The purpose of the present article is to focus on the function concept,
to describe several of its aspects, and to chart some of its appearances in the mathematics curriculum. At the start, we shall rely upon examples;
the modem concept of function will not emerge in its final form until
the end of the chapter, where we enter more deeply into the technical
In human affairs, there seems to be a tendency to swing from one
extreme to another. Often, having swung violently between two
extremes, man finds himself coming to rest somewhere in the middle,
like a pendulum that has stopped oscillating. In mathematics education,
this pendulum-like action has occurred with respect to many
issues. In this chapter we will examine some of those issues.
The main part of our discussion appears under the next four major
headings of this chapter and will be concerned with the undergraduate
training of teachers at Levels I-III with particular attention
to Level III. The first of these four sections, the one immediately
following the Introduction, will describe briefly the situation prior to 1961; the next section will outline the CUPM recommendations;
and the following two will respectively (a) survey the present situation
and (b) discuss deficiencies in the present program and attempt
some predictions as to future developments.
In this chapter, we are going to examine in some measure the
nature of applications of mathematics and how such applications relate
to the teaching of mathematics. The problem is a rather complex
one, and the subject is so beset with various misconceptions and intuitive
prejudgments that we must think our way through the issues
rather carefully. It would, of course, be logical to begin with as precise
a definition of applications of mathematics as is possible. However,
our contact with problems of mathematics education during
recent years has somewhat converted us to the spiral method of
teaching, and this method tends to suffer from attempts at becoming
too precise too early. Our plan, therefore, is to make a modest beginning
and to return to the problem many times.
Our approach to evaluation in this chapter is less structured than Maurice L. Hartung's from several years ago;
however, our interpretation of terminology will be in keeping with
the undefined terms and definitions just identified. We also shall
find it convenient to use the term outcomes to refer to actual (and
hopefully, observable) experiences. Thus, evaluation may be viewed as the process of determining the extent to which outcomes
agree with objectives.
This chapter begins with a general discussion of evaluation, including
the important problem of stating the objectives of mathematics
education in an organized fashion. This discussion is followed by
brief reviews of a number of projects which have been concerned in
one way or another with the evaluation of mathematics programs.
The final portion is a detailed discussion of the variety of evaluation
procedures used by the School Mathematics Study Group (SMSG),
the largest of the mathematics curriculum projects in this country.
This chapter is designed to assist educational workers in achieving
better utilization of available instructional resources in order to more
effectively meet the learning needs of individual pupils. Developing
adequate systems for differentiating or individualizing instruction is
a difficult task, and progress has been somewhat slower than some authoritative
predictions had anticipated. However, there are signs that
indicate a growing capability in many schools to cope with the economic,
managerial, and instructional problems that differentiation of
instruction necessarily incurs. The following remarks summarize the
trends in this area and point to new directions which appear to be
most promising at this time.
Repeated layers of minor revisions of the curriculum had been
applied, year after year, and the time was overdue for a complete
rethinking. As scientists and mathematicians became aware of the
situation, many indicated interest in accepting responsibility and
assisting with the problem. Their resulting involvement in precollege
curriculum improvement assisted both in producing change and in
influencing the character of the change.
One of the characteristics of this movement was that it was discipline centered rather than child centered. The emphasis was upon
updating and reorganizing the academic disciplines which constitute
preparation for college. Thus it attended to the special needs of
middle-class and upper middle-class, college-bound students.
In earlier chapters, and especially in chapters i and viii, we have
indicated something of the growing role that mathematics and its
applications are playing in our society today. Today the areas of mathematical activity are so broad and diverse that it no longer seems adequate to speak simply of mathematics, and
in consequence a new term has arisen: the mathematical sciences.
Arithmetic exhibits some marked contrasts when compared with some
of the other content areas of education. Unlike chemistry, physics; and
the social sciences, its content is not subject to radical changes due to discoveries
such as those involving new elements, forms of energy, or ideas
of social organization.
This chapter does not deal primarily with the place of arithmetic as
assessed by its life values but with its place and relationships in the total
curriculum of the elementary school. The many unmistakable contributions
of arithmetic to life have been convincingly portrayed in various
yearbooks of the National Council of Teachers of Mathematics. It will
be apparent, as the discussion in this chapter proceeds, that the writer
suspects that the proponents of the organized course of instruction in
arithmetic have been too modest and limited, rather than too extravagant
and comprehensive, in their claims.
The arithmetic we teach pupils in school is a way to think about the
numbers of thingsabout quantities, amounts, sizes. It comprises the
questions, "How many?" "How much?" "What part?" and the seeking
and finding of their fitting answers. By means of arithmetic as a way to
think, and in the degree that we have learned this way, we (a) recognize
number questions and (b) determine their answers.
What children say, using number words, is only one of many clues
which the interested parent or teacher should use in making observations
as to the young child's number maturity. As with older children, glib verbal use of number terms should not be mistaken for certain knowledge
of number concepts which a mature person associates with those words.
