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A quaint old teacher has remarked " that when a learner is at first rightly and thoroughly grounded, the rest of the work goes on with readiness, with ease, with speed, and with assurance. When he is ill-grounded, all falls out contrarywise; much labour and much patience of the master, and much diligence and industry of the scholar will hardly be able to rescue him from the mischievous consequences of previous ill-grounding. So powerful is ill habit when it hath once got hold, and so difficult to eradicate, that it is much harder to unteach the wrong than to teach the right.”

Another eminent teacher of modern times has stated the cause of his success in these words: “It has ever been with me a principle that even rudimental teaching must be thorough, and must be founded upon the essential character of the mind, as accretive; and that when once it is rightly proceeded upon, the teaching of any science will not remain quiescent, but will advance to completion and success.

The present state of teaching the Elementary Principles of the Sciences of Number and Space is not satisfactory, as will appear in the following extracts from the Report of the Cambridge Board of the University Examinations, and from the twenty-second Report of the Syndicate appointed to conduct the Examination of Students not members of the University.

The Board of Examinations issued their Report of the Previous Examination of the Freshmen in March, 1880. With respect to the subjects of Elementary Arithmetic, Algebra, and Geometry (which are supposed to be acquired at School) the following remarks appear in the Report :

In the June Examination seven per cent. of the candidates failed altogether and twenty-six per cent. only just passed in the subject of Arithmetic. Questions on Compound Interest, Discount, and Stocks, as well as those on Simplification of Decimals, were answered by very few.

In the December Examination, four per cent. failed altogether and ten per cent. orly just passed. Questions in Stocks and Discount were answered incorrectly by a large number of the candidates, their work shewing ignorance of the meaning of the questions.

In the December Examination in Euclid, one quarter of the candidates did not write out more than half of the bookwork correctly. Many of the candidates had apparently prepared the subjects from text-books in which the order of Euclid is not preserved. By this means great confusion is introduced.

In June, four-ninths of the candidates failed in Algebra ; in December, one quarter. Questions in Ratio, Proportion, and Variation were not well done, and in many cases the candidates did not appear to have read these portions of the subject.

Of the Additional Subjects the Report states :

In Algebra undisguised pieces of bookwork were in general written out correctly, and an easy Problem of a common type was solved by most of the candidates. The Progressions were treated without much thought, the formulæ being often applied at random,


The Regulations for the Previous Examination revised in 1849 required the candidates to be arranged in two classes :

The first class, consisting of those candidates who have passed their examination with credit; the second, of those to whom the Examiners have only not refused their certificate of approval.

The facts stated, and the words employed in the Report suggest a very unsatisfactory state of the acquirements and of the intellectual habits of a large class of Freshmen in their first year of residence at Cambridge.

In the year 1879 the names of 6738 students were entered for the Cambridge Local Examinations, and forty-one per cent. of those who attended, failed to pass the examination. There were 3600 students examined in Pure Mathematics, including Elementary Arithmetic, Algebra, and Geometry.

On the results of the Examination of the Junior Students the Report of the Examiners remarks :

In Arithmetic,

There were fewer cases of confused and bad methods, fewer gross errors in principle ; and the papers were worked in better form and more neatly. But, as compared with last year, improvement is not manifest.

In Algebra,

Although the failures were more numerous this year than last, a fair per centage of the whole number of candidates acquitted themselves creditably.

In Euclid,

Several lost credit by quoting axioms other than Euclid's, and by inattention to Euclid's order, for instance, proving I. 27 by I. 32. More riders than last year were sent up, but still too few of the candidates attempted them.

Of the Examination of the Senior Students the Examiners state :
In Arithmetic,

The work both of boys and girls shewed an improvement on that of last year, as regards both style and accuracy. There were but few candidates, however, of any very great merit. Nearly half the whole number of boys and more than half the girls were but little above the minimum standard required for passing. The girls in many cases used heavy and cumbrous modes of working.

In Algebra,

The Algebra was far from satisfactory. More than half the whole number of candidates failed to pass, and very many could have had no reasonable expectation of passing. The bookwork questions were written out at very great length, but in most cases the important points were slurred over or altogether omitted.

In Euclid,

The propositions of Euclid were on the whole creditably written out, but only the easier riders were solved. The per centage of failures in Euclid was higher in the case of the girls than in that of the boya.

It is more than probable that one of the causes of the unsatisfactory state of elementary knowledge of these subjects is the simple fact that too many subjects of study are prematurely forced on minds of ordinary capacity. In the case of inert and dull minds no sound and exact knowledge is acquired; and in the case of active and precocious minds a loose smattering of many subjects is crammed into the memory which have not passed through the understanding. Whatever

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may be forced into the memory in this way is only held in solution until it has served its purpose, and then it is precipitated.

