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Initial Effect of Calculators in Elementary School MathematicsAuthor(s): Richard J. Shumway, Grayson H. Wheatley, Terrence G. Coburn, Arthur L. White,Robert E. Reys, Harold L. SchoenSource: Journal for Research in Mathematics Education, Vol. 12, No. 2 (Mar., 1981), pp. 119-141Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/748707 .

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Journal for Research in Mathematics Education

1981, Vol. 12, No. 2, 119-141

INITIALFFECTFCALCULATORSNELEMENTARYSCHOOLATHEMATICS

RICHARD. SHUMWAY

The Ohio State University

GRAYSONH. WHEATLEY

Purdue University

TERRENCE. COBURN

Oakland Public Schools

Pontiac, Michigan

ARTHUR . WHITE

The Ohio State University

ROBERT . REYS

Universityof Missouriat Columbia

HAROLDL. SCHOEN

Universityof Iowa

Every principaland mathematics teacher is faced with decisions con-

cerning use of calculators in the schools. Public debate over calculator use

for teaching mathematics is spirited and appears most controversial for cal-

culator use in elementary schools (Shumway, 1976).The Calculator Information Center has collected and abstracted research

reports dealing with calculator effects. In a recent report (Suydam, 1979),over 100 studies were summarized and critiqued. Most of the studies re-

ported suffered from serious design and sampling problems and few valid

conclusions can be drawn.

PurposeThe purpose was to determine the effect the availability of calculators to

students and the availability of calculator-related curriculum resources,consultant resources, and in-service workshops for teachers had on the ele-

mentary school children's attitudes and achievement in mathematics,Grades 2-6.

There were many possible effects one might reasonably expect, such as

changes in parent willingness to make calculators available in the home,differences in teacher acceptance and use of calculators, modification of the

This material is based on research supported by the National Science Foundationunder Grant No. SED 77-18077. Any opinions, findings, and conclusions or recom-mendations expressed in this publication are those of the authors and do not neces-

sarily reflect the views of the National Science Foundation.

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mathematics curriculum, changes in knowledge of basic facts by children,

changes in children's attitude toward mathematics, changes in teacher's atti-

tude toward calculator use, changes in children's general mathematical

achievement, and changes in use of exploratory techniques in learning

mathematics. While information on many of these effects was gathered,principal focus was centered on those effects of immediate concern to par-ents and educators; namely, (a) change in children's attitude toward calcu-

lators and school mathematics, (b) possible interference with children's

growth in knowledge of basic facts and paper-pencil computations, (c)changes in children's scores on standardized achievement tests in mathe-

matics, (d) potential development of additional mathematics concepts re-lated to calculator, and (e) change in computational power of children of all

grade levels (2-6) when using calculators.

Sample

The sampling design included three dimensions: Site, Grade Level, andTreatment. The site dimension had five levels (or states)-Indiana, Iowa,Missouri, Michigan, and Ohio; the grade level dimension had five levels-Grades 2, 3, 4, 5, and 6; and the treatment dimension had two levels-No

Calculator and Calculator.One teacher and his or her class were selected from each site at each

grade level for each treatment. Fifty teachers and their classes were in-

volved in the study, 5 from each of the five sites. Five classes at each sitewere in the No Calculator treatment and five in the Calculator treatment.

The No Calculator and the Calculator groups each included a total of 25

teachers and their classes.A site director in each state selected an elementary school from the local

area. At one site it was necessary to select two schools because the K-3

grades and 4-6 grades were housed in different buildings. The schools se-

lected represent a broad spectrum including large urban, suburban, and ru-

ral consolidated attendance area schools. Table 1 summarizes school char-

acteristics by site.Standardized testing in October using the Stanford Achievement Tests,without use of calculators, is reported in Table 2. These data indicatedmathematics achievement levels ranging from nearly two grade levels belownorm in Grade 6 at Site 3 to more than one grade level above norm in

Grade 6 at Site 4.

Treatments

Teachers and their classes were the subjects. The essential differences be-

tween treatments were on thefollowing

dimensions:(a) availability

of cal-

culators to teachers for their students, (b) teacher workshops on the use of

calculators, (c) availability of calculator-based instructional materials, and

(d) the researchers' nteractionswith teachers as consultants. The treatments

began in October and ended in February, for 67 school days of treatment.

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Table 1Site Characteristics:Descriptive Statistics

Site

Characteristic1 2 3a 4 5

Number of students in 780 348 620 (K-3) 745 420

building 550 (4-6)Attendance rate (Percent) 97 96 92 (K-3) 96 95

93 (4-6)

Average class size (2-6) 27.2 23.2 28.7 25.3 25.5

Percent minority 0 6 50 6 2

Percent bussed 100 19 45 50 18

County population (1970) 109,378 72,127 907,872 80,911 833,249School type Rural Small Large Small City Large City

City Urban Suburban Suburban

Income level Low- Middle Low High Middle

Middle

Building constructed 1968 1954,1963 1957 1964 1955,1958,1968

a Site housed in two buildings K-3, 4-6.

Table 2

Site Characteristics:

Grade Equivalents for October Standardized Testing by Site

SiteGrade

level Test 1 2 3 4 5

Gradea2 Concepts 2.1 2.7 2.0 2.8 2.7

Computations 2.1 2.4 2.0 2.7 2.6

Application 1.9 2.6 2.1 2.9 2.9

Grade 3 Concepts 3.0 3.7 2.7 4.0 3.7

Computations 2.6 3.4 2.8 3.7 3.4

Applicatipn 2.8 3.5 2.9 3.9 3.8

Gradeb4 Concepts 4.3 4.8 3.3 4.7 4.8

Computations 4.2 4.5 3.8 4.8 4.3

Application 4.2 5.0 3.7 4.5 4.9

Grade 5 Concepts 4.9 6.1 4.2 6.4 5.3

Computations 5.4 5.7 4.6 6.5 5.7Application 4.7 5.8 4.0 6.0 5.4

Grade 6 Concepts 6.1 6.5 4.5 7.5 6.5Computations 6.0 6.8 5.1 7.4 6.4

Application 5.4 6.7 4.3 7.1 6.3a Grades 2-3 Stanford Mathematics Achievement Primary IIA.b Grades 4-6 Stanford Mathematics Achievement Intermediate IA.

