BULLETIN
OF THE
UNIVERSITY OF TEXAS
1916: Ne. 58
OCTOBER IS 1916
The Texas Mathematics Teachers' Bulletin
(Volume 2, No. 1, October 15, 1916)
Published by the University six times a month and entered as second-class matter at the postollke at AUSTIN, TEX.AS
Publications of the University of Texas
Publications Committee :
A. C. JUDSON c. HARTMAN
E. c. BARKER J. L. HI<~NDERSON
J. M. BRYANT w. S. HUNTER
G. c. BUTTE J. A. LOMAX
R. H. GRIFI•'ITH
The University publishes bulletins six times a month. 'l'hese comprise the official publications of the University, publications on humanistic and scientific subjects, bulletins prepared by the Department of Extension and by the Bureau of Municipal Research, and other bulletins of general educational interest. With the exception of special numbers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publications should be addressed to the Editor of Univernity Publications, University of Texas, Austin.
A. C. BALDWIN a SONS: AUSTtN
B~36-916-llh
BULLETIN
OF THE
UNIVERSITY OF TEXAS
1916, No. 58
OCTOBER 15 1916
The Texas Mathematics Teachers' Bulletin
(Vol. 2, No. 1, October 1;;, 1016)
Edited by
J. W. CALHOUN,
Adjunct Professor of Pure Mathematics,
and
C. D. RICE,
Associate Professor of Applied Mathematics.
This Bulletin is open to the teachers of mathematics in Texas for the expression of their views. The editors assume no responsibility for statements of facts or opinions in articles not written by them.
CONTENTS.
Methods in Teaching Mathematics .. ........... M. B. Porter . ....... 5
Literal Arithmetic............................c. D. Rice........... S
On Definitions ................................J. W. Calhoun .......11
Seeing Things................................M. B. Porter ... .. ...14
A Reprint. . ...... .... ............ . ... . .. .....r. W. Calhoun . .. . ...18
What Great Men Say About Mathematics ....... . . . ...........22
The Straight Edge.. . ........... .. ....... A. Nonymous .....23
Published by the University Six times a month and entered as
second-class matter at the postoffice at
AUSTIN, TEXAS.
The benefits of education and of useful knowledge, generally dift'usecl through a community, are essential to the preservation of a free government.
Sam Houston.
Cllltivated mind is the guardiaa genius of democracy. It is the only dictator that freemen acknowledge and the only security that freemen desire.
President Mirabeau B, La.Jwuo.
NOTICE
Owing to the fact that only a small number of Vol. 1, Xo. 1 of this Bulletin was printed, and there have been many requests for copirs that conld not be supplied, the articles in that number will be reprinted from time to time during the current year. One article from that number appears in this issue.
ME'l'HODS IN MATHE:\IATICS
By methods we shall mean the methodology of teaching, and here we begin by saying that such methods are as diverse as the capacity of the students and as varied as the personalities of the teachers. Journals concerned with the teaching of elementary mathematics are numerous in all the civilized languag-es, and these swarm with papers large and small on the handlmg of certain snhjects as a whole, and in the t1·eatment of certain topics or theorems, and while of very unequal merit they often contain suggestions that a good teacher will find worth trying. The real teacher '"'ill nse snch material only in so far as it harmonizes with his own individual methods, and only so far as he finds that he can get good results with it. For there is no bf'st method of presenting any particular topic any more than there is any one best method of painting a picture or playing the piano. A skillful teacher is an artist whose material is the most subtile and fluid in existence', namcl~r, the minds and characters of his pupils. It is his delicate and difficult task to induce a certain series of reactions and activities in these minds, and, since they vary between wide limits, his methods must be as varied as the material on which he operat('S. The learning proce!'s is complex and while the meagre data of psycholog-y furnish nsefnl hints. they fall far short of adeqnatc guidance, and only endless experiments and patience can reveal the best methods of gaining the desired end.
Trrn END IN V mw
The teacher of mathC'matics has for his main pmpose to train his pupils to think mathematically. This docs not necessarily imply the memorizing· of snch and snch proofs bnt means the acquirement of certain habits of thought and the mastering of certain processes or 1cays of going about things. The mere ability to state and prove from memory a list of theorems and formulas may imply no mastery whatever of a subject, and the most incessant drill may accomplish nothing more than disgust and
Bulletin of the Um:vcrsity of Te.ras
fatigue, though intelligent drill is indispensable in all good teaching. What is essential is that the pupils' interest in the essential thinking process should be aroused, directed, and developed by the careful choice of material that is within his powers of comprehension, and that he shonld feel steadily a pleasurable sense of effort and power in surmounting difficulties that before seemed insuperable. The feeling of effort is essential. It is poor praise of a teaehcr to say that he makes mathematics easy and the highest praise to say that he helps make the serious efforts to learn it pleasnrable. '!'his is the most he can do, and if he docs it, his students \Yill rise up and call him blessed and even if they do not love him will certainly respect him. There is no such thing as making the real learning of mathematics easy, and it is fortunak that it can never be made so, for then its disciplinary valne, its inculcation of careful, accurate habits of thought would be gone.
