 3 + 5 5 + 3
75
10. If a, b, c, d and e are whole numbers, and if a > d, b < c, e > cand a < b, which is the largest number?
A. a D. d
B. b E. e
C. c
11. Which of these is NOT a factor of 12?
A. 1 D. 9
B. 2 E. 12
C. 4
12. The basic property used in the algorism to theright is illustrated by which of the following?
A. a(bc) (ab)c
B. a + (b +c) a + (c + b)
C. a(b + c) ab 4- az
D. a+b+cc+b+ aE. abc cba
13. Which shaded area represents the largest fraction?
A
76
. , 72
432x 2
460
800864
14. The eqvality. 2 x (3+4) 2 + (3 x.4) is an illustration of whichof the'followtng principles?
A. The associative law of addition
B. The commutative law of addition
C. The distributive law of multiplication with respect to addition
D. 'The commutative law of multiplication
E. None of these
15. The column which is shaded inthe table shows that
0A. n + 0 n
1B. n x n
C. n x2
3D. n n 1
4E. n+r wr+n
0 1 2 3 4\
.........4:::::i.....
AO44W.
i2 3
g: ... pe.2 3 4 5
trtftl!..'
...i:2P 3 4 56,.4:1.:..."§
;Ivo%*31.§l'illud
4
.5
.
674'
...:4..
.
7 8
734. 77hy
PIA-spec/Etccon ten t : Grade 4
74
79
1 Which of these is NOT a factor of 12?
A. 1
B. 2
C. 4
D. 9
E. 12
2. Which number sentence would most apPropriately be solved by division?
A. n x 8 = 24
B. 3 x 8 = n
C. 8 x 3 = n x 3
D. nx8 = 8xn
E. n + 8 + 8 =. 24
3. Which statement about geometric.figures is false?
A. a cube has 8 faces
B. two congruent triangles are similar
C. two perpendicular lines form four right angles
D. if the foci of an ellipsi coincide, the erlipse is a circle
E. a triangle with sides 3, 4, and 5 is* similar to one with sides9, 12, and 15
4. The basic property used in the algorism to theright is illustrated by which of the follouing?
A. a(bc) = (ab)c
B. a + (b +c) = a + (c + b)
C. a(b + c) ab + ac
D. a+b+c=c+b+ a
E. abc'= cba
7580
432x 2
460
800864
5 Geometric figures which are the same size and shape are
A. equal D. symmetric
B. equivalent E. congruent
C. proportional
6. Which shaded area represents the largest friction?
A
If a, b, c, d and e are whole numbers, and if a > d, b < c, e > cand a < b, Vbich is the largest number?
A. a
B. b
C. c
D. d
E. e
8. Which of the following is exact? Finding out the number of
A. .pounds.a boy iiiihs
B. inches in a foot
C. cups of water in a bowl
D. miles between two cities'
E. sluare inches in the areaoi this page
The equality 2 x (34-4) 2 + (3 x4) is an.illustration of which'
of the, following principles?
IL The associative law of addition
B. The commutative law Of additiOn
C. The distributive.law of multiplication.with respect to addition
D. The commutative law of multiplication
E. None Of these
81
;.!
10. Which number sentence represente this story problem:
"Carlos has 40 pennies in his coin collection. He can put 8pennies' on each page of his coin book. How many pages canhe fill!"
11.
A. n x 8 40
B. 8 + n 40
C. 8 x 40 n
AD so WOMB
D. 8 n 40
E. 40 - 8 Is n
The equality of the arrays above illustrates which of the followingproperties?
A. Commutative property of addition
B. Commutative property of multiplication
w. Associative property of addition
D. Associative property of multiplication
E. Distributive property of multiplication with respect to addition
12. Which of the following ratios is not equivalent to the others?
A.
B.
C.
