The status of research on elementary school mathematics

The status of research on elementary school mathematicsAuthor(s): MARILYN N. SUYDAMSource: The Arithmetic Teacher, Vol. 14, No. 8 (DECEMBER 1967), pp. 684-689Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185697 .

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The status of research on elementary school mathematics MARILYN N. SUYDAM The Pennsylvania State University, University Park, Pennsylvania

Marilyn N. Suydam is a research associate at the Center for Cooperative Research with Schools and works with Alan Riedesel, who is our editor for "Focus on Research." Her article grew out of her doctoral dissertation, which she recently completed at The Pennsylvania State University.

JJuring the past few years, an increasing amount of emphasis has been given to research. The need for direction in the teaching-learning process in all areas of the curriculum has been intensified, and the development of curriculum reform move- ments such as that of "modern mathematics" has accentuated that need. Research should be a source of knowledge about curriculum content and methods and a foun- dation for decisions about curriculum in- novations.

Many questions have been asked con- cerning research on elementary school mathematics. They have involved such points as these:

How much has been published in journals?

What types of research have been published?

What topics have been included in these published reports?

How useful is this research to classroom needs?

A study recently completed1 was de- signed to provide some of the answers to such questions. All research reports published in American journals from 1900

1 Marilyn N. Suydam, "An Evaluation of Journal- Published Research Reports on Elementary School Mathematics, 1900-1965" (doctoral dissertation, The Pennsylvania State University, 1967). This study was supported by the U.S. Office of Education, Department of Health, Education, and Welfare, under Grant No. OEG-1-7-068592-0174.

through 1965 were compiled. These were categorized and analyzed on almost a dozen different points:

1. mathematical topic 2. type of study 3. design paradigm 4. sampling procedure 5. sample size 6. statistical procedure 7. grade level 8. duration 9. type of test

10. variables involved 11. major findings In addition, reports of experimental re-

search were evaluated with an instrument developed specifically for this purpose. As a result of the compilation, categorization, and evaluation, answers for the questions at the beginning of this article can be at- tempted.

How much has been published in journals? A total of 799 research reports2 were

found in fifty journals. Of these fifty journals, three published over half (54 per-

2 It was anticipated that more than this number would be found. Previous listings indicate this. How- ever, upon analysis it becomes evident that the criteria for inclusion of articles varied. Frequently articles of interest about research were listed as well as actual research reports. The present compilation includes only the latter.

684 The Arithmetic Teacher



cent) of the reports; ten published 84 percent; thirteen published 89 percent. These journals are:

Arithmetic Teacher 158 California Journal of Educational

Research 14 Child Development 23 Educational Method 16 Educational Research Bulletin 27 Elementary School Journal 132 Journal of Educational Psychology 57 Journal of Educational Research 138 Journal of Experimental Education 30 Journal of Genetic Psychology 37 Mathematics Teacher 36 Peabody Journal of Education 1 1 School Science and Mathematics 35

The remaining reports (11 percent) were published in thirty-seven journals. It is of interest to note that The Arithmetic

Teacher, in publication only since 1954, published one fifth of the total number of reports.

The distribution of these reports by years is:

1900-1910 2 1911-1920 36 1921-1930 89 1931-1940 167 1941-1950 118 1951-1960 165 1961-1965 222

The figure for the last five-year period obviously is greater than for any prior ten-year period, underlining the emphasis being placed on research today.

What types of research have been published and what topics have been included in these published reports?

On Table 1 are two types of information: the frequency by primary mathemat-

Table 1

Frequency of reports by mathematical topic and type of study

Total in- De- Corre- Ex eluding

scrip- Case lation- post Experi- cross-ref- tive Survey Study Action al facto mental Total er enees

A. Educational objectives and instructional procedures 1. Historical development

and procedures 2 2 2 2. Values of arithmetic 2 8 10 11 3. Planning and organizing

