THE UNIVERSITY OF ILLINOIS LIBRARY 370 No. 26-34' rr«ssrfr™^"-™" Digitized by the Internet Arciiive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/teachersresponsi31monr BULLETIN NO. 31 BUREAU OF EDUCATIONAL RESEARCH COLLEGE OF EDUCATION THE TEACHER'S RESPONSIBILITY FOR DEVISING LEARNING EXERCISES IN ARITHMETIC By Walter S. Monroe Director, Bureau of Educational Research Assisted by John A. Clark Assistant, Bureau of Educational Research JAN 2 5 price 50 CENTS PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA 1926 'Hi TABLE OF CONTENTS PAGE Preface 5 Chapter I. The Immediate Objectives of Arithmetic 7 Chapter II. The Processes of Le.^rning and Teaching 26 Chapter III. The Learning Exercises of Arithmetic 33 Chapter I\'. The Learning Exercises Provided by Texts in Arithmetic 46 Chapter \'. The Teacher's Responsibility for Devising and Selecting Learning Exercises in Arithmetic. 56 Appendix A 65 Appendix B 90 PREFACE The research reported in this monograph deals with a very practi- cal problem. Every teacher of arithmetic continually faces the problem, "What learning exercises should I ask my pupils to do?" It is true that few teachers explicitly formulate this question but all must answer it. Incidentally it may be noted that the answer given is a very potent factor in determining the efficacy of the instruction. Attempts to answer questions that ask what should be done may be designated as "complete research" to distinguish such investigations from fact-finding inquiries which may be called "auxiliary research." The work reported in this monograph represents an attempt to carry out a piece of "complete research." In this endeavor the results of a number of "auxiliary" or "fact-finding" studies have been utilized but reference to them has been subordinated to the consideration of the two basic problems. Even the report of the analysis of ten series of arith- metic texts, which represents more than 2500 hours of work. Is made Incidental to the solution of these problems. A critical reader will probably be impressed by the Incompleteness of the data needed for definite answers to the two basic questions. This condition Is due in part to the complexity of these apparently simple problems but the available fact-finding studies relating to them furnish only fragments of the data necessary for detailed solutions. Many more auxiliary studies must be made before we can have what Is commonly called a scientific answer to the question, "What Is the responsibility of a teacher of arithmetic for devising and selecting learning exercises.'" It may even occur to the critical reader that an attempt to answer this question Is not justified at the present time because the answer must be based upon fragmentary data and consequently judgment must be introduced at many places. In reply to this criticism, one may point out that every teacher Is forced to give some answer to the question. Furthermore, If research workers become aware of the Inadequacy of data for dealing with such practical questions, it is possible that they may be stimulated to group their fact-finding studies about certain fundamental problems. The justification of auxiliary studies is based upon the contributions they make to the solution of problems that ask "what should be." Walter S. Monroe, Director Bureau of Educational Research May 14, 1926. [S] THE TEACHER'S RESPONSIBILITY FOR DEVISING LEARNING EXERCISES IN ARITHMETIC CHAPTER I THE IMMEDIATE OBJECTIVES OF ARITHMETIC The problem. The basic problems to be considered In this mono- graph are to determine (1) the nature and extent of the learning exer- cises^ provided by texts in arithmetic and (2) the responsibility of the teacher for supplementing a text in this respect. In order to assist the reader in arriving at a clear understanding of these problems and to provide a basis for their consideration, two subordinate questions are treated: (3) what are the immediate objectives of arithmetic, and (4) what learning exercises are needed for the attainment of these objec- tives. The following pages of this chapter present an exposition of the objectives of arithmetic as a subject in the elementary school. Chapters II and III are devoted to a consideration of learning exercises and their relation to objectives. The explicit treatment of the two basic problems Is given in Chapters IV and V. A general statement of the objectives of arithmetic. The purpose of instruction in arithmetic is to engender in pupils the mental equip- ment needed for responding satisfactorily to certain types of quantita- tive situations which they will encounter in advanced school work and in life outside of school. This "mental equipment" is frequently called "ability in arithmetic." Sometimes the plural, "abilities," is used to in- dicate that the equipment is not a unitary thing but consists of a large number of elements, many of which are independent in the sense that a pupil may acquire certain ones but not others. This general statement, like others which epitomize a group of concepts, will probably not have much meaning for the reader until the nature of the "mental equipment" and the situations for which it is to provide responses are described in some detail. Types of arithmetical ability. Although psychologically all abili- ties have the common characteristic of a "bond" connecting a stimulus or situation and a response, and no sharp lines of demarcation can be ^A learning exercise may be thought of as a request to do something. Examples and problems are prominent as learning exercises in arithmetic but as we shall show later (page 33) there are other types. These problems assume that exercises are to be assigned by the teacher. See page 26. [7] specified, it is possible to identify three general types of ability; (1) specific habits, (2) knowledge, and (3) general patterns of conduct. The classification of a particular ability may not always be apparent. but the recognition of these rubrics will assist the reader in arriving at a clearer understanding of the immediate objectives to be attained by the teaching of arithmetic in the elementary school. Nature of specific habits. If we examine the way in which pupils who have studied arithmetic respond to certain types of situations, we shall note certain distinguishing characteristics. For example, if a fifth- grade pupil is directed to write the numbers being dictated and "eighty- seven" is spoken, he writes the symbols "87" and does so without think- ing, that is. automatically and mechanically. When a number symbol such as "4" is brought to his attention, a meaning- immediately comes into his mind. In other words the response, "meaning of the symbol 4," is connected with visual apprehension of the symbol so that, when the visual apprehension occurs, the meaning comes into consciousness and does so without the pupil making any effort to recall. If a sixth-grade pupil is asked "what is the product of 7 X 6r", 42 comes into his mind as a response. When he is asked "how many feet in a yard:", he an- swers "three." When such words as add, product, divide, multiply, equal, and the like are brought to his attention, either orally or in printed form, a meaning immediately comes into his mind. For situations of the types illustrated in the preceding paragraph, one who has been "educated in arithmetic" possesses a ready-made re- sponse and is able to make it fluently, that is, quickly and with a mini- mum of conscious effort. Such mental equipment is usually designated as motor skills, fixed associations and memorized facts, or more briefl}' as specific habits. The word "specific" is used to indicate that each re- sponse is connected with only one situation and that any given situation requires a certain response. The scope of specific habits in the field of arithmetic. The "tables," addition, subtraction, and so forth represent a number of facts that are to be memorized but the total number of specific habits in the field of arithmetic is much larger than is commonly realized. Investi- gation has shown that 6 -\- 7 and 7 + 6 form the basis of two specific habits instead of a single one. Additional specific habits are required for 17 + 6 and 16 + 7, 27 + 6 and 26 + 7, 36 + 6 and 36 + 7. and so ^There are three general types of meanings for number symbols: a name for a group of objects, a position in the number series and its ratio to other numbers. The meaning which a person associates with a particular number symbol may be a combi- nation of these elemental meanings. [8] forth. One investigator,^ after careful inquiry, has concluded that there are 412 addition combinations which a child "is almost sure to need after he leaves school." This means that, if a child fails to learn any one of these 412 combinations, there will be a corresponding "gap" in his ability to add integers. The specific habits which function in arithmetical calculations have been mentioned first, but they do not constitute all of this type of men- tal equipment. Pupils are expected to learn the meaning of number symbols, both integers and fractions. By implication this includes un- derstanding the decimal system of notation and the ability to read and write numbers in Arabic notation. In addition to the symbols used in expressing numbers, the pupil is expected to learn several such as $, X, ^, and a large number of technical and semi-technical terms includ- ing their abbreviations. The topic of denominate numbers furnishes a large group of such terms, but there are many others such as sum, product, remainder, percent, interest, profit, loss, balance, overdraft, discount, average, bill, rectangle, triangle, buy, sell, at the rate of, and per yard. Quantitative relationships such as the number of feet In a yard, number of quarts in a gallon, the fractional equivalent of 12%%, and the like furnish the basis for another group of specific habits. The nature of abilities designated as knowledge. Specific habits provide controls of conduct for responding to familiar situations. When such situations are encountered, the pupil "remembers" the responses he found to be satisfactory on previous occasions. When "new" situa- tions* are encountered, a pupil's specific habits are inadequate as con- trols of conduct. He must manufacture a response using the ideas, facts, concepts, and principles that the "new" situation suggests to him. This mental equipment is called knowledge^ and the process of using It is designated as reasoning or reflective thinking. It Is not possible to specify a sharp line of demarcation between specific habits and knowledge. The connection between a meaning, concept, or principle, and a given situation may be fixed through repe- tition so that the control of conduct Is changed from knowledge to a specific habit. The degree of the strength of the bond connecting a response with a situation is, however, not the most significant basis of ^OsBURN, W. J. Corrective Arithmetic. Boston: Houghton Mifflin Company, 1924, p. 21. *A "new" situation may, and frequently does, involve familiar elements but the total combination is one to which the pupil has not responded before or one for which he has forgotten the response. '^This definition of knowledge indicates a more restricted meaning than is com- jnonly associated with the term. [9] distinction between specific habits and knowledge. The latter rubric of controls of conduct is characterized by many associations which result in "richness of meaning," and by organization which ties together the items of knowledge so that the recall of one item will tend to bring related items in one's mind. For example, assume that the following problem is "new" to a sixth-grade pupil: "The expenses of running a grocery store amount to 20% of the receipts. How much must a grocer charge for a barrel of flour which costs him $15.00 in order to make a net profit equal to 10% of the selling price.'" Although this problem is "new" to the pupil, it involves familiar elements (words and phrases) to which he responds by recalling ideas, concepts, and principles. These in turn may suggest other items of knowledge. Out of the total ideas, concepts, and principles that are active in his consciousness, the pupil formulates a tentative response to the problem. This is tested and if found unsatisfactory another formulation is made and tested. The solving of the problem is char- acterized by deliberation rather than fluency. This description of knowledge does not constitute a detailed definition but it is sufficient for our present purpose, which is to point out that knowledge is included in the outcomes of arithmetical instruction. General patterns of conduct as mental equipment. Specific habits and knowledge do not sufl&ce as categories for classifying all abilities resulting from the study of arithmetic. Neatness, accuracy, systematic attack, persistence, and the like designate controls of conduct which are sometimes called habits or general habits. However, they differ in sig- nificant respects from "specific habits" described earlier. Accuracy or "habit of accuracy" does not designate a response to a particular situation. It is rather a general mode or pattern of response to many situations. Neatness in calculating is not a response to a particular ex- ample such as "divide 846.84 by 396" but rather a general pattern of response in performing all calculations. In order to emphasize this dis- tinction, the name "general pattern of conduct" is given to such con- trols of conduct as neatness, accuracy, and so forth. Another aspect of the aim of arithmetic. The preceding discussion has pointed out three types of mental equipment which teachers of arithmetic are expected to engender; first, specific habits that function In making calculations and In responding to certain other types of sit- uations; second, knowledge out of which pupils will be able to construct responses to "new" problems which they w^Ill encounter in other school activities and In life outside of school; and third, general patterns of conduct. This analysis of "ability in arithmetic" has added meaning to [10] the general statement of aim with which we began, but the types of sit- uations for which pupils are to be equipped by the study of arithmetic have been indicated only in very general terms. A complete understand- ing of the immediate objectives of arithmetic requires specifications in regard to what specific habits, what items of knowledge and what gen- eral patterns of conduct arc to be engendered by the instruction in this school subject, and the quality of each ability.*' Determination of the particular arithmetical abilities to be engen- dered. A method of determining the specific habits, items of knowledge and general patterns of conduct that should be engendered by instruc- tion in arithmetic is suggested by the statement of the general purpose given on page 7. If this instruction is to engender the Qmental equip- ment needed for responding satisfactorily to certain types of quantita- tive situations'? which will be encountered in advanced school work and in adult life, it appears logical to analyze advanced school work and adult life for the purpose of determining the quantitative situations involved.' With this information at hand, additional analyses should reveal the nature and extent of the arithmetical equipment needed for making satisfactory responses. A number of analyses of adult activities have been made. One of the most elaborate is by Wilson® who collected from adults 14,583 arith- metical problems which they had encountered in their activities. From his analysis of these problems Wilson reached certain conclusions rela- tive to the arithmetical equipment that adults need. For example, the demand for the equipment engendered by the study of the following topics is so slight that he recommends their elimination: 1. Greatest common divisor and least common multiple beyond the power of inspection. 2. Long, confusing problems in common fractions. 3. Complex and compound fractions. 4. Reductions in denominate numbers. 5. Table of folding paper, surveyors table, tables of foreign money. 6. Compound numbers, neither addition, subtraction, multiplica- tion nor division. 7. Longitude and time. *The quality of arithmetical abilities is considered on page IS. 'This method of determining educational objectives is called "job analysis." *WiLsoN, G. M. ''The social and business usage of arithmetic." Teachers College ■Contributions to Education, No. 100. New York: Teachers College, Columbia Uni- versity, 1919. 62 p. [11] 8. Cases 2 and 3 in percentage. 9. Compound interest. 10. Annual interest. 11. Exchange, neither domestic nor foreign. 12. True discount. 13. Partnership with time. 14. Ratio, beyond the ability of fractions to satisfy. 15. Most of mensuration, — the trapezoid, trapezium, polygons, frustum, sphere. 16. Cube root. 17. The metric system. Wilson's study also yielded information relative to the character of the calculations made by adults in solving the problems they encounter. Slightly more than half of the additions involved either one or two place addends and less than two percent involved addends of more than four places. An analysis of a portion of the problems showed that nearly a third (31.2 percent) of the additions involved only two addends and that less than seven percent involved more than six addends. Sub- tractions, multiplications and divisions were also shown to be relatively simple. There were only 1,974 occurrences of common fractions in the 14,583 problems and ten different fractions accounted for in 95.5 per- cent of the cases. ^ In summarizing his conclusions Wilson states: "If to the four fun- damentals and fractions one were to add accounts, simple denominate numbers, and percentage, little would be left for all the other processes, — so little in fact that it seems unfair to give attention to them as drill processes in the elementary schools. Some of them should receive no attention. Others should receive attention only for informational pur- poses or when found necessary in the development of motivated sit- uations." Limitations of the job-analysis procedure. Several other investi- gators employing similar methods,^" have contributed information in re- gard to the arithmetical equipment which adults use in their activities and it may appear that such analyses when sufficiently extended will Tliese fractions in order of frequency are ^/^, %, Yi. %, 73, %, 70. %, % and %. ^"Adams, H. W. "The Mathematics Encountered in General Reading of News- papers and Periodicals." Unpublished JVIaster's thesis, Department of Education, Uni- versity of Chicago (August, 1924). Reviewed by Bobbitt, Franklin K., in Elementary School Journal, 25:133-43, October, 1924. Camerer, Alice. "What should be the minimal information about banking.