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thing happens in a misdirected socialized recitation.
The results of the work of a capable pupil can spread
like wild-fire right under the eyes of the teacher.

There is a splendid social comradery exhibited here
and a very delightful illustration, too, of mutuality.
May this not be after all the real social education we
hear so much about in these latter days? Of course
the teacher could insist upon absolute independence
of work. But the real problem lies deeper. Does a
suggestion to a pupil in difficulty destroy that inde-
pendence insisted upon? Where is the line to be


drawn ? Is not the social principle after all the clutch
which throws the individual into action ? If the class-
room is organized under the instructional ideal with
an insistence upon regimental uniformity, it would
appear that this class in algebra is to be commended
in its resourcefulness in the use of the social principle.

May it not be, also, that one of the primary func-
tions of the public school is to keep the home-fires
burning educationally, so to speak? The good widow,
mother of seven and wage-earner, should be given a
hearing at this point. She complained to the super-
intendent, saying that after the hard day's work and
after the evening work at home she was finding her
educational job rather trying. She said it was dif-
ficult to teach her seven children all the lessons as-
signed them in school by their teachers, now that some
of them had reached the high school. Her proposition
to the superintendent was that if it was agreeable to
him she would be glad to hear her kiddies recite the
lessons if the teachers would teach them in the school-
rooms. This shift of emphasis might work.

It would be a distinct loss if the student failed to
keep the professor educated. On the whole, the re-
sponsibility placed upon the home by our school prac-
tices is good for the home. It serves to keep alive an
interest in education. Parents find it less of a burden
to teach their children or to assist them or to com-
mand them to study their lessons at home than to de-
vise ways of taking care of any marginal free time.
It may be a bit unfair and too severe criticism to in-
sist that the modern home has abrogated its author-
ity. At all events, the home is quite willing that the
school should be exacting enough of the pupils in re-


spect of home study to keep them at some kind of
work during those hours of the day when the respon-
sibility of parents for direct methods of educating their
children would prove a real task. One criticism against
so-called supervised study, mechanically conducted,
is the attempt to delete home study. The good widow
has suggested a far wiser solution. At all events, our
conception of directing activity as the major work of
the teacher will in no sense do away with wholesome
forms of home work for pupils. Parents will still have
an opportunity to participate vitally in the educative
process within the procedure proposed under directing

Directing Pupils in Work. In sharp contrast to the
general practice of assigning a set uniform lesson for
out-of-class preparation and subsequent recitation upon
it, let us study a few situations in which pupils carry
on their work under the immediate direction of the

A. (a) A class of thirty-seven pupils in geometry (loth
grade) began the attack upon some twenty-five original exer-
cises running up into a half-dozen or more rather difficult sup-
plementary exercises. In all, three days were given to this chal-
lenge. During the second and third days the procedure indicated
below was used. The class period was seventy minutes net.
Many pupils were on the job twenty minutes before the class
period formally began.

The pupils were directed to work as rapidly as possible and
to come to the teacher for consultation when they felt sure they
could go no further in their particular exercises. Their work
out of class was a continuation of work begun in class. The
pupil reported with his work (the case method), indicated his
method of work, and pointed out his difficulty if unable to go

The teacher used a pad to jot down just what he said to each


pupil or group of pupils. Each one was on his mark; only such
groupings were formed as were suggested by the teacher during
the procedure for these three days. Occasionally two or three
pupils were directed to go to the board and discuss quietly their

The amount of work, that is, the number of exercises mas-
tered ranged from one to ten or more each day. The circle, in
other words, was described; each pupil was free within it; no
upper limit was set for any one. Some pupils spent the whole
class period on some very difficult exercise (for them) at that

These notes are transcribed from the teacher's pad. They
indicate just what he said to pupils during the last two days of
the challenge. The number of the exercise was noted and the
suggestion or hint or question is recorded. In parentheses, now
and again, the nature of the pupil's difficulty is indicated. The
pupil described his dilemma. The teacher observed the injunc-
tion of not talking too much. The pupil upon the suggestion
went to his seat or to the board and in all these cases below
succeeded in demonstrating his exercises.

