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But more recently new topics have been accepted and other older ones have been receiving a new emphasis, according as such topics are intimately concerned with our welfare. For example, new topics or topics newly emphasized are insurance, stocks and bonds, government revenues and expenditures, the banking business and taxes. These are subjects for children studying somewhat advanced arithmetic. But a similar change is also affecting the problems for younger pupils. Problems dealing with actual situations are more and more in demand for all ages of pupils, such as those dealing with farming, fishing, lumbering, mining, manufacturing, transportation of goods, trade and facts of daily interest.

Knowledge of mental processes is insisted upon as heretofore; but insight into the quantitative conditions of social life is also aimed at through the study of arithmetic. The old style of problems began usually with “if,” being supposed cases, and the pupil was scolded if he worked for the answer. We are now slowly reaching the point where problems are selected for children whose answers are of real interest and, therefore, worth working for; then the children are expected to work for the answer, just as adults always have worked for them. It is the modern doctrine of interest (see ) that has been greatly influencing teachers here. The increased interest in the problem increases the pupils’ concentration of attention, and thus results in a better knowledge of processes and more accurate work in general. It is very difficult for arithmetics to realize this ideal to a great extent, but recent text-books plainly show a movement in this direction. In beginning arithmetic many good teachers make no attempt to follow the Grubé plan, by teaching the four fundamental operations touching one number before considering the next higher number. In fact, many superintendents now make no attempt toward systematic instruction in arithmetic to pupils during the first year of school. The reason for this is that formal instruction in the subject accomplishes little with pupils so young, and they ordinarily have too much formal instruction in other subjects the first school year anyway.

Counting is one of the first kinds of work, such as the counting of objects, “keeping score” in games, etc. Measuring, involving single facts in the table of compound numbers, such as the relation between inch and foot, foot and yard, pint and quart, ounce and pound, etc., simple fractions, such as $$\frac{1}{2}$$ and $$\frac{1}{4}$$, and the symbols of +, −, ×, ÷, may well be taught the first year arithmetic is studied. That is, fractions and various other topics need not be delayed until a certain year is reached; but the pupil should take up whatever facts his interests suggest. The fraction $$\frac{1}{2}$$ is just as naturally used by a six-year-old child as the combination 2 × 2. In teaching addition teachers are not limited to any one device. In explaining a process involving some mental retention of number, as in “carrying,” it is advisable to use sticks in bundles of 10, as is often done, and to adopt, also, such devices as


 * {| style="text-align: right"


 * 26 || = || 20 || + || 6 || || ||width="15%"| || 26
 * 39 || = ||style="border-bottom: solid 1px"| 30 || + ||style="border-bottom: solid 1px"| 9 || || || ||style="border-bottom: solid 1px"| 39
 * || || 50 || + || 15 || = || 65 || || 15
 * colspan="8"| ||style="border-bottom: solid 1px"| 50
 * colspan="8"| || 65
 * }
 * colspan="8"| ||style="border-bottom: solid 1px"| 50
 * colspan="8"| || 65
 * }
 * }

Such devices help greatly to make the steps clear. In general, the use of splints and other objects is very helpful in approaching new facts. They can well be used in the first two years of instruction, along with diagramming and other concrete helps, and also later in the beginning study of fractions. But it should be remembered that these are only temporary helps and that the pupil should soon be able to dispense with such concrete aid. The use of the fingers in counting should be discouraged, because they cannot later be removed entirely from reach when not wanted. In subtraction the “making change” method should be used. For example, if you have 10 cents and buy a pencil for 3 cents, the child should see that you have 7 cents left, because 3 cents + 7 cents = 10 cents. This is the method used at any store, and in business in general. The Austrian method of subtraction is the one that now is most commonly favored. The example just given follows that method, one advantage being that it dispenses with the necessity of learning any subtraction table. An example like 52−27, might be worked as follows:


 * {| style="text-align: right"


 * 52 || = || 50 || + || 2
 * rowspan="3" style="text-align: left; vertical-align: top; padding-left: 1em; padding-right: 1em"| Add 10 to each which leaves the difference the same
 * 50 || + || 12
 * 27 || = || 20 || + || 7
 * style="border-bottom: solid 1px"| 30 ||style="border-bottom: solid 1px"| + ||style="border-bottom: solid 1px"| 7
 * colspan="6"| || 20 || + || 5
 * }
 * colspan="6"| || 20 || + || 5
 * }

No number added to 7 will make 2. But 5 added to 7 makes 12. We have now increased 52 by 10, and we must add 10 to 27, so as not to change the difference. 3 (tens) and 2 (tens) are 5 (tens). Hence, the difference is 25.

The details of such presentations vary greatly, and a teacher should follow the plan that best satisfies her.

In short division it is often advisable to use the “long division” form, showing that the former is only an abbreviation of the latter.

A text-book in arithmetic is hardly desirable before the third year of school.