I dissent from the majority report of the Committee in regard to the
following points:--
_Arithmetic_
1. AS TO FRACTIONS: In teaching arithmetic there does not exist any
greater difficulty in getting small children to grasp the nature of
the fraction as such than in getting them to grasp the idea of the
simpler whole numbers. It is true that the fractions ½, ⅓, ¼, etc., as
symbols, are a little more complex than are the single digits; but as
to the real meaning, when once the fractional idea has been properly
developed by the teacher and the significance of the idea apprehended
by the pupil, it is as easily understood as any other simple truth.
Children get the idea of half, third, or quarter of many things long
before they enter school, and they will as readily learn to add,
subtract, multiply, and divide fractions as they will whole numbers. In
using fractions they will draw diagrams and pictures representing the
processes of work as quickly and easily as they illustrate similar work
with integers. It is, of course, assumed that the teacher knows how to
teach arithmetic to children, or rather, how to teach the children how
to teach themselves. There is really no valid argument why children in
the second, third, and fourth years in school should not master the
fundamental operations in fractions. Not only this, they will put the
more common fractions into the technique of percentage, and do this
as well in the second and third grades as at any other time in their
future progress. There is only one new idea involved in this operation,
and that consists in giving an additional term--per cent.--to the
fractional symbol. When one number is a part of another, it may be
regarded as a fractional part or as such a per cent. of it. A great
deal of percentage is thus learned by the pupils early in the course.
Children are not hurt by learning. Standing still and lost motion kill.
Every recitation should reach the full swing of the learner’s mind,
including all his acquisitions on any given topic. But if the teaching
of fractions be deferred, as it usually is in most schools, the time
may be materially shortened by teaching addition and subtraction of
fractions together. This is simple enough if different fractions having
common denominators are used at first, such as 6/2 + 5/2 = ?, and 6/2 -
5/2 = ? Then the next step, after sufficient drill on this case, is to
take two fractions (simple) of different units of value, as ½ + ⅓ = ?,
and ½ - ⅓ = ? Multiplication and division may be treated similarly.
In decimals, the pupil is really confronted by a simpler form of
fractions than the varied forms of common fractions.
Devices and illustrations of a material kind are necessary to build up
in the pupil’s mind at the beginning a clear concept of a tenth, etc.,
etc., and then to show that one-tenth written as a decimal is only a
shorthand way of writing 1/10 as a common fraction, and so on. He sees
very soon that the decimal is only a shorthand common fraction, and
this notion he must hold to. This is the vital point in decimals. The
idea that they can be changed into common fractions and the reverse
at will establishes the fact in the pupil’s mind that they are common
fractions and not uncommon ones. Fixing the decimal point will, in a
short time, take care of itself.
In teaching arithmetic the steps are: (1) developing the subject till
each pupil gets a clear conception of it; (2) necessary drill to fix
the process; (3) connecting the subject with all that has preceded it;
(4) its applications; (5) the pupil’s ability to sum up clearly and
concisely what he has learned.
2. AS TO ABRIDGMENT: Under this head, I hold that a course in
arithmetic, including simple numbers, fractions, tables of weights
and measures, percentage, and interest, and numerical operations in
powers, does not fit a pupil to begin the study of algebra. That while
he may carry the book under his arm to the schoolroom, he is too poorly
equipped to make headway on this subject, and instead of finishing up
algebra in a reasonable length of time, he is kept too long at it, with
a strong probability of his becoming disgusted with it.
There are subjects, however, in the common school arithmetic that may
be dropped out with great advantage, to wit, all but the simplest
exercises in compound interest, foreign exchange, all foreign
moneys (except reference tables of values), annuities, alligation,
progression; and the entire subjects of percentage and interest should
be condensed into about twenty pages.
Cancellation, factoring, proportion, evolution, and involution should
be retained. Cancellation and factoring should be strongly emphasized,
owing to their immense value in shortening work in arithmetic, algebra,
and in more advanced subjects. Some drill in the Metric System should
not be omitted.
