396 



NATURE 



[January i8, 191 2 



firitt r«cof!ni!>ablf.- su^^ar, which recently obtained strong 

 support from the diwovi-ry in 1907, by Strak(>s<-h. that 

 dfxtro*!!' is the only sugar prestMU in the actual inenophyil 

 of the leaf of the sugar beet, and that cane sugar, which 

 i«i almost the only sugar in the root, first makes its appi-ar- 

 ance, together with l.i'vulose, in the lateral veins of the 

 lamina, and increases in amount in the midrib and petiole. 



From an interesting discussion of the function of cane 

 sugar in plants, with which the paper doses, the following 

 may be quoted : — " Its sp<»cial physical and chemical 

 l>rnfMTties are of interest. It is very soluble and readily 

 < rystallises — more so than the other sugars occurring in 

 plants. It is very easily hydrolysed by acids and by 

 invertase. It siiares with trehalose, alone among the 

 disaccharides, in having no reducing properties. Maltose, 

 lactose, &c., do reduce, and so may be said to have the 

 aldehyde group in their molecule functional. 



" Sucrose may thus have been selected in the higher 

 plants as the chief circulating sugar, partly on account of 

 its non-reducing properties and soluble (mobile) nature, 

 and partly on account of the ease with which it can be 

 hydrolysed into its two components, glucose (dextrose) and 

 tVuctose (Ijcvulose). These hexoses may, as a rule, play 

 listinct parts in metabolism — the glucose more readily 

 li-nding itself to the respiratory needs and the fructose to 

 (.instructive work, such as the building up of the plant's 

 framework. It is also within the bounds of probability 

 that cane sugar itself may take a direct part in the forma- 

 tion of cell-walls. Just as it appears able to be condensed 

 to starch without previous inversion, so it may be trans- 

 formed directly to cellulose in the construction of cell- 

 walls. Kenton's work is interesting in this connection, 

 lie has shown that various kinds of cellulose respond 

 markedly to a special ketose test, and thus concludes that 

 this substance may contain cue or more groups identical 

 with that present in fructose." 



THE DEMOCKATISATION OF MATHEMATICAL 

 EDUCATION.' 



'T'HE work of the Mathematical Association, in connec- 

 tion with its activity in promoting the reform of 

 mathematical teaching in our schools, necessarily involves 

 the expenditure of much time and thought upon the detailed 

 discussion of specific schemes for the improvement of the 

 teaching in special departments of mathematical education. 

 It is, however, well that we should sometimes reflect upon 

 the more general aspects of our work ; and perhaps a presi- 

 dential address afTords the most suitable occasion for 

 reducing some such reflections to an explicit form, even 

 though nothing essentially new can be said upon the 

 matter. 



In making a few brief remarks upon the general 

 character of the reform movement, I propose to emphasise 

 one or two governing principles which I regard as of 

 fundamental importance in relation to mathematical teach- 

 ing. If I venture, in the course of my remarks, to make 

 some suggestions on less general matters, the adoption of 

 such suggestions as parts of the policy of the Association 

 would only be possible after much detailed discussion of 

 the manifold points which would have to (each some degree 

 of settlement before the suggestions could be translated 

 into the domain of practice. 



The modern tendency which has exhibited itself in our 

 time in greater or less degree in all countries in educa- 

 tional policy in general may be described as the tendency 

 towards the democratisation of education. This term, or 

 some synonymous one, has frequently been used to denote 

 the extension of education to wider classes of the popula- 

 tion ; but it is not in this quite general sense that I intend 

 here to employ the expression. I mean J>v it rather the 

 progressive adaptation of educational methods to the 

 ititcllectual democracy; the transformation of the methods 

 of teaching and of the matter of instruction so as to meet 

 the needs of those who are lacking in exceptional capacitv. 

 at least in relation to the particular branch of studv in 

 question ; in other words, the concentration of the attention 

 of the educator, in a much greater degree than formerly, 



i Presidential address delivered to the Mathematical Association on 

 January 10 by Prof. E. W. Hobson, F.R.S. 



