The Teaching of Geometry at Tuskegee
By D. W WOODARD
Inskegee Normal and Industrial Institute
The Teaching of Geometry at Tuskegee
By D. W. WOODARD
Tuskegee Normal and Industrial Institute
Reprinted from School Science and Mathematics, Vol. 13, 1913.
Pages 400-410.
THE TEACHING OF GEOMETRY AT TUSKEGEE.
By D. W. Woodard,
Tuskegee Normal and Industrial Institute.
One day a group of Tuskegee students was engaged in putting
the finishing touches to a new building on the Institute grounds.
To one young man, who was learning the carpenter's trade, was
assigned the work of laying some molding. For quite a while he
pursued his work quickly and successfully. Then his job led
him to a certain corner of the building which deviated consider¬
ably from the usual right angle. What was he to do? Never be¬
fore had he laid molding around such a corner. Now the problem
consisted in finding the angle for cutting the molding so as to
make a proper fit. Failing to think out the problem, the student
by a trial and error method was finally able to make the molding
fit as desired. But this trial and error method involved not only
a loss of time but also a waste of material.
This happened on Friday, a- day on which this particular car¬
penter was engaged wholly in industrial work. According to the
Tuskegee system, a "day" student in a week spends three days at
his trade and three days in academic work, the "trade" days alter¬
nating with the "academic" days. On the day after the incident
mentioned, the student reported the affair to his class in geometry.
After he had attempted to give a statement of his difficulties, the
class visited the scene of the event. Again the student explained
the situation. Finally, a method was worked out on the spot
whereby the molding could be cut without waste. But the method
arrived at, while it eliminated all waste, was unsatisfactory in
that it was very slow in operation. On the next "academic" day
the problem was again discussed and referred for further discus¬
sion to a committee of three carpenters (students) who were
members of this class in geometry. This committee consulted the
instructors in carpentry and all other available sources of informa¬
tion. Altogether three methods were proposed for the task. A
model representing the corner of the room was brought into the
classroom together with pieces of molding, a saw, and the like.
Each method was actually exhibited before the class and the
geometrical principles concerned in each were thoroughly dis¬
cussed.
I have related somewhat in detail the above incident because
it exemplifies the character of the work in mathematics, and par¬
ticularly in geometry, which is attempted at Tuskegee. As has
TUSKEGEE GEOMETRY
401
been stated before, every "day" student at Tuskegee pursues for
six days of the week a prescribed academic course and trade work
on alternate days. "Night" students spend six days in industrial
work and five evenings in the academic department. Such a
system necessitates a close correlation of academic and industrial
activities if the student is not to be keenly conscious of a lack of
continuity in his work. The fact that the trades constantly in¬
volve more or less of the application of mathematical principles,
makes mathematics a subject peculiarly fitted to bring about the
correlation of the two phases of the work. The persistent corre¬
lation of the mathematics with the industrial work causes the
instruction in mathematics at Tuskegee to differ considerably
from the traditional presentation. The differences are found
not only in the character and content of the problems, but also
in the scope of the work, thp distribution of the emphasis in the
instruction, and the method of presentation.
A noteworthy fact in connection with the incident related
above was this: when the student reported his difficulty, the
teacher at once dropped his plan for the day and proceeded to the
solution of the problem. Neither the teacher nor the students
stopped to question whether the situation as reported involved
any immediate application of the lesson assigned for the day, or,
indeed of any lesson they Tiad had so far. The central thought
was: "here is a situation which calls for a solution which is clearly
geometrical in nature. We are studying geometry in order to be
able to solve such problems." As it happened, this problem com¬
pelled the class to work out several new propositions and to re¬
view others. It is significant that, through the interest awakened
by the practical situation and by the actual handling of the
material, these several new propositions were disposed of in a
fraction of the time required for the same propositions under the
usual circumstances. My experience indicates that this is gener¬
ally true for work attempted in this way.
In contrast to the feeling that the student generally has that
the problems solved in class are given for the express purpose
of illustrating some principle already learned, the idea brought
to the front here is that the mathematical principle is developed
to give the student an economical method of adjusting himself
to a new situation, that the principle is worked out for the sake
of the problem. The effect of this point of view upon the atti¬
tude of the student toward the subject and upon the development
of the subject in the course will be discussed later.
