class, all fail. If the assignment is for the mediocre, the dull students do not get anything well and the brilliant ones do not have enough to keep them busy, and they are not given the opportunity of developing to the best of their ability. If the assignment is given to fit the needs of the poor students, neither of the other two groups have enough to do to keep them out of mischief, and the greatest good is not done to the greatest number. It is one of the outstanding faults of our present school system that it neglects the more brilliant students by not giving them enough to do; they are the ones that can and should become the leaders of our country. We cannot change the material of brain matter, we can only develop that material to its highest capacity. When it is all finished, some will naturally be on a higher level than others.

About the fifth week of school this year, after I had discovered the mathematical ability of each student, I divided each of my sophomore geometry classes into three groups. Group I consisted of the bright students, group II the medicore, and group III the dull. I seated group I on one side of the room, group II in the center of the room, and group III on the other side. I use a definite blackboard space for the assignment each day, which is cumulative. Each group does the work of the lower groups plus the assignment for its particular group. The choice of assignment is something on the following order. Group III works all the important theorems and corollaries, group II all the theorems and corollaries plus some of the exercises, and group I all the exercises in the text and in addition many others. This does not mean that group III fails to have the advantage of any of the exercises. The pupils of this group often listen. to the explanations of the other two groups. They are able to follow the explanations, even though they could not possibly have worked them out originally. However, some of the easier exercises might be given to group III. This group would never be able to work out the more difficult ones, even under the former plan of assignment, so why have them attempt something that we know is an impossibility. It is seldom advisable for group I to listen to the recitation of group III. They had better be spending their time on working additional exercises, and it is indeed much more enjoyable on their part. Pupils soon get used to studying while others in the room are reciting.

The plan was thoroughly explained to the pupils at the time of the grouping. They were aware of the fact that some pupils

are naturally more brilliant than others, but they were not used to having a teacher say so. We are in the habit of trying to make pupils think that they can do just as well as anyone else. if they only work hard enough, when down deep in our hearts we know it is not true. A student may move to a higher group when he has done the work of that group for a few days, and he may be put in a lower group when he has failed to do the work of that group for a few days. This moving may be done at any time, although it is often advisable to make moves after a weekly test. No work is accepted that is not done exactly right. The dull students can stay with a thing until they get it without retarding the whole class, while the others go on working at something else.

Most of the teacher's time should be given to groups III and II in the form of supervised study. This does not mean that there must be supervised study time. The regular period may be used to advantage in showing the poorer pupils how to study economically and to the best advantage. The better students do not ordinarily need this training. If pupils are taught how to attack a problem, they will develop into independent workers, and in the end will have gained much more than if the entire time had been spent in reciting something they already know about.

Usually the first thing that is asked concerning this idea is: Does it not result in fatalism? That is, do not pupils become discouraged because they are placed in low groups? They always have the opportunity of moving up or moving down, and this has happened in many cases. I have had many pupils in group III tell me that they liked this plan because they understood what they were doing, and for once in their lives they had the satisfaction of having done something really right. Are there not pupils who overwork in their effort to attain or retain a high group? Yes, but it is no more true than the fact that under the former plan these same pupils would overwork to the same degree in trying to secure high grades. There are pupils who remain in a lower group than their ability calls for, but the number is not nearly so great as the number that made it their business just to get by under the former plan. Their pride is being appealed to by everyone knowing their standing every day, and the result is that almost everyone is doing the best he can and is placed in the group that fits his mentality. The question of "more work for teachers" is of vital interest and concern. This

does require more planning on the part of the teacher, but it requires less manual labor. The teacher finds out the standing of her pupils much easier, which does away with so many test papers to grade. A group at a time may be tested, and in that way there will be fewer papers all at once. Group I needs very little testing. A teacher will spend her time with those pupils who really need help, and her justice in grading is much more accurate. After getting a good start on this plan, there is very little trouble in carrying it through.

We use the A, B, C, D, and F, system of grading in Salina. Pupils in group I comprise the A and B students, group II the C's, and group III the D's and F's. We are all familiar with the fact that the per cent of failures in the required mathematics courses is entirely too high. If we have any faith in the "normal probability curve," which is no doubt as scientific as anything we can mention at present, there is indeed something wrong with either our teaching or our grading. The first semester of last year I had 19 per cent failures in my sopohmore geometry classes. The first semester this year under the group plan I had 9 per cent failures. I am convinced that this difference is due to the pupils knowing their work better.