Availability of calculators. Teachers in the No Calculator group were in-structed to ask students not to use calculators for any mathematics work.

Teachers in the Calculator group were given calculators for themselves andall their students and were asked to assign one calculator to each student. A

registration procedure was suggested that included taping students' namesto their calculators. It was suggested that calculators be kept in individuallyassigned cubby holes in the classroom when not in use by students.

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Teacherworkshops.Two teacher workshops were given by the researcherson classroom use of calculators in mathematics instruction. All teachers ex-

pected to have calculators available sometime during the year. (The No

Calculator teachers expected to have calculators available after February.)The first workshop was given by the researchersbefore treatments had be-

gun and teachers in both the No Calculator and Calculator groups partici-pated. The workshop consisted of a pretest of teacher attitude toward

mathematics and toward calculators, the registrationof a calculator to each

teacher including taping their name to the calculator, basic instruction in

the use of the calculator, 14 pages of activities involving counting, comput-

ing, concept of negative integers, applications problems, and a problem-

solving activity involving approximating the square root of a number to

eight significant digits, 7 sample textbook pages together with suggestions

regarding adapting the pages to calculator use, a list of new concepts likelyto come up through calculator use, a brief examination of the internal

mechanism of a calculator (including the chip), and posttests of attitude to-

ward mathematics and toward calculators. The workshop was 2 hours long.

Immediately following the first workshop, teachers in the Calculator groupwere asked to remain and were given calculators for all their students, two

extra calculators, several extra batteries, and registration forms includinginstructions for students to put their names on their calculators. In an effort

to standardize the workshop across sites, the director gave the workshop to

the researchers, slight modifications were made by the researchers and de-tailed agreement was reached on the conduct of the workshop by each site

director. Each teacher was given a 27-page workbook of the materials to use

during the workshop and to take with them for later reference.The second workshop for teachers was given to Calculator teachers about

2 weeks later, lasted an hour, and included distribution of copies of 45

pages of pupil ready pages of calculator-related mathematics activities and

brief discussion of illustrative examples. The No Calculator teachers re-

ceived no second workshop until after the experiment in February.

Calculator materials. In addition to the 66 pages of calculator activities

from the two workshops, one or two copies of available commercial materi-

als were purchased and placed in each school building. The commercial cal-

culator materials were displayed on a table in the mailroom, library, or

teacher work room for ready access by Calculator teachers for use. (A list of

calculator materials and the workshop materials are available from ERIC/SMEAC, 1200 Chambers Road, Columbus, Ohio 43212.)

Consultants.The site directorsand/or their researchassistantswere in the

school for about 4 hours each day. Mathematics classes were visited daily.The site directors and their research assistants served as mathematics re-

source persons for teachers and responded to requests for help and advice

uniformly for both treatment groups. Consultants identified and provided


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resource materials and gave suggestions to any teacher seeking help. Con-

sultants interacted with children but did not lead or guide instruction. Deci-

sions regarding instruction were always made by the teacher. Every effort

was made by the consultants to treat all teachers in the project the same ex-cept for discussions and suggestions concerning calculator use. The methods

and suggestions for calculator use were included in the discussions the con-

sultants had with the teachers who were in the Calculator group and no

suggestions related to calculator uses were made to the teachers who were in

the No Calculator group.In summary, the treatments differed on four dimensions: calculator avail-

ability, workshops, instructional materials available, and consultant dis-

cussions. The effects of these conditions on teacher and student behavior

will be discussed in later sections.

Potential treatment effects. Teachers together with their students were

subjects, and variability in teacher behavior as well as student behavior was

expected. The treatments, as outlined in the previous section, were stan-

dardized. The amount and nature of calculator use among teachers havingcalculators available as a treatment effect was expected to vary consid-

erably, as it might in a school first adopting calculator use for instruction.Treatment effects could therefore range over variables such as student

achievement on standardized tests, availability of calculators in the home,

amount of teacher use of calculators, administrative position on the use ofcalculators for instruction, and goals for instruction.

Among the array of potential treatment effect variables, the followingstudent variables were chosen to be of highest priority for this initial studyof the effects of calculator availability in elementary school mathematics

programs:(a) student attitude towards mathematics, (b) student attitude to-wards calculators, (c) student knowledge of basic facts, (d) student achieve-ment on a standardized mathematics test of Concepts, Computations, and

Applications, (e) student achievement on an advanced standardized test of

computation while using a calculator, (f) student ability to estimate, and (g)student achievement on a special topics test of potentially calculator related

topics.Other variables observed included amount and type of student use of cal-

culators, teacher attitude toward mathematics and calculators, teacher useof calculators during the instruction, instructional strategies used, student

attendance, availability of calculators in the home, availability of calcu-lators to children in classroom, batteries consumed, calculators replaced,number of children owning their own calculators, teaching styles, goals for

instruction, classroom behavior of children, the introduction of new topics,and children and parent reactions to the use of calculators for instruction.While teacher and classroom activities will be critical variables in further

development and research regardingcalculator use for instruction, the focushere was on student achievement and attitude variables deemed most im-

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portant to teachers, parents, administrators,and researchersas initial calcu-

lator use is considered.

