Harry Lloyd Miller.

Directing study; educating for mastery through creative thinking online

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tute fifths for dogs. (Subsequently substitute for dogs
(the denominator) thirds, sevenths, thirteenths, elevenths,
half, halves, sixths.)

Now express it thus:


Finally express it:

(of my m.) must equal ( ?)

(of my m.) = $100

(of my m.) = ( ?)

^-(ofmym.) = (?)

(There will be need of clarifying the idea of unity.)

Above all, see to it that dogs and fifths are denomi-
nators. (Teachers, I pray you, dwell on the meaning
of denominator.)

Before using the exercises in the book the pupils
should make up their own exercises for two or three
days, using all sorts of imagery in the denominators

Swinging out into a new challenge with a clearing
up of the fundamental principle involved, using simple,
vivid, illustrative material to exemplify the principle,
ought to bring pupils into a better and a more abiding
understanding of mathematical reality than is exhibited
in the endless working of sets of "problems" by formu-
las, or ready-made patterns.

Challenge. A set of exercises in arithmetic, algebra, a foreign,
language, or any other subject in which a number of exercises
in some organizing principle is clearly presented. To illustrate:


Factoring in algebra (8th or gth grade) is the unifying core of
a part of the course. After five minutes of general explanation
of a new phase of factoring in which the whole class participated,
each pupil started into the set of exercises and worked as rapidly
as possible. The teacher checked results, guided procedures,
explained to any one pupil or group of pupils some principle
needing further elucidation, called the whole class to concerted
attention when any fundamental concept could be focussed
upon economically. The exercises in this illustration involved
the differences of squares. Some 60 exercises were listed. In
the course of the remainder of the class period the number of
exercises mastered ranged from 10 to 55. No upper limit was
set. Each pupil worked forward in the challenge at his own best

In a class in geometry, 36 pupils all started at the beginning
of a 7o-minute class period on a set of 19 original exercises listed
at the close of the book on the circle (Wells and Hart). The
pupils worked in pairs at the board. Their work was checked
as rapidly as enough proof was given to indicate mastery.
Five pupils who began 10 minutes before the class period for-
mally began completed the entire set during the class period.
The range was all the way from a mastery of 3 exercises to 19
exercises. The teacher was kept busy checking results and
suggesting modes of attack to pupils in difficulty. The two
boys who finished the entire set 15 minutes before the period
was up assisted the teacher in this work.

The same procedure may be employed in word study, work
on sentences, reading of literature of a type or period, composi-
tion, exercises in foreign language, etc.

Two important matters should be mentioned in this connec-
tion: (i) The organizing principle should be clear. (2) No
upper limit should be set in the number of exercises. Enough
work should be at hand to challenge every ability in the class
group. It is not a uniformity that is sought in true education.
The unity (not uniformity) is gained by all those co-operations
which evolve out of a challenge clearly distinguished as to
some organizing principles and a progressive series of exercises
within the gripping principles.




Challenge. The Circle. (The material, the best modern
texts Book II.)

Procedure. (The procedure in this illustration is based upon
the work of a class of thirty-eight pupils of very wide ranges of
"capacity" and achievement. In fact, a conspicuous minority
of this class had been tried and found wanting pronounced
to be, if not "mathematical idiots," at least mathematically
disinclined or obsessed by defense mechanisms. One purpose
in the experiment was the attempt to demonstrate the feasi-
bility of managing a large section with extremes of "ability"
and attainment, well known at the outset.)

(a) When the challenge was entered upon, a word was men-
tioned about the assignment. The assignment was the circle.
For a period of five or six weeks nothing was said about assign-
ment. No daily assignment was mentioned. The individual
pupils who needed stimulus were operated on as they needed it.

(&) The class period (70 minutes) was developed into a work
period. This class period has never proved too long always
too short. Work begun in class was continued out of class
largely upon the initiative of the pupils. Saturday mornings
were employed for clinic purposes for any pupils who for any
reason (absence mainly) did not make satisfactory progress.
Four hours on the job, steady, makes a difference in any strug-
gling pupil. (See Chapter II, p. 71.)

