REACHING IT, BEFORE PASSING ON TO SOMETHING ELSE.[5]--Apply it to cases
entirely different from those {36} shown in the book, and try to
observe how generally it is applicable. Do not leave it in the
abstract. An infallible test of whether you _understand_ what you have
read is your ability to _apply_ it, particularly to cases entirely
different from those used in the book. An abstract idea or result not
illustrated or applied concretely is like food undigested; it is not
assimilated, and it soon passes from the system. In illustrating, so
far as time permits, the student should use pencil and paper, if the
case demands, draw sketches where applicable, write out the statement
arrived at in language different from that used by the author, study
each word and the best method of expression, and practise to be concise
and to omit everything unnecessary to the exact meaning. Herndon in
his "Life of Lincoln" says of that great man, "He studied to see the
subject matter clearly and to express it truly and strongly; I have
known him to study for hours the best way of three to express an idea."
This kind of practice inevitably leads to a thorough grasp of a subject.
Some of these principles may be illustrated by considering the study of
the algebraical conditions under which a certain number of unknown
quantities may be found from a number of {37} equations. The student
will perhaps find the necessary condition expressed by the statement
that "the number of independent equations must equal the number of
unknown quantities." Now this statement makes little or no concrete
impression upon the minds of most students. They do not understand
exactly what it means, and they can easily be trapped into misapplying
it. To study it, the student should ask himself what each word of the
statement means, and whether all are necessary. Can the word
"independent" be omitted? If not, why not? What does this word really
mean in this connection? Must each equation contain all the unknown
quantities? May some of these equations contain none of the unknown
quantities? What would be the condition of things if there were fewer
equations than unknown quantities? What if there were more equations
than unknown quantities?
This problem too, affords a good illustration of the advantage of
translation into other terms? What, for instance, is an equation
anyway? Is it merely a combination of letters with signs between? The
student should translate, and perceive that an equation is really an
intelligible sentence, expressing some statement of fact, {38} in which
the terms are merely represented by letters. An equation tells us
something. Let the student state what it tells in ordinary
non-mathematical language. Then again, a certain combination of
equations, taken together, may express some single fact or conclusion
which may be stated entirely independent of the terms of the equations.
Thus, in mechanics the three equations _[sigma]H_=0; _[sigma]V_=0;
_[sigma]M_=0; taken together, merely say, in English, that a certain
set of forces is in equilibrium; they are the mathematical statement of
that simple fact. If the equations are fulfilled, the forces are in
equilibrium; if not fulfilled, the forces are not in equilibrium.
Following this farther, the student should perceive, in
non-mathematical language, that an equation is independent of other
equations if the fact that it expresses is not expressed by any of the
others, and cannot be deduced from the facts expressed in the others.
The benefit of translation into common, everyday language, may be shown
by another mathematical illustration. Every student of Algebra learns
the binomial theorem, or expression for the square of the sum of two
quantities; but he does not reflect upon it, illustrate it, or perceive
{39} its every-day applications, and if asked to give the square of 21,
will fail to see that he should be able to give the answer instantly
without pencil or paper, by mental arithmetic alone. Any student who
_fully grasps_ the binomial theorem can give (without hesitation) the
square of 21, or of 21.5, or any similar quantity. With practice and
reflection, results which seem astonishing may be attained.
(_f_) KEEP THE MIND ACTIVE AND ALERT.--Do not simply sit and gaze upon
a book, expecting to have ideas come to you, but exert the mind. Study
is active and intelligent, not dreamy. By this is not meant that haste
is to be practised. On the contrary, what might perhaps be called a
sort of dreamy thinking often gives time and opportunity for ideas to
clarify and take shape and proportion in the mind. We often learn most
in hours of comparative idleness, meditating without strenuous mental
activity upon what we have read. Such meditation is of the greatest
value, but it is very different from the mental indolence of which the
poet speaks when he says:
"'Tis thus the imagination takes repose
In indolent vacuity of thought,
And rests and is refreshed."
{40} This is beneficial to the proper extent; but it is rest, not study.
