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The authors examined the thinking of children who had the opportunity to construct personal knowledge about division of fractions. The authors based this study on a teaching experiment design and used relevant contexts/situations to foster students' development of knowledge. Participants were a group of mixed-ability, 5th-grade mathematics students. They used pictures, symbols, and words to resolve situations and communicate their solutions. The authors analyzed the solutions to describe the students' constructions of division-of-fractions concepts and procedures. All strategies that the students used represented some manifestation of conceptual knowledge about addition and subtraction of fractions and a definition of division. Some students developed formal symbolic procedures, and others developed pictorial procedures; none invented an invert-and-multiply procedure. Through the window of constructivism, this study allowed the authors to glimpse children's constructions of knowledge and provided...

Curiously enough in the discussion of transfer of training one point seems to have been agreed upon by all parties to the con troversy, namely, that there is transfer of identical elements. In early experiments the identical element was an "idea"1 common to both the practice series and the test series, or the identical ele ment consisted of "sensations, images, movements."2 Later by identical elements were meant "mental processes which have the same cell action in the brain as their physical correlates."3 Such an identical element is probably not accessible to experimental investigation. From the outset of the discussion, however, it has been claimed that, "improvement in addition will alter one's ability in multipli cation because addition is absolutely identical with a part of mul tiplication and because certain other processes?for example, eye movements and the inhibition of all save arithmetical impulses? are in part common to the two functions. ' '4 This view is expanded by Gates into a theory of transfer,5 and, since the identical elements of addition and multiplication, subtrac tion and division seem to be objective, this claim has gone far toward the belief that the study of the environment and reactions to it may well replace the study of the mind?one form of current behaviorism. Professor Poffenberger tested the assumption that practice in addition transfers to multiplication, and that in subtraction there is transfer to division because of the partial identity of the processes.

In a previous article1 the writer made an analysis of the examples in long division as found in nine current textbook series in arithmetic. It was found that there was little uniformity in the method suggested for estimating the quotient and in the number and types of examples given in each textbook. This study makes a similar analytical classifi cation of the estimations of the quotient for the different division ex amples found in current arithmetic workbooks. The selection of the workbooks to be included in this study was based on the manner in which they were issued. Each workbook to be selected had to be issued in a series of separate booklets for each grade for Grades III-VIII, inclusive. Since none of the booklets for the third grade con tained long division examples, the booklets scrutinized were for Grades IV-VIII, inclusive. An effort was made to include every series of prac tice booklets by different authors now extant which satisfied the cri terion of having a separate booklet for each grade above the third and extending through the eighth. If an author or a group of authors had written more than one series of workbooks, the edition having the latest copyright date was the only series considered for that author or group of authors. A separate booklet for each grade was essential to make the results obtained from the study comparable. The series had to cover the work for grades four to eight inclusive in order to make the results from this study comparable to those derived from the analysis of textbooks. In the latter case, the series of text books contained the subject matter of arithmetic for grades three to eight inclusive. Using the criterion of having all possible workbook series consist of a separate booklet for each grade from the third to the eighth inclusive, nine different series were selected. In the study of textbooks the method for estimating the quotient was classified under three headings. Two of these methods of estimation are used in this study. They are : First, the apparent method in which

In a previous study by the writer,1 the relative merits of short and long division for a one-figure divisor were analyzed. The data from the former study showed that the long form of division is superior to the short form when the divisor is a one-figure number. Since the long form of division was recommended as the type to be taught, it seemed advisable to make a detailed study of the types of errors and questionable habits of work which pupils made when they divided in the long-form with a one figure divisor. Hence, this study deals with the errors and questionable work habits made by students who took the test used in the former study already cited. A critical analysis was made of errors from a total of 1211 test papers. These tests were given in each grade from the fifth to the fifteenth, in clusive. Grades XIII, XIV, and XV represent the grades for three years of normal school. There were approximately 100 cases for each of the eleven grades sampled. In order to conserve space for publication, only the errors made by pupils from Grades V-VIII, inclusive, are reported in this study. It was assumed that the underlying causes of errors made by students above the eighth grade level contributed little to the phenomena which produce the errors because of the interval elapsing between the initial learning of the process and the time the test was taken. Hence, the data presented in the following pages are taken from a total of 453 test papers. The number of students in each grade is shown in Table I.