Ruth. "Try to use supplementary angles."
Margaret. (A defective figure.) "Draw your figure

with your instruments."
Oscar. "Talk to your figure." *
Tom. "How did you draw line AB?"
Franklin. "Can you see an hypotenuse in your


M. and C. "Work on the size of angles (in de-
H. "Keep one finger on page 62." (A page of

summary directions.)

/., A., and Fr. (Working in a group. Heated de-
bate. Fr. presented one solution of an exercise,
/. and A. another. /. and A. were pointing out

* One pupil developed the habit of drawing a figure and then talk-
ing to it as if it were a kind of personified thing.


the error in Fr.'s reasoning.) "Soft pedal it over

there." *

Halvor. "Can you make any use of exterior angle ? "
Wm. "Supplements?"
A. "Review exercise 120 and try to use it."

Whole Class. (Five minutes.) "Here is an alge-
braic way of working certain situations you will all need
to employ now and again." Explanation and drill.

If (i) a equals c and (2) b equals d and if (3) a plus c
equals b plus d. Then a plus a equals b plus b (by sub-
stituting for c its equal o, etc.).

Then 20, equals 26.

Then a equals b.

Similarly c may be proved equal to d.

"Now apply the principles of this solution to your

L. (Confused as to hypothesis.) "Read your exer-
cise and trace it in your figure with your finger as
you read it."

S. "Apply axiom I to your congruent triangles."

* "Responding to influences from without, life is an unfolding process
from within. This is the conception that is now shaping our methods
of instruction. The old recognized as training and discipline the so-
called voluntary attention which seemed to be mainly the ability to
stare, ox-like, a disagreeable, uninteresting, or unintelligible thing out
of countenance. The new believes in training and discipline that come
from the pupil's effort to follow up from premise to conclusion some-
thing which mightily interests him because of its worthy purpose. The
new values attainment only as it represents a quality of mind that has
acted through its own initiative. The old found satisfaction in a state
of mind that was quietly receptive; the new sees hope in turbulence of
inquiry; and all of these are irreconcilable differences in kind." (Jack-


Lo. "Surely, any side of triangle may be your

T. "Makes with the base an angle? Read it and
trace it in your figure. Dwell on it."

O and H. "Use another fact stated in your hy-
pothesis. Examine all the data given. Plan a
general way of attacking it." *

U. "Tut, tut! You used your conclusion in your
demonstration." (Oh, she says. A very com-
mon expletive in this procedure.)

A. "Try to think exercise 128 and 131 together."
(One is converse of the other.)

M. (Difficulty in seeing related parts in overlap-
ping figures.) "Separate the triangles. Draw
them out aside and look at them."

M. "Try drawing bisector of angle. Go back to
exercise 119."

H. and O. "What did we work out together yester-
day?" (The algebraic way of getting quantities
equal.) "Apply it here."

H. "Keep one eye on page 59. Something on that
page for you. Select two triangles in your figure.
You may draw construction lines, you know."

R. " Where is MN ? I don't see it. Be sure to get
all of the facts in your hypothesis. Read it care-
fully." (Oh, I see.)

K. "What kind of a triangle have you?"

C and F. "Select at once triangles which include

* "It (reasoning) is made easier (i) by systematizing the search; (2)
by limiting the number of classes amongst which the pupil must search
for the right one; (3) by informing him of classes which include the right
one and which he would neglect if undirected; and (4) by calling his
attention to the consequences of membership in this or that class."
(Thorndike, Principles of Teaching, p. 163.)


any parts of your conclusion." (I spent two hours

on this one. I have it.) "Fine."
7. "How many sides has a triangle?"
G. "Rub it out, line HF, and try to use exterior

H. "Why are those lines parallel? Go back to

page 62." (Summary.)
R. "All right so far. Now show what the nature

of angles a and c is."
Fr. "Read your angle there again and point to it

as you read it." (Oh!)
J., M., and /. "Try to apply this principle: a equals

c; b equals d. Then a plus b equals c plus d. Do

you see it now?" (Oh, yes.)
G. "Turn to page 59. There is something there

you can use."
F. "Go to board and draw with instruments the

kind of triangle you want here. Do it quite ac-
R. "Suppose you abandon trying to prove figures

congruent. What are your alternatives now?

