Subject. General Science.

Text.

(a) Text problems solved in full.

(b) Text problems solved in part......
(c) Text problems to be solved by pupils...........

It appears from the above tabulation, that in certain of the subjects which make up the high school curriculum in mathematics and science, pupils are required to solve many problems and that they are given by the texts unequal amounts of assistance in their efforts to master the technique of solving those problems. The time thus spent in other fields than mathematics and science is not great, as is shown in another study not here reported, but will in the opinion of the writer increase in the next decade; for more and more men of today are inclined to be dissatisfied with general statements. There seems to be a growing desire to know not only what, but how much.

THE NEED FOR SPECIAL TECHNIQUE IN PROBLEM SOLVING That the possession of ability to succeed in operations involving one or more of the fundamentals of arithmetic (addition, subtraction, multiplication and division) carries with it a reasonably high degree of success in performing any or all of the other fundamentals is seen from the following results obtained by Dr. Stone in his tests in fundamentals which were given to a sampling of 500 pupils. His results are here quoted:

was +0.98;
was +0.95.

between multiplication and division

The above correlations show close relationships between achievement in division and in multiplication which indicates that the possession of ability in division carries with it almost the same level of ability in subtraction and in multiplication. The correlations here quoted simply bear out facts perfectly well known to teachers of arithmetic.

The results obtained by Dr. Stone through relating the achievements of these 500 pupils in each and all of the fundamentals to the achievements of these pupils in reasoning with arithmetical data do not show as high correlations as were obtained in relat

ing the achievements of pupils in the several fundamental operations.

The relationships which were found by Dr. Stone when expressed as coefficients of correlation as here quoted:

In a study made in the Speyer Junior High School in New York City in 1917, the writer found a correlation of +0.34 between the scores of 125 boys in the Stone Reasoning test and their scores in the Woody Division test.

The low degrees of correlation between fundamentals and reasoning as here quoted indicate that abilities in the fundamentals of arithmetic are different abilities from those utilized in reasoning with arithmetical data. If a high level of achievement in this type of reasoning is desired, more effective teaching methods can be utilized than to provide training in the fundamentals of arithmetic alone.*

Now if ability in dealing with arithmetical situations of the problem sort can be trained, the best practicable method of procedure should be sought. The consensus of opinion of teachers, parents, and business men is that most secondary school pupils have much difficulty and little success in solving those verbally stated problems in which numerical results are required. A part of this failure is no doubt due to errors made in operations involving the fundamental processes, yet the results quoted above show that the attainment of a high degree of accuracy in fundamentals will not in and of itself guarantee success in meeting problem solving situations involving both reasoning and computation.

From two points of view we have seen the need for teaching

*Perfect correlation (or perfect correspondence) is represented by +1.00 indicating a 1:1 relationship between the traits in question; correlation (or chance relationship) by 0; and a strictly opposite relationship by -1.00. A correlation of such as +0.30 indicates a positive but low degree of relationship while +0.80 indicates a high degree of relationship between the measures and therefore between the traits measured.

technique, if pupils are to be trained to meet successfully problem solving situations:

1st. The frequent occurrence of problems in the high school curriculum; and

2d. The fact that a specialized type of teaching is required.

PROBLEM SOLVING ANALYZED.

The concluding portion of this paper will deal with a psychological analysis of the thought processes utilized in the solutions which involve both reasoning and computation and will suggest a technique of solution based on this analysis. Several typical solutions will be presented showing how such an analysis may be embodied in a general method of solution for such problems. The outline here given for analyzing verbal problems results from a critical analysis of my own thinking and experience in teaching pupils to solve problems and from the observation of the work of many teachers and pupils in such situations.

As I recall the matter I was led to make my first analysis of the thought processes involved in problem solving in an attempt to suggest to pupils a definite and helpful method of procedure in their solution of problems in elementary algebra. For some time now the analysis of the thinking done in problem solving which involves reasoning and computation of an algebraic sort has appeared to me to consist in the main in the following mental processes. Each of the six processes here given can, of course, be further broken up into detailed methods of procedure. Further analysis is probably necessary and should doubtless be made. In its present form, the analysis suggested has been used as a basis of discussion with several hundred teachers of mathematics and is here offered for that specific purpose. The analysis follows:

Step I. Identification of the several elements of the given data. Here certain elements will be recognized as known. facts and others will be identified as new or unknown.

Step II. Seeking out and expressing as an equality the quantitative relationship or central thought of the problem. Step III. Representation of the several unknowns in terms

of number symbols.

Step IV. Substitution of known elements and those represented by the number symbols of Step III into the fundamental equality of Step II.

Step V.

Solution of the equation formed in Step IV. Step VI. Checking the results obtained from the solution of Step V.

In somewhat greater detail the above analysis is expanded and expressed in terms of steps necessary to the solution of a problem which involves a quantitative relationship as follows:

Step I. Identification of elements. Reading the problem to identify the several given facts and the required fact or facts.

Step II. Expressing the quantitative relationship. Determining the quantitative relationship (or equality) existing between the several elements of the problem, both known and unknown and expressing this equality briefly in words (or abbreviations) using where possible the sign of equality (=), and the symbols of operation +, −, X, ÷, and the fraction line.

Step III. Representation using symbols. Representation of each unknown number in the problem in terms of a letter or a letter and a number.

Step IV. Substitution. Substituting the numerical values given in the statement of the problem and noted in Step I, for such known quantities as appear in the expressed equality and substituting the symbols obtained in Step III for the unknown quantities of the expressed equality.

Step V. Solution of the equation. Solving the equation thus obtained to get the value of the unknown number which was represented by the letter involved, and where necessary substituting this value in the other number symbols of Step III to find the values of the other quantities identified as unknown elements in the statement of the problem.

Step VI. Check. Checking the work through the substitution of the numerical result obtained for the unknown quantities along with the given values for the known quantities as stated in the text of the problem, or in the equality of Step II, thereby determining whether or not the conditions of the problems are satisfied by the values found for the unknown numbers.

The application of the six steps given above to the solution of such problems as have quantitative relationships which can be expressed by a simple equation will appear in the solution of the following problems:

EXAMPLE I.

The sum of two numbers is 72, and the greater is three times the less. Find both numbers.

Step I. Two unknown numbers are to be found. Their sum is 72. One number is three times the other.

Step II. (Expressing the quantitative relationship)

A stick 120 inches long is to be cut into two pieces, one of which is to be 14 inches more than three times the length of the other. How long is each piece?

Solution.

Step I. (Identification) The stick is 120 in. long, the length of the pieces are to be determined, one is to be 14 in. more than three times the other.

Step II. (Quantitative relationship)

Greater length + less length = 120 in.

Step III. (Representation, using symbols)

Let no. of inches in shorter piece. then 37+14= no. of inches in longer piece.

The value of 30 coins, consisting of nickels and dimes, is $2.60. Find the number of each.

Solution.

Step I. Thirty coins are used, their value is $2.60. The number of nickels and the number of dimes is to be found.

Step II. Value of nickels + value of dimes

=

total value.