The New Student's Reference Work/Arithmetic

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Arithmetic has been greatly influenced by modern educational thought, the same as other studies.  Until very recent years the principal change taking place in the study consisted in a growing willingness to omit topics that had no close relation to our own lives.  For instance, topics now wholly omitted or neglected are the surveyor’s table, apothecaries’ weight and troy weight; G.C.D. and L.C.M. as special topics, complex and compound fractions, except those of a very simple nature; annual interest and most of compound interest; partial payments, except under the United States rule, and with problems involving common amounts, as a principal of $100 with payments like $10 and $25, rather than amounts like $251.42 and $19.79; profit and loss as a special topic; equation of payments; partnership; longitude and time, except problems based on the 15° scheme and a few others; and cube root.  The conviction has been growing that there are too many quantitative matters intimately related to our lives to allow time to be spent on others that lack such relationship.

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 Interest) 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 and , 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 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

26 = 20 + 6 26
39 = 30 + 9 39
50 + 15 = 65 15

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:

52 = 50 + 2 Add 10 to each
which leaves the
difference the same
50 + 12
27 = 20 + 7 30 +  7
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. At least its earlier use tends strongly to make the work too formal.

There is little object in carrying the multiplication table beyond 10 × 10. In compound numbers reduction “ascending” and “descending” should be confined to numbers of not more than three denominations.  The reasons for this are that in practical life we rarely use more than two denominations, as feet and inches or pounds and ounces; and that, if one has learned to perform reduction with two and three denominations, he can easily perform those with more if occasion required.

Quantitative facts are so much more often expressed decimally now than formerly, that much more attention to decimal fractions is in place.

The addition and subtraction of decimals need offer no difficulties. In multiplication the most approved forms are the following:

1. 6.25   
Explanation—Since 5 times hundredths are hundredths, the right hand number of the product is placed under hundredths.  The rest of the work is identical with that of integers, the decimal point going under the others.
2. 6.25 
Explanation—Since hundredths multiplied by tenths is thousandths, the right-hand figure of the product goes in the thousandths place.
3. 6.25  
Explanation—Since hundredths multiplied by hundredths are ten-thousandths, the right-hand figure of the product goes in the ten-thousandths place.

Operations with decimals should be limited to fractions having not over three places, and answers need not be carried beyond three places.

Division of decimals should be taught as suggested in the following Austrian method:

Required to divide 6.275 by 2.5—

Old method

2.5) 6 . 2 7 5 (2.51
5 0
1 2 7
1 2 5
2 5
2 5
“Point off as many places in the quotient as the number of decimal places in the dividend exceeds that in the divisor.”

Common Austrian method

2 . 5 1
2.5) 6 2 . 7 5
5 0
1 2 7
1 2 5
2 5
2 5
Dividend and divisor having been multiplied by such a power of 10 as makes the divisor a whole number, the decimal point in the quotient simply goes above that in the dividend.

The following method is recommended for the early work:

25) 62.75

The entire remainder is brought down each time, and the decimal point is preserved throughout.

In more advanced arithmetic, including the last two or three years of the elementary school, the value of the work must lie largely in the character of the problems, as previously suggested.  By the time a child has reached the sixth year of school, he has usually acquainted himself with the various arithmetical processes, and he is now ready for their various applications to actual conditions in life.  Correlation with geography, manual training and other studies is, therefore, of much importance.

Percentage, formerly a topic by itself, is merely one phase of decimal fractions, and should be so treated.  A large part of business arithmetic involves the finding of per cents, so that the method is continually applied after it is once presented.  The treatment of the subject by “cases,” and the learning of definitions of terms like “amount,” “difference” or even “percentage” may be considered obsolete.  There is need to know what “per cent”, means, namely “hundredths” (“hundredth” or “of a hundredth,” as in 6%, 1%, %), and there is occasionally some value in using the term “base.”  But the two leading problems of the subject are illustrated by two examples not requiring any elaborate vocabulary, namely:

1.  6% of $250 is how much?

2.  If 104% of x = $7.28, what does x equal ?

Practical problems in percentage rarely require any other forms.

The explanation of problems should consist of no carefully learned formula, but should be nothing more than an explanation of the steps involved, with the reasons.  Some use of the equation, with x to represent the unknown quantity, is fully in place.

In general in the study of arithmetic pupils are tempted to “figure” too much, and to allow the formal side to dominate the “thinking” side.  To overcome this difficulty it is well to have much oral work in the solution of problems, without any figuring.  To emphasize the thought side of arithmetic properly, children (1) should often read a problem a second or third time carefully, to get the exact conditions; (2) should then restate the problem in their own words, to make fully sure that they understand its condition; (3) should state the number of steps required for the solution and show the character of each; (4) should then give the approximate answer.  Figuring for the correct answer should often follow; but frequently this fifth piece of work should be omitted.

It is hardly wise to allow children to study their arithmetic and receive help upon it at home.  The reason for this statement is that parents and other home friends usually have different ways of solving problems from those employed at school.  Sometimes these home methods are worse, sometimes better, than those used at school.  But they are almost bound to be a source of confusion.  It is generally best, if home help seems necessary, for the helper to try to understand and follow the school method.

Reference books: Mathematics in the Elementary School, Teachers College Record, Columbia University; Teaching of Elementary Mathematics, D. E. Smith; Special Method in Elementary Arithmetic, C. A. McMurry; The Psychology of Number, McLellan and Dewey.

F. M. McMurry.