Correct. Now which one can you use?"

These are typical hints, helps, questions, direc-
tions, etc.

During the third day in this challenge of twenty-five orig-
inal exercises, Tom and Arnold completed the entire list early
in the class period. They then assisted the teacher, taking down
on a pad just what they said to a pupil at the point of his dif-
ficulty. They did it very well indeed and said they enjoyed
it thoroughly.

Here are a few of their notes on what they actually


H. "Keep your finger on page 62. Try to construct
a line parallel to CF and see what happens."

L. "Look up different ways of finding when a tri-
angle is isosceles."

N. "What are the ways of finding quantities

G. "What do you know about the bisector of the
vertex angle of some kinds of triangles?"

K. "What do you know about a perpendicular
drawn to a line?"

P. "How do you construct a perpendicular to a
line ? " " What is the hypothesis in any theorem ? "
"Do you know what an isosceles triangle is?"

H. "How do you prove two segments equal?"
"Why is BMN a right angle?" "Why is CM
parallel to AB?" "What do you know about
the bisector of an exterior angle of an equilateral

Bear in mind these pupils, Tom and Arnold, were do-
ing this superb work in directing activity in a real
challenge. They can teach all of us a thing or
two. Note the simplicity of their suggestions.
Potential Toms and Arnolds may be realized in
every class. These boys did their assisting with a
quiet dignity. Such work may be made a privi-

* i. In considering such qualities as self -direction, initiative, and
originality, attention is directed to a positive and dynamic meaning of
these traits, such as Thorndike so effectively describes in Teachers' Col-
lege Record 17, p. 405 Jf., 1916. "The view is to think of independence,
not as unreadiness to follow or obey or believe in other men, but as a
readiness and ability to contribute to good causes something more than
is suggested by others; to think of initiative, not as an unreadiness to
wait or co-operate or be modest, but as a readiness and ability to move
ahead, 'speed up,' lead and take promising risks, and as an attitude of


(b) The same class in geometry as in (a) above.

The following exercise was begun in class by all the
pupils with the expectation that each one would have
a chance to do his own thinking. Habits of work were
examined in so far as it was possible to do so. The
teacher sought to discover the particular difficulty
each pupil encountered and to check the work as
rapidly as it was done. The following notes upon each
pupil were gathered in about twenty minutes. An
attempt is made here to record some points about the
work of each pupil.

Exercise. (All on your marks now !)

"// two opposite angles of a quadrilateral are equal, and if the
diagonal joining the other two angles bisects one of them, then it
bisects the other."

The pupils were directed to begin this new exercise at once,
work as rapidly as possible, and come to the teacher at the point
of difficulty. The teacher was active in discovering what the
particular difficulty was, and his procedure was to make only a
suggestion, ask a question, give a hint.

i. William's first difficulty was in knowing what is meant

expecting to create opportunities, and do ten dollars' worth of work
for a dollar. Originality must not mean weakness in doing routine work
in old ways, or any essential dislike of traditional knowledge or customs
as such or any paucity of fixed habits but strength in doing work that
is new or doing it in new ways, an attitude of hoping to change knowl-
edge or practice for the better, an organization of habits that causes
their progressive modification. . . . The dynamic opposite of original-
ity is not efficiency, but stupidity. The dynamic opposite of efficient
routine is not genius, but disorder. . . . Finally, will it not clear the
whole argument somewhat if, in our own thinking about education, we
replace the word 'self-reliance' by reliance on facts ; 'self-direction' by
rational direction; 'initiative' by readiness and ability to begin to think
and experiment; 'independence' by readiness to carry thought or experi-
ment on to its just conclusions despite traditions and customs and lack of


by opposite angles. He stumbled on the distinction between
successive and opposite angles.

