Re: Elements of Mathematics (MS 165)

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NEM 2, pp. 4–6
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PREFACE (2) (94a) (concl.)

Teachers now see that the difficulties of the first steps in mathematics must be divided and conquered, one by one.  The three functions of the mind that are exercised in mathematics are exact reasoning, mathematical imagination, and complex generalization.  The first of these, the logical part, is best acquired in the study of the theory of number, because that subject involves little other difficulty.  It calls for but slight efforts of imagination and of generalization.  Topology, or connective geometry, is the best field for the growth of imagination, demanding little logic and not very much generalization.  Graphics, or projective geometry, carried far enough, affords good training in generalization.  When some familiarity with the business of the mathematician has been acquired by such studies, metrical geometry may be taken up without fear that the student's mind will be confounded by its aggregation of various difficulties.

This volume is intended to contain all the mathematics (except practical arithmetic) which is necessary for a man with a good common school education, and at the same time, to give the thoughts of the student such training as may prepare him for a study of the higher mathematics.

Most of the text books of geometry have contained some algebra.  Euclid's Elements is more than half devoted to that branch.  Benjamin Peirce's work has a brief algebraical introduction.  The present volume gives all the algebra which is indispensable to an ordinary man.

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On Jul 18, 2012, at 8:00 PM, Jon Awbrey wrote:

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> NEM 2, pp. 4–6
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> 
> PREFACE (2) (94a) (cont.)
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> In the fifth place, Georg Cantor and others have succeeded in analyzing the conceptions of infinity and of continuity, so as to render our reasonings concerning them far more exact than they had previously been;  and the fundamental researches that have largely occupied mathematicians of late years, into the theory of functions, do much to render geometrical reasonings more exhaustive and precise.
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> All these intellectual movements ought, in the opinion of very many mathematicians, to have their effects upon the system of teaching the elements of the subject.
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> But this is not all.  Pedagogy is an art which has come in the last sixty years to be based more and more upon modern scientific psychology, and upon modern views of logic.  If diligent and intelligent youths find difficulty in understanding mathematics, teachers no longer deem it becoming to flog them or to objurgate them, as they used to do, in the days when the second theorem of Euclid received the name of the "Asses' Bridge."  On the contrary, they consider the fact that a considerable percentage of the best minds imagine themselves to be utterly unable to comprehend mathematical reasoning to constitute an emphatic condemnation of that old system of teaching which had such a result.
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