^" Third Report of the Committee on Economy of Time in Education. Seventeenth Year- [12] yield a complete and dependable inventory of the arithmetical equip- ment which our schools should endeavor to engender. However, the job-analysis method of determining educational objectives has certain limitations which should be noted. In the first place the functioning of arithmetical equipment is not confined to the solving of problems or the making of calculations. As one comprehends numbers, names of de- nominate quantities, and other items of arithmetical terminology either in listening to a speaker or in reading, he is using elements of his arith- metical equipment. Furthermore, not infrequently one has occasion to estimate magnitudes such as the height of a tree, the number of tons in a pile of coal, and so forth, and to answer thought questions involv- ing quantities but not requiring calculations. In both estimating mag- nitudes and answering quantitative thought questions, one uses arith- metical equipment along with other controls of conduct. A second point is that the present activities of adults do not neces- sarily include all of the uses of arithmetical equipment that should be made. For example, authorities urge that farmers keep a detailed account of their financial activities; that individuals keep personal ac- counts; and that heads of families plan a budget at the beginning of the year and conform to it as closely as possible. However, these activ- ities are not engaged in by all persons to whom they apply. In fact it is doubtful if they are engaged in at all generally. A third point to be noted is that some activities requiring arith- metical equipment are engaged in by practically all adults but other activities are highly specialized. For example, everyone has occasion to count money and to check the making of change by clerks and store- keepers. Most adults have a bank account and should keep the stub of their check book. On the other hand, relatively few adults engage book of the National Society for the Study of Education, Part I. Bioomington, Illinois: Public School Publishing Company, 1918, p. 18-26. MacLear, Martha. "Mathematics in current literature," Pedagogical Seminary, 30:48-50, March, 1923. Mitchell, H. Edwin. "Some social demands on the course of study in arith- metic." Third Report of the Committee on Economy of Time in Education. Seven- teenth Yearbook of the National Society for the Study of Education, Part I. Bioom- ington, Illinois: Public School Publishing Company, 1918, p. 7-17. Noon, Philo G. "The child's use of numbers," Journal of Educational Psychol- ogy, 10:462-67, November, 1919. Smith, Nila B. "An investigation of the uses of arithmetic in the out-of-school life of first-grade children," Elementary School Journal, 24:621-26, April, 1924. Wise, Carl T. "A survey of arithmetical problems arising in various occupations," Elementary School Journal, 20:118-36. October, 1919. Woody, Clifford. "Types of arithmetic needed in certain types of salesman- ship," Elementary School Journal, 22:505-20, March, 1922. [13] TABLE I. OCCUPATIONAL DISTRIBUTION' OF PERSONS TEN YEARS OF AGE AND OVER, 1920 FEDERAL CENSUS, AND DISTRIBU- TION OF PROBLEMS IN FOUR SERIES OF ARITHMETICS WITH RESPECT TO THE ACTIVITY IN WHICH THEY AROSE Occupational Division Percent of population Percent of Problems in Four Texts Agriculture, forestry and animal husbandry Extraction of minerals Manufacturing and mechanical industries. Transportation , Trade , Public service (not elsewhere classified) . . . . Professional service , Domestic and personal service •Clerical occupations , Activities ot the home , Personal activities Activities of school children Total 26.3 2.6 30.8 7.4 10.2 1.9 5.2 8.2 7.5 11. 0, 10. 5. 43. 3. 1.3 0.1 0.1 6.0 17.0 1.6 100.1 in certain vocational activities that provide many arithmetical problems. In the 1920 Federal Census, 572 occupations and occupational groups were used in classifying the persons employed in gainful occupations. The distribution of persons of ten years and over among the nine occu- pational divisions is shown in Table I. The largest percent (30.8) is for "manufacturing and mechanical industries" but a large proportion of those engaged in this division of occupations are listed as laborers or semi-skilled employees. An analysis^^ of the problems of four series of arithmetic texts with respect to their source gave the distribution shown in the last column of Table I. Obviously, "trade" is the principal source of problems although it is engaged in by only about one person in ten. In a more detailed table that is not reproduced here, it is shown that in 1910 approximately 55 percent of our population of ten years of age and over were engaged in occupations to which no arithmetical prob- lems found in the texts examined could be assigned. The facts presented in Table I suggest that the analysis of occu- pational activities for the purpose of identifying the arithmetical prob- ''MoxROE, W.\LTER S. ''A preliminary report of an investigation of the economy of time in arithmetic." Second Report of the Committee on Minimal Essentials in Elementary School Subjects. Sixteenth Yearbook of the National Society for the Study of Education, Part I. Bloomington, Illinois: Public School Publishing Company, 1917, p. 111-27. [14] lems that occur has a very limited value. The need for the arithmetical equipment necessary to meet situations arising in particular occupations may be greater than these facts indicate. When considering educational •objectives one should recognize that the general public may be con- sidered as sustaining a "consumer's" relation to a number of occupa- tions. For example, only one tenth of our adult population is engaged in trade occupations but practically everyone engages in buying and therefore has occasion to check sale's slips, count change, and so forth. It is not possible for us to know in advance just which children will become clerks, which ones farmers, which ones stenographers, which ones machine operators, and so forth. Furthermore, persons engaged in one occupational activity should know something about the work of others. Not only is there considerable transfer of workers from one occupation to another, but social solidarity requires mutual under- standing and respect, and the more the workers in one occupation know of other occupations the greater will be their capacity for understanding and respecting their fellowmen. Conclusion in regard to what arithmetical abilities should be en- gendered. The considerations just noted suggest that job-analysis studies are not likely to yield precise and complete determinations of the particular abilities to be engendered by instruction in arithmetic. Studies already made indicate the elimination of certain abilities form- erly included among the objectives of arithmetic. Other studies have indicated the inclusion of new abilities or increased emphasis on cer- tain abilities already included. Future studies will probably contribute to still further refinements of arithmetical objectives but the limitations noted should not be overlooked. For the present we are able to compile an inventory only in general terms of the arithmetical abilities to be engendered. The quality of arithmetical equipment. Another aspect of the ob- jectives of arithmetic relates to the quality of the controls of conduct to be engendered. In the case of specific habits the quality of an ability is usually described in terms of rate and accuracy. For example, if one describes a pupil's ability to do addition examples of a given type, he specifies the rate at which the addition is done and the degree of ac- curacy of the sums. The idea of both rate and accuracy is frequently combined in the single term "fluency ."^^ ^For a partial specification of fluency see: Herriott, M. E. "How to make a course of study in arithmetic." University of Illinois Bulletin, Vol. 23, No. 6, Bureau of Educational Research Circular No. 37. Urbana; University of Illinois, 1925, p. 10, 11, 29, and 37. [15] Another phase of the quaHty relates to the "degree of permanency." If we assume that it is desirable for a student to acquire a certain abil- ity, how long should he be expected to retain this ability? A pupil may learn a denominate number relation or the meaning of a technical term well enough so that he will remember it for a week. Additional learn- ing will result in his retention of the control of conduct until the end of the school year and if the learning is continued sufficiently the control of conduct will tend to become a relatively permanent acquisition. A description of the specific habit and knowledge objectives of arithmetics^ The preceding discussion has indicated the difficulties en- countered in preparing a complete and detailed inventory of the spe- cific habits and items of knowledge to be engendered by instruction in arithmetic. As yet our information concerning the demands for arith- metical equipment is so limited that such an inventory cannot be form- ulated. However, it is possible to describe in some detail the types of situations which children and adults encounter and to indicate the na- ture of the response to be made. Although such a description will be subject to the limitation that the range of situations within each type is not determined except in a general way, the enumeration of types should lead the reader to enrich his concept of the objectives of arith- metic. No attempt is made to specify the quality (fluency or per- mancy) of the abilities necessary for satisfactory responses, nor to dis- tinguish between specific habits and knowledge. I. Number symbols. These include Arabic symbols 0, 1, 2, 3 9; numbers expressed in the decimal notation, 10, 11, 12 100, 101, 102, 1000, 1001, ; common fractions and mixed numbers; decimals .5, .05, 12.5, .66%, .6666 ; Roman numerals; and verbal expressions of numbers (both oral and written), zero, one, two, one hundred, one-half, first, second, and so forth. Responses to be connected with number symbols. The outstand- ing type of response to be made to number symbols is designated in a general way by the term "meaning" but the nature of this response need not be always the same for the same type of symbol. For "small" integers the meaning should sometimes include an image of a definite group of objects or a rather precise idea of the relations of the integer to other integers. In the case of "large" numbers a less definite mean- ing is expected. Usually a picture or idea of the position of a number in the number system provides an adequate control of conduct. Chil- dren should learn "definite" meanings for the more commonly used 'General patterns of conduct are considered on page 23. [16] fractions, both common and decimal, and "less definite" ones for the fractions that are encountered infrequently. Meaning responses should be connected with the Arabic number symbols, verbal expressions of numbers, oral, printed or written, and with the more commonly used Roman numerals. The conventional oral responses should be connected with the vis- ual apprehension of the various printed or written number symbols and the conventional written responses with the auditory apprehension of spoken number symbols. These two types of equipment are required for reading numbers and for writing them from dictation. Another type of equipment is needed for copying numbers. II. Other arithmetical symbols and technical terms. In addition to number symbols, certain conventional signs such as -f ? — ? X , -^ , == , and % are employed in arithmetic. Closely related to these are the conventional arrangements of the number symbols indicating calcu- lations. For example, numbers written in a column with the right hand margin even indicate addition. The following arrangement indicates the division of 576 by 36. 16 36)576 ^e_ 216 216 The technical terms include (1) those relating to calculation such as add, multiply, sum, remainder, quotient, partial product, "times" as in 6 times 7, "of" as in % of 8, total, and average; (2) names of de- nominate numbers and their abbreviations such as quart, foot, pound, barrel, dollar, and so forth; (3) terms relating to quantitative aspects of certain adult activities such as account, interest, balance, amount, change, profit, premium, rectangle, circle, and area; (4) words and phrases such as how many, how much, and, each, remains, bought, sells, lost, earns, what is, find (the sum, product, etc.), and the like. The terms in the fourth group are not peculiar to arithmetic but in prob- lems they frequently have a technical meaning. Sometimes they are designated as semi-technical terms. Responses to be connected with "other arithmetical symbols and technical terms." The outstanding type of response to be made to this class of situations and stimuli may also be designated as "meaning" but often the meaning of symbols and terms relating to calculation is [17] evidenced by a motor response as in "Find the product of 2894 and 672." The child is expected to respond to this situation by writing 2894 ,„^ and not 672)2894 or some other arrangement of the numbers. 6/2 ^ ^ III. Two or more numbers quantitatively related with one miss- ing. The simplest situations under this head are commonly designated as the "tables" or "basic" combinations such as9-|-3^ , 7 -\- = , 5 (add) 8 (add) 9 (subtract) 6 7-3= , 5-0= , 4 9X5= , 8 (multiply) 4 12-^4= , 9)36. Until recently it has been assumed that the 100 addition combinations^* represented all of the addition situations involving only integers for which responses should be mem- orized. It now appears that the addition situation 6 4-7 cannot be con- sidered as essentially the same as 16 -[- 7, 26 + 7, and so forth, and therefore 300 or more "higher decade" combinations must be added to the 100 "basic" combinations. A limited number of "higher decade" subtraction combinations occur in short division. There are no addi- tional combinations in multiplication. A feature of the "higher decade" combinations in addition and subtraction is that one of the numbers may be an "inner stimulus," an idea, and not something seen or heard. For example, in adding the column of figures shown on the right, one sees the 7 and 3 but as he adds up the column he does not see the partial sums 10, 15, 24, and 31 to which the numbers 5, 9, 7 8 and 8 respectively are to be added. In division it appears likely 7 that responses should be "learned" for all situations having as 9 divisors 1, 2, 3, 4, 5, 6, 7, 8, or 9 and dividends ranging from 5 to those that are 10 times the divisor. ^^ This means that 3 17-f-2^ , 13-^4^ , and 52^-9^ , represent combina- 7 tions as well as 16^-2= ,12-^4= and 54 -^9= . \\'hen considered in this way, division affords 360 additional combinations and each one can be expressed in two ways such as 17 -4- 2 = , and 2) 17.^** In addition to the three-number relationships described in the pre- ceding paragraphs, there are a number of situations involving only two ^■"Investigation has revealed that 6 + 3 = and 3 -f- 6 ^ cannot be considered identical situations. Similar statements can be made with reference to subtraction, multiplication, and division. Hence there are 100 "basic" combinations in addition, subtraction and multiplication, and 90 in division. "OsBURX, W. J. Corrective Arithmetic. Boston: Houghton Mifflin Company, 1924, p. 20. "The combinations described in this paragraph may be called "secondary"' to dis- tinguish them from the '"basic" ones commonly referred to as ''the tables." [18] numbers; equivalent fractions, such as y2 = %, % ^ %> -5 = •!/2> 3^ = .12% and denominate number relations such as 3 ft. == 1 yd., 1 bu. = 2150.42 cu. in. Responses to be connected with quantitative relationships with one number missing. The outstanding response to be made to quanti- tative situations of the types described in the preceding paragraphs is the supplying ot the missing number. Sometimes this response is to be expressed in written form; on other occasions it is to be partially written (e. g., units written and tens carried); and in column addition and multiplication the response may not be expressed but functions as an element in the next situation. IV. Examples. The term "examples" is used to designate explicit requests to add, subtract, multiply, divide or extract a root, the num- bers being given. Examples differ from "requests for the missing num- ber in a specific quantitative relationship" in respect to the manner in which the response is given. In the latter class of exercise, the pupil is expected to memorize the response and when two numbers are given he is expected to "remember" the response. There is no calculation. In the case of examples, the pupil responds to elements of the request and builds up the response to the total situation. In addition to those given in explicit form, examples are created in solving verbal problems (see page 21) when a decision has been reached in regard to the calcu- lations to be performed. Response to examples. A fluent response is to be made to ex- amples, that is, one needs to be equipped so that he can perform the specified calculation accurately and with reasonable speed. In making this response one utilizes his ability to respond to the basic and sec- ondary number combinations. Sometimes one is expected to be able to make a special response, that is, employ a short cut or use a calculating device such as an interest table. V. Questions, usually implied, concerning functional relation- ships. A question concerning a functional relationship^^ is implied in the statement of a problem. Consider the problem, "If a quart of paint covers 9 sq. yd. of floor surface, how much paint is required for the floor of a porch 12 ft. by 20 ft.:" This problem implies the question, "How is the area of the porch floor in square yards to be calculated from the dimensions 12 ft. and 20 ft?" The problem, "Find the value "A functional relationship is a statement of the quantitative relation between certain general quantities such as base, altitude and area of a rectangle, or the face of a note, time, rate of interest, and amount due. [19] of 52.3 bu. of wheat at $1.27 per bushel," imphes the question, "How is the value of a number of units (in this case 52.3 bu.) calculated when the number of units and price per unit are given?" A search through the literature relating to the objectives of arith- metic has failed to reveal any attempt to determine the particular ques- tions concerning functional relationship which children should learn to identify in problems and problematic situations and to answer. In an an- alysis of the second and third books of ten three-book series of arithme- tics 333 types of questions were recognized.^* A few illustrations are given here and a suggested minimum essential list is given as Appendix B. page 90. What calculation must be made: To find the total, given two or more items, values, and so forth. To find the amount, or number needed, given a magnitude and the number of times it is to be taken. To find how many when reduction ascending is required, given a magnitude expressed in terms of two or more denominations. To find the total price, given the number of units and the price per unit of another denomination. To find the return percent on an investment, given the net profit or net income and amount invested. To find the rate of profit, given the cost of goods, and the expenses and losses. The reader should bear in mind that verbal problems implying the same question concerning a functional relation may vary greatly in form of statement. For example, the question, "How may the number of units bought or sold be calculated when the price or value per unit and the total price or value are known," is implied in each of the following problems:^" 1. How many yards of silk at $1.50 per yard can be bought for $7-50? 2. The silk for a dress cost $7.50. How many yards were purchased at $1.50 per yard: 3. At $1.50 per yard, how many yards of silk does a woman get if the amount of the purchase is $7.50? 4. At the rate of $1.50 per yard my bill for silk was $7.50. How many yards were purchased? 5. How many yards of silk at $1.50 a yard does a bill of $7.50 represent? 6. When silk is $1.50 a yard, a piece of silk costs $7.50. How many yards in the piece? 7. At $1.50 a yard how many yards of silk does a merchant sell if he receives $7.50 for the piece? 8. Mrs. Jones purchased silk at $1.50 a yard. The entire amount paid was $7.50. How many yards were bought? 9. Silk was sold at $1.50 per yard. A check for $7.50 was given in settlement. Find the number of yards bought. "This analysis is described on page 41 and the 333 types of questions are given in Appendix A. '*The problems of this list were suggested by analysis of several texts. [20] 10. At $1.50 per yard, how many yards can be bought for $7.50? 11. A merchant sells a number of yards of silk for $7.50. The price being $1.50 for each yard, how many does he sell? 12. I invested $7.50 in silk at $1.50 per yard. How many yards did I buy? 13. When silk is $1.50 per yard, how many yards can be bought for $7.50? 14. When silk is sold for $1.50 for each yard, what quantity can be bought for $7.50? 15. At the rate of $1.50 per yard, how many yards can be bought for $7.50? 16. Silk is selling for $1.50 per yard, how many yards should be sold for $7.50? 17. At a cost of $1.50 a yard, how many yards can be bought for $7.50? 18. Silk was bought at a cost of $1.50 per yard. At that rate, how many yards can be bought for $7.50? 19. At $1.50 a yard a piece of silk cost $7.50. How many yards in the piece? 20. How many yards of silk at $1.50 can I buy for $7.50? 21. $7.50 was paid for silk at $1.50 per yard. How many yards were bought? 22. Find the number of yards; cost $7.50. Price per yard $1.50. 23. The cost of a piece of cloth is $7.50 and the cost per yard $1.50. How many yards are there in the piece? 