While achievement in arithmetic or any other school subject is no
longer so important in determining grade promotion as it was a generation
ago, the middle grades still remain critical in the teaching of arithmetic.
Some aspects of arithmetic in the junior-senior high school need reconsideration.
This need is called to the attention of those interested in the
program of the junior-senior high school, not only by the evidence of
mathematical illiteracy exhibited by our armed forces during the last war
but also by the performance of the graduates of our public school systems
in the business world.
The teaching of arithmetic today shows extreme variation. Markedly
different procedures are employed even in classrooms that are considered
superior, to say nothing of differences in methods between classrooms
representing superior and inferior instruction. Attention will first be
given to procedures that are generally considered to be good, those most
popular in school systems rated as being good or superior.
The present chapter is directed to teachers and supervisors rather than
to psychologists. The aim of the writer will be to make a sensible interpretation
of psychological theory and experiments as they affect the
arithmetic program. Illustrations will be drawn from the teaching of
arithmetic to make the interpretations as clear as possible.
The success of a meaningful program in arithmetic depends, in a large
degree, upon methods (see chap. vii) and materials of instruction. There
is no one method or any single type of instructional material which will
suffice in all situations. The skillful teacher selects methods and materials
in terms of the outcomes to be achieved and of the needs and the interests
of the children. If instruction in arithmetic is to insure a steady growth
in understanding number relationships, a wide variety of instructional
materials must be used to enrich and to supplement the learner's experiences.
A discussion of testing instruments and practices in relation to present
concepts of teaching requires identification of the major characteristics
of teaching. Five such characteristics are listed and briefly described.
This chapter deals with the academic and professional preparation of teachers of arithmetic.
A curriculum designed for the training of elementary-school teachers,
in common with the rest of our educational program, is a subject for debate.
The writer, by virtue of his position that is concerned in part with
teacher education in the state of New York, is aware of the force with
which proponents of rival philosophies can suggest changes in the curriculum.
It is very difficult, and perhaps impossible, to construct a teacher-training program that will receive any unanimity of acceptance. However,
such chapters as the present one must be written in order that specific
proposals may be set up for purposes of discussion and trial.
teachers often carry on modest experimentation with techniques and procedures
on their own initiative and without any particular encouragement
of administrative help. Some teachers try to find new and better
ways even at the risk of incurring official displeasure. There is reason to
believe, therefore, that if supervision is conceived as a process of mutual
study and investigation by the staff of a school or a school system under
competent leadership, far more will be gained in the improvement of instruction
than if a series of directives on content and method are merely
passed out by a supervisory authority.
Giving names, tallying, comparing and combining actual
groups, and countingthese, it is clear, must have been evolved to serve
human purposes. Primitive men can hardly be said to have invented or
discovered their arithmetic; they lived it.
The social value of arithmetic, that is, its human serviceability, is no
less evident in the historical period.
This chapter consists of two parts. First, there is a brief consideration
of some problems of research as they relate to arithmetic; and, second,
there is a group of specific proposals for research submitted by some
twenty members of the profession who have an active interest in the
teaching of arithmetic.
Researches dealing with arithmetic were few prior to 1900. While
this section is concerned only with the influences of research, a brief
consideration of the status of arithmetic at that time will serve as a
useful basis of comparison.
Part One of this volume consists in a systematic treatment of the general field of arithmetic. Part Two is devoted to research relating to this subject. In this part of the Yearbook the Committee is presenting two types of material. The first of these, contained in Chapters II and III, is a classification of the entire body of published research in arithmetic arranged under the nineteen techniques which have been employed in such investigations. The second type of material consists in a series of new research studies relating to several aspects of arithmetic.
Exact and valid knowledge concerning the many problems of teaching and learning in arithmetic is to be had only from fundamental research studies which are carefully planned, painstakingly prosecuted, and wisely interpreted. In the last analysis the central issue in such studies consists in the development of a technique adapted to the type of problem which is to be investigated.
No attempt will be made in this chapter to present a detailed summary of the research studies in arithmetic which are now in print. Such detailed summaries have already been published and may be procured easily. The chapter will deal with the nature and findings of investigations in arithmetic without attempting to mention all the persons who have made contributions.
The conventional arithmetic textbook, a book intended to be placed in the hands of third-grade children, begins with the presentation of a gaudy picture in which certain priggish children, impossibly clean in appearance, are occupied with hoops, marbles, dolls, or whatnot of the apparatus of play.
Recent developments in the techniques for analyzing and constructing instructional materials in arithmetic include two methods related to procedures for studying the content of drill in fractions. These methods make it possible to construct practice exercises in fractions according to specifications that can be set in advance, as can now be done in the work with whole numbers. It is the purpose of this paper to point out the fundamental difference between the two methods that have been proposed by Knight and his co-workers and by Brueckner for analyzing the content of drill materials in fractions. The discussion will be limited to a consideration of the procedures as applied to the subtraction of fractions.