(_g_) WHEN YOU MEET WITH DIFFERENCES OF OPINION UPON A SUBJECT, REFLECT
UPON THE REASONS WHICH MAY CAUSE INTELLIGENT MEN TO ARRIVE AT DIFFERENT
CONCLUSIONS.--These reasons are:
1. One or both may fail to grasp all the pertinent facts, or even the
problem itself, or may assume, as true, facts or principles which are
really erroneous. This should easily be ascertainable.
2. One or both may reason incorrectly even from accurate premises.
This also should be discoverable.
3. One or both may see facts out of proportion--may lack a true mental
balance or perspective.
4. One or both may illustrate the inherent stubbornness or
imperviousness of the human mind.
Whether the student can discover the last two sources of error will
depend upon his own mental characteristics. He must not forget,
however, that on many matters no definite demonstrable conclusion is
possible, and that the result must remain more or less a matter of
opinion.
(_h_) REMEMBER THAT A STATEMENT IS NOT A PROOF. MANY STUDENTS THINK
THEY PROVE A STATEMENT BY MERELY REPEATING IT IN DIFFERENT WORDS. YOU
DO NOT UNDERSTAND A CONCLUSION UNLESS YOU CAN SEE THE STEPS IN ITS
LOGICAL DEMONSTRATION.
{41}
It is quite surprising how many students commit this error. For
instance, if I am asked why can I see through glass and I reply,
because it is transparent, I am giving no reason at all, for
transparent means what can be seen through, so I am simply saying that
I can see through glass because I can see through glass. The same
error often occurs in arguments or syllogisms. For instance, suppose I
make the following statements:
No unsportsmanlike act should be done;
Smith's act was unsportsmanlike;
Therefore, Smith's act should not have been done.
Now, this of itself is not correct reasoning, for the reason that the
word "unsportsmanlike" simply means something which no sportsman should
do. The conclusion, therefore, is simply a repetition of the second
statement. The real thing to be proved in this case is whether Smith's
act was or was not unsportsmanlike.
[1] "General ideas and great conceit are always in a fair way to bring
about terrible misfortune."--_Goethe_.
[2] "I tell you earnestly and authoritatively (I know I am right in
this) you must get into the habit of looking intensely at words, and
assuring yourself of their meaning, syllable by syllable--nay, letter
by letter."--_Ruskin: Sesame and Lilies_.
"Neither is a dictionary a bad book to read--it is full of
suggestions."--_Emerson_.
Benjamin Franklin, writing to a lady who asked him to give her advice
about reading said:
"I would advise you to read with a pen in your hand, and enter in a
little book short hints of what you find that is curious or that may be
useful ... and as many of the terms of science are such as you cannot
have met with in your common reading, and may therefore be unacquainted
with, I think it would be well for you to have a good dictionary at
hand to consult immediately when you meet a word you do not know the
precise meaning of. This may at first seem troublesome and
interrupting, but it is a trouble that will daily diminish, and you
will daily find less and less occasion for your dictionary, as you will
become more acquainted with the terms; and in the mean time you will
read with more satisfaction because with more understanding."
[3] "A man who has no acquaintance with foreign languages, knows
nothing of his own."
[4] "The Principles of Argumentation" by Baker and Huntington, is
another excellent book, not treating of formal logic, but discussing
the general principles which should govern the preparation of a paper
or argument, the principles of evidence, and the logical fallacies in
reasoning. It is recommended to readers. This book is, or has been,
used in the course in English at Harvard University, and similar books
are used in other colleges. A thorough training in English under a
good teacher is a good training in logic, for clear and logical writing
requires clear and logical thinking. Nevertheless, the writer strongly
advocates the study of formal logic also.
[5] "It is not enough to know, we must also apply; it is not enough to
will, we must also do."--_Goethe_.