There are still some people old-fashioned enough to believe that the mind can be trained, and some still more old-fashioned that believe that. mathematics lends itself admirably to such training. The writer is one of these and while he believes that the so-called vocational value of mathematical knowledge is of little value out_<;ide of the careern of science and engineering he believes that it furnishes an indispensable key to the comprehension of the so-called exact sciences. There is hardly a law of Physics or Chemistry that is clearly understandable without some mathematics; not mathematics as a mysterious and incomprehensible tool, hut as a process of thoug:ht, a way of thinking.
We do not mean to imply by this that our text-books should swarm with problems of Physics and. Chemistry. The data of such problems are usually incomprehensible until these subjects have been carefully studird, but the understanding of either of them postulates mathematical training·.
The most precious resnlts of the training we have in mind is the power of inrlependent thought. This is often confused with the knack of invention, bnt the two things are really distinct, for we have some pnpils with good accurate minds who are largely devoid of this power of imag·ination. Both classes must he cared for by the eonscientions teacher. He mnst give to each
The Texas Mathematics Teachers' Bulletin
according to his capacity and special aptitude. Both classes may be taught the meaning and essence of the method used, but the first class only will clearly apply it to new problems. The reader may ask what has become of methods; what method would you apply in teaching such and such a topic? We hope that it has been made clear from the foregoing that there is not one, but there are many methods. A good teacher always has a good method, no matter what it is, and a poor one has a bad method no matter what authority he can cite to prove its excellence, just as the best paint, brushes, and canvas will not make a painter out of a dauber, while some artists have done creditable work with the most primitive tools. Sense, sympathy, and observation cannot be replaced by any method. no matter how plausible, and given these attributes with the necessary knowledge of the subject, a good method with surely result. A teacher gifted with these attributes will always be a learner and while he may not greatly increase his fund of information in the subject taught, he will certainly be always learning something new concerning the complex learning process, important truths given in no text book, individual peculiarities and limitations of his pupils analogous to the constitutional peculiarites of a doctor's patients. And this is one of the principal justifications of the teacher as a useful member of society. If he cannot deal with the individual he is no better than a text-book for self-instruction, and his function degenerates at most into that of a policeman whose whole duty it is to see that his pupils learn "what is in the book." Fortunately there is much to be learned that is not in the book and the good teacher is here wonderfully helpful.
Bulletin of the Un·iversity of Texas
LITERAL ARITHMETIC
In the last bulletin some notion was given in regard to the manner in which students may use formulas and statements already g.iven in letters. It was seen that all rules of Arithmetic may be given in the "short-hand" statements with letters. It is now our purpose to suggest a method of proceeding so that pupils may learn to use this new language or mode of expression and hence think in letters instead of numbers. To gain such a power on the part of the pupil some little time and training will be necessary. It must come through practice and drill hy the teacher. Nothing will be so valuable here to the pupil as a skillful teacher in regular outlined lessons extending over a period of time sufficient to enable the pupile to gain this power of thought expression. One or two recitations per week during the last year before Algebra proper is begun is suggested.
After a sufficient amount of drill has been given in the substitution of numerical values for letters in formulas, the nature and purpose of literal statements are more clearly seen by the pupil and in this way he will begin to realize that he is learning to write and form general statements which may have many particular applications. The teacher will find also that the pnpil gains in power in making general expression, just in the same proportion that particular llpplications are understood.
In regard to the signs and symbols of operation, we find the signs of addition, subtraction, etc., are now taught in the earlier grades and most pupils understand their meaning and use, especially in the earlier literal notation. The multiplication of the two numbers represented by a and b by the expression ab will be taught in the first work with formulas.
Following a sufficient practice in making numerical substitutions in well known rules expressed by letters, some work could be given by asking the pupils to write out in this "shorthand" method rules written in words or given orally by the teacher. 'l'hus the pupil could be asked. for instance, to write out a rule:
(1)
for the area of a rectangular floor,
(2)
for the length of a floor when the area and the width are given,
(3)
for reducing dollars and dimes to cents,
(
4) for finding the volume of a room,
(5)
for finding the total interior surface of a room exclusive of the floor,
(
6) for finding the height of a room when the volume and area of the floor are given,
(7)
for finding the principal at interest when the time, rate and per cent are given,
(
8) for finding the rate when base and percentage are given,
(9)
for finding the radius when the area of a circle is known.
The Texas Mathematics Teachers' Bulletin
In this manner every rule of Arithmetic may be used in different ways and studied from different viewpoints in this new way of writing them.
