Objects for Counters.

Each Primary room should be provided with suitable objects for the use of the pupil in examining and illustrating for himself the nature and relations of numbers.

Half-inch cubic blocks are very convenient, as they will remain in position when arranged in groups on the pupil's slate, thus:

With these blocks the pupil should be required to solve every exercise given in his book in grouping and in Addition, Subtraction, Multiplication, and Division, where objects are referred to. He should also be required to illustrate definitions and principles in the same manner.

This device consists of a frame containing a series of grooves in which objects made with a tongue which fits into the grooves can be placed at pleasure. Its use has the following advantages:

(.) Nothing need ever be placed before the pupil's eye, in any illustration, except the objects necessary to make clear to his mind the point presented. Hence the attention is secured upon just what the teacher wants to fix in the pupil's mind.

(b.) The variety of objects, signs, and figures which accompany the frame are such as will enable the teacher to represent objectively all the elementary processes of Arithmetic. This will be seen by examining the above cut. Observe that on the left the numbers 111 and 122 are represented by squares, and on the right the number 243. In the centre of the cut the Numeration Table is shown, and at the bottom Multiplication and Division are presented objectively by the use of balls, etc.

(c.) No objects are fixed in position on the frame. They can all be removed, and arranged when in use in any order the teacher may desire. Hence the teacher can exercise tact and judgment in adapting illustrations to the peculiar point in hand and the condition of the class.

This device can be used with excellent results in all drill exercises in Notation, Numeration, Addition, Subtraction, Multiplication, and Division, both in Integers and Fractions. The teacher will find full and definite instructions for the use of the Example-frame in each of these subjects in the Hand-book accompanying it. The following indicates in short some of the advantages resulting from its use.

1. It affords the most perfect facilities for an indefinite number of rapid sight exercises in each of the fundamental rules.

In presenting sight exercises, it is of the first importance that nothing should be before the eye but the numbers on which the pupil is to operate, and that these numbers should be instantly removed as soon as the work is performed. Both of these conditions are perfectly secured by the Example-frame.

2. No time is wasted in pronouncing examples or in writing them upon the blackboard.

The pupil's whole time can thus be spent in working for results. In this way sufficient practice can be given to make him expert in operating with numbers, without interfering with his time for other work.

3. Examples can be presented so as to make it impossible for the pupils to copy each other's work.

Thus, three or more different examples can be presented in one view, and the pupils numbered in groups, one, two, three, one, two, three, and so on, corresponding with the number of examples presented, and the same example given to each pupil whose number is one, then another example to each pupil whose number is two, and so on.

Observe that by this course the pupils working the same example will be removed some distance from each other, and hence copying, the chief hindrance to progress, is made impossible.

NUMBERS FROM ITO 10

METHOD OF PRESENTATION.

1c. Cautions which should be regarded.

1. Remember that TACT is a PART of METHOD.

The study of the course which should be pursued in presenting a subject can never make a successful teacher where tact is wanting. Tact enters as a necessary element into the method of every first-class teacher.

2. Remember that the method of presentation indicated in this or any other book can only be an outline guide.

Hence, to produce good results, the filling out of such outline must be the spontaneous product of your own inner experiences. You must never forget that to be successful you must work in your own harness.

3. Remember that what the pupil may at first appear to know, or to say or do, is not to be taken as a sure exponent of his real inner consciousness or experience.

A large proportion of your pupils will state and do mechanically what their book states or requires done, or what their teacher or classmates have stated or done, without having any consciousness of the reality. Hence, whatever your method, be sure and hold the pupil to his work, and probe him until he describes everything from his own consciousness of the real.

1d. What should be taught.

1. The perception of the succession of numbers from 1 to 10, and the use of the words one, two, three, and so on to ten.

2. The use of figures in representing numbers, or the change of the ear language, one, two, three, and so on, into the eye language, 1, 2, 3, 4, and so on.

3. The analysis of each number from 1 to 10 into two parts in all possible ways, and the addition of any two numbers whose sum is not greater than ten.

4. To count or name the consecutive numbers one, two, and so on, in their order both forwards and backwards, or to count by addition and subtraction, and to discriminate the number from the thing numbered. 5. The use of the sign of addition (+), the sign of equality (=), and the sign of interrogation (?) or question-mark.