Design

The basic design was experimental including two treatment groups withpre- and posttesting. Figure 1 is a summary of the time and sequence of

events. The experiment was conducted from the second week in October

1977 through the second week of February 1978. The pretests were giventhe first week in October 1977 and the posttests were given during the third

week of February.The treatment occurred during an 18-week time period which included

regular school holidays and days missed due to heavy snowfalls and bliz-

zard conditions. The treatments were in effect for an average of 66.6 school

days, ranging from 56 days at Site I up to 74 days at Site 4. This is equiva-lent to 13.32 weeks of school time.

A follow-up period was included in the project to allow for the remedia-

tion of any detrimental effects, should they occur. The follow-up period ex-

tended from the middle of February through the first week in May.

Instruments

The testing included responses from students in five categories. The cate-

gories were attitudes, basic facts, mathematics achievement, estimation and

special topics. Classroom observations were made throughout the treatmentperiod.

Attitudes.The attitude scales were two six-item semantic differentialswith

five response options. One scale was used to measure mathematics attitude

and the other to measure calculator attitude of the children. The six pairs of

terms used were bad--good, sad--happy, boring--exciting, jump in--holdback, hard--easy, and more--less. These terms were selected as a result of

consultation with teachers and children. The instruments were piloted and

children interviewed to determine how valid the scales were for reflectingthe students' attitudes. The same scales were used for all grades (2-6). Theinstructions and the scales were read to Grade 2 and Grade 3 students but

not to Grade 4 through Grade 6 students.

The scales were scored by assigning scores of 5 through 1 to the student

responses. If the student response was in the space adjacent to a term signi-

fying a positive attitude a score of 5 was assigned; if the response was next

to a term signifying a negative attitude a score of 1 was assigned. For ex-

ample:

---score---(1) (2) (3) (4) (5)bad -_ - :good

The scores for the six items were added together resulting in an attitude


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score with a maximum of 30 and minimum of 6. The internal consistencyreliability estimates of these scales ranged from 0.82 to 0.92 for Mathe-matics Attitude (MAT) and from 0.66 to 0.88 for Calculator Attitude

(CAT).

Basicfacts. The four basic facts tests (ADD, SUB, MUL, DIV) were eachmade up of 20 randomly selected items (10 easy, e.g., 4 x 2, and 10 hard,e.g., 8 x 7, basic fact combinations). These were read to students at all

grades (2-6) with a 5-second delay between items for responding. Calcu-lators were not used.

The student responses were scored right or wrong. The scores on eachbasic facts test could range from 0 to 20. The internal consistency reliabilityestimates were from 0.55 to 0.89 for Addition, 0.88 to 0.90 for Subtraction,

0.84 to 0.94 for Multiplication, and 0.75 to 0.95 for Division. The low relia-bility estimates for Addition were for Grade 6. The Addition basic factstest was very easy for Grade 6 and the majority of errors were careless er-rors resulting in low internal consistency estimates.

Mathematics achievement. The Mathematics Tests of the StanfordAchievement Tests (1972-1973) were used to test Concepts, Computations,and Application. A summary of the Stanford Achievement Test schedule isincluded in Figure 1. Grades (2-3) were given primary level tests. The Pri-

mary Level II A tests were used as pretests and the PrimaryLevel III A testsas posttests. Grades (4-6) were given intermediate level tests. The Inter-mediate Level I A tests were used as pretests and the Intermediate Level IIA tests as posttests. The internal consistency reliability estimates were deter-mined for each of the 50 classes on each of the achievement tests and

ranged from 0.72 to 0.93.

Calculator-relatedmathematicsachievement. At posttest time Grades 2-3were given the Intermediate Level I A Computations test and Grades 4-6

were given the Advanced Level A Computations test. The students weregiven calculators to use while taking these tests. The tests which involvedthe use of the calculator are identified with an asterisk in Figure 1.

The Estimation test (EST) was a 12-item multiple choice test designed bythe researchers.The test was administered to Grades (2-6) in a fashion sim-ilar to the basic facts tests. The items were read by the teachers with a 5-

second delay between items for students to respond. Calculators were notused by the students. The student response sheet included the item and the

response choices. The students were asked to circle their responses. For ex-

ample:Teacher says, "two hundred eight divided by ninety-eight is about"

(circle one)

Response Sheet: 208 + 98 2 20 200 2000


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The responses were scored as right or wrong. The scores could rangefrom 0 to 12. The internal consistency reliability estimates for the Estima-

tion test ranged from 0.29 to 0.68. The low reliabilities were partially due to

the extreme difficulty of the items resulting in random responses from the

students.Two levels of Special Topics posttests were designed by the researchers.

The primary level (SPCPA), Grades (2-3), consisted of two parts (I, II).Part I consisted of 12 items and was taken by students without he use of cal-

culators. Part II consisted of 6 items and was taken using the calculators.An example of a Part I item is:

(circle one)

8. Which number is smaller? -6 -3

The teacher was instructed to read the item as follows: "Number 8. Whichnumber is smaller?"The teacher was instructed not to read the options.

An example of a Part II item is:

(circle one)

17. Mr. Jackson has 1253 2274 4780 4,419,331 28,148,443cars to load on his boat.

Each car weighs 3527

pounds. How many pounds

will be added to the boat?

The teacher asked all children to turn on their calculators to see that the

calculators were working just prior to beginning Part II. For all Part II

items the teacher was instructed to read the item but not the options. The

total number of items answered correctly on Parts I and II was used as the

score on the primary level Special Topics test. The maximum possiblescore was 18.

The Special Topics test for the intermediate level (SPCIA), Grades (4-6)consisted of two

parts (I, II).Part I consisted of 12 items and was taken

bystudents without the use of the calculators. Part II consisted of 11 items andwas taken using calculators.

An example of a Part I item is:

5. Write the number for fivethousand seven hundred sixty- 5.three and sixty-five thousandths.

The student was instructed to write the answer in the space provided. Theteacher did not read the

questionsto the students.

