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DOCUMENT RESUME
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Hoffman, Joseph R.; Tardif, Pobert F.A Plan for Improving Mathematics Instruction inCalifornia Elementary Schools. Final Feport of theMathematics Education Task Force.California S.-tate Dept. of Education, Sacramento.7663p.Publications Sales, California State Department ofEducation, P.O. Box 271, Sacramento, California 95802($1.25 plus 6% sales tax)
MF-$0.83 Plus Postage. HC Not Available from EDRS.*Educational Planning; Elementary Education;*Elementary School Mathematics; _Thstruction;*instructional Improvement; Mathematics Education;Research; *Si-atewide Planning; *Surveys*California
In a project designed to improve elementary schoolinstruction in mathematics, the California Department of Educationcollected achievement and profile data from 67 elementary schools.Schools were classified according to size, socioeconomic status,minority representation and mobility of students, city size, and typeof community. Profile data were related to instructional practice,teacher characteristics, leadership activities, and communityinvolvement. Data were analyzed to identify profile elements whichdistinguished between high and low achievement schools in the sameclassification. These data affirmed 60 hypotheses whose implicationsare discussed in this document. These hypotheses led to plans forworkshops in leadership training and for implementation workshops inthe schools. The purposes, characteristics, and planning of theseworkshops are discussed. Appendices provide copies of the surveyinstrument, information on making school profiles, instruments forevaluating workshops, and transparency masters for use in workshops.(SD)
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U 5 DEPARTMENT OF HEALTH.EDUCATION & WELFARENATIONAL INSTITUTE OF
EDUCATION
THIS DOCUMENT HA; BLEN REPRO-OUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANiZATION ORIGIN-ATING IT POiNTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRE-SENT OFFICIAL NATtONAl INSTITUTE OFEDUCAT'ON POSITION OR POLICY
CIQNi'Mari yn J. Butts
Pu-57 Distri. Manage!
A Plan for ImprovingMathematics InstructionIn Colifornia Elementary Schools
Final Report
PA L ICARMI A CTATC rieClA CrrIlfleRrr fie emir. A-rinIt n
0.1,t1fr:,
r44.4:11411i,
',-4t'''41111-4024
A Plan for ImprovingMathematics InstructionIn California Elementary Schools
Final Report of the Mathematics Education Task Force
Prepared by
JOSEPH R. HOFFMANNROBERT F. TAROIFConsultants in Mathematics Education
3
Copyright (d) 1976by
California State Department of Education
This publication, which was edited and prepared in metric measurefor photo-offset production by the Bureau of Publications, Cali-fornia State Department of Education, was published by theDepartment, 721 Capitol Mall, Sacramento, CA 95814.
Printed by the Dffice of State Printingand distributed under the provisions
of the Library Distribution Act
1976
Copies of this pubhcation are available for $1.25 each, plus 6 percent sales tax forCdlifornia residents, from Publications Sales, California State Department ofEducation, P.D. Box 271, Sacramento, CA 95802. A list of other publicationsthat are available from the Department, Selected Publications of the CaliforniaState Department of Education, may be obtained by writing to the same address.Two other publications of particular interest to mathematics educators areMathematics Framework for California Public Schools (1975) and Inservice Guidefor Teaching Measurement, Kindergarten Through Grade Eight: An Introductionto the SI Metric System (1975). Both are available at $1.25, plus 6 percent salestax for California residents.
Fore Is.ord
Although the subject of this publication is the improvement 14,Clkalieltaty schoolmathematics, the underlying theme of the document is the achieveils`knt or excellence ineducation. Those in the Mathematics Education Task Force were a'1.(e(1 to
excellence determines whether or not we make lasting improvemea:achers, parents, students, and the total community. Their willif tglless to work For0 s in duality of
1"1".Ovk.11 insiriletion,
identilY the
it takes people. It takes a commitment to excelleimprove the quality of education. As vital as each of these may be to
nce on the part ol administrators,1.11
)41 nd out, as Othersfeatures of successful mathematics programs in California. And theYhave also discovered, that more than reform and innovation nd l!lurley are needed to
education.Excellence starts with all of us -with the standards we set for tho 1-)Qrf-rmance or 011r
duties and the standards we establish for student achievement. .
i et
outcomes.
higher education. We must lead the fight to establish relevant standart;s1 'ben most seethat every student, minority or majority, rich or poor, goes to the 'university qualified, not simply qualifiable.
Excellence depends upon our acceptance of responsibility ror
We must take the lead in maintaining high qualifying standards for t'll
,13 b ,t-
resu,,s
)"I. inarket or the
Excellence depends upon our willingness to put the needs of the chil Lore the 0the profession or the institution.
If we want to restore excellence as the hallmark of our societY \ye must restoreex....:ellence as the hallmark of education.
Excellence, in the final measure, depends upon our willin,nessc., IP hQ iudg,ed by the
quality and effectiveness of our work.
improvement in the quality of education:
The members of our Mathematics Education Task Force have devel°t1:1diesa
4. Those procedural and attitudinal changes that make a difference3. Priorities must be established.
5. There must be a commitment to excellence on the part of everyovk
2. Responsibility for action must be identified.
If we are to have excellence, we should keep in mind these five a ns much lead to
I. A conscious decision must be made to change- to improve
and the students. Everyone of us must care about excellence.
and evaluating the results of our actions.
educationthe taxpayers who provide the resources, the legislative,.
grouped within an appropriate framework of goals, with provision
those resources, the policy-making bodies, the administrators, the
but it must not be change simply for change's sake.the duality
ele yy
0 ao ti
clio
unicloc
the Pltoring
tool
tt,de.lers,that
nriniproved qthat will help you who are involved in mathematics programs bring aboliin your chissrooms. I encourage you to take advantage of this plan for Or' (-1Vin
n,
r-nLaryg e'eemathematics programs.
c)
In about 42 RI. IubIiIius Syrus said, "It take,: a long time to bring excellence tomaturity.- Rut neither we nor our children will understand or appreciate the henelits ofthat maturity it' we never take the time to search for excellence ii we never take the time togive a deeper, higher meaning to our lives.
Superintendent id Public histriicli()It
6
Pre face
In 11)73 the California Legislatur.' responded to public concern about declining scores inmathematics achievement. It instructed the Department of Education to conduct a studyidentifying the salient characteristics of successful mathematics programs in Catiforniaelementary schools and to disseminate the results to all elementary schools in the state. Theempirically derived results of the study were to be incorporated into elementary schoolmathematics programs so that pupil achievement in mathematics would he improved.
The Department referred the legislative mandate to the Department's MathematicsEducation Task Force. The Task Force attempted to identify the characteristics ofsuccessful mathematics programs by observing what certain high-achieving elementaryschools do that certain low-achieving schools fail to do and vice versa. Then the Task Forceexamined the characteristics by type of school, using basic demographic variables that mightinteract with the characteristics. Finally, it developed an implementation plan that wouldallow a close examination of the results of the study and provide opportunities forimplementation at the school and classroom level.
The members of the Mathematics Education Task Force conducting the study wereRobert Tardif (Manager), Joseph Hoffmann, Delmer Lansing, and Fred Lorenzen. Valuableassistance was provided to the Task Force by Leonard Maraseuilo, University of California,Berkeley. and several mem of the Office of Program Evaluation and Research, St:IteDepartment of Education.
Special commendation is due to the hundreds of teachers and principals in the 67 sampleschools in which the Task Force conducted its study. (These schools are listed on page vi.)Without their honest responses, this study would be worthless. Their honesty gives witnessto the strength and vigor of the education profession and gives promise that the objectivesof this study will soon he accomplished. This publication, A Plan for hnprovingMathematics Instruction in CaliJOrnia Elementary Schools, should be a valuable aid inattaining those objectives.
WILLIAM E. WEBSTERDeputy Superintendentfor Programs
7
J. WILLIAM MAYA ssistan t Superintendent
and Director. Office of Curriculum Services
Elementary Schools Sampled in Mathematics Study
Ada S. Nelson Elementary School, WhittierAvalon Gardens Elementary School, Los AngelesBarnard Elementary School, San DiegoBennett Valley Elementary School, Santa RosaBroad Avenue Elementary School, WilmingtonBuena Vista Elementary School, Walnut CreekCajon Park Elementary School, SanteeCarlton Hills Elementary School. SanteeCarrisa Plains Elementary School, Santa MargaritaCastle Heights Avenue Elementary School, Los AngelesCentralia Elementary School, AnaheimCorswell Elementary School, El MonteCrescent Elk Elementary SchooL Crescent CityDavid Reynolds Elementary School. FremontDel Mar Elementary School, El CerritoDiamond Heights Elementary School, San FranciscoFair Oaks Elementary School, SunnyvaleFieldbrook Elementary School. ArcataFlorence E. Flanner Elementary SchooL La PuenteFranklin Avenue Elementary School. Los AngelesFranklin Elementary School, AnaheimGrand Oaks Elementary SchooL Citrus HeightsHacienda Elementary School. La HabraJackson Elementary School, San DiegoKathryn Hughes Elementary School. Santa ClaraKing Elementary School, Long BeachLaguna Road Elementary School. FullertonLangdon Avenue Elementary School. SepulvedaLi Tijera Elementary School, InglewoodLichen Hementary School, Citrus HeightsLinda Verde Elementary School. LancasterLockhurst Drive Elementary School. Woodland I lillsLoma Vista Elementary School. WhittierLos Alamos Elementary School. Los Alamos
vi
8
Lynda le Elementary School. San JoseMagnolia Elementary School. ReddingManchester Elementary School, FresnoMarion Elementary School, NovatoMcCandless Elementary School. Redondo leachMendocino Elementary School, MendocinoMiller Elementary School, EscondidoMurwood Elementary School, Walnut CreekOso Avenue Elementary School. Woodland HillsOverland Avenue Elementary School, 'Los AngelesPalo Verde Elementary School, Palo AltoParadise Elementary School. ModestoRhoda Street Elementary School, EncinoRichmond Street Elementary School. El SegundoRiley Elementary School, San DiegoRio Vista Elementary School, North HollywoodRound Valley Elementary School, BishopSan Pasqua] Elementary School, EscondidoSanta Lucia Elementary School, CambriaSanta Monica Bmilevard Elementary School. Los
AngelesShepherd Elementary School. HaywardSkycrest Elementary School, Citrus HeightsSpreckels Elementary School. Spreckels'rwin Hills Elementary School, SebastopolVallemar Elementary School. PacificaVictorine Klein Elementary School. Mountain ViewVista Mar Elementary School. Daly CityWarner Elementary School, Warner SpringsWashington Elementary School. San GabrielWaugh Elementary School. PetalumaWest Hollywood Elementary School. West liollywoodWestmont Elementary School, PoinomiWestwood Elementary School, Santa Clara
ForewordPreface
Contents
Chapter 1 In troductior to the StudyRationale for the StudyMethod of Research
Chapter 2 Research ActivitiesResearch StrategySelection of Sample Schools 4Characteristics of Sample Schools 4Research Instruments 7
Chapter 3 Verification of Hypotheses 8Use of the School Profile KeyGeneral Classroom Teaching Styles 10Specific Teaching TechniquesInstructional Materials 14Teacher/Pupil Interactions 15Instructional Leadership 17Parent/Community Involvement 19
Chapter 4 Implications of the Hypotheses 20
Chapter 5 Implementation of a Plan for Improvement 30Workshops in Leadership Training 30Implementation Workshops 31Recommendations for Further Research 32
Appendixes 35
Appendix A Observation Instruments Used in the Study 36Appendix B Cover Letter and Information Needed to Make a School Profile 45Appendix C Evaluation of Workshop in Leadership Training 48Appendix D Transparency Masters 50Appendix E Evaluation of Implementation Workshop 55
ChapterIntroduction to the Study
In the past decade substantial criticism has beendirected against mathematics education. Many per-sons have written and telephoned their complaintsabout "modern math" progrz.ms to members of the
lifornia State Department of Education and thecornia Legislature as well as to editors of
newspapers and periodicals. In response to thepublic concern about the quality of mathematicsprograms, the California Legislature in 1973 autho-rized increased funding for the mathematics unit ofthe State Department of Education and directedthe Department to identify those features ofschool mathematics programs which account forpupil success.
The Department of Education assigned this taskto a newly created Mathematics Education TaskForce, a Department unit consisting of four consul-tants. The task force was instructed to design astudy that would reveal the information requeste,I,conduct the study, and report the findings. It wLsfurther stipulated that the report of the studyshould be suitable for immediate use by individualteachers in developing programs for the improve-ment of mathematics instruction.
Rationale for the StudySome research studies conducted prior to the
study described in this report were designed toidertify mathematics programs that had demon-strated pupil success. As a result several attemptswere made to transplant "successful" programsinto other settings. Unfortunately, the transplantedprograms were found to be unsuitable when certainquestions were asked:
Do the teachers play an important role inselecting the new program?
Does the program meet the special needs ofpupils and teachers in the school?
Do teachers believe the program will succeed?Do teachers have a personal (or emotional)
investment in making the program work?
It was the position of the Mathematics Educa-tion Task Force that inattention to such questionswas sufficient to jeopardize even the best efforts inprogram development in mathematics. Therefore,the study was conducted to equip teachers withprogram development information that teacherscould readily manipulate and incorporate into amathematics program. With the control of programdevelopment in the hands of teachers, the responsi-bility for creating the best possible programs wouldbe theirs also.