6. Practice in making figures and drill exercises.

1e. The perception of numbers, how presented.

1. Test and sharpen the child's perception of one or unity by causing him to handle and point out single objects of various kinds.

Thus, first let the teacher ask the pupil to show him one ball, one book, one boy, and so on. Second, let the teacher present the objects and call upon the pupil to name how many balls, books, desks, and so on.

2. In presenting the numbers from two to ten inclusive proceed first by analysis, then by synthesis, comparing in each case the new number with the number immediately preceding it.

The following example with the number three will suggest the method for all numbers up to ten.

(a.) Place for example where they can be seen by the class, two books and three books. Ask the class to notice carefully each pile of books. Remove one of the three books and ask the class how many books now in each pile. The answer will be, "Two books." Put back the book, and ask how many books in the larger pile. The answer, if they do not know the word three, will be "Two and one." In this case give the name three, and then ask the questions, "What do you mean when you say two books? When you say three books?" The answer to this question, based on the previous analysis, will be," Two books and one book."

Continue this drill, using various objects, until the perception that three is two and one is fixed sharply in the mind. Pursue the same course in presenting higher numbers up to ten.

(b.) After the perception of the relation between each new number and the number immediately it is given, let it be fixed in the mind of the pupil by a drill: first, in naming the number at sight of the objects; second, in showing the objects as the teacher names the number. Let this drill include each time all the numbers already mastered, and let it be conducted in a spirited manner.

1f. The use of figures, how presented.

Observe, the work here to be done is simply to change the sound or ear language into a brief eye language; hence the following course:

1. If the child can read print or script, place the words one, two, three, and so on up to ten on the board. Let the pupil read the words and at the same time show the teacher the corresponding number of objects.

2. Let the teacher point to the numbers written on the board in irregular order, and call upon the pupils to exhibit a corresponding number of objects, directing their attention to the fact that their eyes tell them each time how many objects to exhibit.

3. Place the figures under the words on the board, thus: two, three, four, five,

one,
1,

etc.

Ask the pupils to observe how much easier and quicker the figures

can be written than the words, and hence how much better it will be to use them. Proceed with the drill thus:

(a.) Point to the figures in irregular order, and let the pupil show the corresponding number of objects. After a short exercise, erase one, two, etc., and continue the drill by using the figures.

(b.) Point to the figures rapidly, and call upon the pupils in turn to pronounce the names.

(c.) Assign for work at the seats, copying the figures on the slates as directed page 25. If figures are put on the board to copy from, let them be well made.

4. If the pupil cannot read the words one, two, three, and so on, present one, two, three, etc. objects, and let him pronounce the number, while the teacher writes on the board the figure representing the number pronounced. Drill in this way until the figures 1, 2, 3, 4, and so on, are associated in the mind of the pupil with the sounds one, two, three, four, and so on. Then proceed with the drill as directed in 3a and b.

1e. The analysis of numbers, how expressed. This exercise may be conducted as follows:

1. Take for example 8 books or other objects, and place them in one pile, where they can be seen by the class. Separate them into 7 books and 1 book, and let each pupil state orally or write on his slate, 8 books make 7 books and 1 book. Present in the same way the fact that 8 books make 6 books and 2 books; make 5 books and 3 books, and so on.

2. Let each pupil be required out of class to use objects and analyze each number and record the results on his slate, as indicated for the class exercise. Let these results be read as a review in class.

1g. Counting, how presented.

1. Observe, counting proper may be defined as the process of adding one at a time and naming the results, thus giving the consecutive numbers, 1, 2, 3, 4, 5, and so on. The term is now, however, applied to adding or subtracting the same number consecutively.

For example, we count by addition by twos from 1 to 23 thus: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23. Observe that in this case, commencing with 1, we add consecutively 2 until the sum is 23.

In the same manner we count by subtraction by subtracting consecutively the same number. Thus, commencing with 28 and counting by subtracting three, we have, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1.

2. Observe that when a group of objects are placed together, thus:

and counted by pointing to an object and saying "one," then to another