24. A woman paid $7.50 for a piece of silk that cost her $1.50 per yard. How many yards were there in the piece? 25. A woman had $7.50 and bought silk at $1.50 a yard. How many yards did she buy? 26. A quantity of silk at $1.50 per yard cost $7.50. What was the quantity? 27. Silk is $1.50 a yard and I bought $7.50 worth today. How many yards did I buy? 28. A woman's bill for silk was $7.50. If each yard cost $1.50, how many yards were bought? This list does not exhaust the types of statements of one-step problems which imply this question. It is also implied in combination with other questions in many problems involving two or more steps. However, the list illustrates something of the variety of situations (prob- lem statements) to which the response "divide the total price (cost, value, etc.) by the price (cost, value, etc.) per unit and the quotient will be the number of units" is to be given. Response to be given to questions concerning functional relation- ships. The type of response to be given to questions concerning general quantitative relationships is described in the preceding paragraph. The reader should note that the answer specifies certain calculations to be made. VI. Verbal problems. Verbal problems have been referred to in the preceding pages but when they constitute occasions for manu- facturing a response by reflective thinking rather than recalling a ready- made response, that is, when they are "new" and are really problems for the pupil, they justify recognition as an additional type of situation for which arithmetical equipment is needed. It is not possible to give an objective definition of the line of demarcation between the problems to which one responds by reflective thinking and the "problems" that [21] he "solves" by recalling a ready-made response. Almost any problem may come under the second type provided a person encounters it or very similar exercises sufficiently frequently so that the bond connect- ing the required response with the situation represented by the verbal statement of the problem has become fixed. When this happens the problem ceases to be a "problem" for the person in question, that is, it is not a situation requiring reflective thinking. For example, a sev- enth-grade pupil may think reflectively in solving an interest problem but a banker would respond to it in much the same way as the pupil responds to a request to multiply 846 by 52. A verbal problem in the sense the term is used here is a new situ- ation, that is, one for which the person does not have a ready-made re- sponse. Thus when we state that one of the objectives of instruction in arithmetic is to equip the pupils to solve verbal problems, we mean that they are to be equipped to respond satisfactorily to situations to which they have not responded previously, that is, to answer questions they have not answered in their study of arithmetic. A 7iew situation is not necessarily new in all its elements. In fact the opposite is usually true. A new problem will usually involve many familiar words and phrases. The implied questions relative to general quantitative relations will usually be familiar. The total situation, how- ever, is new either because unfamiliar elements are introduced or be- cause familiar elements appear In a new combination. Response to problematic situations. As indicated in the preceding paragraphs, the nature of the response one makes is the distinguishing characteristic of a problematic situation. The response is complex. In so far as the situation is familiar, the elements of the response belong under other types of situations. Meanings are connected with words and symbols; the implied question concerning a functional relationship is identified and answered; denominate number facts are recalled; num- bers are read and copied. However, the response cannot be adequately described by enumerating the responses to such elements. Reflective thinking is involved. It should be noted that the total response to a verbal problem includes the determination of the calculations to be performed plus the response to the example formulated. Reflective thinking Is Involved In only the first phase of the total response. VII. Informational questions about business and social activities. Adults find occasion to answer a number of Informational questions re- lating to such activities as banking, transportation, transmitting money, taxation, Insurance, manufacturing, construction, and so forth. The following are typical: How Is money transmitted: What is a promisory [22] note? What is a sight draft? How does a city secure funds for pav- ing streets? How are taxes levied? What is board measure? What is overhead? How is postage computed on parcels? What conditions affect fire insurance rates? The questions may be asked in explicit terms but frequently they are implied in a general request or need. The range of such questions for which instruction in arithmetic is expected to engender equipment has not been determined but it is ob- vious that other school subjects, especially geography and civics, must assume some of the responsibility for equipping pupils to answer in- formational questions relating to business and social activities. Response given to informational questions relating to business and social activities. The general nature of the response to informa- tional questions relating to business and social activities is implied by the illustrative questions in the preceding paragraph. However, it may be noted that usually precise and definite answers are required, VIII. "Practical experiences." Under the head of "practical ex- periences" we group a number of types of situations such as (1) United States currency and other collections of objects to be counted, (2) magnitudes to be estimated or measured in terms of some unit, (3) business forms (sales slips, checks, money orders, etc.), catalogue lists, proposals for bond issues, newspaper quotations, and so forth to be comprehended, (4) situations in which arithmetical problems are to be identified and formulated. The engendering of the arithmetical equipment required for re- sponding to the situations enumerated under the head of "practical ex- periences" represents important objectives. The need for counting ob- jects, estimating or measuring magnitudes, and comprehending busi- ness forms is generally recognized but the need for identifying and form- ulating the arithmetical problems arising in practical situations is even more important. With few exceptions adults seldom need to solve a verbal problem stated by another person. Their problems are encount- ered in their "practical experiences" and before the solution is begun the problem nmst be formulated, at least mentally. General patterns of conduct as objectives in arithmetical instruc- tion. As stated on page 10 a general pattern of conduct does not provide a response to a particular situation but it exercises a general control of one's responses to many situations, the range depending upon the ex- tent of the generalization of the pattern. Accuracy or the "habit of accuracy" is usually listed as an objective of arithmetical instruction but it is different from the specific habits [23] which function in performing calculations. The latter designate definite responses to definite situations. A "habit of accuracy" is a general pattern of conduct which controls responses to a variety of situations which in this case are calculations. This control may result in per- forming the calculation a second time, checking, inspecting the work for errors, judging the answer with respect to reasonableness, and the like. A person who has attained the "habit of accuracy" tends to give one or more of these responses to any calculation situation. The pres- ence of the word "habit" indicates that the response always tends to be made and is made skillfully. Much the same idea is expressed by the statement that a person who possesses the "habit of accuracy" knows what to do in order to attain accuracy and how to do it. and derives satisfaction from doing what is necessary. Other general patterns of conduct listed among the objectives of arithmetic are neatness, honesty, initiative and resourcefulness in solv- ing problems, perseverence, and systematic procedure. A general pat- tern of conduct which may be designated as a "problem solving atti- tude" is implied in some of the statements of the aim of arithmetic. Its central element appears to be the belief that the way to respond to a new situation, that is, a problem, is to ascertain what is known about it and precisely what question or questions are to be answered, and then to focus one's resources upon the problem in an attempt to manufac- ture a response by formulating solutions (hypotheses) and testing them until a satisfactory one is found. Usually there is coupled with this belief, confidence in one's own ability to solve the problem. The absence of these phases of a "problem solving attitude" is evidenced when a pupil searches in his text for the solution of a similar problem or re- stricts his efforts to recalling the solution of a similar problem. Another significant phase of the "problem solving attitude" is in- volved in the pupil's concept of what it means to solve a problem. One point of view is that to solve a problem is to get the answer given in the text or one that will be accepted by the teacher. The "problem solving attitude" requires that one think of the solving of a problem as a case of reflective thinking in which the fundamental objective is to conform to the requirements of good thinking. Summary. Although the preceding discussion of the objectives of arithmetic has filled several pages, it has doubtless been apparent to the reader that the items of mental equipment (specific habits, knowledge and general patterns of conduct) to be engendered by the instruction in [24] arithmetic have not been specified at all completely ."° However, the enumeration of eight types of situations to which pupils are to be equipped to respond and the consideration of general patterns of con- duct should lead the reader to attach more meaning to the first state- ment of the purpose of instruction in arithmetic (see page 7). The teaching of arithmetic is expected to engender the specific habits, knowledge, and general patterns of conduct needed for responding in a satisfactory way to the following general classes of situations: I. Number symbols. II. Other arithmetical symbols and technical terms. III. Two or more numbers quantitatively related with one miss- ing. IV. Examples. V. Questions (usually implied) concerning functional relation- ships. VI. Verbal problems. VII. Informational questions about business and social activities. VIII. Practical experiences: (1) collections of objects to be counted, (2) magnitudes to be estimated In terms of some unit, (3) business forms, catalogue lists, newspaper quotations, and so forth to be comprehended, (4) situations in which arithmetical problems are to be Identified and formulated. ■*The situations for which abihty to respond should be engendered have been described only in terms of types and no attempt has been made to specify the quality of the several abilities. [25] 1/ CHAPTER II THE PROCESSES OF LEARNING AND TEACHING The discussion of the objectives of arithmetic in Chapter I furn- ishes a description of what pupils should learn during their study of this subject in the elementary school. The problem of this chapter is to describe certain phases of the learning process and the teaching pro- cedures that are essential to the attainment of these objectives. ^ Learning an active process. In discussing the work of the teacher, we commonly use verbs such as "impart," "communicate," "present," "explain," and "instruct," which not infrequently appear to imply that in the process of educating children the teacher transmits specific habits, items of knowledge or general patterns of conduct to the pupil whose mind may be receptive or even eager to receive or may be indifferent or hostile. Although no one who is informed in regard to modern psy- chology would support such a theory of learning, the reading of current educational literature and the observation of classroom procedures sug- gest that many teachers, in planning lessons and in conducting recita- tions, do assume that their function is to transmit ideas, facts, rules, and other items of knowledge to their pupils. The statement, "learning is an active process," is commonplace but its significance is far-reaching. What a child learns is the product of his own activity, physical, mental, and emotional. A child who is not active does not learn. In order to learn the multiplication combi- nations a child must enage in certain types of activity;^ in no other way can he acquire the necessary specific habits (fixed associations). Assignment of exercises required as a basis for attaining arith- metical objectives. Acceptance of the thesis, that learning is an active process and that the acquiring of certain abilities requires participation in certain types of learning activities, raises the question: What means must the teacher employ to secure the pupil activity that will lead to the attainment of the objectives described in Chapter I? A child learns as the result of his activity outside of the school such as playing games; doing errands for his parents, including the making of purchases; read- ing newspapers, magazines and books; constructing toys, playhouses ^The reader should not interpret this statement to mean that all pupils must go through the same activities. The activities of one pupil engaged in learning the multi- plication combinations may differ in certain respects from those of another pupil work- ing toward the same end but the activities of the two pupils will have certain common characteristics. For example, in this case both will involve repetition. [26] and the like; being a member of an organization such as the Boy Scouts or Campfire Girls; observing the activities of adults, and the like. How- ever, the establishment of schools is evidence of the recognition of the fact that the abilities resulting from participation in such activities dur- ing the period of childhood would seldom if ever constitute adequate equipment for meeting the demands of adult life. If our schools were abolished, many necessary abilities would not be acquired and the qual- ity of others would be unsatisfactory. Prior to the inclusion of arith- metic in the curriculum of the "public schools," very few children ac- quired ability to "cipher" and those who learned arithmetic did so as the result of attending a special school in which it was taught. Participation in efficient educative activities in school is not secured by the teacher making the direct request, "Be active" or "Do some- thing." In response to such commands or requests the pupils may be- come active. In fact children are by nature inclined to be active and unless restrained will "do something," but "spontaneous" activity is very unlikely to contribute greatly to their mental equipment, at least in the field of arithmetic. Observation of the teaching of arithmetic justifies the statement that most of the educative activity in this field is in response to definite exercises" assigned by the teacher. Those who believe in the project method will probably take issue with the thesis of the preceding paragraph. It is true that needs which furnish a basis for efficient educative activities do arise in the attempts of pupils to realize their own immediate purposes both in and out of school. It is also true that the number of such needs can be increased by skillful encouragement and manipulation of conditions by the teacher, but it does not appear that in general the project method can be de- pended upon to stimulate the learning activities necessary to produce the mental equipment specified by the objectives of arithmetic. In most teaching situations there must be explicit assignment of exercises which will function as the basis for much, and in many instances practically all, of the educative activity in the field of arithmetic.^ Types of learning activities resulting in the acquisition of specific habits and knowledge. Before consideration is given to the types of exercises assigned by teachers of arithmetic, it will be helpful to note the types of learning activities in which children engage in acquiring "The types of exercises employed in arithmetic will be considered later (seepage 33). ^For a more complete discussion of the "project method" versus the "assignment method" see: Monroe, Walter S. "Projects and the project method." University of Illinois Bulletin, Vol. 23, No. 30, Bureau of Educational Research Circular No. 43. Urbana: University of Illinois, 1926. 20 p. [27] specific habits and knowledge. In attempting to analyze mental activ- ity, psychologists have identified such processes as s^nsgljon, jiexcep- tion, conception, imagination, niernory, association, analysis, generaliz a- tion, and reasoning. Recognition of these mental processes is helpful for certain purposes but a somewhat different analysis of learning ap- pears to provide a more practical basis for considering the technique of teaching. The "types of learning activity" enumerated in the follow- ing paragraphs represent a pedagogical rather than a scientific analysis of the learning process. 1. Direct or perceptual experiencing occurs in learning arithmetic when a pupil counts objects, measures the length of the room, handles weights or money, steps off a distance, and the like. In direct experi- encing there is perception and hence the functioning of one or more of the sense organs. Perceptual experiences are required as a basis for the other rubrics of learning activity. Much of the necessary experiencing will take place outside of the school but when additional experiencing is required the teacher must provide opportunity for this type of learning activity. Perceptual experiencing occurs in measuring and counting ob- jects, dramatizing adult activities, visiting business concerns such as banks, grocery stores, department stores, and the like. 2. Vicarious experiencing occurs when one listens to or reads an account of the perceptual experiences of another, provided the listener or reader comprehends the terms used in the description. For example, a pupil may experience vicariously or second-hand, without visiting and observing Its activities, the operation of a building and loan associa- tion. When a person observes an activity such as an athletic contest his experience Is a combination of direct and vicarious. His playing of the game is vicarious but his seeing of the players and the spectators is direct. A foundation of direct or first-hand experience Is an essential prerequisite for vicarious experiencing. 3. Generalizing experience* is used as a name for analyzing, com- paring, organizing, and abstracting experiences, both direct and vicari- ous. The products of these activities are called concepts, rules, prin- ciples, generalizations, and abstractions. Words like sum, multiplica- tion, interest, premium, volume, and fraction represent concepts. They are sometimes called abstract and general meanings. 4. Comprehending the products of thought expressed in terms of words, phrases or sentences designates a type of learning activity which In some respects Is the reverse of generalizing experience. In the ^''Inductive development" has also been used as a name for this type of activity. [28] course of the history of the race a number of terms have been developed for use in describing arithmetical calculations and in stating problems. A pupil encounters such words as addition, multiplication, interest, pre- mium, numerator, percent, and the like. Before he has learned from his experiencing the meaning they represent he faces the necessity of comprehending or understanding the products of the thinking of other persons. A similar statement can be made with reference to rules and principles. 