In the first problem "milk" could be changed to "water," "syrup," or any other liquid without changing the problem. Furthermore, "bottles" could be changed to "cans" or "jars." Hence, there is no significant connection between the problem and any adult activity. A similar conclusion applies to the other problems.

Thus it appears that there are two general classes of problems: A, Operation Problems, those not identified with a particular activity or identified with an activity that does introduce a technical terminology peculiar to that activity; B, Activity Problems, those identified with a definite activity of children or adults which introduces a technical terminology.

Within each of these two general classes of problems,19 a further differentiation may be based upon the implied question concerning the functional relationship. (See page 19.) All problems that ask the same question may be considered to form a problem type which may be described by designating the quantities given and the one to be found. The question concerning functional relationship is: What calculations are to be performed upon the given quantities in order to obtain the one to be found?

An elaborate study of the problems provided by texts2o resulted in the identification of 52 problem types in the field of "operation problems" and 281 problem types under "activity problems."21 Descriptions of representative problem types are given in the following pages. A complete list of all problem types is printed in Appendix A. See also pages 90-92.

Al To find totals by addition, given two or more items, values, etc.

A3 To find the amount, or number needed, by multiplication, given a magnitude and the number of times it is to be taken.

"A verbal problem in arithmetic is a description of a quantitative situation or condition plus a question that usually requires a numerical answer. The solving of this requires the determination of the calculations to be made in order to obtain the answer. The basis of the determination of the calculations to be performed in the solving of a problem is the general quantitative relation which connects the quantities of the problem. For example, consider this problem: "An agent sells goods on a commission of 10 percent. How much does he remit to his principal for sales amounting to $1150?"

The quantities of this problem, proceeds (amount remitted to principal), rate of commission and amount of sales are related as follows: Proceeds = amount of sales - the product of amount of sales and rate of commission. In order to solve this problem rationally, that is, by reasoning, it is necessary that one answer the question, "How is the amount to be remitted to the principal calculated from the amount of sales and the rate of commission?"

20See page 48 for a description of this study. The "problem types" are used in explaining the process of solving verbal problems. See page 21.

"A few types of learning exercises are included that do not require calculation. See Appendix A, A24, A25, B5, B5h, B5i, B5j, B5k, B51.

A5 To find how many times a stated quantity is contained in a given magnitude, given the quantity and the magnitude.

6 To find how many when reduction ascending is required, given a. a magnitude expressed in terms of a single denomination.

b. a magnitude expressed in terms of two or more denominations. 18 To find a dimension, given the area of a rectangle and one side.

3 To find a difference, given denominate numbers of different denominations.

5 To find the ratio of one number to another, given the two numbers.

6 To find a part of a number, given the ratio of the part to the number and the number. (The fraction may be in terms of fractions or decimals.)

1 Buying and selling," simple cases.

a. To find the total price:"

1. given the number of units and price" per unit.

2. given the number of units and the price per unit of another denomination. b. To find the number of units:

1. given the total price and price per unit.

2. given the total price and the price per unit in another denomination.

3. received in exchange of commodities, given an amount of each commodity and the unit for each.

4. given the price per unit of each of two commodities, the total price of both, and the ratio of the number of units of one to the number of units of the other.

5. given the margin2 per unit and the total margin.

c. To find the price per unit:

1. given the total price and the number of units.

2. given the total price and the number of units in another denomination.

3. in exchange of commodities, given the number of units of each commodity and the price per unit of one.

4. given the number of units of each, the combined price of both, and the ratio of the price of the one to that of the other.

d. To find the amount to be received for several items, given the price of each. e. To make change, given an amount of money and the price of a commodity. f. To find the margin or loss given the cost price and the selling price.

g. To find the total margin or total loss:

1. given the number of units and the margin or loss per unit.

2. given the unit cost, the unit selling price, and the number of units.

h. To find the margin or loss per unit, given the total margin or loss and the number of units.

"Descriptions of quantitative relations given below are expressed in terms of buyIn some cases changes in terminology would be necessary if the activity were to considered from the standpoint of selling.

23Total price" is used to designate the amount received for several units of the he commodity rather than the amount received for several commodities.

"Price is used to designate the quantity taken as a basis of computation. Usu- "price" refers to the value or worth of a unit rather than a specified number of ts. "Price" is often limited by the qualifying terms cost, selling, marked, and list. "Margin is a term used to represent the difference between the cost price and the ing price and therefore is a substitute for the words "gain" and "profit" as they commonly used.

B2 Buying and selling, more complex types.

a. To find the selling price:

26

1. given the rate of discount or loss, and the price.

2. given the rate of advance or margin and the price.

3. given the rate of two or more successive discounts and the price.

4. given the price, rate of advance or margin, and rate of discount or loss.

5. given the rate of commission, discount, margin, or loss and the amount of commission, discount, margin, or loss.

6. given the price and the amount of commission or discount.

b. To find the amount of margin, loss, commission, or discount:

1. given the total price and the rate of margin, loss, commission, or discount. 2. given two or more successive discounts and the total price.