402 SCHOOL SCIENCE AND MATHEMATICS
Reference has been made to the fact that a problem under
consideration was referred to a committee of three members of
the class, who by their trade were particularly concerned. This
grouping of the students into committees for the study of special
problems is a distinguishing feature of the method employed at
Tuskegee. The idea arose in this way: Two students belonging
to the same trade differed in regard to a matter which had been
reported to the class. When the teacher appealed to the other
members of the class following the same trade, it was found that
the disagreement was quite general. Thereupon a group was
formed consisting of all members of the class belonging to the
trade, for the purpose of studying the matter and making a final
report to the class. Not only did this group of students clear up
the matter under discussion, but, to the great surprise of the
teacher, they searched their trade practice and brought in all the
problems involving geometry that they could find in the short
period of time given them in which to make their report. From
that time, whenever a student reports to the class a matter which
he thinks calls for a geometrical solution, the whole problem, un¬
less the solution is obvious, is referred to a group of students. In
this group are placed students in connection with whose trade
the problem has arisen, together with at least one student whose
trade is different from that of the other members of the group.
Of this group the student reporting the affair is the chairman.
The group meets at his call and he informs the teacher of the
progress of the work of the group. The purpose in associating
with the group a student whose trade is different from that of the
other members of the group is to compel the students to translate
the situation from the trade jargon into language intelligible to
other members of the class. The educative value of this fea¬
ture is obvious.
The greater part of the work of the class is carried on by
means of the reports of these groups. The plan of procedure is
as follows: (1) Report of the problem by student; (2) assign¬
ment of certain students to a group under the chairmanship of
the student reporting; (3) meeting of the group to consider the
problem; (4) if the problem is not solved at the group meeting, a
preliminary report is made to the class where the matter is dis¬
cussed and the problem more definitely formulated from the
standpoint of geometry; (5) further meetings of the group
until the problem is solved and reported to the class in finished
form. Frequently a group will hold as many as a half dozen
TUSKEGEE GEOMETRY
403
meetings and make quite as many reports to the class before
being dissolved. I have known a group to have a matter under
consideration as long as five months. Every student belongs at
all times to one or more of these groups. Exceptionally bright
students are attached to several groups. If a student becomes
interested in a problem under consideration by a group to which
he is not fegularly assigned, he may of his own will attach him¬
self thereto. Every student is assigned to a group which works
out systematically the geometry involved in the operations of his
own trade. In this system, too, opportunity is given the teacher
to distribute the work according to the ability of the individual
members of the class.
Frequently students of different trades are grouped together
because of a community of interests in trade work. Thus, car¬
penters and brick masons were associated together to consider the
construction of the brick arch because the carpenters make the
forms about which the masons build the arches. Blacksmiths
and wheelwrights work on the building of the same vehicle and
hence are mutually helpful when assigned to the same group.
Unexpected points of contact between the trades often appear
as the following incident will show. When in the recitation the
teacher called for new matter, one student, a wheelwright, spoke
of the method employed in his shop in "squaring up" the shafts
of a buggy. He wanted to know why the method always gave
the desired result. His chief explanation of the mechanical proc¬
ess was not very clear to the class. Further dicussion was
postponed until the class could visit the shop and actually see the
operation described by the student. As soon as the wheelwright
sat down, a carpenter arose and said that there were "squaring
up" processes in his trade, especially in connection with win¬
dow frames, and so on. A 'group to consider in detail such prob¬
lems was thereupon formed consisting of carpenters and wheel¬
wrights with the wheelwright mentioned above as, chairman.
When the class visited the wheelwright shop, not only did the
student explain the operation of "squaring up" the shafts, but he
also showed several other operations connected with the con¬
struction of the vehicle which involved geometry. This has been
my almost invariable experience. Whenever the class, has been
taken to a shop to have explained an operation there carried on,
almost without exception unlooked for geometrical problems
have come to light. This is one reason, aside from another which
I shall give later on, why classes should be taught as far as
404 SCHOOL SCIENCE AND MATHEMATICS
possible at places where the operations under discussion are
taking place.