5. Using one's knowledge in manufacturing a response to a new situation^ is commonly called "problem solving" or reflective thinking. These terms, however, are used somewhat carelessly and for this reason the writer has chosen a descriptive phrase which explicitly specifies that the learner is engaged in responding to a new situation. It is generally assumed that reflective thinking occurs when a pupil responds to a ver- bal problem. This is not true because the pupil may remember how the problem or a similar one was solved In the text or by the teacher, or he may search his text for the solution of a similar problem. When this constitutes his activity he Is not thinking reflectively, that Is, he Is not manufacturing a response to a new situation. He Is simply searching for a ready-made response. Random guessing represents another type of activity that Is not Included under the caption used here, "Using knowledge In manufacturing responses to new situations" is an Import- ant type of learning activity. Its occurrence is not confined to solving verbal problems. Thought questions that do not Involve arithmetical calculations also furnish a basis for using knowledge. Whenever a student encount- ers a difficulty or Is asked a question which he Is unable to answer, he has a problem to solve. If he manufactures a solution for it he engages In reflective thinking and consequently engages in learning activity. 6. Tracing the thinking of another person by listening to an oral description of it or by reading a printed record constitutes a sixth type of learning activity. It occurs In arithmetic when a pupil listens to an explanation of the solution of a problem given by the teacher or by an- other pupil. 7. Expressing one's ideas is educative, particularly when attention Is given to their evaluation and organization. A pupil learns by ex- plaining the solution of a problem but the amount of learning may be- come almost negligible if he merely follows a definite formula, as was frequently required in the teaching of mental arithmetic. Expression of ideas also occurs In responding to thought questions. "See page 22. [29] 8. "Prolonging, repeating and intensifying one's experiences,"® represents a type of learning activity which is very prominent in arith- metic. "Drill" or '"practice" clearly comes under this type of learning activity. "Living over" perceptual experiences, recalling what has been read or heard, thinking through the solution of a problem, retracing an explanation given by the teacher, reconnecting a meaning with an ab- stract or general term, and the like are also illustrations of "repeating experiences." 9. Learning activities resulting in the acquisition of general pat- terns of conduct. A description of the activities that result in the acquisi- tion of general patterns of conduct is difficult, but It appears that the production of this class of outcomes tends to be governed by subtle fac- tors. The specific habits necessary for responding to quantitative rela- tionships in which one number is missing" can be engendered by having the pupil engage in appropriate practice. On the other hand, the "habit of accuracy'' does not result from engaging in any certain activities. Two pupils may apparently engage in the same learning activities and one will acquire the ''habit of accuracy" while the other will not. General patterns of conduct have been described as by-products which may be produced in the acquiring of specific habits and knowl- edge. This appears to be a valid description but it should be noted that the statement is "may be produced" rather than "are produced." The engendering of general patterns of conduct Is Incidental but not accidental. They result when the conditions are right; they do not when the necessary conditions are not secured. However, our knowl- edge of this phase of the learning process is not yet sufficient for us to specify in detail what conditions are necessan.' for the engendering of a general pattern of conduci. Studying and reciting frequently involve a combination of types of learning activities. The preceding analysis of learning activity Is not intended to imply that each type occurs separately and independently. Frequently the total activity of the pupil Is a combination of two or more types. Perhaps a more accurate statement would be that the learner, in doing a school exercise, may shift from one type of activity to another. For example, in attempting to solve a verbal problem one may "trace the thinking" of the teacher by listening to an explanation *This phrase is used by: BoBBiTT. Fr.\n"klix. How to Make a Curriculum. New York: Houghton Mifflin Company, 1924, p. 57. The word "experiences" in this expression has a somewhat broader meaning than on page 28. "See page 18. [30] of certain phases of it or "comprehend the meaning of a new abstract term" in addition to thinking reflectively about the problem. Relation between types of learning activity and rubrics of abili- ties. In the preceding pages we have described the types of learning activity in which children engage and the types of abilities that are pro- duced as outcomes. Perhaps the reader has already raised the ques- tion: just how are the different types of activity related to the different kinds of ability; or more specifically, if a pupil engages in a specified kind of learning activity, what will be the nature of the resulting out- comes. Although the relation between mental activity and the resulting outcomes cannot be stated in terms of precise laws, such as have been formulated in chemistry and physics, it is possible to state certain gen- eral laws. Any ability (specific habit, knowledge, or general pattern of conduct) may be thought of as a response connected with a stimulus and its quality depends upon the strength of the connection as well as upon the response. For example, 6x7= is a stimulus, and 42 the response. The strength of the connection between 6x7= and 42 is a very important element in a pupil's ability to respond to 6 X 7. In each of the types of learning activity there is the exercise of connections between stimuli (situation) and responses. In direct ex- periencing the connection is between the stimulus apprehended by means of a sense organ and resulting percept. In "problem solving" there is a sequence of stimuli and responses, both of which usually are ideas, meanings, concepts, and the like. In solving a problem a pupil manufactures a "new response" (the solution) which is connected with the problem. Laws governing the effectiveness of learning activities. The effec- tiveness of any learning activity in producing mental equipment (spe- cific habits, knowledge or general patterns of conduct) is governed by certain laws'^ which may be stated as follows: I. If other factors affecting learning remain unchanged, the strength of a modifiable connection between a situation and a response is strengthened as it is exercised and up to a certain limit the strength of the connection increases with the amount of exercise but not in a constant ratio. ^These "laws of learning" are based on formulations by: Thorxdike, E. L. Educational Psychology, Vol. II. Xew York: Teachers Col- lege, Columbia University, 1913. 452 p., or Educational Psychology, Briefer Course. New York: Teachers College, Columbia University, 1914, Part II. Gates, Arthur I. Psychology for Students of Education. New York: The Mac- millan Company, 1925, Chapter X. [31] II. Other conditions being equal, the more recent the exercise of a modifiable connection between a situation and a response, the stronger the connection is. This implies that a connection which is not exercised gradually grows weaker. III. The effect of the exercise of a modifiable connection between a situation and a response depends upon the degree of satisfaction that accompanies or follows the activity. Other conditions being equal, when "a satisfying state of affairs" prevails the connection is strengthened; when a state of dissatisfaction or annoyance prevails, the connection is weakened. IV. The strengthening effect of the exercise of a modifiable con- nection between a situation and a response depends upon the distribu- tion of the exercise, and other things being equal the maximum effect is obtained by distributing the exercise rather than by concentrating it. \'. The subject's capacity to learn (commonly called general intel- ligence) contributes to the effect of the exercise of a modifiable connec- tion between a situation and a response. Predicting activity necessary for attainment of specific objectives. When the teacher has formulated her immediate objectives, that is, cer- tain specific habits, items of knowledge or general patterns of con- duct to be engendered, she then must predict the learning activities in which it will be necessary for her pupils to engage in order to attain these objectives. For example, suppose the immediate objectives are the fixed associations designated as the multiplication combinations, the teacher's problem is: 'Tn what activities must I get my pupils to en- gage in order to learn these fixed associations?" Our knowledge of the relations between learning activity and outcomes enables us to con- clude that perceptual experiencing is necessary to provide a founda- tion for the concept of multiplication. There must also be generalizing of these experiences and of course many repetitions of the exercise of the connections between the products and the numbers whose products are beinR learned. [32] CHAPTER III THE LEARNING EXERCISES OF ARITHMETIC The problem of this chapter. It is the problem of this chapter to identify and describe the kinds of exercises which may be assigned by teachers as the basis for the activity necessary for the attainment of the objectives described in Chapter I. No attempt will be made to determine the relative effectiveness of the several types of exercises or to indicate when each should be used. Classes of learning exercises. The analysis of learning activity in the preceding chapter might be used as a basis for a corresponding list of types of exercises but it has seemed desirable to recognize other fac- tors than the general type of ensuing mental activity. For example, counting objects and measuring linear distance involve perceptual ex- periencing but they are sufficiently different to justify listing them as two separate types of exercises. The following list is not offered as a complete enumeration of the classes of requests^ that teachers of arith- metic make of their pupils but it will serve to indicate their range. 1. Requests to count objects such as the children in the class, the windows in the classroom, marks on the blackboard or in the textbook, and the like. 2. Requests to measure magnitudes such as length of desk or room, the water in a pail, and the like. 3. Requests to estimate physical magnitudes. 4. Projects and construction exercises including those involv- ing cooking, sewing, gardening, and the like. These involve implied requests to detect and formulate needs for measuring, computing, keeping records, and the like. 5. Games involving quantitative activities such as keeping score. 6. Requests to visit such places as a retail store or bank in order to observe adult activities. 7. Requests to describe perceptual experiences relating to arithmetic. ^An analysis of two of the classes, examples and verbal problems, is given, be- ginning p. 35. [33] 8. Accounts of experiences or other descriptions to be listened to or read. 9. Requests to generalize experiences. (Usually these requests are not direct. See page 28.) 10. Dramatization of an adult activity such as farming, man- ufacturing, banking, or keeping store. 11. Dictation exercises in which numbers and other arithmet- ical symbols are to be written. 12. Requests to copy numbers and other arithmetical symbols from the text or blackboard. 13. Requests to read orally numbers and other arithmetical symbols. 14. Requests to repeat orally or to write certain groups of symbols or facts. This includes counting by 2's, 3's, and so forth as well as groups commonly designated as "tables." 15. Requests to memorize, that is, to repeat lists of facts, tech- nical terms, abbreviations and the like without specific exercises calling for repetitions. 16. Requests, both oral and written, for the missing number in a specific quantitative relationship, such as 7 -(- 5 = ?, 27 + 6 = .^ 24 ^ 4 = .', 47 ^ 9 = ?, .1214 = ?, 1 yd. = r ft. 17. Explicit requests to perform specified calculations, com- monly called "examples."- (For types of examples see page 35.) 18. Verbal problems.'' (For types of verbal problems see page 41.) 19. Requests to explain performed calculations or solutions of problems. 20. Requests to read or listen to an explanation. 21. Requests to check calculations. 22. Requests to inspect and verify solutions of problems. 23. Fact questions other than requests to supply the missing number in a quantitative relationship. (Questions concerning functional relationships would be included here when the pupil does not find It necessary to think reflectively in answering them.) 'For definition of example see page 19. ^Usually a verbal problem is to be solved but a pupil may be requested to esti- mate the answer. [34] 24. Thought questions.* (These include questions concerning general quantitative relations and problems without numbers.) 25. Requests to read and comprehend descriptions, definitions, rules and abstract terms." 26. Requests to read, or reproduce, business forms such as checks, notes, sales slips, and so forth. 27. Requests to collect or formulate problems. 28. Requests to collect quantitative information such as prices or other items in regard to business practices. 29. Graphs to be read. 30. Groups of data to be represented graphically. 31. Requests to use tables and other calculating devices. Variations within the classes of learning exercises. Each of the classes in the preceding list includes learning exercises that differ in certain respects. Some of these differences are significant but others are not. The exercises included in the first class, "Requests to count objects," differ with respect to the kind of objects to be counted. Such a difference has little or no significance because the counting of objects is essentially the same as a learning activity regardless of the nature of the objects counted.^' On the other hand, in the second class the meas- urement of linear distance differs in a significant way from the meas- urement of mass because the instruments and units are different. "Specific requests to perform certain calculations" (examples) and "verbal problems" represent very complex classes of learning exercises. Since "examples" and "problems" are used extensively as bases of learning activity in the field of arithmetic, it will be helpful to note the types of exercises included in each of these classes. Types of examples. The term "example" is used as the name for an explicit request to add, subtract, multiply, or divide," the numbers being given. The example may call for two or more of these opera- tions to be performed but in all cases the request is explicit. The re- quest for the calculation may be expressed in terms of symbols, such ■'For a general discussion of types of thought questions, see: Monroe, Walter S., and Carter, Ralph E. "The use of different types of thought questions in secondary schools and their relative difficulty for students." Uni- versity of Illinois Bulletin, Vol. 20, No. 34. Bureau of Educational Research Bulletin No. 14. Urbana: University of Illinois, 1923. 26 p. ^Formulae may be added. This statement is not intended to imply that a pupil's activity is essentially the same in all cases. The point made is that, other things being equal, the nature of the objects counted is not significant. 'The extraction of roots may be added as a fifth calculation process. [35] as 694 -(- 27 ^ r, 37)848, or technical terms may be used, as ''Find the product of 87 and 64," "Divide 694 by 27." When considered as learning exercises, it is obvious that examples which differ in any way do not afford the basis for identical mental activities. The connections exercised by responding to 646 X 23 are different from those exercised by responding to 646 X 67. However, the difference is dissimilar to that existing between the responses to "sub- tract 746 from 9286" and "divide 18% by P/f" In the case of the re- sponses to 646 X ^^ snd 646 X 67, we may say that they are similar in the sense that each involves the exercises of multiplication and addi- tion combinations. This condition of similarity is expressed by saying all examples calling for the multiplication of a three-place integer by a two-place multiplier constitute a type. Examples such as 387 X 6 pro- vide a sufficiently different learning activity to justify recognition as another type. A request to multiply 412 X 4 constitutes a third type since no carrying is involved. Recognition of differences of the kind illustrated in the preceding paragraph raises the the question, "\\'hat are the significant types of arithmetical examples:" Those involving only one calculation process fall naturally into four general groups; (1) addition, (2) subtraction, (3) multiplication, and (4) division. Within each of these groups three subdivisions are created by the three types of numbers; integers, com- mon fractions and decimals. Each of these twelve divisions obviously includes examples that differ in certain respects. Some of these differ- ences have been recognized in arithmetic texts for many years by ex- plicit "cases" such as "short multiplication." "subtraction with borrow- ing" and the like. However, the "cases" usually mentioned do not appear to constitute a complete enumeration of the types of examples. In addition a long column of figures (12 to 15) appears to constitute a different type of example from that furnished by a short column of fig- ures (3 to 5). In multiplication the presence of zeros in the multiplier appears to create at least one separate type of example and possibly two. We have little experimental evidence concerning the differences that must exist between two examples in order to require that they be listed as belonging to separate types. The following lists are intended to be conservative. Several of the types include examples which differ in certain respects and it may be that the differences are sufficiently sig- nificant to justify the recognition of subordinate types. However, the enumeration of the types given here will serve to show the general char- acter of the learning exercises which are commonly called examples. [36] I. ADDITION OF INTEGERS' 1. Short column addition," 3 . to 5 addends: 8 3 9 4 2 4 7 4 S 3 2. Long column addition. (Two or more sub-types may be recognized by making divisions on basis of length.) Frequently columns of more than 7 to 9 addends con- stitute a situation different from that furnished by an example of 5 to 7 addends be- cause the "span of attention" is increased beyond the normal length. 3. Addition with carrying. 4. Addition of numbers of different lengths. II. SUBTRACTION OF INTEGERS 1. Subtraction of a number of one digit from a number of two digits. (Such subtractions may be considered as additional combinations corresponding to the ''higher decade combinations" in addition.) 2. Subtraction of numbers of two or more digits involving "borrowing," but no zero in either subtrahend or minuend. 3. Subtraction of numbers of two or more digits involving "borrowing" and one or more zeros in the minuend. 840 507 1000 602 73 184 63 276 4. Subtraction of numbers of three or more digits with at least one zero in the subtrahend. 896 383 170 207 III. MULTIPLICATION OF INTEGERS 1. Short multiplication with carrying." 2. Long multiplication without carrying. 3. Long multiplication with carrying. 4. Multiplications involving one or more zeros in multiplicand. 8350 70S 92 37 5. Multiplications involving one or more zeros in multiplier. 4736 845 805 30 IV. DIVISION OF ONE INTEGER BY ANOTHER 1. Short division: divisor 1 to 9, no zeros in quotient, with or without remainder. *An example may be expressed in two or more wavs: 18 18 + 33 + 187= . 33 Add 18, 33, and 187. 187 Find the sum of 18, 33, and 187. Such variations in form are not considered here. "As the term has been used in the preceding pages (see page 19) the basic and secondary combinations do not constitute examples. "Short multiplication without carrying might be listed as a separate type but such examples are essentially only groups of fundamental combinations. [37] 2. Short divisicn: divisor 1 to 9, one or more zeros in quotient, with or without remainder. 