THE TEXTBOOK IN GEOMETRY
In considering the nature of the textbook in geometry we need to bear in
mind the fact that the subject is being taught to-day in America to a
class of pupils that is not composed like the classes found in other
countries or in earlier generations. In general, in other countries,
geometry is not taught to mixed classes of boys and girls. Furthermore,
it is generally taught to a more select group of pupils than in a
country where the high school and college are so popular with people in
all the walks of life. In America it is not alone the boy who is
interested in education in general, or in mathematics in particular, who
studies geometry, and who joins with others of like tastes in this
pursuit, but it is often the boy and the girl who are not compelled to
go out and work, and who fill the years of youth with a not
over-strenuous school life. It is therefore clear that we cannot hold
the interest of such pupils by the study of Euclid alone. Geometry must,
for them, be less formal than it was half a century ago. We cannot
expect to make our classes enthusiastic merely over a logical sequence
of proved propositions. It becomes necessary to make the work more
concrete, and to give a much larger number of simple exercises in order
to create the interest that comes from independent work, from a feeling
of conquest, and from a desire to do something original. If we would
"cast a glamor over the multiplication table," as an admirer of Macaulay
has said that the latter could do, we must have the facilities for so
doing.
It therefore becomes necessary in weighing the merits of a textbook to
consider: (1) if the number of proved propositions is reduced to a safe
minimum; (2) if there is reasonable opportunity to apply the theory, the
actual applications coming best, however, from the teacher as an outside
interest; (3) if there is an abundance of material in the way of simple
exercises, since such material is not so readily given by the teacher as
the seemingly local applications of the propositions to outdoor
measurements; (4) if the book gives a reasonable amount of introductory
work in the use of simple and inexpensive instruments, not at that time
emphasizing the formal side of the subject; (5) if there is afforded
some opportunity to see the recreative side of the subject, and to know
a little of the story of geometry as it has developed from ancient to
modern times.
But this does not mean that there is to be a geometric cataclysm. It
means that we must have the same safe, conservative evolution in
geometry that we have in other subjects. Geometry is not going to
degenerate into mere measuring, nor is the ancient sequence going to
become a mere hodge-podge without system and with no incentive to
strenuous effort. It is now about fifteen hundred years since Proclus
laid down what he considered the essential features of a good textbook,
and in all of our efforts at reform we cannot improve very much upon his
statement. "It is essential," he says, "that such a treatise should be
rid of everything superfluous, for the superfluous is an obstacle to the
acquisition of knowledge; it should select everything that embraces the
subject and brings it to a focus, for this is of the highest service to
science; it must have great regard both to clearness and to conciseness,
for their opposites trouble our understanding; it must aim to generalize
its theorems, for the division of knowledge into small elements renders
it difficult of comprehension."
It being prefaced that we must make the book more concrete in its
applications, either directly or by suggesting seemingly practical
outdoor work; that we must increase the number of simple exercises
calling for original work; that we must reasonably reduce the number of
proved propositions; and that we must not allow the good of the ancient
geometry to depart, let us consider in detail some of the features of a
good, practical, common-sense textbook.
The early textbooks in geometry contained only the propositions, with
the proofs in full, preceded by lists of definitions and assumptions
(axioms and postulates). There were no exercises, and the proofs were
given in essay form. Then came treatises with exercises, these exercises
being grouped at the end of the work or at the close of the respective
books. The next step was to the unit page, arranged in steps to aid the
eye, one proposition to a page whenever this was possible. Some effort
was made in this direction in France about two hundred years ago, but
with no success. The arrangement has so much to commend it, however, the
proof being so much more easily followed by the eye than was the case in
the old-style works, that it has of late been revived. In this respect
the Wentworth geometry was a pioneer in America, and so successful was
the effort that this type of page has been adopted, as far as the
various writers were able to adopt it, in all successful geometries that
have appeared of late years in this country. As a result, the American
textbooks on this subject are more helpful and pleasing to the eye than
those found elsewhere.
The latest improvements in textbook-making have removed most of the
blemishes of arrangement that remained, scattering the exercises through
the book, grading them with greater care, and making them more modern in
character. But the best of the latest works do more than this. They
reduce the number of proved theorems and increase the number of
exercises, and they simplify the proofs whenever possible and eliminate
the most difficult of the exercises of twenty-five years ago. It would
be possible to carry this change too far by putting in only half as
many, or a quarter as many, regular propositions, but it should not be
the object to see how the work can be cut down, but to see how it can be
improved.
What should be the basis of selection of propositions and exercises?