An example of a Part II item is:

(circle one)

14. Which number is bigger? 3/7 5/13

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The teacher asked all children to turn on their calculators to see that thecalculators were workingjust prior to beginning Part II of this test. The cor-rect answers for Part I and II were combined to obtain the students' scoreon the Special Topics test (SPCIA). The maximum possible score was 23.

The internal consistency reliability estimates for the Grade (2-3) and Grade

(4-6) Special Topics tests (SPCPA and SPCIA) ranged from 0.59 to 0.90.

Observationaldata. A classroom observation schedule was established forthe 18-week time period during which the treatment occurred. Observa-tional data was collected every 11 school days by the site directors at eachof the five sites. Both the Calculator and No Calculator classes were ob-served.

The purpose of these observations was to document the degree and type

of calculator use as well as other important characteristicsof the classroomactivities and mathematics instruction. Two observation forms were pre-pared to guide the observers. The site directors observed the entire mathe-matics instruction time for each class (10 per site) once every 11 days for atotal of 6 observations throughout the project. Each site director also ob-

served each class at a time other than the mathematics instruction time todocument other uses of the calculator.

In addition to these scheduled observations, each site director made dailyvisits to the schools. Time was spent visiting classes and interacting with

teachers and students informally on a daily basis. The site directors keptlogs for recording observations and interactions throughout the project.

Results

The results include the analysis of pretest and posttest data relevant to

the hypotheses stated earlier. The purpose of the pretest analysis was to

identify the level of achievement and attitudes of students prior to treat-

ments and determine whether or not pretest differences would require co-

variance procedures in subsequent analyses.

Pretests

The pretests were given 3-6 October 1977. Tables 3 and 4 give marginmeans for Grade and Treatment contrasts and summarize the multivariateand univariate analyses of variance for the pretest measures. The data forGrades 2-3 were analyzed separate from the data for Grades 4-6. Since no

interaction effects were significant, the interaction contrast was omitted

from the summary tables. In the primary grades (2-3) the expected gradelevel differences were found on basic facts and mathematics achievement

while no attitude differences were found between grades. The treatmentgroups were assumed to be equivalent prior to the beginning of the experi-ment since no evidence of treatment group differences were found.

The pretest data for Grades (2-3), Table 3, result in the Grade Level dif-

ferences that are generally expected for ADD, SUB, and MUL basic facts.


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The third-grade students do better than the second-grade students. The

same effect exists for the Stanford Achievement Tests. The means for the at-

titude measures by Treatment, and Grade Level (2-3) are given in Table 4.

The attitudes of second- and third-grade children towards calculators (X =

27.8) is significantly higher (p ! .001) than their attitudes toward mathe-matics (X = 22.6).

The pretest data for Grades (4-6), Table 5, reports no multivariate statis-

tical significance (p ! .05) for Grade Level attitude differences, therefore

the univariate test for CAT (p _ .036) is not considered significant. In-

spection of the Calculator Attitude means from Tables 3 and 5 show some

indication of lower attitudes toward calculators in Grade 6 than in Grades

(2-5). The mean attitude scores by Treatment, and Grade Level (4-6) are

given in Table 6. The pretest attitudes of fourth-, fifth-, and sixth-grade

children towards calculators (X = 27.2) was statistically significantly higher(p ! .001) than their attitudes toward mathematics (X = 21.2). The grade

Table3MarginMeansandSignificanceLevels

for GradeLevelandTreatmentContrastsorPretests-Grades2-3

Grade level Test Treatment Test

U M U M2 3 p< p< NC C p< p<

Attitude .724 .808MAT 22.9 22.4 .651 22.5 22.8 .801CAT 27.6 28.0 .509 27.6 28.0 .530

Basic facts .023* .626ADD 14.6 18.6 .001** 16.4 16.8 .726SUB 12.2 16.2 .004** 13.7 14.7 .423MUL 2.6 6.7 .013* 4.7 4.5 .914DIV 1.0 4.7 .039* 2.7 3.1 .780

Mathematicsachievement .004** .706

CNP2A 18.3 25.6 .001** 21.7 22.3 .702CMP2A 18.3 25.3 .001** 21.6 22.1 .739APP2A 14.5 19.6 .001** 16.5 17.6 .402

Note. NC-No Calculator; U-univariate; C--Calculator; M-multivariate.

*p < .05.

**p< .01.

Table 4Cell Means,MarginMeans,and SignificanceLevels

for Mathand CalculatorAttitudePretestsby Treatment or Grades2-3

Attitude

Treatment grade MAT CAT

Second 23.7 27.1NC Third 21.3 28.1

Combined 22.5 27.6

Second 22.2 28.1C Third 23.4 27.9

Combined 22.8 28.0

Overall 22.6 27.8 p .001

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Table 5

Margin Means and Significance Levelsfor Grade Level and Treatment Contrasts for Pretests--Grades 4-6

Grade evel Test Treatment Test

U M U M4 5 6 p< p< NC C p< p<

Attitude .145 .965MAT 21.2 20.9 21.6 .733 21.1 21.3 .788CAT 27.5 27.6 26.5 .036(*) 27.2 27.2 .915

Basic acts .001** .997ADD 19.3 19.6 19.3 .620 19.4 19.4 .944SUB 18.2 18.4 18.4 .974 18.3 18.4 .927MUL 13.3 18.1 18.5 .001** 16.6 16.6 1.000DIV 10.0 14.7 16.5 .001** 13.8 13.6 .908


CNI1A 13.9 18.4 21.6 .001** 18.3 17.6 .632CMI1A 15.4 23.9 28.6 .001** 22.7 22.6 .991APIlA 23.1 27.5 30.9 .009** 27.5 26.8 .724

Note. NC-No Calculator; U-univariate; C--Calculator; M-multivariate.

