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DEVOTED TO THE SCIENCE, ART, PHILOSOPHY AND
LITERATURE OF EDUCATION.
CONFERENCE REPORT ON MATHEMATICS.
SUPT. J. M. GREENWOOD, KANSAS CITY, MO. The parts of this report that I do not concur in may be arranged under two tolerably distinct sub-divisions :
(1) abridging, enriching, overlapping, and transferring ; (2) the low conception of arithmetic as a science. The report as a whole is divided as follows:
1. General Statement of Conclusions.
1. Abridging, Enriching, Overlapping, and Transferring.
The limited time at my disposal will be devoted almost exclusively to that part of the report pertaining to the subject of arithmetic. The language of the report must be its own interpretation as to the views the conference entertained on the subjects under discussion. Judged as a production and of the actual state of the schools in which arithmetic is now taught, it is evident that the gentlemen who made the report are not engaged in teaching arithmetic, or if they are they are not familiar with the best methods in vogue in many schools of our country.
As a whole the report is suggestive, but is decidedly weak. By this is meant that in the opinion of the conference, the children are less able to work in the elementary branches of mathematics than in any other department of elementary or secondary education.
This fact is brought more prominently into view by comparing the magnificent course in geography, for instance, with the lean course in arithmetic. The geography conference treats the children as great philosophers, while in arithmetic they are regarded almost as weaklings. The mathematical report is turned around so that the conclusion comes first, and it virtually starts in with the assertion that the course in arithmetic should be abridged and enriched, — "abridged by omitting entirely those subjects which perplex and exhaust the pupil without affording any really valuable mental discipline; and enriched by a greater number of exercises in simple calculation and in the solution of concrete problems.”
“ Among the subjects which should be curtailed, or entirely omitted, are compound proportion, cube-root, abstract mensuration, obsolete denominate quantities, and the greater part of commercial arithmetic. Percentage should be rigidly reduced to the needs of actual life. In such subjects as profit and loss, bank discount and simple and compound interest, examples not easily made intelligible to the pupil should be ommitted. Such complications as result from fractional periods of time in compound interest are useless and undesirable. The metric system should be taught in applications to actual measurements to be executed by the pupil himself; the measures and weights being actually shown to, and handed by, the pupil. This system finds its proper application in the course which the conference recommends in concrete geometry.”
This extract may be received in two ways,—(1) abridgment;(2) enrichment. Under the first are the subjects of compound proportion, cube-root, duodecimals, abstract mensuration, obsolete denominate quantities and the greater part of commercial arithmetic.
Duodecimals expired by the statute of limitation long ago, except in college and university examination questions. As to compound proportion and cube-root, ten recitations are ample time for teaching both subjects. Compound proportion when treated analytically, or as “cause and effect,” has a very high educational value as well as a wide application beyond arithmetical computation.
Hundreds of thousands of pupils never go to high school and have no opportunity to study algebra, but from the recommendation of the conference, they should not have anything to do with cube-root.
The kindergarten children handle globes, cubes, cylinders and cones, not to mention other simpler material forms, and yet, notwithstanding the necessity for such knowledge, a pupil must not learn how to find the edge of a cube till he studies algebra, lest the labor should exhaust his mental powers. Should algebra never be studied, what then? Starve the mind !
As a means of definite information and strengthening the mental faculties, unless it be the critical analysis of sentences in English Grammar, there is no other kind of work more interesting and satisfactory to pupils than the extraction of square and cube root, illustrated by means of geometrical forms made by the pupils themselves. These processes embrace the most elaborate and connected chain of reasoning that the school boy or girl meets with in the grammar school course. The discipline is greater and equally as valuable, or more valuable, than the extraction of these roots by the "algebraic method.” No one who has ever taught these subjects properly and intelligently will contend that they are difficult for boys and girls to understand or to apply.
What meaning the conference attaches to the term “abstract mensuration” is not clear. If by it is meant that volumes, areas and lines are always to be measured first by the pupils before computation should be made, then the hypothesis becomes impossible if not contradictory. But if, at the beginning of a subject, the pupil shall make his own measurements, the position is not only tenable but educationally correct. However, when the pupil has once obtained a definite idea of an inch, foot, yard, rod, mile, etc., future measurements need not be performed but assumed. The gathering of “field notes” is well enough at first, but a surveyor or an astronomer does not measure, in a physical sense, all the lines and angles used in making a calculation.
