Practical. 1. Provide a dinner on paper for 8 people. You may spend any amount up to 3s. 8d. 2. Measure your paper and tell me how many times it will go into the length of the board. 3. What is the height of the girl sitting next to you? Who is the taller, your partner or yourself? By how much? 4. Weigh any suitable and available article as accurately as the weights provided will allow. Write name of article and weight upon your paper. 5. How many times does the serviette ring turn in rolling across the table? Find out in any way you please. 6. Count the girls in the class, and tell me how many are away, and who those girls are. How many times could each girl have attended school this week? If we were cent. per cent., how many attendances would have been made altogether? APPENDIX V. SCHOOL E (BOYS'). TERMINAL TEST SET TO THE HIGHEST 1. How much money was realised by selling out 1,500l. of Railway 4 per cent. stock at 87 ? 2. Express in £ s. d. the value of 2989375 of 1007. 3. Find the amount of 871. 168. 3d. for 2 years at 3 per cent. (simple interest) 4. A boy wrote 61 instead of 61; express in decimals the amount of the error. 5. Find the value of 3 of 1/91 of 11. 4s. Od. +0375 of 158. and express the result as the decimal of 51. TERMINAL TEST SET TO THE HIGHEST CLASS IN 1910. Practical. 1. Describe a circle which shall have as nearly as possible 12 square inches of area. 2. Find as nearly as you can the area of the irregular field on a scale of 1 cm. = 20 yards. [This was a figure of irregular outline, drawn on paper ruled finely in millimetre squares.] 3. A cistern can be emptied by a tap in 3 hours and filled by another in 5 hours. If the cistern be empty and the filling tap be allowed to run and after 2 hours the emptying tap be set going, how much of the cistern will be filled at the end of 4 hours from the time of setting the filling tap running? 1907., and 2 horses and 1 cow cost 807., 4. If 3 horses and 5 cows cost find the cost of a horse and also of a cow. 5. Find a line which shall be 125 cm. long. 6. Draw a graph to show the path of a body moving as follows: Time taken in minutes Distance travelled in yards 2 4 5 8 9 10 - 150 250 325 525 650 700 7. A and B run a race of 500 yards. A starts from scratch and finishes the race in 4 minutes. B has a start of 50 yards but commences a minute later at the rate of 120 yards a minute. Which shall win? 8. Two steamers lie E. and W. of a certain point 5 miles apart. The westerly one sails in a N.E. direction at 10 miles an hour, the easterly one sails due N. at 8 miles an hour. and at what time. Draw a graph showing where they meet 9. Plot the points (6, 8) (6,-8) (-2, 2) (-2,-1). Join the 4 points and find the number of units of area in the quadrilateral thus formed. Theoretical. 1. A walks at 4 miles an hour and starts at 10 a.m. B starting from same place at 10.45 and going in same direction, walks at 5 miles an hour. How far will B have walked when he catches A ? 2. Find the number of degrees between the hands of a clock at 20 minutes past 6. How long after this will the minute hand cross the hour hand? 3. Two trains start from the same point and run on parallel rails in the same direction. One travels at 35 miles an hour and the other at 30 miles an hour. After what time will the faster train be 4 miles in front of the other and what distance will the faster train have travelled? 4. One tap lets 50 gallons per hour into a cistern while 10 gallons leak out every hour. If the cistern holds 1,000 gallons when full and the tap is set running when the cistern is empty, how much water will have run into the cistern by the time it is full ? 5. Explain what is meant by stock at par, stock at a discount, 5% stock at 120. How much shall I have to pay for 2,500l. stock at par and for 1,2001. stock at discount? 6. A cylinder 3" radius and 10" deep is filled with water. A ball of iron diameter 3" is gently dropped in. How much water will overflow and how much will be left in the vessel? 7. Find the cost of gravel at 98. a cubic yard required to cover to a depth of 3 inches a path 5 feet broad round a circular pond of diameter 50 feet. 8. A map is made on a scale of inch to a mile. How many acres on a map will be covered by a penny piece; diameter 14 inches? 9. There are 3 casks of wine. One contains 50% water, the second 7% water and the third 12% water. One gallon of the first is mixed with 2 gallons of the second and 4 gallons of the third. Find the percentage of water in the mixture. 10. Find the value of the following if x stands for 5s. and y for 2s. 6d. (3x+2s. 6d. 5y — 2x ̧ x) 7 + 30 Y) APPENDIX VI. SCHOOL F (GIRLS'). TERMINAL TEST SET TO THE HIGHEST CLASS IN 1910. NOTE:-Ten sums are to be done out of 12, but those marked with a must be done. 1. Find the value of xy + yz when x = 2. Express 51. 78. 34d. as decimal of 11. division in £ s. d. 3, y= 4}, z= 512. * Divide it by 12 and verify by 3. The area of a triangle is 50 square cm. and the base is 85 mm. Find the height. 4. A boy obtains 165 marks out of a maximum of 185. Bring this to a percentage. 5. Draw an equilateral triangle ABC. Side 3 inches. Bisect each side in P, Q, and R respectively. Join P, Q, R and find area of triangle P,Q,R. 6. Express the difference between 17. 2s. 6d. and 1l. 68. 8d. as a percentage of 11. 2s. 6d. 7. Find the contents in gallons of a tank 3 feet long, 2 feet 8 inches wide and 1 foot 9 inches high. 8. Add 18 to 24l. 12s. 6d. 9. A man buys goods for 5l. 10s. 6d. and sells at a profit of 27%. Find the selling price. 10. Draw 3 straight lines, length 5 cm., 7 cm. and 4 cm. respectively. Draw a fourth line whose length is to 4 cm. as 7 cm. is to 5 cm. 11. If 10 eggs cost 18., how many can be bought for 38. 