3. Long division:^' trial quotient true quotient, no zeros in quotient, no carrying in multiplications, no borrowing in subtractions and no remainder. (This is the simplest type of long division example.) 4. Long div'ision: trial quotient true quotient, no zeros in quotient, and no re- mainder. (This differs from the preceding by permitting carr>'ing in the multiplications and borrowing in the subtractions.) 5. Long division: trial quotient true quotient, no zeros in quotient but with remainder. 6. Long division: trial quotient not true quotient, no zeros in quotient and no remainder. 7. Long division: trial quotient not true quotient, no zeros in quotient, but with remainder. 8. Long di\'ision: zeros in quotient. (This may be considered a composite of several types. WTien a zero occupies units place in a quotient the example Is probably different from those in which it appears in an interior position. Differentiations might also be made in respect to the trial quotient and the remainder.) V. ADDITION OF FRACTIONS'^ 1. Addition of two or more fractions with common denominators, the sum being non-reducible, that is, in its lowest terms and less than unity. 2. Addition of two or more fractions with common denominators, the sum being reducible. 3. Addition of two or more fractions, the denominators not being common. (The examples under this type may be divided according to the reducible quality of the sum.) 4. Addition of mixed numbers. 5. Addition of an integer and a fraction, the sum to be expressed as an improper fraction. VI. SUBTRACTION OF FRACTIONS L Subtraction of fractions having common denominators.'^ 2. Subtraction of fractions not having common denominators. 3. Subtraction of a fraction from a mixed number, requiring borrowing. 4. Subtraction of one mixed number from another, not requiring borrowing. 5. Subtraction of one mixed number from another, requiring borrowing. 6. Subtraction of a fraction from an integer. VII. MULTIPLICATION OF FRACTIONS 1. Multiplication of an integer greater than unitv bv a unit fraction such as ^ or 1/5. _ 2. Multiplication of an integer greater than unity by other proper fractions. 3. Multiplication of one unit fraction by another unit fraction. 4. Multiplication of two fractions neither of which is a unit fraction. Product may be either reducible or non-reducible. "The possibility of a large number of types of long division examples is apparent from the fact that five conditions are specified in defining this type. The list given here includes only what appears to be the most significant types of long division examples. ^For a much more elaborate list of types of examples In the addition of frac- tions see: Kallom, Arthur W'. '"Analysis of and testing in common fractions," Journal of Educational Research, 1:177-92, March. 1920. "The character of the difference is not considered a differentiating factor. If this were done additional types would be found. [38] 5. Multiplication of a mixed number and a fraction. Fractional product may be reducible or non-reducible. 6. Multiplication of a mixed number by an integer. 7. Multiplication of two mixed numbers. VIII. DIVISION OF FRACTIONS" As stated on page 36 the list of types presented here is not intended to include all possible ones and some of those given obviously include examples which differ in certain respects. 1. Division of an integer by a unit fraction. 2. Division of an integer by other proper fractions. 3. Division of one fraction by another. 4. Division of a fraction by an integer. 5. Division of a fraction by a mixed number. ■ 6. Division of a mixed number by a fraction. 7. Division of a mixed number by a mixed number. 8. Division of a mixed number by an integer. 9. Division of an integer by a larger integer. IX. ADDITION OF DECIMALS" 1. Addends form addition example with right hand margin even and no zeros to the left of the last significant figure.^" .6 5.08 .4876 .5 1.26 .8428 .8 7.31 .9371 .3 12.83 .8476 2. Addends form addition example with right hand margin uneven but no zeros to the left of the last significant figure. .6 17. .346 .8942 .15 .327 .3 1.25 "It would be possible to increase the number of example types under this and the other groups by listing all of the possible combinations of the conditions affecting the example. In the case of the division of fractions, the dividend may be (1) a unit fraction, (2) other proper fractions, (3) an improper fraction, (4) a mixed number or (5) an integer. The same possibilities exist in the case of the divisor. The quo- tient furnishes another basis of differentiation. It may be (1) a proper fraction in lowest terms, (2) an improper fraction in lowest terms, (3) a proper fraction but not in lowest terms, (4) an improper fraction not in lowest terms, or (5) an integer. An indication of the number of possible types of examples under division of fractions is given In an article by: Knight, F. B. "A note on the organization of drill work," Journal of Educa- tional Psychology, 16:108-13. February, 1925. In this article the division of fractions is divided into 55 units of skill. *The significant differences between addition of integers and addition of deci- mals have not been determined. It appears to be a reasonable hypothesis that the differences are confined to (1) placement of the decimal point in the sum, (2) the possible uneven right hand side of the addition example, and (3) the possible presence of zeros between the decimal point and the first significant figure of the addends. It does not seem that the presence of integers in the addends either separately or in combination with a decimal should constitute a significant characteristic. A similar statement may be made in the case of subtraction. ^*The "last" figure is the one farthest left. [39] 3. Addends have zeros to the left of the last significant figure. .05 .05 .0082 .06 .075 .04 .04 .03 X. SUBTRACTION OF DECIMALS 1. The right hand figure of the subtrahend is written under the right hand figure of the minuend and no zeros to the left of the last significant figure in either decimal. 12.5 .75 8.2 .42 2. The right hand figure of the subtrahend farther removed from the decimal point than the right hand figure of the minuend, but no zeros to the left of the last significant figure in either decimal. 1. .75 .25 .125 3. The right figure of the minuend farther removed from the decimal point than the right hand figure of the subtrahend, but no zeros to the left of the last sig- nificant figure In either decimal. .875 .5 4. Zeros to the left of the last significant figure in at least one of the decimals. .0025 .875 .0042 .012 XI. MULTIPLICATION OF DECIMALS 1. Multiplication of a decimal by an Integer. .75 .875 1.25 5 64 8 2. Multiplication of an integer by a decimal. 845 950 837 .06 .7 1.5 3. Multiplication of a decimal by a decimal. XII. DIVISION OF DECIMALS'" 1. Division of a decimal by an Integer with no remainder. 2. Division of a decimal by an Integer with a remainder that may be completely expressed by additional decimal places in the quotient. 3. Division of a decimal by an Integer with a remainder, quotient to be carried to a specific number of decimal places. 4. Division of an Integer by a decimal. (It is possible that sub-types are formed by divisors such as .6, .06. .0006, 1.06.) 5. Division of one integer by another with a remainder, quotient to be carried to a specific number of decimal places. "See: Monroe, Walter S. "The ability to place the decimal point In division," Elementary School Journal, 18:287-93, December, 1917. This investigation Indicated that many pupils do not place the decimal point in division by applying a general rule but use a special rule or device for difTerent cases. If a special rule or device were employed for all possible combinations of dividend and divisor, the number of types of examples would be large even If the quantities were restricted to relatively few decimal places. Only a few of the more significant types are given here. [40] 6. Division of one decimal by another when the number of decimal places in the dividend equals or exceeds those of the divisor, with no remainder. ("Xo remainder" implies that if the decimal point w'ere removed from both dividend and divisor, the former would be larger than the latter.) 7. Same as the preceding except with a remainder. 8. Division of one decimal by another when the number of decimal places In the dividend is less than those In the divisor. Types of problems. When we examine the problem Hsts in current arithmetics, we find a conspicuous lack of uniformity in the captions by which these lists are designated in different texts. Formerly, most of the problems given in an arithmetic were listed under such captions as: "Rule of three, direct," "Rule of three. Inverse," "Partnership," "Alligation," "Barter," "Practice," "Profit and Loss," "Trade discount," "True discount," "Partial payments," "Exchange," and so forth. Changes In business practices and in the activities of adults apart from the carrying on of business have created new "applications" of arithme- tic. Many of the titles formerly used as captions for problem lists have been discarded and new ones substituted. The result Is that at the present time we have no generally recognized plan for classifying the problems of arithmetic. The buying and selling of commodities, borrowing money, con- structing houses and other buildings, insuring property, carrying on a business, and the like create many arithmetical problems. This sug- gests that the sources of problems be used as a basis for their classi- fication^^ but such a plan will not give groups which approximate homo- geneity with respect to the activity required in solving the problems. Furthermore, an examination of the problems In our arithmetics will reveal a number of problems whose source is not easily identified. In some cases the problem does not appear to be connected with any par- ticular activity or the suggested adult activity might be changed with- out affecting the problem. The following are typical: 1. "A pail of milk holding 2 gallons is to be poured into quart bottles. How many bottles will be needed?" 2. "Henry caught three fish. The first weighed 12 ounces, the second 10 ounces, and the third 15 ounces. What was the total weight of the three.'" 3. "On a vacation trip Robert walked 6^ miles the first day, 7 miles the second day, and 5% miles the third day. Find the total distance traveled." 4. "The length of an iron rod was 95i%6 inches. After It was heated Its length was found to be 96%2 inches. How much was the length increased by heating.^" ^"The writer has employed this method of analysis. See: Monroe, W.-^lter S. "A preliminary report of an Investigation of the economy of time in arithmetic." Second Report of the Com.mittee on Minimum Essentials In Elementary School Subjects. Sixteenth Yearbook of the National Society for the Study of Education, Part I. Bloomington, Illinois: Public School Publishing Company, 1917, p. 111-27. [41] In the first problem "milk" could be changed to "water," "syrup," or any other liquid without changing the problem. Furthermore, "bot- tles" could be changed to "cans" or "jars." Hence, there is no signifi- cant 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 term- inology. Within each of these two general classes of problems, ^^ a further difi"erentiation 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 de- scribed 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 texts-° resulted in the identification of 52 problem types in the field of "operation prob- lems" and 281 problem types under "activity problems."-^ Descrip- tions 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 an- swer. The basis of the determination of the calculations to be performed in the solv- ing 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 com- mission 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?" ^See 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 tvpcs of learning exercises are included that do not require calculation. See Appendix A, A24, A25, B5f, B5h, BSi, B5j, B5k, BSl. [42] AS To find how many times a stated quantity is contained in a given magnitude, given the quantity and the magnitude. A6 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. A8 To find a dimension, given the area of a rectangle and one side. A13 To find a difference, given denominate numbers of different denominations. A15 To find the ratio of one number to another, given the two numbers. A16 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.) Bl Buying and selling, ^^ simple cases. 1 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 denonination. 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 margin^^ 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 commodty 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 num- ber of units. ^^Descriptions of quantitative relations given below are expressed in terms of buy- ing. In some cases changes in terminology would be necessary if the activity were to be considered from the standpoint of selling. ^'"Total price" is used to designate the amount received for several units of the same commodity rather than the amount received for several commodities. "Price is used to designate the quantity taken as a basis of computation. Usu- ally "price" refers to the value or worth of a unit rather than a specified number of units. "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 selling price and therefore is a substitute for the words "gain" and "profit" as they are commonly used. [43] B2 Buying and selling, more complex types. a. To find the selling price: 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: i. 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 dupli- cations. This is to be expected because it is theoretically possible for any question concerning a quantitative relationship listed under oper- ation 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 intro- duced which tended to make them different from other problems re- quiring the same calculations but not identified with the same activity. "Rate may be expressed in terms of percent or as a fraction. [44] 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 im- possible to classify many problems now found in arithmetic texts. A comparison of certain of the groups of problem types under act- ivity problems (e.g., Bl, 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 ten- tative formulation representing the judgment of the writer and his as- sistants. However, this tentative formulation should prove useful be- cause it emphasizes that an arithmetical problem asks a question con- cerning a quantitative relationship which the solver of the problem must identify and then answer. Furthermore, it provides a workmg 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 re- spects, from that occurring in a pupil's response to any other type. Hence, this analysis of the learning exercises of arithmetic is a neces- sary prerequisite for a consideration of the teacher's responsibility for ^y devisine and selecting exercises. [-^5] CHAPTER IV THE LEARNING EXERCISES PROVIDED BY TEXTS IN ARITHMETIC Three types of content in arithmetic texts. Arithmetic texts in- clude three general types of content; (1) statements of what pupils are to learn (facts, rules, definitions, and principles); (2) illustrations, ex- planations, descriptions, and the like which imply learning exercises in- volving 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 con- J tent of arithmetic texts. The information relating to provisions for the first type of learning exercise is taken from studies reported by other investigators. The mformation 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 ex- amples or the number of learning exercises of this type does not con- stitute 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, 4:415-23, 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 Educa- tional Psychology, 16:108-17, February, 1925. [46] texts agree that the provisions for practice on the different combinations vary greatly, and that pupils who do all of the examples provided by a given text will receive more practice on the easier combinations than on the more difficult ones.* For example. Clapp^ reports the following fre- quencies of combinations In Book II of a certain series of arithmetics: 1 + 1 = ,434 times; 2 + 1 = ,444 times; 1 + 2 = ,299 times; 4+1= ,447 times; 7 + 5= ,76 times; 7 + 6= ,74 times; 8 + 7= , 102 times; 6 + 8= , 62 times; 7 + 9 = , 74 times. Os- burn reports that ''one hundred and eighty out of a total of 1,325 com- binations do not occur at all in the book considered*^ while some easy ones occur more than 300 times." Clapp reports the following coeffi- cients of correlation between the difficulty of combinations and the fre- quency of their appearance in textbooks: addition — .452 ± .054; subtraction —.329 ±.061; multiplication — .384 ± .057; division — .421 ±1 .061. These results are for Text A. Similar coefficients of correlation are given for Text B. Since all of the coefficients are nega- tive and "large" in comparison with the probable error, they mean the more difficult the combination the less frequently it occurs." As might be expected, when texts are compared with reference to their provisions for practice on the combinations, there Is a conspicu- ous lack of similarity In their example content. With the possible ex- ception of texts published since the results of the first analyses have been available. It appears certain that the practice a pupil receives upon the combinations of arithmetic, both basic and secondary, will not be adjusted to the difficulty of the combinations, and that the amount of practice upon the different combinations will depend upon the text he studies. Thorndike, Edward L. The Psychology of Arithmetic. New York: The Mac- millan Company, 1922, Chapter VL ^■"Example content" Includes both examples as defined on page 19 and requests for the fundamental combinations. *Since "easy" combinations are those which pupils respond to with the fewest errors and the "difficult" combinations are those which pupils know least well, one might Insist that the combinations found to be "easy" possessed this quality because the texts provided much drill on them and that the "difficult" ones were not known so well be- cause the pupils were not given as much opportunity to learn them. However, a care- ful study of the available data does not support this hypothesis. It appears that cer- tain combinations are inherently more difficult than others. 'Clapp, Frank L. op. at. *This is described as Book I of a widely used series of arithmetics. 'An analysis of the practice exercises prepared by Courtis and by Studebaker re- veals similar conditions. See: OsBURN, W. J. "A study of the validity of the Courtis and Studebaker Practice Tests In the Fundamentals of Arithmetic," Journal of Educational Research, 8:93-105, September, 1923. [471 The distribution of practice. In considering the learning exercises provided by a series of arithmetics, it is important to note the distribu- tion of practice as well as the nature of this practice. Thorndike* has shown that, in certain texts which are probably representative, the prac- tice is distributed in a way that appears to represent inefficient instruc- tion. Investigations in the psychology of learning indicate that in learning the combinations of arithmetic there should be a reasonably large number of repetitions during the first learning period and a grad- ual decrease in their number during subsequent periods which should occur at gradually increasing intervals. Thorndike found that the amount of practice on 5 X 5 in the first two books of a three-book series increased as the pupil advanced through the series. He suggests that the distribution of practice in this combination ''would be better if the pupil began at the end and went backwards." Problem content of arithmetic texts. In order to determine the nature of the problem content^ of arithmetic texts, the list of 333 prob- lem types described in Chapter III was used as a basis for analyzing the second and third books of ten three-book series of arithmetics.^" Each problem in these books was read and a decision made in regard to the question it Implicitly asked concerning a functional relationship.