(c) The challenge (the circle) is the organizing principle,
and the exercises and propositions furnish the differentials.
(See Chapter IV.) The big challenge was broken up into four
or five major organizing principles around which and with which
discussion could be carried on in a profitable manner. Recita-
tion work, as we ordinarily find it, was deleted.

The following organizing means were employed as principles
to think with:

1. A radius perpendicular to a chord.

2. A radius drawn to a tangent at point of contact.


3. Measuring angles. Angles measured by the same number

of arc degrees.

4. Parallels intercepting equal arcs.

5. Loci problems a few clarifying principles.

These, or some aspects of them, were written on the board
from time to time to guide the thinking of the pupils.

(<T) No effort was made to keep the class together. Goal
ends were mentioned from time to time. "Now, don't you think
we had better be prepared to discuss measuring angles next
Monday?" Or, "Would it not be a good idea for you to have
mastered our challenge by next Saturday night before 10 o'clock,
so that we can all start on loci problems one week from to-day ? "
No two pupils were ever found at the same place in any part of
the challenge. Responsibility was sought in many ways. The
great majority of pupils in a working laboratory do not need to
be prodded and supported in their work. After working into
some part of the challenge, after mastering some of the work-
ing tools, and after learning the game, as it were, by individual
guidance and checking of results, the whole class enter into
vigorous discussion and snappy response to rapid-fire questions.

The aim in this work procedure is to bring the whole class to a
concerted "attention" only when some organizing principle can
be economically cleared up for all members of the class by a
single drive, or when enough work has been accomplished in a
given part of the big challenge to make a class discussion profita-
ble for (practically) every member of the class, because every
one has actually turned out some work of his own in the part of
the challenge under discussion, or (concerted attention) when the
aim is gripped up in the contest or game spirit, wherein it be-
comes a matter of vigorous rivalry among individuals or between
groups in which true sportsmanship sweetens the competition
of life.

(e) In the procedure individual achievement is focal. Out of
individual activity class co-operations are developed. Partner-
ship teaching is employed. There must be no lesion of the
social sense in this drive for individual mastery and responsi-
bility. For one very important type of teacher-activity in this
work period, the reader is directed to Chapter IV, p. 120 jf. It
will be recalled that the teacher is now a directing genius, never
sitting apart engaged in any sort of busy aimlessness, such as


occupying the furtive pulpit in a supervisory capacity, correct-
ing papers, visiting somebody who comes to "see" the imponder-
ables. The new teacher is first of all master in the challenge,
ready to give a hint in any part of it. For some pupils will be
plunging ahead; others will bring in new material. A live class
will require a teacher alert in many directions.

Referring again to Chapter III, let us examine the teacher's
task at the point of the learner's real difficulty. This is no cram-
ming, memorizing school now. We are not interested in de-
veloping the mirror-minded pupil. That can be done. Tens of
thousands of pupils have absorbed enough geometry to pass it.
They have pursued it with no confident hope of overtaking it.

The habit of writing on a pad just what the consulting expert
(the new teacher) says to each pupil (or group of pupils two
or more) at the fork of the road in the dilemma which each pupil
sets forth that habit is stressed because we feel sure that
teachers talk too much. They have so much to impart !

" What are your data ? State each point in your hypothesis."

"Draw your figure with your instruments."

"Trace the angle with your finger as you read it."

"Where is the vertex of an inscribed angle?"

"Express the arc degrees. 360 arc AC. Now try it."

"Make sure of your proposition here."

"Which way do you think the author intended to solve this

"Look at the board. Angles are equal if "

"You need a plan, don't you?"

"Have you examined all the facts in your hypothesis?"

"Oh, but is that a central angle?"

"Talk to your figure."

"Have you used all your hypothesis?"

There is no end to this kind of personal, intimate suggestion
with a group of pupils at work in a challenge with no upper
limit for any one. These directing hints are given to the pupil
in his puzzled state, to two or more pupils working at the board
together, to the whole class now and again.

