A Diagnostic Study of Whole Number Computations of Certain Elementary Students.

Transcription

1 Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1972 A Diagnostic Study of Whole Number Computations of Certain Elementary Students. Leslie Clyde Ellis Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: Recommended Citation Ellis, Leslie Clyde, "A Diagnostic Study of Whole Number Computations of Certain Elementary Students." (1972). LSU Historical Dissertations and Theses This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact gradetd@lsu.edu.

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3 72-28,337 ELLIS, Leslie Clyde, A DIAGNOSTIC STUDY OF WHOLE NUMBER COMPUTATIONS OF CERTAIN ELEMENTARY STUDENTS. The Louisiana State University and Agricultural and Mechanical College, Ed.D., 1972 Education, theory and practice University Microfilms, A XEROX C om pany, A nn Arbor, Michigan

4 A DIAGNOSTIC STUDY OF WHOLE NUMBER COMPUTATIONS OF CERTAIN ELEMENTARY STUDENTS A D is se rta tio n Submitted to the Graduate F aculty of the Louisiana S ta te U n iv ersity and A g ric u ltu ra l and Mechanical College in p a r t i a l fu lfillm e n t of the requirem ents for the degree of Doctor of Education in The Department of Education by L e s lie Clyde E llis B.A., S outheastern L ouisiana C ollege, 1963 M.Ed., L ouisiana S ta te U n iv e rsity, 1966 May, 1972

5 PLEASE NOTE: Some pages may have in d is tin c t p r in t. Filmed as received. U n iv ersity M icrofilm s, A Xerox Education Company

6 ACKNOWLEDGMENTS The w rite r wishes to express h is a p p re cia tio n to those who guided and aided him during the period of study. He is e sp e c ia lly g ra te fu l for the guidance provided by h is major p ro fesso r, Dr. G.C. Gibson, and fo r the u n se lfish a ss is ta n c e rendered by Dr. Sam Adams. The w rite r a lso wishes to thank Superintendent of Schools Marvin C u rtis of L ivingston P a rish, the p rin c ip a ls, and teachers whose cooperation made the study p o ssib le. F in a lly, to h is w ife, fam ily, and frien d s whose encouragement and a ssista n c e were in v alu ab le, the w rite r is humbly g ra te fu l. ii

7 TABLE OF CONTENTS Page ACKNOWLEDGMENTS... LIST OF TA B LE S... LIST OF FIGURES... ABSTRACT... i i v v i v i i Chapter 1. INTRODUCTION... 1 THE PROBLEM... 1 Statem ent of the Problem... 1 D elim itatio n... 2 D e fin itio n o f Terms... 2 Importance of Study REVIEW OF RELATED LITERATURE METHOD OF PROCEDURE DEVELOPMENT OF INSTRUMENT UTILIZATION OF INSTRUMENT PRESENTATION AND ANALYSIS OF DATA SCREENING TESTS A d d itio n S u b t r a c t i o n M u ltip lic a tio n D i v i s i o n DIAGNOSTIC T E S T S A d d itio n iii

8 iv Chapter Page S u b t r a c t i o n M u ltip lic a tio n I M u ltip lic a tio n I I D ivision I D ivision I I SUMMARY AND CONCLUSIONS SUMMARY CONCLUSIONS LITERATURE C IT E D APPENDICES APPENDIX A - Screening T ests and D iagnostic T ests APPENDIX B - Sequence of D iagnostic Tests APPENDIX C - Perm ission from Marvin E. C u rtis, Superintendent of L ivingston P arish School Board to conduct study VITA 107

9 LIST OF TABLES Table Page 1. Screening T est R esu lts in A ddition According to Frequency and Percentage of Scores in Each Performance Group Screening T est R esu lts in S u b tractio n According to Frequency and Percentages of Scores in Each Performance Group Screening T est R esults in M u ltip lic a tio n I According to Frequency and Percentages o f Scores in Each Performance G ro u p Screening T est R esults in M u ltip lic a tio n I I According to Frequency and Percentages of Scores in Each Performance Group Screening T est R esults in D ivision I According to Frequency and Percentages of Scores in Each Performance Group Screening T est R esults in D ivision I I According to Frequency and Percentages of Scores in Each Performance Group A nalysis of E rrors on A ddition T e s t s A nalysis of E rrors on S u b tractio n T ests A nalysis of E rro rs on M u ltip lic a tio n I T e s t s A nalysis o f E rrors on M u ltip lic a tio n I I T e s t s A nalysis of E rrors on D ivision I T e s t s A nalysis o f E rro rs on D ivision I I Tests v

10 LIST OF FIGURES Figure Page 1. D istrib u tio n of In c o rre c t Answers in A ddition According to Type of E x e r c i s e D is trib u tio n of In c o rre c t Answers in S u b tractio n According to Type o f E x e r c i s e D istrib u tio n of In c o rre c t Answers in M u ltip lic a tio n I According to Type of E x e r c i s e D is trib u tio n of In c o rre c t Ansers in M u ltip lic a tio n I I According to Type of E x e r c i s e D istrib u tio n of In c o rre c t Answers in D ivision I According to Type of E x e r c i s e D istrib u tio n of In c o rre c t Answers in D iv isio n I I According to Type o f E x e r c i s e vi

11 ABSTRACT This study had the follow ing as i t s purposes: (1) to determ ine the types and frequency of e rro rs committed by se le c te d elem entary stu d en ts in th e ir computations w ith whole numbers, (2) to devise a d iag n o stic instrum ent through which data could be c o lle c te d for the study, and (3) to determ ine i f s p e c ific types of e x ercises in each o p eratio n were e sp e c ia lly d i f f i c u l t. The study included each of the whole number o p e ra tio n s ad d i tio n, s u b tra c tio n, m u ltip lic a tio n, and d iv is io n. The study was lim ited to the six th -g rad e stu d en ts of the public schools in L ivingston P a rish, L ouisiana. E rror a n a ly sis was lim ited to exam ination of w ritte n work on a s e rie s of te s ts devised by the w rite r. Data were c o lle c te d from te s ts adm inistered to 690 stu d en ts during the l a s t month of the school sessio n. Each stu d en t was given the Screening T est in Whole Numbers to place him in an a p p ro p riate group fo r fu rth e r study. The screening t e s t had six su b -p arts which examined the stu d en ts in the four operations w ith whole numbers. Based on the re s u lts of the scores on each of the six su b -p arts of the Screening Test in Whole Numbers, each student was placed in one of th ree performance groups. Group One was composed of students who performed w ith 100 p ercen t accuracy. Group Two had stu d en ts who dem onstrated an understanding of the o p eratio n but ex h ib ited some evidence of d if f ic u l ty. Group Three included those students who could not accur a te ly complete any of the e x e rc ise s. Students who were placed in Group Two on any su b -p art of the vii