3. AS TO MENTAL ARITHMETIC: Till the end of the fourth year the pupil
does not need a text-book of mental arithmetic. So far his work
in arithmetic should be about equally divided between written and
mental. At the beginning of the fifth year, in addition to his written
arithmetic, he should begin a mental arithmetic and continue it three
years, reciting at least four mental arithmetic lessons each week. The
length of the recitation should be twenty minutes. A pupil well drilled
in mental arithmetic at the end of the seventh year, if the school age
begins at six, is far better prepared to study algebra than the one who
has not had such a drill. There are a few problems in arithmetic that
can be solved more easily by algebra than by the ordinary processes of
arithmetic, but there are many numerical problems in equations of the
first degree that can be more easily handled by mental arithmetic than
by algebra. To attack arithmetical problems by algebra is very much
like using a tremendous lever to lift a feather. Those who have found a
great stumbling-block in arithmetical “conundrums” have, if the inside
facts were known, been looking in the wrong direction. A deficiency of
“number-brain-cells” will afford an adequate explanation.
4. REARRANGEMENT OF SUBJECTS: There should be a rearrangement of the
topics in arithmetic so that one subject naturally leads up to the
next. As an illustration, it is easily seen that whole numbers and
fractions can be treated together, and that with U. S. money, when the
dime is reached, is the proper time to begin decimals, and that when
a “square” in surface measure first comes up, the next step is the
square of a number as well as its square root, and that solid measure
logically lands the learner among cubes and cube-roots. When he learns
that 1728 cubic inches make one cubic foot he is prepared to find the
edge of the cube. What is meant here is pointing the way to the next
above. All depends upon the teacher’s ability to lead the pupil to see
conditions and relations. My contention is that truth, so far as one is
capable of taking hold of it when it is properly presented, is always a
simple affair.
5. AS TO ALGEBRA: If algebra be commenced at the middle of the seventh
year, let the pupil go at it in earnest, and keep at it till he has
mastered it. Here the best opportunities will be afforded him to
connect his algebraic knowledge to his arithmetical knowledge. He
builds the one on top of the other. The skillful teacher always insists
that the learner shall establish and maintain this relationship between
the two subjects. To switch around the other way appears to me to be
the same as to omit certain exercises in the common algebra, because
they are more briefly and elegantly treated in the calculus. It is
admitted that a higher branch of mathematics often throws much light
on the lower branches, but these side-lights should be employed for
the purpose of leading the learner onward to broader generalizations.
Unless one sees the lower clearly, the higher is obscure. Build solidly
the foundation on arithmetic--written and mental--and the higher
branches will be more easily mastered and time saved.
_History of the United States._
In teaching this branch in the public schools, there does not appear,
so far as I can see, any substantial reason why the pupils should not
study and recite the history of the Rebellion in the same manner that
they do the Revolutionary War. The pupils discuss the late war and
the causes that led to it with an impartiality of feeling that speaks
more for their good sense and clear judgment than any other way by
which their knowledge can be tested. They may not get hold of all the
causes involved in that conflict, but they get enough to understand
the motives which caused the armies to fight so heroically, and why
the people, both North and South, staked everything on the issue. Just
as the men who faced each other for four years and met so often in a
death grapple will sit down now and quietly talk over their trials,
sufferings, and conflicts, so do their children talk over these same
stirring scenes. They, too, so far as my experience extends, are
singularly free from bitterness and prejudice. It is certainly a period
of history that they should study.
_The spelling-book._
In addition to the “spelling-lists,” I would supplement with a good
spelling-book. So far, no “word-list,” however well selected, has
supplied the place of a spelling-book. All those schools that threw
out the spelling-book and undertook to teach spelling incidentally or
by word-lists failed, and for the same reason that grammar, arithmetic,
geography, and other branches cannot be taught incidentally as the
pupil or the class reads Robinson Crusoe, or any similar work. It is an
independent study and as such should be pursued.