2. Margaret was on the whole the best thinker in the class.
She had written four perfect "examinations" in the first eight
weeks of her geometry, and her daily work was invariably ex-
cellent. Her difficulty in this exercise was in the antecedent of
them. She carried it back to the first dependent clause. When
the teacher asked her to relate her pronoun to some other pos-
sible antecedent, she found the solution perfectly easy.

3. Frances, a very good pupil, read the exercise and began
the demonstration by drawing an equilateral triangle. "Quadri-
lateral" was translated "equilateral." When she discovered
her initial error she sailed on without difficulty. She probably
discovered her error in trying to draw diagonals of an equilateral
triangle. Was her difficulty a failure to read? Hardly. She
caught lateral in quadrilateral by the tail of her eye and did
what every one who really reads does: she filled in meaning
out of her head. Did she think? The fact is she perhaps did
think too much.

4 and 5. Kenneth read the exercise four times and remained
wholly innocent of the meaning of it all. When prodded to
draw a figure which seemed to be suggested by these words of
telegraphic brevity he got under way. Jim had to have four or
five additional social starters before getting to the point of un-
derstanding what it was a'l about. By that time more than
half the class had made a complete demonstration of the exer-

6. Henry met his Waterloo on the word diagonal.

7. Mary wrestled with the two dependent clauses, and as
soon as it dawned upon her that each one gave her the basis for
a statement in her hypothesis in terms of her figure, the rest
of it was very quickly done.

8. Mamie read it and represented it in a figure as she read it
and solved it without hesitation.

9. 10, n, 12, and 13 indicated experience similar to that of
Mamie. ,

14. Oscar, a very cautious thinker, grew a bit timid in at-
tacking the triangles formed by the diagonal. He was perfectly
clear in his intellectual method at the point of hesitation. He
had in mind two alternatives. He wanted assurance in his next


step. When told that either alternative would bring him safely
to a correct conclusion, he became confident of his ability to go

15. Loraine made a false application of one of the ways of
proving triangles congruent. A hint, and she corrected her

16. Lorna slipped on the meaning of an included angle.

17. Tom had missed a corollary on account of absence. He
was directed to turn to it and master it then and there. He did
so and made use of it within the first ten minutes of work on the

18. Arnold did the work quite accurately, but he made of it
a very long proof. A suggestion at one point in his proof, and he
at once made a short-cut proof.

19. Pearl was dazed before the array of conditions, apparently
unable to grip the thing at any angle. She merely got started
in the time allotted for this experiment.

20. Mildred, a very dependable thinker, was not quite sure
of homologous angles in her figure. By "a stroke of the eye,"
as it were, her difficulty cleared away.

21. Helen, a rather silent partner in the procedure, responded
when asked how she was getting on, that she felt sure of her
method of attack.

22. Melvin seems perfectly happy in bearing lightly the sor-
rowful burden of human knowledge. He is content to be a mir-
ror mind, carelessly reflecting what he picks up in haphazard
work. As soon as the solution of the exercise was presented, he
absorbed it and was prepared to give it back just as he received
it. He manifested practically no initiative, even though he read
it several times like a good little boy obedient to authority.

And so on for thirty-seven pupils in this particular

The striking fact about this type of analysis of the
habits of work of any group of pupils is lack of uni-
formity of achievement. No two pupils needed the same
treatment. It would have been absurd to call the at-
tention of the entire class to the difficulty which


Frances encountered in the word quadrilateral. Why
fuss the other members of the class with the particular
difficulty of a single pupil ?

Yet that is precisely what is done in the recitation
system. The attention of the entire class is arrested
by some unique response of the pupil called on to recite
or to relate his particular progress and difficulty.
There is a time for class discussion. Again, any ex-
tended explanation in the situations just cited could
hardly be justified.

Difficulties do not come to any class group by the clock.
Difficulties are for the most part individual. The same
individual does not respond with a high degree of uni-
formity from day to day. The teacher who has de-
veloped the experimental attitude of mind may find
some such study of habits of work as this one a profit-
able departure on many occasions in directing activity
in teaching. Such an exercise conducted in a controlled
environment enables the pupil to do independent think-
ing, or rather, let us say, rationally dependent think-
ing. May it not be a real beginning in creative or
scientific thinking?