12 screening t e s t were given a more d e ta ile d d ia g n o stic te s t in the same viii o p eratio n. The d iag n o stic te s t s were scored, and each in c o rre c t exercise was analyzed in order to determ ine the e rro r which caused the in c o rre c t response. Each type of e rro r and i t s frequency was enumerated. The number of types of e rro rs fo r each group of te s t s was as follow s: Add itio n, 17; S u b tractio n, 20; M u ltip lic a tio n I, 14; M u ltip lic a tio n I I, 16; D ivision I, 15; and D ivision I I, 16. R elated types of e rro rs were combined in broad c ate g o rie s such as com binations, procedure, renaming, re la te d o p eratio n s, and m iscellaneous. A to ta l of 9172 e rro rs were analyzed. Based on the percentage of in c o rre c t e x ercises on the diagnost i c t e s t s, D ivision was the most d i f f i c u l t o p e ra tio n. S u b tractio n was the second most d i f f i c u l t w hile A ddition proved to be le a s t d i f f i c u l t. The follow ing conclusions seemed to be w arranted: 1. A nalysis of e rro rs on w ritte n work can be a p ro fita b le means of gath erin g data fo r planning group and in d iv id u al in s tru c tio n. 2. A s u b s ta n tia l number of e rro rs of undetermined o rig in emphasized a need fo r more thorough a n aly sis through in d iv id u al conference and work-aloud se ssio n s. 3. The v a ria tio n among the scores on the screening te s ts in d i cated th a t some fa c u ltie s of schools were more p ro fic ie n t in developing whole number com putational s k i l l s than o th ers although the same b asic program was u tiliz e d throughout the system. 4. Numerous types of e rro rs in d icated a need fo r in d iv id u a liz e d, d ia g n o stic a lly o rie n te d in s tru c tio n in a l l o p eratio n s. 5. The o p eratio n of d iv isio n had not advanced to a le v e l where a m ajo rity of the stu d en ts could fu n ctio n p r o f ic ie n tly.

13 6. The tr a n s itio n from the su b tra c tiv e method to the conventional method in d iv is io n had not been completed by many students and contrib u te d to the d i f f i c u l t i e s experienced by many who had made the tr a n s itio n. 7. Evidence showed th a t more emphasis should be placed on m astery of the b a sic combinations in each o p eratio n. 8. P rocedural e rro rs in whole number computations freq u en tly re fle c te d an inadequate understanding o f the number system.

14 Chapter 1 INTRODUCTION Whole number com putational s k ills is one of the major areas of the elem entary school mathematics curriculum. M astery of the fundamental processes is no panacea, but Buswell (1926) ably pointed out i t s sig n ific a n c e when he s ta te d : "While a m astery of the fundamental does not guarantee a b i l i t y to solve problems, a lack of knowledge of the fundamentals very se rio u sly in te fe re s w ith problem -solving" (1926:195). W ithin the p ast decade many innovations in elem entary mathematics have been applied in classroom s itu a tio n s. D espite the emphasis on m athem atics, the w rite r had o ften heard the com plaint th a t many students were g e n erally in e f f ic ie n t in whole number com putations. Some of h is colleagues were prone to place the blame on the "new math" program. Brownell (1956) suggested a so lu tio n fo r the problem re la te d to balance between meaning and s k i l l. He sta te d th a t competence in computa tio n must be given i t s r ig h tf u l place and th a t a rith m etic meanings w ill c o n trib u te to g re a te r com putational s k i l l s. P ra c tic e should be h eld to a minimum u n til the student has progressed to meaningful h a b itu atio n ; then s u f f ic ie n t p ra c tic e should be provided to assure m astery of the s k i l l s. THE PROBLEM Statem ent of the Problem The p rin c ip a l in v e stig a tio n sought answers to two q u estio n s.

15 What types of e rro rs do stu d en ts make in com putation of whole numbers? What is the frequency of each type of e rro r? A th ird q u estio n was concerned w ith the sequence of the program. Are there s p e c ific areas w ith in the development of each o p eratio n which tend to be e sp e c ia lly d if f ic u l t? D elim itation - j The study was lim ited to the p u b lic school system of L ivingston P a rish, L ouisiana. The e n tir e six th -g rad e enrollm ent of the system was included. The stu d en ts of two schools p a rtic ip a te d in th a t phase of the study which developed the instrum ent. The rem ainder of the stu d en ts was involved in the in v e stig a tio n of the major problem s. The study a lso lim ited i t s e l f to the a n a ly sis of e rro rs evident in stu d e n ts' w ritte n work on d iag n o stic te s ts adm inistered by the in v estig a to r. D e fin itio n of Terms Computational e r r o r. Any procedure th a t co n trib u te d to an in c o rre c t answer. Computational competence. The a b i l i t y to work w ithout e rro r a l l. ex ercises on the screening t e s t w ith in a given o p e ra tio n. Excessive com putational r e ta r d a tio n. The i n a b ility to work c o rre c tly any e x ercises on the screening t e s t w ith in a given o p eratio n. Importance of Study Teaching, by i t s very n a tu re, must be d ia g n o stic to be e ffe c tiv e.

