290 SCHOOL SCIENCE AND MATHEMATICS

SECONDARY MATHEMATICS.

BY R. L. SHORT.

Mathematical Editor for D. C. Heath & Co.

Until some five years ago little had been done by the math-ematical world toward either the pedagogy of mathematics orthe correlation of related subjects. For the most part teacherswere content not to progress. Authors had strengthened uptheir text-books to some extent, some demand had arisen formore concrete problems and a few teachers believed in the graph.It took Prof. Perry to set in motion a scheme which has largelychanged our ideas of mathematical teaching. The promotersof his ideas swung too far and urged upon teachers a plan whichwould only succeed under ideal conditions. These men prac-tically urged the abandonment of operative algebra and the sub-stitution of the workshop and measuring rod. This agitationaccomplished much that was, good. It aroused thinking, pro-gressive teachers to action, aroused antagonism, brought manynew and sane ideas into teaching of mathematics, brought to-gether the science and mathematics teacher, took the varioussubjects out of their several compartments and correlated them.And now the result? These prime movers, with one or two

exceptions, have settled down to a conservative basis, especiallyso far as algebra is concerned. The algebra of to-day meansmore to the student. Transposition is no longer a name. It isa process dependent on familiar axioms. The fundamentaloperations are the operations of arithmetic, generalized. Theequation is the central thought for which the operations preparethe way. The x, y, z, of a decade ago have lost their exclusive-ness and the physics teacher complains less than formerly thathis student cannot solve for t. And as for the graph, the manwho now stands up and shouts that the graph does not belongto algebra, shouts alone.One danger still confronts the algebra teacher. The pupil is

so interested in the practical side, the pictorial side, the geometricside if you please, that this often causes neglect of the operativeside, i. e., formal algebra. The development of the thoughtprocess is necessary, the applications of algebra are necessary.Thought processes are carried on in various subjects throughoutthe high school course. Are they to weaken formal algebra?Practically all the mechanics of algebra must be taught during

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the first year of ’the high school. A large amount of drill workis necessary during this first year. How much time shall wegive up to other things ?Up to this time geometry has been little disturbed. For this

there have been two reasons. The colleges, technical schoolsand the physics teachers have not complained that the studentis deficient in geometry. Why? The student’s mind is moreoften geometric than algebraic and the student needs no geometryin his work. A little mensuration, mostly learned in the grades,and a few additional theorems picked up in the secondary schoolamply serve his purpose. Some attempt is being made, andrightly, too, to reach geometry in a less formal way, to take awaysome of the interdependence of theorems and to leave the studentmore to his own resources in the development of theorems. Thishas two effects, partially good,, partially bad: The beauty andcontinuity of the subject are impaired, the student knows fewertheorems, has more thought power and can use his theorems togreater advantage.The tendency to favor algebra at the expense of geometry will

probably have a marked effect on our courses in mathematics.Some go so far as to declare that the year and a half each nowdevoted to algebra and geometry will soon give way to a coursein secondary mathematics, similar to those now offered in Ger-many and Italy. Such course would doubtless give us studentsbetter prepared for college, better prepared for business and withless dislike for mathematics. It should produce more mathe-matics teachers, of which there is now a dearth, and should conse-quently improve mathematical- conditions in both college andsecondary school.

FACTORING THE TYPE px’+qx+r.By BYRON E. TOAN.

Boulder’, Montana.

Of all types this is the mo’st important, not alone because of itsfrequent occurrence but also because it includes those commonforms ^-{-px-^-q, xfi-\-2xy-\-yi and x2’�Vs. In fact, it is the gen-eral form of the product of two binomials, each of which has aterm of the first degree in x. Because it is the most importanttype met in factoring, its proper presentation to beginning classes