Method of ResearchThe orientation of this publication and the
study reported in it is to provide informationuseful in the devdopment of mathematics pro-grams that are likely to produce pupil success. Thedefinition of an instruction! program as used in theresearch project is as follows:
An instructional program is the entire set of eventsthat lead up to and include the actual learning experi-ence of pupils the classiLom. Thus, an instructionalprogram includes the foPowing: the styles and strategiesof the teachers and the school administration; theinstructional leadership that influences classroominstruction and management; the concern and planningrelated to pupil-teacher interactions; the nature ofparent/community support fm the school program; thequality and use of instructional materials, supportmaterials, and media; and the climate in the classroom.
The research was conducted in 67 public ele-mentary schools in California during the spring of1974 by the members of the Mathematics Educa-tion Task Force. The sample schools were broadlyrepresentative in several demographic variablessuch as school size, ethnic characteristics, and themean socioeconomic status of tht school popula-tion. Half of the target schools had exhibited apattern of high achievement in mathematics, andthe other half had exhibited a pattern of lowachievement.
10
The researchers spent at least a full day ;it CdC11st..11001 site observing the ongoing program. inter-viewing teachers and principals. and securing. :indarranging for the acw.iisition of data on the entireschool mathematics program. In es .ence the aggre-gate of data for all schools (about 5(H) pieces ofdata on each school) kvas utilved to test anextensive array of hypotheses about the impor-tance of certain program-related variables. Statis-tical treatment of the data allowed the identifica-tion of 60 hypotheses most related to pupil
2
I t
,d.10.TSti. 111()W
(Ids report.Although the laypotheses 111;ly ht: referred to ;IN
recommendations for program development, theare still hypotheses: i.e., scientific plesses; aboutthose variables which seem best to account forsuccessful! achievement patterns in Calif'irnia ele-mentary schools. Nevertheless, in the nands ofteachers and program developers, the hypothesesrepresent a scncrntnhic Yam.' for designing or revisingsuccessful school mathematics programs.
ChaplerResearch ztcuil d ies
The Department of Filucation study WA'.(!esigned to identify in elementary mathematicsprograms those salient program ch;iracteristics thataccount for consistently low test performance. In aprevious study conducted by the Department.grade levels at which deficiencies in mathematicsachievement first appeared were identified, as werethe specific content areas that accounted for thedeficiencies. In that study the ComprehensireTests of Basic Skills ((IBS): Arithmetic and theSequential Tests of Educational Progress (STEPII): Computation Subtest and Basic GmceptsSubtest were administered to 3000 pupils in classesselected by a stratified random sampling schemefrom grade levels four through eight. The findingsof the study indicated that deficiencies in achieve-ment appeared as early as grade four but weremore intensified in grades five through eight thanin grade four.
Research Strategy
The mathematics section of the ComprehensireTests of Basic Skills was the measurement instru-ment selected to gauge the pupils' mathematicalskills and concepts, as prescribed for instruction inthe Mathematics Framework for California PublicSchools.' The results of the CTBS tests adminis-tered to all sixth graders in each of the previousfour years were assumed to be adequate indicatorsof achievement strength and weakness. Using threeyear-to-year differences in CTBS mean raw scoresfor schools over a four-year period, the researchersfound that all the schools in the study fell into oneof the categories which were established to suggestthe quality of mathematics programs:
1 -bowl Report of the Special Mathematics Arhievement Study.Prepared by the Office of Program Evaluation and Research.Sacramento: California State Department of Education, 1973.
2Mathematics Framework fry Cablbrnia Public Schools.. Kinder-garten Through Grade Twelve. Sacramento: California State Depart-ment of Education, 1975.
1 23
I. Three increases in mean raw scores2. Two increases and a net increase overall ( from
the first year to the fourth)3. One increase and a net increase overall4. Two increases and a net decrease overallS. One increase and a net decrease overall6. No increase
The target schools in the study were sampledfrom two groups of schools:
High test-perfOrmance sclu,ols: (a) the upperIS percent of all schools for 19(0 1972
(categories 1 through 0: and (I)) the upper SOpercent of all schools in 1972 that achieved anet increase in raw scores greater than sixpoints (categories I through 3)
2. Low test-per/in-mance schools: (a) schoolswith one increase and a net decrease greaterthan 14 points (category 5); and ( b) schoolswith no increase and a net decrease greaterthan 10 points (category ())
The use of trends in CTBS scores for iden ti tyingtarget schools is justifiable because a trend overfour years could indicate either a concerted effortfor improvement or a consistent lack of programsuccess. The use of the results of a single testing ofgroups of pupils would have provided a less reliableindication of program success.
The study did not attempt to evaluate allot' thegoals in mathematics that 'Jhould be achieved byCalifornia pupils. For example, it would have beendifficult to use simple test-score trends to assessthe ability of pupils to apply mathematical skills inreal-life situations, long-term retention, desire tolearn independently, or similar goals. However,CTBS scores did provide a reliable indication of themathematical skill under investigation in thisstudy; namely, computational skill. The definitionof high-performance mathematics programs used inthe study must therefore by interpreted in thisrestricted perspective because 48 of the 98 CTBS
test items were purely computational. In that sensethe CTBS mathematics test measured computa-tional ability well.
Selection of Sample SchoolsA total of 72 schools representing a wide range
of demographic characteristics were selected forthe research sample. The basic demographic charac-teristics used for categorizing the sample schoolswere (a) socioeconomic status; (b) minority repre-sentation; and (c) school size (enrollment).
Schools that were determined to be alike for allthree of these characteristics were compared. Theywere judged to be similar for a characteristic ifthey were in the same third of the distribution ofthat characteristic value for the sample schools.After the schools had been selected, they wereclassified further on the basis of three moredemographic variables to provide a more completedescription of each schooL These three variaNeswere (a) city size (population); (b) type of commu-nity (environment); and (e) pupil mobility.
The first three variables were used in the originalsample selection in a way that minimized inter-dependence among the variables. Specifically, theschools were divided into three levels for each ofthree variables so that 27 categories of schoolsresulted. For example, one category containedschools in communities that ranked low in socio-economic status, minority representation, andschool size (enrollment). Another category con-tained schools in communities that ranked mediwnin socioeeonomic status and low in minorityrepresentation and enrollment. Finally, a thirdcategory was composed of schools in communitiesthat ranked high in socioeconomic status and lowin minority representation and enroHment. Withineach group, two of the schools were randomlyselected for the high-performance sample andtwo for the low-performance sample. lt is thispairing of similar schook that is the basis of thisstudy.
The 72 sample schools were selected from the27 categories, and the last three variables wereimposed on these schools after the selection. Noguarantee of the independence of these variablegoups was present. To have categorized theschools on all six variables prior to selection wouldhave produced 36, or 729 categories, many ofwhich would have contained only one school.Five schools were dropped from the study becauseof time limitations and scheduling difficulties. Atotal of 67 schook were examined as a researchsample.
4
Characteristics of Sample SchoolsEach group of schools was represented by a
profile based on the six variables, and the majorityof California elementary schools could comparewith at least one profile. Each profile provided thebasis for a separate analysis because the importantcharacteristics that might discriminate betweenhigh and low schook in one profile might beinsignificant in another. The following are detaileddescriptions of the demographic factors used todescribe and categorize the pool of sample schook.
Socioeconomic Status
An estimate of the socioeconomic status of eachsample school community was undertaken by theone individual who was probably the best qualifiedfor the task.--the teacher. The estimates were madewhen zdl California public school pupils in gradestwo and three were being tested in reading as partof the California Assessment Program.3 Eachpupil's answer sheet was marked by that pupil'steacher for background information, ncludingsocioeconomic status. The teacher chose obe of thefollowing as indicating the family backgi-ound ofthe pupil:
I. Executives, professionals, and managers2. Semiprofessionals, clerical and sales workers,
and technicians3. Skilled and semiskilled employees4. Unskilled employees (and vxlfare recipients)5. Unknown
The five choices were condensed to three levels forstatistical analysis:
Level onechoices 4 and 5Level twochoice 3Level threechoices I and 2A single statistic for the socioeconomic variable
was obtained for each school by computing themean for the three levels in the item. The.rangc ofthe variable had a minimum of 1.00 and amaximum of 3.00. For the purpose of dividing thetotal sample into three levek with approximatelythe same number of schools in each level, the limitsof the variable were set as shown in Table 1 (seealso Figure 1).
3Nofi les of School District Performancc, /973.74, Prepared hythe Office of Program Fva Illation and Research. Sacramento:California State I>epartment of Education, 1974.
1 3
TABLE. 1
Distribution of School Enrollmentsby Socioeconomic Status
Group Limits
Sample schools
Respondineelementary
schools
Nur. Per-h. cent
Nuns-ber
Per-cent
LowMed:urn ..Ilieh
IMO 2.082.09.-2.402.41 --3.00
232222
34.332.832.8
2258861
1696
46.917.935.2
Total 67 99.9 4815 100.0
10 _
5
0
,o
0 Schools in sampleE Schools statewide
2.0II3.0
Department of Education in the fall of 1972, oneyear prior to the study described in tlUs report.An on-site examination of enrollment at thesample schools revealed some discrepanciesbetween the enrollments reported and the enroll-ments observed. However, the discrepancies werein no case large enough to change the classificationof a sample school. The distribution of schools byschool size is presented in Table 3 (see also Figure 3).
City Size
City size (pop'ulation) for each school in theresearch project was determined at the time of the
TABL 2Distribution of School Enrollments, by Minority
(Nonwhite) Composition
(1roupLimits
( percent)
Sample schools
Respondingelementary
schools
Num-her
l'cr-cent
Nuns-her
Pc r-cent
LowMediumlligh
0 910 2829 100
1,__2421
32.835.831.3
76011451116
25.237.936.9
lotal 67 99.9 3021 100.0
20 ---Mean socioeconomic status ( interval: 0.1)
Fig. 1. Graphic distribution of school enrollments, bysocioeconomic status
15Minority Representation
Minority representation was determined for eachschool from the ethnic school survey which wasadministered in all California public schools in1972, one year prior to the mathematics studyreported in this publication. In the survey eachprincipal reported the number of the school'spupils from each of the following ethnic uroups:Asian, black, Chicano, Native American, and whiteand other.
By the use of the data provided by schoolprincipals, the total percent of nonwhite pupils wasascertained for the schools. The schools were thendivided into three groups according to their minor-ity representation as shown in Table 2 (see also
i uure 2).
10
0
0
0 Schools in samplt,Schools statewide
11061.450 100School Size
Percent of minority (nonwhite) school enrollmentsThe school size (enrollment) for each school was (interval: 5 percent)
taken from the elementary school questionnaire Fig. 2. Graphic distribution of school enrollments, byas reported by school principals to the State minority (nonwhite) composition
1 4 5
'FABLE 3 Small (ity (or mmcity) less than 50 000Distribution of Schools, by Size of School Crescent City
Arcata-Eureka
Group Limits
Sample schools
(area)Responding Santee-El Cajon (area)elementary
schools Escondido-San Marcos (area)Num-
herPer-cent
Num-ber
Per- Santa Rosacent Pacifica
LowMedium ..lligh
31 349350 560561 1298
2311
21
34.332.832.8
209921492942
29.2 Medium city 50 000 to 200 00029.9 Citrus Heights-Carmichael-Orangevale (area)40.9
San Gabriel-Alhambra-Monterey Park (area)rota! (m7 99.9 7190 100.0 Fresno
10
5
500
1: Schools in sample1M Schools statewide
Number of pupils enrolled (interval: 50)
Fig. 3. Graphic distribution of schools, by size of school
visit to the school. The information was recordedduring the interview with the principal and waslater modified as necessary after all the data hadbeen examined. The city size categories were smallcity (or noncity), medium city, and large city. Forgeographically isolated cities and towns, the cate-gory distiflctions were based on simple populationstatistics. When two or more cities were clusteredgeographically, the observer had to make a subjec-tive choice on the basis of apparent commercialand political spheres of influence. The populationof the surrounding area was used to determine citysize if any one of the following three conditionswas applicable:
I. The school district boundaries included thesurrounding area.
2. Some connecting streets were nonrural; i.e.,commercially or residentially developed.
3. The city administrative bodies were inter-dependent.
Examples of' the city size categories of some ofthe schools in the study arc as follows:
6 5
Large city -over 200 000I ngle wood- Le n nox-1 lawthorne (area)Anaheim-Fullerton-Buena Park (area)San JoseSan DiegoSan FranciscoLos Angeles
The distribution of sample schools by city size isshown in Table 4.
TABLE 4
Distribution of Sample Schools, by Size of City
G r ou p Limits
Sample schools
Number Percent
Low Less than 50 000 27 40.3Medium 50 000 200 000 14 20.9Iligli Over 200 000 20 29.9
6 1" 91.1
JOr the 67 schools in the sample. Si\ were not categorized Ly citysize heeause of insufficient information.