16 This is e sp e c ia lly tru e in the teaching of s k i l l s in elem entary mathem atics. The study should provide u sefu l inform ation fo r the classroom teacher in h is e f f o r ts to lo c a liz e s p e c ific d i f f i c u l t i e s. Inform ation re la te d to s p e c ific e rro rs common to many students is v aluable in p lan ning fo r the c lass as a whole and fo r groups w ith in the c la s s. Awareness of le ss common type e rro rs is im portant in analyzing a l l work and in in d iv id u a liz in g in s tru c tio n. The in v e s tig a to r f e l t th a t those teach ers who took an a ctiv e p a rt in the study became more d ia g n o stic a lly o rie n te d as a r e s u lt of th e ir ro le. The data c o lle c te d in the study presented a comprehensive re p o rt on the s ta tu s of th is p a r tic u la r phase of the elem entary mathematics program in L ivingston P a rish. Supervisors of in s tru c tio n, p rin c ip a ls, and curriculum study groups should be able to u t i l i z e the findings in form ulating th e ir plans fo r improving the elem entary mathematics program. S tudies by Buswell (1926) and R oberts (1968) rep o rted some of the b asic inform ation sought in the p resen t study. They d iffe re d w ith the p resen t study in the method of data c o lle c tio n and in grade le v e ls stu d ied. The p resen t study, w ith i t s focus on a p a r tic u la r system, w ith i t s own instrum ent fo r d iag n o sis, and w ith stu d en ts who had been taught by one of the modern mathematics programs, combined some of the stre n g th s of the former s tu d ie s. A b e tte r understanding of the stu d en t, h is problems in com putational s k i l l s, and the com putational phase of the mathematics program was gained.

17 Chapter 2 REVIEW OF RELATED LITERATURE A review of re la te d l i t e r a t u r e revealed th a t in te r e s t in diagnost i c stu d ie s of e rro r a n aly sis was evident in the second decade of the p resen t cen tu ry. Buswell and Judd (1925) summarized a rith m e tic stu d ies and l is te d th irty -o n e d iag n o stic stu d ie s in a rith m etic from 1909 to Of the th irty -o n e s tu d ie s, twenty were devoted to the a n aly sis of e rro rs. Not a l l of the twenty were concerned w ith whole number com putations, but sev eral s ig n ific a n t ones were included. Some of the stu d ie s were of only one o p eratio n although sev eral included a l l four o p eratio n s. I n te r e s t in study of e rro rs continued through the 1930's but a t a steady d e c lin e. In volumes devoted to mathematics and scien ce, the Review of E ducational Research (1931, 1933, 1935, 1937, 1941, 1942) had sp e c ia l sec tio n s devoted to stu d ies of e rr o r s. A bibliography of d o cto ral d is s e rta tio n s by Summers (1961) and Summers and Stockl (1961) r e ite r a te d the fa c t, th a t stu d ies in the area were lim ited. Although the number of in v e stig a tio n s had declined by the 1940's, th ere was y et much in te r e s t in the a re a. A tte n tio n was placed more on r e medial work to overcome the d i f f i c u l t i e s, than on stu d ie s of e rro rs. An example of the emphasis was an a r t i c l e by Holland (1942) which was not based on a p a r tic u la r study b u t u tiliz e d the findings of previous ones. One of the e a rly stu d ie s of e rro rs was made by Phelps (1913). His study was lim ite d to a d d itio n and emphasized speed and accuracy in working

18 w ith the b asic com binations. Two hundred seventy eig h t-g rad e students were examined. Each stu d en t was given tw en ty -fiv e one-minute combination te s ts spread over a fiv e-d ay p erio d. The in v e stig a tio n undertook to answer four q u estio n s: How o fte n are e rro rs made on com binations? What percentage of the group make them? How do the com binations rank in order of d if f ic u lty? What kinds of m istakes are made? Some of the findings of P h elp s' study were re le v a n t to the study being rep o rted. I t was found th a t c e r ta in "type" e rro rs were p rev alen t throughout a l l the schools. One such e rro r was m u ltip lic a tio n which occurred mainly in the ones and in some of the doubles, such as 3-3 = 9. Another was in such e x ercises as = 18. Phelps disagreed w ith C ourtis on two p o in ts as sta te d in h is " In stru c tio n s fo r Making and Recording Scores" in giving the C ourtis Standard T e s t. Phelps found th a t th ere was not a close p o s itiv e c o rre la tio n between speed and accuracy. Secondly, Phelps found few est e rro rs in the zero com binations. Smith (1916) stu d ied v a ria tio n s in a rith m e tic among elem entary stu d en ts in grades th re e through eig h t in the Elementary School of the U n iv ersity of Chicago. A rithm etic t e s t s in the Cleveland Survey were given, and each ch ild was asked to work the ex ercise s o ra lly so th a t s p e c ific e rro rs could be lo cated. The e rro r types were categ o rized under broad headings including in c o rre c t com bination responses, in a b i l i t y to c arry on two or more p ro cesses sim ultaneously, one process in te r f e r r in g w ith another, and lack of knowledge of the p ro cess. Carrying and copying fig u re s were o th er

19 6 sources of e rro rs. A second purpose of the study was to determ ine the b est means of co rre c tin g the e rro rs a f t e r they were lo c ated. The conclusion reached was th a t the b est method was c la ss d r i l l supplemented by in d iv id u al a ssista n c e on the p o in t of weakness as diagnosed by the t e s t. D iagnostic te s tin g in a rith m e tic was fu r th e r developed by Woody (1916) w ith the Woody A rithm etic S c a le s. Although the scales were designed to measure achievement, the author expressed the b e lie f th a t p erhaps the d ia g n o stic power of the t e s t was i t s most v aluable a s s e t. L a te r, Woody (1920) analyzed 1500 e rro rs in d iv isio n made by seventh and eighth-grade stu d en ts. He found n in e te en ty p ic a l types of e rro rs and gave the percentages fo r each. E rro r a n aly sis in s u b tra c tio n, m u ltip lic a tio n, and d iv isio n was undertaken by G ist (1917). He examined 812 papers of students who took th e C ourtis T ests in the S e a ttle, W ashington school system. The students were in grades four through e ig h t. G ist did not attem pt to c ate g o riz e the e rro rs in to numerous types. He was very conservative in th a t area w ith only th re e types in m u ltip lic a tio n, s ix in s u b tra c tio n, and nine in d iv isio n. He a lso found consid e ra b le constancy of e rro rs in the e x ercises on the same paper. A study conducted by Myers (1924) u tiliz e d stu d en ts of education during th e ir p ra c tic e teach in g. Each stu d en t teach er worked w ith one or sev e ra l c h ild re n. A record was kept of o ra l and w ritte n e rro rs in whole number computations made by the c h ild re n over a period of sev eral weeks. The w rite r was not as concerned w ith c ate g o riz in g e rro rs and l i s t i n g th e ir frequency as he was w ith how the c h ild responded to the same example on o th er occasions. The c h ild re n who p a rtic ip a te d in the study

