tests by questions and answers were the order of school life. Mental development was subordinate to the acquisition of knowledge, and results were therefore tested by the formal expression of knowledge rather than by the mental movements involved in its acquisition.

The fundamental principle of the conversational method of teaching is that it is psychological rather than logical. It adapts itself to mind rather than to matter to be acquired. Formerly the test of a method was in its being logical, in its presenting a subject in a systematic order determined by adult minds; now its test lies in its being psychological, in its presenting a subject in an order required by the mode of thought of the pupils. Then mind was forced to conform to the artificial formulation of knowledge; now knowledge must be adapted to the natural order of the developing mind. This order can only be determined by informal, interesting conversation.

The greatest advantage of this method is its encouragement of what has been well called "thinking aloud." The whole thought process is thus revealed, and not only the conclusions given; consequently if a conclusion is wrong it is easy to locate the point of deviation in the train of thought. The one characteristic which made the text-books of a few years ago so dry and unintelligible was that they gave only conclusions. We do not wonder that children had trouble with these conclusions when the processes of development were omitted. The conversational method aims to reach no conclusions until the pupils have passed through every stage of the mental process necessary, for them, to reach it. The mental processes involved in reaching a given principle may be very different with different pupils-a fact recognized by the method under discussion, but overlooked by most others.

In the informal discussion of a subject it is not touched only at isolated points-as is necessarily the case in using the "method of questions and answers"-but it is covered completely and connectedly. It is not a mere figure of speech to speak of "turning" a topic under consideration. For by this method it becomes, as it were, a sphere in a state of neutral equilibrium; every touch causes it to turn somewhat and reveal

some new aspect; each person approaches it from a different view-point, and therefore brings something new to the general conclusion. But in a recitation dominated by the teacher, the view-point is always his own and may not be at all intelligible to the pupils.

By the conversational method the initiative is given to the pupil. He starts the discussion at a point which is to him the natural point of attact, and from this beginning the development of the subject takes a trend determined by his own psychic movements. Any mind works most freely and rapidly along a line starting from its own view-point and directed by its own characteristics, intellectual or emotional. An attempt by a teacher to force such a mind to follow a line of development prescribed by him is almost certain to end disastrously. A wise teacher will, therefore, always allow the trend of the recitation to be determined by the "mental movements" of the class. This leads us to mention another advantage of the conversational method; namely, that it is extremely flexible and permits a change of plan in the conduct of the recitation whenever such change is seen to be desirable.

Again, this method leads to careful, coherent thinking. When a pupil is asked to "turn" a topic under discussion he is expected to express whatever comes into his consciousness concerning it. Knowing that his utterances will be subject to the comment and criticism of others, he will endeavor to think coherently and consistently, to exclude irrelevent things, and to avoid a repetition of what has been already said. At any rate, that is what we should expect from the method in the service of a good teacher.

It may not be adding anything to what has already been said to state that the conversational method is in harmony with the principle of evolution; but it will, at any rate, be stating the truth in another form. The pupil is encouraged to start with what he knows of a subject, and is led gradually and naturally out into that which was before unknown, by the unfolding or evolving of his own powers. That which was vague, crude and isolated, by numerous turnings, comparisons and changes of view-point, becomes distinct, finished and related to other knowledge. Just as mind itself becomes more specialized with

growing maturity, so does the knowledge of any principle become more exact and comprehensive, and its applications more general and adequate with successive turnings.

The bearing of this method in training in social co-operation should not be forgotten. Each pupil desires to make some contribution to the general fund of information concerning the topic which is being discussed, and on the other hand learns to give consideration and deference to the ideas of others concerning it. This participation in the formation of a body of knowledge and acceptance of it prepare children for later participation in the making of laws, the formation of public sentiment, and obedience to them.

In conclusion, the conversational method is psychological because it conforms to a well-known law of life. All life may be said to consist of a series of actions on the individual by his environment and reactions by the individual on his envi

This unending series of actions and reactions forms the whole content of consciousness, and by it alone is development possible. In a recitation conducted by the conversational method, a presented object or principle acts upon the mind of every pupil, and calls forth from each a characteristic reaction, every touch or turn of the subject by teacher or pupil causes it to present a somewhat changed aspect and hence to produce a new reaction in each mind. By this continued interplay of actions and reactions, the subject gradually develops and assumes definite form and relationships, and at the same time is so related to the mode of thought of each child as to be available for his use throughout life.

Some Suggestions on the Teaching of

Elementary Algebra

ERNEST B. LYTLE, UNIVERSITY OF ILLINOIS

In a book by a well-known author and teacher, x=-1 is given as a root of the equation

In a recent algebra examination the students were asked to prove that

Over fifty per cent of the answers read by the writer gave as a proof the solution of equation (3) by the method of completing the square which gave x equal to the above two values (2). These facts, together with some notions of equations observed to be common among my students, have led to this discussion.

Algebraic equalities may be divided into two mutually exclusive classes, identities and equations. An algebraic identity is an equality between two algebraic expressions (functions) which is true for all values of the variables for which the two expressions are defined. An identity then, may be said to be a declarative sentence stating a fact to be proved. The two expressions in an identity may be transformed into the same identical forms. An algebraic equation is an equality between two algebraic expressions which is true for only a finite number of values of the variables. An equation is, therefore, an interrogative sentence which asks for the values of the variables which make the two expressions equal, that is, which satisfy the equation. Solving an equation consists in finding by any method whatever, the values of the variable which will satisfy that equation.

Proving an identity and solving an equation are quite distinct processes, hence it is of fundamental importance that these two classes of equalities be carefully distinguished. That these notions of identity and equation are not sufficiently emphasized by teachers is evident from the frequency with which we see students trying to solve identities, and from the nonsensical things done by students in working with trigonometrical identities which is a common place of difficulty. The writer believes it tends to greater clearness to present equalities as mathematical sentences in which the equality signs are the verbs; an identity is a declarative sentence and an equation is an interrogative sentence. In this way mathematics becomes a language and symbols are used to express thoughts, which tends to overcome senseless juggling in algebra.

In teaching equations not enough stress is placed upon the point that a root or solution must always satisfy its equation. In solving an equation it is not always sufficient to go through a series of operations which lead to certain explicit values of x, for very often these values will not satisfy the original equation. A solution is complete when the values for x are shown by substitution to satisfy the original equation in x. This substitution is usually termed "the check," but it is really an important part of the solution, unless the theory of reversible processes is developed. Equation (1) above, illustrates the importance of this step. If we follow the usual method of solving fractional equations, multiply (1) by x2-1, transpose the x-terms to the first member, and other terms to the second member, and divide by the collected co-efficient of x, we certainly get x=—1. But -1 does not satisfy equation (1), for by substitution we get the unequal expressions

in which the first two terms have no value whatever, for division by zero is excluded from algebra. Often such expressions

2

as are said to be "equal to infinity," but infinity is not a

0

number as generally defined, and this is just a technical and