then when he came to solve it, the chances were that he would not see the relations in the right way, and so his work would be erroneous. When I became interested in him, I asked him at the outset to read and interpret orally every problem which had been assigned him by his teacher. Out of six problems to be solved the first day I worked with him, he understood only one correctly. He had not learned to read problems with sufficient attention to each factor. In an arithmetical problem every word is significant, while in the ordinary reading lesson it is quite different. This boy could get impressions from his reading lessons readily and with sufficient accuracy to meet all requirements; but in arithmetic this may not be enough.

The first work which was required of him was to read one problem at a time, and to explain in his own words what the relations in the problem were, and to supplement his interpretation by diagrams whenever possible, which could be easily done in all problems requiring the determination of lengths, areas, volumes, etc. In problems involving operations in weights, measures, money, etc., the boy was asked to illustrate actual relations by using the proper units. In this way he constructed each problem concretely. After a month's work of this kind, there were very few problems encountered, the con

ditions of which the boy could not illustrate concretely in some way. In the last resort he could by gesture indicate the relations described in his problems.

After a few weeks' work of this sort, the boy's improvement in his arithmetic was occasion for remark on the part of his teacher and his classmates. He is not yet always accurate in his fundamental operations; but he has made the beginning of a habit of finding out definitely what a problem means before he attempts to solve it. He can now be set six problems a day to solve, and he will write out in his own words the relations described in each problem. It may be remarked in passing that he has gained much from this experience besides efficient thinking in his arithmetic. It has been a good training for him in reading, for he was required to read with such care and attention in order to grasp the significance of every phrase and even every word that it has given him ability in accurate interpretation that he did not have before.

Of course, reading of this sort could be carried too far so far that it would arrest the child's freedom in his reading in non-mathematical subjects. But every individual in his daily life needs, at least occasionally, to read selections where the minutest detail must be appreciated with mathematical exact

ness.

How can this ability be gained more effectively than in the accurate reading of arithmetical problems? The writer feels that a teacher should require children from the third grade on to read and interpret problems every day, the aim being to have them acquire a habit of determining precisely what is stated in any given problem.

Verbal study of weights and

measures as an

example

Let us look at the results of mere verbal teaching in other phases of arithmetical work. I have been observing in a certain schoolroom in which the teacher has been endeavoring to teach weights and measures to children who have had experience outside with few if any of the units which they have been studying in the school. And how are they studying these units? The teacher had the pupils learn by heart the table of linear measure, as an instance. So far as one could tell, these children had not handled a foot ruler. They certainly had only a dim idea of what a yard measure was; and a rod and a mile might mean the same thing to them. But they had said over and over again that "twelve inches make a foot", "three feet make a yard", "five and a half yards make a rod", and "three hundred twenty rods make a mile". The teacher frequently gave problems like these for drill: "How many inches in one-third of a foot?"

"In three yards?" and so on. "How many yards in two rods?" etc. In this work the effort of the children was to remember their tables, and to perform correctly the multiplication processes given by the teacher. They were not actually imaging any of the situations which the teacher presented. She apparently never once thought of asking them to indicate the length of a yard, either on the board, or by extending their hands, or in any other way. It did not occur to her as necessary that she should have the children draw a line an inch long, a foot long, and so on. Her principal aim was to have her pupils remember verbal statements they had learned, and apply these in the special, formal situations which she presented. But these situations really did not involve any knowledge of actual measurement, only a knowledge of words.

The teaching of weights or measures of any sort which does not require pupils to deal with the actual units, so that when they solve a problem they do not think of the result in terms of actual distance, or area, or weight, or size of the measure employed, is defective. Of course, if the pupil has had the actual experience of using units of measure, and constructing higher out of lower units, then he should have experience in solving rapidly problems in which he simply performs the process involved

without attempting to visualize each factor and the results. But care must be taken in the early years to make his arithmetical thinking definite and concrete, so that he can translate correctly, when need be, the results reached in the solution of his problems. As an aid in this work, he should be required to make diagrams to illustrate all problems involving weights and measures, unless such diagrams should be too complex, which would rarely be the case. In estimating areas, for instance, the pupil should at the outset always make his diagram according to scale. In finding cubical contents this can be done. Children in the third and fourth grades can easily make drawings illustrating the ratio and size of the different units in dry and liquid measure. When a pupil is required to do this work he gains a comprehension of the meaning of his processes which he can never do if he simply learns tables, and then tries to apply them to situations in which he does not have actually to construct anything.

No teacher of arithmetic could fail to be interested in reviewing the changes which have been

Clear thinking

and useful problems in arithmetic

made in text-books on this subject during the past twenty-five years. Perhaps the most significant change which has occurred

relates to the character of the problems which pupils