Evidently the selection must include the great basal propositions that
are needed in mensuration and in later mathematics, together with others
that are necessary to prove them. Euclid's one hundred seventy-three
propositions of plane geometry were really upwards of one hundred
eighty, because he several times combined two or more in one. These we
may reduce to about one hundred thirty with perfect safety, or less than
one a day for a school year, but to reduce still further is undesirable
as well as unnecessary. It would not be difficult to dispense with a few
more; indeed, we might dispense with thirty more if we should set about
it, although we must never forget that a goodly number in addition to
those needed for the logical sequence are necessary for the wide range
of exercises that are offered. But let it be clear that if we teach 100
instead of 130, our results are liable to be about 100/130 as
satisfactory. We may theorize on pedagogy as we please, but geometry
will pay us about in proportion to what we give.
And as to the exercises, what is the basis of selection? In general, let
it be said that any exercise that pretends to be real should be so, and
that words taken from science or measurements do not necessarily make
the problem genuine. To take a proposition and apply it in a manner that
the world never sanctions is to indulge in deceit. On the other hand,
wholly to neglect the common applications of geometry to handwork of
various kinds is to miss one of our great opportunities to make the
subject vital to the pupil, to arouse new interest, and to give a
meaning to it that is otherwise wanting. It should always be remembered
that mental discipline, whatever the phrase may mean, can as readily be
obtained from a genuine application of a theorem as from a mere
geometric puzzle. On the other hand, it is evident that not more than 25
per cent of propositions have any genuine applications outside of
geometry, and that if we are to attempt any applications at all, these
must be sought mainly in the field of pure geometry. In the exercises,
therefore, we seek to-day a sane and a balanced book, giving equal
weight to theory and to practice, to the demands of the artisan and to
those of the mathematician, to the applications of concrete science and
to those of pure geometry, thus making a fusion of pure and applied
mathematics, with the latter as prominent as the supply of genuine
problems permits. The old is not all bad and the new is not all good,
and a textbook is a success in so far as it selects boldly the good that
is in the old and rejects with equal boldness the bad that is in the
new.
Lest the nature of the exercises of geometry may be misunderstood, it is
well that we consider for a moment what constitutes a genuine
application of the subject. It is the ephemeral fashion just at present
in America to call these genuine applications by the name of "real
problems." The name is an unfortunate importation, but that is not a
matter of serious moment. The important thing is that we should know
what makes a problem "real" to the pupil of geometry, especially as the
whole thing is coming rapidly into disrepute through the mistaken zeal
of some of its supporters.
A real problem is a problem that the average citizen may sometime be
called upon to solve; that, if so called upon, he will solve in the
manner indicated; and that is expressed in terms that are familiar to
the pupil.
This definition, which seems fairly to state the conditions under which
a problem can be called "real" in the schoolroom, involves three points:
(1) people must be liable to meet such a problem; (2) in that case they
will solve it in the way suggested by the book; (3) it must be clothed
in language familiar to the pupil. For example, let the problem be to
find the dimensions of a rectangular field, the data being the area of
the field and the area of a road four rods wide that is cut from three
sides of the field. As a real problem this is ridiculous, since no one
would ever meet such a case outside the puzzle department of a
schoolroom. Again, if by any stretch of a vigorous imagination any human
being should care to find the area of a piece of glass, bounded by the
arcs of circles, in a Gothic window in York Minster, it is fairly
certain that he would not go about it in the way suggested in some of
the earnest attempts that have been made by several successful teachers
to add interest to geometry. And for the third point, a problem is not
real to a pupil simply because it relates to moments of inertia or the
tensile strength of a steel bar. Indeed, it is unreal precisely because
it does talk of these things at a time when they are unfamiliar, and
properly so, to the pupil.
It must not be thought that puzzle problems, and unreal problems
generally, have no value. All that is insisted upon is that such
problems as the above are not "real," and that about 90 per cent of
problems that go by this name are equally lacking in the elements that
make for reality in this sense of the word. For the other 10 per cent of
such problems we should be thankful, and we should endeavor to add to
the number. As for the great mass, however, they are no better than
those that have stood the test of generations, and by their pretense
they are distinctly worse.
It is proper, however, to consider whether a teacher is not justified in
relating his work to those geometric forms that are found in art, let us
say in floor patterns, in domes of buildings, in oilcloth designs, and
the like, for the purpose of arousing interest, if for no other reason.
The answer is apparent to any teacher: It is certainly justifiable to
arouse the pupil's interest in his subject, and to call his attention to
the fact that geometric design plays an important part in art; but we
must see to it that our efforts accomplish this purpose. To make a
course in geometry one on oilcloth design would be absurd, and nothing
more unprofitable or depressing could be imagined in connection with
this subject. Of course no one would advocate such an extreme, but it
sometimes seems as if we are getting painfully near it in certain
schools.