. for teaching 6 4 5 91438 62 4. Attitude and climate 13 15 2 3 24 30 5. Specific procedures

a) Drill and practice 2 3 11 24 31 43 b) Problem solving 6 13 2 10 4 22 57 84 c) Estimation 1 12 3 d) Mental computation 3 1 5 9 13 e) Homework 1 1 4 6 6 /) Review 1 1 1 Z) Checking 1 1 3 h) Writing and reading

numerals 2 13 3 6. Foreign comparisons 12 3 17 23 26

B. Topical placement 1. Pre-first-grade concepts 18 1 19 29 2. Readiness 15 2 2 10 25 3. Logical order 0 6 4. Quantitative under-

standing 5 2 7 16 5. Content to be included

in grade 7 9 2 3 3 24 46 6. Time allotment 2 1 3 10

December 1967 685



Total in- De- Corre- Ex eluding scrip- Case lation- post Experi- cross-ref- tive Survey Study Action al facto mental Total erences

C. Basic concepts (and methods of teaching them) 1. Counting 6 1 7 13 2. Number properties and

relations 3 1 4 12 3. Whole numbers 2 2 4 34

a) Addition 4 4 8 16 36 b) Subtraction 6 1 3 6 16 26 c) Multiplication 14 1 3 9 23 d) Division 10 7 1 1 6 25 40

4. Fractions 5 1 2 8 20 a) Addition 1 1 8 b) Subtraction 1 17 c) Multiplication 2 2 7 d) Division 1 13 5 7

5. Decimals 2 1 2 5 13 6. Percentage 12 1 2 6 6 7. Ratio and proportion 0 0 8. Measurement 13 1 9 23 43 9. Negative numbers 111

10. Algebra 0 6 11. Geometry 3 3 7 12. Sets 11 2 5 13. Logic 1 3 1 5 5 14. Our numeration system 2 13 8 15. Other numeration systems 2 2 4 7 16. Probability and statistics 4 4 4

D. Materials 1. Textbooks 21 1 1 2 25 56 2. Workbooks 1 12 3 3. Manipulative devices 2 1 9 12 19 4. Audiovisual devices 2 1 1 4 8 12 5. Programmed instruction 1 1 1 8 11 18 6. Readability and

vocabulary 6 5 1 3 15 24 7. Quantitative concepts in

other subject areas 2 2 15 6 E. Individual differences

1. Diagnosis 2 19 2 3 1 4 31 55 2. Remediation 2 1 12 9 2 1 8 35 52 3. Enrichment 1 2 1 3 8 15 19 4. Grouping procedures 2 12 1 3 10 28 36 5. Physical, psychological,

and /or social characteristics 7 1 11 10 23

6. Sex differences 17 8 8 7. Socioeconomic differences 3 14 6

F. Evaluating progress 1. Testing 3 7 4 4 5 23 44 2. Achievement evaluation 17 1 3 12 1 34 76 3. Relation to achievement 2 3 2 7 11

a) Age 3 3 6 7 b) Intelligence 11 15 3 20 37

4. Effect of parental knowledge 1 2 3 3

5. Effect of teacher background 1 1 4 6 9

G. Studies related to learning theory 1. Transfer 1 12 13 24 2. Retention 3 2 7 12 25 3. Generalization 1 12 5 4. Organization 2 1 2 5 15 5. Motivation 3 1 4 8 20 6. Piagetian concepts 2 9 1 3 9 24 32 7. Reinforcement 3 3 12

Totals 107 230 18 63 56 79 246 799




ical topic and the frequency by type of study. The definitions for types of studies are these:

Descriptive. - research in which the researcher reports on records which may have been kept by someone else; includes reviews, historical studies, textbook comparisons

Survey. - research which attempts to find characteristics of a population by asking a sample (questionnaire, interview); includes also status study, which investigates a group as it is, to ascertain characteristics

Case study. - research in which the researcher describes in depth what is hap- pening to one designated unit, usually one child

Action. - research which uses nominal controls; generally teacher or school orig- inated; may note procedures of actual practice

Correlational. - research which studies relationships between or among two or more variables; uses correlational statistics

Ex post facto. - research in which the independent variable (treatment) or variables were manipulated in the past; the researcher starts with the observation of a dependent variable or variables and then studies the independent variables in retro- spect for their possible effects on the dependent variables

Experimental. - research in which the independent variable or variables are manipulated by the researcher to quantitatively measure their effect on some dependent variable or variables

The number of reports of experimental research was 246, a figure almost equaled by the 230 reports of surveys that were found. The distribution of reports gives some indication of the concern for various topics, as well as an indication that some topics lend themselves more readily to one type of research. For instance, readiness is most easily ascertained through surveys, while case studies were most frequently

used to depict individualization techniques, particularly for remediation.

