Research in the teaching of elementary school mathematics

Research in the teaching of elementary school mathematicsAuthor(s): KENNETH E. BROWN and THEODORE L. ABELLSource: The Arithmetic Teacher, Vol. 12, No. 7 (NOVEMBER 1965), pp. 547-549Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185239 .

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Research in the teaching of elementary school mathematics

KENNETH E. BROWN and THEODORE L. ABELL U.S. Office of Educatiorij Washington, D.C. Dr. Brown is specialist in mathematics education and Mr. Abell is research assistant in mathematics education in the U.S. Office of Education.

J. o obtain information about the research in mathematics education, the U.S. Of- fice of Education, with the assistance of the National Council of Teachers of Mathematics, sent a questionnaire to 1,049 colleges that offered graduate work in mathematics education, or whose staffs or students had made contributions to previous studies. Replies were received from 645 colleges. Many reported no research in mathematics education in the calendar years 1961-62, but requested a report of the survey. Approximately 50 investigations were reported in the area of elementary school mathematics, Grades 1-8.

Research in the teaching of elementary school mathematics was centered around experimentation in teaching advanced concepts to elementary school children. Many familiar topics were studied, such as kinds of grouping which provide the most effective learning of the new concepts in mathematics, the pupils' attitudes toward the new programs, the use of aids in teaching mathematics, and effective ways of teaching problem solving.

More mathematics can be developed in the primary grades The research seems to indicate that con-

siderably more mathematics can be taught to young children than the traditional programs include. In one study in which the emphasis was on structural concepts, the experimental group achieved signif-

icantly above the grade placement norms for the posttest. In another investigation, it was found that the second-grade children were ready for third-grade work in arithmetic. No studies were reported on the desirability of teaching more advanced concepts in the primary grades.

Grouping for instruction may or may not result in increased achievement In one study, randomly grouped pupils

made significantly greater gains in computation and reasoning than did pupils specially grouped within grade levels. In a nongraded school, teacher-prepared materials for pupil self-instruction gave promise for use with achievement grouping. Subgrouping within a classroom with emphasis upon meaningful instruction produced significant gains. One study in- dicated that removal of the top group might be beneficial for the less able students. A survey conducted by questionnaire revealed that a majority of a group of 246 junior high school pupils thought that the top and low groups of students benefit most from ability grouping. Per- haps grouping is helpful with certain types of pupils on specific kinds of learning ac- tivities. No research was reported on this phase of the problem.

The use of manipulative materials is popular in arithmetic instruction The use of counting devices, pictures,

games, and supplementary materials

November 1965 547



shows a decided increase in the last two decades with more materials used in the primary grades.

The use of the " bundles77 method and the "hundred boards" method by first- grade children in learning place value seemed to supplement each other. The use of a commercially produced expensive set of number aids was found to be no more effective than inexpensive materials se- lected by the teacher. The use of a variable base abacus for counting in numeration systems other than base ten produced no significant gains over the use of the chalk- board alone.

Many questions arise in the study of the research. Are there certain manipulative devices that lend themselves better to different methods of instruction? Will a device help one child and hinder another? Will the learning pattern of the child indicate the device that may help him in learning a specific concept? These and similar questions were not answered.

Pupil attitude affects pupil achievement One investigator found that the correla-

tion between pupil attitude and achievement was higher for arithmetic than it was for spelling, reading, or language. The student's general ability to learn seems to be associated with his liking for arithmetic. Teacher observation appears to be inadequate as a method of appraisal of students7 attitudes toward mathematics.

Methods of teaching fractions There are probably too many variables

in instructional situations to find the one most effective method of teaching fractions. The use of a flannel board with felt disks and automated devices produced significant achievement in teaching the multiplication of fractions. A comparison of the mechanical with the meaningful method in the division of fractions produced no significant difference in achievement, but the meaningful method pro-

duced greater retention. In one experi- ment, teaching computation with decimal fractions on the basis of an orderly ex- tension of place value prior to any formal computation with common fractions produced significant gains over the usual procedure. During the past decade greater emphasis has been placed on deriving and proving the rules that are used in per- forming the operations with fractions.