Type of Community
The type of community surrounding each schoolin the study was recorded by the observer duringthe interview with the school principal. Two of thefollowing three descriptions could be chosen bythe in terviewdi-:
Type I (low) --rural, farmingType 2 (medium)suburban, residentialType 3 (high ) inner city, commercial, industrial
So that the size of each of the three groups inthe sample could be made as similar as possible, thedefinition of Type 2 schools, the largest group, wasnarrowed. If the school's geographic communitycontained any rural or farming property, regardlessof the extent of residential or industrial develop-ment, the community was cateLlorized as Type I. Ifthere were no local farming property, then it wasclassified as Type 2 or Type 3, depending on the
nature of the commercial development in thecommunity. Type 3 was chosen when the follow-ing features characterized the community:
I. Old apartment, commercial, or industrialbuildings (more than 40 years old)
2. Low-income or substandard dwellings3. Predominantly commercial-zone property
The distribution of sample schools by type ofcommunity is shown in Table 5.
TABLE 5
bistribution of Sample Schools, by Typeof Community
Group Lim itS
Sample schools
Number l'erc.mt
Low Rural/rarming 18 26.9Medium Suburban/residentiol 29 43.3High Inner city/
commercial/industrbl
15 22.4
Total 62" 92.6
"Of the 67 schools in the sample. five were not categc rized by typeof community because of insufficient information.
Pupil N.;
A pup.. mobility index for each school wascomputed from data collected at the time of thevisit by the observer. The principal (or, in mostcases, the secretary) of each school gave thenumber of pupils added to the enrollment betweenOctober 1, 1973, and May 1, 1974, and thenumber deleted during the same period. Themobility index was then derived by dividing thesum of these two numbers by the actual enroll-ment on October 1 . All mobility indices were
16,
divided into three groups of approximately thesame size as those shown in Table 6.
.1-ABLE 6
Distribution of Sample Schoolsby Mobility Index
Group Limits
Sample schools
Number Percent
LowMediumHigh
0- 2223 -37
Over 37
171716
25,425,423.9
Total 50" 74.7
"Of the 67 schools in the sample, 17 were not cdtegorized bymobility index because of insulTicient information.
Resear ch In s t rumen ts
Seven;i instruments were devised to assess thecharacteristics of the mathematics programs inthe sample schools. The instruments includedan anonymom teacher-principal opinionnaire, ateacher interview guide, a principal interview guide,and two classroom observation inventories. Theprincipal's interview included factual and policyitems pertaining to the curriculum, the school, thestaff, the district, and the community. Theteacher's interview involved items pertaining toattitude, training, pedagogical tactics, instructionalsupport materials, testing, and classroom climate.(See Appendix A.)
The teacher observation procedure involved ameeting with the teacher or team leader before ascheduled class period to ascertain general informa-tion about the lesson to be taught and then a seriesof five-minute observations of the class. A five-minute interval occurred between observation ses-sions. A meeting after the class with the teacherconcluded the series of interviews and observations.
7
Chapter
3 Verification of Hypotheses
No absolute rules exist for successful teach-ingonly hypotheses about what works best. Edu-cators must form hundreds of hypotheses daily inplanning lessons and teaching; these hypothesesinvolve all aspects of the instructional program. Inthe research study analyzed in this report, peda-gogical hypotheses were devised and categorized asfollows:
I. General classroom teaching styles2. Specific teaching techniques3. Instructional materials4. Teacher/pupil interactions5. Instructional leadership6. Parent/community involvement
Most teachers' hypotheses are never articulatedexplicitly; nevertheless, educators hope that their
assumption that an hypothesis is verified andgeneralized if it is used more frequently in schoclswith high pupil achievement than in schools withlow pupil achievement.4 It was further supposedthat the hypotheses would be verified for sometypes of schools and not for others. Each hypo-thesis accordingly has been identified for thecategories of schools for which it has been verified.Each hypothesis has also been identified to indi-cate whether it has been verified for all schools ingeneral. The analyses of all the hypotheses led toverification of 60 hypotheses for three or moreschool demographic characteristics.
The absence of a black square in one of the cellsdoes not imply that the characteristic was unim-portant in a particular type of school. Rather,there was apparently so little difference in the use
Example
General Classroom Teaching Styles (Continued)
7. Teachers perceive mathmatics as a useful and impor-tant learning activity for pupils.
This black square indicates that the program featuredescribed by this hypothesis was a significantly morefrequent element of programs in high-achieving schools thanin low-achieving schools regardless of the background charac-teristics of the school.
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1Thisiblack square indicates that in ehools in which pupilsfrom minority groups constitute between 10 and 28 percenof the school population, high-achieving schools exhibitedthe program feature described by the hypothesis significantlymore than did low-achieving schools.
hypothetically based actions will result in pupilachievement. The implicit form of each hypothesisused in the study is that pupils tend to exhibithigher achievement patterns if action X occurs.
Hundreds of hypotheses were considered by theresearchers and were exposed to verifications in thesampling of schools. The study was based on the
of this characteristic by high- and low-achievingschools that it had no effect on achievement.
4A 90 percent confidence interval was used for significancetesting. The researchers wished to include all hypotheses that heldpromise for improving programs and to have hypotheses submittedfor experimental testing later.
8
17
L.A. of the Sdiool Profile Key
A printed card is provided with this report sothat the results can be easily analyzed and hypoth-eses applied to each school. You will need a paperpunch to make the key operational.
Each school will have a unique "profile," andone key should serve the entire staff in a school.Elowever, additional keys can be produced from 3x 5-inch cards.
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90. Hypothesis
School Profile Key
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A. Punch holes in the appropriate boxes for yourschool according to the definitions of thedemographic variables. (In borderline casespunch two holes.)
B. Place the key over the corresponding part ofany item identified in the study.
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Schwol name
Date
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C. Note how many holes in the key are superim-posed over a black square. A black squareindicates that the item is a significant discrimi-nator between high and low test performanceschools for schools similar to yours. The moreblack squares revealed, the more important thatitem is to the improvement of the mathematicsprogram in your school.
1 8 9
General Classroom Teaching Styles
Discovery and inquiry are emphasized in teaching;limited use is made of "traditional- methods.
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2. The teaching reflects "traditional concerns; limiteduse is made of discovery and inquiry techniques.
3. Learning activities make provisions for participation byall pupils; i.e., activities which appeal to a variety ofinterests and abilities.
4. Mathematics lessons are prepared thoroughly.
5. The improvement of pupil attitudes is a major empha-sis in classroom teaching.
6. The faculty recognizes, discusses, and attempts toremedy disruptive pupil behavior.
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General Classroom Teaching Styl^s (Continued)
7. Teachers perceive mathematics as a useful and impor-tant learning activity for pupils.
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8. The teaching focuses on the content of mathematics.M
9. The learning of computational skills and concepts isvalued highly and is emphasized as an important goal.
10. The teaching activities allow pupils to refine theirunderstanding of concepts, and a convergent mode ofthinking is used.
1 I. The classroom activities are managed in a way thatpromotes orderly and quiet pupil behavior.
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12. A classroom esprit de corps is promoted so that pupilstake pride in their behavior and achievement.
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Specific Teaching Techniques
13. Instruction is activity-centered, and techniques such asgames, learning centers, and active learning situationsare used.
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14. Learning assignments are differentiated for the partie-ular needs of individual pupils.
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15. The teacher diagnoses. understands, and acts uponeach pupil's learning difficulties.
16. Pupils are involved in planning their own activities.
17. Pupils have opportunities to enizage in independentlearning projects,
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18. Stimulating learning activities are planned for pupils;e.g., game-like activities, lascinatinu materials, andlively discussions,
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2112
Specific Teaching Techniques (Continued)
19. Teachers use audiovisual materials or visual demonstra-tions in a formal classroom atmosphere.
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20. The teacher gives clear verbal instructions in a well-controlled classroom atmosphere.
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21. The instruction is based on intensive, responsive verbalpresentations in which pupils are told specifically whatthey are expected to know.
11. The chalkboard is used frequently.
23. The overhead projector is used frequently.
14. The teacher alid aides (where available) supervise andtutor the pupils as they study.
2 2
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Instructional Materials
25. Teachers are given special trainintz in effective tech-niques for presenting instructional materials.
26. Assignments and directions are clear.
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27. Additional instructional materials are acquired andused in the classroom.
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28. A broad range of audiovisual instructional materials areacquired and utilized in the classroom.
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29. Teachers are encouraged to reduce dependence ontextbook presentations.
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30. Manipulative aids are used in the learning of skills andconcepts at all levels of ability.
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Instructional Materials (Continued)
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31. Mathematics is integrated with other subject areas.
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32. Pupils are allowed to accelerate individually or to2eth-er to the next higher grade level of instructionalmaterials.
33. Frequent enrichment instruction is provided for capa-ble pupils.
34. Mathematically gifted pupils explore abstract concepts,logic, and mathematical applications as part of theirclassroom learning.
35. A special mathematics program is offered to giftedpupils.
Teacher/Pupil Interactions
36. Teachers are aware of and are sensitive to the abilities,needs, and interests of individual pupils.
2 1 15
Teacher/Pupil Interactions (Continued)
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37. Teachers have a positive feeling toward pupils and afeeling of being well-liked by them in return.
38. Teachers avoid abusive criticism of pupils and refrainfrom scolding or threatening pupils or arguing withthem.
39. Teachers consistently clarify pupil misconceptions,correct pupil errors, and resolve difficulties promptly.
40. Teachers allow pupils to discover their own errors andbe self-directed.
41. Teachers are sympathetic and tolerant and allow pupilsto express their own Opinions.
42. Teachers praise and encourage pupils by providingsupport, optimism, or an appreciation of pupilresponses.
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Teacher/Pupil Interactions ( Continued )
43. Teachers accepi and use the ideas of their pupils.
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44. The instructional program is focused on pupils andtheir needs.
45. The instruction is restricted to presentations, demon-strations, and selection of topics by the teacher alone.
46. The teacher controls the classroom scene with inter-esting teacher-directed activities.
47. Teachers hold informal, relaxed conferences with theirpupils to allow free exchange of opinions and discus-sion of needs.
Instructional Leadership
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48. The school's mathematics program is made a matter ofhigh priority.
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Instructional Leadership (Continued)
49. The school's mathematics program :s updated toreflect new trends in mathematics education.
50. Schoolwide instructional objectives are developed forall subject areas.
51. Instructional objectives are developed for the mathe-matics programs.
52. A consultant assists in the development and mainte-nance of a coherent mathematics program.
53. Teachers feel confident in their ability to teachmathematics.
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54. Teachers attend mathematics workshops and courses. M
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Instructional Leadership (Continued)
55. A district-level emphasis on mathematics education isinitiated and publicized.
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56. The school principal supports teachers who findmathematics textbooks inadequate.
Parent/Community Involverr cot
57. The mathematics program supports pareltal views ofwhat should be taught.
58. The school staff is involved in a community relationscampaign to develop parental interest in the school'smathematics program.
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59. Teach-n-s confer with the parents of each pupil at leastthree times annually regarding the pupil's learningprogress and problems.
60. The mathematics program is appropriate for thecultural character of the community and realisticallyreflects the community's needs. izoals, and interests.
19
ChapterImplications of the Hypotheses
Each of the 60 hypotheses was used in thestudy in the following context: Pupils tend toexhibit higher achievement patterns if.... (hypoth-esis). For types of schools in which a particularhypothesis was not 'found to be an importantcontributor to pupil success, the reader should notassume that pupils tend to exhibit lower achieve-ment patterns if .. . (hypothesis). Therefore, thisstudy provides no basis for discontinuing anactivity which has been designed to enhance themathematics program in a school. Rather, thestudy provides a basis for establishing priorities forfuture program planning and program analysis.
The study indicated, for example, that exceptfor schools in a wealthy or suburban community orin a large city, the use of stimulating learningactivities (hypothesis 18) is not a characteristicwhich distinguishes high-achievement schools fromlow-achievement schools. Nevertheless, the use ofstimulating learning activities should not be discon-tinued; the study does not assert that the use ofstimulating learning activities produces lowachievement.
The same interpretation applies to hypothesesfor which only one or two black squares are shownfor a given type of school. On the other hand,when three or four black squares are shown for agiven type of school, that hypothesis deservescareful consideration for implementation. Thehypothesis is especially important if the blacksquares shown for a given school type are the onlyones given for that hypothesis. When five or sixblack squares are shown for a given school type,that hypothesis is an essential ingredient forsuccessful mathematics program planning, and itsimplementation should receive the highest priority.
The following discussions of the implications ofthe hypotheses should provide the reader withuseful ideas and directions for developing imple-mentation plans. When a set of hypothecos hasbeen identified for implementation in a lvenschool, all of the implications should be considered
and a general implementation pattern sought. Itmust be remembered that the hypotheses aredesigned as the basis for future progam develop-ment and as such serve as focuses for the planningactivities of teachers, administrators, and membersof the community. Those who create the programwill believe in its ultimate success.
1Discovery and inquiry are emphasized inteaching; limited use is made of "tradi-tional" me t hods.
Encouragement of pupil discovery and inquiryin mathematics represents one of the most univer-sally accepted aspects of the mathematics revolu-tion of the 1960s. Most educators agree that pupilsshould think about what they are learning; thatpupils should relate what they are learning to whatthey have already learned; and that mathematicalconcepts deserve exploration by learners of allabilities. The research study led to some revealingobservations about discovery and inquiry activities.To begin with, such activities must be carefullyplanned and executed; they tend to lose theireffectiveness if overused as a motivational device.Furthermore, effective discovery activity need notinvolve the entire class at one time, and discoveryshould not always be used to introduce everytopic. In most cases effective discovery and inquirylearning activities are created by inspiration. Theseactivitiesand their inspirationshould be sharedliberally by teachers.