Cross-referencing adds more depth, for in many instances the topic that was cited first was selected arbitrarily. The totals within each mathematical category shift somewhat as all references are counted, as can be noted from the final two columns on Table 1. The topics under which the largest number of all types of research were categorized are:

problem solving 84 achievement evaluation 76 planning and organizing for teaching 62 textbooks 56 diagnosis 55 remediation 52 content to be included in grade 46 testing 44 drill and practice 43 measurement 43 division of whole numbers 40

The research which is experimental was also categorized by design type or paradigm. Of the 246 experimental studies, 39 involved no control group; 150 involved possible sampling errors. Thus, sampling and/ or the way in which a researcher re- ported the sampling for his experiment was a point of great variability and ambiguity.

How valid are the findings of this research?

Since research efforts vary widely in quality, the question of how much con- fidence can be placed in the findings of a study is one of considerable importance. The attempt to evaluate is a significant characteristic of the present compilation.

An "Instrument for Evaluating Experi- mental Research Reports" was developed and tested for reliability. In one study with three judges, the interrater agreement was found to be .91, while the intraclass reliability, or the reliability which could be expected from any one judge, was .77. In a second study with twelve judges, the

December 1967 687



interrater reliability was found to be .94, with an intraclass reliability coefficient of .58. Thus, the coefficient for agreement between judges is satisfactorily high.

Analysis of the qualitative values that resulted from evaluation of the experimental research with this instrument shows a range from 13 to 44 (of a possible 9 to 45). Ratings of satisfactory or better for each of the nine questions on the instru- ment3 are tabulated below:

1. How practically or theoretically significant is the problem? 73.5 percent

2. How clearly defined is the problem? 72.3 percent

3. How well does the design answer the research question? 50.7 percent

4. How adequately does the design control variables? 29.7 percent

5. How properly is the sample selected for the design and purpose of the research? 27.7 percent

6. How valid and reliable are the meas- uring instruments or observational techniques? 53.3 percent

7. How valid are the techniques of analysis of data? 44.4 percent

8. How appropriate are the interpreta- tions and generalizations to the data? 59.8 percent

9. How adequately is the research re- ported? 65.0 percent

Two questions, those involving control of variables and sampling, were rated especially low. The improvement of research possibly depends on increasing the researcher's awareness of the need to consider these especially carefully. The evaluation with the other seven points on the instrument also indicates that more careful planning and reporting of research proj- ects are needed. The Instrument for Evalu- ating Experimental Research Reports may serve as a guide in planning as well as in

3 The instrument includes these questions plus sup- porting "key points" to focus attention on specific aspects of the reports.

evaluating the finished product. There is a need to develop similar instruments to evaluate types of research other than experimental.

Only eighty reports of the 246 in the experimental category were considered satisfactory or better on total scores. This would seem to indicate a need to improve the reporting of research, and possibly research procedures as well.

When these eighty studies are considered, no possible summary can be made, either because there was only one study in a category or because the studies were aimed at diverse phases. In other cases, inconsistency is evidenced. For instance, consider some of the studies on problem solving, on which the greatest amount of research has been done. Burch,4 studying formal analysis, and Hudgins,5 testing the effectiveness of requiring specification of steps, found no relation to problem solving effectiveness. Falk and Landry,6 on the other hand, found that a systematic approach was slightly superior to the textbook approach. Irish,7 Pace,8 Riedesel,9 and VanderLinde10 identified some speci- fied procedures which aided problem solving - though the procedures were diverse. Some specific help is provided for the classroom teacher - and this is the ultimate purpose of any curriculum research - but there is no clear and well-defined pattern evidenced from journal-published research.

