A mathematics course for prospective elementary school teachers

A mathematics course for prospective elementary school teachersAuthor(s): JERRY SHRYOCKSource: The Arithmetic Teacher, Vol. 10, No. 4 (April 1963), pp. 208-211Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186748 .

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A mathematics course for prospective elementary school teachers*

JERRY SHRYOCK Western Illinois University, Macomb, Illinois Dr. Shryock is a member of the department of mathematics at Western Illinois University.

What mathematical topics are appro- priate for study in a background course for prospective elementary teachers of arithmetic? Are mathematics educators in agreement as to which topics should be introduced in such a course? Are textbooks suitable for such a course currently available? The intended purpose of this report is to answer these and related ques- tions.

A lack of agreement as to what topics should be included in a background course in mathematics for elementary school teachers has been in evidence since before World War II. Morton and Judd,1 writing in 1935 in the Tenth Yearbook of the National Council of Teachers of Math- ematics, reported on a study which indicated a great diversity of topics in teacher-training courses in arithmetic among members of the American Associa- tion of Teachers Colleges. On reporting the results of a questionnaire regarding teacher opinions on time allotments to various topics in a three-hour course in the teaching of arithmetic, Morton and Judd wrote :

It will be seen that those who would give less time outnumbered those who would give more time to equations, devices, history objectives, problem solving, and the fundamental operations, although the vote is very close in the case

of problem solving. On the other hand, those who would give more time than was suggested in the questionnaire decidedly outnumbered those who would give less time in the case of drill exercises, common fractions, decimal fractions, percentages, interest, and mensuration. Evidently, many consider one hour inadequate for these topics.

Some teachers suggested additional topics for inclusion in a course in the teaching of arithmetic. These included: banking, partial pay- ments, the metric system, ratio and proportion, painting and paper-hanging, graphs, checks and drafts, workbooks, budgets, investments and saving, insurance, tests, constructions, taxes, and others.2

There is at present much concern about the improvement of the arithmetic pro- gram in the elementary schools of our na- tion. Most often advocated changes in- clude: (1) teaching for pupil discovery of mathematical relationships, and (2) a de- sire to impart more insight into the struc- ture of mathematics, including not only arithmetic, but algebra, geometry, and other topics once considered much too ad- vanced for the elementary child. As an example of such advocated changes, the School Mathematics Study Group ma- terial3 may be considered.

An examination of the literature shows that writers, with surprising unanimity, maintain that elementary teachers who know arithmetic as learned from their own elementary school background are not adequately prepared to teach the subject in its newer and more meaningful aspects.

* This article is adapted from a section of the author's doctoral dissertation "A Study of Mathematical Background Courses for Prospective Elementary School Teachers," State University of Iowa, 1962.

1 R. E. Morton and R. D. Judd, "Current Practices in Teacher-Training Courses in Arithmetic," in The Teaching of Arithmetic, Tenth Yearbook of the National Council oř Teachers of Mathematics (Washington, D.C.: The Council, 1935), p. 157-72.

» Ibid., p. 169. * School Mathematics Study Group, Mathematica for the Elementary School (Ann Arbor, Mich.: Edwards Brothers, Ino., 1961).

208 The Arithmetic Teacher



With the same degree of unanimity, writers point out that more mathematics in teacher-training programs will prove to be a solution to the problem of obtaining adequately prepared teachers of arithmetic.

A recent writing which supports this contention that more mathematics should be included in elementary teacher-training programs is one by Stone4 in an article concerning fundamental issues in the teaching of elementary school mathematics. He maintained that to improve the teaching of arithmetic it is necessary to give much better training to these future teachers. It was pointed out that because the turnover of elementary teachers is high, better training can have a large scale effect in a short period of time. He also observed that too many mathe- matically ill-prepared teachers, many of whom dislike the subject, cause their pupils to fear and hate mathematics. For a goal of the new mathematics training Stone wrote:

The new mathematics training which may be developed for future grade-school teachers should give them an appreciation and understanding of the subjects they may be expected to teach and as a result should inspire them with a certain degree of respect and admiration for mathematics, if not a real liking of it.6

On the other hand, an examination of the literature in this area indicates a paucity of reports pertaining to a clarification of what the elementary teacher should know in the field of mathematics. In fact, writers are in agreement that there is an imperfect understanding of what arithmetic teachers ought to know, to say, and to do if pupils are to progress with maximum achievement in mathematics.

In a recent article in The Arithmetic Teacher, Sparks6 recognized this need

* M. H. Stone, "Fundamental Issues in the Teaching of Elementary School Mathematics," The Arithmetic Teacher, VI (October, 1959), 177-79. « Ibid., p. 179. e Jack N. Sparks, "Arithmetic Understandings Needed by Elementary School Teachers," The Arithmetic Teacher, VIII (December, 1961), 395-403.

for clarification of what mathematical understandings elementary teachers should have. He wrote that educational research to date has been unable to offer any sub- stantial assistance in planning programs of teacher education.

Procedure

In order to gather pertinent data for the problem, the writer examined eight col- lege textbooks published for use in a mathematics background course for prospective elementary teachers. Only recent textbooks were included in this analysis with the earliest copyright date being 1956. The authors represent a group of mathematics educators who, because of their professional attainment in this field, would seem to' be adequately prepared to provide sound counsel regarding the ap- propriateness of topics.