Although I have only been using this plan for one semester, I am well satisfied with the results. I believe it is far the best plan to use in a required course in mathematics, where there are all types of pupils in the classes and the division with reference to mentality has not been provided for otherwise.

IS THE TEACHING OF MATHEMATICS AN EASY TASK? BY G. H. JAMISON,

Professor of Mathematics, State Teachers College, Kirksville, Mo. A writer in a recent mathematical journal expressed the thought that there is no subject in the high school curriculum easier to teach than algebra if a standard text is used. also frequently hears the same utterance with respect to any high school mathematics from speakers who have not had any experience in teaching mathematics and from those who think they are teaching mathematics when in reality they are not teaching at all.

There is much so-called teaching which is nothing more than presiding over a group of students, reading lists of exercises for them, pointing out errors, checking up the final results by

the aid of an answer book, assigning so many problems for the next day and repeating all this to the end of the term. To teach arithmetic, algebra and geometry in this manner is perhaps easier than to teach any other subject unless it be spelling or reading when taught equally as poorly. Suppose we do not call this teaching and try to describe real teaching.

Real teaching of high school mathematics involves a knowledge on the teacher's part of the foundations of the science being taught. The teacher will feel or appreciate the logical development of the subject. The value of the part to be played by definitions and assumptions will be known. In genuine teaching, the art of questioning will be highly developed. It is easy to ask a poor question and often very hard to ask a good question. The good teacher will answer most questions by asking other questions. Good teaching will hardly be exemplified by the one who does all the work and all the thinking. Good teaching insists on the why and how as well as the results. It will make the work in mathematics hard for students because it makes them think. Good teaching will be difficult to do because it requires so much alertness, such skill in marshalling the forces, so much knowledge and such quick reaction to the varying needs of students.

Let us now see if this popular contention about the teaching of algebra being easy is true.

If this subject is properly taught, the class in the beginning will realize that there is but little if any difference from the arithmetic unless it be more of generalization. For many days no new elements will be introduced. Principles long used in arithmetic will be discovered and stated. The student will see that the beginning of algebra is in part the study of a new language. But this is not wholly new for in arithmetic, while studying the topic of mensuration, the rules were often stated in the language of algebra, that is, symbols were employed. The matter of substituting numerical values for letters and evaluating expressions is a topic which causes panics for many students. This has been done in the formulae of mensuration in arithmetic. The teacher's task is to prepare the way for this generalization which differentiates in one respect at least the algebra from the arithmetic, so that the student will see that he is simply going farther than he did in arithmetic. This generalization will one day cause the class to try to subtract any two numbers. Here will be developed a need for the negative number.

To introduce properly the negative number and the operations on it is no easy task. At this point the teacher will need to know mathematical history, as well as the importance of definitions and assumptions. What can be done with these new numbers? Nothing can be done until certain definitions are made. There is need for going slowly but surely at this point. Students need to see that they are being naturally and gradually led into new number worlds, and that they are now placing the foundations, as far as they are concerned, for the science.

Algebra can be made a thought study or it can be made a subject of endless, wearisome drill. If the teacher demands thought her task is hard. If she be indifferent as to whether her students are gaining power to meet new number relations, then her task is easy enough.

Suppose we look at the subject of factoring. To teach this so that power is gained the learner must know the meaning of type expressions. If any teacher has a recipe for teaching types easily, request is hereby made for it. What does it mean to say that ax2+bx+c or x2+bx+q is a type? Under which type should 4x2+3x+8 be placed and why? It looks as if it would not be hard to get students to understand the type a2—b2. As long as the two terms are squares of monomials the student does well, but let one or two of the terms be squares of polonomials and then watch results.

Does the average teacher of high school mathematics who finds the subjects so easy to teach, understand the fundamental laws or assumptions as applied, first, to integral exponents and then fractional, negative and zero exponents? Does the teacher build up the science or force the pupils to do as rules say to do? Mathematics can be taught, and it is the easy way to teach it, so that students can work problems by the wholesale without knowing how to work them. A student taught by this plan was asked how to reduce 2/3 to twelveths. She replied, "Divide 12 by 3 and multiply the result by 2. This gives 8/12." Good

It will insist on the pupil

teaching will not allow such a result. knowing how to do things as well as possessing ability to get results. The how and the why of mathematics requires thought on the part of the student and difficult work on the part of the teacher. The results or answers to problems can be had in many ways. From the educational point of view it is just as important, if not more so, to know how to get a result than to be able to get it. Ask a class how to add fractions, clear an