Now I wish to call attention to a point which, I think, shows
an important phase of these experiences. While in the wheel¬
wright shop, not only did the class under the leadership of the
student wheelwright see performed a useful operation, but the
students received training in analyzing a real situation and form¬
ulating the geometrical propositions connected therewith. They
did real thinking, thinking in direct relation to a concrete situa¬
tion. This analysis of a situation and this sharp definition of its
problematic character finally leading to the formulation of the
geometrical propositions involved constitute, in my opinion, a
most valuable part of the training of the students, valuable alike
from the standpoint of geometry and from the standpoint of
their command of the theory and practice of their trades.
In this connection I might mention that, when a problem re¬
ported by a student requires the presence of the class at a cer¬
tain place, the details of the arrangements of the trip are left
as largely as possible to the student. He is made to feel that
the success of the venture depends in great measure upon his
efforts.
The attempt is made to build up the geometry about the situa¬
tions which arise in the daily work of the students at their
trades. There is no lack of such situations. In fact I have
found it a matter of impossibility to work out in class all of the
real problems reported by the students. There is always a period
during the recitation which is devoted to the hearing of new
problems. So eager are the students at times to have their prob¬
lems brought before the class that the teacher is frequently at a
loss, in the face of many such requests, as to which problem to
consider first. On such occasions an appeal is made to the mem¬
bers of the class and they decide what is to be the order of the
recitation. In great part the class teaches itself. The teacher is a
guide, a friend who is interested in everybody's problem.
It can be seen from the foregoing that the instruction in geom¬
etry at Tuskegee is by no means confined to the ordinary class¬
room exercises. When a student in his experience meets that
which he thinks has the remotest connection with geometry, he
will, if possible, without consulting the teacher bring the actual
material to the classroom. On one occasion a student brought
to the classroom a good part of the gearing of a buggy. I do
not know to this day how he managed to get it there. Whenever
TUSKEGEE GEOMETRY
405
it is inconvenient or undesirable to bring the necessary material
to the classroom, the recitation is held where the material is to
be found or the operation under consideration is being carried
on. It may be in a shop or at a new building in the course of
construction. Problems are solved in their natural setting. The
atmosphere of the classroom, just because it is a classroom with
the traditional furnishings, militates against the proper apprecia¬
tion of a concrete situation in many instances. As I have found
to my cost, it is one thing to sit down in a classroom and imagine
the manner in which a piece of work is being done and quite an¬
other to be on the spot and attempt to explain an actual operation
surrounded by the conditions under which it is usually effected.
In line with this thought, I might further add, that I have been
amazed to see the increased fluency on the part of a student when
he is in his own shop explaining his own work.
Just as the method of grouping the students for the investiga¬
tion of certain problems arose by accident, so another important
feature worked itself out without any special design on the part
of the teacher. One of the students asked the teacher for per¬
mission to consult him about a problem that had arisen in some
of his trade work. The teacher set aside a Tuesday evening for
the conference and casually remarked in class that all other stu¬
dents who were interested in the problem might meet at the
same time. When the appointed time came, the room was full
of students. This meeting was so successful that it was agreed
to meet on the following Tuesday. From that time this weekly
conference has been held. Attendance upon this meeting is ab¬
solutely voluntary. The students are free to come and go as
they please. The fact that a considerable number of the students
taking the course in geometry will voluntarily spend more than
an hour each week in such work bespeaks more than anything
else their interest in the subject. These meetings are devoted
almost wholly to the discussion of real problems. As has been
said before, there are so many problems reported to the regular
class that it is impossible to discuss them all adequately in the
recitation period. The weekly meeting, now known as the
Geometry Club, affords an opportunity for the discussion of these
surplus problems. There is no prearranged program, no particu¬
lar order of business. The conduct of the meeting is largely in
the hands of tfo» students, the teacher taking his place as one of
them. There is no formality, no hurry. When a student has a
matter to preser % he virtually takes charge of the discussion. In
406 SCHOOL SCIENCE AND MATHEMATICS
spite of the fact that there is no formal-plan of procedure, I have
never known a meeting in which there were not helpful discus¬
sions. I believe that some of the most valuable work of the
course is done in this meeting. Certainly the most enjoyable work
is here done.