^^ *Thorndike, Edward L. The Psychology of Arithmetic. New York: The Mac- millan Company, 1922, Chapter \TII. ^"Problem content" does not include explicit requests for a definite calculation, such as 'What is 7 percent of ^7400.00.'" or '"Reduce 2 miles to yards." Such exer- cises were considered examples. "The series analyzed are: Anderson, Robert F. The Anderson Arithmetic. New York: Silver, Burdett and Company, 1924. Alexander, Georgia, and Dewey, John. The .Alexander-Dewey .Arithmetic. New York: Longmans, Green and Company, 1921. Drushel, J. Andrew, Noonan, Margaret E., and Withers, John W. .Arith- metical Essentials. New York: Lyons and Carnahan. 1921. Hamilton, Samuel. Hamilton's Essentials of Arithmetic. Higher Grades. New York: American Book Company. 1920. HoYT, Franklin S., and Peet, Harriet E. Ever^'day .Arithmetic. New York: Houghton MifBin Company, 1920. Lennes, N. G.. and Jenkins, Frances. Applied .Arithmetic. Tlie Tliree Essen- tials. Philadelphia: J. B. Lippincott Company, 1920. Stone, John C, and Mii.lis, J.^^mes F. New Stone-Millis .Arithmetic. New A'ork: Benjamin H. Sanborn and Company, 1920. Thorndike, Edw.ard Lee. The Thorndike .Arithmetics. New A'ork: Rand, Mc- Nally and Company, 1917. Watson, Bruce M.. and White, Charles E. Modern .Arithmetic for Upper 'Grades. New A'ork: D. C. Heath and Company, 1918. Wentworth, George, and Smith, D.wid Eugene. Essentials of .Arithmetic. New York: Ginn and Company, 1915. "This work was done by Ollie Asher during the year 1924-25 under the immediate supervision of John .A. Clark, Assistant in the Bureau of Educational Research. [48] TABLE II. NUMBER OF PROBLEMS IN THE TEXTS EXAMINED Text Number of Problems Book II Book III Total A 899 1441 824 1339 1134 1052 1269 1240 1186 1070 876 1775 B 1482 2923 C 678 1502 D 1953 1246 3292 E 2380 F 1008 2060 G H I 1336 1184 1417 1363 2605 2424 2603 J 2433 Total 11454 12543 i 23997 It was then classified under the problem type described by that ques- tion. (See page 41.) Some of the problems asked relatively simple questions but in other cases the question was complex in the sense that its answer involved the specification of an extended series of calculations. Analysis of such ''complex" problems revealed that in most cases they might be consid- ered as consisting of a sequence of two or more simpler problems. Since it soon became apparent that unless some such policy were adopted, the number of problem types would be increased indefinitely, the more "complex" problems, amounting to slightly more than one-fourth of the total numbers, were classified as consisting of a sequence of two or more simpler problems. This procedure is illustrated by the following problems whose classification is given in the left-hand margin.^- Bllal(a) It cost Robert ^4.25 to grow the corn. He figures that it cost Al him 9 hours labor in selhng the corn. Counting the labor of selling the com at 8 cents an hour, what was the total cost of growing and seUing the 234 ears? A6a2 The children of the Mullanphy School collected in two months Blals 11325 lb. of old newspapers and 2550 lb. of magazines. They received Al $1.25 per 100 lb. for the newspapers and $2.75 per 100 lb. for the magazines. What was the total received for old paper? Bllal(a) Andrew's father worked for a farmer. He received $50 a month for A3 the 12 months of the year, free house rent worth $23 a month, 12 bu. Blala of potatoes worth $1.30 a bushel, and 365 qt. of milk worth 8c a Al quart. What he received was equivalent to what money wages for the year? "For a description of the problem types indicated by the symbols used, see Ap- pendix A. [49] TABLE III. FREQUENCY OF OCCURREN'CE OF PROBLEM TYPES Book II Book III Total Total frequency of problem types occurring in simple problems Total frequency of problem types occurring in "complex" problems 9107 11801 8445 18655 17552 30456 Total 20908 27100 48008 Blcl What percent of the cost does a newsboy make on papers that he Blf buys at the rate of 3 for 4c and sells at 2c each? What percent of B2c2 the selling price does he make? What percent of the selling price does AIS^ the news dealer receive? What percent of the selhng price does the newsboy receive? The total number of problems in each book analyzed is shown in Table II. According to Table III, 17,552 of the 23,997 problems were classified under some one of the 333 problem types. The remaining problems were considered "complex" and were classified as represent- ing a combination of two or more problem types. The fact that 6,442 problems represent a total frequency of 30,456 problem types indicates that most of them were very "complex." Results of the analysis of problem content. A detailed summary of the results of analyzing the ten series of arithmetic texts is given in Appendix A in the following form. Al To find totals by addition, given two or more items, values, etc. A82 B112 C62 D66 El 32 F43 G49 H82 146 J75 749 A416 B495 C489 D440 E532 F375 G290 H537 1387 J419 4380 Ble To make change, given an amount of money and the price of a commodity. A26 B23 C6S D15 E13 II Jl 147 A28 B36 C96 D33 E36 G3 H3 111 J12 258 The first line of the frequencies gives the number of occurrences of the problem type when not combined with another type, that is, in "simple" problems. The second line gives the total occurrences of the problem type in both "simple" and "complex" problems. The number of occurrences in "complex" problems may be found by subtracting the upper number from the lower. Each of the letters, A, B, C, D, E, F, G, H. I, J, indicates a particular text. The detailed summary given in Appendix A should be studied in order to secure a clear idea of the nature of the problem content of the arithmetics analyzed. Only sixty out of 333 problem types appear in all of the ten series of arithmetics, and only twenty-five, ten or more times in all texts. A number appear in only one or two of the texts. [50] TABLE IV. FREQUENCIES OF PROBLEM TYPES AND NUMBER OF PROBLEM TYPES IN EACH TEXT Operation Problems Activity Problems Total Text Total frequency of problem types Number of problem types Total frequency of problem types Number of problem types Total frequency of problem types Number of problem types A 2602 3298 2479 3046 2449 2046 2782 3793 2661 2613 44 35 39 38 38 35 40 37 44 37 1900 1998 1564 2912 1842 1805 2042 1969 2059 2148 129 133 122 140 179 95 137 110 112 125 4502 5296 4043 5958 4291 3851 4824 5762 4720 4761 173 B 168 C 161 D 178 E 217 F 130 G 177 H 147 I 156 T 162 Total 27769 387 20239 1282 48008 1669 Twelve types^^ have total frequencies over 1000. The sum of the twelve frequencies is 29,964 or slightly more than three-fifths of the sum of all frequencies. Table IV gives the number of problem types in each text and the sum of the frequencies. In interpreting this table, the reader should bear in mind that there are 52 problem types under operation problems and 281 under activity problems. The fact that all problem types do not appear in all texts is apparent from Appendix A. Table IV shows that two texts, A and I. include 44 of the 52 problem types under oper- ation problems and that 35 in text F represents the lowest number of problem types. A somewhat more analytical summary of the operation problem content of the several texts is given by Table IV A. It is clear that the texts differ in respect to the problem types included and also in the frequency of the occurrence of the types. For example, A14 ap- pears in all of the texts but in text B its frequency is 1 and in text H, 50. The variability among the ten series of arithmetics is even greater in the case of the activity problems. Text F includes only 95 of the 281 problem types. The greatest number of problem types found in a single text is 179 in text E. The extent of the variability is more clearly indicated by Table IVB. It is obvious in all of the texts that there are "These with their frequencies are: Al, 4380; A2, 4074; A3, 3532; A5, 1101; A6a, 1556; A7a, 1148; AlOa, 1115; A15, 2172; A16, 2644; A19, 1364; Blal, 4472; B4a, 2366. [51] w P3 O o < w Ow oz uu wO CO *C cut-' to O :§ > < « o fe 4380 4074 3532 859 1101 1878 1365 340 ■«< I-- -♦ VO O vO — — — ■* •—1 b ■*Tj.m — — — .-12;;! .c^,^- O — U-. — On-J- to 3 H ^ _H — — . — (V, r^ 00 ■-^f^m : ; — 1^ to - b. t^ -). ^ vo -f r^ t^ 1^ ■ — -I" CM OO — ro -t -^ W1 t^ . uo — to- VO • 3 H „_ — __0,0'OT»-^ .- : :-- Ov pa fa lnoot<^vDOOOt^"^ ■*-Ot-^ — — — ootor^ ^H . fMfN oo OS to H ^ — — — — r^r^vo ■ -*rof^ to to < -1- fa \0 "-1 to OS — O 00 — ^HaN-HioaNO<^ro tOVO- • o\ — i^ 0\to-*>0-* — — OOONOO-*- — CNO 1 f^ rt„„ — — c»)r^O • NOfO vo : : — _ E „ Oh , ^^--r^^r. C O fat-' O — r-i r^" ■*' i/^ NO l~-.' 00 a- <<<<<<<<< O — t-^co ■*LovOr^ooc>o — r- <<<<<<<<<<<<< ^ cr 6.^ OhH [52] Q < W PQ O CL, >H H H O < OW z JDS *• "-1 1^ r'l 00 h — i-*-*rJO— l^ .-^I^Lnt^r- ui " ij-, 00 00 "-> CM ^^ On 00 t^u-jvO^O-^ ON H r^ X PL, OOmOO^r^c-^^Of^O^^OC^-* f*i — « m 00 f*-i ON NO -^^^r^r>l-H ON NO H Oooooi^MO\vCrJ■?-- "• of another ratio equal to the first. (.Inverse : A49 B181 C"6 D116 E80 F:>: A114 B211 Ci:4 D145 E95 F"4 GIjI Hi5:> To find the ratio of items to total, given a series of items. Bl HI Al Bl El Gl H7 atio and one member I12I TllO Iloi T152 Jl 982 1364 To compare pairs of quantities by ratios, given the pairs of quantities. C5 E14 G3 H40 15 J6 73 Al C20 D4 £15 Fl Gil H46 17 J12 117 To find the largest quantity which will be contained equally in two or more given quantities. B7 7 B7 7 To find the least quantity which will contain exactly each of two or more quantities. B9 B9 To draw to scale, or to represent graphically in tables. AS B14 C39 D42 E23 F31 H^" 131 18 243 A12 B:: C68 D60 E27 F33 H4^ I3i J9 311 To interpret tables, graphs, or diagrams, given completed graphs, tables, oi^ diagrams. A26 B19 C61 D23 E30 F26 GI2 H82 17 J7 293 A30 B21 C"^ D26 E31 F2S G20 H91 19 J14 34S B. ACTIVITY PROBLEMS Buying and selling,^ simple cases. a. To find the total price:* 1. given the number of units and price* per unit. A67 B92 C51 D206 E90 F29 GIO" Hl"9 1212 J162 A391 B4"3 C348 D625 E433 F289 G4:2 H424 1440 J577 2. given the number of units and the price per unit of another denomii A20 D3 E8 A25 B3 CI D7 E8 b. To find the r-umber of units: 1. given the total price and price per uait. A9 B60 C42 D71 E15 Fll A14 B83 C6f D82 E22 Fi: G15 G2: GlO 015 H7 HI 5 H33 H55 19 117 133 138 JIO J27 J36 J56 1195 4472 ition. 72 130 320 442 I ■Descriptions of quantitative relations given below are expressed in terms of bu^yiag. In some cases changes in terminology would be necessary if the activity were to be considered from the standpoint of selling. *"Total price" is used to designate the amount received for several units of the^ same commodity rather than the amount received for several commodities. '"Price" is used to designate the quanrity taken as a basis of computarioi Usually "price" refers to the value or worth of a unit rather than a specified numbei of units. "Price" is often limited by the qualifying terms cost, selling, marked, ant list. [68] 2. given the total price and the price per unit in another denomination. Jl 1 Jl 1 3. received in exchange of commodities, given an amount of each commodity and the unit for each. C2 2 C2 2 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. El 1 El 1 5. given the margin,'' per unit and the total margin. El ~ 1 Al B3 C5 D2 E2 Gl II 15 c. To find the price per unit: 1. given the total price and the number of units. A19 B22 CIO D19 E20 F7 G12 H88 19 JIS 221 A63 B93 C59 D48 E62 F19 042 H163 127 J67 643 2. given the total price and the number of units in another denomination. Al Gl 2 A3 Fl G2 Jl 7 3. in exchange of commodities, given the number of units of each commod- ity and the price per unit ot one. B2 2 B2 2 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. El 1 El 1 d. To find the amount to be received for several items, given the price of each. A7 B3 C3 D4 E19 F8 G3 H13 II T3 64 A22 B22 C15 D37 E72 F43 G28 H28 112 J51 330 e. To make change, given an amount of money and the price of a commodity. A26 B23 C68 D15 E13 II jl 147 A28 B36 C96 D33 E36 G3 H3 111 J12 258 f. To find the margin or loss given the cost price and the selling price. Al Bl Fl H3 Jl 7 A64 B102 C75 D89 E13 F27 G46 H32 133 J77 558 g. To find the total margin or total loss: 1. given the number of units and the margin or loss per unit. A6 Dl H3 Tl 11 A15 B6 C4 Dl E12 F2 G8 H3 14 J8 63 2. given the unit cost, the unit selling price, and number of units. Al BIO CI D2 E3 Fl G5 II J3 27 A3 B14 CI D2 E6 Fl G6 H2 13 J9 47 h. To find the margin or loss per unit, given the total margin or loss and the number ot units. Dl 1 A2 Bl C3 D30 E2 F3 Gl H13 55 ^Margin is a term used to represent the difference between the cost price and the selling price and therefore is a substitute for the words "gain" and "profit" as they are commonly used. [69] B2 Buying and selling, more complex types. a. To find the selling price: 1. given the rate* of discount or loss, and the price. A70 B18 C5 D83 E13 F52 G119 H106 141 J14 521 A82 B25 C14 D86 E31 F64 G128 H140 162 J35 667 2. given the rate of advance or margin and the price. All B19 CI DIO F6 G8 H25 18 J5 93 A28 B44 Cll D13 E8 F8 017 H39 112 J14 194 3. given the rate of two or more successive discounts and the price. A6 B12 C2 D25 E5 F47 037 H15 II J12 162 A9 B13 C7 D36 ElO F49 038 H16 111 J17 206 4. given the price, rate of advance or margin, and rate of discount or loss. Al B2 D2 01 J3 9 A2 B2 D2 El G2 J3 12 5. given the rate of commission, discount, margin, or loss and the amount of commission, discount, margin, or loss. Bl 01 II J2 5 Bl 01 II J3 6 6. given the price and the amount of commission or discount. E2 2 A6 B18 C9 D14 E4 F4 09 H5 II Jll 81 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. A49 B26 C5 D104 E6 F16 029 H34 19 JIO 288 A91 B76 C30 D140 E28 F30 053 H70 125 J50 593 2. given two or more successive discounts and the total price. Dl 1 D4 01 5 3. given the total price and the selling price. Al 1 A3 3 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. A65 Bll C8 D2 E14 F17 G3 HI 12 J5 128 A122 B78 C63 D93 E30 F36 O30 H14 123 J60 549 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. 116 J2 18 A2 121 J3 26 3. given the total price and the selling price. A24 Cll D61 ElO F29 062 H24 T26 247 A32 B2 C13 D61 Ell F29 065 H24 J29 266 d. To find the price: 1. given the selling price and the rate of discount or loss. Al B17 C2 D2 E5 F13 04 13 J6 53 A2 B30 C8 D2 E7 F14 04 15 J20 92 2. given the amount of margin, loss, commission, or discount and the rate of margin, loss, commission, or discount. A8 B19 C7 D2 01 H5 JIO 52 A12 B34 C8 D2 E2 G3 H5 12 J12 80 'Rate may be expressed in terms of percent or as a fraction. [70] 3. given the selling price and rate of margin. A2 B24 D2 E4 F29 G6 19 J6 82 AlO B37 C5 D2 E5 F30 G7 116 J17 129 4. given the selling price and two or more successive discounts. ^Cl F2 12 5 Bl CI El F2 12 7 e. To find the amount due the agent or agents, given the number of units, the price per unit, and the rate of commission. (Agent purchases commodity.) Al BIO C3 G2 12 18 Al B12 C4 02 12 21 f. To find the equivalent single discount in percent, given two or more successive rates of discount. CI El 2 CI El 2 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. Jl 1 Jl 1 B3 Carrying on a business. Note: Types listed under "carrying on a business" are similar in certain respects to those found under the activity ol "buying and selling," but in general the follow- ing distinction prevails. The problems under Bl and B2 are those in which a pur- chaser is explicitly involved and in which he may be expected to be interested, at least to the extent of checking the solution by the seller. The problems under B3 are those which in general only the one carrying on the business will encounter. The degree of magnitude of the quantities of the problem and the terminology were also used as criteria. In cases where the distinction is not obvious, a footnote indicates similarities or differences. a. To find the selling price:^ 1. (a) given the cost price, rate of net profit, and rate of overhead. (Profit and overhead are figured on the cost price.) HI 1 HI 1 (b)given the cost price, rate of net profit, and rate of overhead. (Profit and overhead are figured on selling price.) El 1 El I 2. (a) given the cost price, rate of net profit, and the overhead. (Profit is figured on cost.) A4 Dl Gl 6 A6 B4 D4 G2 Jl 17 (b) given the cost price, rate ot net profit, and the overhead. (Profit is figured on selling price.) Bl CI El 3 b. To find the total receipts, given the total cost and the net profit. El I c. To find the overhead, given the cost price or selling price and the percent of overhead. E6 G4 10 CI E17 G9 II 28 d. To find the rate of overhead, given the cost price or selling price and the overhead. G4 Jl 5 ^General terminology and "overhead" are the factors which distinguish this classification from B2a. [71] e. To find the net profit or loss: 1. given the cost price, overhead, and selling price- El H3 4 A5 B7 CI Dl E20 F2 G6 HIO 16 58 2. given the cost price, rate of overhead, and selling price. (Overhead is figured on the cost.) DIO 10 Bl C2 DIO E3 Gl 17 3. given the cost price, the rate of overhead, and the selling price. (Overhead is figured on the selling price.) E4 4 A2 Bl CI E4 F3 04 15 4. given the total costs and •■otal receipts. El Gl HI 3 A8 B8 C9 D17 E25 Fl G21 H8 14 JIO 111 5. given the itemized costs and total receipts. Al B3 E3 HI 8 A4 B9 C6 D12 E12 F2 H2 12 49 6. given the rate of gross profit, the rate of overhead, and the selling price. (Overhead and gain are figured on the selling price.) G5 5 El G5 6 7. given the rate ot gross profit, the overhead, and the selling price. (Gross profit is figured on the selling price.) G2 2 G3 3 8. given the gross profit and the overhead. G2 Jl 3 f. To find what percent the net profit is ot the cost price or selling price, given the cost price, overhead, and the selling price. Al E2 G3 6 A5 E4 G4 Jl 14 g. To find what percent the net profit is of the cost price or selling price, given the net profit, and the total receipts, original outlay, or amount invested. Ell F38 G2 51 B4 CI D8 E31 F41 G6 Jl 92 h. To find what percent the profit or loss is of the cost price or selling price, given the profit or loss and the cost price or selling price.' Bl H2 II 4 A3 B7 C4 Dl E9 Fl G7 H6 II 39 i. To find the gross profit: 1. given the cost price and total receipts. ElO Gl 11 2. given the cost price or selling price and rate of gross profit. G4 Jl 5 j. To find the rate of gross profit: 1. given the wholesale price (cost price) and the retail price (selling price). El Jl 2 E2 . Jl 3 2. given the total receipts, total amount invested or total costs, and the amount of gross profit or gross income. Bl E6 G2 9 ^This classification differs from B2cl in size of quantitative terms and in termi- nology. [72] k. To find the amount invested: 1. given the itemized costs. El 1 Dl E3 Jl 5 2. given the rate of profit on the investment and the net profit. Bl Fl II 3 Bl El F2 Gl 12 7 I. To find the profit on an investment, given the amount of the investment, the rate of profit per unit of time and the time. Bl Fl 138 40 B2 El F2 Gl 141 47 m. To find the profit per person on the basis of investment, given the amount invested by each person and the profit on the total investment. Bl CI 2 B2 CI 3 n. To find the commission: 1. given a series of commodities, the number of units in each series, the unit price of each commodity, and the rate of commission. CI El Gl 3 A5 CI El Gl HI 9 2. given the total sales, the expenses, and the rate of commission charged. Bl 1 Bl HI 2 3. given the total cost, the expenses, and the amount remitted to the agent Dl 1 Dl 1 4. given the proceeds remitted by the agent, the rate of commission, and expenses. Al 1 Al 1 o. To find the vnlue of goods to be sold, given the rate of commission and the amount of commission to be earned. Al B3 El J2 7 Al B4 E2 J5 12 p. To find the amount remitted to the agent given the selling price, the rate of commission and expenses. Gl 1 Al CI Dl El G3 Jl 8 q. To find the value of goods sold (selling price), given the net proceeds, the rate ot commission and the expenses. B2 Fl J3 6 B2 Fl J3 6 r. To find the net proceeds. (Wholesaler's point of view.) 1. given the amount of the sales and the rate of commission. B2 