A summary of the tabulations for each of the topics found in the eight mathematics background textbooks is given in Table 1.

Interpretation of Table 1

A tentative list of topics was obtained by first examining several of the textbooks and listing those topics found and then adding to the list as the other textbooks were subsequently examined. For purposes of conciseness each textbook was assigned an identifying letter. Each topic found, indication of textbook presenting that topic, and total number of textbooks presenting that topic are included in Table 1. It should be noted that the extent of treatment of a topic varies among the textbooks presenting that topic.

Analysis of Table 1

Table 1 indicates a lack of agreement among mathematics background textbooks on topics presented.

Among the textbooks examined, only three topics were included in each of the eight textbooks. Each of these three topics were from the area of arithmetic : basic

April 1963 209



Table 1 Topics presented in eight textbooks employed in mathematics background courses for prospective elementary teachers

Textbooks Topics •

ABCDEFGH N

Topics from arithmetic Basic operations

with natural numbers XXXXXXXX 8 Basic operations

with common fractions XXXXXXXX 8 Basic operations

with decimal fractions X XX XXX 6 Percent X XX XXX 6 Linear measurement XXXXXXXX 8 Area measurement X X X X X X 6 Volume measurement X X X X X 5 Metric system X X X X X X 6 Ratio, proportion X XXXXXX 7 Scale drawing XXX X 4 Interest on money X X 2

Topics from algebra Negative integers XXX XXXX 7 Equation solving XX X 3 Inequalities XXXX 4 Graphs of functions X X X X 4 Irrational numbers XX XXXX 6 Complex numbers X X X 3 Logarithms X X X 3 Slide rule XX 2 Trigonometry X X X X 4 Variables X XXXX 5 Substitution in formulas XX XXXX 6

Topics from number theory Ancient systems of notation XX X XX 5 Exponential notation XX XXXXX 7 Different scales of notation XX XX XX 6 Prime and composite numbers X XX XXX 6

Topics from approximate computation XX XXX X 6

Topics from statistics Mean, median, mode X X 2 Probability XX 2 Statistical graphs XX X X 4

Topics from elementary logic Set terminology X XXXX 5 Nature of proof XX 2 Countably infinite set X XX 3

Topics from informal geometry XXX 3

operations with natural numbers, basic operations with common fractions, and linear measurement. Only three topics were introduced in each of seven of the

eight textbooks examined: ratio-proportion from the area of arithmetic, negative integers from the area of algebra, and exponential notation from number theory.

210 The Arithmetic Teacher



ment is found for those topics from the fields of algebra, statistics, elementary logic, and informal geometry.

3 Not one of the examined textbooks included all of the topics thought to be important by the authors taken as a group. On the other hand, the writer's selection of two particular textbooks would provide a treatment of twenty- nine topics - from the area of arithmetic: basic operations with natural numbers, basic operations with common fractions, basic operations with decimal fractions, percent, linear measurement, area measurement, volume measurement, metric system, ratio and proportion, and scale drawing; from the area of algebra: negative integers, equation solving, inequalities, graphs of fune tions, irrational numbers, variables, and substitution in formulas; from the area of number theory: ancient systems of notation, exponential notation, different scales of notation, and prime and composite numbers; topics from the area of approximate computation; from the area of statistics: mean-median- mode, probability, and statistical graphs; from the area of elementary logic: set terminology, nature of proof, and the countably infinite set; and topics from the area of informal geometry. The writer feels that if capable instruc- tion, together with attention given to teaching methods, could be provided in presenting these topics, the students would have adequate background for understanding topics involved in the current programs for the elementary school.

4 In the writer's opinion, there is an ur- gent need for studies to determine if knowledge of such topics actually results in better teaching of arithmetic.

April 1963 211

"To look is one thing. To see what you look at is another. To understand what you see is a third. To learn from what you understand is still something else. To act on what you learn is what really mattere."

Fifteen of the thirty-four topics listed in Table 1 appeared in only four or fewer textbooks: scale drawing and interest on money from the area of arithmetic; equation solving, inequalities, graphs of functions, complex numbers, logarithms, slide rule, and trigonometry from the field of algebra; mean-median-mode, probability, and statistical graphs from the area of statistics; nature of proof, and the countably infinite set from the area of elementary logic; and topics from informal geometry.

An examination of Table 1 indicates that textbooks E and H presented twenty- seven topics. None of the other six textbooks presented more of the thirty-four topics than this. On the other hand, textbooks A and С presented the least number of topics: fifteen.

In summary, the lack of agreement as to which topics should be introduced in a mathematics background course for elementary teachers, indicates that authors of such textbooks are not in accord on this subject. Table 1 points up the consid- erable variation in topics offered in the eight textbooks examined. In general, the greatest agreement is indicated for those topics from the fields of arithmetic, number theory, and approximate computation. The least agreement is found for those topics from the fields of algebra, statistics, elementary logic, and informal geometry.

Concluding statements

1 Authors of such textbooks are not in accord as to which topics should be introduced in a mathematics background course for elementary teachers.

2 The greatest agreement as to which topics should be introduced is indicated for those topics from the fields of arithmetic, number theory, and approximate computation. The least agree-



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A mathematics course for prospective elementary school teachers