Natural Sciences and Mathematics || Mathematics in the Elementary School

Mathematics in the Elementary SchoolAuthor(s): J. Fred Weaver and E. Glenadine GibbSource: Review of Educational Research, Vol. 34, No. 3, Natural Sciences and Mathematics(Jun., 1964), pp. 273-285Published by: American Educational Research AssociationStable URL: http://www.jstor.org/stable/1169404 .

Accessed: 28/06/2014 18:53

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Educational Research Association is collaborating with JSTOR to digitize, preserve and extendaccess to Review of Educational Research.

http://www.jstor.org

This content downloaded from 193.0.146.7 on Sat, 28 Jun 2014 18:53:38 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=aera

http://www.jstor.org/stable/1169404?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


CHAPTER II

Mathematics in the Elementary School J. FRED WEAVER and E. GLENADINE GIBB

RESEARCH REPORTED during the calendar years of 1961, 1962, and 1963 reflected some perennial areas of investigation as well as some newer areas of research interest. The selected contributions to be reviewed are presented under these major headings: summaries and bibliographies, children's learnings, instructional media and programs, and teacher education. The concluding section of this chapter is devoted to needed research.

Summaries and Bibliographies

During the past three years, there have been several important bib- liographic contributions concerning research in mathematics education as well as at least one important analytic survey of this research. Brown and others (1963) analyzed the research on the teaching of mathematics at the elementary, secondary, and college levels for 1959 and 1960. Hartung (1961, 1962, 1963) prepared annual selected annotated bibliographies that included research studies. Summers (1961) and Summers and Stochl (1961) listed doctoral dissertations completed during the period from 1918 to 1960. The annual research bibliographies prepared by Weaver (1963a) included references to studies summarized in Dissertation Ab- stracts, most of which have been or will be reported in other sources.

Children's Learnings A major objective of elementary school mathematics instruction has been

to help each child, regardless of his intellectual and mathematical ability, to acquire an understanding of mathematics and a level of skill in using mathematics in accordance with his abilities. Specific attention has been given to (a) concept formation, (b) attainment of specific mathematical concepts, (c) factors that contribute to general arithmetic achievement and positive attitudes toward mathematics, (d) ways of providing for individual differences in learning, (e) effects of classroom organization and environment on the learning situation, and (f) influence of new programs and media in the improvement of existing mathematics curriculums.

Concept Formation

A problem of much concern to those planning and implementing an effective mathematics curriculum is establishing a sequence of learning experiences in appropriate contexts in order that students may acquire

273



REVIEW OF EDUCATIONAL RESEARCH

quantitative understandings and knowledge of specific mathematical tasks and skills. There is still much to be discovered about the way in which the learner forms abstractions and makes generalizations of mathematical ideas as well as about the factors that influence learning and retention.

Gagn6 and others (1962) focused attention on ability to recall and on integration of certain theory-relevant features of a learning program for addition of integers; they attempted to measure the effects of the program by comparing the performance of students of low and high intellectual ability. Gagne and others found that the acquisition of tasks at successively higher stages of learning was dependent upon the mastery of subordinate tasks, that achievement of higher-level tasks indicated amounts of transfer, and that there was no evidence of a difference in performance that could be attributed to categories of low and high levels of ability. In a study of retention of some topics of elementary nonmetric geometry, Gagne and Bassler (1963) found that measures of retention for the group given the smallest variety of task examples were significantly lower than those of the group exposed to other treatments.

Suppes and McKnight (1961) and Suppes and Ginsberg (1962) reported a series of experiments concerned with children's mathematical concept formations. Among their findings were the following:

1. Learning was more efficient if the child who had made an error was required to make a correct response in the presence of the stimulus to be learned.

2. Although much reliance had been placed on incidental learning of mathematics for young children, this type of learning did not appear to be effective.

3. Conditions that focused the child's attention upon the stimuli to be learned enhanced learning.

4. Transfer of concept was more effective if the learning situation required the learner to recognize the presence or absence of a concept in a number of stimulus displays than if it required him to match activities that involved a number of possible responses.

5. Prior learning of one concept did not improve the learning of a related concept.

Wohlwill and Lowe (1962) noted the limitations of an experiment in which too marked restrictions were placed upon the learning sequence. In attempting to answer some questions concerned with children's acquisition of the concept of conservation of number, Wohlwill and Lowe designed their study to force the learning of an empirical rule instead of providing for the intended opportunity to understand a general principle.