4 Robert L. Burch, "Formal Analysis as a Problem- Solving Procedure," Journal of Education, CXXXVI (November 1953), 44-47, 64. 5 Bryce B. Hudgins, "Effects of Group Experience

on Individual Problem Solving," Journal of Educational Psychology, LI (February 1960), 37-42.

6 Charles J. Faulk and Thomas R. Landry, "An Approach to Problem-Solving," The Arithmetic Teach- er, VIII (April 1961), 157-60.

7 Elizabeth H. Irish, "Improving Problem Solving by Improving Verbal Generalization," The Arith- metic Teacher, XI (March 1964), 169-75.

8 Angela Pace, "Understanding and the Ability to Solve Problems," The Arithmetic Teacher, VIII (May 1961), 226-33. 9C. Alan Riedesel, "Verbal Problem Solving: Sug-

gestions for Improving Instruction," The Arithmetic Teacher, XI (May 1964), 312-16.

10 Louis F. VanderLinde, "Does the Study of Quantitative Vocabulary Improve Problem-Solving?" Elementary School Journal, LXV (December 1964), 143-52.




The knowledge of the educator must be used to supplement the findings.

It seems obvious that more research needs to be done on many topics. Re- searchers should consider the points on which more knowledge is needed, since the 799 studies seemed almost randomly dis- tributed to topics. When the evaluation

criteria were applied, few studies were found to be completely valid. Careful and precise planning of research is vital. Equal- ly careful and precise reporting would be helpful. The possibility of using research as a means of developing a theory of instruction should be carefully and thought- fully pursued.

Searching for the unknown! [Continued from p. 683]

extremely difficult. Actually, children love problems of this type and will work with them until they discover the answers. Some pupils

1 may think like this: "- will fit into what num-

8 1 1

ber - times? - . What number is the square 2 16

1 1 1 root of - ? -. Therefore, x = -." The pupils

16 4 4 are always asked to solve for the variable, not the variable squared. Because no formal algorism is introduced, the pupils will think of a variety of highly imaginative ways to discover the variable. The teacher should, of course, encourage this approach.

After an equation had been solved in my class, I always encouraged the pupils to share the different methods that they had used to discover the unknown number. I was always amazed at the creative and variety of ways that pupils used. They often used methods that I had never thought of when I made up the equations! However, when I made up equations, I always made sure that the pupils had the necessary skills to solve them.

Although no formal algorism was taught to solve the equations, the students were permitted to use any methods that worked and were mathematically accurate and logical. Therefore, many students used their knowledge of mathematics to discover algorisms that worked for solving many of the equations. The pupils knew that factor times factor equals product. When given an example of this type:

1

3 - = 2, n

the students knew that 2 and n were factors,

1 and that - was the product. Many students

would solve this example by dividing - by 1 1

2. - -i- 2 = -. Other students simply thought, 3 6

1 1 "What number will fit into - twice? -." Either

3 6 method was acceptable.

After the teacher has made up equations for several days, the pupils should then be asked to make up equations for the entire class to solve. I always reminded my pupils that they should have their answers in mind when they composed equations. While some students copied patterns of equations that I had used earlier, most of them composed highly creative and challenging equations. One of the most creative equations composed by a pupil in my class was:

x + jc2 + X* + x* + jc5 + x* + 1 = 1 .5

A unit on equations is an excellent review unit because it can review all of the funda- mental concepts covered during the year in a novel and creative fashion. A review of both decimals and common fractions is represented

.5 in this equation: - - A. Most students would

x 5 1

probably think of - as -, in order to discover 10 2

1 that - will fit into .5 four times. Mathematical

8 reasoning is developed, and previously acquired skills are strengthened. The pupil's enthusiam on his search for the unknown is a joy for the teacher! - James A. Banks, Graduate fellow and assistant, Michigan State University, East Lansing, Michigan. (Former fifth-grade teacher, Francis W. Parker School, Chicago, Illinois.)

December 1967 689



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The status of research on elementary school mathematics