To what extent do pupils use short division in examples with single digit divisors? What are some of the factors which influence the selec tion of the method used? W^hich method is selected by brighter pupils? Is long division used with greater accuracy than short division? The customary order of procedure in teaching division of numbers to elementary school pupils is to teach a child first to know a specific number of simple division combinations; secondly, to work examples with single-digit divisors and with varying numbers of digits in the dividend by a short division method; and, thirdly, to divide numbers containing divisors with two or more digits by a long division process. In short division the pupil mentally carries out the steps in the process, writing only the answer; in long division the child places each step of his calculations on paper. Several writers have contended, and some evidence has been produced to prove/ that long division should precede short division and that the latter should be taught as a short-cut to the former. The evidence and arguments center largely in the problem of the order of introducing the two processes. Few individuals seem to have considered the question whether we should continue our present insistence on pupils' using the short division method with single-digit divisors. If we knew to what extent pupils throughout the different grades regularly use the short method under the condition prescribed and also the comparative accu racy with which the short and long processes are employed, we might even see reasons for contending that our present stress on short division is not worthwhile. By thus abandoning our insistence on short division, we could eliminate a field of considerable worry for the elementary school teacher. With the purpose in mind, therefore, of determining the extent to which and also the accuracy with which pupils throughout the grades use the short division method with single-digit divisors, the present study was undertaken.

This study was made to determine the constancy of an error to a basic fact in the four fundamental operations in arithmetic. There is some experimental evidence to indicate that many errors made in arith metic are sporadic and due to chance rather than to a faulty knowledge. An error due to chance is known as a chance error and it is not constant. An error due to a faulty knowledge should be persistent and constant. A test containing only one response to a combination would not differen tiate between a sporadic and a constant error. One of the first studies to show that an error is not always constant was made by Brueckner and Ewell.1 This study used fractions instead of basic combinations. The results indicated that at least three examples of the same type had to be offered in a test to see if an error were constant. Brueckner2 further showed that errors to examples in division of fractions from a test taken six different times were largely due to chance. The senior writer3 showed that a multiplication fact in division had to occur at least three times before the effect of chance could be determined in diagnosing the results. Ruch showed that a pupil's score made on a test of basic facts in addi tion is a variable. He arranged the one hundred addition facts in random sequences on five different forms and gave these tests to thirty pupils in Grade III. There was not one case in which a pupil made the same score on all of the five forms.4 This variation must have been due to chance errors. The investigation reported in the following pages is a more ex haustive study than that reported by Ruch for each of the four funda mental operations.

In a previous study by the writer,1 it was found that errors of com binations had the greatest frequency of any type of error made in long division when the divisor is a one-figure number. In this same study, it was shown that many errors in division were caused by lapses in attention. From an observation of many errors on the test papers from students in Grades V-XV, inclusive, (Grades XII, XIY, and XV are in normal school), the writer concluded that frequent incorrect products in multi plication resulted from the fact that multiplication was used in division. In other words, if a response had been made to a multiplication fact iso lated from the division process, the pupil would have been able to give the correct product. However, if the same fact occurred in the division pro cess, there was a possibility of an incorrect product being given to the mul tiplication fact. This hypothesis can be substantiated by an illustration, _7_ 7)38 . In the given illustration, the pupil estimated the number of 7's

In research work it often happens that there is a need for the division of many numbers by the same divisor. In case these numbers are made up of not more than two digits, Holzinger's tables prove to be a fairly rapid means of making the divisions. To use these tables when the dividend possesses more than two digits involves several operations. For example, to find the value of 341/14, it would be necessary to find either 300/14 + 41/14 or 340/14 -f 1/14. Since the tables are ar ranged according to the size of dividends rather than the size of divisors, such divisions as these must be obtained by referring to two or more sections of the tables, sometimes on widely separated pages. This is time consuming. If a calculating machine is at hand this will be found to be a very economical method of making such divisions. On several occasions, how ever, when it has been necessary to divide many numbers of three or more digits by a common divisor of two digits, it has been found eco nomical to construct a table from which quotients can be read directly. If there are several hundred divisions to be performed, such a table can be constructed and the divisions made by its use in less time than they can be made with Holzinger's or Crelle's tables or with some calcu lating machines. In the hope that the method of constructing such tables may prove useful to others, it will be illustrated and briefly explained. Suppose we have several hundred such numbers, each to be divided by 14. By use of a calculating machine, Holzinger's tables or another method, divide each number from 0 to 13, inclusive, by 14, carrying the division to as many decimal places as is desired in each of the quotients. Holzinger's tables give quotients to four decimal places. Arrange these quotients across the top of a sheet of squared paper as shown by the numbers in heavy face type in Table I. On a table written by hand, it is well to have these decimals in red ink.