Status of Concepts Other studies were concerned with the nature of children's concepts on

specific topics at given times. Piaget's analyses of children's mathematical

274

Volume XXXIV, No. 3



June 1964 MATHEMATICS IN THE ELEMENTARY SCHOOL

concepts have been of particular interest to elementary school mathematics teachers. Coxford (1963) summarized some of Piaget's findings on the attainment of concepts of number and measurement by children at different age levels.

Studies of the number concepts of preschool children were made by McDowell (1962) and Dutton (1963). Mascho (1961) studied children's familiarity with measurement as well as the influence of factors of age, socioeconomic status, and mental ability on degree of familiarity.

Developing Specific Concepts

Over the years, attempts have been made to identify variables that contribute to effective problem solving. During the past three years, there were several noteworthy studies. Pace (1961) investigated the effects of understanding the processes of addition, subtraction, multiplication, and division upon problem solving. She found that, although students made gains either when they were presented with many problems to solve or when they were given systematic instruction in understanding the processes, greater gains were made through use of the latter approach.

Faulk and Landry (1961) observed no significant difference in success in problem solving between (a) children taught by a method in which emphasis was placed on developing a vocabulary, talking and thinking through the total situation, making simple diagrams, and estimating and (b) children taught by a method in which emphasis was placed on labeling, checking, making original problems, and selecting key words.

Within the limitations of a study in which a small sample was employed and in which reading ability was not controlled, Schell and Burns (1962) found that the take-away situations in contrast to comparison situations represented problems that more obviously were to be solved by subtraction. As demonstrated by drawings made by children to interpret problem situations, children did not understand the difference in the three situations-take-away, how many more, comparison-from the point of view of visual manipulation.

In his study of formal analysis as a way of diagnosing difficulty in solving problems, Chase (1961) provided evidence to support the conclusion that formal analysis is not successful in identifying differences between good and poor sixth grade problem solvers.

Among three groups of children of high, average, and low intellectual ability, Klausmeier and Loughlin (1961) sought to identify differences in efficiency of method, in persistence, and in mode of attack while the children were in the process of solving problems. Evidence from this study supported the authors' conclusion that noting and correcting mis- takes independently, verifying solutions, and using a logical approach are associated with higher rather than lower intellectual ability. Lacking persistence, giving an incorrect solution, and using random processes were

275




associated with lower rather than higher intellectual ability. Differences among individuals as well as among intellectual groups were great.

Interested in children's difficulties with fractions, Scott (1962) found that many regrouping errors in subtracting fractions seemed to be associated with the tendency of children to rely on concepts that are related directly to the decimal scale of the numeration system. In a study incor- porating traditional teaching, Capps (1962b) discovered that either the common-denominator or the inversion method of dividing one fraction by another could be taught with a similar degree of success. However, the first method does not provide extra practice in multiplication as does the second.

Achievement and Attitudes

Several investigators attempted to identify factors that contribute to children's success or failure in mathematics and to their positive or nega- tive attitudes toward mathematics. In general, some of these factors are intellectual ability, sex, socioeconomic background, emotional disturbance, length of class, motivation, method of instruction, and context in which materials are presented.

Muscio (1962) reported that achievement on measures intended to represent quantitative understanding of both the numeration system and the system of measurement was closely related to achievement on measures of arithmetic computation, arithmetic reasoning, mathematics vocabulary, general reading ability, and intellectual level as well as to sex (in favor of boys).

Lyda and Morse (1963) noted not only changes in attitudes toward arithmetic when meaningful methods of teaching were used but also significant gains in arithmetic computation and reasoning associated with methods and attitudes. Jarvis (1963) found that longer (60-minute) class periods in arithmetic were substantially related to arithmetic achievement whether pupils were average, dull, or bright.

Rose and Rose (1961) attempted to assess the relation of intelligence, sibling position, and sociocultural background to arithmetic performance. Although one must regard with caution the treatment of data, the analysis indicated the following: (a) a significant relationship between level of intelligence and success or failure in arithmetic, (b) a greater significance between intelligence and arithmetic performance when children were involved in a homogeneous classroom situation than when they were in a learning situation that was socioculturally heterogeneous, and (c) a lack of a significant relationship between sibling position and success in arithmetic.