The injudicious solicitude of some teachers to develope prematurely the mental faculties of youth will only result in furnishing illustrations of the consequences which flow from a disobedience to the laws of nature. The teacher ought not to forget that the organs of the brain equally with those of the body have their predetermined periods of growth and development. Any attempts to interfere with these cannot be made with impunity. How seldom do those who in youth have been nurtured in the forcing hot-bed exhibit in after life superior intellectual powers. They more commonly fall beneath than exceed the average in talent. But even if the mental capacity be enlarged, it is at the expense of the corporeal energies. These are not displayed in the strongly knit and active limbs, in the well-formed and robust frame. The brain may have grown, but it is almost invariably accompanied by a feeble and imperfectly developed body, which, in general, prematurely breaks up. If nature has given a superiority of mind, the less interference there is with the laws which regulate its development the more ample and gratifying will be the results. If she has withheld these conditions which are necessary for the manifestation of talent, it is not only in vain to endeavour to create what she has denied, but injudicious and fraught with danger. The vegetablo kingdom illustrates the justice of these views.

In the use of ordinary language it is implied that there exists some sort of analogy between the bodily and mental faculties. As the mysterious processes of digestion and assimilation are necessary for the healthy development of the body, so also, unless the food of the mind be inwardly digested, it cannot contribute to the like development of the intellectual powers. The reception of knowledge into the mind has been described under the comparison of good seed sown in good soil or bad soil; and it is written that 'men do not gather grapes of thorns nor figs from thistles.'

As, therefore, the physical development of the organs and functions of the body take place gradually according to the order of fixed laws; it would appear that the development of the mind must, in a similar way, be subject to the same order. The mental acquirements suitable for youth cannot be put off with advantage till the approach of manhood. The consequences of neglect in early mental training will in due time become as manifest as the lack of proper food and care in the healthy growth of the body. As the body for food, so the mind for knowledge has its hungerings and thirstings. The craving appetite in both cases implies a process of assimilation. As the mental food assimilated will affect the character of the mind itself, only wholesome mental food should be supplied in the right order and quantity suitable for healthy and vigorous intellectual growth.

The study of the exact sciences is one of the most effective means of cultivating and developing the reason. Geometry is the Science of Space, and Arithmetic with Algebra in its character of Universal Arithmetic is the Science of Number. As all our knowledge of the external world must be subject to the conditions of Space and Number, the elementary portions of these sciences are from their nature better adapted than any other to form the habit of fixity of


attention, of distinctness in the conception of ideas, and of precision in expression. The language, like the subjects, is fixed and definite, and does not admit of the same ambiguity and uncertainty as the language employed on other subjects. The reasonings are always conclusive and exact, expressed in terms whose meaning cannot change from the sense in which they have been defined. In one of his recent charges, the present Lord Chief Justice made a passing remark to the effect that Euclid's Elements were a mental training second to none. Teachers who need a hint or two on this subject may learn something from the experience of the Rev. Stephen Hawtrey, late Assistant Master at Eton, as stated in his “Short Introduction to the Elements of Euclid.”

It may be remarked that persons of the highest acquirements in any science do not always become the best teachers. Experience has shewn that such persons may be utterly incompetent to adapt their knowledge to the capacity of minds of a lower character than their own.

If a teacher has not skill to make the subject of his instruction interesting, and tact to adapt his knowledge to minds of different capacities, he does not possess one of the essential requisites of a teacher. It is a delusion to imagine that correct habits of thinking can be created or exact knowledge acquired by the mere passive attendance on lectures, however excellent. Class-teaching may be useful or useless, or even worse than useless. There are many youthful minds so constituted that they must be taught individually, if ever they are to be able to draw inductions from facts, or comprehend principles and apply them with success. In dealing with the misapprehensions and mistakes of learners, the teacher should observe how the learner was led into error, and by suitable questions lead him to perceive his mistake and to make the correction for himself. By this method the mind of the learner is brought into active exercise, and he will be less likely to repeat the mistake than if the correction were received passively from the teacher. If the learner exhibit a listless inattention or a positive dislike to the subject of study, the efforts of the most judicious teacher are in vain.

It may also be remarked that implicit obedience in the learner and the love of knowledge are also necessary conditions of improvement. A sense of duty rather than a desire of surpassing others constitutes the right motive of the learner. If a morbid appetite for praise or an eager strife for pre-eminence be encouraged, it may grow, and at length become the ruling passion, and create envy and hatred of every successful rival, and generate a feeling of discontent which may become a fatal obstruction to all mental and moral improvement.

These few words to teachers and learners may be concluded with the expressive words of the late Dr. Whewell:

“The object of a liberal education is to develope the whole mental system of man:-to make his speculativo inferences coincide with his practical convictions :—to enable him to render a reason for the belief that is in him, and not to leave him in the condition of Solomon's Sluggard, who is wisor in his own conceit than seven men that can render a reason.'

R. P.

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