2The teaching reflects "traditional" con-cerns; limited use is made of discovery andinquiry techniques.
A "traditional" mathematics program may meanmany things to many people; but for most teachersit probably means using a program which preceded"new math." One of the characteristics of thatearlier curriculum was deliberate attention to thedevelopment of computational skills. In the "new
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math" this attention was often diverted to devel-oping an understanding of "modern" topics inearly grades. Another characteristic of "new math"was the unfortunate requirement that teacherspresent textbook topics with which they were notfamiliar. The implications of the study are thatmore attention should be given to computationalskills and that unfamiliar topics should not betaught until the teacher understands themthoroughly.
Learning activities make provisions JOr par-ticipation by all pupils; i.e., activities whichappeal to a variety of interests and abilities.
Teachers must recognize the importance ofparticipation by all pupils. Learning activities aresimpler to plan if the activities involve only oneability level or one interest area; but clearly suchan approach is not adequate for meeting all of thepupils' needs. If selection and purchase of instruc-tional materials are based on the needs of all typesof pupils, an improved instructional approach canbe undertaken. Then, for every lesson, methods forinvolving all pupils in the activities must bedeveloped.
3
4 Mathematicsoughly.
lessons are prepared thor-
Planning and preparing mathematics lessons arenot accomplished to the same degree in all schools.Sometimes, in those schools of low mathematicalskill performance, extenuating circumstances mayhave prohibited adequate lesson preparation. buteducators must realize that thorough lesson pre-paration will lead to improved performance in alltypes of schools. Lesson development may requireeliminating the prohibiting circumstances of theschool and developing creative plans to be sharedamong teachers, possibly including team teachingopportunities.
5 The improvement of pupil attitudes is amajor emphasis in classroom teaching.
During the interviews the researchers askedteachers which instructional emphasis was con-sidered more important: how pupils felt towardlearning (attitudinal) or what pupils learned (cog-nitive). Teachers' responses were classified accord-ing to whether emphasis was all attitudinal, allcognitive, or somewhere in between. Although anemphasis on attitude to the exclusion of cognitivelearning is not to be recommended, schools in theidentified categories should consider increasing theemphasis on attitude. Most teachers indicated that
3 0
they were concerned about both areas of learningand that both were essential.
6The faculty recognizes, discitsses, andattempts to remedy disruptive pupilbehavior.
In some locations teachers and principals rankedpupil attitude toward school as a major impedi-ment to the success of their mat, ema tics programs,even though the pupils may have attained highachievement in mathematics. The result indicatedthat in those schools the faculty had recognizedthe problem, openly discussed it, and prepared aplan for resolving it. Most schools had the problemof poor pupil attitude, and more teachers neededto be aware of the problem and concerned aboutit. The research study affirmed that taking stepstoward resolving the problem can improve mathe-matics instruction.
7 Teachers perceive mathematics as a use fidand important learning activity for pupils.
An understanding of the purposefulness ofmathematics learning is a characteristic of high-performing pupils and teachers. T. , hypothesiswas corroborated by the questionnaire responsesfrom teachers and principals, who apparently feltthat their programs were designed to serve usefulpurposes and that pupil time was not being wasted.The hypothesis might, however, merely reflectteacher attitudes toward mathematics in general. Ineither case improvement in a school's mathematicsprogram could be expected if purposeless andtime-wasting exercises were eliminated.
8 The teaching JOcuses on the content ofmathematics.
The research study indicated that some schoolscould expect improvement in their mathematicsprogram if they would provide more structure,including well-written objectives, mastery-levellearning criteria, pupil achievement profiles, andminimum learning requirements. Pupils in a struc-tured program would have progress charts or asimilar record of achievement. In general, emphasisin a structured program would be placed onlearning the subject matter rather than on pursuingthe interests of individual pupils.
The learning of computational skills andconcepts is valued highly and is emphasizedas an important goal.
Computational skills may be mastered by theuse of manipulatives, games, and other interesting
9
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activities and need not involve heavy drill exercises.Some schools may make effective use of pupil goalsetting by emphasizing computational skills as animportant goal or by associating rewards withachievement of these skills.
10The teaching activities allow pupils torefine their understanding of concepts, anda convergent mode of thinking is used.
A convergent mode of thinking is used inteaching when pupils may be easily confused byambiguity or distraction. For example, the con-vergent mode of thinking is illustrated by the useof a single model for the addition concept; i.e., xy means taking x things and y other things alltogether. Ambiguities are avoided in the conver-gent mode of thinking, such as when the numberline model or "block" arithmetic is used. Distrac-tions are avoided also, such as in determiningwhether the model works when some of the xthings are the same as some of the y things. Ingeneral, in the convergent mode pupils look fo: asingle correct response; discovery and inquirytechniques arc not emphasized.
The classroom activities are managed in a11 way that promotes orderly and quiet pupil
behavior.
Orde,'iness and quiet are characteristic of' thetype of classroom management in which discussionis carefully controlled by the teacher, pupil assign-ments are completed quietly and intently, andpupils understand and accept the ground rules foracceptable behavior.
A classroom esprit de corps is promoted so12 that pupils take pride in their behavior andach ie ve men t.
A formal atmosphere was associated with highperformance in several types of schools, a resultthat must be interpreted as to how pupils respondwhen visitors are present. The formalism couldhave been based on a desire to perform well andplease the visitor, or it could have demonstratedan appreciation for the visitor taking time towatch the class. In either case the behavior indi-cated an esprit de corps reflecting pride in theclass accomplishments and comprehension by thepupils of standards of classroom behavior. Theresults of the research study should not be inter-preted as encouragine continuous formal classroomstandards.
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13Instruction is activity-centered, and tech-niques such as games, learning centers, andactive learning situations are used.
The teacher may be in a dominant role inactivity-centered instruction, but the pupils are notpassive. The pupils are challenged as individuals.They participate actively and learn as a result ofwhat they are doing. The results of the studyindicated that activity-centered instruction is effec-tive when lecturing, recitation, drill, and discussionare not effective. Activity-centered instruction isdifferent from the independent learning project,which does not involve the teacher to the extentthat activity-centered instruction does.
14 Learning assignments are differentiated JOrthe particular needs of individual pupils.
Differentiated pupil assignments can be used inany classroom to provide an opportunity for someindividualized instruction. Assignments can beplanned for two or more groups of pupils in a classso that each group's assignment provides the typeof learning activity beneficial to the members ofthat group. The assignments may differ accordingto the interests of the pupils, the mathematicsachievement levels, the class projects, or therewards given for performance. Differentiatedassignments should allow some pupils to exploremathematical concepts in depth and others tospend more time improving retention of basicskills. So that equality of achievement can bemaintained, different amounts of study may berequired.
15 The teacher diagnoses, understands, andacts upon each pupills learning difficulties.
For schools with a medium or large minorityrepresentation, it is particularly important thatteachers know their pupils' learning difficulties.Ethnic barriers can be overcome by a consciousprogram of diagnosing learning difficulties. Themode of diagnosing can bc either formal (makinguse of special testing programs) or casual (teacher-perceived). In either case the teacher takes time toprescribe learning activities that will assist the pupilto overcome the difficulties.
16 Pupils are involved in planning their ownactivities.
Teachers and principals in many types of schoolsfind that involving pupils in the selection oflearning activities tends to produce higher pet-for-
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mance. Arrangements vary greatly. Two -olesare differentiated assignments and individt, ,in-ing contracts. The common characteristic oi theseplans is the provision of a choice of severalalternative learning activities from which pupils(individually or c011ectively) may select whatappears to them most beneficial. That some pupilswill try to take advantage of this instructionalmethod is understood. But the many advantages ofthe method merit its serious consideration. First ofall, the opportunity to select what one doesincreases the interest in the selected activity.Furthermore, one learns to be responsible formaking decisions. Teachers find the creativityneeded in planning for alternatives to be exhila-rating and rewarding. Sharing by teachers becomesa necessity and opens new avenues of teaching.Finally, selection of learning activities can be usedas a reward f or achievement and can be initiated ona small scale in any classroom.
17 maw, hare opportunities to engage iiiindependent learning projects.
The use of independent learning projects as ateaching technique is difTicult for the teacher but isoften successful in improving mathematics learningfor the pupils. The project approach has beenproposed by leading educators, particularly bythose in the Nuffield Project in England.' Thetechnique often succeeds because pupils engage inappealing and meaningful activities that require athorough comprehension of mathematical con-cepts.
18Stimulating learning actirities are plannedJOr pupils:e.g.. game-like activities. Jascinat-ing materials, und lirely discussions.
In sonie classrooms the observers noted verylively, stimulating learning activities. Teachers wereplaying games with the pupils, presenting fasci-nating materials, or encouraging discussion. Incontrast, in some classrooms the presentationsappeared to be dull and uninteresting, most of thepupils seemed bored, and the teacher did not seeminterested. If teachers are enthusiastic. are not toodependent on textbooks, and are flexible towardnovel learning activities, they will make the class-room an exciting place for learning.
5Begtrimngs. Nuffield Foundation Mathematics Ptoject. NewYork: John Wiley & Sons, Inc.. 1967. See ako Mathematics: PreFirst Three Years. Nuffield Foundation Matherualics Prolect, NewYork: John Wiley & Sons, Inc., 1972.
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19Teachers use audiorisual materials or visualdemonstrations in a fin7nal classroomatmosphere.
Teachers were observed using audiovisual mate-rials or visual demonstrations more often in high-performance schools than in low-performanceschools. These learning activities sometimesinvolved pupils actively and sometimes passively.The study revealed, however, that the use ofaudiovisual materials and demonstrations dependedfor its effectiveness on a formal classroom atmo-sphere in contrast to a casual atmosphere. Con-straints must be placed on the way in whichdemonstrations and audiovisual materials may beused to improve performance. That is, pupils mustunderstand that, although the material presented isattractive, they are expected to consider thematerial as matter for study.
20 The teacher gives deur rerbal instructionsin a well-controlled classroom atmosphere.
Classroom observations of teachers disclosedfrequent use of the lecturing technique -logical,clear, and matter-of-fact expository teaching. Thisstyle of teaching was effective in classrooms inwhich teachers tended to dominate the learningenvironment.
The instruction is based on intensive,
21 responsive verbal presentations in whichpupils are told spectlieully what they areexpected to know.
In classroom observations of teachers, a tech-nique was observed that differed from lecturing inseveral ways. The technique was characterized byintensity and succinctness and amounted to theissuance of authoritative statements of fact. Whatwas to be learned and remembered was made clear.Teachers using this technique emphasized verbal-ization in their teaching and in evaluating learning.
22 The chalkboard is used frequently.According to the research study, teachers and
principals considered the chalkboard as an ef fectiveteaching tool. The value of having pupils work atchalkboards is based partially on providing animmediate check of answers and on the cross-pollination of ideas that occur as pupils watch eachother work. Teachers found that examples anddirections written on the chalkboard helped pupilsorganize and clarify their understanding.
23
23 The m'erhead projector is used frequently.
The use of overhead projectors was found to bevaluable for the same reasons chalkboards werefound valuable. Other valuable features were theconvenience of facing the class: the saving ofmaterial for later review: and the magic-likeprojection from table to screen which tended tofascinate learners.
24The teac'her and aides (where arallahle)supervise and tutor the pupils l/S theystudy.
A great deal of class time was used for seat-work. Pupils completed assignments independentlyor received individual tutoring. As the teachermoved about the classroom, a personal contact wasmade with each pupil who needed assistance. If theteacher could not keep up with the demands, aidesand other pupils were directed to assist in thetu toring.
25Teachers are giren special training in effec-tive techniques fin- presenting instructionalmaterials.
The extent of inservice training in mathematicswas a factor which was observed to differentiatebetween high-performance and low-performanceschools regardless of whether the training was incontent or in methods. Nearly all teachers indi-cated that inservice training had been beneficialand should be designed to meet specific teachingneeds. Some of the topics suggested by elementaryteachers were the following:
Motivation Development ofMetrics materialsLaboratory Problem solvingSlow learners Calculators in theDifferent learning classroomstyles Applications
Mathematics games Mathematics forEn rich men t ma terials consumers
26 i ssignments and directions are ckar.
Clear explanations and directions for mathe-matics assignments were especially critical iii sometypes of schools. New teachers often have diffi-culty in explaining and giving directions. Theability to explain and give directions clearlyshould, therefore, be a prerequisite for teachershired to teach in these schools.
24
27 Additional instructional materials MT ac-quired and used in the classroom.
Teachers in some schools attributed their pupils'high performance to acquisition of additionalinstructional materials. One may assume that thematerials were carefully selected to enhance thetotal mathematics program. The actual amount ofexpenditure was unavailable, but it should benoted that the schools involved were not wealthy.Improvement in a school's mathematics programcan, therefore, often be expected to acconmany anincrease in budget priority.
broad range of waliorisual instructionalmaterials are acquired and utilized in theclassroom.