{42}
III
THE THIRD ESSENTIAL FOR A PROPER METHOD OF STUDY IS SYSTEM
(_a_) DISCOVER THE FUNDAMENTAL IDEA OF THE SUBJECT.--Strip off the
detail and get down to the root of the thing. See the really important
point. Then, after this has been clearly perceived and mastered,
arrange the details in their proper relations to the fundamentals. The
subject will thus have a skeleton, and upon this the details will be
placed. A subject of study thus viewed may be compared to the human
body, with its bony skeleton or framework, and all the various organs
and parts supported by it; or to a tree, with its trunk, branches and
leaves. Thus to consider the relative importance of facts, to sift out
the essential ones, will train the power of mental discrimination and
cultivate the judgment.
When this is done, subsequent facts relating to the subject can be
correlated with what is already known, and will in this way be easily
retained by the memory. Remember and observe Jacotot's maxim, "Learn
something accurately, and refer {43} the rest to that." Unessential
facts, or those of secondary importance, may be passed over in the
first reading, and left for a second or later reading, for a proper
method of study _always involves re-reading_, perhaps many times.
You cannot possibly know everything even of a single subject, hence the
importance of knowing the fundamental things about it and knowing them
thoroughly. Even if you gain but an elementary knowledge of a subject,
that knowledge may be thorough and should include fundamentals.
Thorough elementary knowledge must not be confused with _a smattering_.
The latter is worse than useless, and is marked by vagueness,
uncertainty, and failure to grasp fundamentals. But elementary
knowledge, if clear and definite as far as it goes, is valuable, and
the first step toward more complete knowledge. Many students deceive
themselves and others into thinking that they know something of a
subject, because they have looked into it, while their knowledge may be
entirely superficial and valueless.
When the fundamental principle or fact is perceived, study this
carefully until it is thoroughly mastered. One who knows how to study
properly will thus pick out the sentence or the paragraph which
contains the key to the {44} subject--the fundamental fact or
principle--and will read and re-read this many times until its full
meaning is clearly grasped. When this is done it is sometimes
remarkable how quickly the rest of the chapter or subject may be
mastered, for it will often be found to consist of discussions or
illustrations, which will be obvious once the fundamentals are clearly
in the mind. The ordinary student, however, does not do this. He does
not see the fundamental principle, and each illustration is like a
separate problem, different from the others, which has to be studied by
itself, and is never fully mastered, because the underlying fundamental
principle is not grasped.
(_b_) BEFORE YOU BEGIN TO STUDY A SUBJECT, THINK IT OVER CAREFULLY AND
FIND OUT WHAT YOU ALREADY KNOW OF IT OR WHAT YOU CAN ARRIVE AT BY YOUR
UNAIDED EFFORTS.--Try also to perceive what you expect to get out of
the study of the subject, and how it is related with what you have
already studied, and how it is to find application.[1] The historian,
Edward Gibbon, states in his autobiography that before reading any
book, he made it a rule to reflect {45} upon the subject, arranging and
classifying what he already knew of it.
This method may be followed to different degrees, depending on the
subject. A student beginning the study of a new science which he has
never studied before, can do comparatively little; but at least he can
insist upon getting a clear idea of what the subject or problem is, its
extent, what its objects and methods are, how it is related to other
subjects, what its uses are, and how other studies will find their
application in it.
(_c_) CLASSIFY AND ARRANGE WHAT YOU HAVE LEARNED.--When you have
finished part of a subject, stop and think over the ground that has
been covered, and arrange the various points made. Draw up a topical
index and compare it with the table of contents. Note the correlation
or interdependence of facts and link them together. By the principle
of association the retention of facts and principles in the memory will
be much facilitated. Note down concisely the steps of an argument in
your own words, and see if the conclusion is justified. Close the book
from time to time and go over in your mind what you have learned.
The importance of systematic classification is very great. The minds
of many students are {46} like a library without arrangement or
catalogue; the books may be there, but cannot be found when wanted, and
so are valueless for use.[2]
[1] "We must keep carefully that rule of Aristotle which teaches that
the best way to learn anything well which has to be done after it is
learned, is always to be a-doing while we are a-learning."--_Richard
Mulcaster_.