Shine (1961) reported a definite positive relationship between the performance of kindergarten children who completed items on the Revised

276

REVIEW OF EDUCATIONAL RESEARCH Volume XXXIV, No. 3




Stanford-Binet Intelligence Scale and their achievement in arithmetic. In a study designed to examine characteristics of children who were accel- erated or retarded in arithmetic, Capps (1962a) found that retardation tended to be related to personal adjustment and that a considerable portion of the cause seemed to be in the area of instruction. Callahan (1962) indicated that bright underachievers could benefit from a good diagnostic and remedial program in mathematics. In the determination of the relationship between intelligence scores and measures of gain in arithmetic reasoning, social studies, and science achievement as well as the relationship between gains in reading and gains in these same three measures, Scott (1963) noted that the highest correlation was between arithmetic reasoning and intelligence.

In their attempt to investigate the relative effectiveness of selected manipulative materials for teaching, Harshman, Wells, and Payne (1962) indicated the difficulties of controlling the many variables in such studies. Nevertheless, limited evidence was provided to support the hypothesis that there may not be significantly greater achievement on the part of children involved in programs that have too great an expansion in content and rapidity in instruction than on the part of those exposed to programs that are somewhat barren in content. It may be concluded that there is danger in overloading a program as well as in making it too meager.

Klausmeier and Check (1962) found that, regardless of ability level, children who received learning tasks graded appropriately to their levels of achievement retained and transferred these learnings equally well to new situations of appropriate difficulty. Furthermore, there was no difference in measures of retention in arithmetic when the learning task was graded to each child's achievement level.

Finley (1962) observed that achievement of both normal and educable retarded children may be affected by the context in which materials are presented. For mentally educable children, concrete test items were more difficult than were pictorial or symbolic tasks. For normal children, pictorial items were significantly easier than were concrete or symbolic materials. Since no procedure was used to eliminate order effects, one may question whether the objects themselves were more difficult than were other stimuli or whether the factor of adjustment to the test situation influenced performance.

Bassham (1962) found that the level of the teacher's understanding of basic concepts in mathematics was significantly associated with a pupil's efficiency in learning and that this efficiency was correlated positively with level of pupil intelligence.

Weaver (1962a) not only directed attention to the effect of the particular standardized test used upon the measured level of a pupil's achievement but also emphasized intrapupil variability in achievement level for different aspects of arithmetic ability. Subsequently, Weaver (1962b) published readers' reactions to his paper about standardized tests.

277




REVIEW OF EDUCATIONAL RESEARCH Volume XXX1V, No. 3

Individual Differences and Organization for Instruction A persistent problem in implementing any mathematics program for all

children has been finding effective ways of providing for individual differences. Most recommendations have been for some type of grouping.

Primarily interested in the relationship of grouping to change of attitude toward arithmetic, Lerch (1961) found no significant difference in achievement between intraclass grouping and nongrouping procedures. Moreover, results of a study by Davis and Tracy (1963), who employed the Joplin plan and random grouping, did not lend support to the feasibility of ability grouping.

Hart (1962) compared the arithmetic achievement of 50 beginning fourth grade pupils who had spent three years in a nongraded primary school with that of 50 matched pairs who had spent three years in a graded primary program. The difference, which favored the nongraded sample, was significant at the .02 level.

Guggenheim (1961) used matched pairs of third grade pupils in order to study arithmetic achievement in two different classroom climates: (a) a "dominative" climate, in which most of the behaviors and interactions observed within the classroom were teacher centered, and (b) an "integra- tive" climate, in which there was reciprocity in pupil-teacher interaction. An analysis of variance yielded no significant differences in arithmetic achievement for each of three levels: (a) high achievers from the two classroom climates, (b) average achievers from the two classroom climates, and (c) low achievers from the two classroom climates.

Will children learn more effectively in a situation in which special teachers are used for instruction in selected content areas than in the familiar self-contained (one-teacher) classroom situation? Gibb and Matala (1961, 1962) investigated aspects of this question and of related questions in connection with the use of both special teachers of science and special teachers of mathematics in grades 5 and 6. Findings were based on data from 48 classes (three sets of four classes in each of four school systems); some classes participated in the study for one year, others for two years. Premeasures and postmeasures of concepts, abilities, and skills in mathematics, science, and social studies were made. An interest inventory also was used. There was no evidence that children learned mathematics more effectively in one situation than in another; however, there was some reason to believe that the two instructional plans were not equally effective for pupils of different levels of intellectual ability. The findings for science differed somewhat from those for mathematics. The investigators concluded that, in general, good teachers are effective regardless of the nature of classroom organization.