The highest achievement is to develop minds capable of analyzing
problems in the light of facts. This procedure is aimed at that
goal in every stage of its development.

Work is done in note-books and checked. Not all exercises


are written out in full by every pupil. Some work is checked
from the blackboard. Oral explanation is accepted in many
instances. Often a clear-headed pupil may simply schematize
the proof for the teacher. Pupils rising to their mathematical
heritage "as if to the manner born" (always by hard work,
for any "talent" here is a task) are given the privilege of assist-
ing the teacher in checking work which is being turned off at
high speed a thing which happens very frequently when pupils
get their "second wind" in the challenge.

No contribution, in my judgment, to better teaching and
thinking has been made than the introduction of the idea of
having the pupil work with a plan in mind. (Mr. Hart, Uni-
versity of Wisconsin, has made this procedure perfectly clear.)
In reality it is the essence of the scientific mode of thinking.
The goal (conclusion) is set out; it is the city yonder toward
which the road is to be built. Too often the learner is allowed
to think that the road determines the direction of the city. Not
so. The city determines the general direction of the road.
The particular, immediate direction the road takes at any given
time in its construction is influenced by circumstances. No
matter, if, for the nonce, the road seems to be veering off to the
right or left, or, for that, directly opposite from the city (goal)
provided only, the city is in the surveyor's mind as the objective.
The pupil in a situation in which creative (scientific) thinking
is possible ought to be given this rare privilege of building his
road to the city. He is both the surveyor and builder.

So, to every pupil in this work procedure: "What is your goal
in this and that part of the challenge?" Now: "What is your
plan, your (intellectual) method, by which you expect to build
the road toward your city?" The consulting expert (the new
teacher) will understand the significance of circumstances which
deflect the mind of the pupil from an air-line construction of the
road to the city. "Try it. It may work." "Make the ad-
venture. You may find that your plan will lay golden eggs for
you." Such remarks are not idle. They may prove to be
encouragers. These builders need many encouragers along the
way. Judicious praise should not be spared.

(/) In the various forms of concerted classroom activity,
the reader is again directed to Chapter III. One thing more


seems fruitful. The uses that may be made of the half-
dozen, or more, very difficult original exercises in the challenge
on the circle have not been exhausted. They furnish excellent
material for the emerging masters of the challenge in the class.
It is a refreshing, exhilarating emotion to hear high-school boys
and girls say : "I spent four, six, or eight hours on that exercise,
and I'm going to get it." That is a shocking thing to hear in
these days of "soft" pedagogy and the process of "painless"
information and "movieized" education!

An additional use of the most difficult original exercises is
suggested. (Four pupils in the experiment on the circle had
emerged triumphant at a certain station in the journey. In
the old rural school they could have been sent to the spring a
half-mile away to carry the pail of water to their thirsty class-
mates one of the real boons in that old school, and, by the
way, an excellent device for getting rid of "bright" pupils for
a weary hour.) These four pupils, who had mastered a certain
set of supplementary exercises, were given a chance during the
class period on this day to draw a big circle on a piece of paper
and to put into that circle all manner of lines. The chart was
exhibited, and the pupils were challenged to formulate as many
conclusions from the complicated figure as possible. More than
fifty were suggested an excellent review, by the way.

Now, with these four out of the way, and protected against
being bored, let us suggest the additional use of the very difficult

"All of you, let us draw a circle." "Read carefully exercise
so and so." Inscribed hexagon, not a regular one. "What
data (facts) are given?" Two pairs of opposite sides are
parallel. Conclusion The two remaining sides are parallel.

Now for a plan. (All working on this exercise, except the
four who have it.)

"How do we prove lines parallel?"

"We have a right to do anything we choose or will to our

A diagonal is hit upon. The plan is set forth. Pupils dis-
cover it themselves.

"Keeping the plan (blueprint) before you, let's work by it.'*

"Examine every item in your hypothesis." (Each pupil,
on his own mark now, writes all he can, using his hypothesis.)


Certain arcs are found to be equal by using the given facts.