3. given the total price and the selling price.

c. To find the rate of margin, loss, discount, advance, or commission:

1. given the total price and the amount of margin, loss, discount, advance, or commission.

2. given the cost price in terms of two successive rates of discounts and the list price, and the selling price in terms of a single rate of discount and the list price.

3. given the total price and the selling price.

d. To find the price:

1. given the selling price and the rate of discount or loss.

2. given the amount of margin, loss, commission, or discount and the rate of margin, loss, commission, or discount.

3. given the selling price and rate of margin.

4. given the selling price and two or more successive discounts.

e. To find the amount due the agent or agents, given the number of units, the price per unit, and the rate of commission.

f. To find the equivalent single discount in percent, given two or more successive rates of discount.

g. To find one of two or more successive discounts, given the list price, one or more of the successive discounts in percent, and the net price.

Limitations of the list of problem types. A comparison of the problem types appearing under the caption, operation problems, with those listed as activity problems reveals a number of apparent duplications. This is to be expected because it is theoretically possible for any question concerning a quantitative relationship listed under operation problems to be implied in a problem clearly identified with some activity. When this occurs the problem has been classified as belonging to a type under activity problems.

The recognition of two overlapping groups of problems appeared to be justified by the fact that many problems found in arithmetic texts could not be assigned to an activity and that when they were clearly identified with a particular activity such as "borrowing, lending or saving money" or "insurance" a technical terminology was introduced which tended to make them different from other problems requiring the same calculations but not identified with the same activity.

"Rate may be expressed in terms of percent or as a fraction.

Failure to group problem types under some such heading as "activity problems" would suggest that the problems of arithmetic were abstract. The absence of a list of "operation problems" would have made it impossible to classify many problems now found in arithmetic texts.

A comparison of certain of the groups of problem types under activity problems (e.g., B1, B2 and B3) will reveal a type of duplication caused by the fact that two bases of differentiation were recognized; first, the general character of the activity in which problems occur, and second, the question concerning functional relations which a problem implies. It was decided that the first basis (general character of the activity) should have precedence over the second.

The writer and his assistants were compelled to exercise judgment on a number of other points. Consequently, the list of problem types should not be accepted as final. Especially, the conclusion that there are exactly 333 problem types should not be drawn. This total would have been different if different decisions on a number of minor points. had been made.

Value of the list of problem types. Although the list of problem types has been evolved after much careful thought and has been used as a basis in analyzing ten series of arithmetic texts, the enumeration of problem types given in Appendix A must be considered only a tentative formulation representing the judgment of the writer and his assistants. However, this tentative formulation should prove useful because it emphasizes that an arithmetical problem asks a question concerning a quantitative relationship which the solver of the problem must identify and then answer. Furthermore, it provides a working basis for considering the problem content of arithmetics.

Conclusions in regard to learning exercises. The most significant conclusion to be drawn from this description of the learning exercises of arithmetic, especially the types of examples and problems, is that the number of types of exercises is large. Each type of exercise constitutes a basis for a learning activity which is different, at least in some respects, from that occurring in a pupil's response to any other type. Hence, this analysis of the learning exercises of arithmetic is a necessary prerequisite for a consideration of the teacher's responsibility for devising and selecting exercises.

CHAPTER IV

THE LEARNING EXERCISES PROVIDED BY TEXTS

IN ARITHMETIC

Three types of content in arithmetic texts. Arithmetic texts include three general types of content; (1) statements of what pupils are to learn (facts, rules, definitions, and principles); (2) illustrations, explanations, descriptions, and the like which imply learning exercises involving tracing (see numbers 8, 20, and 26, page 34); and (3) explicit learning exercises. Examples (explicit requests to perform specified calculations) and verbal problems make up the majority of this third type of material.

The problem of this chapter. The problem of this chapter is to present certain information relative to the example and problem content of arithmetic texts. The information relating to provisions for the first type of learning exercise is taken from studies reported by other investigators. The information concerning the problem content is based on an original investigation conducted under the direction of the writer.

The example content of arithmetic texts. Since the principal function of examples is to provide practice on the combinations (basic and secondary), the most significant information relative to the example content of arithmetic texts is the number of occurrences of each of the combinations. A statement of the amount of space devoted to examples or the number of learning exercises of this type does not constitute a very significant description. An analysis of the examples with respect to the operations involved is a little more helpful but it still leaves one with only a very vague notion of the nature of the learning exercises which the texts provide.

Writers who have analyzed the example content of arithmetic

'For an illustration of an analysis of this type see:

SPAULDING, F. T. "An analysis of the content of six third-grade arithmetics." Journal of Educational Research, 4:413-23, December, 1921. The investigator presents a count of the examples and problems in six third-grade arithmetics. He found that the ratio of examples to problems varied from nearly 5 to 1 to approximately 2 to 1, the average being a little more than 3 to 1.

CLAPP, FRANK L. The number combinations: their relative difficulty and the frequency of their appearance in textbooks." Bureau of Educational Research Bulletin No. 2. Madison, Wisconsin: University of Wisconsin, July, 1924. 126 p.

KNIGHT, F. B. "A note on the organization of drill work," Journal of Educational Psychology, 16:108-17, February, 1925.