Instructional Media and Programs During the past few years, there has been an intense and extensive

interest in "modern," or "contemporary," mathematics. Numerous attempts 278





have been made to include such content in mathematics programs from kindergarten through the sixth grade. Interest also has been centered on certain media and materials of instruction as applied to the development of modern, or contemporary, content. However, no definitive research in this area has been reported.

Contemporary Content Emphases

Studies by Suppes and Ginsberg (1962) and by Suppes and McKnight (1961), mentioned earlier in this chapter, were concerned with the development of ideas associated with sets and numbers among children in the primary grades.

In an investigation at the Minnesota National Laboratory (1963), the achievement for 10 classes using the fourth grade School Mathematics Study Group (SMSG) text was compared with the achievement for 10 classes using conventional fourth grade texts. Relative to scores on the mathematics test of the Sequential Tests of Educational Progress, no statistically significant difference in mathematics achievement was observed between experimental and control groups. In a similar type of study, Weaver (1963b) reported that fourth and fifth grade pupils who used SMSG texts during a particular school year made mean gains in both reasoning (problem solving) and computation that were equal to, or higher than, the normally expected gains in terms of grade equivalents on a standardized arithmetic test.

Peck (1963) reported no statistically significant difference in arithmetic reasoning and in arithmetic computation (as measured by a standardized achievement test) between a sample of sixth grade pupils who had studied selected contemporary topics and a matched sample of pupils whose instructional program had not included these topics.

Ruddell (1962) observed that those seventh grade children in an accel- erated mathematics program who studied modern content earned scores on a standardized achievement test as high as or higher than those of a comparable group who studied traditional content.

These studies lend support to the hypothesis that pupils who study modern or contemporary mathematics do not sacrifice gain in either computational skill or problem-solving ability.

Programed Instructional Materials

Banghart and others (1963) compared achievement of children who were using a programed fourth grade text (including some contemporary content) with that of pupils who were using a conventional text. From an analysis of data derived from a standardized arithmetic test, it was observed that the difference in achievement was significantly in favor of

279





the programed-text group for comprehension, but that no significant difference between groups existed for problem solving. Kalin (1962) observed the difference in achievement of two groups of intellectually superior fifth and sixth grade pupils on a two-week unit on equations and inequalities. One group was given work with programed text materials; the other group was taught with conventional methods and text materials. The difference in achievement between the two groups was not statistically significant, although the programed-text pupils did complete the unit in 20-percent less time, on the average, than did the conventionally taught group.

Special Manipulative Materials

Several studies were reported on the use of Cuisenaire rods and similar manipulative materials. Lucow (1963) examined the difference in achievement of two groups of Manitoba children in the third grade. One group of classes adhered to the Cuisenaire program of instruction; the other group of classes followed the regular Manitoba curriculum. When growth in achievement in multiplication and division over a six-week period (with the effects of prior knowledge held constant) was employed as the criterion, the Cuisenaire method was judged effective; however, there was some doubt concerning its general superiority over traditional methods of instruction. Passy (1963) compared the achievement of third grade children who followed an eclectic program involving use of Cuisenaire materials with the achievement of third grade children in two other programs, neither of which involved experience with such materials. It was found that the sample using Cuisenaire materials did receive significantly lower scores on a standardized arithmetic test than did either of the other samples.

Brownell (1963) conducted an investigation in English and Scottish schools that were employing Cuisenaire, Dienes, and traditional programs of arithmetic instruction. Data relating to mathematical understanding, computational proficiency, and problem-solving ability were collected by an interviewing-testing technique that was applied to approximately 1,500 children (with 40-70 minutes devoted to each child). For the reported subsamples of 86 and 89 children from 12 schools in a single English city, Brownell observed what he judged to be greater similarity than dis- similarity between the Cuisenaire- and traditional-program children in their progress toward maturity of mathematical understanding, computational proficiency, and problem-solving ability. He felt that his observations and data did not support the claims that had been set forth concerning the effectiveness of programs in which Cuisenaire rods had been employed. Two noteworthy aspects of Brownell's investigation should be cited: (a) the nature of the research procedure and (b) the effect of this procedure in casting an entirely new light upon both instructional objec- tives and classroom practices for those who participated in the investigation.