46
6
9
9
12
E. 18
27
13. What number property explains why themultiplication table is symmetric aboutthe otted line?
A. n x 0 0
B. nxl n
C. n n 1
D. nxrmirxn
E. none of these
\ 0 1 2
0 %0\
0 0 0 0
1 0\1 2 3 4
2 0 2 \4\ 6 8
3 0 3
-
6
N
\
4 0 4 8 12\16\
L
14. Which of the follawing pairs of sentences are equivalent?
(I) 8 + n "A 13 and 13 - 8 n(II) 7 + n 10 and 10 - n m 7
(III) n + 3 9 and 119 - 13 n
A. only I
B. only I and II
C. only / and III
D, only /I and III
B. /, // and III
15. Which figure below does NOT showe line of symmetry?
A
Goo
Basic Mathematics
Questions 16-35
16. Which of the following number sentences are equivalent to the sentencewhose model is shown on this number line?
7
0 I 2 4 4 5 6 7
(a) N 7 + 5
(b) N a. 7 - 5
A. Only (a) and (b)
B. Only (b)
C. Only (b) and (c)
79
(c) N + 5 = 7
(d) 7 - N = 5
D. Only (b), (c) and (d)
Z. All of these
85
17. If x and y are positive even integers and if p.and q are positive oddintegers, which of the following is FALSE?
A. x. + y is a poiitive even integer
1. p + q is a positive odd integer
C. xy is a positive even. integer
D. pq is a positive odd integer
E. px is a positiile.even integer
Questions 18 and 19 refer to the addition exercise below:
672 6 hundreds + 7 tens+ 2 ones4.. 834 8 hundreds 4 P tens + 4 ones
Q hundrede'+10 tens + 6 ones
R. thousands + S hundreds + T tens +.0 ones
S T U (Final answer)
18. P
A. 8
B. 3
C. 83
D. 34
E. 834
19. In the final answer, RI S T U represents:
A. The product of the nuabers R, SI T' and U
B. The sum of the numbers R, S, T and U
C. (R x 1000) + (S x'100) + (r x1.0) + (U x I)
D. (IR x 1) + (S x 10) + (Tx 100) + (11 x 1000)
E. Eleven thousand, five hundred six
20. In order to find the product 6 x 9 we might:
A. Form the union of two disjoint setuwith six and nine members
B. Form a six by nine array
C. Use the division algorithm
D. Notice that (6 x 91-base 10 m ten x Othree x 9threeY"
E. .Draw a triangle with sides 6" and 9" long.
86 80 GPO 1127-794-7
I4 x 376 = (4 X 300) + 0 + (4 x 6) 1
21.
22.
5
4
20
Which of the following checks of the computation above
makes direct use of the commutative property?
4A. 4 1-2i B. 5 Y5 C. 5 D. 4 E. 4
20 20 5 4 -20 0 5 4 20
5 4
20 4
20
What number goes in place of 0 above?
A. 12
B. 24
C. 28
D. 240
E. 280
23. Which of the points A, B, C, D or E on this number line cin be named bythe fraction 5 ?
8
0 A B E C D 1
24. Choose the best way to complete this statements Standard
units of measurement ire used because
A. it is important for people to use the.same unit in.
dealing with each other.
B. standard units give more accurate measurements than
units which are not standard.
C. people have always used them.
D. they are all related to base 10 numeration.
E. the English sYstem is more natural than the metric
system.
81 87
25.
12 4
15
k9
30 9
If the algorism above is based on interpreting division as successivesubtraction, then k
A. 2
B. 3
C. 5
D. 6
E. 9
26. Suppose I is a binary operation with
1 1 1 - 0
2 1 2 - 3
5 6 29
What is 6
A. 6
B. 3
C. 9
.1_3 ?