Formulas of a more general type may now be taken np. Examples should be taken from facts familiar to pupils. The algebraic or literal setting of any concrete fact or set of facts is possible only when a real knowledge of those facts is in the possession of the pupil. Here the skillful teacher must make the selection of the material to be used. The following examples arc offered as suggestions only-the teacher with a knowledge of the ability of his class may be able to use to greater advantage other problems.
(
1) A boy begins working for a dollars per month and has his salary increased by d dollars every month. What will be his salary in t months?
(2)
Write the value of the nth term of the seri<:.s.
(a) 3, 5. 8, 11. .... .
(b) 7, 11, 15..... .
(c) 1.3, 1.6, 1.8, 2.2 . .... .
(3)
A tank of V gallons is fed by two pipes one supplying 1.5 gallons per minute and the other 2.5 gallons per minute. Give formula for finding
(a)
the time tit will take the first pipe to fill the tank
(b)
the time tit will take both pipes to fill the tank
(
4) A tank of V gallons is fed by a pipe supplying 1.5 gallons per minute and is emptied by a waste pipe that carries away 2.G
B1tlletin of tlie University of 1'exas
gallons per minute. If the tank is full and both pipes begin to run, find the time t it will require to empty the tank.
(5) Two automobiles start from the same place and travel in opposite directions at speeds of a and b miles per hour. Find the formula for expressing
(a)
the distance d they are apart after t hours
(b)
the time t required in order that they may be the distance d apart
The following is taken from Prof. Nunn 's book on the Teaching of Algebra: ''In connection with this topic it is hardly possible to lay too much stress upon the importance of cultivating a neat and orderly way of setting down the steps in an arithmetical or algebraic argument. A piece of algebraic symbolism should be as capable of straightforward and continuous reading as a passage from a newspaper. To achieve this end the teacher will find it a sound rule never to permit a line to contain more than two expressions connected by the sign of equality and to insist upon the pupil 's setting the signs of equality, in successive lines of the argument, directly underneath one another. 'rhus such expressions as
V Bh=32.7 X12.4= 405.48 C. Cm
should be excluded from the exercise book and from the black
board in favor of the arrangement:
V=Bh
=32.7X 12.4
=405.48 C. Cm
ON DEI<'INITIO~S
As a small boy the writer reealls J'('nding i11 ~IcGntfoy's Pourth Reader that "words enclosed in a parenthesis shoulrl be read in a low tom• of voice and may be left out entirely without obscuring the sense.''
An examination of a good ir students convinces him that definitions in geometry are pretty generally regarded by teachers and students in this i;;arne light. They might be left out entirely without having tlwir absence noted.
This is exceedingly unfortunate. In th<' study of geometry definitions are of prime importance. There can be no real and orderly progress till this is realizepectively to the three sides of the other. (Reference to this theorem: 3 side..<>=3 sides.)
Theorem
If two parallel lines are cut by a transversal, the alternateinterior angles are equal. (Reference to this theorem: Alt.-int.< ofl!lines.)
Theorem
When two lines in the same plane are cut by a transversal, if the alternate-interior angles are equal, the two lines are pnrallcl.
(Reference to this theorem: Alt.int.< are=.)
'I'he following cnts are published by permission of the author, Professor II.. A. Morrison.
Bulletin of the University of Texas
STATEMENT OF PROBLEM
Elements Given Construction
Proof -----·----------------·------
Argument Reason
7.
7.
8.
8.
---------------!------------·------·
---L -----·----------!-L -----·----------··---·
lfl. 10.
-------------!------·--··-----------------
!o~'. ·· '=-=-="--'-·---"--~~~~
--.~·-,··~=···~==-c~-"""'====~~~-'-~-~-·-· =
[ 1:1 ·1
The Texas Mathematics Teachers' Biilletin
THEOREii
Hypothesis or Given Conditions I Construction
2.
t. ~~~itL ~
c,
.. 3•
.
To Prove
~«PB
1.
\
CM r shows its traces in the comprehension of the development of civilization and the ability to pal'ticipate in the further tasks of civilization.
Unterrictsbliitter f iir Jfa.theniatik 11 nd Naturwissenschaft.
The Texas Mathematics Teachers' Bulletin
THE STRAIGH'l' EDGE
Do your students think mathematics dnll ? l\faybc it is not the mathematics.
* * * * *
Do yon think mathematics in need of being "vitalized" 1 Maybe so, but a lot of ns teachers need it worse.
* * * * * *
Do the students in yonr school do better in everything· else than mathematics? That is not the fanlt of either the stndeut:-; or mathematics.
* * * *
The teacher of English knows where his snbject leads, ditto the science and domestic eeonomy teachers. How far beyond the high school course in mathematics have yon explored the snbject 1
* * * * * * *
Have you polished np your i