Which are rec- J Trr ^ tangular prisms (shaped like a box or beam)? 2. Use the equa- tions printed O below to find the volume of n, each solid and the area of its Base of g Base o( h surface. In a cylinder, volume = alt. X area of base. In a cone, volume — % ^^^- ^ ^^^'^ ^f b'^^^- In a sphere, volume = % irr^. In a pyramid, volume = % alt. X area of base. In a cylinder, surface =£ X {area of base) + (alt. X ^irr). In a cone, surface = area of base + {% slant height X circumf. of base). The pictures show what the slant height is. In a sphere, surface = 4''^''^^ • In a pyramid, surface = area of base + (^ slant height X perimeter of base). 197 198 The New Methods in Arithmetic UNDUE USE OF "CRUTCHES" Many teachets use "crutches," or methods that are easily- explained and learned but must sooner or later be supplemented by methods which are better for eventual use. Adding and subtracting by counting on the fingers, writing the number to be carried, and writing the sign + or — or X to guide you in n 568 568 568 . -r computing, m cases like , ooi _q21 VS21' ^'^^ familiar illustrations. They would agree to the principle "Other things being equal, form no habits that will have to be broken." The question is about the inequality of the other things. They would defend the use of the crutch on the ground that it saved more time in learning than it cost later for replacement by the regular habit. In some cases methods are taught for permanent use which other teachers regard as valuable, if at all, for temporary use. The question then is whether or not the habit is one that must be broken. Teachers are tempted to be shortsighted, sacrificing the future for the present. In order to evade some difficulty at the outset, they are tempted to involve the teachers in later grades in worse difficulty. Every school, therefore, will profit by a definite policy with respect to each such real or alleged crutch. The tendency is to use too many of them, and use them too long. Let us, therefore, consider some of the most popular ones. Counting forward by ones as a crutch in adding integers may be used for a few days in deriving the additions and for a few weeks as a check to verify additions. Then it should dis- appear. Counting backward by ones should not be used as a crutch in subtraction. The reason is that any child capable of learning arithmetic at all can learn the addition facts and sub- traction facts and will save much more than the time required to learn them in a short time thereafter. These counting Some Common Mistakes 199 crutches are popular with very few teachers, but with many pupils. Adding and subtracting by reference to some familiar com- bination (as 9+7 = 16 by "10+7 would be 17, 9 is 1 less than 10, so 9+7 = 16"; or 11-5 = 6 by "10-5 would be 5, 11 is 1 more than 10, so 11 — 5 = 6") may be called intelligent wastes of time. _ They are wastes of time because children who can do such reasoning could very quickly learn the combinations direct. They may be called intelligent because they replace a more mechanical process by a more thoughtfiol one. They do little harm, because they tend to "telescope" into the direct process. It is to be doubted, however, whether they really are as easy to teach and to learn as the direct processes. Using +, — , or X as a sign of what you are to do in com- 596 putations like on the blackboard and in books, and teach- ing the child to write +, — , or X in his own computations, are popular practices with teachers, but seem surely inadvisable. It seems much better to write at the head of the page, row, or column, "Find the sums," or "Find the differences," or "Find the products," than to attach a sign to each pair of numbers. It seems much better for the pupil to think what he is to do as he would in ordinary life. For, other things being equal, we should form a habit in the way in which it is to be used, pre- senting the situation as life will present it, and requiring the response which life will require. It seems that the practice of using the signs +, — , and X was adopted partly as a crutch to save teaching the meianings of add, sum, subtract, difference, multiply, and product, arid partly as a crutch to save the pupil the work of remembering what he had decided to do with cer- tain numbers. Experiments are needed for a final decision, but they will almost certainly show that neither saving is large enough to warrant the custom. 200 The New Methods in Arithmetic Writing the number that is to be "carried" in addition is perhaps the most popular crutch. Perhaps it is not a habit that must be broken, but should be permitted in life as well as in school. Perhaps, though to be abandoned later, it is worth employing in the lower grades. It is cne of the better crutches, since it does not introduce new sources of error, or much mar the appearance of clerical work, and makes checking quicker, though a bit less trustworthy. On the other hand, children can learn to add without it even in the third or second grade, and nobody has demonstrated that it is of very great help in early training. It is a sample of a question which we might debate endlessly without decision. Experiments are needed to measure what good it does and what harm it does. Writing the numbers as changed in subtraction seems almost indefensible except for a few times at the beginning 49 1 to show the procedure. Whether the form used is $50 . 