THE CONDUCT OF A CLASS IN GEOMETRY
No definite rules can be given for the detailed conduct of a class in
any subject. If it were possible to formulate such rules, all the
personal magnetism of the teacher, all the enthusiasm, all the
originality, all the spirit of the class, would depart, and we should
have a dull, dry mechanism. There is no one best method of teaching
geometry or anything else. The experience of the schools has shown that
a few great principles stand out as generally accepted, but as to the
carrying out of these principles there can be no definite rules.
Let us first consider the general question of the employment of time in
a recitation in geometry. We might all agree on certain general
principles, and yet no two teachers ever would or ever should divide the
period even approximately in the same way. First, a class should have an
opportunity to ask questions. A teacher here shows his power at its
best, listening sympathetically to any good question, quickly seeing the
essential point, and either answering it or restating it in such a way
that the pupil can answer it for himself. Certain questions should be
answered by the teacher; he is there for that purpose. Others can at
once be put in such a light that the pupil can himself answer them.
Others may better be answered by the class. Occasionally, but more
rarely, a pupil may be told to "look that up for to-morrow," a plan
that is commonly considered by students as a confession of weakness on
the part of the teacher, as it probably is. Of course a class will waste
time in questioning a weak teacher, but a strong one need have no fear
on this account. Five minutes given at the opening of a recitation to
brisk, pointed questions by the class, with the same credit given to a
good question as to a good answer, will do a great deal to create a
spirit of comradeship, of frankness, and of honesty, and will reveal to
a sympathetic teacher the difficulties of a class much better than the
same amount of time devoted to blackboard work. But there must be no
dawdling, and the class must feel that it has only a limited time, say
five minutes at the most, to get the help it needs.
Next in order of the division of the time may be the teacher's report on
any papers that the class has handed in. It is impossible to tell how
much of this paper work should be demanded. The local school conditions,
the mental condition of the class, and the time at the disposal of the
teacher are all factors in the case. In general, it may be said that
enough of this kind of work is necessary to see that pupils are neat and
accurate in setting down their demonstrations. On the other hand, paper
work gives an opportunity for dishonesty, and it consumes a great deal
of the teacher's time that might better be given to reading good books
on the subject that he is teaching. If, however, any papers have been
submitted, about five minutes may well be given to a rapid review of the
failures and the successes. In general, it is good educational policy to
speak of the errors and failures impersonally, but occasionally to
mention by name any one who has done a piece of work that is worthy of
special comment. Pupils may better be praised in public and blamed in
private. There is such a thing, however, as praising too much, when
nothing worthy of note has been done, just as there is danger of blaming
too much, resulting in mere "nagging."
The third division of the recitation period may profitably go to
assigning the advance lesson. The class questions and the teacher's
report on written work have shown the mental status of the pupils, so
that the teacher now knows what he may expect for the next lesson. If he
assigns his lesson at the beginning of the period, he does not have this
information. If he waits to the end, he may be too hurried to give any
"development" that the new lesson may require. There can be no rule as
to how to assign a new lesson; it all depends upon what the lesson is,
upon the mental state of the class, and not a little upon the
idiosyncrasy of the teacher. The German educator, Herbart, laid down
certain formal steps in developing a new lesson, and his successors have
elaborated these somewhat as follows:
1. _Aim._ Always take a class into your confidence. Tell the members at
the outset the goal. No one likes to be led blindfolded.
2. _Preparation._ A few brief questions to bring the class to think of
what is to be considered.
3. _Presentation of the new._ Preferably this is done by questions, the
answers leading the members of the class to discover the new truth for
themselves.
4. _Apperception._ Calling attention to the fact that this new fact was
known before, in part, and that it relates to a number of things already
in the mind. The more the new can be tied up to the old the more
tenaciously it will be held.
5. _Generalization and application._
It is evident at once that a great deal of time may be wasted in always
following such a plan, perhaps in ever following it consciously. But, on
the other hand, probably every good teacher, whether he has heard of
Herbart or not, naturally covers these points in substantially this
order. For an inexperienced teacher it is helpful to be familiar with
them, that he may call to mind the steps, arranged in a psychological
sequence, that he would do well to follow. It must always be remembered
that there is quite as much danger in "developing" too much as in taking
the opposite extreme. A mechanical teacher may develop a new lesson
where there is need for only a question or two or a mere suggestion. It
should also be recognized that students need to learn to read
mathematics for themselves, and that always to take away every
difficulty by explanations given in advance is weakening to any one.
Therefore, in assigning the new lesson we may say "Take the next two
pages," and thus discourage most of the class. On the other hand, we may
spend an unnecessary amount of time and overdevelop the work of those
same pages, and have the whole lesson lose all its zest. It is here that
the genius of the teacher comes forth to find the happy mean.