NO. 2203, VOL. 88] 



on the work of developing the mind* of the average many 

 and not solely of tho»r of the exceptionally gifted few. 

 The progress of democratization of education, in this sense, 

 haH b«fen perhaps more marked in the case of mathematical 

 instruction than in other departments. In our own country 

 thf Mathematical .\sso<-iation has been conspicuous as an 

 agent in furthering the demixratisation of mathematii .li 

 education. It is v<-ry certain that no such democratisation 

 could be effected without more or les» radical changes being 

 made both in the methods of teaching and in the selection 

 of the matter taught. It would b«> of but little avail that 

 the attention of the teacher should be concentrated in a 

 greater degree than formerly on the average many if the 

 methods of teaching and the material taught remained 

 unreformed. 



With a view to the formation of some estimate of the 

 profit and loss due to the changes which have taken place 

 of late years in the teaching of mathematics in our schools, 

 let me briefly glance at some of the differences, both in 

 theory and in practice, which distinguish the older and the 

 newer methods from one another. Any exaggeration of 

 which 1 may be thought guilty must find its excuse in the 

 fact that 1 am attempting to indicate only the more salient 

 features in a continuously progressive movement. 



In accordance with the older and traditional treatment 

 of mathematical instruction in our schools, geometry was 

 treated in a purely abstract manner, the idea being that 

 Euclid, as a supposed model of purely deductive logic, 

 should be studied entirely with a view to the development 

 of the logical faculty. Any knowledge of space relations 

 which might have been imparted by this study w-as reduced 

 to a minimum by the excessive insistence on all the details 

 of the syllogistic form, the whole attention of the pupils 

 being engrossed by the effort to commit to memory a long 

 chain of propositions in which the actual geometrical con- 

 tent was exceedingly small. On the other hand, algebra, 

 and to a great extent arithmetic, were taught without any 

 regard to their logical aspects, but mainly as affording 

 discipline in the purely formal manipulation of symbols in 

 accordance with prescribed rules, little or nothing being 

 said as to the origin of such rules. The teaching of 

 mechanics was assimilated, so far as possible, to that of 

 geometry, the true position of the subject as a fundamental 

 part of physical science being almost wholly obscured. 

 That the average boy or girl is not by nature appreciative 

 of formal logic or of the interest and meaning of abstract 

 symbols was thought to be a reason why the subjects 

 so treated should be especially insisted on. 



In fact, the notion of mathematical teaching was that 

 it should be in the main medicinal and corrective. Its 

 advantages consisted largely in calling forth the use 

 of faculties which are the rarest in the average boy or girl, 

 and were therefore thought to be in special need of develop- 

 ment. It was thought to be by no means wholly a dis- 

 advantage that these subjects, so treated, were found hard 

 and repulsive by the majority. It was thought that the 

 hard discipline involved in the attempt to assimilate them 

 developed a kind of mental grit, and involved a certain 

 species of moral training, even when the intellectual results 

 were small. A certain strengthening of faith, to be 

 acquired in the process of hard work spent on subjects of 

 which neither the aim nor the utility was obvious to th< 

 pupil, was thought to be highly beneficial. 



It is unnecessary for me to enlarge upon the defects of 

 this system, and on the inadequacy of the ideals underlying 

 it. The existence of the Mathematical .Association is a 

 warrant of the widespread dissatisfaction with these 

 methods, both in their results and their aims. The system 

 as it existed in our schools was condemned by its failure. 

 It failed to attain even its own narrow ideals, except in 

 the case of a very select few among the pupils. The 

 many rejected the material which was for them wholly 

 indigestible mental food. The system was, in the sense in 

 which I have used the ^erm, undemocratic. The results 

 obtained in the case of the vast majority were deplorable : 

 and it needs indeed a strong faith in the anti-democratic 

 principle to imagine that this failure was compensated by 

 the effect of a hard and bracing training on the few who. 

 by mental constitution, were enabled in some degree to 

 profit by it. Even the chosen few suffered severely from 

 the effects of the narrow conception of education which 

 lay at the base of the methods of instruction ; for the 



