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"base Avarice of the Infinite, and in blind imagination
"of it?
In counting of minutes, is our arithmetic ever
solicitous enough? In counting our days, is she ever
severe enough"?*

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Here, then, we obtain authority for our hope that our training is to help in the training of a good and capable citizen of his country. Stress is laid then upon one or two points in the general teaching

(1.) The children are taught to formulate rules for themselves by working out several examples from first principles, and when the rule is formulated to use it immediately to shorten their work, e.g., a child works several sums such as, "Find the cost of 12 things at 3d. each, 4d. each, &c.," and hence formulates for himself the rule that" the number of shillings per dozen is the same as the number of pence apiece," this leads to the habit of investigation so essential to the higher mathematician.

(2.) We insist also upon concentration of thought throughout the lessons which range in duration from 20 minutes at first to 25 minutes in the last year; during that time attention and concentrated thinking are required; the children generally have an easy lesson, such as handicrafts or writing, to follow so that their brains are rested after the effort expended.


To every teacher of this subject it is now clear that the historical presentation of the subject is the easiest and most natural, i.e., that it is to be presented to the child as it presented itself to the race; beginning with the concrete and working back to the abstract generalisation; and having as far as possible a practical bearing on matters of everyday life. Thus much is clear, but what is not always kept so distinctly before the teacher is that the concrete in our case is simply the means and not the end: the function of the concrete is to be simply a preparation for the abstract, or a means of symbolic illustration; it must therefore be discontinued as soon as possible. find that the children get worried and bored by counters and beads, and work drags wearily on, always the same counters and beads, until the children's attention and interest have both vanished, and the lesson is actively productive of harm. A variety of objects is necessary in these lessons with the concrete, a child may, for instance, learn all about the number 10 from certain cubes of wood, but when the cubes being absent or something else in their place, he is asked about the number 10, all knowledge of it has vanished; 10 pertains specially to cubes and does not exist without them. Once the concrete has been passed, it is better not to go back to it for assistance; it thus becomes just a stage on the way to something fuller, and not a prop to be constantly leaned upon.

*loc. cit., PP. 135-137.

We often hear that sums dealing with interesting things like oranges or tops, or dolls, should not be given to young children, as they are apt to fix their attention on the tops or dolls and not on the numbers; this we are inclined to think is entirely the teacher's fault; though the question is always approached by means of a problem, a beginner can soon be taught that for the working out the numbers are the primary things; this is made easier by writing in arithmetic books only the figures involved, until the answer is obtained, e.g., “I went out with 6d. and spent 4d. on biscuits; I then met a friend, who gave me another 6d., how much did I come home with? The sum would be stated

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The answer must be noted as pennies and hailed as an important and interesting conclusion by some remark such as "So I came home richer than I went out."

There are of course several excellent text-books providing intelligent examples and lucid explanations; but at these very early stages of the subject a text-book can be almost entirely dispensed with; rather than consult text-books the teacher must use her ingenuity, must utilize other lessons and walks, &c., to point her definitions and amplify her instruction.



Now that Practical Geometry has become so important a part of the Elementary Mathematics in the Junior and Middle School, leading on as it does to the more logical deductions, it is well to find what useful correlation can be made with other subjects in the curriculum. Many Schools connect it with Elementary Physics, though this merely requires the accurate measurement and drawing of lines and areas and brings in little else. But the meaning of the word Geometry, i.e., earth measurement, suggests a subject more nearly cognate, viz., Geography, and it is the aim of this paper to show in some ways how the correlation may be made, using Elementary Geometry as a basis for Scientific Geography which can be introduced to pupils about twelve and a half years of age.




As soon as the units of length have been explained in the mathematical lesson, lines can be measured, either as between two points A and B, or between two towns on a map, e.g., London and Birmingham, Manchester and Southampton. Incidentally this accustoms the class to finding places in Following on this as quickly as possible comes the idea of scale, the representation of large lines by small, and vice versa. results obtained previously from the map in inches or centimetres can now be translated into actual distances by means of the given map scales. Interest will be aroused if the results are checked as to reasonable accuracy by means of railway timetables, though if this method is adopted care must be taken to select towns connected by a direct railway line. Problems on the converse can also be worked: the class take their maps and, with a given town as base point, answer such questions as, What is the port 150 miles away? Find a lake 50 miles away, &c. This introduces the use of the compasses in determining the position of the free end of a given finite straight line, the other extremity of which is fixed. Arithmetical problems may be correlated with these lessons as the parliamentary fare gives a rough clue to the distance between two towns, and much useful information will thus be absorbed incidentally.