6d. ? Work (a) graphically on squared paper and (b) arithmetically. * 12. Plot the points A (10.10), B (20.20), C (30.10), D (20.0). Join the points. This figure represents the plan of a cube. Find the volume. Illustrate wherever possible. THE TEACHING OF ELEMENTARY MATHEMATICS IN ENGLISH PUBLIC ELEMENTARY SCHOOLS.* The object of this memorandum is to give a brief historical sketch of recent changes in the teaching of Elementary Mathematics in an English public elementary school and to indicate (a) what, in the opinion of an elementary school teacher in daily contact with the realities of the schoolroom, are the main difficulties and the most serious defects in the present-day teaching of this subject, (b) some principles and methods by which Elementary Mathematics may be made of increased value even under present conditions, with specimen syllabuses for Infants', Senior, and Higher Elementary or Central Schools, and (e) to make some suggestions of a practical and definite character for applying the principles advocated to the mathematical work of the Higher Elementary or Central Schools. The school period of the English worker's child may be said to extend from 4 to 14, or possibly 15 years of age. The first four years of this period are spent in the preparatory or infant school, from which the child passes at the age of 71⁄2 to 8 years to the upper or senior department, where he remains until the leaving age of 14 to 15 is reached. In a fair number of cases, however, especially in town schools, it will happen that the scholar, on account of more than average ability, finds his way, at the age of 11 or 12, to an upper standard or to a secondary school. In this review the secondary school has no place and will not receive further notice, but the Higher Elementary or Central Schools are now so numerous, and play so important a part in our elementary school system that it is impossible to omit them from any survey of English elementary education. Great changes amounting almost to revolution, have taken place in the aims and methods of our schools during the last decade, and particularly in mathematical teaching. The old practice of labouring mechanically through a number of complicated arithmetical processes, imperfectly understood, is gradually giving place to objective teaching, visualisation, and the actual individual handling of objects; to problems within the range of the child's intellect and experience; and to progressive and systematic training on the purely intellectual side. But this development is by no means complete. So rapidly have changes come that the older teacher, the man of 20, 30, 40 years' experience, has had difficulty in adapting himself to new demands, and still feels the hampering effects of the old conditions and methods. The acknowledgments of the writer are due to Mr. John Scott, of the Ashburnham School, Fulham, and to the staff of the Bloomfield Road School for help in the preparation of this paper. Space does not permit of any extensive survey of these conditions obtaining, say, prior to 1900, but some short review is necessary for our purpose. Suffice it to say that schedules defining in the baldest manner the work to be done in Arithmetic were issued yearly by the Education Department as instructions to teachers, that formal examinations were held annually covering these schedules, and that money grants were made conditional upon the results obtained--the famous system of payment by results." Each year's work was compartmentally distinct. Previous work was rarely revised, the pupil spending his year upon the exact portion of work allotted to that year. 66 The character of the old schedules will be made clear by one or two extracts from them, e.g. Standard III. (age 9-10).--Notation and numeration to 1,000,000; long division; compound addition and subtraction. Standard IV. (age 10-11).-Compound rules and reduction of money; common weights and measures. Two criticisms of the time are quoted from Government Blue Books" to illuminate the results achieved: And 66 "The progress made in Arithmetic is out of proportion small in comparison with the extraordinary labour bestowed upon it." 66 'I come across children [of 11, 12, 13] who can add and subtract intricate vulgar fractions, but are incapable of writing down from dictation a sum in simple addition, which a class of well-taught infants would do easily. In a class of 58 children in Standard V. only 11 could encounter with success the difficulties of a sum in simple addition." A change was obviously inevitable, and a schedule known as Scheme B." was issued in 1894. This, though in some respects defective, was a notable advance upon the older schedules involving the manipulation of large numbers, and, as it still forms with various modifications the basis of work in many schools, it is given in detail in Appendix I. to Mr. Ballard's paper in this series (see p. 26). The change begun in 1894 was completed a few years later by the withdrawal of all schedules of work, and the abolition of annual examinations for payment of money grant-a step which almost immediately began to give a new complexion to our mathematical work and methods, in that each school had to make its own syllabus. The change has, up to the present, perhaps, been more one of aim than of practice. Schemes of work in different schools are found to present wide variations both in matter and treatment, and many faults incidental to an experimental stage have naturally arisen, so that, notwithstanding * Reports of the Committee of Council on Education, 1878-79 (p. 522), and 1879-80 (p. 396). |