280




MATHEMATICS IN THE ELEMENTARY SCHOOL

Teacher Education

Current emphasis upon modern, or contemporary, content has brought into sharp focus certain needs in both preservice and in-service education. Too frequently, teachers have not studied the kind of mathematical content that they are being called upon to teach. Consequently, there is growing concern for including appropriate work in content mathematics in the preservice program for elementary school teachers. There is a similar growing concern for including appropriate work in content mathematics in programs of in-service education for elementary teachers.

Various facets of both preservice and in-service education were investigated during the period covered by this chapter. Sparks (1961) summarized and discussed much of the significant research that appeared in the professional literature prior to 1961.

Preservice Education

Making use of six groups of students of elementary education, Gross- nickle (1962) compared scores on a mathematics examination that had been taken upon entrance to college and again four years later. Intervening work in content mathematics and in methods of teaching arithmetic had taken place during the college years. He found statistically significant gains in test scores for each group.

In his study of arithmetic comprehension and of the attitudes toward arithmetic of 55 undergraduate students of elementary education, Dutton (1961) observed statistically significant gains in comprehension during the semester in which these students were enrolled in a mathematics content course. In another investigation, Dutton (1962) found that the attitudes toward arithmetic of 176 undergraduate students were virtually identical with the attitudes of a comparable group that had been studied eight years previously.

Scrivner and Urbanek (1963) noted what appeared to be greater improvement in mathematical understanding and skill among undergraduates who served as teachers' aides than among a matched group that did not serve in such a capacity, although statistical significance was associated with only the .08 level. Groff (1963) reported that among 645 student teachers from six different California colleges, arithmetic ranked second only to reading as the subject which they felt best prepared to teach. Shryock (1963) found a substantial lack of agreement on topics included in eight mathematics textbooks intended for use in undergraduate courses for prospective elementary teachers.

Nelson and Worth (1961) compared the mathematical competence of 468 Canadian (University of Alberta) prospective elementary teachers with that of two American samples studied in earlier investigations. Hypotheses were advanced to account for the significantly better per-

281

June 1964




formance of the Canadian sample. The discussion of these hypotheses by the investigators contributed to an appropriate interpretation of their findings.

In-Service Education

Roehr (1963) reported observations on locally devised in-service programs involving approximately 185 teachers who were using prelimi- nary editions of the SMSG texts for grades 4-6.

Houston, Boyd, and DeVault (1961) studied the effectiveness of different media of presentation (closed-circuit television, lectures, question- discussion sessions, and written materials) in connection with a program of in-service education for 252 participating teachers from a city school system. Findings from the study indicated the need to individualize in-service mathematics education programs for elementary teachers.

It is commonly accepted that a positive relationship exists between a teacher's mathematical understanding and that of the children he in- structs. Two studies pertinent to this hypothesis were reported. One of these, by Bassham (1962), is mentioned in a preceding section of this chapter. In the other, Houston and DeVault (1963) observed that, during a program of in-service education, both the participating teachers and their pupils made significant gains in mathematical achievement.

Investigations such as these, however, leave unanswered the question concerning cause and effect. Existing evidence is consistent with the hypothesis that teacher change begets pupil change of like kind in mathematics. Nevertheless, one must look to the future for research designed specifically to test this hypothesis.

Needed Research

As one looks to future research for guidelines to the improvement of mathematics instruction in the elementary school, one kind of broad investigation appears to be more needed than any other. Much more must be known about the genetic development of mathematical ideas and abilities among children of different psychological characteristics under different instructional conditions and in different mathematical environments. Such key parameters as personality aspects of children, methods and materials of instruction, school organization, motivating forces, and sequence and level of mathematical content are so intertwined that it is not particularly fruitful to study any one of these characteristics when others are equated or held constant. In fact, results from univariate rather than multivariate designs may be misleading.