*p < .05.**p <.01.

Table 6Cell Means, Margin Means, and Significance Levels

for Math and Calculator Attitude Pretests by Treatment for Grades 4-6

Attitude

Treatmentrade MAT CAT

Fourth 20.6 27.6Fifth 21.3 27.3

NC Sixth 21.5 26.7Combined 21.1 27.2

Fourth 21.8 27.3Fifth 20.5 28.0

C Sixth 21.6 26.2Combined 21.3 27.2

Overall 21.2 27.2 p- .001

level differences that are generally expected for MUL and DIV basic factsare found in Table 5. The students at the higher grade levels do better thanthose at the lower grade levels. A similar interpretation can be made con-

cerning the Stanford Achievement Test mathematics scores.

Posttests

The posttests were given 13-17 February 1978. The basic facts (+,-,x,+)and mathematics achievement (Concepts, Computations, and Applications)

tests were analyzed for pre-posttest differences. The same basic facts testwas used for Grades (2-6) and for pre- and posttests. These scores were an-

alyzed directly. The mathematics achievement scores for Concepts, Compu-tations and Applications were obtained from different batteries of the Stan-ford Achievement Tests depending on the Grade Level of the students and


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Testing Time (pre or post). In order to compare scores on the different bat-

teries of the Stanford Achievement Tests, the scores were converted to

scaled scores (Madden, Gardner, Rudman, Karlsen, & Merwin, 1973, p.

13). These scaled scores allow comparisons, within a single subtest area,

from grade to grade, battery to battery, and form to form.A Treatment by Grade by Site by Testing Time (pre-post) four-way mul-

tivariate analysis of variance was computed using the four basic facts scores

and the three mathematics achievement test scores as dependent variables.

The results were as follows:

1. No significant effects for

(a) Treatment by Grade by Testing Time interaction

(b) Treatment by Grade interaction

(c) Treatment by Testing Time interaction(d) Treatment

2. Significant effects for

(a) Grade by Testing Time interaction (Multivariatep ! .001) (Uni-variates-ADD p ! .001, SUBp < .006, MULp ! .001, DIVp _.001)

(b) Grade (Multivariate p _ .001) (Univariates each p _ .001)

(c) Testing Time (Multivariatep! .001) (Univariates each p _ .001)

The Grade by Testing Time interaction was due to the basic facts scores.The lower grade students had relatively low pretest scores on the basic facts

tests and showed expected growth on the posttests. The upper grade levels

had high basic fact pretest scores and did not show the same proportion of

growth since they tended to be nearer the maximum scores on the pretests.The means by Grade Level and Testing Time are plotted in Figure 2.

The Testing Time effects were all significant at p _ .001 for all tests. Thisresult indicates that the students in the study made significant growth onbasic facts and mathematics achievement from October 1977 to

February1978.In addition, the result of no significant difference for the Treatment by

Testing Time effects, gives no evidence that the use of calculators was detri-mental or facilitative for basic facts or mathematics achievement.

The mean scores for the pretest and posttest basic facts by treatment

group are given in Table 7.The scaled Stanford Achievement Test mathematics scores for Concepts,

Computations, and Applications showed significant gains from pre- to post-testing. There was no significant interaction effect for testing time by treat-ment. This result provides no evidence of detrimental or beneficial effects ofcalculator use. The grade equivalents for pre- and posttest Concepts, Com-

putations, and Applications scores are given in Table 8.The posttests were given 13-17 February 1978. Tables 9 and 10 summa-

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20 20

6.-

5"

16 16

5

12 12

4

8 8

3

4 42 MUL DIV

Pre Post Pre Post

20 5 _ 204

--6 6

16 16 :3

12 12 2

8 8

4 4ADD SUB

0o I 0

PrePost Pre Post

Figure 2. Basic facts grade by testing time.

rize the margin means and the multivariate and univariate analyses for

posttest difference for Grades 2-3. No interaction effects were found; there-


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Table 7Basic Facts Means Across Grade Levels by Treatment by Testing Time

No Calculator Calculator

Pretest Posttest Pretest Posttest

ADD 18.2 18.8 18.4 19.0SUB 16.5 17.9 16.9 17.8MUL 11.8 14.6 11.8 14.5DIV 9.4 12.4 9.4 12.4

Table 8Grade Equivalents for Mathematics Achievement

by Treatment by Testing Time

No Calculator Calculator

Pretest Posttest Pretest PosttestConcepts 4.4 5.4 4.4 5.5

Computations 4.3 5.0 4.3 4.9

Applications 4.2 5.2 4.1 5.2

Table 9

Margin Means and Significance Levels forGrade Level and Treatment Contrasts for Posttests-Grades 2-3

Grade evel Test Treatment Test

U M U M2 3 p< p< NC C p< p<

Attitude .595 .569MAT 23.1 22.4 .591 22.3 23.2 .444CAT 27.0 27.2 .635 27.1 27.0 .805

Basic acts .001** .509ADD 17.2 19.0 .009** 17.7 18.5 .210SUB 15.0 18.2 .001** 16.7 16.6 .880MUL 4.4 13.0 .001** 8.6 8.8 .915DIV 2.5 9.7 .001** 5.9 6.4 .770


CNP3A 13.3 19.4 .001** 15.8 16.9 .515CMP3A 13.3 21.2 .001** 17.1 17.4 .840APP3A 13.8 19.1 .004** 16.1 16.8 .671

Calculator-relatedmathematicsachievement .002** .258

EST 3.6 4.7 .001** 4.0 4.2 .508SPCPA 3.6 6.6 .001** 4.5 5.7 .091CMIlA 20.3 27.8 .001** 23.8 24.3 .748

Note. NC-No Calculator; U-univariate; C-Calculator; M-multivariate.*

p < .05.**p < .01.

fore, only the grade level and treatment contrasts are reported in the sum-

mary. There were no treatment effects identified for Grades 2 and 3 as evi-

denced by the significance tests reported in Table 9. None of themultivariate tests for differences between treatment groups were significant.