"Angles may be proven equal. How?" (Perhaps all the
ways previously developed will be proposed. Here is the
teacher's opportunity to ask what the author perhaps had in
mind in this exercise. So, it may be necessary to point to one
of the organizing principles in the challenge just a physical
gesture to measuring angles, etc.)

The rest of the solution is a manipulation of equations, and
it is necessary, it seems, to give a short class drill on handling
simple equations.

In this concerted class work it is important to impress this
point: "William, are you paying attention?" "Yes," he replies.
"Then quit it!" Yea, verily, quit it. It is highly probable
that the habit of "paying attention" is from the medulla
oblongata down, not up. The drive of attention forward into
new difficulties at the point of crisis calls for eternal teaching

We have used a very difficult original exercise as a means of
clarifying the use of data (given) in developing a plan (or intel-
lectual method) of attack.

The most difficult exercise may be utilized also as a basis of
review a thoroughly sound practice in which the simpler
elements are caught up in a new synthesis.

The so-called "bright" or "clever" pupils (always emerging
in and through work) are not required to listen to what they
know perfectly well. They can refine their thinking by partici-
pating in the development of a technic of attack. They enjoy
working out "the rules of the game."

The "poorer" thinkers in the class have not suffered. They
have something to reach up to. It is not a confusing situation
to them. They may not have been able unaided to solve the
exercise. That is, in this point of view, a secondary considera-
tion. The difficult exercise has been used to the benefit of the
entire class. Every individual in the group has found in it
something worth while.

(g) Observations. (i) Out of the last point, first: Is it not a
futile cry to try to determine the native capacity of pupils
ready (for one reason or another) for the great adventure (into
geometry, physics, Latin, agriculture, what not), and thereupon
to classify alleged abilities? This class of thirty-eight boys


and girls exhibited as far-reaching differences in achievement as
could be found in any group. No one was held back because
of a "mentally delayed" classmate. The circle is big enough
and flexible enough for a challenge to every ability. The circle
is a function of the radius. A "short radius" can describe a
complete circle; a "longer radius" can describe a complete circle;
a "very long radius" can describe a complete circle. The big
circle is not scandalized by being associated with the little cir-
cle. The small circle need not be humiliated by a big one. Two
essential matters emerge. The circle offers the unifying, organ-
izing principle; all sorts of transmissions ahead are provided in
the endless variety of materials utilized in the challenge. No
two radii need to be the same length except in the same or equal
circles. No two individuals could conceivably be the same; no
two individuals could possibly have identical environmental fac-
tors. In fact, two children in the same home may receive from
their father diametrically opposite training. We are in dire need
of a careful social diagnosis in our efforts to appraise the reactions
of pupils to tests of all kinds. It is supererogation to add that
the "radius" is not a constant in the same individual.

The silly administrative twaddle we are hearing these days,
to the effect that there are those who cannot learn geometry,
and that there are those who can do no more than memorize a
few propositions with a full demonstration included, is only
another method of dodging responsibility. To parade now in
the "livery of science" by classifying our potential mathema-
ticians, physicists, etc., in terms of their I. Q.'s (intelligence
quotients) is evidence of another good idea done to death by
educators having a penchant for fads. It is ridiculous to main-
tain that the boy with an I. Q. of 77.77 cannot profit by a study
of geometry (or any other study for that).

(2) It may be urged that the "poorer" pupils in this class
are unduly tempted when taken up into the high mountains
the high peaks of the difficult original exercise. The objectors
and doubting Thomases are perhaps influenced by the Biblical
account of his Satanic Majesty and the temptation scene, but we
hasten to assure them that ours is only a decided leaning toward
"prescribed" temptations to excellence. The "poorest" pupil in
the class needs to be lifted up where he too, now and again, may
catch something of the vision and perspective of the mountain-


climbers. What he does on his "level ".will soon begin to take
on a new significance by the fact of having taught asg>'<- .'ar'*
higher reaches, even though assisted in the climbing. . VB sJoii

(3) Chapter IV is illustrated in this challenge. The organ-
izing principles and differentials are admirably delineated.
Provision is made for individual differences. No minimum
essential is ever allowed to degenerate into a maximum necessity.