On occasion, seemingly contradictory research findings indeed may not be in conflict when relevant factors are considered. For example, contra-

282

Volume XXXIV, No. 3 REVIEW OF EDUCATIONAL RESEARCH




dictory findings regarding the influence of prior learning upon subsequent learning may continue to arise; however, if factors such as mathematical content and learners' ages are taken into account, it may be that there is no lack of congruence in the findings reported.

Imaginative research designs must be developed that will enable one to investigate the interaction of factors that both facilitate and interfere with mathematical learning among children of elementary school age. Such studies cannot be conducted most effectively by individual researchers working more or less independently. A form of cooperative research on a large scale over an extended period of time is necessary. Undoubtedly, this is the intention of the five-year National Longitudinal Study of Mathematical Abilities, initiated during 1961 and 1962 (Cahen, 1963). In addition, the bases for numerous hypotheses for future investigations were suggested in the report of the Cambridge Conference on School Mathematics (Educational Services Incorporated, 1963).

There is much to be learned about effective instruction in mathematics for children from kindergarten through the sixth grade. How rapidly and how well basic knowledge is attained will depend greatly upon the extent and quality of research efforts in the years ahead.

Bibliography

BANGHART, FRANK W., and OTHERS. "An Experimental Study of Programmed Versus Traditional Elementary School Mathematics." Arithmetic Teacher 10: 199-204; April 1963.

BASSHAM, HARRELL. "Teacher Understanding and Pupil Efficiency in Mathematics- A Study of Relationship." Arithmetic Teacher 9: 383-87; November 1962.

BROWN, KENNETH E., and OTHERS. Analysis of Research in the Teaching of Mathe- matics: 1959 and 1960. U.S. Department of Health, Education, and Welfare, Office of Education, Bulletin 1963, No. 12. Washington, D.C.: Government Printing Office, 1963. 69 pp.

BROWNELL, WILLIAM A. "Arithmetical Abstractions-Progress Toward Maturity of Concepts Under Differing Programs of Instruction." Arithmetic Teacher 10: 322-29; October 1963.

CAHEN, LEONARD S. "The National Longitudinal Study of Mathematical Abilities." Science Education News. Miscellaneous Publication No. 63-4. Washington, D.C.: American Association for the Advancement of Science, 1963. pp. 5-6.

CALLAHAN, LEROY. "Remedial Work with Underachieving Children." Arithmetic Teacher 9: 138-40; March 1962.

CAPPS, LELON R. "A Comparison of Superior Achievers and Underachievers in Arithmetic." Elementary School Journal 63: 141-45; December 1962. (a)

CAPPS, LELON R. "Division of Fractions: A Study of the Common-Denominator Method and the Effect on Skill in Multiplication of Fractions." Arithmetic Teacher 9: 10-16; January 1962. (b)

CHASE, CLINTON I. "Formal Analysis as a Diagnostic Technique in Arithmetic." Elementary School Journal 61: 282-86; February 1961.

COXFORD, ARTHUR F., JR. "Piaget: Number and Measurement." Arithmetic Teacher 10: 419-27; November 1963.

DAVIS, O. L., JR., and TRACY, NEAL H. "Arithmetic Achievement and Instructional Grouping." Arithmetic Teacher 10: 12-17; January 1963.

DUTTON, WILBUR H. "University Students' Comprehension of Arithmetical Concepts." Arithmetic Teacher 8: 60-64; February 1961.

283





DUTTON, WILBUR H. "Attitude Change of Prospective Elementary School Teachers Toward Arithmetic." Arithmetic Teacher 9: 418-24; December 1962.

DUTTON, WILBUR H. "Growth in Number Readiness in Kindergarten Children." Arithmetic Teacher 10: 251-55; May 1963.

EDUCATIONAL SERVICES INCORPORATED. Goals for School Mathematics. Report of the Cambridge Conference on School Mathematics. Boston: Houghton Mifflin Co., 1963. 102 pp.

FAULK, CHARLES J., and LANDRY, THOMAS R. "An Approach to Problem-Solving." Arithmetic Teacher 8: 157-60; April 1961.

FINLEY, CARMEN J. "Arithmetic Achievement in Mentally Retarded Children: The Effects of Presenting the Problem in Different Contexts." American Journal of Mental Deficiency 67: 281-86; September 1962.

GAGNE, ROBERT M., and BASSLER, OTTO C. "Study of Retention of Some Topics of Elementary Nonmetric Geometry." Journal of Educational Psychology 54: 123-31; June 1963.