CI D30 El F7 II 42 B3 C3 D32 E3 FIO 12 53 2. given the amount of sales, rate of commission and expenses. Al B4 E2 Gl J2 10 A2 B16 C7 D3 E4 G6 HI J3 42 3. given the commission, rate of commission and expenses. Al Bl 2 Al Bl 2 s. To find the rate of commission: 1. given the cost price or selling price, the expenses, and the amount remitted to the agent. Al Dl 2 [73] 2. given the proceeds and the amount of sales. El I E2 Fl 3 3. given the amount remitted to the owner or dealer, the amount of sales, and the expenses. Al 1 Al Dl 2 B4 Borrowing, lending, and saving money. a. To find the interest or discount, given the amount loaned, the rate of Interest or discount, and the time or term. A183 B68 C57 D343 E70 F394 G234 H168 1346 J251 2114 A195 B98 C79 D361 E86 F426 G245 H232 1374 J270 2366 b. To find the total interest received, given the rate of Interest, the amount loaned, the term for compounding the interest, and the total time. B2 CI Dl E9 J4 17 A13 B2 C4 Dl E9 J4 33 c. To find the exact interest, given the date the loan was made, the date the loan was due, the loan, and the rate of interest. All B5 C2 D2 E8 F31 J6 65 All B5 C3 D2 E8 F31 J6 66 d. To find the amount loaned: 1. given the interest or discount, the rate, and the time. Al B2 C2 D5 ElO FIO H5 15 J3 43 Al B4 C2 D5 ElO F12 G3 H7 15 J3 52 2. given the amount due, the time, and the rate of interest. B2 E5 F2 Jl 10 B2 E5 F2 Jl 10 e. To find the face of a note, given the rate of discount, the proceeds, and the term of discount. E2 J5 7 E2 J6 8 f. To find the amount due: 1. given the amount loaned, the rate of Interest, and the time. A2 B48 C17 D120 E66 F6 G18 H6 110 J13 306 A9 B70 C43 D122 E71 F9 G26 H19 139 J29 437 2. given the amount loaned, the rate of discount, and the time. E7 F8 J2 17 CI E7 F9 H6 J2 25 3. given the rate of Interest, the amount loaned, the term for compounding the interest, and the total time. A5 D3 E19 F22 G5 HIO 121 Jl 86 A5 B2 C3 D3 E20 F25 G6 HIO 121 Jl 96 4. at a given time, given one or more deposits, the date of each deposit, the rate of interest, and the term for compounding the interest. AlO Bl Cll D20 E3 Fl G15 12 63 AlO B3 C12 D21 E3 F2 G15 13 69 g. To find the balance due, given the amount loaned, the time of interest pay- ments, the partial payments, the total time, and the rate of interest. A6 C2 D5 Gl HI 16 113 34 A6 B3 C2 D5 Gl HI 16 }l4 38 h. To find the proceeds, given the face of the note, draft, or trade acceptance, term of discount, rate of discount and time. All B4 CI D21 E20 F15 G28 163 115 178 A17 B20 C15 D34 E22 F17 G29 172 J20 246 [74] i. To find the balance due at a given time, given a series of deposits, the time of each deposit, withdrawals, rate of interest, and the term for compounding the interest. A3 C3 Dl 7 A3 C3 Dl 7 j. To find the rate of interest or discount, given the amount loaned, the amount of interest or discount, and the time. CI D9 E14 F61 H8 16 ]2 101 CI DIO E15 F63 H14 18 }3 114 k. To find the time, given the amount loaned, the rate ot interest, and the amount of interest. (Reductions of time elements.) Al D9 E9 Fl II 21 Al B3 D9 Ell Fl II 26 B5 Keeping accounts. a. To find the total of a bill or invoice, given an Item or series of items, the number of each, the price of each, and the terms. A4 B13 C7 D15 E41 F7 G3 H9 124 J2 125 A17 B16 C15 D30 E46 F31 G3 H9 125 J5 197 b. To find the total value, given an inventory, and value of each item. ^ Dl El H2 4 D2 El H2 5 c. To make out a bill or invoice, given an item or a series of items, the number of each, the price of each, the names of the purchaser and seller, and the terms. A2 B7 C4 D5 El F14 G20 HI 145 J2 101 A14 B35 C15 D51 E8 F38 G38 H2 163 J14 278 d. To make out a bank deposit slip, given two or more checks, an amount of bills, and several coins. A4 C2 E4 F6 G3 18 27 A4 C2 Dl E4 F6 G3 111 31 e. To make a monthly statement, given the items bought, the credits allowed, the purchaser, and the seller. C3 II 4 C3 D3 II 7 f. To make a contract, given the agreement, the consideration, the parties con- cerned, and the witnesses. C2 El 3 C2 El 3 g. To make out an inventory, given a series of items, the number in each series, and the value. HI II 2 HI II 2 h. To receipt a bill when paid, given the bill and the payment. Jl 1 A8 B27 CI D39 E7 Fl G14 HI 112 J13 123 i. To write a receipt, given the amount for which payment was received and the name of the payer. A4 B3 C12 D6 E5 HI 17 38 A4 B5 C15 D6 E5 HI 18 44 j. To write a note or trade acceptance, given the amount, rate ot interest, payee, payer, and time. A7 B4 C3 D2 E2 F8 Gl H4 II 32 AlO B6 C8 D4 E2 F14 G3 H4 115 J5 71 k. To write a check, given the name of the bank, the amount of the check and the payee. A5 Bl C5 D4 G2 17 A7 Bl C17 D17 El G4 H4 II 52 [75] 1. To write a draft, given the amount of the draft, the name of the person in favor of whom the draft is drawn, of the bank on which the draft is drawn, and the bank drawing the draft. A2 C3 D4 El G3 13 J3 19 A2 C3 D4 E2 G4 17 J4 26 m. To keep a stub of a check book: 1. given an original deposit and a series of checks. GI 1 Gl HI 2 2. given an original deposit, a series of checks, another deposit, and another series of checks. E3 G6 9 E3 G6 9 n. To keep a cash book, given receipts and expenditures. Bl C2 Gl HI 14 A5 B2 C8 D4 E2 F6 G8 H2 122 J4 o. To keep an account, given purchases and payments, simple accounts. CI F6 G3 p. To indorse a check, given a check. (Drafts and notes included.) A2 CI D2 F2 Gl A6 CI D6 El F8 G4 19 J4 q. To find the balance of a cash book, given expenditures and receipts. A2 Bl C6 D15 E22 G7 H2 AS B2 C12 D19 E26 Fll G15 H3 122 J2 r. To balance an account, given purchases and payments. 1. simple accounts. E2 Fl G6 CI D3 E3 F7 G9 2. complex accounts. G5 G5 s. To balance a bank account, given an original balance, a series of deposits, and a series of withdrawals. A2 CI Dl F5 G7 H7 110 Jl 34 A2 CI D3 F5 G7 H7 114 Jl 40 t. To find a balance, given the exchange of commodities. Jl 1 Jl 1 B6 Construction. Note: Problems involved in the following activities were included in this classi- fication: woodworking, sewing, cooking, building construction, and fencing. Costs of construction materials were included. a. To find how many times a given pattern, border, design, or length is con- tained in a given length. A7 B15 C7 D7 E16 G38 H13 19 J3 115 A26 B27 C15 D45 E18 F4 G51 H20 118 ]3 111 \ b. To find the amount of fencing, given the number of wires to be used in a dimension of the area. Gl Bl Gl c. To find the total number ot units: 1. given the dimensions of the unit, and the dimensions of the whole A24 B17 C12 D21 E22 F9 G48 H8 18 J27 196] A37 B28 C20 D44 E37 F30 G84 H13 117 J59 3691 9 63 10 39 55 120 9 23 5 5 [76] 2. given the number of wholes and the dimensions of each whole. Gl HI 2 B2 G8 HI J5 16 3. given the dimensions of the whole and the size of the unit. Al G9 Jl 11 Al B3 Dl E6 G14 II J6 32 4. given the dimensions of the whole, the size of the unit, and the allowance. (Allowance for openings, waste, matching, etc.) 13 3 B3 Dl El G3 110 J6 24 5. given the dimensions of the whole, the dimensions of the unit, and allow- ance for waste, etc. F3 II 4 CI D3 El F3 15 Jl 14 d. To find the number of shingles needed: 1. given the number of shingles used per square or a given area, and the dimensions. El F2 12 5 Dl El F2 12 Jl 7 2. given the number of shingles used per square and the area to be covered. El 1 El Jl 2 e. To find the total number of board feet, given a mill bill. A2 D42 H25 18 77 A5 B3 CI D49 E4 F7 H36 113 J23 141 f. To find the amount of paint needed to cover an area, given the area covered by a unit measure of paint and the total area to be covered. Bl Gl 2 B2 E4 Gl 7 g. To find the number of rolls of paper needed, given the dimensions of the room, and the allowance for openings. F5 5 D6 F5 11 h. To find the rim speed, given the number of revolutions per minute, and the diameter. H14 12 16 H14 12 16 i. To find the number of revolutions per minute, given the rim speed, and the diameter. Gl H3 4 A3 Gl H3 II Jl 9 j. To find the total cost of construction, given the cost per unit and the number of units. R 1 H 1 16 8 A23 B47 C4 D50 E27 F19 G33 H4 124 J34 265 k. To find the cost per unit of construction, given the total cost and the number of units. Al Bl El Gl 13 7 A3 Bl Dl El F3 Gl HI 13 14 1. To find the number of units, given the total cost and the cost per unit. E4 4 E4 4 m To find the number of units, given the size of the whole, and the size of the unit. Bl HI 2 Bl E9 HI II [77] B7 Travel, transportation, and communication. Note: This type of problem includes travel by any means such as automobile, train, etc. It also includes transportation by truck, train, express, parcel post, or by any other means. Communication of any type may be included here, such as mail, telephone, telegraph, or radio. a. Travel. 1. To find the distance: (a) given the time and the rate. Bll C2 D8 E13 F5 G4 H19 16 Jl 69 A2 B23 C4 D8 E17 F9 G4 H29 19 J6 111 (b) between two places, given the rate of travel, the time taken to travel the distance, the number of stops and the time for each stop. Dl 1 Dl 1 2. To find the distance traveled per unit of time: (a) given the total distance and the total time. A2 B3 C5 D13 E7 F14 G16 H20 12 J2 A3 B7 Cll D17 ElO F30 022 H25 18 J3 (b) given the distance between two places, the time taken to travel the distance, and the time spent for stops. 01 84 136 01 HI HI n 3. To find the time: (a) given the distance and the rate. B4 C4 D2 ElO F5 05 H5 18 Jl 44 B7 C4 D2 E17 F7 05 H8 18 J2 60 (b) between two stations, given one station in one time belt and another station in another time belt. (Eastward travel.) Bl 1 Bl 1 (c) between two stations, given one station in one time belt and another station in another time belt. (Westward travel.) Bl 1 Bl 1 b. Transportation. 1. To find the amount hauled by the same power over a good road, given the power, the amount hauled on a poorer road, and the ratio of the amount hauled on the poorer road to that which can be hauled on a better road. El 1 El 1 2. To find the number of trips needed to haul a given amount over a good road with the same power used on a poorer road, given the power, the amount to be hauled, the amount hauled per load on the poorer road and the ratio of that load to the load hauled on the better road. E2 2 Dl E3 4 3. To find the cost: (a) of sending a commodity or commodities by parcel post, given the rate of the article for a given zone, and the weight. A26 B7 C27 D22 El F22 01 ]3 109 A27 B7 C28 D22 El F22 Ol 13 J16 127 (b) of shipping commodities by express, given the rate, the weight, and distance. A6 B2 C2 D2 Ell H 24 A9 B4 C7 D5 El 2 Fl HI II J2 42 [78] (c) of shipping small commodities, given the cost per pound, weight, or size. A4 4 A4 4 (d) of hauling bulk commodities, given the total number of units and the cost per unit. 11 1 Bl CI Dl El 15 9 (e) of carrying a load of equal weight over a good road, given the cost of power per mile on a poorer road, the distance traveled, and the ratio of the load the same power can haul on the good road. El 1 El 1 (f) To find the cost per unit of hauling or transportation, given the total cost and the number of units. Dl 12 3 A3 C3 Dl 15 12 4. To find the total cost of an article sent by parcel post, given the weight, the rate for the zone, the value of the article or articles, and the rate of insurance. C2 El 3 C15 El 16 5. To find the freight rate, given the amount of freight charges, and the weight. CI 1 CI Dl 2 c. Communication. 1. (a) To find the amount charged lor collection of a draft, given the face value, and the rate charged. D3 E2 J5 10 D3 E5 J7 15 (b) To find the rate of exchange, premium, or discount, given the face value of a money order or draft, and the total cost. 12 2 12 2 (c) To find the proceeds, given the amount ol the draft, money order, or bill, and the rate charged for collection. E2 II 3 E8 119 2. To find the cost: (a) of a money order or draft, given the amount sent, and the rate charged. CI DIO E12 F6 H2 145 116 92 A17 CI DIO E19 F6 Gl H2 145 J26 127 (b) of mailing letters, newspapers, etc., given the rate of postage per unit and the number ot units. (Unit may mean letters or weight.) A14 D2 F2 18 A20 Dll F4 35 (c) To find the cost of sending a telegram or telephoning, given the number of units, a rate lor a given number ot units, and an added rate for each additional unit. Bl D3 4 Bl D3 4 B8 Municipal and federal activities. (Excluding taxation.) a. Municipal activities. 1. To find the per capita expense of a community activity, given the total cost and the population. (Total number of persons.) E4 F12 Gl H2 II 20 CI E6 F12 Gl H2 II 23 [79] 2. To find the number of lives saved, given the death rate, the decrease in percent (due to an applied remedy) and the population. El 1 El 1 3. To find the number of lives saved, given the death rate at one period, the death rate at a later period, and the population at each period. El 1 4. To find the per unit cost of a community activity, given the total cost, and the number of units. A2 G7 II 10 A2 Dl F4 G7 II 15 5. To find the per capita loss, given the valuation of property destroyed, and the total population. El 1 6. To find the total cost of a community activity, given the number of units or the total population, and the cost per unit or per person. EI G2 3 E2 02 4 7. To find the death rate per a given number, given the number of deaths, and the total number of persons. CI E3 Fl 5 C17 E3 Fl 21 b. Federal activities. 1. To find the number of years of peace needed to pay for a year of war, given the amount saved during a year of peace, and the total amount spent during a rear of war. HI 1 B9 Insurance. a. To find the premium: 1. given the face value of the policy, the rate of insurance, and the term. A4 B5 D22 F4 G2 H9 136 Jl 83 A19 BIO CI D29 El F13 G2 H12 137 J9 133 2. given the valuation of the property, the ratio of that value which was accepted for insurance, rate and term. Al Bl C3 Dl E5 Fl Gl H2 14 J3 22 AS Bl C5 D2 E6 Fl G5 H2 15 J5 37 3. given the face value of the policy, the original rate of insurance, the per- cent of decrease due to the installation of protection devices. El ■ 1 b. To find the total premium, given itemized values and respective rates, and an added rate for an additional risk. El 1 El 1 c. To find the amount of insurance, given the rate and the premium. B3 CI Gl 114 J8 27 B3 C2 El Gl 116 J12 35 d. To find the rate of insurance, given the premium, the face value of the policy, and the term. Al D3 112 J3 19 A2 Bl D6 H2 114 J4 29 BIO Personal investments such as life insurance, real estate, stocks and bonds. (Stocks include investments in building and loan associations.) a. Life insurance. 1. To find the premium on a life insurance policy, given the table of annual premiums based on $1000.00, the kind of policy and time. A18 Dl E3 F4 G6 19 Jl 42 A21 B2 C6 Dl E3 F5 G6 111 Jl 56 [80] 2. To find the difference in the amount paid in and the amount received, given the age at which the policy was taken, age at maturity, kind of policy, and table of premiums. B5 El F4 G3 13 B7 El F4 G3 15 3. To find the cost of protection, given the face value of the policy, the premium per year, the number ot years, and cash surrender value. G2 2 G2 2 b. Real estate. 1. To find the profit, given the original cost, the selling price, other necessary costs, receipts, and the time. D2 F2 H2 6 D6 El F2 H2 11 2. To find the loss, given the original cost, the selling price, other necessary costs, losses, receipts, and the time. Al El 2 3. To find the rate of profit or loss, given the cost price, the selling price, and expenses. (Selling price includes receipts. Expense includes added cost or losses.) Bl Dl E2 Fl 5 B2 D6 E3 F2 13 4. To find the cost per front foot, given the total cost and teet of frontage on street. El 1 E4 4 5. To find the rate of profit, given the cost, the rent, and the expenses and losses. B3 D5 E2 Fl Gl II 13 A3 B3 CI D6 E2 F3 Gl 12 21 6. To find the amount of rent necessary to make a given rate on an invest- ment, given the amount of the investment, and the expenses. Al B2 CI Dl El G2 11 Al B2 CI D4 E3 Fl G2 14 7. To find the total price, given the rent, expenses, and rate of profit on the investment. El 1 E2 2 8. To find the net income on an investment, given the amount invested, the profit, and the expenses. Dl Fl 2 c. Stocks and bonds. 1. To find the dividend, given the amount of the bonds, or stock, the interest period, and the rate ot interest. AlO Bl C3 D15 E14 G18 HIO 17 J8 86 A16 B2 C4 D20 E16 Fl G28 H55 19 J18 169 2. To find the dividend, given the total cost of bonds, rate of dividend, and the quotation. Gl J2 3 Gl J2 3 3. To find the amount of dividend, given the number of shares or bonds, the par value per share or bond, the rate of dividend, and time. A3 D14 E5 12 24 A4 CI D14 E5 H7 13 34 4. To find the cost of bonds or stock, given the quotation, par value per share, brokerage, and number of shares. C3 D23 E14 G6 H15 II J14 76 Al C3 D23 E15 G6 H15 II J17 81 [81] 5. To find the cost of a bond or bonds, given the amount of the bond or bonds, the interest, the time, quotation, and brokerage. a: G1 J3 6 a: Gil J3 16 6. To find the cost of bonds, giv-en the dividend, the rate of dividend, broker- age, and rate of premium. Gl 1 Gl 1 7. To find the total cost or amount of stock, given the number of shares and quotation or par value per share. A6 Bl C: D: H3 19 JIO 33 A8 Bl C3 D9 E2 H12 110 J30 75 8. (a) To find the profit or loss, given the number of shares of stock, the brokerage, the quotation at which it was bought, and the quotation at which it was sold. A4 C3 D6 E2 G2 16 Tl 24 A4 C3 D6 E2 G2 H2 16 jl 26 (b) To find the profit or loss, given the number of shares of stock, the brokerage, the quotation at which it was bought, the quotation at which it was sold, and the rate ot dividend received. Al CI Dl EI Gl 5 Al CI Dl E2 Gl 6 9. To find the profit, given a cost of stock or bonds, the amount of dividend received, and the selling price. HI 1 HI 1 10. To find the number of shares or bonds, given the amount of dividend, the rate of interest, and the par value per share or bond. Al G2 T4 Al G2 "19 12 11. To find the number of shares or bonds, given the total cost of stocks or bonds, the quotation, and brokerage. C3 T7 10 Al C3 G2 J8 14 12. To find the number of shares or bonds, given the total cost or total amount of bonds, and the quotation, or par value per share or bond. CI D3 13 T5 12 A2 CI D3 El H4 13 J"l9 33 13. To find the amount received for bonds or stocks, given the amount of bonds or stocks, the quotation and brokerage. G6 ]5 11 G6 Jll 17 14. To find the amount of bonds or stocks, given the percent of dividend, and the amount of the dividend. Bl El G3 II 14 10 Bl EI G3 II j9 15 15. To find the amount of stocks or bonds, given the total cost, the quotation, and brokerage. T2 2 J3 3 16. To find the amount received (amount remitted by agent after deducting his brokerage) given the number of shares or bonds, the quotation, and brokerage. CI D4 E2 II 8 A2 CI D4 E2 II 10 [82] 17. To find the percent of profit, given the quotation, percent of dividend, brokerage and time. Bl C6 F20 G4 14 35 Bl C6 F20 G4 14 35 18. To find the rate of profit, given the total cost of bonds or stocks, and the amount of profit. Al H2 Jl 4 A2 El Fl H8 J3 15 19. To find the percent of profit or loss, given the amount of profit or loss, and the amount invested. CI El 2 20. To find the rate of dividend, given the amount ot dividend, and the amount ot bonds or stocks. Bl Gl HI Jl 4 A2 Bl Gl H2 12 J3 11 21. To find the amount of brokerage, given the total cost of bonds or stock, and the rate of brokerage. HI 1 HI J2 3 22. To find the cost or par value per share or bond, given the total cost and the number of shares or bonds. C3 H2 Jl 6 C3 H2 J2 7 23. To find the proceeds, given the quotation, rate of interest, and brokerage A9 B2 D4 15 A9 B2 D4 15 Bll Personal activities involving wages and salaries. a. Wages. 