(1943). Kinds of Errors in Division of Decimals and Their Constancy. The Journal of Educational Research: Vol. 37, No. 2, pp. 110-117.

It ?s possible to have research on a purely academic basis, and while it may not be valuable positively, it can have a certain amount of negative value. For example, it is possible to study the errors which pupils make in division of decimals at upper grade levels. It is interesting to note2 that 70% of the errors which do occur in the division of decimals are "peculiar to decimal situations". It is interesting to note that practically every type of error which can occur, does occur. It is interesting to note that decimal errors, grades 6-9 inclusive, can be grouped into twenty-one classifications; that errors resulting from placing the point or shifting the point are high and reasonably constant throughout these grades; that zero errors increase at the ninth grade level; and that all told, results are so poor as to suggest that something very radical should be done.

It is generally believed that excellent achievers differ from poor achievers in certain skills and abilities involved in two-figure division, but as yet, no one has attempted by systematic study to discover the characteristics, if any, that distinguish these two kinds of achievers. If it can be shown that excellent achievers differ significantly from poor achievers in certain skills and abilities, the information will be of value in diagnosis. It will also throw light on how to differentiate methods of instruction, practice ma terials, and courses of study in division according to the characteristics of the learner.

Such studies as those by Bond,1 Fendrick, 2 and Russell3 have shown the difference between pupils who are good and poor in some particular subject. There is no published report to indicate if there is any distinguish ing difference between pupils who are good and poor in arithmetic. This study is an attempt to find if there is a quantitative difference between pupils who are good and poor in a certain restricted function in arithmetic. A group of each kind was selected on the basis of success in learning to divide with a two-figure divisor. The measure of success in learning to divide with a two-figure divisor is the number of errors made. A pupil who learned the process and made no errors, or a very few, is considered good in division. On the other hand, a pupil who learned the process and made many different kinds of errors is considered poor in division. The pupils considered in this study were used in another investigation by the writer.4 A record was kept of the number of different kinds of errors which each pupil made from the time of the beginning of the in vestigation until its completion, or a total of 76 school days. Elsewhere the writer5 showed that most of these errors were sporadic and not constant.

In a former study by the writer1 it was shown that fewer errors were made when the long form of division for a one-figure divisor was used than when the short form was used. The results for the investigation were taken from approximately 200 pupils, divided equally into two compar able groups, for each grade from the fifth to the fifteenth inclusive. Grades XIII, XIV, and XV constituted the first, second, and third years, respectively, of normal school. The test in division forming a basis for the study consisted of 30 examples, of which 20 contained three figures in the quotient and the remaining 10 examples contained four figures in the quotient. As previously stated, more errors were made by the group which used the short form of division than by the other group. The number of errors is one index by which to measure the relative merits of the two methods. Another vital factor which affects the two forms of procedure is the point of incidence of error. This study was made to see if there was much va riation between the two forms in the point of incidence of error. That is, an effort was made to determine if there was a tendency for the first error in the quotient in a particular grade to occur in the first, second, third, or fourth figure of the quotient. The percentages of initial error for each of the four possible places in the quotient for short-and-long-division are shown in Tables I and II. The points of inception of error for short-and-long-division were slightly different for the first quotient figure as is shown by Tables I and II. When the long form of division was used in Grades V-VII, in clusive, the percentage of error in the first figure was greater than for the short form for the same grades. This situation may be explained by the fact that the use of the long form of division for a one-figure divisor was unfamiliar to the pupils tested.2 Under such conditions, many errors would result in the first estimation.