(4) A group mediocrity is not desired. In fact, the more
highly selected the class group the greater the possible ranges of
achievement. Endless differentiation is possible where endeavor
is negotiated on a life basis. If anybody actually wanted a
bona-fide regimental uniformity in things intellectual, moral, and
spiritual, the way to get it is to assemble for the study of ge-
ometry a class of functionally near-imbeciles. There would be,
no doubt, a high degree of uniformity in such a group. It is not,
however, essential to have approximate equality of "capacity"
in any normal group for the pursuit of any subject in the cur-

(5) Testing for mastery may be conceived in many ways.
In this class the challenge was closed (never finished) with all
on their marks for a class period, writing on as many of some
dozen parts (exercises) in the big challenge as they could do.
Twenty-five per cent was given for each one of the twelve
mastered in the class period. Some pupils earned as high as
300 per cent. For those who fell below 100 per cent, a Saturday
morning was set apart for a second or third try-out with all
the time the pupil in difficulty wanted to use four hours or
more. A dozen hands were up to volunteer to coach a class-
mate in difficulty, preparatory to the Saturday-morning oppor-
tunity class. The test for mastery in the Saturday-morning
situation was similar to the one just described. The boy, a
victim of defense reactions, finally got the belt on his generator
in a Saturday-morning class after about an hour's fussing the
spinal cord, and actually got down to hard work and made
175 per cent in the test for mastery. The new teacher will re-
fuse to regiment adolescents under time-and-space-efficiency
methods. The law of chance needs to be distributed more equita-
bly than happens in any test by the clock under the hammer.

(6) What we have indicated in this elaborate presentation of


six to eight weeks' work on the circle is applicable to almost
her subject in the curriculum. The reader is urged to
.^ain the illustrative exercises, pp. 120 Jf., 130 jf., i6ojf., in
the body of our discussion. To be sure, each course must em-
ploy its own special technics. There is no general method,
universally applicable as a method, such as the enthusiasts for
the "project" level of teaching would seem to imply.

(7) The tired, the inert, the mechanical teachers (made such
by the system), and all others who enjoy poor pedagogical health,
may not have the courage to make the adventure upon the
challenge procedure.

(8) "They say": All this could be done if we had teachers of
dynamic personality. The answer to this honest scepticism is
by way of analogy. The old practitioner in medicine, let us
assume, is a wholesome, radiant, dynamic personality a lova-
ble man who kisses all the babies in the neighborhood. Across
the street is a physician-surgeon who has mastered the tech-
nic of modern medicine. His personality is not particularly
charming or virile, but he knows modern medicine and surgery.
To which one is a man going for an operation ? The initial act
is bound to be far-reaching. The system employed does make
a tremendous difference. The ideal is a new scientific human-

(9) And "they" will ask: How do you know these thirty-
eight pupils have done any better than they would have done
under the recitation system ? Frankly we don't know. It is a
manifest impossibility to compare the same pupils in two different
systems. We could crawl among the dust of figures, piling up
the "averages of the averages," and, perhaps, make out a case.
But we maintain the proposition that the common practice of
resorting to the popular psychology of arithmetic, believing
that an argument backed up by cold figures must carry certi-
tude, is a fallacious practice. It may be mere rationalizing
just a method of arraying evidence to support a belief already
accepted. The essential matter lies deeper. The drive is
headed up in the direction of building minds capable of analyzing
problems in the light of facts. The mind, conceived as a truth-
finding apparatus, is held to be an aim far superior to that of
making the mind a truth-testing apparatus. It is the difference


between education as a creative, productive process and educa-
tion as an assimilative, reproductive process. Suffice it to say
that these boys and girls took to their work as ducks to water.
That, also, seems to be a worthy measure of educational prac-

(10) There were no "failures" in this class. The goal set

Online LibraryHarry Lloyd MillerDirecting study; educating for mastery through creative thinking → online text (page 2 of 28)