A pupil has a passing interest in geometric design. He should learn to
use the instruments of geometry, and he learns this most easily by
drawing a few such patterns. But to keep him week after week on
questions relating to such designs of however great variety, and
especially to keep him upon designs relating to only one or two types,
is neither sound educational policy nor even common sense. That this
enthusiastic teacher or that one succeeds by such a plan is of no
significance; it is the enthusiasm that succeeds, not the plan.
The experience of the world is that pupils of geometry like to use the
subject practically, but that they are more interested in the pure
theory than in any fictitious applications, and this is why pure
geometry has endured, while the great mass of applied geometry that was
brought forward some three hundred years ago has long since been
forgotten. The question of the real applications of the subject is
considered in subsequent chapters.
In Chapter VI we considered the question of the number of regular
propositions to be expected in the text, and we have just considered the
nature of the exercises which should follow those propositions. It is
well to turn our attention next to the nature of the proofs of the basal
theorems. Shall they appear in full? Shall they be merely suggested
demonstrations? Shall they be only a series of questions that lead to
the proof? Shall the proofs be omitted entirely? Or shall there be some
combination of these plans?
The natural temptation in the nervous atmosphere of America is to listen
to the voice of the mob and to proceed at once to lynch Euclid and every
one who stands for that for which the "Elements" has stood these two
thousand years. This is what some who wish to be considered as educators
tend to do; in the language of the mob, to "smash things"; to call
reactionary that which does not conform to their ephemeral views. It is
so easy to be an iconoclast, to think that _cui bono_ is a conclusive
argument, to say so glibly that Raphael was not a great painter,--to do
anything but construct. A few years ago every one must take up with the
heuristic method developed in Germany half a century back and containing
much that was commendable. A little later one who did not believe that
the Culture Epoch Theory was vital in education was looked upon with
pity by a considerable number of serious educators. A little later the
man who did not think that the principle of Concentration in education
was a _regula aurea_ was thought to be hopeless. A little later it may
have been that Correlation was the saving factor, to be looked upon in
geometry teaching as a guiding beacon, even as the fusion of all
mathematics is the temporary view of a few enthusiasts to-day.[35]
And just now it is vocational training that is the catch phrase, and to
many this phrase seems to sound the funeral knell of the standard
textbook in geometry. But does it do so? Does this present cry of the
pedagogical circle really mean that we are no longer to have geometry
for geometry's sake? Does it mean that a panacea has been found for the
ills of memorizing without understanding a proof in the class of a
teacher who is so inefficient as to allow this kind of work to go on?
Does it mean that a teacher who does not see the human side of
geometry, who does not know the real uses of geometry, and who has no
faculty of making pupils enthusiastic over geometry,--that this teacher
is to succeed with some scrappy, weak, pretending apology for a real
work on the subject?
No one believes in stupid teaching, in memorizing a textbook, in having
a book that does all the work for a pupil, or in any of the other ills
of inefficient instruction. On the other hand, no fair-minded person can
condemn a type of book that has stood for generations until something
besides the mere transient experiments of the moment has been suggested
to replace it. Let us, for example, consider the question of having the
basal propositions proved in full, a feature that is so easy to condemn
as leading to memorizing.
The argument in favor of a book with every basal proposition proved in
full, or with most of them so proved, the rest having only suggestions
for the proof, is that the pupil has before him standard forms
exhibiting the best, most succinct, most clearly stated demonstrations
that geometry contains. The demonstrations stand for the same thing that
the type problems stand for in algebra, and are generally given in full
in the same way. The argument against the plan is that it takes away the
pupil's originality by doing all the work for him, allowing him to
merely memorize the work. Now if all there is to geometry were in the
basal propositions, this argument might hold, just as it would hold in
algebra in case there were only those exercises that are solved in full.
But just as this is not the case in algebra, the solved exercises
standing as types or as bases for the pupil's real work, so the
demonstrated proposition forms a relatively small part of geometry,
standing as a type, a basis for the more important part of the work.
Moreover, a pupil who uses a syllabus is exposed to a danger that should
be considered, namely, that of dishonesty. Any textbook in geometry will
furnish the proofs of most of the propositions in a syllabus, whatever
changes there may be in the sequence, and it is not a healthy condition
of mind that is induced by getting the proofs surreptitiously. Unless a
teacher has more time for the course than is usually allowed, he cannot
develop the new work as much as is necessary with only a syllabus, and
the result is that a pupil gets more of his work from other books and
has less time for exercises. The question therefore comes to this: Is it
better to use a book containing standard forms of proof for the basal
propositions, and have time for solving a large number of original
exercises and for seeking the applications of geometry? Or is it better
to use a book that requires more time on the basal propositions, with
the danger of dishonesty, and allows less time for solving originals? To
these questions the great majority of teachers answer in favor of the
textbook with most of the basal propositions fully demonstrated. In
general, therefore, it is a good rule to use the proofs of the basal
propositions as models, and to get the original work from the exercises.