20 were in grades two through six. Several conclusions were drawn by Myers from h is study. He f e l t th a t a m istake was a very serio u s m atter because one cannot be sure th a t the same m istake w ill not be repeated in exactly the same manner. He also concluded th a t an in c o rre c t answer is as d e fin ite as a c o rre c t one. This would dim inish the ro le of what is o ften lab eled "ca re le ss m istakes. The author a lso concluded th a t there were s p e c ific e rro rs made in the method of teaching the com putational s k i l l s. He f e l t th a t reasoning was a v i t a l p a rt of developing the b asic concepts, but th a t a ch ild should n o t be required to reason out every basic f a c t which is u tiliz e d in h is com putations. There should always be m astery of p rev io u sly p re sented com binations before new ones are undertaken. In lig h t of the conclusion th a t an in c o rre c t answer is as p o s i tiv e a response as a c o rre c t one, the author advocated a re v ised system of marking which would deduct the number of in c o rre c t from the number of c o rre c t responses. The system would discourage the form ulation of in c o rre c t responses. One of the most comprehensive stu d ie s of e rro rs to i t s date was c a rrie d out by Osburn (1924) in W isconsin in The study was d i rected by Osburn w ith the aid of school su p erin ten d en ts, su p erv iso rs, and te a c h e rs. The Woody-Theisen P a r a lle l Tests in A rithm etic were given to stu d en ts in grades three through s ix, and answers were l i s t e d and analyzed. The work included 21,548 wrong answers in the four fundamental o p e ra tio n s. The number of wrong answers in each operation was as follow s a d d itio n 6107, s u b tra c tio n 6459, m u ltip lic a tio n 4551, and d iv is io n 4431.

21 Osburn summarized the ty p ic a l sources of e r r o r of the four o p eratio n s as follow s (Osburn 1924, 38-39) : 1. Trouble w ith zero com binations in each of the four p ro c e sse s. 2. F a ilu re to deal w ith number fa c ts when presen ted in the equation forms. 3. D if f ic u ltie s in column (higher-decade) a d d itio n. 4. Trouble in s u b tra c tio n when a d ig it in the subtrahend is g re a te r than the d i g i t ju s t above in the minuend. 5. In te rfe re n c e between what is required and what is already known (harm ful t r a n s f e r ). 6. Ignorance of the com binations in a ll of the processes. 7. E stim ating the q u o tie n t in long d iv isio n. 8. C arrying in a d d itio n. 9. C arrying in m u ltip lic a tio n. 10. Borrowing. 11. Copying. 12. B ringing down in long d iv is io n. 13. F a ilu re to complete the e x ercise even when the time is s u f f ic ie n t. In an e f f o r t to gain a b e tte r understanding of e rro rs committed by stu d en ts, Osburn devised a d ia g n o stic t e s t in d iv is io n which he adm in istered to 500 stu d e n ts. Seven hundred six teen e rro rs were analyzed. Of the to ta l, 39 p ercen t were due to e rro rs in estim atin g the q u o tie n t. E rro rs in m u ltip lic a tio n and su b tra c tio n combinations accounted fo r 21 p ercen t each, or a t o t a l of 42 p ercen t of the e rro rs. The remaining

22 19 percent were due to e rro rs in carry in g, borrowing, copying, bringing down, and f a ilu r e to complete the e x e rc ise, A work which may be considered a c la s s ic in d iag n o stic stu d ie s was conducted by Buswell and John (1926). That comprehensive in v e stig a tio n included a study o f eye-movements made in column ad d itio n, time analyses of the four fundamental o p e ra tio n s, and d e ta ile d diagnoses of in d iv id u al cases. The diagnosis of in d iv id u al work in the fo u r fundamental operatio n s proved to be e s p e c ia lly s ig n if ic a n t. One of the prim ary purposes o f th a t phase of the study was to compile an ex ten siv e c atalo g of e rro rs and fa u lty h a b its e x h ib ite d by stu d en ts in th e ir work w ith whole number com putations. In d iv id u al work sessio n s were h eld w ith the in v e s tig a to r 's questioning the stu d en t where c l a r i f i c a t i o n of the method of work was needed. Buswell and John devised th e ir own instrum ent fo r d iag n o sis. The re v ise d form had each o p eratio n broken down in to v ario u s step s of in c re a s in g d if f ic u l ty. Each of the step s had two e x e rc is e s. D iagnostic te s tin g was conducted on 414 stu d en ts in a d d itio n, 373 s u b tra c tio n, 329 in m u ltip lic a tio n, and 300 in d iv is io n. The grade le v e ls included grades th re e through six. The in v e s tig a tio n re s u lte d in the c ata lo g in g of h a b its which co n trib u ted toward in c o rre c t or in e f f ic ie n t work. The number of h a b its ex h ib ited in each o p eratio n were as follow s: a d d itio n, th ir ty - th r e e ; su b tra c tio n, tw enty-seven; m u ltip lic a tio n, fo rty -o n e ; and d iv is io n, fo rty - one. One had to co n sid er the f a c t th a t not a l l h a b its were p resen t in each of the four d if f e r e n t grades included in the study.