The high rating, by teachers and principals, ofthe use of audiovisual instructional media inmathematics programs was observed in severalparts of the research study. This supportive evi-dence makes consideration of such media essentialfor certain types of schools. Audiovisual mediaincluded a broad range of materials and a broadrange of modes of presentation, from passive toactive pupil participation.
28
29 Teachers are encouraged to reduce depen-ele;:e.e on te.vthook pre.ventations.
The responses by teachers and principals demon-strated that much support existed for the use ofnontextbook, nonworkbook learning activities andteacher-designed follow-up activities alter comple-tion of textbook assignments. Independence fromtextbook learning calls for a creative approach toteaching.
Manipulative aids are used in the learning"II of skills and concepts at all levels of alfility.
The study revealed that the use of manipulativeaids in mathematics instruction was receiving new,widespread interest. Their use was noted particu-larly in schools in which the staff was activelypursuing a program of' improvement. Furthermore,their application appeared at all levels of pupilability: they were not limited to remediation.manipulative aids used were often inexpensiveteacher-made materials created from objects com-mon in the pupil's environment.
Mathematics is integrated with other sub-t..P ject areas.
In tegration of mathematics instruction in toother subject areas has been given increased atten-
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tion recently. One way this integration may beaccomplished is through individual pupil projects,which May be undertal,:en sequentially as coursesevolve and as mathematical concepts arise in thevarious course subjects. Numerous ideas for pupilprojects are suggested in the teacher guides forsome mathematics textbooks.
32Pupils are allowed to accelerate indiritheallyor together to the ne.vt higher grade keel ofinstructional materials.
Accelerated programs may refer to opportunitiesfor pupils to complete the materials assigned totheir grade level at a Caster pace and continue withmaterials normally assigned to the next highergrade level. Because of complications with crossingover grade lines, accelerated learning may alsomean probing deeper into an assigned topic orexploring an independent project. Accelerationshould not mean giving a pupil more practice intopics already mastered.
33 Frequent enrichment instruction is pro-vided for capable pupils.
In the research study the term enrichmentinstruction was used to refer to the offering offollow-up activities for all pupils who were capableof performing them. Enrichment instruction canhelp in narrowing the gap between high- andlow-ability pupils: enrichment stimulates teachersand high-ability pupils intellectually and motivatesother pupils to learn mathematics.
34Mathematically glited pupils explore ab-stract concepts, logic, and mathematicalapplications a.s. part of their classroom
Mathematically talented pupils can be found innearly every school. The complete mathematicsprogram will provide opportunities for them.Those opportunities should go beyond acceleratedand enrichment instruction (see hypotheses 32 and33) by providing greater challenge and difficulty.However, such opportunities should never burdenpupils with repetitive practice in topics they havealready mastered.
35 A special mathematic's pmgrum is offeredto gifted pupils.
A startling result that appeared in the study thatprovided nearly perfect discrimination betweenhigh- and low-performance schools was a successfulprogram of special instruction for gifted and
high-achieving pupils. Low performance was foundto accompany inadequate instruction for giftedand high-achieving pupils.
Programs for the gifted are usually designed toprovide special instruction for pupils with abilitiesthat meet particular criteria. Special funds areoften available for such programs. In some pro-grams for the gifted, the pupils were excused fromtheir regular mathematics class and permitted toattend one with other gifted pupils. Every schoolshould consider a program for pupils in the upperID percent of the pupil population. The abilities ofthat group may, however, vary greatly from schoolto school, and each program should be tailoredaccordingly.
Mathematics programs for gifted pupils entailexploring abstract concepts, learning to playmathematical games that involve logic, and apply-ing mathematical understanding to other subjectareas such as science: the social sciences: art(patterns and geometric models): and e,..onomics(see hypothesis 34). The gifted program mayinclude field trips that are of partieular interest tothe pupils.
36Teachers are aware of and are Se'llsitire toabilities, needs, amul interests of individual
The research study findings supported the basicprinciple of education that the successful teacher issensitive to the individual pupil's abilities, needs,and interests. Suggested plans for implementingthat principle are as follows:
a.A program ol intraschool visitation nhght bea starting point to help teachers become aware ofwhich of their classroom behavior-, reflect sensi-tivity and Ixhich do not. The visitations must benonthreatening and nonevaluative: they must beconsidered exploratory and therapeutic. Moreover,although a teacher might be considered quitesensitive to pupil needs, the experience of observ-ing another teacher can help observers considertheir own teaching and can lead to an understand-ing of professional strengths :Hid weaknesses. Afterthe visitation each teacher-observer should be ableto prepare a list of specific ways in which he/shewill try to be more sensitive to pupil needs. A-buddy system- whereby teachers observe andrecord attempts to implement the new teacherbehaviors would help improve teaching throughrepetition and reinforcement. Educational researchhas shown such a plan to be effective in modifyingteaching behavior.
34.25
b. A second plan miOt be considered as analternative or as an extension of the previous one.In the second plan each teacher assesses each pupilin terms of objective tests, subjective reports,interviews, and parent conferences. From theseobservations the teacher prescribes a program ofinstruction for each pupil (possibly with theassistance of a team including other teachers) andimplements that program for a prearranged lengthof time. At the end of the program, the gains areassessed for each pupil, and the program for thatpupil is rated by the teacher on its effectivenessand is modified accordingly.
Teachers have a positive Jeeling towardpupils and a _feeling of being well-liked bythem in return.
Teacher evaluations of the quality of theirinteractions with pupils may be used as a measureof how well they relate to their pupils or, moresimply, how well pupils seem to like them.Improvement in this area might be accomplishedby a "buddy system" in which two teachersobserve each other in all contacts with pupils. Theevaluation tool could be a printed form on whichto record the number of positive and the numberof negative communications from teacher to pupil.The Flanders interaction analysis technique is auseful. effective model for this type of ohserving."
TeaMers avoid abusive criticism of pupilsand reJthin from scolding or threateningpupils or arguing with them.
In some schools avoidan;:e of abusive treatmentof pupils is critical. Although pupils may beresponsible for certain classroom problems. abusivecriticism by the teacher increases frustration andharms the learning processes. Alternatives to angryreactions to pupils are expressions of disappoint-ment, redirection of activitic5, discriminatoryapproval of properly behaving pupils, Gr. whenpossible, removal of pupils causing problems (seediscussion of hypothesis 37).
Teachers consistently clarify pupil miscon-9 ceptions, correct pupil errors, and resolve
difficulties promptly.In some schools teachers should be constantly
alert to pupil difficulties and respond promptly.Lessons need to be planned for maximum immedi-
37
38
3
6Ned A. Flanders, 4na/y:ing Teaching Behavior. Reading, Mass.:Addison-Wesley Publishing Co., Inc., 1970.
26
ate assessment of pupil understanding (contrary tohypothesis 40).
40 ems, pupils to discover their ownerrors and be selFdirected.
ln some schools prompt attention to pupildifficulties was found in the study not to beeffective in improving performance (see hypothesis39). In these schools pupils should be encouragedto check their own answers. During discussions theteacher should encourage pupils to respond to thecontributions of others.
41 Teachers are sympathetic and tolerant andallow pupils to) express their Own opinions.
The teaching style described in hypothesis 41reflects a confident, self-assured teacher whoaccepts the feelings of pupils. That type of teacheris often referred to as a "good listener who neverhesitates to draw out a pupil's feelings and opin-ions in order to understand that pupil's problems.
42Teachers praise and encourage pupils hyproviding support, Optimism, or an appre-ciatm of pupil responses.
Praise and encouragement can be easily seen in 1nod, a smile, a touch, a word or two, or anelaborate reward system. In their classroom obser-vations the researchers were particularly attentiveto the till btle expressions rather than the elaborate,-ewards. Teachers can improve their skills in thisarea by using one of the plans described inhypothesis 37.
43 Teachers accept and use the ideas of their
Possibly the greatest praise that pupils Ca rireceive is to have their ideas accepted by theteacher and used to help their classmates. Teachersneed to have flexible lesson plans and he alert touseful pupil ideas. Sometimes the idea can he usedin a simple way, perhaps as a clarification of astatement. At other times the idea may be used torebuild an entire lesson or learning activity. Thesource of an idea should be broadly publicizedwithin the class, to parents, and to other teachers.
44 The instructional program is JOcused onpupils and their needs.
Hypothesis 44 is contrary to hypothesis 8,which focuses on content. When the mathematicsprogram focuses on the learner, more time will bespent in considering the pupil's interests an,.;
3 5
ambitions. Furthermore, teaching will be flexibleto accommodate pupil suggestions. Progress will beevaluated on the basis of objectives which weremutually selected rather than preselected by theteacher or textbook.
45The instruction is restricted to presenta-tions, demonstrations, and the selection oftopics by the teacher alone.
In teacher-dominated classrooms, the pupilsmake little contribution to the lesson. The teacherselects the lesson topic and the time schedule forit. The pupils seldom interrupt with ideas andsuggestions, and their questions are answered bythe teacher with authoritative finality.
46 The teacher cwitrols the classroom withinteresting teacher-directed activities.
In this type of classroom control, the teachermanipulates pupils to provide learning activities.Pupils perform on cue so that concepts can beillustrated and learned by classmates. The lessonmay be lively and interesting, but the teacher is theprimary center of attention. Misbehavior is sup-pressed by intensive focus on activities.
47Teachers hold infOrmal, relaxed confer-ences with their pupils to allmy freeexchange of opinions and discussion ofneeds.
Most teachers talk frequently with their pupilson an individual basis, but making appointments toconfer with pupils is quite a different form ofteacher-pupil interaction. The primary advantageof prearranged conferences is the free exchange ofneeds and opinions under nonstressful circum-stances. That is, when a conference is held at aprearranged time, there is more of a positiveclient-counselor relationship and less of a negativechiid-adult relationship. When the reason for con-ferring is immed ate (i.e., not prearranged), theconference may serve as incitement for an undesir-able behavior pattern (namely, misbehavior, dis-approval, embarrassment). In a prearranged confer-ence, however, this pattern need not occur, as bothpupil and teacher can share goals and concernsfrom positions of mutual acceptance.
4,8 The school's mathematics program is madea matter of high priority.
Many schools have a cycle of priority setting forsubject areas so that the mathematics program isperiodically given specific attention. For some
schools, however, the mathematics program shouldreceive hnmediate and continuing attention.
The school's mathematics program is up-49 dated to reflect new trends in mathemathw
education.
High pupil performance in mathematics may insome schools result from teacher belief that theirmathematics program has been updated. Therefore,to attain higher pupil performance, a school shouldexamine and consider adopting a new mathematicsprogram. The addition of a mathematics specialistor mathematics resource materials may also beeffective. Teachers should be made to feel thattheir mathematics program is being improved inconformity 1.vith new trends.
50 Schoolwide instructiomil objectivesdeveloped for ull subject areas.
Schoolwide instructional objectives serveartibroader purpose than providing uniform guidelinesfor learning: they stimulate communication be-tween and within grade levels; they refresh thememories of experienced teachers and curb theexcesses of' the novices; they allay some of thefears of the parents; and they provide a finitehandle on an infinite task. Instructional objectivesare not meant to be a description of all the learningexpected of all the pupils. Rather, they serve as amap, providing enough detail to help travelers seewhere they are and how to get where they wouldlike to be. The development of instructionalobjectives is a major effort that should not beundertaken without careful preparation. The entireschool staff, interested parents, and communityrepresentatives should participate,
51 Instructional objectives are developed forthe mathematics programs.
Instructional objectives should be developed forcoordination of the mathematics program betweenand within grade levels. The objectives serve asorderly guidelines for teachers to use when select-ing topics. Without guidelines, unnecessary repeti-tion or serious gaps are more likely to occur inpupils' mathematics education (sec the discussionfor hypothesis 50).
il consultant assists in the development andmaintenance of a coherent mathematicsprogram.
A surprisingly large number of schools operatewithout the services of a mathematics specialist. Insome types of' schools, the use of' a consultant
52
3 6 17
could definitely improve mathematics perfor-mance. Schools should consider the followingmethods of acquiring consultant help: hiring adistrict or interdistrict consultant; hiring a part-time consultant to serve several neighboringschools, releasing a key teacher to provide leader-ship; using the services of a graduate assistant froma nearby college; discovering a volunteer in thecommunity with expertise; and soliciting help fromthe office of the county superintendent of schools.
53 Teachers feel «mfident in their ability toteach mathematics.
Teacher confidence in relation to the teaching ofmathematics is a factor which should be consideredcarefully in designing inservice training. Developingconfidence in one's teaching ability is particularlydifficult when pupils perform poorly. As a startingpoint to overcome lack of confidence, teachersshould search out and identify all the good aspectsand successes of their programs. These positiveelements should then be expanded. Thereafter,causes of failures should be identified and remedied.
54 Teachers attend mathematics workchopsand courses.
Implicit in this hypothesis is the assumption thatteachers are given opportunities and encourage-ment to study mathematics. Moreover, opportu-nities should be convenient and attractive, andworkshop leaders should he both knowledgeable inmathematics and responsive to learners.
Courses should be designed to accommodate thewide range of background and ability of theteachers. The teachers should be shown relation-ships between the mathematical concepts theyteach and the broader structure of mathematicalsystems. Encouragement to attend such specialmathematics courses is given real credibility by theschors arranging for substitutes, staging specialassemblies for the pupils of released teachers, andhaving the principal teach classes for the teachers.Encouragement calls for real effort on the part ofthe principal.