[2] "There's a vast difference between having a carload of
miscellaneous facts sloshing around loose in your head and getting all
mixed up in transit, and carrying the same assortment properly boxed
and crated for convenient handling and immediate delivery."--_Lorimer:
Letters from a Self-made Merchant to his Son at College_.
{47}
IV
MENTAL INITIATIVE
It will become evident from the foregoing that a fourth essential for
proper study is mental initiative. The student must have a definite
purpose, and must do what is the proper thing without it being
suggested to him. He must not simply do as he is told. If he have not
initiative and cannot develop it, he will probably never study
intelligently, nor gain a thorough understanding of what he reads, but
will merely memorize.
Memory is a most important faculty; it is not, however, a _substitute_
for thought, but should be based upon it. Thinking is essential in
order to decide what to memorize. Memory, however, is often made the
sole factor in study. Fundamental principles should frequently be
memorized, so that by numberless repetitions they may be permanently
impressed upon the consciousness, and can be repeated verbatim as a
guide in any concrete case where they are to be applied.
Some suggestions may be useful as to the use and cultivation of mental
initiative.
{48}
(_a_) CULTIVATE AN INTEREST IN WHAT YOU ARE STUDYING, AND SOME IDEA OF
WHAT IT LEADS TO.--Without interest your study will be perfunctory and
of little use to you. Make yourself believe that for you, at that
time, it is the most important thing in the world. It is of course
true that in most schools students are required to study definite
subjects according to a curriculum arranged by the faculty. In some of
these subjects a student may take little interest; indeed they may be
so foreign to his natural tastes that he is not able to cultivate any
interest in them. In such a case his study of them will be of little
value to him. If, relying upon the judgment of those who prescribe the
curriculum as necessary or desirable for the object which he has in
view, he cannot persuade himself that they have value for him or make
himself take an interest in them, it would probably be better for him
to drop them even though he may thereby become a special student in the
school or lose his degree. A degree which simply means slipshod,
unintelligent and uninterested study of a considerable number of
subjects embraced in the curriculum, is verily a "scrap of paper" not
worth having. If you wish to concentrate your entire attention upon
certain subjects in which {49} you take an active interest you may
become proficient in those, but you may become very narrow minded and
altogether lacking in that all-around breadth of view which comes from
the cultivation of other subjects which well informed men consider
necessary.
(_b_) INSIST UPON FIRST CLEARLY FORMULATING THE PROBLEM, IF ONE IS
BEFORE YOU.--Many students literally do not know what they are doing,
because they neglect this injunction, which is a necessary corollary of
the necessity of forming definite ideas. Do not proceed to endeavor to
solve the problem until it is clearly formulated, no matter how long it
may take. See what the data of the problem are, whether definite or
not, and what is required. See also how variations of the data, if
indefinite, would affect the result.
(_c_) WORK INDEPENDENTLY OF OTHERS.--Solve your own difficulties and
welcome them. Do not expect things to be easy. You will never gain
strength by being shown, but only by the exercise of your own unaided
powers. Therefore, do everything for yourself, so far as possible.
Seek only _suggestions_ from your teacher, when you need help, except
in regard to mere matters of fact, which you could not be expected to
{50} reason out. Let the suggestions be as slight as possible.
If you have problems assigned, solve them entirely by yourself, even if
you make mistakes. Then, when those mistakes are pointed out, consider
them with great care and discover the causes for them, and _remedy
them_, so that you will not again make the same mistake or one
analogous to it. You should delight in discovering difficulties which
give you an opportunity to test and increase your strength and so avoid
future errors. In the same way, examinations should be welcomed, not
dreaded. The teacher does not mark you--you mark yourself; the teacher
merely records the mark. Even if you fail in the examination, that
should indicate to you what you lack, and so be a benefit. Indeed, it
is better to fail than to scrape through.[1] There must be a line
somewhere. The man just above the line passes, and the man just below
the line fails. The former may not be as capable as the latter, but,
having passed, he does not remedy his faults; while the man who has
failed is required to remedy his. Huxley said that the next best thing
to being right is to be completely and wholesomely wrong.