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Table 10Cell Means, Margin Means, and Significance Levels

for Math and Calculator Attitude Posttests by Treatment for Grades 2-3

Attitude

Treatment rade MAT CATSecond 23.0 26.6

NC Third 21.6 27.6Combined 22.3 27.1

Second 23.2 27.3C Third 23.3 26.8

Combined 23.2 27.0

Overall 22.8 27.1 p : .001

Only one variable out of the 12 measured showed even the slightest in-

dication of a statistical difference and that was SPCPA, the Special Topicstest (p < .091). The scores for the Calculator classes showed some sign of

being greater than those of the No Calculator classes. The difference be-

tween Calculator and No Calculator classes in experience with negativenumbers could have accounted for such a difference.

Across the basic facts measures and all achievement measures Grade 3

scores were significantly higher than Grade 2 scores (p-

.01), which is to be

expected.The grade equivalents associated with the Calculator Aided Computa-

tions (CMIlA) were 5.0 and 6.0 for Grades 2 and 3, respectively. This levelof computation performance, when the calculator was used while taking the

test, was not dependent on the treatment condition. The children learned to

use the calculator for calculations with ease. Assuming that second graderswould be expected to have grade equivalent scores of 2.5 and third gradersscores of 3.5 on the Computations test without calculators in February, the

Calculator Aided Computations scores of 5.0 and 6.0 were found to be sig-nificantly higher (p

-.001) than would be expected for both second- and

third-grade children.

The second- and third-grade children did not do well on the Estimationtest. The mean score was 4.1, while the chance score was 3.0. Similar differ-

ences between attitudes toward mathematics and calculators observed from

the pretest scores were also observed in the posttest scores. A breakdown of

the attitude score means by Grade Level and Treatment are given in Table10.

There were no Treatment effects identified for Grades 4-6 as evidenced

by the significance tests reported in Table 11. None of the multivariate tests

for differences between treatment groups were significant.

Statistically significant Grade Level effects were identified for the mathe-matics achievement measures. These differences reflect an increase in per-formance level from Grade 4 to Grade 6 which is to be expected. The grade

equivalents associated with the Calculator Aided Computations (CMADA)were 5.8, 7.0, and 7.8 for Grades 4, 5, and 6, respectively. This level of com-


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Table 11

Margin Means and Significance Levels forGrade Level and Treatment Contrasts for Posttests-Grades 4-6

Grade level Test Treatment Test

U M U M4 5 6 p< p< NC C p< p<

Attitude .065 .569MAT 21.7 20.6 21.4 .512 21.3 21.2 .916CAT 26.6 25.3 24.8 .020(*) 25.8 25.3 .261

Basic facts .140 .940ADD 19.4 19.6 19.2 .488 19.5 19.3 .619SUB 18.7 18.6 18.7 .971 18.7 18.6 .834MUL 17.7 18.8 18.8 .122 18.6 18.3 .559DIV 15.1 17.4 17.4 .082 16.8 16.4 .685

Mathematicsachievement .023* .223

CNI2A 16.4 20.3 23.6 .008** 20.3 19.9 .825CMI2A 18.4 24.6 29.4 .002** 25.0 23.2 .427API2A 18.2 23.6 27.3 .005** 23.5 22.6 .651

Calculator-relatedmathematicsachievement .006* .840

EST 4.9 5.9 6.3 .053 5.7 5.6 .851SPCIA 5.2 8.1 10.8 .002** 7.9 8.2 .817CMADA 16.5 21.0 24.6 .002** 20.5 20.9 .777

Note. NC-No Calculator; U-univariate; C--Calculator; M-multivariate.*

p < .05.**p < .01.

putational skill when the calculator was used while taking the test was not

dependent on the treatment condition. Performance was virtually the same

for Calculator and No Calculator groups. It was assumed that the fourth

graders would be expected to have grade equivalent scores of 4.5, the fifth

graders scores of 5.5, and the sixth graders scores of 6.5 on the Computa-tions test without use of calculators in February.

The Calculator Aided Computations scores of 5.8, 7.0, and 7.8 for Grades

4, 5, and 6, respectively, were found to be significantly higher than would be

expected for fourth graders (p < .001), fifth graders (p -< .01), and sixthgraders (p < .01).

The Estimation scores show increased skill from fourth grade to sixth

grade. The means range from 2 to 3 points above chance. This is lower than

expected.The multivariate Grade Level effect for attitude approached significance

(p < .065). On further inspection of the univariate tests of significance it can

be seen that the Calculator Attitude scores were primarily responsible for

this effect (CAT, p < .02). Inspection of the means across Grades (4-6) re-

veals a decrease in positive attitude from Grades 4 through 6. The older stu-dents still viewed the calculator as something they liked, but perhaps theylearned that the calculator does not solve all of the problems for them.

Mathematics still requires thinking even if you use a calculator. This effectwas not dependent on treatment.

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The differences between Mathematics Attitudes (MAT) and CalculatorAttitude (CAT) observed from the pretest scores were also observed in the

posttest scores. A breakdown of the attitude score means by Grade Leveland Treatment are given in Table 12. For Grades 4, 5, and 6 the overall

mean of the MAT posttest scores was 21.2, and the overall mean of theCAT scores was 25.5. These means were significantly differentatp _ .001.

The classroom observations were tabulated and the following statementssummarize these data:

1. Games were more prevalent in Calculator classes than in No Calcu-lator classes (p < .05).

2. When calculators were used by the Calculator classes during mathe-

matics instruction, approximately 60%of the students were using them.