55 .21 district-lcrel emphasis on mathematicseducation is initiated and publicized.
The degree of emphasis given mathematics bythe district administration is an important consid-eration in pupil performance. It has been found,for example, to be in the best interests of thepupils in one school of a district to increase theemphasis on mathematics in the other schools ofthe district. Specific proposals delineating mathe-
28
matics prograin goals and plans should be carefullydesigned for each district.
The principal .s.upports teachers 15,ho find31.P mathcmatieS frXthooks inadequate.
It was found in the study that in those schoolsin which principals felt that the mathematicstextbooks were inadequate, the teaching was suchthat pupils performed well. This finding reveals theworth of mathematics programs which are notdependent on textbook teaching and furtherreveals the important role of the principal inSchools With such programs.
The mathematics pmgram supports paren-c-, d tal views of st'Ilat should he taught.
In the schools in which the teachers felt thatparents views were reflected in the mathematicsprogram, pupil performance was better. The staffin those schools may have sensed parental satisfac-tion with the program and may have actuallypromoted that satisfaction in every contact withparents. Teachers in such schools may use parentalevaluation as a source of program planning forindividual pupils as well as for the entire school.
The school staff is inyolred in a communityrelations campaign to derelop parentalinrolrement in the school's mathematicsprogram.
Parental involvement is a way to improve aschool's mathematics program. Developing theinterest and involvement of the parents is a
frequently recurring theme in teachers' meetingsbecause it is recognized as an important factorin pupil performance. Parents usually becomeinvolved not as the result of the efforts of teachersto do something individually but rather as theresult of a schoolwide community relations pro-gram similar to a political campaign. Strategiesmust involve current community issues, grass-rootssupport and work, and advertisement of theschool's potential role in helping to produce abetter life in the community. Parent-teacher associ-ations, the school board, and parent advisorygroups are valuable instruments in such acampaign.
58
59Teachers confr with the parents of eachpupil at least three times annually regardingthe pupil's learning progress and problems.
Many teachers in high-performance schools indi-cated that they conferred with parents of eachpupil at least three times per school year. The
3 7
teacher-parent conferences included in the researchstudy were only timse in which a pupil's progresswas discussed. The setting for most conferenceswas after school in the classroom, but manymeetings were held in the homes or pupils orsomethnes in unplanned meeting places.
The mathematics program is appropriate
60 for the cultural eluiracter of the commll .nify and realistically reflects the commu-nity's needsgoals, and interests.
An important relationship between the successof a school's mathematics program and its cultural
relevance in the community was disclosed by Iheresearch study. ln schools where teachers andprincipals felt that their program addressed itself tothe cultural character of the community, pupilsperformed better. The research study was based onthe assumption that different programs must bedesigned for different communities to maximi/epupil performance, and the results corroboratedthat position. Clearly, a program to improvethe performance of pupils in a given school mustbe based on the application of alternatives accord-ing to the needs and goals relevant to Or,: commu-nity.
3 8
29
charier
5 implementation of a Plan for improlement
So that the results of this study can be put intothe hands of teachers for the greatest effect, atwo-stage dissemination plan has been included asan integral part of an overall implementation plan.In the first stage of the plan, the results of thestudy will be presented to school teams consistingof a principal or an administrator and a classroomteacher. In the second stage each school team willconduct a program for implementing the results ofthe study in the school's mathematics program.Both stages of the dissemination plan are based ona workshop format. The workshop for the firststage is designed for leadership training, and theworkshop for the second stage is designed forclassroom implementation.
Workshops in Leadership TrainingThe first stage of the plan for improving
elementary mathematics programs will consist ofleadership training workshops. The suggestionsgiven here are directed to a district, county, orstate supervisor, administrator, or consultantresponsible for the elementary mathematics curric-ulum in more than one school. The workshops can,however, be integrated into any inserviee trainingprogram concerned with mathematics teaching atthe elementary level. The steps to be taken inconducting a leadership training workshop are thefollowing:
Arrange for the participation of school teamsfrom 20 to 40 local schools approximately sixweeks in advance of the workshop. Reserve theuse of a suitable, centrally located meeting roomon a convenient day for a three-hour workshop.An assistant will be useful at the workshop forproviding hospitality and for helping partici-pants analyze their school's mathematics pro-gram profile. The motivation for attendinif theleadership workshop should be that participantswill identify ways of improving their mathe-matics programs.
2. Send a copy of Appendix 13 of tlik publicationto each participating school approximately twoweeks in advance of the workshop, including thecover letter explaining the importance of collect-ing the requested information prior to theworkshop.
3. Collect the tbllowing materials, one copy foreach participating school tThe materials may beduplicated from this publication, or additionalcopies may be purchased from PublicationsSales, California State Department of Education,P.O. Box 271, Sacramento 95802. The price isS 1.25 plus ti percent sales fax ('or Californiaresidents.):
a. Printed materialsImplementation workshop outlineItypotheses for improving elementary
mathematics programsImplications of hypothesesSchool profile key (card)Evaluation of workshop in leadership train-
ing (see Appendix (')
b. Transparencies ('or use on overhead projector(see Appendix D)
Selection of sample schools for this studyInstruments used in study of schoolsSample hypothesisCategories of hypothesesSchool profile key
4. Conduct the workshop, using the followingguidelines (total time-180 minutes):a. Part One (50 minutes)
RegistrationPremeetingDistribution of name taif.sWorkshop introductionOverview, charge, and instructions
List of main topics in these guidelinesExplanation of purpose and responsibilitySuggestion: note-taking
3 930
Discussion of the it'SC I
Sidi'Lli011 t1 salliple SchOOls nansparency;ill types of schools identified; derinitionhigh-achieving and low-achicving schools.:ind sampling procedureInstruments used in study (transparency I.on-site observation or teaching; interviewsof teachers and principals; and anonymousquestionnaireslypot heses (transparency): 500 hypo-
theses tested; (10 hypotheses verified statis-tically; and hypotheses checked by type orschoolSchool prorile key (trinsparency methodof recording school information on key andusing the key to select applicable hypo-theses (demonstration or sample hypo-theses transparencyCategories or hypotheses (transparency):examples of each category
Questions about the research methods ( ten-minute maximum)
b. Part I'm, 140 minutes)School profile key preparation
TransparencyIndividual assistanceCompletion by every le:1111
dell tiflea 0011 or applicable hypothesesUse of school profile kevRecording or number of black squaresshown for each hypothesis (mark onlythose hypotheses with tliree Or more blacksquares)('areful reading of applicable hypotheses
Workshop break (included in ;11c 40-minuteperiod)
c. Part Three (60 minutes)Implications of hypotheses
lypotheses with most black sq- !res to herirst itemsHypotheses to he read :Ind correspondingimplication and illustrative remarks to bemade as time permitsDiscussion or one category or hypothesesat a timekxplanation or hypotheses as likely hut notguaranteed ways of improving programsApplication of hypotheses to known situa-tions in participating schoolsNot all hypotheses to be discussed
d. Part L'Hur 1 30 mimites)Discussion or implementation workshops
Outline of workshops to he reviewed
5
\Vorksliop be held within two to fourweeksdvance publicity to begin immediatelyQuestions and problems; concluding. re-marks
I valuation or leadership training, workshopsPurpose: to assist State Department orIducation in planning future workshopsConipletnm or suggested Form Isee Appen-dix (')
Include in the h)llow-up:
Telephone calls to aid in ilunallging workshopsQuery made to find whethYr evaluation formswere turned inIssaluation sent rrom leadership trainingworkshops
Implementation WorkshopsImmediately after a leadersInp training work-
shop has been completed. planning should hebegun ror implementation workshops at the schoollevel. The pesons involved in the planning will bethe members of the school teams that participatedin the leadership training workshop, The steps toInc ta:.;en ill couducting all implementation work-shop are the following:
pnmipay. A two-hour planning meeting rorall teachers administrators. resource personnel .
and twiructionai aides in the school should bescheduled two to rour weeks in advance of theworkshop, (Much of the value of the leadershiptraining workshop could be lost if the imple-mentation workshop is held [Imre than fourweeks alter the leadership workshop.) If possi-ble, the workshop should he scheduled oil theafternoon of ZI shortened day. Time should beallowed for teachers to take care or lesson plans,reports. and clerical duties prior to the work-shop.
The productivity of the workshop will be,...nhanced ir the meeting place dwily fromschool and adinniistrativc ,%brices, A nearbyhusiness ort ice. church, home. restaurant. orother racility can provide a pleasant and relaxedatmosphere with few dist.ractions, interruptions.and reminders of frustrations.
2. Publicize the workshup. Pulnlieity %kill helpestahlish expectations and relieve anxieties. Theteacher on the school leadership team shouldtake primary responsibility for publiciiing andpresenting the workshop. Caeater productivityand program improvement often result %kiwi]reform is initiated by classroom teachers.
4 0 31
Posters in the teachers' room, announcementsin the daily bulletin, and other media shouldreflect the team's creativity and the s,:hoorsspecial needs. Some questions that might beconsidered are: (a) What are we doing aboutmathematics? (b) How effective is the mathe-matics program in our school? (c) Is ourmathematics program up-to-date? and (d \ Whycan't Johnny add?
3. Prepare the workshop training Inaterials. Prepa-ration requires listing the hypotheses (withoutthe black squares) in the order of importance tothe improvement of the mathematics program.The list for each of the six categories ofhypotheses should be on a separate page. andone copy should be made for each teacher. Anex)lanatory sheet should explain briefly thatthe hypotheses apply specifically to this typeof school and that if teachers can find waysto implement the hypotheses. improvement inthe pupils' mathematics achievement can beexpected. One copy of Appendix E should beprepared for each participant.
4. conduct the workshop according to specificguidelines. The workshop should provide ampleopportunity for open discussion about the appli-cable hypotheses. Teams or three to five personsshould then prepare implementation plans andensure that the plans are carried out.a. Introduction and explanation (20 minutes)
Explain the need for improvement in theschool's mathematics program.Explain the hypotheses verification proce-dure used in the research.
b. Discussion of hypotheses (60 minutes)Seek ideas for implementing the hypo-theses.Discourage negative, pessimistic opinions.Use implications printed in the report tostimulate discussion.Take notes on each idea for implementa-tion, including who made it.Look for plans that implement more thanone hypothesis.
c. Assignments on implementation teams (30minutes)
Select plans with high potential or interest.Name the plans and start a sign-up sheet foreach one.Allow time for each teacher to select twoteams on which to participate.Set an initial meeting time for each team.See that the leader serves on all teams.
32
d. Evaluation of implementation workshop (10minutes)
Purpose: Assist State Department of Educa-tion personnel in planning future work-shops.Complete the form in Appendix E.
5. Carry out JOIlow-up activities:
a. Remind teachers of their team meeting.b. Seek a leader for each team.c. Publicize the plans developed by each team.d. Provide liaison between teams.e. Seek cooperation between teachers and
administrators.1. Send evaluations to the coordinator of the
leadership training workshop.
Recommendations for Further ResearchA number of ideas emerged during the planning
stages or the study that deserve further examina-tion. The most obvious additional research neededin ,.nmection with the study is follow-up in thefollowing areas:
Which hypotheses created the most profoundchanges in mathematics programs?
o Which types of schools were helped the most?How much improvement in pupil achievementoccurred as a result of implementation of thevarious hypotheses?
Furthermore, a school-by-school survey ofimplementation plans developed by school staffcould be a valuable resource. A comparison orteacher-perceived effectiveness and the actualeffectiveness of the approach to program improve-ment suggested in this study might also be made.
Replication of this study for another sample ofschools in California or for schools in another statewould be useful. Verification could be accom-plished by the use of different observation instru-ments, interview procedures, and questionnaires. Inparticular, more subtle interview questions mightprovide better insight into such teacher characteris-tics as sense of mission, empathy, rapport, percep-tion of individual pupil needs, ability to listen, andinnovativeness. Interview questions with structuredresponse choices might be designed to providemore uniformity of interpretation. The ques-tionnaires might also be modified to ensure unifor-mity of interpretation, and open-ended questiontechniques might be utilized. The observationinstruments could be simplified and checked forobserver reliability.
4 1
As to the demograPhic variables used in thestudy for categorizing sample schools, data areavailable for a different basis for measuring eachschool's socioeconomic status and pupil mobility.Other variables might be considered that statisti-cally or logically provide better categories thanthose selected in this study. The additional vari-ables could include mean family income, varianceof the distribution of family incomes, educationalexpenditure per pupil, educational achievement forparents, pupil-teacher ratio, age and type of schoolbuilding, and variables that affect learning but liebeyond the direct control of a school's staff.
The study might be improved by excludin4those schools from the sample for which certainmajor changes in demographic variables haveoccurred during the period used for establishingthe trend of mathematics achievement. Likewise,some assurance should be present that the mathe-matics program scrutinized in the study at eachsample school is basically the same one whichproduced the trend in achievement at that school.At least the factors that ac.;:ount for the trendsshould be identified.
