MATHEMATICS.- IV by J. H. Haughey

Training-School Pub. Ass'n, Ltd. · Battle Creek, Michigan · May 1900

FromTraining School Advocate, Vol. 2, No. 5.

MATHEMATICS.— IV.

By J. H. Haughey

BY J. H. HAUGHEY.

Theory and Practice.

The progress of knowledge is from the known to the unknown, from the concrete to the abstract, from the unit to the universal; but just as one should not lose sight of the known in his search after that of which he is as yet ignorant, or of the one in his knowledge of the many, just so he should not permit the relation between the abstract and the concrete, in passing from the one to the other, entirely to leave the mind. There is always an objective point toward which the mind is moving in its search after truth; and this should be clearly apprehended in its relations to the physical, mental, or moral conditions imposed. More than this, not merely the knowledge, but the utility of the knowledge, to be gained, should form the constant incentive in its pursuit.

The small as well as the large child naturally asks why as well as what and how; and he is not satisfied to receive for a reply any mere hollow, impractical, empty thing. The right answer to this question arouses the desire to know; and desire to know is a necessary condition to knowing. Truth is revealed to those who long for it. Indeed, knowledge and wisdom, repentance and righteousness, are given only to those who hunger and thirst for them. But what one longs for, he will also seek after; so the Lord himself says, “Ye shall seek me, and find me, when ye shall search for me with all your heart.” The first and continuous effort, therefore, of every successful teacher is to arouse and sustain the interest of the pupil in the subject taught by directing the mind toward some good to be obtained as the result of the intellectual efforts made.

Mystery, or the fact of one's ignorance, or of a truth's being, to the student, in the regions of the unknown, also acts as a powerful incentive to some minds, towards a desire for knowledge. But even here the essence of goodness must be seen to permeate the mystery, or the result will be a vain philosophy growing out of mere curiosity or vague imagination. It is because the “mystery of godliness” is the gospel of salvation, that men are attracted to its study, and that angels desire to look into it. It is seeing the infinite value of the soul in the light of Calvary that expands and enlarges the mind's powers to grasp truth to an extent so exceedingly beyond that derived from the solution, or attempted solution, of any other problem. Yet the correspondence between the laws of creation and the laws of redemption are such as to lead the mind step by step into a clearer conception of this great truth. He who studies mathematics, therefore, or any other science, only in the abstract, as pure reason, and merely for the intellectual benefit to be obtained from wrestling with difficult problems, no matter how complex, can as yet hardly be said to have taken the first step in the divine progress of what he may be prone even to consider his favorite science.

Scientists and educators are advocating and urging more and more that theory and practice should go hand in hand in the teaching of the higher mathematics as well as in the teaching of arithmetic. The well-known astronomer, R. A. Procter, in his little work on the calculus, gives a very suggestive title: “Easy Lessons in the Differential Calculus: Indicating from the outset the Utility of the Processes called Differentiation and Integration.” And in the preface, speaking of the time after he left college, he adds, “What I have had occasion to study since, respecting the differential calculus, the calculus of variations, and higher matter, I have dealt with as occasion required.—the only really effective way of studying mathematics.”

Another writer, Truman H. Safford, Professor of Astronomy in Williams College, in his brief work on mathematical teaching, says: “Just so far as the teaching [of arithmetic] is based upon the concrete, so far is it profitable; but when the book-makers begin to make it too abstract, as they very often do, it becomes a torture to both teacher and learners, or, at best, a branch of imaginary knowledge unconnected with real life.” Again, on another page he says: “Now, in arithmetic, throughout, time must be taken away from riddles, puzzles, operations with enormous numbers, long sums on the slate or blackboard, and given to solid work in the elements of numbers, especially to mental work, to extemporalia, to practical applications of arithmetic which really mean something.” And still again in the closing paragraph we read with respect to all the mathematics: “Much of what has been taught in our schools must and will be dropped as artificial and ugly, to be replaced by that which is natural, beautiful, and useful.”

Now from what has been said in this article it is not intended to convey the idea that every problem or theorem has an immediate or direct application to some practical law or relation between man and nature, man and man, or man and God. It may constitute only an intermediate step in the solution or demonstration of a more general problem or theorem. For instance, in the study of algebra, every problem solved becomes a general formula, or rule, for the easy and rapid solution of every other problem of its kind. Thus problems are universalized and classified into distinct species, which may be recognized just as readily as those of the plant and animal creations. Again, in geometry, that useful and necessary truth enunciated in the Pythagorean proposition, and so familiar to every carpenter in its practical application, is reached in our elementary works on geometry through the demonstration of twelve other propositions, as is determined by tracing the process of reasoning back to the fundamental axioms and definitions. It has been the writer's privilege to examine

upwards of thirty different demonstrations of this proposition, some of which are very simple and easy. This illustrates that revelations of truth should never be restricted to one process for all persons.

But the object of this brief article is to call attention to the importance of not divorcing theory and practice in the study of mathematics. Where shall practical problems be found? Search your works on arithmetic and bookkeeping: look in your physiology, your botany, your geography; consult your physics, chemistry, and astronomy; look in your Bible for examples in chronology, history, and prophecy; study the measurements in connection with the tabernacle, the ark of the testimony, and the altars; the temple of Solomon and its courts; the New Jerusalem and its walls. Then consult the carpenter, the farmer, the orchardist, the gardener, the merchant, the minister, the school-teacher, or any other man in any other legitimate profession or business. Draw them out. They all have something for you.