220 SCHOOL SCIENCE A^D MATHEMATICS

A DISCUSSION" OF THE REPORT OF THE COMMITTEE ON

ALOEBRA.^

BY W. H. WILLIAMS,

State Normal School, Plattemlle, Wis.

The widespread agitation in respect to the teaching and con-tent of secondary mathematics has, I believe, already resultedin considerable improvement in the teaching of the subject.There are, however, some dangers in the agitation. It givesa splendid opportunity to the man of one idea, though, fortu-nately, we have few of these teaching mathematics. There is no

great harm in this. But a man has worked along a new line an’dhis faith and enthusiasm in the departure he is making have madehis work eminently successful. Others following in his tracksbut without his faith or enthusiasm make a failure. I mentionthis because a report that is likely to be adopted in whole or inpart by^ a large number of teachers ought to be conservativeand to contain few, if any, suggestions of changes that are notlikely to prove successful in the hands of the average teacherusing the text-books at his disposal.The present report is to be commended in this respect. The

committee has evidently given the subject careful considerationand suggested only changes that have met the approval of alarge number of teachers. The committee very wisely makes itclear that the most of the recommendations are suggestive only.What may be best for one class is not for another. What isbest for one teacher is not for another, and were it possible tohave classes exactly alike, the wise teacher will, from year toyear, vary both the mode of presentation and, to a less degree,the subject-matter.

I am in hearty sympathy with most of the suggestions made.I am inclined to disagree with the main purpose of the firstcourse. As Professor Cajori points out, early 4n the last centurytoo much emphasis was laid upon the solution of problems andin the reaction the pendulum swung too far the other way, andnow, as it is swinging back, there is danger that the place ofthe problem will be over magnified. The problem should havea larger place in the elementary course, but, in my opinion, notthe chief place. I agree with Professor Stone when he says.

*Read before the Mathematics section of the C. A. S. & M. T., November 28, 1908.

DISCUSSION OF ALGEBRA REPORT 221

"The power to interpret fully the meaning of the general num-

bers of a formula or of an identity is of greater value than the.power to solve a problem."

Some may object that this is too difficult for the beginningpupil. He must of necessity begin to interpret at the com-mencement of his study of algebra, and this phase of the workshould be kept in mind throughout the course, and his .progressis measured largely by his progress in this direction. It is -truethe problem gives excellent drill in this work, and in other re-

spects it has great disciplinary value, and if problems are sochosen and arranged that the algebraic principles are developedfrom them, as Professor Slaught suggests, the place of the prob-lem will be greatly enhanced. Nevertheless its place is, in mvopinion, a subordinate one.

It must not be lost sight of that much of the mechanical workought to be done at as early an age as possible, hence a sec-

ondary aim in the first course must be the acquiring of skill inthe mechanical work. If the work is carried out wisely, I be-lieve it has much more disciplinary value than is generally cred-ited to it.How much demonstration work would best be given will de-

pend upon the class; as a rule, not much, but just what themajority of the class can get -hold of without undue waste oftime. The teacher should not insist too strongly on the pupilgiving formal proofs, but, as the report suggests, he should beled to see the reasonableness of the principles.

I agree with the committee quite well as to the topics thatmay be omitted in the first course. If square root of polyno-mials is taken, I prefer to give a little work in cube root to showthat the method of .extracting this root is developed in the sameway as that of square root. The teacher may find it wise attimes to take up one or more of these topics in a first yearcourse but, as a rule, I believe it best to omit all of them.

I question the wisdom of teaching multiplication and divisiontogether. It seems best to me to give a considerable drill onmultiplication, introducing division incidentally that the pupilmay be prepared when he comes to it. Then when long divisionis studied, by all means bring out its relation to multiplication.

I have the same thought in regard to teaching exponents andradicals. I prefer to teach exponents first, and then, in thestudy of radicals, make free use of the fractional exponent and

222 SCHOOL SCIENCE A^D MATHEMATICS

call attention to the fact that the radical is unnecessary; and

that were it not that the radical was in use before the exponent,we probably would not have the radical form.

I am glad for the suggestion of the early introduction of the

quadratic equation. The solution by factoring immediatelyafter factoring indicates a use for ’the latter and a use that de-

lights the pupil. In addition to this, it introduces a method ofsolving equations that is applicable so far as it is possible to

find linear factors. We often find the solution by factoringeither omitted altogether or barely noticed after a considerabledrill in completing the square has been given. The early intro-

duction makes the more thorough mastery of the quadratic equa-tion probable.

In the selection of problems, the more satisfactory problemswe can get from the sciences, from geometry, and from every-day life and experience, the better. Our problem sets unques-tionably need enriching; but care must be taken to see that theproblems chosen are not too long in statement and that the termsused are understood by the pupils. The-old, time worn problemthat tells its story briefly and to the-point is preferable to themodern problem that tries the patience of the pupil to read itthrough.Many of the formulae of physics and geometry can be given

and should be solved for each of the letters used, but in choos-ing problems from the physics, I should be even more cautiousthan the report suggests. I realize that under ideal conditionsfor such an experiment the physics and algebra may be workedout side by side successfully, but choosing problems from thephysics before the pupil has studied the subject is likely to resultin loss of time- and may tend to create a dislike for both physicsand algebra.

In closing I wish to say that I am in hearty sympathy withthe movement to unify secondary mathematics. A larger useof arithmetic and algebra in geometry and of geometry in alge-bra are steps in the right direction and will prepare the way fora still closer union of algebra, geometry and trigonometry. Ihave faith to believe that a course in secondary mathematics canbe worked out that will have many advantages over our presentcourse and few if any disadvantages.