Despite the wide range of program characteris-tics selected prior to visits at the sample schools.many more characteristics exist, some of whichmay be significant for progi-am improvement; e.g.,the kind of homework assignments and the amountof emphasis placed on them; the type of reportcards; and the quality of interaction between theprincipal and the pupils. Other characteristicsmight be revealed by researchers concentrating onone type of school, comparing le:Jures of high-achieving school programs with features of low-achieving school programs. Characteristics thusfound might be quantified so that their occurrencecould be recorded for one type of school in thesample. After verifying or rejecting the varioushypotheses regarding features of mathematics pro-grams for that type of school, the researcherswould move on to another type of school andrepeat the process.
Many of the recommendations for mathematicsprograms made in this study can also be applied to
4 2
the teaching of other subjects, particularly reading,spelling, and grammar; that is, subjects that utilizemental processes and skills similar to those used inthe study of mathematics. Furthermore, achieve-ment scores for these subjects are available fromthe same source as the mathematics achievementscores.
This research study has provided useful informa-tion for organizing and implementing changes inthe mathematics program of a given school. Thehypotheses are verified for different types ofschools on the basis of total school scores. It isunlikely that a single teacher could implement allthe hypotheses that apply to that teacher's school,nor could a single teacher expect all the applicablehypotheses to be effective in improving achieve-ment in one classroom. An additional useful studymight be undertaken in which the classroom ismade the basic unit.
The number of control variables would have tobe expanded cowiderably in a classroom-levelstudy beyond We demographic variables used inthis school-level study. Descriptors of the school-wide mathematics programs would become vari-ables which the individual teacher would haveno control, hut these variables would neverthelessaffect the mathematics program in each classroom.Such a classroom-level study would aid individualteachers in a given school setting independently ofthe success or failure of other teachers in the sameschool. his strategy would be particularly advan-tageous in a school with weak internal communica-tion channels, uncooperative factions, or an influ-ential group of unadaptive teachers. A classroom-level study might also be used effectively in thevarious departments and divisions of secondaryschools, grades seven through fourteen.
In summary, the model for educational researchproposed in this report has many possible varia-tions and implications for improvement in instruc-tional programs at all levels in all subject areas.Hopefully, educational researchers will use thisnew mathematics program model to provide front-line teachers with guidelines for "what to do whenthe students arrive on Monday morning.
33
Appendixes
Appendix A. Observation instruments Used in the Study
Appendix B. Cover Letter and Information Needed to Make a School Profile
Note: The material contained in Appendix B should be sent to those invited to attend aleadership training workshop. The material should be sent well in advance of the workshop.and those invited should be reminded of the location and time of the workshop. (Seeinstructions for workshops in leadership training on page 30.)
Appendix C. Evaluation of Workshop in Leadership Training
Note: The evaluation form is to be completed at the conclusion of the workshop. Eachparticipant should be given an evaluation form. (See instructions for the evaluation ofworkshops in leadership training on page 31.)
Appendix D. Transparency Masters
Note: A complete set of transparencies should be made from the masters provided. (See theoutline of the presentation of the transparencies on page 31.)
Appendix E. Evaluation of Implementation Workshops
Note: The evaluation form is to be used at the conclusion of the implementation workshop.Copies may be provided to workshop leaders for their use when they conduct implementa-tion workshops in their own schools. Or a single c.opy may be given to each leader withinstructions to make copies for the teachers at his/her school. (See reference toimplementation workshops on page 32.)
43
35
Appendix
A Observation histruments Used in the Study
Mathematics Program Inventory (School Level)A. Description of Your School's Mathematics Program
Check the items which best describe your school's mathematics program:ExemplaryExperimentalRelevantContent-centeredPupil-centeredTeacher-centeredAcceleratedRemedialTeam effortInnovative
O TraditionalO Contemporary
CreativeO New programO Same old programO High priorityO Medium priorityO Low priorityO Differentiated staffingO Differentiated pupil assignments
Modular schedulingHomogeneous groupingHeterogeneous groupingReflects teachers' viewsReflects administration viewsReflects parental viewsReflects societal needsBased on modern curriculumprogram
Schoolwide perfornmnce objectives
Other
B. Teaching Methodology Used in Your School's Mathematics ProgramCheck the items which best describe the teaching methodology used in your school's mathematics program:
Much individualizationSome individualizationLittle/no individualizationDiscovery approachIndividually prescribed instructionDiagnosis of learning difficultiesIndependent projects/inquirySome enrichment instructionSome gifted instructionManipulative aids
Multimedia approachPerformance contractsLearning packagesTask cardsRole playingMathematics gamesPeer tutoringTeacher-pupil conferencesFrequent practice/drill
Emphasis on computational skillsand concepts
Textbook-orientedWorkbook-orientedWorksheet-orientedChalkboardOverhead projectorClassroom learning centersResource center outside of roomSmall-group projects
Other
C. Description of the Teaching Style in Your School's Mathematics ProgramRank numerically ( I , 2, 3, and so forth) the teaching modes in the order of usage:
Lecture/demonstrationDiscussionQuestion/answer (not,
discussion)
Recitation/drillIndividualized instructionSmall-group instruction
_ Peer tutoring
Demonstration_ Tutoring_ Programmed instruction
Other
36
44
D. Appraisal of the Pupil Levels in Your School's Mathematics Program
Indicate the approximate number of pupils in your school (or room) mathematics program for each characteristic:Achievement level: Under At grade levelMotivation level: Low AverageAbility groups (if used): Low Middle
OverHighHigh
What is the grouping criterion.'
E. Improvement of Your School's Mathematics Program
Rank numerically ( , 2, 3, and so forth) the ten items which you feel describe the most effective ways which have beenused to improve your school's mathematics program:
_ Additional faculty Workshop topicsAdditional aides Higher salariesAdditional resource personnel Smaller classes
_ District mathematics consultant Better ethnic balanceDistrict involvement _ Additional instructional materialsAdministrative support and aidsMathematics workshops Bet ter classrooms (better
physical environment)
Other
_ Open classroomsGreater emphasis on reading skills
_ More ability groupingLess ability groupingAdequate textbook guideGreater parental interest
and involvement
F. Identification of Factors Limiting Program Success
Rank numerically (1, 2, 3, and so forth) the characteristics which you perceive as limiting the success of your school'smathematics program:
Faculty abilities in mathematics Ethnic compositionFaculty attitudes toward De facto segregationmathematics Class size
Pupil attitudes toward school Condition of buildingsPupil attitudes toward mathematics Program irrelevancyPupil motivation levels Lack of concern for individualSchool's traditionalism pupil needs
_ Diluted programShortage of aides
_ Ability grouping scheme_ Mobility/absenteeism rate_ Resistance to change and innovation
Lack of program objectives_ Inappropriate textbook_ Ineffective use of existing resources
Other
4 5
37
G. Your Opinion of Your School's Mathematics Program
Check the ap,mopriate box of the scale which best describes your opinion of your school's mathematics program:
I. Sensitivity to individualabilities, needs,and interests
2. Purposefulness of mathe-matics tasks and assign-men ts
3. Encouragement of pupildiscovery and inquiry inmathematics
4. Quality of teacher/pupilinteraction
5. Inservice training inmathematics content
6. Inservice training inmathematics methods
7. Planning and organizationof mathematics program
8. Instructional objectivesfor mathematics program
9. Planning and preparationof mathematics lessons
10. Provision for participationby all students
11. Diagnostic and prescriptivemethods for learning
12. Extent that pupils areinvolved in selectingmathematics activities
13. Special instruction forgifted and high achievers
38
Superior
NeedsSatis- improve-fac tory men t
El
El
El
El
El
El
El
El
El
El
El
El
El
0
El
El
El
0
El
0
El
El
0
El
El
o
0
El
El
El
El
El
El
El
El
El
El
El
14. Extent mathematics pro-gram is integrated withother subjects
15. Relevancy of mathematicsprogram for culturalcharacter of the school
16. Extent mathematics pro-gram has been updated
17. Adequacy of basic mathe-matics textbook
18. Extent of independencefrom textbook toworkbook
19. Clarity of explanations anddirections in mathematics
20. Use of audiovisual mediain mathematics
21. Use of manipulative aidsin mathematics
Use of small-group workingsituations
23. Opportunities forindependent mathe-matics projectsand experiments
24. Use of activity-centeredmathematics instruction
25. Teacher's confidence withmathematics
4 6
Satis-Superior factory
0
Cl
0
Cl
0
Cl
CI
CI
Needsimprove-
ment
Mathematics Teacher Inventory
Personal
Sex Age level (decade) Ethnicity Socioeconomic statusDaily teaching load (in hours)
Education
College major College minorRecency of formal mathematics training (year)Number of college courses in mathematics methods Number of college courses in mathematics contentNumber of inservice mathematics courses Number of workshops in mathematics
Experience
Total experience (in years) District experience (in years) Motives in choosing career
Satisfaction with grade level Satisfaction with socioeconomic status, ethnicitySatisfaction with school Satisfaction with school districtSatisfaction with community Type of school (socioeconomic status, ethnicity, community) preferred ideally
Choice of teaching career again?
School Goals
Does the teacher have a perception of school goals?View of school goals, purposes
View of school prioritiesView of current school objectives/priorities attainedView of school problemsView of textbook Supplementary materialsAdherence to textbook Page by page9Variance of mathematics program/strategy from day to day (or week to week)
Encourages alternate methods in pupil problem solvingExtended assignments (hour/week) Number of monthly use of niedi:! .(nirces: Overhead
projector 16mm film ITV Tape Filmstrip OtherAwareness of pupil weaknesses/strengthsRadical changes in teaching methods/organization (past four years) In what area?
Individualized instruction Ability grouping Manipulative aidsLess emphasis on textbooks Activity centers Other changes
Preference/avoidance of certain mathematics topicsFlexibility in developing/using new strategiesPreference/avoidance of certain mathematics topicsSource of new podagogical ideasTechniques used to stimulate interest and motivate pupils
Relationships/Committees/Improvements
View of effectiveness of teacher/parent communicationView of parent support of mathematics programView of parent support of teacher innovation/experimentationActual participation on committees/curriculum planningOpportunity for participation on committeesNumber of teacher/parent conferences (this year)Standard mathematics achievement tests administered and frequency
4 7 39
Are test data available to teacher?Do classroom physical facilities, equipment, furniture appear to be adequate?Does teacher experiment with new strategies?Does textbook restrict the way the teacher teaches mathematics"Is class time efficiently used?Number of aides Part-time Full-timeAide used in tutoring? In recordkeepine
In drill? In testing"In centers? Other?
Is aide efficiently used (observe)?Is primary emphasis of instructional objectives aimed at: Attitudinal change
Cognitive change Motivational changeDoes teacher use a variety of strategies?Are teaching strategies consistent with objectives?
Mathematics School Inventor's,
Principal Variables
Sex Age level (decade) Ethnic originWho plans mathematics inservice workshops?Greatest parent contribution to mathematics program in past yearPrediction of greatest contribution parents might make to mathematics program in next year
Number of visitations to each mathematics classroom per yearSupport/encouragement of mathematics program (nonvisitation)View of the need for evaluation of mathematics programActions/programs to improve mathematics achievement test scoresUnderstanding of thr potentials/limitations of teacherEncouragement of teacher self-improvementExtent of encouragement of teacher innovation/experimentation in mathematics programFaculty ability to identify with school/communityInvolvement with school/community/parent committee
School Variables
Mathematics consultant in district?Other curriculum consultant working in mathematics?Consultant role in planning of the mathematics curriculum/programRating of mathematics program in district (low, medium, highsNumber of mathematics workshops this year Last yearWoikshop topicsPriority level of mathematics in district Number of teachers with M.A. degreeNumber of teachers in probationary statusSchool reputation in district 00w, medium, high) Number of disadvantaged pupilsSchool prioritiesSchool problemsFaculty strengths/weaknessesIs mathematics program relevant to social needs? To pupil needs"Which needs of pupils are not met by program?