7 _LI 13
4 L 4 15
9 1 2 17
27. The inverse operation for addition is:
A. addition
B. subtraction
C. multiplication
D. division
E. none of these
28. A point P lies 4.03 inches from the center of a circle with a radius of 4.3inches. Which of the following is true?
A. P Ues outside the circle D. P lies somewhere on thecircumference
B. P lies at the center
2. It cannot be determined whetherC. P lies inside the circle, but
not at the centerP lies inside, on, or outsidethe circle
88
29. 3
3 1
9
6 2
3
3
0 , 4
Which of the following numberline diagrams illustrates the subtractionoperations depicted in the division alorism Above?
A.
B.
D
E.
Al"1,e0"11e-N0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12 '
0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
30. If r < 0, s > p, s <q, and 0 < p, whiCh of the following number linesshows the correct relative positions of p, q, r, and s?
A
B
C 4
a
0
s q
D.
B.
s 2I I
r q sI 1
S.
31. If xo yo ,s are numerals in base five notation, which of the following arecorrect?
(a) x+yny+ x
(b) x (y + z) xy + xs
(c) if x < y and y < z then x < z
A. Only (a)
B. Only (b)
C. Only (a) and (b)
D. Only (a) and (c)
E. (a), (b), and (c)
32. /f A and B are two sets which can be matched (put in one-to-one correspondenceithen:
A. A and B must both be sets of numbers
B. A and B must have the same number of elements
C. Neither A nor B can be the empty set
D. A and B must both be sets of points
B. A and B must each have one member
33. Which of the following is the multiplicative inverse of 4.0?
A. 0.04
B. 0.25
C. 0.4
D. 0.5
E. 1.25
34. Which ona of the following division examples will result in the quotient 25.2?
A. 5) -7016
B. .0-771s
C. .00-71716
90
D. .0057-777.5
B. .0005)-772.5
35. A football team starts the following 'series of plays from iti awn20-yard line: gains 8 yards, loses 2 yards, loses 1 yard, gains 6yards. On what yard line is the ball now?
A. 11 D. 34
B. 26 E. 37
C. 31
When you have finished, please check to make sure you have answered
all questions on'the questionnaire and on this test. Then hand in
this booklet along with your honorarium form.
Thank you for taking part in this project.
ss 90.
Appendix FAdditional Statistical Tables and Figure
86
Tab
le 1
-FSa
mpl
e M
ean
Scor
es b
y T
reat
men
t by
Site
(O
btai
ned
from
Ave
rage
s ov
er G
rade
)
SIT
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rban
Site
1V
aria
ble
Sam
ple
PIA
Con
trol
Ave
.
Urb
an S
ite 2
Sam
ple
PIA
Con
trol
Ave
.
Rur
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ite 1
Sam
ple
PIA
Con
trol
Ave
.
Rur
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ite 2
Sam
ple
PIA
Con
trol
Ave
.
Ave
rage
a
All
Site
sPI
A C
ontr
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Kno
wle
dge:
Pre
test
sB
asic
Mat
hb12
.93
11.3
612
.15
10.3
09.
369.
8310
.63
10.0
910
.36
13.6
610
.28
11.9
711
.88
10.2
711
.08'
Oth
er P
rete
st I
tem
sc11
.57
10.9
311
.25
10.0
29.
139.
579.
759.
589.
6611
.48
8.76
10.1
210
.70
9.60
10.1
5T
otal
24.5
022
.29
23.4
020
.32
18.4
919
.40
20.3
819
.67
20.0
225
.14
19.0
422
.09
22.5
919
.87
21.2
3
Kno
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dge:
Pos
ttest
PIA
-Spe
cifi
cd9.
528.
348.
938.
907.
598.
247.
888.
808.
349.
757.
208.
489.
017.
928.
47B
asic
Mat
hb14
.02
13.6
713
.84
9.70
10.7
910
.24
11.7
212
.40
12.0
615
.10
12.4
813
.79
12.6
312
.33
12.4
8
Gai
n: B
asic
Mat
h1.
092.
311.
70-.
601.
43.4
11.
092.
311.
701.
442.
201.
82.7
52.
061.
41
Atti
tude
: Pre
-que
stio
nnai
reE
njoy
men
te8.
938.