3. Calculators were used for instruction by the Calculator classes, on theaverage, about 40% of the class time.

Summaryof Resultsby Hypotheses

Hypothesis L No evidence was found to support the hypothesis that useof calculators influences student attitudes toward mathematics.

Hypothesis II. Evidence was found to support the hypothesis that stu-dents have a more positive attitude toward calculators than toward mathe-

matics. These effects were observed at Grades (2-3) and Grades (4-6) (p.001) independent of calculator use.

Hypothesis III. Evidence was found to support the hypothesis that stu-

dents with and without calculators show gains for basic facts (+, -, x, +:p

- .001) and for mathematics achievement (Concepts, Computations, and

Applications, p-

.001). These gains were independent of calculator use.

HypothesisIV. No evidence of effects of calculator use on student knowl-

edge of basic facts or on student mathematics achievement was found.

Table12CellMeans,MarginMeans,andSignificanceLevels

forMathandCalculatorAttitudePosttestsbyTreatmentor Grades4-6

Attitude

Treatment grade MAT CAT

Fourth 21.7 27.5Fifth 20.6 24.7

NC Sixth 21.6 25.3Combined 21.3 25.3

Fourth 21.7 25.7Fifth 20.7 21.2

C Sixth 21.2 24.3Combined 21.2 25.2

Overall 21.2 25.5 p .001


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Hypothesis V. No evidence of effects of calculator use on student achieve-

ment of estimation skills or special topics of mathematics was found.

Hypothesis VI. Evidence was found to support the hypothesis that the use

of the calculator on computations tests would increase student computa-tions test scores. This effect was observed at Grades [2-4, (p < .001)] and

Grades [5-6, (p-

.01)] independent of Treatment group.

Limitations

It is important to identify limitations and assess potential impacts such

limitations may have on results and future work. Limitations were placed in

three general categories: Breadth, Duration, and Power.

Breadth. (a) The standardized achievement tests do not necessarily reflect

specific classroom activities and valued mathematics outcomes but are testsof general achievement.

Because current evaluation of school mathematics programs by policy

making groups such as administrators,school boards, and parents is almost

always made with standardized tests, standardized testing, while of generalnature, needed to be a major component of the testing. It is not envisioned

that many policy making groups will change policy regardingcalculator use

without standardized test data. The tests of basic facts and attitudes used do

reflect specific goals and are less susceptible to this limitation.

(b) There was a broad range of calculator use across teachers rather than

a systematic controlled calculator use.

All site directors could envision activities which, in their view, would

have significantly multiplied both the time and quality of calculator use in

the classroom. However, we were attempting to define and evaluate the first

level use of calculators which typical elementary schools might consider. It

is our view that the first level use must be examined before more extensive

treatments are considered. The question of massive, general debilitating ef-

fects for calculator use in elementary schools was of highest priority.The purpose was to determine the most prominent general effects of first

level calculator use. The treatments and testing were necessarily limited and

general in nature.

Duration. The actual treatments were approximately 67 school days in

length. For a year long study, 67 days would appear to be shorter than nec-

essary. There were several factors which reduced the available time. Given

the general public view that calculator use in schools would cause serious

debilitation of students' mathematical achievement, it was important to of-

fer ample time to recover from such debilitation and/or share the benefitsof calculator use for the children in both treatment groups. The posttestswere given in February of the school year to allow for careful testing and

time for recovery from potential disadvantages. Even though significant

growth was found for Calculator and No Calculator groups clearly the 67-

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day treatment over 41 months may have been too short to produce potentialdebilitations or potential advantages.

It is clear once initial fear over debilitating effects for calculator use is re-

lieved, longer term studies need to be conducted to determine the effects of

regular calculator use over several years time. We accepted the limitationsof an 18-week treatment in order to protect children from possible debilita-tions and obtain quick careful information concerning the initial effects ofcalculator use.

Power. Any study designed to determine whether or not a device or treat-ment causes feared debilitations must expect to present data describing the

power of the statistical tests used to support the strength of the inferencethat if no differences were detected then there were no differences. Because

power arguments are very infrequently reported, it is important to note thelimitation is not that such an inference cannot be made or supported by

probability statements. We have used the statistical technique of poweranalysis to appropriately support the inference that no significant differ-

ences detected implies no differences existed. The limitation relative to

power is that our planned variability among schools necessarily reduced the

power of the experiment. Significantly greater power would have been pos-sible by reducing cell variance due to school. However, for example, if wehad chosen all schools of a similar type the power would have been in-

creased,but at the

expenseof

limiting generalizabilityto schools of

onlyone type. Our choice was to obtain generalizability to most elementaryschools and consequently sacrifice some statistical power.

A rigorous standard of a = 0.05 and power = 0.95 was adopted for this

study. The data in Tables 13 and 14 give the raw score differences (critical

differences) which would have to exist between the means of the Calculator

group and the No Calculator group if statistical significance at the 0.05 level

would be found. The probability of finding such a difference if in fact it ex-

isted is the power (0.95).For instance in Table 13 a SUB difference of 2.76 items between Calcu-

lator and No Calculator group posttest means would be found to be signifi-cant at the 0.05 level and if such a difference in fact exists it will be detected

95 times out of 100. Since no significant difference between Treatments was

found for SUB in this study then it is unlikely that such a difference (--3items) exists. The nearer the critical difference approaches or exceeds a

meaningful difference in raw scores the less confident we are that no differ-ences exists.