“Obsolete denominate quantities” must refer to old tables common in English Arithmetics prior to, and immediately after, the Revolutionary war.
The recommendation that all business or commercial arithmetic should be handed over to the tender mercies of the commercial colleges, under the plea that the children in common schools are two young and immature to understand these subjects, is “a dream theorem of words-words.”
Children hear very much more of banks, stocks, insurance, assessments, taxes and such things, around the fireside than they do of scientific instruments, or of such problems as “computing the quantity of coal which would have to be burned in order to heat the air of a room from the freezing point to 70°.” How this can be very much simpler to a child of twelve years than the idea of shares in a corporation, is a “sphinx riddle.”
No doubt, it is because the conference has said so! That the child can have no idea of business till brought face to face with it
may be true from the standpoint of learned professors, but certainly not with the average American boy “who catches on without the second telling.”
The conference recommends also that practice in quick and accurate reckoning in the fundamental rules, including operations with vulgar fractions and decimals, should be made an essential feature. I agree most heartily with this except that the adjective
vulgar” has a flavor of Thomas T. Smiley’s “ Federal Calculator,” published in 1825, and which volume certainly made a much deeper impression on the conference than any arithmetic of recent date.
Children in first, second and third grades are doing these things now. Come west and see !
The time devoted to algebra is an instance of extreme lengthening out beyond anything I have ever known in mathematical instruction. The pupil is to study algebra a year, then he is to spread it out two years longer, giving as much time to it for two years as he did the first year, that is to say, five hours in recitation each week for the first year and then two and a half hours a week for the second and third years. Two solid years on algebra is tough on the books, but tougher still on the boys and girls.
There is not a college algebra published in the United States, except Stoddard and Henkle's University Algebra, that pupils will not study through and review in a year and a half. When a pupil begins algebra why should he not work at it in earnest ?
For the average American Algebra fifteen months, at the farthest, is ample time.
It is an educational mistake for children to lay aside common school arithmetic to dip a little bit into algebra. A boy or girl will be far stronger in mathematics after a good drill in mental
arithmetic than if mental arithmetic be omitted, and elementary algebra, in a half hearted way, be substituted. Until a pupil can handle quadratic equations well, algebra is little help in the solution of arithmetical problems. Many equations of the first degree are more easily and elegantly handled by arithmetic than by algebra. A boy with a smattering of arithmetic and of algebra is not half so well equipped, either for life or for progress in mathematics, as he is if thoroughly grounded in arithmetic, both practical and mental.
I have yet to see the man, woman, or child that was good in algebra that was not also good in arithmetic. Jumping around never counts for much in either of these branches. The better the learner is in arithmetic, the easier algebra is. The outlook that elementary algebra can give is quite limited.
While the report is full of suggestions about what should be taught, the nearest approach as to how any particular subject should be taught, is a statement that the children should see and handle some measures. Had the gentleman been entirely familiar with the methods of presentation employed in nearly all the schools of the country and the appliances provided by boards of education or by the teachers for teaching concrete arithmetic, a considerable part of the report would never have been written. It, in this respect, is a verbal contest with a straw man that certainly has no existence in the progressive schools of the country. Portions of the report, in my judgment, should have been leveled at the methods of teaching certain subjects rather than at the subjects themselves.
Nearly all the subjects recommended to be dropped occupy but a few pages in our modern school arithmetics. I have before me Dr. Milne's Standard Arithmetic, published in 1892. Compound Proportion covers three pages ; Cube-Root, including pictures, explanations and applications, seven pages. Duodecimals not in the book ; True-Discount, Banking and Bank Discount, six pages; Insurance - property and personal — three pages ; Partial-Payments, three pages ; Equations of Payments and Accounts, four pages. These subjects cover twenty-six pages in a book of 428 pages, and can be easily mastered in twenty-six recitations, yet this is what is called “abridging and enriching” the course in arithmetic.
I will venture to suggest certain subjects that may very profit