44 
NATURE 
[Marcu 14, 1912 
Reaching the South Pole, discovering the end of 
Ross’s Great Ice Barrier, and making the first land- 
ing on King Edward VII. Land is a remarkable 
triple achievement, and the Norwegian expedition has 
certainly gained results of first-rate geographical 
value. Dr. Nansen is to be congratulated on the 
latest success of his school of polar heroes. 
THE TEACHING OF MATHEMATICS. 
Pee papers before us on ‘‘ The Teaching of Mathe- 
matics in the United Kingdom” are published 
by the Board of Education as special reports on educa- 
tional subjects. Each paper of the series (eleven papers 
are now before us) is written by an expert on the 
particular subject he treats, and their substantial 
. agreement on educational principles shows the revo- 
lution which has taken place in the last decade, and 
is still taking place in mathematical education. 
Last century the subject was taught on the most - 
conventional lines. Few thought of comparing the 
values, for either mental discipline or knowledge, of 
different portions of the subject or of different methods 
of teaching. Such books as the “Inventional Geo- 
metry’ of Herbert Spencer’s father proves the exist- 
ence of occasional thoughtful men; but in the deadness 
of the time such books were lost sight of until re- 
discovered to-day. 
The reformers of the later nineteenth century dealt 
with rigour of proof and completeness of logical 
development. They aimed at doing for other branches 
of mathematics what Euclid had done for geometry. 
A system of mathematics in which the whole subject 
develops by irrefragable reasoning from a_ small 
number of assumptions is a lofty ideal and is an 
entrancing occupation for certain mature minds; but 
the school is no place for it. The examination in 
recent years of attempts at such a system, Euclid’s 
included, leads to the view that no system can do 
more than approximate roughly to the ideal; the 
statement of the preliminary assumptions cannot be 
made complete or the logical development rigorous. 
This conclusion has added strength to the arm of the 
band of reformers who hold that this ideal, even if 
attainable, is out of place in the school. 
These reformers recognise that the boy’s mind is not 
the adult mind writ small, that reasoning power 
develops from an approximate zero in the infant to 
something far short of perfection in the adult; per- 
fection of reasoning not being attained even in the 
greatest mathematicians. Consequently they replace 
this ideal of logical perfection by the ideal of a course 
suited at every age to the mental development of that 
age, both in matter and in method of presentation. 
1 Board of Education. Special Reports on Educational Subjects. ‘‘The 
Teaching of Mathematics in the United Kingdom,” being a Series of Papers 
prepared for the International Commission on the Teaching of Mathematics. 
(x) “‘ Higher Mathematics for the Classical Sixth Form.” By W. Newbold. 
Pp. 14. Price 1a. 
(2) ‘The Relations of Mathematics and Physics.” By Dr. L. N. G. 
Filon. Pp. ii+g. Price rd. 
(3) ‘The Teaching of Mathematics in London Public Elementary 
Schools.” By P. B. Ballard. Pp. ii+28. Price 2d. 
(4) ‘The Teaching of Elementary Mathematics in English Public 
Elementary Schools.” By H. J. Spencer. Pp. 32. Price 24d. 
(5) ** The Algebra Syllabus in the Secondary School.” By C. Godfrey. 
Pp. 34. Price 24. 
(6) ‘* The Correlation of Elementary Practical Geometry and Geography.” 
By Miss H. Bartram. Pp. ii+8. Price 1a. 
(7) “The Teaching of Elementary Mechanics.” By W. D. Eggar. 
Pp. ii+13. Price rd. 
(8) ‘* Geometry for Engineers.” By D. A. Low. Pp. iit+-15. Price 14d. 
(9) ‘“‘ The Organisation of the Teaching of Mathematics in Public Second- 
ary Schools for Girls.” By Miss Louisa Story. Pp. ii+-15. | Price 12d. 
(ro) ‘Examinations from the School Point of View.” By Mr. C. 
Hawkins. Pp. iit+104. Price 9d. 
(x1) ‘* The Teaching of Mathematics to Young Children.” 
Stephens. Pp. iit+1o9. Price 14d. 
NO. 2211, VOL. 89| 
By Miss Irene 
The matter must in the earliest years be entirely con- 
crete, and must gradually become more abstract with 
the increasing age and power of the pupil. It should 
never become entirely abstract, to the exclusion of the 
concrete, for even in its highest developments mathe- 
matics is merely a tool for ultimate application to 
concrete problems. It is true that it is an economy 
of labour to have a few mathematicians who work 
chiefly in the abstract and improve the tool for others 
to use; but even for these few some knowledge of 
concrete problems has value for the proper direction 
of their efforts. 
The method of presentation must likewise have 
regard to the age of the learner. At first there is 
little reasoning, the teacher’s object being to provide 
in connection with concrete material the abstract 
ideas for later reasoning, as well as to give precision 
to such abstract ideas as the pupil already possesses. 
In the earlier stages evidence is chiefly experimental 
and intuitional. By appropriate training and increase 
of years the mind develops and demands more logical 
evidence. The evidence, suited always to the needs of 
the pupil, and restricted to the kind which he asks 
and can grasp, gradually approaches that Euclidean 
form at which the nineteenth century aimed. 
The choice of material out of the various branches 
of mathematics is important in two ways. The first 
and obvious criterion is that, other things being equal, 
the branch which has a direct use in after life, a 
“‘bread-and-butter” value, is to be preferred to the 
branch which has not. The other is that the branch 
which is the better mental gymnastic is to be chosen. 
Fortunately these two criteria generally indicate the 
same branches, the bread-and-butter subject by its 
relation to life exciting an interest which. goes far 
to give it the preference as mental gymnastic. 
The above views run through most of the eleven 
papers now under review. The battle was first fought 
in the secondary school, and has been won there as 
far as the principles are concerned, the questions now 
at issue being the working out of courses founded 
on them. The principles are being brought to bear 
even on the classical boy, naturally enough the last to 
be affected by a reform in mathematics. In the first 
paper of the series, ‘‘Higher Mathematics for the 
Classical Sixth Form,’’ Mr. Newbold shows how, in 
place of the dull committing to memory of Euclid’s 
propositions, such a Form has, by a discussion of 
problems of everyday life, been given a real and use- 
ful grasp of the ideas of the infinitesimal calculus. 
In the universities the battle for the new principles 
is beginning, and Dr. Filon, in his paper on ‘‘The 
Relations of Mathematics and Physics.’’ does yeoman 
service. As evils requiring regulation he names 
“(7)  mutual misunderstanding due to _ over- 
specialisation; (2) the accumulation of uninterpreted 
material in physics and of abstract concepts in mathe- 
matics; (3) the neglect of applied mathematics.” 
It is unfortunate for the mathematical students at 
Cambridge that in the rearrangement which admitted 
physics to a position of consequence, that subject was 
placed in a tripos distinct from mathematics. Since 
this estrangement between the two subjects, Cam- 
bridge has produced no mathematicians to compare 
with giants like Kelvin, Stokes, Clerk Maxwell, and 
Sir J. J. Thomson. Recently a move in the right 
direction has been made in the attempt to combine 
the early training of mathematicians, physicists, and 
engineers; but the success of such a scheme requires 
more than the revision of regulations. 
The third and fourth papers are on “The Teaching 
of Mathematics in Public Elementary Schools.” In 
these schools the position is somewhat disappointing. 
The teachers are slow to avail themselves of the free- 