25 or 19.67 $50.25 20 7, the crutch seems to do much more harm by confusion 19.67 ■ to the eye and mind, distraction and loss of time in writing the figure, and interference with the later habit's formation, than it does good by delaying a difficulty until pupils are older and abler. Since the change is always —1 or +1, there is no burden of remembering what it is, such as there is in addition. To remember that the change is to be made requires only that the pupil should keep in mind what he did in the previous step ; and this seems reasonable training for a pupil in Grade 3. The same procedure used with carrying in multipHcation seems much worse. Writing the numbers to be carried in this case, though permitted in many excellent schools, seems a clear case of shortsightedness in teaching. If the habit is Some Common Mistakes 201 not broken before "long" multiplication is studied, we have computations appearing as shown below in which the eye is distracted by irrelevant figures and in which there is a possi- bility of frequent error. 389 .47 276 685 I $325.00 55 32 2334 2740 66 5100 2723 53 11 4795 778 Whereas in addition you write one "crutch" figure to represent a more or less long series of operations, here a crutch figure is written for every multiplication except those few resulting in products under 10 and those at the end of each partial prod- uct. There is thus an appreciable waste of time in writing the numbers and in looking at them so as to add them. The crutch is practically valueless in checking, since there would be no advantage in a checking back of the products themselves. The "regular" habit of holding the number in mind, and adding it mentally not only must be acquired seme time; it also offers a form of specific training in keeping things in mind which seems well suited to pupils in Grade 3 and Grade 4. The regular procedure is in no sense harder to imderstand, or less indicative of the general rationale of the process of multi- plication. It is harder only in so far as it requires the pupil to keep more facts and relations under control. Finally, it is very hard to replace the crutch habit by the regular habit or in any way transform the former into the latter. Some crutches easily grow into the regular habits, either by mere dropping out of steps or by dropping a Uttle at one place and adding on a little at another; but if a pupU has learned not to think of what he is to carry, but to write 14 Subtract 3 973 2 qi 5 18 5 8 •'•OT^ 9 4 202 The New Methods in Arithmetic it, he has to drop off just what he learned to put on, and put on just what he learned to drop off. It used to be almost universal to teach pupils in adding or subtracting unlike fractions to rewrite them as reduced to a common denominator somewhat after this fashion: Add 51 9f 8| _6| 29f For the sort of additions and subtractions that used to be taught, this was necessary. Indeed it was often necessary also to do much computing to find some convenient common denominator for them. When rare collections of fractions are to be added together, such as -^s, -^s, and -5-s, or — s, -5-s, -^s, and — s, we should encourage any reasonable written work that will insure accuracy. Modem good practice, however, puts almost all of its time upon securing mastery with the additions and subtractions that will be used. Its "regular" method in adding s, and 2 —rs, or — s and -5-s, or -j-s and -^s, or even -jr-s, — s, and — s, is to think of them as reduced to -j-s or — s or -x-s, as 4 D o the case may be, but not to rewrite them. The problem then arises whether what used to be the regular method shall be retained as a crutch. It is an interesting problem, which I shall leave the reader to answer, first calling his attention to certain facts. One answer might be right for addition and the opposite answer for subtraction. When there are only two fractions, Some Common Mistakes 203 both ones in common use, the choice of a common denominator, the mental reduction of the two fractions to it, and the sub- traction using these remembered fractions are not a very hard task. Surely nobody would advocate the crutch in these four cases: 12f 9i n 31 m 17f 6i When, on the other hands, there are a dozen ntimbers in — s, -j-s, and -^s to be added, the task is much increased. A 'compromise crutch of the type shown below might be worth considering : 61 3 8i 2 7| 5 91 6 5i 4 2f 1 21 The use of the crutch hinders the learning and use of direct fraction combinations such as: 1 1 4 4 1 \ 1 4 3 4 3 4 \ 3 8 5 8 \ 1 4 1 2 \ \ 1 2 \ \ From the discussion of these different crutches, it should be clear: first, that crutches vary very greatly in merit or demerit; second, that the objection to the objectionable ones is not that it is childish to write instead of think, but that on the whole they waste more time than they save or weaken the learner more than they facilitate learning. 204 The New Methods in Arithmetic EXERCISES 1. Consider this definition and this rule: Numbers applied to the same unit are called like numbers. Thus $9 and $43 are like numbers, $9 and 43 cents are unlike num- bers, 9 dollars and 43 boys are imlike numbers. Only- like numbers can be added, and the sum is like the addends. Try to follow the rule in solving this problem: "Mr. Jones has 7 horses, 9 cows, and 23 sheep. Mary gave each one a name. How many names did she give in all?" 2. What useful purpose does the rule serve? What harm does it do? 3. Consider this rule: The multiplier must be thought of as abstract, and the product is like the multiplicand. What useful purpose does the rule serve? What harm does it do? • 4. How must you think of this equation: "The number of volts times the nimiber of amperes = the number of watts," in order to make it fit the rule quoted in 3? 5. Give other practices approved in science which seem incorrect according to the rule. 6. What is the purpose of the following exercises? I Let r =tlie number of miles per hour traveled. Let