The questions might properly be asked: In the attempt to build
up the geometry around concrete situations, what becomes of the
rigorously logical treatment which is supposed to be associated
with courses in geometry? When a student has finished such a
course as described, what conception of geometry as a science has
he attained? How does his conception of the subject compare
with that of the student trained according to traditional methods ?
To answer these questions, let us consider again the disposition
of the practical problems reported to the class. In the first place,
no problem is considered solved until every proposition has been
reduced to the conventional mathematical language. Then, too,
from the system of handling the problems, it results that a certain
minimum of propositions is in possession of all the members of
the class; but members of certain groups may have problems as
yet unreported to the class that have compelled them to work
out and study new propositions in many instances considerably in
advance of the minimum already mentioned. In one of the best
reports prepared by any group, it was found that the students
had formulated correctly the fundamental proposition connected
with their problem, but, after much groping about, discovered
that this proposition necessitated the proof of several propositions
unknown to them and to their class. These propositions they had
numbered and set down in their proper sequence, showing that
they fully appreciated what is meant by the logical order of
propositions in geometry. Of course, few reports are so thor¬
oughly worked out prior to their presentation before the class.
But finally all reports are cast into this shape. I do not feel
that there is any loss in such rigor as might be expected of an
elementary course in geometry, but, on the contrary, the critical
attitude developed by the analysis of concrete situations in all
their complexity and their abstract geometrical formulation, is, I
believe, of enduring advantage to the student. Ordinarily few stu¬
dents in elementary geometry courses receive any training what¬
ever in the formulation of propositions from given data.
When the student has finished such work, he has a sequence
of fundamental propositions, and above all these propositions
are not only connected in his mind in logical order, but they are
TUSKEGEE GEOMETRY
407
associated with his vital interests. They have an interest and
value outside of themselves. They represent a source of methods
of doing things. They are classed with his saw, his spirit level,,
his trowel, his lathe and other tools and machines. From this
close association of geometry with his life experiences, the stu¬
dent gets the notion that the same kind of common-sense reason¬
ing used in ordinary situations is to be brought into play inrr
geometry. He becomes acquainted with the subject in an inti¬
mate way; it loses its character as a very unusual procedure in¬
vented to call into action some particular brand of mental power
never used before nor afterwards.
In this work the student builds up his own geometry. He sees,
each bit of the structure put into place before his very eyes.
At certain intervals, the whole edifice so far constructed by the
class with emphasis upon the dependence of its various parts is
discussed in class. If a proposition arises from a matter reported'
by a student, it is often named after him, as, for instance, the
Jones proposition. In a word, the subject becomes properly and
helpfully orientated in the mental life of the student. One of
the most gratifying experiences of the writer as a teacher was
to have one of his students who had graduated from the insti¬
tute, on a recent visit relate to the geometry class how in a
given instance in his work as a brickmasqn he had been able to
use his geometrical knowledge at a critical time.
While we, are speaking of the sequence ,in which propositions,
arrange themselves in conformity to the demands of the practical'
problems brought before the class, it may be of interest to recount
the manner in which the subject of the circle was approached1
during the current school year. I did not say, "On the next day
we shall begin the study of the circle. Study page so-and-so,""
but this, "One of our students, a carpenter, wishes to take us.
to see some repair work on a circular wooden pillar of the library-
building. The class will meet there instead of in the usual recita¬
tion room." The essential part of the operation for us as a.
class consisted in finding the diameter of the circular pillar from1
the outside. The proposition that justified the method used was,
the diameter of a circle is equal to the distance between two-
parallel tangents. The point that I wish to make is, that the-
consideration of the circle was begun, without any preliminary-
preparation, with a discussion of a proposition concerning tang¬
ents. But, under the stimulus of the real situation the necessary-
notions were immediately developed in an interesting and prac-~
408
SCHOOL SCIENCE AND MATHEMATICS
tical manner. The rapidity with which the class in the face of
a real problem has covered the ground necessary to its solution
has been a source of great astonishment to me, brought up under
the usual methods. Under ordinary circumstances, perhaps, I
should not have had the temerity to begin the.formal considera¬
tion of the circle at a point not sanctioned by the traditional
classroom procedure, but I am fully convinced that at least in
this kind of work any real problematic situation involving the
idea to be discussed may be used as a first point of contact with
the particular topic. At Tuskegee interesting experiences of this
type may be trusted to develop a sequence of propositions which
invest the geometry with a most desirable significance for the
student.