23 10 An exam ination of the r e s u lts of the six th -g ra d e stu d en ts r e vealed sev eral p e rtin e n t f a c ts. With the exception of m u ltip lic a tio n, the most frequent h a b it in each o p eratio n was e rro r in com binations. Although th ere were numerous h a b its in each o p eratio n as c ite d above, 90 p ercent of the t o t a l number of h a b its ex h ib ited were accounted fo r by approxim ately 50 p ercen t of the h a b its. The work by Buswell and John was a most s ig n ific a n t c o n trib u tio n in d iag n o stic stu d ie s in a rith m e tic. One of the r e p o r t's most worthy c o n trib u tio n s was the il l u s t r a t i o n s of h a b its of work as discovered through the in v e stig a tio n (Buswell and John, ). Anyone concerned w ith d iag n o stic work w ith the fundamental o p eratio n s could p r o f it by a study of the i l l u s t r a t i o n s of h a b it. Lazar (1928) in v e stig a te d a rith m e tic d i f f i c u l t i e s in the fundament a l o p e ra tio n s. The study group was composed of fo r ty -fiv e six th -g rad e stu d e n ts. Hie in v e s tig a to r used the Monroe D iagnostic T ests to lo cate p a r tic u la r d e fic ie n c ie s. In d iv id u al s tu d e n t's e rro rs were stu d ied by exam ination of t e s t papers, observation of stu d en ts as they worked, and by o ra l exam ination. An inventory of e rro rs fo r each o p eratio n was compiled. Hie re p o rt did not give s p e c ific numbers and percentages fo r each type of e r r o r, but they were l i s t e d in the order of frequency. The e rro rs in a d d itio n were categ o rized in to nine groups. in the com binations were lis te d as the most frequent d i f f i c u l t i e s. E rro rs C arrying d i f f i c u l t i e s, m isreading sig n s, and m isreading or f a ilin g to read word d is tin c tio n s were l i s t e d as freq u en t sources of e rro rs. Less f r e quent e rro r types included irr e g u la r methods of adding, f a ilu r e to place

24 addends In proper columns, m isreading the crutch as p a rt of the answer, carelessn ess in w ritin g the answer, and ille g ib le fig u re s. 11 S u b tractio n e rro rs were lis te d under seven b asic ty p es. Most frequent were e rro rs in com binations. L isted as very frequent were borrowing e rro rs. Frequent e rro rs involved m isreading of signs and word d ire c tio n s. Less frequent sources of e rro rs were confusing minuend and subtrahend, m isreading the crutch as p a rt of the answer, and e rro rs due to ille g ib le fig u re s. Seven types of e rro rs were a lso l is te d fo r m u ltip lic a tio n. The same p a tte rn p re v a iled w ith e rro rs in combination and carrying being the most freq u en t. D if f ic u ltie s w ith p a r tia l products very freq u e n tly contrib u te d to the number of e r r o r s. were a lso areas of d i f f i c u l t i e s. M isreading signs and word d ire c tio n s C arelessness and ille g ib le fig u res completed the l i s t of e rro rs in m u ltip lic a tio n. L a z a r's study included both sh o rt d iv isio n and long d iv isio n. Only the e rro rs in long d iv isio n were summarized due to the n atu re of the p resen t study. E rro rs in com binations, d iv is io n, m u ltip lic a tio n, and s u b tra c tio n proved to be the g re a te s t sources of e rro rs in long d iv isio n. D iffic u lty in e stim atin g the q u o tie n t fig u res was another frequent type of e r r o r. Various types of zero d i f f i c u l t i e s were resp o n sib le fo r a larg e percentage of e r r o r s. C areless placement of q u o tien t fig u res caused many d i f f i c u l t i e s. D if f ic u ltie s w ith bringing down, f in a l r e m ainder, and placement of products were o th er sources of e rro rs. Clemens and Neubauer (1928) made a study of e rro rs in m u ltip lic a tio n. Two thousand stu d en ts in grades four through eig h t p a rtic ip a te d. The in v e stig a to rs devised th e ir own te s ts which were coded to id e n tify s p e c ific types of s itu a tio n s in the process of m u ltip lic a tio n. The

25 12 study was a lso concerned w ith rem edial procedures and r e te s tin g. From th e ir fin d in g s, the authors concluded th a t the causes of f a ilu r e in m u ltip lic a tio n could be c la s s if ie d under seven general causes. The sin g le g re a te s t c o n trib u to r toward f a ilu r e was the lack of m astery of h igher decade a d d itio n. From 45 p ercen t to 55 percent of the e rro rs were a ttrib u te d to th a t cause. C arelessn ess, according to the in v e s tig a to rs, accounted fo r from 15 p ercent to 25 percent of the f a ilu r e s. Lack of m astery of the m u ltip lic a tio n combinations made up from 5 p e r cent to 15 p ercent of a l l e rro rs. Five o th er types of e rro rs made up the o th er causes. They were as follow s: om ission of carry in g fig u re s, confusion of number, tra n s p o sitio n of fig u res in w ritin g product, additio n in stead of m u ltip lic a tio n, and d isreg ard of the zero e n tir e ly when in the te n s ' p lace. A secondary phase of a study on methods of teaching d iv isio n by John (1930) was concerned w ith the type of e rro rs made by fifth -g ra d e stu d en ts in th e ir i n i t i a l encounter w ith d iv is io n. The conclusions could npt be gen eralized due to the sm all size of the sample. The work of ten stu d en ts was analyzed. Two groups of te s ts were used; one group contained o n e -d ig it d iv iso rs and the o th er tw o -d ig it d iv is o rs. One of the s ig n if ic a n t c o n trib u tio n s of the study, in r e la tio n to the p resen t in v e s tig a tio n, was the l i s t i n g of the vario u s types of e r r o r s. The group taught only by the lo n g -d iv isio n method had a to ta l of six te e n types of e rro rs in d iv isio n w ith one-place d iv is o rs. Of the six te e n types, only fiv e types c o n trib u ted 5 p ercen t or more to the to ta l number of e rr o r s. E rro rs in d iv isio n combinations made up approxi-