The fourth division of the hour should be reached, in general, in about
ten minutes. This includes the recitation proper. But as to the nature
of this work no definite instructions should be attempted. To a good
teacher they would be unnecessary, to a poor one they would be harmful.
Part of the class may go to the board, and as they are working, the rest
may be reciting. Those at the board should be limited as to time, for
otherwise a premium is placed on mere dawdling. They should be so
arranged as to prevent copying, but the teacher's eye is the best
preventive of this annoying feature. Those at their seats may be called
upon one at a time to demonstrate at the blackboard, the rest being
called upon for quick responses, as occasion demands. The European plan
of having small blackboards is in many respects better than ours, since
pupils cannot so easily waste time. They have to work rapidly and talk
rapidly, or else take their seats.
What should be put on the board, whether the figure alone, or the figure
and the proof, depends upon the proposition. In general, there should be
a certain number of figures put on the board for the sake of rapid work
and as a basis for the proofs of the day. There should also be a certain
amount of written work for the sake of commending or of criticizing
adversely the proof used. There are some figures that are so complicated
as to warrant being put upon sheets of paper and hung before the class.
Thus there is no rule upon the subject, and the teacher must use his
judgment according to the circumstances and the propositions.
If the early "originals" are one-step exercises, and a pupil is required
to recite rapidly, a habit of quick expression is easily acquired that
leads to close attention on the part of all the class. Students as a
rule recite slower than they need to, from mere habit. Phlegmatic as we
think the German is, and nervous as is the American temperament, a
student in geometry in a German school will usually recite more quickly
and with more vigor than one with us. Our extensive blackboards have
something to do with this, allowing so many pupils to be working at the
board that a teacher cannot attend to them all. The result is a habit of
wasting the minutes that can only be overcome by the teacher setting a
definite but reasonable time limit, and holding the pupil responsible if
the work is not done in the time specified. If this matter is taken in
hand the first day, and special effort made in the early weeks of the
year, much of the difficulty can be overcome.
As to the nature of the recitation to be expected from the pupil, no
definite rule can be laid down, since it varies so much with the work of
the day. In general, however, a pupil should state the theorem quickly,
state exactly what is given and what is to be proved, with respect to
the figure, and then give the proof. At first it is desirable that he
should give the authorities in full, and later give only the essential
part in a few words. It is better to avoid the expression "by previous
proposition," for it soon comes to be abused, and of course the learning
of section numbers in a book is a barbarism. It is only by continually
stating the propositions used that a pupil comes to have well fixed in
his memory the basal theorems of geometry, and without these he cannot
make progress in his subsequent mathematics. In general, it is better to
allow a pupil to finish his proof before asking him any questions, the
constant interruptions indulged in by some teachers being the cause of
no little confusion and hesitancy on the part of pupils. Sometimes it is
well to have a figure drawn differently from the one in the book, or
lettered differently, so as to make sure that the pupil has not
memorized the proof, but in general such devices are unnecessary, for a
teacher can easily discover whether the proof is thoroughly understood,
either by the manner of the pupil or by some slight questioning. A good
textbook has the figures systematically lettered in some helpful way
that is easily followed by the class that is listening to the
recitation, and it is not advisable to abandon this for a random set of
letters arranged in no proper order.
It is good educational policy for the teacher to commend at least as
often as he finds fault when criticizing a recitation at the blackboard
and when discussing the pupils' papers. Optimism, encouragement,
sympathy, the genuine desire to help, the putting of one's self in the
pupil's place, the doing to the pupil as the teacher would that he
should do in return,--these are educational policies that make for
better geometry as they make for better life.
The prime failure in teaching geometry lies unquestionably in the lack
of interest on the part of the pupil, and this has been brought about by
the ancient plan of simply reading and memorizing proofs. It is to get
away from this that teachers resort to some such development of the
lesson in advance, as has been suggested above. It is usually a good
plan to give the easier propositions as exercises before they are
reached in the text, where this can be done. An English writer has
recently contributed this further idea:
It might be more stimulating to encourage investigation than to
demand proofs of stated facts; that is to say, "Here is a
figure drawn in this way, find out anything you can about it."
Some such exercises having been performed jointly by teachers
and pupils, the lust of investigation and healthy competition
which is present in every normal boy or girl might be awakened
so far as to make such little researches really attractive;
moreover, the training thus given is of far more value than
that obtained by proving facts which are stated in advance, for
it is seldom, if ever, that the problems of adult life present
themselves in this manner. The spirit of the question, "What is
true?" is positive and constructive, but that involved in "Is
this true?" is negative and destructive.[41]
When the question is asked, "How shall I teach?" or "What is the
Method?" there is no answer such as the questioner expects. A Japanese
writer, Motowori, a great authority upon the Shinto faith of his people,
once wrote these words: "To have learned that there is no way to be
learned and practiced is really to have learned the way of the gods."
FOOTNOTES:
[41] Carson, loc. cit., p. 12.