At this stage it is important to practise making rough estimates of linear dimensions. Boys probably have a rough unit of length in the cricket pitch, but many girls have not this advantage. Each member of the class should know

the length of his or her pace, should measure out in the playground a yard, a hundred yards, &c. The principle of linear measurement can be applied to the sphere. Two places on the surface of the sphere are connected by a piece of thin string drawn as tightly as possible, and the distance between the two places can then be measured. In the Geography lessons it can be explained why ships plying between ports do not always take the shortest route as found on the model sphere. Again, if a tracing wheel (such as is used by dressmakers) be employed, the principles of linear measurement can be applied to curved and irregular lines, e.g., the length of a river from source to mouth, the extent of coast line of a country, &c., can be measured by counting the number of revolutions that the wheel makes as it traces out the line, and then. multiplying the distance traced out in one revolution by the number observed.


The idea of an angle and its measurement must enter at an early stage in order to lead on to more practical problems. As soon as the pupils have made a protractor in cardboard, with the chief angles marked upon it, they are ready to study the points of the compass. Their early lessons in Geography have given them ideas, probably vague, of the four cardinal points, and they can crystallise these by constructing a compass card accurately with the help of their protractors, and from the division of the right angles, either by folding or by geometrical construction, they obtain the new points, N. E., N.N.E., &c. The use of the compass needle can be shown practically by letting the pupils make a map of their route to and from school. In the first instance the plan should only be asked as regards direction, but later on it may be made more accurate by taking the pace as a unit of length and drawing all lines to scale.


When the measurement of lines and angles is thoroughly understood, the making of a map follows. In the Geometry lessons, dictated maps can be drawn, beginning with simple problems such as: A man walks five miles east, then two miles south, and then returns home. Draw a map of his walk. In what direction does his last road run? These lead on to more difficult maps. The pupils should be taught to draw their first line in all directions so that a horizontal line in their note books does not always represent one from west to east. At the same time, in the Geography lessons, practical map-making can be taken. First the pupils learn how to determine the south, either by finding the position.

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of the sun at noon, or by the help of the mariners' compass.* When this has been determined, the class should go to a level piece of ground which is fairly high, (a flat roof is excellent), and make a map, or rather panorama, of the surrounding district, purely as regards direction. A piece of paper is placed on the ground with a fixed line on it pointing to the south, and lines are drawn on the paper pointing to prominent places in the neighbourhood, e.g., the church, a school, a factory, a pond, &c. The angles made by these lines with the base line are then measured, and on the return to the class-room the pupils draw a map of their observations.

Next linear and angular measurement can be combined. A ground plan of the class-room should be made first, the actual measurement of the walls and corner angles being made, and then reduced to a convenient scale for the note books. The class should be made to understand thoroughly that a reduction of scale only affects the lines and not the angles. As a rule, class-rooms are rectangular, so it is advisable to proceed to the playground, if that be more irregular. Here, two other methods of drawing a map may be introduced, that of triangulation, which brings in much elementary Geometry, and that of the field book used in surveying, the base, line and offsets. The class will be interested in seeing that any one of the three methods produces the same results as the other two, and later on they find that the third method enables them to calculate the area of the map most conveniently. The data for making various maps can be dictated, either method of measurement being adopted. If these maps are of real places, and most guide books contain ground plans of churches, old castles, &c., the plans become more living to the pupils, and there is a certain stimulus of fascination in wondering what the design is going to be. The class can now solve quite easily problems such as:-London is 40 miles

* Some teachers to determine the south use the following roughand-ready rule: hold a watch horizontally, with the hour hand pointing to the sun, then the line bisecting the angle between the hour hand and the figure 12, measured counter clockwise, will point to the south. But this method is inaccurate and should only be used for such practical purposes as determining the aspect of a room, or the general direction of a road, where the variation of a few degrees is immaterial. It may, however, be of interest for the pupils to find out how this rough rule arises. They know that the sun appears to take 24 hours to go from south on one day back to south again, a revolution, while the hour hand of a watch revolves completely in 12 hours, i.e., the hour hand moves twice as quickly as the sun. Next let them observe the shadow of a stick during six hours, say 9 a.m. to 3 p.m., it describes a quadrant of a circle while the hour hand describes a semi-circle. Thus again the watch hand moves twice as quickly as the sun. Hence the rule appears to hold good. But by closer observation they find that the shadow does not describe the same angle in each hour, which shows that the angular motion is not uniform. They see then why they cannot assert that at any given time the watch hand has gone twice as far as the sun in its angular motion, and that the rule only gives the approximate position of the south.

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