{51}
(_d_) DRAW YOUR OWN CONCLUSIONS, WHENEVER POSSIBLE, BEFORE YOU KNOW
THOSE OF THE WRITER You ARE STUDYING.--When you read, "From the above
it is evident," stop, close the book, and see if you can state what is
evident. When you have written this down, compare with the result
reached by the writer. Practise such exercises in whatever form they
present themselves. If your conclusions are different from those of
the writer, in kind or in character, see which is right, or whether
both are right. If you are right, why did the writer not reach your
conclusion? Was it because it was not pertinent to his problem? Is it
simply a difference of expression?
The process of investigating any subject is a process of question and
answer. The student must first propound to himself a question, and it
must be the proper question. He must be able to perceive what the
proper question is, under the circumstances. Then he must give to
himself the proper answer out of all the possible answers that are
verbally correct, namely, the answer that affords a new vantage ground
from which another question may be asked; and so the problem may be
gradually unravelled.
Then again, many questions are indefinite, and {52} can only be
answered indefinitely; but to all questions a correct answer can be
given, and the student must give the most definite answer the case
admits of, and must gain the ability to qualify his answer or classify
possible cases in such manner as may be necessary.
(_e_) IF YOU CANNOT SEE HOW THE AUTHOR REACHES A STATED CONCLUSION,
BECAUSE HE DOES NOT INDICATE THE PROCESS WHICH HE FOLLOWS, DO NOT SPEND
TOO MUCH TIME TRYING TO FIND OUT HOW HE DID IT, BUT RATHER SEE IF YOU
CAN COME TO A CONCLUSION IN YOUR OWN WAY, THUS CULTIVATING YOUR OWN
POWER AND INITIATIVE RATHER THAN FOLLOWING THE AUTHOR.--A good textbook
should not make things too clear, or relieve the student of the
necessity of exerting himself.
(_f_) LEARN TO GENERALIZE.--Draw the most general conclusion possible
from the premises. Try to see if a general principle can be laid down.
This is a most important faculty to acquire. At the same time, avoid
the prevalent fault of hasty generalization, based on insufficient data.
(_g_) GO BEYOND THE BOOK.--Regard the book as suggestive and not final,
as the assistant to your own powers that you are for the moment
employing. Pursue the subject as much farther {53} as you have time
for. In this way you may develop a faculty for independent thinking.
(_h_) VISUALIZE YOUR RESULTS SO FAR AS POSSIBLE.--Train the imagination
by perceiving results in your mind, in concrete form, and in imagining
applications of facts and principles. Remember that use is the object
of study, and try to see the use that may be made of what you have
acquired.
We have seen that there are four main requisites for proper study,
viz.: (1) Mental courage; (2) Understanding; (3) System; (4)
Initiative. In addition to these may be mentioned (5) Proper habits
and methods of work, under which head a number of minor but important
suggestions may be made.
[1] "The greatest piece of good fortune is that which corrects our
deficiencies and redeems our mistakes."--_Goethe_.
{54}
V
PROPER HABITS AND METHODS OF WORK
(_a_) SELECT THE BEST BOOK FOR YOUR PURPOSES AND STUDY IT
THOROUGHLY.--The best book for your purposes will depend upon
circumstances. If you are beginning a subject, do not start with the
most complete book, but take a more elementary one. Remember that
elementary knowledge is not the same thing as superficial knowledge,
but may be quite the reverse. A knowledge of fundamental elementary
principles is essential for the understanding of any subject. These
should be obtained first from some elementary book, and made to form a
skeleton or framework, upon which the more elaborate portions of the
subject may be hung in their proper places. In large books there will
be found too great detail for the beginner, and he will be discouraged
by having too many things thrust upon his attention at once.
Elementary knowledge, thoroughly assimilated, is essential. Begin,
therefore, with the best elementary book there is, one which will make
you {55} think, weigh, understand, test and discriminate; and get from
it the kernel of the subject; and gain, if possible, a stimulation to
go beyond to a more elaborate treatise.