Conclusions

Widespread concern over the use of calculators by young children

prompted a broad, year-long study of the impact of calculators on elemen-

tary school mathematics learning. In the context of the level of calculator

use elementary teachers are likely to implement in the first year with calcu-


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Table 13Critical Treatment Differences Requiredfor 0.95 Power in Grades 2-3 (a = .05)

Possible raw

score range Pre PostMAT 6-30 4.47 4.53

CAT 6-30 2.32 1.97

ADD 0-20 3.75 2.28

SUB 0-20 4.65 2.76

MUL 0-20 5.58 5.58

DIV 0-20 6.31 6.28

CNP2A 0-35 5.88 NG

CNP3A 0-32 NG 5.97

CMP2A 0-37 6.20 NG

CMP3A 0-36 NG 5.97

APP2A 0-28 4.71 NGAPP3A 0-28 NG 5.92

EST 0-12 NG 1.07

SPCPA 0-18 NG 2.63

CMIIA 0-40 NG 5.61

Note. NG-Test not given.

Table 14Critical Treatment Differences Requiredfor 0.95 Power in Grades 4-6 (a = .05)

Possible rawscore range Pre Post

MAT 6-30 3.49 3.51

CAT 6-30 1.68 2.34

ADD 0-20 1.30 1.21

SUB 0-20 2.57 2.17

MUL 0-20 - 2.33

DIV 0-20 5.96 4.33

CNIIA 0-32 6.83 NG

CNI2A 0-35 NG 7.82

CMI1A 0-40 - NG

CMI2A 0-45 NG 10.21API A 0-40 8.57 NG

API2A 0-40 NG 9.50

EST 0-12 NG 1.94

SPCIA 0-23 NG 5.36

CMADA 0-45 NG 7.39

Note. NG-Test not given.

lators available for all children and limited supplementary materials for stu-

dent use, the following conclusions seem warranted:

1. Children's attitude towardcalculators is more

positivethan their atti-

tude toward mathematics.

2. Children grow significantly on basic fact and mathematics achieve-

ment tests taken without the use of calculators regardless of whether or not

calculators were used during instruction.

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3. The use of calculators increases children's computational power withlittle instruction.

4. Accepting power arguments associated with the needed differences re-

ported in Tables 13 and 14, there is no evidence of measurable detrimental

or positive effects for initial first-yearuse of calculators for teaching mathe-matics.

Observed growth in basic facts and mathematics achievement argues forevidence that feared debilitations did not occur. Lack of differences on theEstimation and Special Topics tests is most likely explained by insufficientattention to these topics by teachers in either treatment.

In summary, children enjoyed calculators, increased their computationalpower with little instruction when using calculators, and did not develop

any of the feared debilitations when tested without calculators because ofcalculator use for instruction.

FutureDirections

Recommendations for future directions are divided into those for re-

search, those for school practices, and those of a philosophical nature need-

ing widespread discussion and study.

Research

Almost 100 studies on theeffects of calculators have been conducted andmost conclusions indicate no measurable detrimental effects associated with

the use of calculators for teaching mathematics (Suydam, 1979). This re-search is consistent with these findings. We believe the next step is to exam-ine specific effects for specific calculator activities. Such work should be

coupled with continued monitoring of the longer range effects of classroomuse of calculators. Specific potential advantages for calculator use suggestedby our experiences included:

1. increased number and variety of examples of mathematical concepts and

computations;2. facilitation of the introduction of new topics such as decimals, metric

measure, negative integers, and number theory earlier in a student'smathematical training.

3. improvement in student and teacher attitude towards mathematics.

Careful determination of whether or not these suggestions are, in fact, po-tential benefits of calculator use needs study.

We believe the existing body of work on calculator effectsjustifies further

work in specific areas and longer range study of calculator effects.SchoolPractices

Programs of calculator use similar to those examined here can be consid-ered and implemented without fear of automatic, dramatic debilitation of


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students' mathematical abilities as usually measured. The researchdoes not

guarantee no debilitation due to calculator use but simply gives evidence

that debilitation need not occur. We endorse school use of calculators pro-vided there is appropriate monitoring of effects and significant time is allo-

cated to thoughtful consideration of appropriate activities for calculatoruse.

We believe the most common error made in decisions about calculator

use is the failure to use calculators at times when they could be used effec-

tively.A second common erroris one of double standards. Children are encour-

aged to use calculators but are tested without calculators. Testing programsshould accurately reflect expected student outcomes. Consequently, most

testing should be conducted with calculators available. Calculator use can

support drill and practice activities including basic facts, estimation, andmental arithmetic. Testing of these skills should be done without calculator

use. Other testing should reflect calculator use as expected in the classroom.

In most cases, the school uses of standardized test data can be adapted

quickly to the new "norms" generated by calculator use through mainte-

nance of system norms and cooperative sharing of such data with other rep-resentative schools.

PhilosophicalQuestions

Although we did not attempt to do so, it would seem possible to use cal-culators in such a way that algorithms such as the long division algorithmneed not be taught and would not be learned. The impact of such deletions

from the curriculum is unknown to us but should be considered. Societal

need of such algorithms would be one factor, mathematics curriculum need

of such algorithms another factor, and psychological need in learningmathematics yet another factor. The replacement of these traditional al-

gorithms by others should be considered. Such issues cannot be resolved ef-

fectively by experimental research alone but rather in conjunction with sur-

vey, philosophical, and clinical research. We endorse such efforts as aneeded next step.

REFERENCES

Madden, R., Gardner, E., Rudman, H., Karlsen, B., & Merwin, J. StanfordAchievement Test,Intermediate Level II Battery:Norm Booklet Form A. New York: Harcourt Brace Jovano-

vich, 1973.

Shumway, R. J. Hand calculators: Where do you stand? ArithmeticTeacher, 1976, 23, 569-572.

Stanford Achievement Tests. Mathematics Tests. New York: Harcourt Brace Jovanovich,1972-1973.

Suydam,M. N. The Use

ofCalculators in

Pre-CollegeEducation:A

State-of-the-Art-Review.Columbus, Ohio: Calculator Information Center, May 1979.

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