DISCUSSION. R23 
;he center, under pressure, and it develops in the direction of the diameter, and we will 
stick to that point. In the circle there is always a central force. 1 would then say that 
in the elementary school the child must learn to analyze these forms and show the rela- 
tion of these things to the creation of art. Then, at a subsequent stage, perhaps in the 
so-called grammar period, the child may well learn, as he learns the general laws of 
arithmetic, each form connected between special arithmetic and algebra. He may learn 
sonstructive geometry, a geometry in which he learns to discover certain inceptional 
laws ; and which, on the other hand, teaches him to make up in definite, accurate, 
sxact ways, with the help of an instrument, if you choose, certain forms in which he 
may learn to draw the plancs that are conceivable to him, If he then comes to the 
secondary school, it seems to me, this question of which shall we take first, algebra or 
geometry, will almost solve itself. We will have to take them together. If we take 
geometry we must have quantitative knowledge. 
Let us take alway geometric study in that form in which it is open to us by the algebra 
which the child possesses. Whether or not the Euclidian geometry would answer that 
ourpose I do not know. It seems to me it is a logical study rather than mathematical. 
It is conceivable that a student may go through all the books of Euclid without going 
to form. There is a phase which I have found that makes the Euclidian geometry a 
oarticularly unfortunate form for the child to use. For instance, in drawing I have the 
children in the elementary departments draw the life of the plants, and the life of 
‘he animal, thus telling me what they see ; and they give me a living representation 
of the animal. . 
I leave the question as open as it was before, simply because I cannot solve it. I 
aave attempted to place before you certain considerations which should guide us in this 
oroblem, that for a long time must remain a problem for us all. 
> 
1 
+ 
1 
v 
e 
1 
e 
d 
@ 
y 
B 
Kg 
r 
3 
3 
e 
ud 
1c 
i= 
mn 
Dn 
«(1 
1e 
J 
Amy Ravson, of New York: I have had some experience in both methods of educa- 
ion, both in the United States and England. In the United States we always take, 
0 far as I have observed it, algebra for two years and geometry for one. In England, 
where for some nine or ten years I have been engaged in preparing pupils for the uni- 
versities, we study both together. In my own experience I am inclined to the idea that 
hat is probably the most thorough plan, because it gives a longer time for the mathe- 
natical ideas to enter the pupil’s mind. To get a due amount of theory distinctly it 
seems to me the time must be allowed. In a meeting in New York, about three months 
wgo, after a paper by Professor Stafford was read, a great many of the mathematical 
teachers agreed that the preparation given to the pupils was not sufficient. . The corree- 
jon for this was not that more subjects should be studied, but that a more thor- 
ough course should be taken. I think there should be a longer time. I would sug- 
vest three years. You might begin algebra before geometry. Take algebra first for 
she half year, or a whole year, and then begin geometry, and carry on the algebra. 
What makes me incline to this opinion is that children’s minds move so very slowly. 
But this would give them, taking the two together, more time for the ideas of both to 
vet into their heads. As Emerson says. ¢ Knowledge comes, but wisdom lingers.” 
Proressor H. H. Spayp, Principal of Schools, Minersville, Pa. : I believe in the uni- 
ication of the studies in the schools of our country. Call it by whatever name you 
slease. I have repeatedly asked teachers of geometry in the smaller towns in Pennsyl- 
vania whether their pupils comprehended what they were studying, or whether they 
simply memorized. Many of the teachers confessed to me, professionally, that a great 
leal of it was memory work, The pupils did not see the life and soul of that which 
-hey were studying. They saw merely the form, and they scarcely saw the form. They 
:ommitted the demonstrations, I was pleased to hear from both speakers the idea of 
\nventional geometry. I hope the day will come when we shall have unification on this 
subject ; when we shall draw from nature, and geometry will come hand in hand with 
sther studies. 
[ have noticed again and again that the drawing we taught in our public schools was 
entirely ignored when the boys went to college. Now, I hope some mathematician will 
ake up this subject and give us a little book on the line of Spencer and Hill. I am 
waiting for that Look. I am also waiting for a book on algebra that will more fully 
cover the subject and begin a little lower down. I think the day will come when we 
shall be compelled, not only in mathematics but in all our studies, to see them as a 
rrand whole, and not ten or fifteen, in the school machinery. It is a sad confession 
shat many of our pupils leave our schools who have been taught so much and know so 
ittle. 1 speak from sad experience—how much I was taught and how little I know !