A short list of propositions so formulated by the students may
be of interest.
OCCASION
1. A carpenter was compelled to
construct an arch under a stairway
where the floor prevented his get¬
ting the center in the usual way.
He contrived an instrument giving
rise to the proposition to the right.
2. Justifying the use of the cen-
trolinead, an instrument used by the
students in the course in architec¬
tural drawing.
3. Justifying a part of the proc¬
ess used in the blacksmith shop in
connection with the adjustment of
the gearing of a vehicle.
4. Justifying a method used in
getting the cut on molding laid
around non-rectangular corners.
PROPOSITIONS
1. If an angle is moved so that
its sides constantly pass through
two fixed points, its vertex de¬
scribes the arc of a circle.
2. (a) If line No. 1 is bisected
at right angles by line No. 2 so that
one-half of line No. 1 is a mean
proportional between the segments
which it cuts off on line No. 2, then
line No. 2 is the diameter of a circle
passing through the extremities of
line No. 1.
(b) Consider the angle formed by
joining the extremity A of line No.
2 to the extremities of line No. 1.
If this angle is moved so that its
sides constantly pass through the
extremities of line No. 1, the bisec¬
tor of the angle will constantly pass
through the extremity B of line No.
2.
3. If in a quadrilateral one pair
of opposite sides consists of equal
lines and the diagonals are equal,
the remaining two sides of the fig¬
ure are parallel.
4. The diagonals of a parallelo¬
gram bisect the angles from whose
vertices they are drawn when the
distance between one pair of oppo¬
site sides is equal to the distance
between the other pair.
TUSKEGEE GEOMETRY
409
A long period of study and many reports on the part of a
group of students were required before the propositions given in
the" case of the centrolinead were correctly formulated and
proved. This selecting of the conventional terms of geometry
and the fitting of the terms to a concrete situation insures the full
grasp of the meaning of these terms on the part of the student.
The student's "mental definition" mentioned by Prof. Dewey and
the true mathematical definition are more likely to agree than
under the usual presentation. It is a truism that the test of a
student's understanding of a term consists in his ability to use
the term.
The points that I have tried to emphasize here have been:
1. The grouping of the students according to their practical
interests in problems affords a method for keeping the instruc¬
tion in geometry in constant contact with the real life of the
student.
2. The student creates his own geometry. Propositions are
gathered into a logical sequence in response to a felt need.
3. The abstract formulation of concrete situations is a valu¬
able feature of the work.
4. This sort of work in geometry gives the student a grasp
of certain theoretical and practical aspects of his trade not other¬
wise possible. His geometrical knowledge is a source of help
in situations where ordinary rules break down. Instances could
be furnished of this fact.
5. And last but not least, the work is interesting and enjoyable
both to the teacher and, so far as can be observed, to the student.
In closing I wish to make two quotations which I think relevant
to the above discussion from an article by Prof. Dewey on "The
Psychological and the Logical in the Teaching of Geometry"
(Educational Review, April, 1903).
"But he (the student) also needs the habit of looking at defi¬
nitions and propositions with reference to the real experiences
which they express. More than any other one thing it would
seem as if the high school pupil, in particular, were at the point
where his greatest need is neither merely intuitive nor strictly
demonstrative geometry but rather skill in moving back and
forth from the concrete situations of experience to their abstracts
in geometric statement."
"The serious problem in instruction in any branch is to acquire
the habit of viewing in a two-fold way the subject-matter which
is taught day by day. It needs to be viewed as a development
410 SCHOOL SCIENCE AND MATHEMATICS
out of the present habits and experiences of emotion, thought
and action; it needs to be viewed also as a development of the
most orderly intellectual system possible. These two sides, which
I venture to term the psychological and the logical are limits of
a continuous movement rather than opposite forces or even inde¬
pendent elements."