578.
757.
847.
267.
558.
147.
727.
937.
949.
188.
568.
218.
188.
20C
onfi
denc
ee6.
116.
866.
487.
006.
156.
586.
286.
076.
175.
856.
606.
236.
316.
426.
36
Atti
tude
:. Po
st-q
uest
ionn
aire
Enj
oym
ente
8.90
8.96
8.93
7.75
7.15
7.45
7.80
8.79
8.30
8.46
9.33
8.89
8.23
8.56
8.39
Con
fide
ncee
8.45
8.02
8.24
7.60
7.00
7.30
7.96
8.79
8.38
7.50
6.95
7.23
7.88
7.69
7.78
Sam
ple
Size
- S
choo
ls20
1838
54
95
510
88
1638
3573
Tea
cher
s69
7814
754
3892
3536
7121
1839
179
170
349
eAll
aver
ages
are
unw
eigh
ted
arith
met
ic m
eans
.
b20
item
s.
c30
item
s.d15
item
s.eF
or p
urpo
ses
of a
naly
sis
and
coi-
tpar
abili
ty, t
he e
njoy
men
t and
con
fide
nce
vari
able
s fr
om th
e qu
estio
nnai
res
wer
e tr
ansf
orm
ed s
oth
atea
ch h
ad a
ran
ge f
rom
1 =
low
to 1
3 =
hig
h.
Tab
le 2
-FW
eigh
ted
Mea
n U
niva
riat
e A
naly
sis
of V
aria
nce,
with
Gra
de a
s R
epea
ted
Mea
sure
With
in S
choo
l
Sour
ce
1 =R
esul
tsPI
A-S
peci
fic
2B
asic
Mat
hem
atic
sD
irec
tion
If R
elev
ant
NIS
rM
SF
2M
ean
110
473.
98--
---
-23
444.
22A
mon
g Sc
hool
sa72
Am
ong
Cel
ls7
Am
ong
Site
s3
4.46
40.
99.4
078
.654
9.55
.000
1*T
reat
men
tb1
29.9
756.
66.0
13*
1.75
80.
21.6
5PI
A h
ighe
rT
reat
men
t x S
ite3
11.7
422.
61.0
612
.034
1.46
.23
With
in C
ells
656.
040
1.34
.13
13.6
291.
65.0
3*W
ithin
Sch
ools
66A
mon
g G
rade
s Sp
ecif
ic to
Site
s4
7.00
71.
56.2
02.
525
0.31
.87
Tre
atm
ent x
Gra
des
Spec
ific
to S
ites
42.
452
0.54
.70
7.96
80.
97.4
3Sc
hool
s x
Gra
des
With
in C
ells
584.
501
---
8.24
0--
-
*Sta
tistib
ally
sig
nifi
cant
with
.05
leve
l cri
teri
on.
aAm
ong
Scho
ol a
naly
sis
conf
orm
s to
cla
ssic
al A
mon
g U
nits
ana
lysi
s fo
r "r
epea
ted
mea
sure
s" a
naly
sis.
bRem
oved
aft
er s
ite e
ffec
ts r
emov
ed;
assu
mes
Tre
atm
ent x
Site
eff
ects
are
zer
o.
i
co 4:T
able
3-F
Est
imat
es o
f Pa
ram
eter
s: K
now
ledg
eV
aria
tions
in C
ovar
ianc
e M
odel
(Site
x T
reat
men
t Eff
ect A
ssum
ed Z
ero)
Para
met
er
Var
iabl
e
PIA
-Spe
cifi
cB
asic
Mat
h (p
ost)
Gai
n: B
asic
Mat
h
Est
imat
eS.
E.a
(Est
.)S.
E .
aE
stim
ate
(Est
.)E
stim
ate
s.E
.b(E
at.)
Cov
aria
tes
(Reg
ress
ion
Coe
ffic
ient
s)B
asic
Mat
h Pr
etes
t (A
vera
ge)
.380
.065
.807
.072
-.18
6.0
71B
asic
Mat
h Pr
etes
t x T
reat
men
t-.