No "best" method of estimating the quotient A computational analysis of the division

of 44,550 dividends by 81 two-digit di- visors disclosed that the "round-off" method gave the true quotient on the first estimate the greatest percent of the time. However, in a study with fifth-grade children, the "round-off/7 "increase-by- one,77 and "apparent77 methods were equally effective in improving children7s abilities in estimating quotients.

In division by one- and two-place decimals, children preferred to use the caret sign to show that the divisor and dividend were multiplied by a power of ten. The research seems to indicate that there is no one best method for estimating the quotient.

Problem solving Using vocabulary exercises, discussing

problem situations, diagramming the problem, estimating the answer, and writing solutions produced a slight gain in favor of the experimental group. Both above- and below-average problem- solvers benefited from the use of 30 problem-solving lessons, each written at two levels of difficulty, and each followed by an optional difficult problem titled, "How7s your P.Q.?77 The heuristic method of teaching the solution of verbal prob- lems in algebra was found to be more effective than the textbook method. The use of a specific textbook series produced no significant changes in patterns of think- ing when measured by problem-solving competence.

548 The Arithmetic Teacher



Follow-up to televised instruction One investigator sought to measure the

value of an introduction and follow-up to televised arithmetic instruction. One class merely viewed the program. Another class receiving teacher help in an introduction and follow-up to the viewing made greater gains during the first half of the experi- ment, but no significant difference was found for the second half. More research is needed to determine the effectiveness of television and follow-up instruction in mathematics.

Summary The studies seem to indicate that many

young children can learn more mathematics than has been expected of them. The crucial question seems to be, "What mathematics should children learn at an early age?" The evidence indicates that the answer is not the same for all children. It may be desirable for some pupils to ex- plore nondecimal systems of numeration to understand the decimal system better. Other students may know so little about the decimal system that such a procedure would lead only to confusion. Very little research is available to help the educator answer the question, "What mathematics for what children?"

The use of aids was considered in several research studies. Most of the aids that were advocated seemed to help some pupils. If children differ and if they learn in different ways, we would expect that some techniques would be more effective with one child than with another child. In

harmony with this philosophy, many teachers for years have tried many aids in the hope that one device would stimulate learning on the part of some of the pupils. Research seems to show that multisensory aids are helpful to some pupils, but research has given little direction as to which students will be helped and with which concepts an aid should be used.

Likewise, the grouping of children seems to increase learning of certain topics with some pupils. Research has done little to answer the question, "When and how can grouping be helpful?"

Several research studies on the best method of teaching a particular skill or concept were made. This type of research has not yielded great returns. Perhaps there is no one best method for estimating the quotient for all pupils. Maybe there is no most effective method of teaching fractions. There may be a very effective method for teaching certain pupils. Re- search has given little information on methods for different types of pupils. Until research gives us more information, no doubt, successful teachers will con- tinue to use many methods in an attempt to clarify a mathematical concept.

A more detailed report of the research will be available in the fall from the U.S. Office of Education. For the reader who is interested in mathematics education research completed in other calendar years, the Review of Educational Research, June, 1964, is recommended. The U.S. Office of Education has summaries for 1955-56, 1957-58, and 1959-60.

Effective instructional practices threaten the conception of teaching as a form of maieutics. If we suppose that the student is to "exercise his rational powers," to "develop his mind," to learn through "intuition or insight," and so on, then it may indeed be true that the teacher can- not teach but can only help the student learn. But these goals can be restated in terms of ex- plicit changes in behavior, and effective methods of instruction can then be designed. - From 11 Why Teachers Fail" in Saturday Review, October 16, 1965.

November I960 549



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Research in the teaching of elementary school mathematics