26 m ately o n e-th ird o f the t o ta l number of e rro rs. E rro rs in m u ltip lic a 13 tio n co n trib u ted approxim ately 20 p e rc en t. Almost 15 percent were due to su b tra c tio n e r r o r s. Problems w ith remainders and zeros in the q u o tie n t were the major causes of e rro rs. There were eighteen types of e rro rs made by the same group when they worked w ith two and th r e e - d ig it d iv is o rs. Only four types contrib u te d 5 percent or more to the t o t a l number of e rro rs. M u ltip lic a tio n e rro rs were the g re a te s t c o n trib u to rs and accounted fo r over 40 percen t of the t o t a l. D ivision combination e rro rs made up approxim ately 20 p ercent w ith su b tra c tio n e rro rs near 10 p ercen t. The use of the wrong number of d ig its in the dividend to estim ate the q u otient was the o th e r major cause of e rro r. Burge (1932) conducted a study of e rro rs in s tu d e n ts' work in m u ltip lic a tio n. The work involved over 2000 students in grades fo u r through s ix. Two methods were u tiliz e d in determ ining s p e c ific e rro rs and q u estio n ab le work h a b its in working w ith m u ltip lic a tio n. The stu d e n ts' t e s t s were stu d ied, and interview s were held w ith stu d en ts whose work needed more in te n siv e a n a ly s is. The m ajo rity of the work was conducted on the interview b a s is. The re s u lts of the in v e stig a tio n revealed an exceedingly large number of e rro rs and questionable h a b its. A to ta l of s ix ty -e ig h t e rro rs and h a b its were l i s t e d. Of the t o t a l, 58 percent were e rro rs and the rem aining 42 percent questionable h a b its. The study in d icated th e exte n t to which one may c arry such an a n a ly s is. The findings were c la s s if ie d under seven major headings and presented in ta b u la r form. M ajor areas of e rro rs were com binations,

27 14 carry in g, and zero d i f f i c u l t i e s. In an unpublished study re p o rte d by W ilson (1940), E arly in v e stig a te d sources of d i f f i c u l t i e s in long d iv isio n. The study in volved 236 students in grades fiv e through e ig h t. The e rro rs were c la s s if ie d in to seven types w ith s u b tra c tio n e rro rs and e rro rs in estim ating q u o tie n t accounting fo r 50 p ercen t of the t o t a l. Other stu d ies have shown th a t e stim atio n of the q u o tie n t was one of the major problems. One such study rep o rted by Brueckner and Melbye (1940) was re la te d to d iv is io n w ith tw o-figure d iv is o rs. Three general types of d iv is io n examples were categorized, and th e ir re la tiv e d if f ic u l ty was determined according to the mental age of the stu d en ts. Students had le a s t d if f ic u l ty w ith examples in which the apparent quotie n t was the tru e q u o tie n t, most d if f ic u l ty w ith examples in which the apparent q u o tie n t was not the tru e q u o tie n t, w ith the d i f f i c u l t y of examples w ith zeros in the q u o tie n t f a llin g between the two. According to the fin d in g s th e re was a g re a te r degree o f re la tiv e d if f ic u l ty between examples w herein th e s ig n ific a n t fa c to r was th e tru th fu ln ess of the apparent q u o tie n t than between examples w ith one-figure and tw o-figure d iv is o rs. One of the more recen t stu d ie s in diagnosis of com putational e rro rs was conducted by Roberts (1968). His method involved item - a n aly sis of the th ird -g ra d e com putation sectio n of a Stanford Achievement Test given 766 th ird -g ra d e s tu d e n ts. Roberts did not attem pt to c la s s if y e rro rs se p a ra te ly w ith in the four o p e ra tio n s. He divided the stu d en ts in to four q u a r tile s according to g ra d e-lev el score and randomly sele c ted th ir ty - f iv e to f o r ty papers from each group. The e rro rs were divided in to fo u r c ate g o rie s

28 wrong o p eratio n, obvious com putational e r ro r, d e fectio n algorithm, and 15 random response. Obviously th e re were a v a r ie ty of s p e c ific e rro r types w ith in each of the broad c a te g o rie s. The an aly sis yielded sev eral in te r e s tin g r e s u lts. P urely comp u ta tio n a l e rro rs were among the low est, 18 p ercent of the t o t a l, and were r e la tiv e ly c o n siste n t throughout the v ario u s a b i l i t y ranges. Wrong o p eratio n s accounted fo r the same p ercen t of e rro rs fo r the t o t a l but was g re a te r in the lower a b i l i t y groups. Random responses had a sim ila r p a tte r n of a g re a te r percentage as a b i l i t y decreased and accounted fo r 28 p ercen t of the t o t a l number of e rro rs. The g re a te s t c o n trib u to r to f a ilu r e was d e fectiv e algorithm. T h irty -e ig h t percent of the to ta l number of e rro rs were due to th a t cause. Smith (1969) studied the e rro rs of th ir d and fo u rth -g rad e s tu dents in su b tra c tio n of whole numbers. He found a p o s itiv e c o rre la tio n e x iste d between performance on su b tra c tio n and understanding of place v alu e in the decimal system. The e rro rs which stu d en ts committed most fre q u e n tly were re la te d to b a sic su b tra c tio n com binations, su b tra ctin g the minuend from the subtrahend, and w ritin g zero as an answer in stead of borrowing. With regard to the type of instrum ent used in d ia g n o stic s tu d ie s, a u th o ritie s g e n erally agree th a t instrum ents designed e s p e c ia lly fo r d ia g n o stic work are su p erio r to inform al sampling of the s tu d e n ts' work and achievement t e s t s. In re feren ce to h is t e s t s, Brueckner (1931) f e l t th a t th e ir design enabled a teach er to lo cate the s p e c ific area of d if f i c u l t y in a given process and a lso provided a b a sis fo r in d iv id u al a n a ly sis in order to discover the source of e rro r.