1. To find the amount of wages: (a) given the number of units, and the wage per unit. A6 Bll CI D6 E16 F3 G3 H32 15 J14 97 A50 B37 C62 D27 E47 F22 G24 H95 153 J38 455 (b) given the price per unit for a given number of units, a higher price for added units, and a still higher price for more added units, and the total number of units. HI 1 Bl CI E3 H7 12 (c) To find the amount of wages earned in a given time, given an amount earned in a different length of time at the same rate. ElO 10 ElO 10 2. To find the wages earned per unit, given the number of units and the total wage. Al Bl D3 Ell Fl G5 H4 II J6 33 A8 B7 D5 E18 Fl Gil H8 14 J7 69 3. To find the number of units, given the total wages earned, and the wage per unit. Al B3 G3 H2 12 11 Al B3 G3 H2 J4 13 4. To find how much a group of men can earn in a given time at a given rate per hour, given a different number of men, and the total amount earned at the same rate per hour. D2 2 D2 2 5. To find the wages earned by each person, given the total wages earneil by the total number ot persons, and the time each worked. D4 12 6 U4 12 6 [83] 6. To find the amount of advance or decrease, given an original wage or pay- roll, and the percent of advance or decrease. D4 4 Al Bl D4 El 7 7. To find the wage or amount of payroll, given an original wage or an original payroll, and the percent of advance. B4 D7 G2 H6 19 B5 D7 El G2 H6 Jl 22 8. To find the smallest number of coins and bills necessary for a payroll, given a number of workmen, the wage per unit, and the number of units. E2 2 E2 2 9. To find the rate of advance or rate of reduction, given an original wage, and the wage after the advance or reduction. A4 D8 12 A4 D8 12 b. Salaries. 1. To find a salary, given an original salary and rate of increase. Bl Dl 2 Al Bl D4 15 11 2. To find the total salary, given the number of units, and the salary per unit. HI 1 CI D2 E2 Fl Hll II J9 27 3. To find the salary per unit, given the total salary and the number of units. HI II 2 E2 HI II 4 B12 Taxation, municipal, state, or national. a. To find the amount of tax: 1. given the rate of taxation and assessed valuation, or quantity. (Note: Duties and poll tax included in this item.) A14 B28 CI D50 E8 F17 G31 H16 156 J29 250 A39 B65 C43 D77 E22 F85 037 H29 166 J55 518 2. given the real value, the ratio of assessment to real value, and the rate of taxation. B2 D3 Fl 6 A3 B3 C2 D9 Fl 1« b. To find the rate, given the tax and assessment. A16 Bll CI D12 G9 II JS 58 A20 B15 C2 D18 E2 F3 G12 11 J12 85 c. To find the total amount of assessment or quantity taxed, given the rate, and amount of tax or duty. A2 B3 Dl E2 Gl 12 J9 20 M B4 Dl E2 Gl 12 J12 24 B13 Determining economy of two or more procedures. (This classification includes problems involving difference, saving, choice, and comparison.) a. Difference. 1. To find the difference in unit costs, given different unit costs of two com- modities. (This includes two qualities of the same commodity.) E5 5 Al B3 Ell Gl H2 18 2. To find the difference in price per unit, given the number of units and the total price of one quality of a commodity and the number ot units and the total price of another quality of the same commodity. Dl E2 Jl 4 Dl E3 Jl 5 [84] 3. To find differences in amount, given two different unit costs for the same commodity or different commodities, and the total number of units. A2 B2 E8 Fl G3 H8 12 J3 29 A3 B3 Dl E12 Fl G4 H8 13 J5 40 4. To find the difference in amount, given a total cash payment, and a given number of installments at a given pavment each. El ' 1 5. To find the difference in the amounts of a bill, given different successive discounts, or different terms. J2 2 J2 2 6. To find the difference in number of units purchased for the same amount of money, given the amount of money and different prices for each of two qualities of the same commodity. El 1 El 1 7. To find the difference in units of time between two places, given two distances of different lengths, and the rate of travel. E2 2 E2 2 8. To find the difference in the rate of discount, given the marked price and the selling price of one commodity and the marked price and the selling price of another quality ot the same commodity. El 1 El 1 9. To find the difference In the rate of travel, given the distance between two places, and the total time for each of two means of travel. El 1 El II 2 10. To find the difference, given a selling price with a discount and a different selling price with a different discount. Al E2 3 Al E2 HI 4 11. To find the difference in amount of profit, given the amount received for a given number of units before spraying, and the amount received for a given number of units after spraying, and the cost of spraying. D4 ' Gl ' 5 12. To find the difference In amount of profit, given an amount of money invested in real estate with the cost, time, necessary expenses, rent per month, and selling price; and the same amount of money, drawing interest at a given rate for the same length of time. CI D2 Gl H2 6 Al CI D3 Gl H2 8 13. To find the difference in amount of profit, given an amount of stock with rate of dividend, and the same amount invested in a bond and mortgage with rate of interest. Dl 1 14. To find the difference in interest due, given an amount of money, for a given time, at a rate of simple interest, and the same amount of money, for the same time, at the same rate but compounded. Dl El 2 CI Dl El 3 15. To find the difference in amount of interest due, given an amount of money drawing interest for a given time at a given rate, compounded at a given period; and the same amount of money drawing interest for the same time at the same rate, but compounded at a given shorter period. El HI 2 El HI 2 [85] 16. To find the difference in amount of interest, given the amount, time, rate of interest; and the same amount of money for the same time, but at a different rate of interest. CI El H3 II 6 CI El H4 II 7 17. To find the difference in the cost of shipping crated and uncrated articles, given the rate of the crated article and the rate of the same article shipped uncrated. El 1 18. To find the difference in cost, given the total number of units traveled, the total cost by one method, and the cost per unit per person by another method, and the number of persons. Dl 1 Dl " 1 19. To find the difference in cost of sending an amount of money, given the amount sent, the rate or charges by one method, and the rate or charges by another method. Dl E2 3 A2 Dl E2 5 20. To find the difference in premiums, given two buildings of equal value, different material, and different insurance rates. El 1 21. To find the difference, given an amount of a ten-year endowment policy, a twenty-year endowment policy and a table ot annual premiums. ' ' El 1 El 1 22. To find the amount of difference, given a salary, plus a commission on sales over a certain amount; or a higher commission on all sales and no salary, and the total amount of sales. Dl 1 Dl 1 23. To find the difference in tax, given rate and assessed value in one area or at a given time and rate and assessed value in another area or at another time. El HI 2 El HI 2 24. To find the difference in wages, given one wage and hour per day schedule, and another wage and hour per day schedule, and the time. E3 b. Saving. 1. To find the amount saved: (a) given the saving per unit and the number of units. CI E2 Gl 12 Al B2 CI E6 G2 H2 17 (b) given the number of units saved, and the price per unit. Gl A3 Dl El G4 (c) given two different unit costs and the number of units. Bl D2 F2 G2 HI B4 C61 D3 E4 F3 G5 HI (d) given the itemized list of original accounts and the itemized list of the same accounts reduced. Jl 1 J2 2 (e) given the cost per unit, a smaller cost per unit for a larger lot, and the number of units bought. HI 1 Bl HI Jl 3 [86] Jl J4 7 25 Jl 1 10 J2 J4 10 85 (0 given a price per unit, a smaller proportionate cost for a larger lot, and the number of units. A6 B3 CI D5 El G4 H8 17 J3 38 AS B3 C6 D6 El Fl G6 H15 18 J3 57 (g) given a cost price or selling price, or total quantity, and the percent or fractional part saved. G3 HI 12 J2 8 A2 B4 E2 G6 H5 17 J2 28 (h) given an amount of insurance on a building, a rate; a lower rate due to installation of a safety device, and the term. Fl 1 El Fl 2 (i) given the amount of insurance, the term, a number of policies at a given rate for a long term, and a larger number of policies at a given rate for a shorter term. E2 2 E2 2 (j) given the cost price or selling price of one or more articles in each of two invoices, and the terms. El G4 5 E2 G4 6 (k) given an amount paid down, an amount paid in installments, the time and rate of interest; and the same amount down, with different amounts of installments for a different time, and no interest. HI 1 HI 1 2. To find the percent saved, given the amount saved and the basic price. Al B3 C22 Fl G5 Jl 32 3. To find the amount saved if the commodity is home made: (a) given an itemized list of commodities and the cost per item without cost of labor; and given the total cost for the complete job including the cost of labor. A6 El G2 9 A8 Bl CI E2 G2 HI Jl 16 (b) given an itemized list of commodities and the cost per item with the cost of labor allowed for; and given the total cost for the complete job. Al El 2 Al Bl El Gl 4 4. To find the saving in premium, given the value of a building, the rate for a policy for one year, a cheaper rate for more than one year, and the time. Gl Jl 2 Al Fl Gl Jl 4 5. To find the time saved, given the total time by one method of travel, and the total time bv another method of travel. Bl 1 Bl 1 Choice. 1. To find the most economical purchase, given the cost of a large unit, a larger proportionate cost of a smaller unit, and a still larger proportionate cost of a still smaller unit, and the number of units. B2 Dl Gl HI B5 Dl Gl HI 2. To find the more economical procedure: (a) given the selling price of a commodity, and the discount or two or more successive discounts. Al B2 D2 Gl Al B2 D2 El G2 [87] 12 12 Jl 7 11 oice of a single i.' 7 9 (b) given the total number of units, the cost per unit of the whole com- modity; or the cost per unit of one quality and the cost per unit of the other remaining quality of the commodity. Bl 1 Bl 1 (c) given an amount of money borrowed for a given period of time at a given rate of interest; or a part of the amount borrowed paid in cash by the lender, and the remainder for the given time at a given rate of interest. HI 1 (d) given one cost and selling price and a different cost and selling price. Bl 1 Bl 1 3. To find the better selling price of the same commodity, given the selling price and discount in one case and a lower selling price in the second case. B2 Gl 3 B2 Gl 3 4. To find the economical method of shipping, given the weight of an article and the rate charged for shipping in each of two or more modes of trans- portation. C4 4 C4 4 5. (a) To find the better investment, given the amount invested in each case, the interest in each case, and the time. 02 HI 3 Dl G2 HI 4 (b) given one investment with amount of profit, and another investment with an amount of profit. Bl 1 (c) given the amount invested, the rent, expenses, and time; and the same amount invested, for the same time at a given rate of interest. Bl CI Dl 3 B2 CI Dl 4 6. To find the better wage: (a) given one wage and hour per day schedule, and another wage under another hour per day schedule. Al 1 Al 1 (b) given a wage per unit of time, the time, and a lower wage per unit of time for a longer time, and the time. Al 1 Al 1 7. (a) To find the better salary, given a salary per small unit of time, and another salary per larger unit of time. Dl HI 2 (b) given one salary at one time, another salary at a different time, and the ratio of the value of a dollar at the one time to the value at the other time. Bl H2 3 B6 H2 8 8. To find the more economical purchase: (a) given a number of units of one quality (width, material, etc.) of a commodity and the price per unit; and a different number of units of a different quality of the same commodity, at a different price per unit. Bl Dl G2 HI 5 Bl Dl G2 HI 5 [88] (b) given different total prices, with different successive discounts. (May be the same total prices.) C3 D2 5 C3 D2 5 9. To find the better offer, given an amount of insurance, the rate for a short term, the rate for a longer term, and the time. Al 1 Al B3 .4 Comparison. 1. To compare the results of two or more procedures, given the procedures. A2 B4 C2 D3 El F2 GIO H31 111 Jl 67 A5 B4 C2 D7 E4 F2 Gil H33 115 Jl 84 2. To compare areas, given the dimensions of each. El G2 H3 6 Bl El G2 H3 7 3. To compare costs of two or more commodities or items, given the cost of each per unit and the number of units. C6 Gl HIO 17 C6 Gl HIO II 18 4. To compare pairs of quantities by ratios, given the pairs of quantities. (Ratios may be in whole numbers, percents, or fractions.) H14 Jl 15 C3 H14 J3 20 5. To find how many times as much it costs to send a given number of small money orders, at a given rate per small money order, than to send one large money order equivalent to the small ones, at a given rate per large money order. H2 2 H2 2 [89] APPENDIX B The following questions concerning functional relationships are suggested for recognition as minimum essentials. The list is based upon the analysis of ten series of arithmetics but represents in part the judg- ment of the writer. In each case the question is : What calculations must be performed in order to find the quantity named, given the quantities specified? Al To find totals by addition, given two or more items, values, etc. A2 To find the difference, given two items, values, etc. A3 To find the amount, or number needed, given a magnitude and the number of times It is to be taken. A4 To find the size of a part of a magnitude, given the magnitude and the number of parts into which it is to be divided. A5 To find how many times a stated quantity is contained in a given magni- tude, given the quantity and the magnitude. A6a To find how many when reduction ascending is required, given a magni- tude expressed In terms of a single denomination. A6b To find how many when reduction ascending Is required, given a magni- tude expressed In terms of two or more denominations. A7a To find how many when reduction descending Is required, given a magni- tude expressed in terms of a single denomination. A7b To find how many when reduction descending Is required, given a magni- tude expressed In terms of two or more denominations. A8a To find a dimension, given the area of a rectangle and one side. AlOa To find the area, given dimensions of a square, rectangle, or parallelo- gram. AlOb To find the area, given the base and altitude of a triangle. AlOc To find the area, given the diameter of a circle. Alia To find the perimeter, given one side of any equilateral figure. Allb To find the perimeter, given two adjacent sides of a rectangle or parallelo- gram. Allc To find the circumference, given the diameter or radius of a circle. A12bl To find the cubic contents, given the three dimensions of a rectangular solid, such as room, bin, woodpile, etc. A12b2 To find the cubic contents, given the area of one surface of a rectangular solid and the depth or altitude. A14 To find the average, given a series of Items. A15 To find the ratio of one number to another, given the two numbers. A16 To find a part of a number, given the ratio of the part to the number, and the number. A18 To divide a quantity Into parts having a given ratio, given the quantity and the ratio. A19 To find a member of a ratio, given two members of one ratio and one member of another ratio equal to the first. (Inverse ratio Included.) A20 To find the ratio of items to total, given a series of items. Blal To find the total price, given the number of units and price per unit. Bla2 To find the total price, given the number of units and the price per unit of another denomination. [90] To find the number of units, given the total price and price per unit. To find the price per unit, given the total price and the number of units. To find the amount to be received for several items, given the price of each. To make change, given an amount of money and the price of a commodity. To find the margin or loss, given the cost price and the selling price. To find the total margin or total loss, given the unit cost, the unit selling price, and number of units. To find the selling price, given the rate of discount or loss, and the price. To find the selling price, given the rate of advance or margin and the price. To find the selling price, given the rate of two or more successive discounts and the price. To find the selling price, given the price and the amount of commission or discount. To find the amount of margin, loss, commission or discount, given the total price and the rate of margin, loss, commission or discount. To find the rate of margin, loss, discount, advance or commission, given the total price and the amount of margin, loss, discount, advance or commission. To find the rate of margin, loss, discount, advance, or commission, given the total price and the selling price. To find the price, given the selling price and the rate of margin. To find the net profit or loss, given the cost price, overhead, and selling price. To find the net profit or loss, given the total costs and total receipts. To find the net profit or loss, given the itemized costs and total receipts. To find what percent the net profit is of the cost price or selling price, given the net profit, and the total receipts, original outlay, or amount invested. To find what percent the profit or loss is of the cost price or selling price, given the profit or loss, and the cost price or selling price. To find the interest or discount, given the amount loaned, the rate of interest or discount, and the time or term. To find the amount due, given the amount loaned, the rate of interest, and the time. To find the balance due, given the amount loaned, the time of interest payments, the partial payments, the total time, and the rate of inter- est. To find the total of a bill or invoice, given an item or series of items, the number of each, the price of each, and the terms. To find the balance of a cash book, given expenditures and receipts. To balance a bank account, given an original balance, a series of deposits, and a series of withdrawals. To find how many times a given pattern, border, design, or length is contained in a given length. To find the total number of units, given the dimensions of the unit, and the dimensions of the whole. To find the total number of units, given the dimensions of the whole and the size of the unit. To find the total cost of construction, given the cost per unit and the number of units. To find the cost per unit of construction, given the total cost and the number of units. [91] B7al(a) To find the distance, given the time and the rate. B7a2(a) To find the distance traveled per unit of time, given the total distance and the total time. B7a3(a) To find the time, given the distance and the rate. B7b3(a) To find the cost of sending a commodity or commodities by parcel post, given the rate of the article for a given zone, and the weight. B7c2(a) To find the cos t of a money order or draft, given the amount sent, and the rate charged. B7c2(b) To find the cost of mailing letters, newspapers, etc., given the rate of postage per unit and the number of units. (Unit may mean letters or weight.) BlObl To find the profit, given the original cost, the selling price, other necessary costs, receipts, and the time. (Real estate.) BlObS To find the rate of profit on real estate, given the cost, the rent, and the expenses and losses. B10b6 To find the amount of rent necessary to make a given rate on an invest- ment, given the amount of the investment, and the expenses. BlOcl To find the dividend, given the amount of the bonds, or stock, the inter- est period, and the rate of interest. Bllal(a) To find the amount of wages, given the number of units, and the wage per unit. Blla2 To find the wages earned per unit, given the number of units and the total wage. Bllb2 To find the total salary, given the number of units, and the salary per unit. B13bl(c) To find the amount saved, given two different unit costs and the number of units. B13bl(0 To find the amount saved, given a price per unit, a smaller proportionate cost for a larger lot, and the number of units. B13b3(a) To find the amount saved if the commodity is home made, given an item- ized list ot commodities and the cost per item without cost of labor; and given the total cost for the complete job including the cost of labor. B13dl To compare the results of two or more procedures, given the procedures. J.a:j 2 5 :c77 [92]