Textbook publisherView of textbook Supplementary materialsWhen adopted
4 840
Workshops critically needed
Enrollment (ADA) District expenditure per pupilSchool expenditure per pupil Length of school day (in minutes)Number of full-time faculty Part-time ratio Number of male facultyTeacher male/female ratio Number of instructional aides: Full-time_ Part-time
Paid Volunteer Parent OtherNumber of new pupils added since October 1 Number of pupils leaving school since October 1Pupil mobility rate (annual)Physical environment/neighborhood characteristics: Inner city Commercial Industrial
Suburban Rural Farming Econonfically depressed Sinai! cityMedium city Large city Luxury housing Custom housing Tract housingApartments Substandard housing
Number of blacks Number of Chicanos Number of AsiansNumber of Native Americans Percent of pupils bused for de facto segregationDoes the community favor busing'> Ethnicity index (percent white)Socioeconomic status index: Low MediumNumber of AFDC families Teacher mobility (last year)School needs assessment conducted? Describe.Temporary state/federal funding affecting mathematics programSourceNumber of resource/consulting personnel Title I personnelNumber of pupils receiving free lunchTitle I mathematics projects affecting schoolFunding levelTitle III mathematics projects affecting schoolFunding levelNumber of teachers who are very talented in mathematicsNumber of teachers who show high interest in mathematicsNumber of teachers whose classes score high on district mathematics achievement testsPercent of pupils obtaining free lunch Number of special mathematics programsPercent of pupils in remedial reading programsPercent of pupils in remedial mathematics programs
Iligh
4 941
Observed Classroom Behaviors and Organization
Verbal Behavior
Teacher
I. Accepts feeling1. Praises/encourages3. Accepts/uses pupil ideas4. Asks questions5. Answers pupil questions6. Lectures (explains, tells)7. Corrective feedback8. Gives directions9. Criticizes pupil/justifies authority
Pupil
10. Pupil-initiated talkII. Pupil response12. Pupil questions
Nonverbal Behavior
Teacher
13. Teacher demonstration/AV usage14. Teacher supervision/monitoring15. Teacher clerical activity
Pupil
16. Pupil-directed practice/activity17. Pupil inattentiveness/confusion
None Some Much
O 0O 0
ElEl
Room groupingNumber
of pupils Leader
Large-groupSmall-groupSmall-groupSmall-groupSmall-groupIndividual
A P
A P
A P
A P
A I'A I'
'reaching style
O Lecture0 Question/answerO DiscussionO SupervisionO Overhead projector
O Demonstration0 Testing0 TutoringO Chalkboard
Room layout/equipment
O Self-containedO Open spaceO Mathematics tableO Mathematics bulletin
boardO Row/column layoutO Cluster layout
O Individual layoutO DesksO TablesO Learning centersO Individual AV setupsO Manipulative aids (group)O Manipulative aids (pupil)
Strategies
O Discovery0 DidacticO Drill
0 Rule/exampleO Mathematics gamesO Worksheet activity
Notes:
5 0
42
Classroom Observation Seale
Pupil Behavior
I . Apathetic 1 2 3 4 Alert 5. Lacking in enthusiasm I 2 3 4 Enthusiastic2. Obstructive 1 2 3 4 Responsible 6. Compliant 1 2 3 4 Self-directed3. Uncertain I 2 3 4 Confident 7. Lacking in participation 1 2 3 4 Full participation
Teacher Behavior
8. Partial I 2 3 4 Fair (impartial) 26. Unenthusiastic 2 3 4 Ently,iastic9. Autocratic / 3 4 Democratic 27. Impatient 2 3 4 Patient
10. Aloof 1 1 3 4 Responsive 28. Uncontrolling 2 3 4 Controlling11. Restricted I 2 3 4 Understanding 29. Inhibited 2 3 4 Uninhibited
Harsh (cold) I 1 3 4 Kindly 30. Unprepared 2 3 4 Prepared13. Dull 1 2 3 4 Stimulating 31. Inefficient 2 3 4 Efficient14. Stereotyped 1 2 3 4 Original 32. Lacking in humor 2 3 4 Sense of humor15. Apathetic 1 1 3 4 Alert 33. Ignores pupils / 3 4 Listening ability16. Unimpressive 1 2 3 4 Attractive 34. Frustrated 2 3 4 Happy17. Evasive I 2 3 4 Responsible 35. Unimaginative 2 3 4 Resourceful18. Erratic I 1 3 4 Steady 36. Subjective 3 4 Objective19. Excitable 1 2 3 4 Poised 37. Coarse 2 3 4 Refined20. Uncertain 1 2 3 4 Confident 38. Uncooperative 2 3 4 Cooperative/1. Disorganized 1 1 3 4 Systematic 39. Lacks ambition / 3 4 Ambitious
Inflexible 1 / 3 4 Adaptable 40. Inconsiderate / 3 4 ConsideratePessimistic I 1 3 4 Optimistic 41. Worried 2 3 4 BuoyantImmature 1 1 3 4 Integrated (mature) 41. Unreliable 3 4 Reliable
25. Narrow (rigid) 1 2 3 4 Broad (accepting) 43. Dependent l Self-reliant
Classroom Environment
44. Teacher-centered I 2 3 4 Pupil-centered 54. Disorderly 1 2 3 4 Orderly45. Structured 1 1 3 4 Unstructured 55. Quiet 1 1 3 4 Noisy46. Passive 1 / 3 4 Activity-oriented 56. Didactic (telling) 1 / 3 4 Disaivery-oriented47, Grouped (15) 1 1 3 4 Individualized(<3) 57. Flexible materials 1 2 3 4 Text book-oriented48. Lacks manipulation / 3 4 Manipulative 58. Rigid layout 1 2 3 4 Flexible layout49. Lacks media 1 / 3 4 Multimedia 59. Uniform tasks I 2 3 4 Differentiated tasks50. Aimless tasks 1 / 3 4 Purposeful tasks 60. Convergent thinking 1 1 3 4 Divergent thinking51. Formal / 3 4 Casual 61. Teacher-selected tasks 1 2 3 4 Pupil-selected tasks52. Unattractive 1 2 3 4 Attractive 62. Nondiagnostic I 2 3 4 Diagnostic53. Restricted movement 1 2 3 4 Free movement 63. Self-contained 1 2 3 4 Open
Notes:
5 143
Appendix
Dear Friend:
Col er Letter and Information Nee(IedTo Make a School
ICOVER LETTER)
You recently received an invitation to attend a special workshop to be presented in your area. In anticipation of yourattendance, we would remind you to make all the necessary arrangements and to obtain the information requested below.
The workshop is based on a study completed by the Mathematics Education Task Force of the California StateDepartment of Education. To apply the results to your school, you must classify your school according to six variables:
Socioeconomic statusMinority representationSchool size (enrollment)
City size (population)Type of. communityPupil mobihty
Each of these variables has three classifications: low, medium, and high.
The information on the attached sheets describes the variables and defines the limits of the three classifications for eachvariable. Before the workshop presentation, current information and statistics should be collected for your school. On thebasis of this information, a profile key card will be punched at the workshop and used to determine which recommendationsapply to your school. In the case or incomplete or inconclusive informatkm, decisions about the classifications will be madeat the workshop before the profile key card is punched.
Sincerely yours,
5 2
45
Information Needed to Make a School Profile
Six demographic variables are used to analyze the results of this study. Every school ranks in one of the three levels of'each variable: low, medium, or high. The limits of those levels, along with instructions, are as follows:
Socioeconomic Status
The level of a school's socioeconomic status is determined by the following computation:Three (3) points for each percent of all pupils whose supporting parent is an executive. a professional or semiprofessional,
a manager, a clerical worker, or a college-trained technicianTwo (2) points for each percent of all pupils whose supporting parent is a skilled or semiskilled employeeOne (1) point for each percent of all pupils whose supporting parent is an unskilled employee, is unemployed, or is
receiving welfare assistance
Scale: High 24 I -300Medium 209 240Low 100 208
Example: 20 percent professional 60 points70 percent skilled 140 points10 percent unskilled 10 points
Totals: 100 percent 2-1-0 points
Socioeconomic status level: Medium (between 20) and 240)
Minority Representation
The minority representation level of a school is determined by combining the percent of Asian, black, Chicano, and NativeAmerican pupils enrolled:
Scale: High 29 100 percentMedium 10 28 percentLow 0 9 percent
School Size (Enrollment)
The enrollment .igures should be taken from the most recent records available. If significant changes are anticipated, theprojected enrollmen; figure should be used,
Scale: High Over 560Medium 350-560Low Under 350
City Size (Population)
City size is determined by the combining of population figures for residentially or commercially connected communities.Scale: High Over 200 000
Medium 50 000-200 000Low Under 50 000
Exampk: High AnaheimFullerton Buena Park(area) Los Angeles
Medium Citrus HeightsCarmichael(area)
Low SanteeEl Cajon (area)Santa RosaPacifica
Type of Community
The classifications for the community environment are not quantified, but the "high," "medium," and "low" labels areretained for convenience. If a school's geographic community contains any rural or farming industry, regardless of the extentof residential or industrial development, it falls in the "low" category. If there is no farming but there are old commercial orindustrial buildings, substandard dwellings, or predominantly commercial zoned property, then the community falls into the
46
"high" category. Other communities are suburban residential and are labeled "medium." The communities from whichbused pupils come are not considered in this classification.
Scale: High Old commercial or industrial buildings, substandard dwellings. conunercially zoned (no farming)Medium Suburban, residential (no farming, no industry)Low Rural, farming
Pupil Mobility
The pupil mobility index used in this study is the sum of the pupils added to or deleted from the school's enrollmentbetween October 1 and May I divided by the actual enrollment on October 1 and expressed in percent. A projected sum forthe seven-month period may be used if a trend has been established.
Scale: High Over 37 percentMedium 23-37 percentLow Under 23 percent
5 4
47
AppendixEvaluation of Workshop in Leadership Training
Name (optional) Title of positionSchool Telephone numberSchool address CityCounty ZIP codeWorkshop leade- Date
H
Record of schoolprofile key:
Please mark all items which indicate your opinion.
1. Discussion of the Research:
0 Clear 0 Interesting D Too short 0 Important0 Unclear E Borii:g 0 Too long 0 UnimportantComments
2. Marking of Applicable Hypotheses:
0 Clear 0 Interesting 0 Too short 0 Important0 Unclear 0 Boring 0 Too long 0 UnimportantComments:
3. Implications of Hypothess:
0 Clear 0 Interesting 0 Too short 0 Important0 Uncle:, 0 Boring 0 Too long 0 Un impor tan tComments:
5 548
4. Discussion of Workshop in Leadership Training:
0 Clear 0 Interesting 0 Too short 0 Important0 Unclear 0 Boring 0 Too long 0 UnimportantComments.
S. General Rating of the Workshop Experience:
0 Helpful 0 Needed 0 Interesting 0 Pleasing0 Useless 0 Unneeded 0 Uninteresting 0 DisappointingComments.
6. Plans for Implementation Woricshop in Your School:
O Already scheduled (dateO Intend to set date promptly; no problems.O Intend to set date, but there are problems. Explanation:
O Not likely. Explanation:
7. Anticipated Attitude of Teachers to Implementation of the Study in Your School:O Very helpful 0 Somewhat helpful 0 Probably helpful 0 HarmfulComments-
Please hand this evaluation to the leader before leaving. Thank you.
5 649
AppendixTransparency Masters
Selection of Sample Schools for Mathematics Study
High-Achievement
Schools
CaliforniaE lementary
Schools
Medium-Achievement
Schools
Low-Achievement
Schools
HighAchievement
LowAchiteLnent.7
Socioeconomic Status
50
5 7
Instruments Used in Mathematics Study at the School Level
1. Teachers' and principals' selection of descriptors for their mattmatics program*
2. Teachers' and principals' ranking of ten descriptors for tillmathematics program*
3. Teachers' and principals' ranking of ten methods used to improtheir mathematics program*
4. Teachers' and principals' ranking of ten problems that affected tlsuccess of their mathematics program*
5. Teachers' and principals° rating of their success in attaining certaobjectives in their mathematics program*
6. Observers° rating of classroom attitudes, behaviors, and atmospheion the Likert Scale of Opposites
7. Observers' count of types of classroom interactions and techniqu(
8. Information about schools obtained in interviews with the princip;
9. Information about teachers obtained in personal interviews
*Anonymous questionnaires
r o
Sample Hypothesis
Pupils tend to exhibit higher achievement patterns if... .
The principal makes frequentclassroom visits.
52
Pupils are involved in planningtheir own activities.
5 9
Categories of Hypotheses
General classroom teaching styles
Specific teaching techniques
Instructional materials
Teacher/pupil interactions
instructional leadership
Parent/community involvement
6 0
53
School Profile Key
Low Medium High
Socioeconomic status 100-208 209-240 241-300
Minority representation 0-9 10-28 29-100
School size (enrollment) Under 350 350-560 Over 560
City size (population) Under ROM 50M-200M Over 200M
Type of community Rural Suburban Industrial
Pupil mobility Under 23 23-37 Over 37
54
School Profile Key
H
M
L
School name
Date
6 1
1
AppendixEvaluation of Implementation Workshop
Name (optional) Title of positionSchool Telephone numberSchool address CityCounty ZIP codeWorkshop leader Date
Please mark all items which indicate your opinion:
I. Explanation of Workshop Purpose and Plan:
0 Clear 0 Interesting 0 Too short 0 Important0 Unclear 0 Boring 0 Too long 0 UnimportantComments:
2. List of Hypotheses:
0 Clear 0 Interesting 0 Too short 0 Important0 Unclear 0 Boring 0 Too long 0 UnimportantComments:
3. Discussion of Implementation of Hypotheses:
0 Clear 0 interesting 0 Too short 0 Important0 Unclear 0 Boring 0 Too long 0 UnimportantComments:
4. General Rating of Workshop Experience:
0 Helpful 0 Needed 0 Interesting 0 Pleasing0 Useless 0 Unneeded 0 Uninteresting 0 DisappointingComments:
5. Anticipated Effect of This Plan on Improvement of the Mathematics Program in Your School:El Very helpful 0 Somewhat helpful 0 Probably helpful Harmful
6. Additional Information
List the hypotheses by number which seem to you to be most important for improvim: e mathematics program inyour school:
List the hypotheses which you will be most concerned with on your implementation teams:
If you are the leader of a team, (a) circle the numbers of the hypotheses listed above which your team willimplement; ancl (b) briefly describe the plan for implementation as you see it. Use the other side of this sheet.
75-33 03-0300-300 5-76 12MPlease return this evaluation form to the workshop leader.
6 255
CA
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