29 16 There are two major uses of a n a ly tic a l te s tin g. One use, according to Brueckner (1935), is c la s s d iag n o sis. When the process under exam ination is broken down in to i t s c o n stitu e n t elem ents and each elem ent is te s te d, i t is p o ssib le to lo cate s p e c ific areas of d if f ic u lty fo r the c la s s in g eneral. The second use is to analyze in d iv id u al work to d e te r mine the n atu re of the e rro r made. Brueckner l i s t e d four general methods fo r a n aly sis of e rro rs ; one method was the a n a ly sis of w ritte n work. He pointed out one major weakness of the procedure which was the in a b ility to account fo r a l l e rro rs w ithout more inform ation on the thought process involved. Even w ith th a t inherent weakness, Brueckner f e l t th a t as a f i r s t step in d iag n o sis, a n aly sis of w ritte n work should be used. The study by Buswell and John was based on data gathered by d iag n o stic te s ts devised by the au th o rs. The t e s t s divided each operatio n in to v ario u s p h ases, each d if f e r e n t from the preceding one in some s ig n if ic a n t manner. Each stage of development w ith in the o p eratio n s was te s te d by two e x e rc ise s. In summary, the s c ie n tif ic movement in education had a s i g n i f i cant impact upon d ia g n o stic stu d ie s in a rith m e tic. The study of e rro rs in whole number com putations was only a small p o rtio n of a much broader movement. The scope of the stu d ie s v aried considerably. Some in v e stig a tio n s were concerned w ith only one o p eratio n, w hile others involved a l l fo u r. Each o p eratio n was the exclusive o b ject of a study. The study of e rro rs in d iv isio n seemed to have claimed the most a tte n tio n. There was a lso a wide range in the number of types of e rro rs l i s t e d by the v ario u s in v e s tig a to rs. Of the stu d ie s reviewed, a m ajo rity

30 17 of the authors grouped the e rro rs Into f a i r l y broad c a te g o rie s. A le s s e r number of in v e s tig a to rs attem pted to c ata lo g as many d iff e r e n t types of e rro rs as p o ssib le. B urge's in v e s tig a tio n of m u ltip lic a tio n, in which s ix ty -e ig h t e rro rs and q u estio n ab le h a b its were l i s t e d, was an example of such an approach. Where s tu d ie s l is te d many types of e rro rs, i t g en erally was tru e th a t a larg e m a jo rity of th e to ta l number of e rro rs was caused by a small percentage of the to ta l number of types of e rr o r s. One cannot assume th a t types of e rro rs which account for only a small percentage of the number of e rro rs are unim portant. Their cum ulative e ffe c t can have a g reat impact and is a source o f r e a l concern for the in d iv id u als who have th a t p a r tic u la r d if f ic u l ty. D iagnostic methods g e n erally f e l l in to one of two procedures. The stu d ie s reviewed were approxim ately evenly d iv id ed between the methods of analyzing w ritte n t e s t r e s u lts by exam ination of the papers and by analyzing t e s t r e s u lts by having the stu d en t work aloud. Each method had i t s own stre n g th s and weaknesses. There was l i t t l e doubt th a t the o ra l method was su p e rio r in d elv in g in to the thought processes involved in the com putational e x e rc ise s. The method required a co n sid erable amount of time and u s u a lly re s u lte d in a sm all sample being stu d ied or the involvement of many in v e s tig a to r s. The exam ination of w ritte n work was not as accurate in assessin g the cause of e rro r, b u t many of the major causes were determ ined. The a n aly sis o f w ritte n work made i t p o ssib le to study la rg e r samples of work.

31 C hapter 3 METHOD OF PROCEDURE Two f a c to r s, a c c e s s a b ility and siz e of sample, were instrum ental in th e s e le c tio n of the su b jects fo r the study. The L ivingston P arish School system met both requirem ents. There were approxim ately 875 six th -g ra d e stu d en ts e n ro lle d in the p u b lic schools of L ivingston P a ris h, L ouisiana. As an employee of the system, the author was able to work w ith in the schools and have the complete cooperation of the p ro fe ssio n a l s ta f f s of the v ario u s schools. The i n i t i a l step was to secure perm ission from the P arish Superin ten d en t of Schools to conduct research in the schools of the system. A second phase o f the p re p a ra tio n was to co n tact each school p rin c ip a l p e rso n ally and fa m ilia riz e him w ith the stu d y. While in each school, the author was perm itted to confer w ith the teachers whose stu d en ts were included in the study. This was done to involve the teach er in the research and to conduct the te s tin g in as n a tu ra l a s e ttin g as poss ib le. DEVELOPMENT OF INSTRUMENT The in v e s tig a to r determ ined th a t i t would be of value to design h is own instrum ents fo r g ath erin g data in the study. The school system in which the research was conducted had used one b a sic mathematics te x t book sin ce the p resen t six th -g ra d e stu d en ts were in the f i r s t grade. 18

32 19 I t was decided th a t the te s ts would be based on the whole number opera tio n s as presented in the b a sic program. The screening t e s t and d iag n o stic te s ts were designed to p a r a lle l the sequence of the b a sic program. A prelim inary ste p was to examine c lo se ly the textbooks in order to determ ine the developmental sequence. Textbooks for grades fo u r, fiv e, and six were stu d ied r e la tiv e to the p re sen ta tio n of whole number o p eratio n s. The vario u s lev els w ith in each of the four operations were lis te d in order of p re s e n ta tio n. From the inform ation gathered from a study of the textbooks, each o p eratio n was divided in to se q u e n tia l step s of development. (See Appendix B.) The sequence was su b jec tiv e on the p a rt of the author although i t c lo se ly p a ra lle le d the textbook p re s e n ta tio n. An e f f o r t was made to have each step d if f e r from the preceeding one in a s ig n ific a n t manner. The f in a l r e s u lt of the sequence study yield ed a comprehensive a n aly sis of the development stag es of whole number com putations. Additio n was divided in to twelve d is tin c t sta g e s. There were eig h t stages in the development of s u b tra c tio n. M u ltip lic a tio n was more complex and had to be divided in to th e two major c ate g o rie s of o n e -d ig it and two- d ig it m u ltip lie r s. The o n e -d ig it m u ltip lie rs were developed in six steps and the tw o -d ig it m u ltip lie r s in seven ste p s. In a sim ila r manner d iv is io n had two major c a te g o rie s of o n e -d ig it d iv iso rs and tw o -d ig it d iv is o rs. Each was developed in six ste p s. (Appendix B is a complete l i s t i n g of the se q u e n tia l development of the four o p e ra tio n s.) The next task was to develop the instrum ents needed fo r the study. Three types of te s t s were needed: a screening t e s t, d iag n o stic te s t s, and b a sic fa c ts t e s t s. The screening t e s t was composed of a t le a s t one