Unless we preserve these model proofs, or unless we supply them with a
syllabus, the habit of correct, succinct self-expression, which is one
of the chief assets of geometry, will tend to become atrophied. So
important is this habit that "no system of education in which its
performance is neglected can hope or profess to evolve men and women who
are competent in the full sense of the word. So long as teachers of
geometry neglect the possibilities of the subject in this respect, so
long will the time devoted to it be in large part wasted, and so long
will their pupils continue to imbibe the vicious idea that it is much
more important to be able to do a thing than to say how it can be
done."[36]
It is here that the chief danger of syllabus-teaching lies, and it is
because of this patent fact that a syllabus without a carefully selected
set of model proofs, or without the unnecessary expenditure of time by
the class, is a dangerous kind of textbook.
What shall then be said of those books that merely suggest the proofs,
or that give a series of questions that lead to the demonstrations?
There is a certain plausibility about such a plan at first sight. But it
is easily seen to have only a fictitious claim to educational value. In
the first place, it is merely an attempt on the part of the book to take
the place of the teacher and to "develop" every lesson by the heuristic
method. The questions are so framed as to admit, in most cases, of only
a single answer, so that this answer might just as well be given instead
of the question. The pupil has therefore a proof requiring no more
effort than is the case in the standard form of textbook, but not given
in the clear language of a careful writer. Furthermore, the pupil is
losing here, as when he uses only a syllabus, one of the very things
that he should be acquiring, namely, the habit of reading mathematics.
If he met only syllabi without proofs, or "suggestive" geometries, or
books that endeavored to question every proof out of him, he would be in
a sorry plight when he tried to read higher mathematics, or even other
elementary treatises. It is for reasons such as these that the heuristic
textbook has never succeeded for any great length of time or in any wide
territory.
And finally, upon this point, shall the demonstrations be omitted
entirely, leaving only the list of propositions,--in other words, a pure
syllabus? This has been sufficiently answered above. But there is a
modification of the pure syllabus that has much to commend itself to
teachers of exceptional strength and with more confidence in themselves
than is usually found. This is an arrangement that begins like the
ordinary textbook and, after the pupil has acquired the form of proof,
gradually merges into a syllabus, so that there is no temptation to go
surreptitiously to other books for help. Such a book, if worked out with
skill, would appeal to an enthusiastic teacher, and would accomplish the
results claimed for the cruder forms of manual already described. It
would not be in general as safe a book as the standard form, but with
the right teacher it would bring good results.
In conclusion, there are two types of textbook that have any hope of
success. The first is the one with all or a large part of the basal
propositions demonstrated in full, and with these propositions not
unduly reduced in number. Such a book should give a large number of
simple exercises scattered through the work, with a relatively small
number of difficult ones. It should be modern in its spirit, with
figures systematically lettered, with each page a unit as far as
possible, and with every proof a model of clearness of statement and
neatness of form. Above all, it should not yield to the demand of a few
who are always looking merely for something to change, nor should it in
a reactionary spirit return to the old essay form of proof, which
hinders the pupil at this stage.
The second type is the semisyllabus, otherwise with all the spirit of
the first type. In both there should be an honest fusion of pure and
applied geometry, with no exercises that pretend to be practical
without being so, with no forced applications that lead the pupil to
measure things in a way that would appeal to no practical man, with no
merely narrow range of applications, and with no array of difficult
terms from physics and engineering that submerge all thought of
mathematics in the slough of despond of an unknown technical vocabulary.
Outdoor exercises, even if somewhat primitive, may be introduced, but it
should be perfectly understood that such exercises are given for the
purpose of increasing the interest in geometry, and they should be
abandoned if they fail of this purpose.
=Bibliography.= For a list of standard textbooks issued prior
to the present generation, consult the bibliography in Stamper,
History of the Teaching of Geometry, New York, 1908.
FOOTNOTES:
[35] For some classes of schools and under certain circumstances courses
in combined mathematics are very desirable. All that is here insisted
upon is that any general fusion all along the line would result in weak,
insipid, and uninteresting mathematics. A beginning, inspirational
course in combined mathematics has a good reason for being in many high
schools in spite of its manifest disadvantages, and such a course may be
developed to cover all of the required mathematics given in certain
schools.
[36] Carson, loc. cit., p. 15.