GAGNE, ROBERT M., and OTHERS. "Factors in Acquiring Knowledge of a Mathematical Task." Psychological Monographs: Vol. 76, No. 7 (Whole No. 526), 1962. 21 pp.

GIBB, E. GLENADINE, and MATALA, DOROTHY M. "A Study of the Use of Special Teachers in Science and Mathematics in Grades 5 and 6." School Science and Mathematics 61: 569-72; November 1961.

GIBB, E. GLENADINE, and MATALA, DOROTHY C. "Study on the Use of Special Teachers of Science and Mathematics in Grades 5 and 6: Final Report." School Science and Mathematics 62: 565-85; November 1962.

GROFF, PATRICK J. "Self-Estimates of Ability To Teach Arithmetic." Arithmetic Teacher 10: 479-80; December 1963.

GROSSNICKLE, FOSTER E. "Growth in Mathematical Ability Among Prospective Teach- ers of Arithmetic." Arithmetic Teacher 9: 278-79; May 1962.

GUGGENHEIM, FRED. "Classroom Climate and the Learning of Mathematics." Arith- metic Teacher 8: 363-67; November 1961.

HARSHMAN, HARDWICK W.; WELLS, DAVID W.; and PAYNE, JOSEPH N. "Manipulative Materials and Arithmetic Achievement in Grade 1." Arithmetic Teacher 9: 188-92; April 1962.

HART, RICHARD H. "The Nongraded Primary School and Arithmetic." Arithmetic Teacher 9: 130-33; March 1962.

HARTUNG, MAURICE L., compiler. "Selected References on Elementary School Instruc- tion: Arithmetic." Elementary School Journal 62: 165-67; December 1961. 63: 170- 72; December 1962. 64: 169-71; December 1963.

HOUSTON, W. ROBERT, and DEVAULT, M. VERE. "Mathematics In-Service Education: Teacher Growth Increases Pupil Growth." Arithmetic Teacher 10: 243-47; May 1963.

HOUSTON, W. ROBERT; BOYD, CLAUDE C.; and DEVAULT, M. VERE. "An In-Service Mathematics Education Program for Intermediate Grade Teachers." Arithmetic Teacher 8: 65-68; February 1961.

JARVIS, OSCAR T. "Time Allotment Relationships to Pupil Achievement in Arithmetic." Arithmetic Teacher 10: 248-50; May 1963.

KALIN, ROBERT. "The Use of Programed Instruction in Teaching an Advanced Mathematical Topic." Arithmetic Teacher 9: 160-62; March 1962.

KLAUSMEIER, HERBERT J., and CHECK, JOHN. "Retention and Transfer in Children of Low, Average, and High Intelligence." Journal of Educational Research 55: 319-22; April 1962.

KLAUSMEIER, HERBERT J., and LOUGHLIN, L. J. "Behaviors During Problem Solving Among Children of Low, Average, and High Intelligence." Journal of Educational Psychology 52: 148-52; June 1961.

LERCH, HAROLD H. "Arithmetic Instruction Changes Pupils' Attitudes Toward Arith- metic." Arithmetic Teacher 8: 117-19; March 1961.

Lucow, WILLIAM H. "Testing the Cuisenaire Method." Arithmetic Teacher 10: 435- 38; November 1963.

LYDA, WESLEY J., and MORSE, EVELYN CLAYTON. "Attitudes, Teaching Methods, and Arithmetic Achievement." Arithmetic Teacher 10: 136-38; March 1963.

MCDOWELL, LOUISE K. "Number Concepts and Preschool Children." Arithmetic Teacher 9: 433-35; December 1962.

284





MASCHO, GEORGE. "Familiarity with Measurement: A Study of Beginning First-Grade Children's Concepts." Arithmetic Teacher 8: 164-67; April 1961.

MINNESOTA NATIONAL LABORATORY. "Evaluation of SMSG Text-Grade 4." Reports. Newsletter No. 15. Stanford, Calif.: School Mathematics Study Group, 1963. pp. 8-10.

MuscIO, ROBERT D. "Factors Related to Quantitative Understanding in the Sixth Grade." Arithmetic Teacher 9: 258-62; May 1962.