11 2
.065
.050
.072
Bas
ic M
ath
Spec
ific
to P
IA.2
68.0
46.8
57.0
50W
ithou
t PIA
.492
.1 2
2.7
56.
.135
---
Tar
get P
opul
atio
n M
ean
8. 2
7c.2
91
2.94
d.3
31.
84.3
0
Site
Con
tras
tsU
rban
-Rur
al.4
2.7
9-1
.49
.88
-1. 5
3.8
7U
rban
Site
1-U
rban
Site
2-.
16.5
61.
7 6
.62
1.77
.61
Rur
al S
ite 1
-Rur
al S
ite 2
.3 2
.59
-.36
.66
-.4
2.6
5
Gro
ss P
IA I
ncre
men
te3.
1 28
1 . 5
25
-2. 0
391
. 693
-.89
.39
Ave
rage
PIA
Inc
rem
ente
.658
.373
-.93
1.4
15-.
89.3
9
adf
= 6
3.
bdf
= 6
4.
cOut
of
15 it
ems.
dOut
of
20 it
ems.
eThe
"gr
oss"
incr
emen
t is
the
diff
eren
ce if
the
Bas
ic M
ath
Pret
est S
core
is z
ero;
the
"ave
rage
" in
crem
ent i
s th
e in
crea
se a
t the
mea
n Pr
etes
t Sco
re o
f 11
.1; e
.g.,
the
gros
s do
es n
ot a
llow
for
var
iatio
n in
NA
eff
ect b
y Pr
etes
t Sco
re.
Tab
le 4
-FE
stim
ates
of
Para
met
ers:
Atti
tude
Var
iatio
ns in
Cov
aria
nce
Mod
el(S
ite x
Tre
atm
ent E
ffec
t Ass
umed
Zer
o)
Para
met
er
Var
iabl
e
Enj
oym
ent o
f T
each
ing
Mat
hC
onfi
denc
e in
Tea
chin
g M
ath
Stan
dard
Err
orE
stim
ate
(Est
imat
ed)
Est
imat
eSt
anda
rd E
rror
(Est
imat
ed)
Cov
aria
tes
(Reg
ress
ion
Coe
ffic
ient
s)E
njoy
men
t Pre
test
.586
.131
.222
.119
Con
fide
nce
Pret
est
.1 2
0.1
08.4
05.0
98
Tar
get P
opul
atio
n M
eana
8.53
5.2
917.
554
.269
Site
Con
tras
tsU
rban
-Rur
al-.
754
.884
-.30
1.8
01U
rban
Site
1-U
rban
Site
2.7
66.6
17.7
31.5
59R
ural
Site
1-R
ural
Site
2-.
225
. 65
21
. 309
.590
PIA
Inc
rem
ent
-.26
0.3
79.4
64.3
44
aFor
pur
pose
s of
ana
lysi
s, th
e en
joym
ent a
nd c
onfi
denc
e va
riab
les
from
the
ques
tionn
aire
wer
e tr
ansf
orm
edso
that
eac
h ha
d a
poss
ible
ran
ge f
rom
1 =
low
to 1
3 =
hig
h.
98
PI(1)(1
InA(1
1
Urban Urban Mural Rural
101-
IN-
A-Speollio 7-I Items)
Sr-
3-
1
illit
.
1
'r,
r ,>!
---t
77;
ne T
.%
111).-xl
s 4:0 :I,..
11
i.
., ...
.
1
t
X I 0111111
ii1
20MO 15ithomallas111,-10 Item) 14-
4-2-
--
fk't
ti,
: pi;
fV
.
//.
(1z,,//, z
10b-
11-
1Ioyment II-Toaohing 7,Ithmtioolow, , S
91'1110) 4
2
-4
u.1
V
%
7-.