A year after taking their geometry some pupils were
requested to make a frank statement about the gen-
eral procedure illustrated in this class. Two pupils
responded as follows, fifteen and sixteen year old
pupils. These statements are decidedly original and
first-hand. They express very clearly the procedure.
No apology is offered for including them. Pupils are
not lectured to about any particular intellectual

"I like your system of 'challenges' very much. It gives the
student responsibility and a greater opportunity for initiative.


Class work rather than recitation makes individual help possible;
for instance, student A may understand certain principles very
well, while student B does not. Then the instructor has time
to help B out of his difficulty without holding A back. Perhaps
next time, vice versa. By a class recitation held, say once in
two weeks, the work of the completed challenge can be summed
up, thus testing each student. This system also develops com-
petition, and 'emulation among students incites to industry.' "
-(L. O.)

"Each person has a different method of learning a subject.
In geometry one person may learn by repeated application,
another by finding the reason behind each theorem. The teacher
has to follow the system and thoughts of each individual in the
class. The challenges offer a splendid opportunity. In every
geometry* class there will be a certain number of pupils who
learn by application. Part of these will be able to work ahead
by themselves when a certain goal is given them. The other
part will not grasp the subject so easily, and the teacher can
give them help individually or as a group without keeping back
the first group. The same will apply to the group which has
to find the reason before they can apply the theorem. In this
way every person is progressing as rapidly as possible without
retarding another person. Then, when a certain part of the
subject has been studied, to have a general discussion clears up
every point, and every one is ready for a new phase of the work."
-(R. N.)

B. Class in biology, twenty-six pupils, nth and 12th. grades.

In preparation for this experiment the class (juniors and
seniors) had worked out a set of experiments on osmosis and
digestion (covering four days' work) so that they fully under-
stood the following definitions: "Osmosis is the interchange of
liquids of different density that are separated by a plant or an
animal membrane (cell-walls). In the process of osmosis the
greater flow is always from the less dense to the more dense."

"Digestion is a chemical change whereby soluble food sub-
stances are made ready to pass through cell-walls or made ready
to be used in cells."

The first question was given, answered by each on paper.


No class discussion followed. Students were then asked to write
out thought processes.

The second question (more difficult than the first) was given
with the understanding that they were to analyze their proc-

As a teacher, it is helping me greatly to realize how my in-
dividual students think. It will help me in directing their
thought processes in the future. (Teacher, L. W.)

a and c in each case below are the answers to the questions,
b and d the pupils' analysis of their intellectual method. The
questions are not repeated. Only six typical reports are in-
cluded; one record of a college senior participating in the class
is included. Answers are not edited.

I. (a) Why will dried raisins and prunes become fitted when
you put them into water?

They will become filled because water will pass
through a membrane, and the water seeps into the
cells through the cell-wall.

(6) The first thought I thought was whether the raisins and
prunes were cooked or not. They were not. If they
weren't cooked the cell-walls must still be there. Water
will pass through a membrane, and cell-walls are mem-
branes; therefore water must pass into the cells, or, in
other words, water passes into the raisins and prunes
and fills them up.

(c) In order to cook meat to obtain rich soup or broth how would

you prepare it ?

I would pound the meat first and then cook it so
that the broth could escape from the meat.

(d) I first thought what makes broth. It must be the sub-

stance contained in the cells. These would burst any-
way in heating, but if I broke the cells first more of
the substance could escape in the time allowed for the
meat to cook. Then I thought: "Why wouldn't the
water seep into the cells, as the water is less dense than
the substance contained in cells and the flow is always
from the less dense to the more dense?" This could
not happen, because for the process of osmosis a cell-
wall or membrane is necessary, and I had broken the


cell-walls by pounding. Therefore, my answer is

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Online LibraryHarry Lloyd MillerDirecting study; educating for mastery through creative thinking → online text (page 10 of 28)