to the process of abstraction. The sort of memory involved is a volitional recall. With mature students this may merge into an habitual reaction to situations as they are presented.

We have seen also that words and symbols are the chief instruments of abstraction and that the effective use of these is wholly dependent upon the recall of their interpretations and applications. Sheer recall will again function. here in the initial attempts to make use of these instruments, but it is only through habituation of the recall that they can. be utilized to the best advantage.

MEMORY IN RELATION TO SPECIFIC TYPES OF ALGEBRAIC

THOUGHT.

In the first chapter of this paper we set forth several specific types of thought processes which are involved in the study of algebra. Similarly, in the second chapter we identified a number of particular types of memory. In view of the aim of this paper, it seems appropriate at this point to bring these two elements together, and to discover

what

TABLE I.

correlation

exists

between them. This might be done by taking each type of thought independently and discussing it in connection with each of the types of memory; or each type of memory might be taken up in turn, and the different types of thought examined in order to determine in which ones the various memory types find application. For two reasons, however, neither of these methods will be followed. In the first place, the quantitative limitations of this discussion preclude the prolixity which would be inevitable in either case. In the second place, each type of thought and each type of memory has been discussed already, and either of the methods we have mentioned would necessitate a large amount of repetition. We shall resort to tabulation, then, and reserve our comment for the inferences which may be drawn from the tabulated findings.

The table I is an attempt to show the correlation between algebraic thought and memory types. The spaces which are marked with squares represent close correlations which are almost certain to exist; those marked with triangles indicate only a probable correlation. It should be borne in mind that the correlations indicated are but the result of a subjective analysis. There is no way of objectifying it, but this will not save it from criticism on this score. The reader is urged to examine it carefully, keeping in mind its subjective nature, and after a thorough perusal, to make such modification of it as may seem to him justified.

An examination of Table I shows that all the types of thinking involve several of the memory types, the strong correlations predominating in some cases, and the possible or weak ones in others. This data is collected and compiled in Table II. This table shows that there is not one of the thought types which does not involve almost certainly one or more of the types of memory, while most of them show a distinctly greater number of strong correlations and a still larger number of probable correlations. On the average, a single type of thinking almost certainly involves three types of recall and probably involves four others.

Table I furthermore reveals the fact that certain types of memory are found in algebra, while certain other types find relatively little application. For example sheer recall is found in all the types of thought, conceptual recall, reflective, and rational-associational recall in all but two, while mere repetitive verbal recall is involved in only one. These correlations are

This table should be read as follows: Sheer recall is a certain and important factor in eight of the thought types, and is probably involved in 5 others, thus giving it a possible application in 13 of the types into which algebraic thinking has been separated. Examination shows that certain of the memory types appear to be of much more importance than others. For

example, sheer recall finds 13 applications, reflective, rationalassociational, and conceptual recall eleven each, and so on down the list to a type which finds only one possible application. This difference is further emphasized by considering that only 4 memory types have 8 or more definite correlations each, while 4 of them have only from 2 to 4 such applications, and the remaining 5 have none at all.

We may now profitably confine our attention to those types of memory which find specific and definite application to algebraic thought. Table III shows that there are 8 of these. Two of this number (reflective, and rational-associational recall) are identical in their application and we shall reduce the number to 7 by considering them as one. Reference to Table I shows that every type of thought process made use of in algebra definitely involves some form of memory to a greater or less degree. From an aggregation of the table we find that in 91 possibilities of application there are 38 actual definite correlations. This would seem to establish beyond question the importance of the memory factor in the study and teaching of algebra.

The matter of the particular application of the different types of recall to the great variety of algebraic situations is a problem for the teacher and the supervisor. We shall endeavor, in the following sections of this discussion, to discover the effect of a consideration of the application of memory to algebraic thought processes in the formulation of a teaching technique and a supervisory program or policy.

(To be concluded.)

A COMPOSITE COURSE FOR SEVENTH AND EIGHTH GRADE MATHEMATICS.

BY M. J. NEWELL,

High School, Evanston, Ill.

A report by the Articulation Committee for Mathematics of the Lake Shore Division of the Illinois Teachers' Association.

Foreword. The territory of the Lake Shore Division of the Illinois Teachers' Association comprises some half dozen counties in the northeast corner of the state exclusive of the city of Chicago. A considerable portion of its population is suburban to the larger city and practically all of its business activities center there. Its educational experiences and problems are not peculiarly its own for all of its communities are vitally Ameri

can.

It does admit a large foreign element in its population but this is well distributed and eagerly striving to become American. But its compact territory, good transportation, and alert school authorities have all served to keep it well forward in striving for the solution of educational problems.

Organization of Township High Schools. With one or two exceptions all the secondary schools within a radius of seventyfive miles of the city of Chicago are organized under the Township High School Act. This plan of organization is very popular in Illinois with nearly four hundred such districts already in existence and the number increasing everywhere through the state with the coming of paved roads.

Drawbacks of Township High School. But the township high school tends greatly to magnify the differences in organization, and especially, in methods of instruction that exists everywhere in this country between the elementary school and the secondary school.

(1) The elementary school and the township high school are under distinct school boards, with distinct and absolutely independent teaching forces and superintendents. The high school is not responsible to the elementary school nor is the elementary school in any legal way responsible to the high school.

(2) The township, which is the high school district, usually includes several independent elementary school districts, each with its own school board, its own independent superintendent, and its own methods of meeting educational problems. If one elementary district in a given township has a reasonable population and considerable wealth, it cannot help but work out its problems in a better manner than its neighbor district in the same township, with larger population and less wealth. But under the Township High School Act the secondary school must accept the diplomas from every grade school in its territory and at their face value.

Efforts to Meet Drawbacks. To bring about a better co-ordination between the grades and the high school, many township principals have adopted the plan of making some instructor who is well-known and popular throughout the high school district chief liaisson officer-his duty being to visit all the teachers of the higher elementary grades in their classrooms, study their difficulties at first hand, and present the problem of deficiency in preparation for the upper school. Of course this can only be done with the full approval and consent of the grade superintendent.