2e
10
milldam 8
'reaching 7MimeticHow, 5Boihigh) 4
3-2-1-
-
-
F.
/
/.
x 'S.
.
)4 Is
.
P
7
/1 /.1
I
17 °X
F3Experimental Group: Sept. 1970xperimental Group: May 1971Control Group: Sept. 1970
12 Control Group: May 1971
Fig. 1-F. Performance levels on knowledge and attitudevariables for experimental and control teachers:Sample data.
References
Braswell, J. Patterns in Arithmetic field test-ing. Grade 2. 1967-68. Wisconsin Researchand Development Center for Cognitive Learn-ing, Working Paper No. 17, 1969a.
Braswell, J. Formative evaluation of Patternsin Arithmetic, Grade 5, 1967-68. Wiscon-sin Research and Development Center forCognitive Learning, Working Paper No. 18,19691).
Braswell, J. The formative evaluation of Pat-terns in Arithmetic. Grade 6. using itemsemolina. Wisconsin Research and Devel-opment Center for Cognitive Learning, Tech-nical Report No. 113, 1970.
GPO 127494-0
92
Braswell, J. S., & Romberg, T. A. Objectivesof Patterns in Arithmetic and evaluation ofthe telecourse for Grades 1 and 3. Wiscon-sin Research and Development Center forCognitive Learning, Technical Report No.67, 1969.
Educational Testing Service. Sets and SystemsAchievement Tests I-V. Princeton, N. J.:E.T.S., 1964.
School Mathematics Study Group. SMSGFilm Text Evaluation Studv MathematicsInventory. Stanford, California: The Boardof Trustees of the Leland Stanford JuniorUniversity, 1963.
99
National Evaluation Committee
Helen BainImmediate Past PresidentNational Education Association
Lyle E. Bourne, Jr.Institute for the Study of Intellectual BehaviorUniversiy of Colorado
Jeanne S. ChallGraduate School of EducationHarvard University
Francis S. ChaseDepartment of EducationUniversity of Chicago
George E. DicksonCollege of EducationUniversity of Toledo
Hugh J. ScottSuperintendent of Public SchoolsDistrict of Columbia
H. Craig SipeDepartment of InstructionState University of New York
G. Wesley SowardsDean of EducationFlorida International University
Benton J. UnderwoodDepartment of PsychologyNorthwestern University
Robert J. WisnerMathematics DepartmentNew Mexico State University
Executive Committee
William R. BushDirector of Program Planning and Managementand Deputy Director, R & D Center
Herbert J. Klausmeier, Committee ChairmanDirector, R & D Center
Wayne OttoPrincipal InvestigatorR & D Center
Robert G. PetzoldProfessor of MusicUniversity of Wisconsin
Richard A. RossmillerProfessor of Educational AdministrationUniversity of Wisconsin
James E. WalterCoordinator of Program PlanningR & D Center
Russell S. Way, ex officioProgram Administrator, Title HI EWAWisconsin Department of Public Instruction
Faculty ef Principal Investigators
Vernon L. AllenProfessor of Psychology
Frank H. FarleyAssociate ProfessorEducational Psychology
Marvin J. FruthAssociate ProfessorEducational Administration
John G. HarveyAssociate ProfessorMathematics
Frank H. HooperAssociate ProfessorChild Development
Herbert J. KlausmeierCenter DirectorV. A. C. Henmon ProfessorEducational Psychology
Stephen J. KnezevichProfessorEducational Administration
Joel R. LevinAssociate ProfessorEducational Psychology
f33
L. Joseph LinsProfessorInstitutional Studies
Wayne OttoProfessorCurriculum and Instruction
Thomas A. RombergAssociate ProfessorCurriculum and Instruction
Peter A. SchreiberAssistant ProfessorEnglish
Richard L. VenezkyAssociate ProfessorComputer Science
Alan M. VoelkerAssistant ProfessorCurriculum and Instruction
Larry M. WilderAssistant ProfessorCommunication Arta