NELSON, L. DOYAL, and WORTH, WALTER H. "Mathematical Competence of Prospec- tive Elementary Teachers in Canada and in the United States." Arithmetic Teacher 8: 147-51; April 1961.

PACE, ANGELA. "Understanding and the Ability To Solve Problems." Arithmetic Teacher 8: 226-33; May 1961.

PASSY, ROBERT A. "The Effect of Cuisenaire Materials on Reasoning and Computa- tion." Arithmetic Teacher 10: 439-40; November 1963.

PECK, HUGH I. "An Evaluation of Topics in Modern Mathematics." Arithmetic Teacher 10: 277-79; May 1963.

ROEHR, GEORGE L. "A Report on In-Service Education." Reports. Newsletter No. 15. Stanford, Calif.: School Mathematics Study Group, 1963. pp. 25-36.

ROSE, ALVIN W., and ROSE, HELEN CURETON. "Intelligence, Sibling Position, and Sociocultural Background as Factors in Arithmetic Performance." Arithmetic Teacher 8: 50-56; February 1961.

RUDDELL, ARDEN K. "The Results of a Modern Mathematics Program." Arithmetic Teacher 9: 330-35; October 1962.

SCHELL, LEO M., and BURNS, PAUL C. "Pupil Performance with Three Types of Subtraction Situations." School Science and Mathematics 62: 208-14; March 1962.

SCOTT, CARRIE M. "The Relationship Between Intelligence Quotients and Gain in Reading Achievement with Arithmetic Reasoning, Social Studies and Science." Journal of Educational Research 56: 322-26; February 1963.

SCOTT, LLOYD. "Children's Concept of Scale and the Subtraction of Fractions." Arith- metic Teacher 9: 115-18; March 1962.

SCRIVNER, A. W., and URBANEK, R. "The Value of 'Teacher-Aide' Participation in the Elementary School." Arithmetic Teacher 10: 84-87; February 1963.

SHINE, AILEEN. "Relationship Between Arithmetic Achievement and Item Perform- ance on the Revised Stanford-Binet Scale." Arithmetic Teacher 8: 57-59; Febru- ary 1961.

SHRYOCK, JERRY. "A Mathematics Course for Prospective Elementary School Teach- ers." Arithmetic Teacher 10: 208-11; April 1963.

SPARKS, JACK N. "Arithmetic Understandings Needed by Elementary-School Teach- ers." Arithmetic Teacher 8: 395-403; December 1961.

SUMMERS, EDWARD G. "A Bibliography of Doctoral Dissertations Completed in Ele- mentary and Secondary Mathematics from 1918 to 1952." School Science and Mathematics 61: 323-35; May 1961.

SUMMERS, EDWARD G., and STOCHL, JAMES E. "A Bibliography of Doctoral Disserta- tions Completed in Elementary and Secondary Mathematics from 1950 to 1960." School Science and Mathematics 61: 431-39; June 1961.

SUPPES, PATRICK, and GINSBERG, ROSE. "Experimental Studies of Mathematical Con- cept Formation in Young Children." Science Education 46: 230-40; April 1962.

SUPPES, PATRICK, and MCKNIGHT, BLAIR A. "Sets and Numbers in Grade One, 1959-60." Arithmetic Teacher 8: 287-90; October 1961.

WEAVER, J. FRED. "Disparity in Scores from Standardized Arithmetic Tests." Arith- metic Teacher 9: 96-97; February 1962. (a)

WEAVER, J. FRED. "Readers' Reactions to 'Disparity in Scores from Standardized Arith- metic Tests.'" Arithmetic Teacher 9: 342-43; October 1962. (b)

WEAVER, J. FRED. "Research on Elementary-School Mathematics-1960, 1961, 1962." Arithmetic Teacher 8: 255-60, May; 301-306, October 1961. 9: 287-90; May 1962. 10: 297-300; May 1963. (a)

WEAVER, J. FRED. "Student Achievement in SMSG Classes, Grades 4 and 5." Reports. Newsletter No. 15. Stanford, Calif.: School Mathematics Study Group, 1963. pp. 3-8. (b)

WOHLWILL, JOACHIM F., and LOWE, ROLAND C. "Experimental Analysis of the Development of the Conservation of Number." Child Development 33: 153-67; March 1962.

285




Documents

Natural Sciences and Mathematics || Mathematics in the Elementary School