33 ex ercise from each of the v ario u s stages o f development w ith in the four 20 operations on whole numbers. The t e s t was a to ta l of fifty -tw o exerc is e s. The purpose of the screen in g te s t was to determ ine who needed to take s p e c ific follow -up d iag n o stic te s t s. The d iag n o stic t e s t s were more comprehensive than the screening t e s t. There were th re e e x ercises fo r each stage of the development in each of the four o p e ra tio n s. Thus, A ddition had t h i r ty - s ix e x e rc ise s, S u b tractio n had tw enty-four, M ultip lic a tio n I had eig h teen, M u ltip lic a tio n I I had tw enty-one, and D ivision I and I I had eighteen ex ercises each. were made up of a l l f a c ts w ith in each o p eratio n. The b a s ic -f a c ts te s ts A tape recording was made to pace the stu d en ts taking the fa c ts te s t s. The tape c a lle d out each fa c t and the stu d en ts recorded the answers as they followed on th e ir copy of the t e s t. To t e s t the in stru m en ts, the examiner se le c te d two schools w ith in the p a rish which c lo se ly p a ra lle le d the g eneral school population of the system. Southside Elementary had a stu d en t body which was re p re se n ta tiv e of both the small town and the r u r a l p o pulation. French Settlem ent was ty p ic a l of the French influenced southern p o rtio n of the p a ris h. school had a r a c ia l com position sim ila r to the p a ris h as a whole. Each The two schools had a six th -g rad e enrollm ent of 105 s tu d e n ts. The te s tin g required two d iff e r e n t ad m in istra tio n s. Complete d ata were c o lle c te d on only n in e ty -th re e stu d e n ts. A good d iag n o stic instrum ent must be v a lid, r e lia b le, and u sab le. The n atu re of the t e s t insured content v a lid ity. The c o rre la tio n between the screening t e s t and corresponding d ia g n o stic te s ts, which was a form of concurrent v a lid ity, was determ ined by th e Pearson product- moment c o e ffic ie n t. The c o e ffic ie n ts of c o rr e la tio n were as follow s:

34 21 A ddition.59, S u b tractio n.74, M u ltip lic a tio n I.80, M u ltip lic a tio n I I.78, D ivision I.77, and D ivision I I.80. R e lia b ility c o e f fic ie n ts o f the d iag n o stic te s ts were determined by Kuder-Richardson formulas to estim ate te s t r e l i a b i l i t y. The r e s u lts were the follow ing: A ddition.88, S u b tractio n.92, M u ltip lic a tio n I.93, M u ltip lic a tio n I I.93, D ivision I.94, and D ivision I I.92. The c r i t e r i a fo r judging the u s a b ility of the instrum ents were length o f time fo r a d m in istratio n and c o n trib u tio n to the study. Two a lte r a tio n s were made as a r e s u lt of the u s a b ility study o f the p re lim inary a d m in istra tio n. The timed te s t on the b a sic fa c ts was d eleted as a p a rt of the te stin g program. I t was determined th a t i t did not make a d i s t i n c t c o n trib u tio n to the program under in v e stig a tio n. I t was found th a t some stu d en ts who scored poorly on the timed fa c ts did very w ell on the screening and d ia g n o stic te s t which were not tim ed. A second change was in the number of d iv isio n ex ercises in each stag e of the d ia g n o stic te s ts. In order to reduce fatig u e and keep w ithin single c la ss periods each was lim ited to f i f t y m inutes. In order to sta y w ithin the time lim its i t was necessary to reduce the number of examples in each stag e of d iv isio n to two. UTILIZATION OF INSTRUMENT The f i r s t p a rt o f the data c o lle c tio n was to adm inister the Screening T est in Whole Number Computations to a l l six th -g rad e stu d en ts. The number of students in th is phase of te s tin g was 690. In order to discourage in e f f ic ie n t methods, the stu d en ts were told th a t the time req u ired by each to complete h is work would be recorded on h is paper.

35 Emphasis was placed on the f a c t th a t the t e s t was n o t a speed t e s t, but the stu d en ts were encouraged to work a t a ra te which insured both speed 22 and accuracy. The p relim in ary te s tin g o f the instrum ent had determined th a t f i f t y minutes was s u f f ic ie n t time fo r most stu d en ts who were capable of perform ing the o p e ra tio n s. A fte r f i f t y m inutes, time was c a lle d by the examiner. The te s tin g was c arrie d out in as normal a s itu a tio n as p o ssib le. The re g u la r classroom teacher was p resen t and a s s is te d the examiner. The t e s t s were adm inistered in the stu d e n ts' own classroom ; th e re was no combining of c la sse s during the te s tin g period. A ll work was done on the t e s t i t s e l f w ith no scratch paper allowed. The screening te s ts were scored to determine which follow -up te s ts were needed by each stu d en t. Each of the six su b -p a rts of the t e s t was considered se p a ra te ly ; the o v e ra ll screening te s t score was of no s ig n ific a n c e in the study. Several students may have had the same to ta l score, but some may have had a l l e rro rs in one o p eratio n whereas others may have had th e ir e rro rs d is trib u te d throughout a l l o p eratio n s. Based on h is score fo r each sub-part each stu d en t was placed in one of three c a te g o rie s. One category c o n siste d of a l l students who displayed comp u ta tio n a l competence. rem ainder of the study. Those stu d en ts did not p a r tic ip a te in the A second group was made up of those students who ex h ib ited excessive com putational re ta rd a tio n. Normally, such stu d en ts required in ten siv e in d iv id u a l counseling in order to determ ine th e ir p a r tic u la r problem s. For th a t reason they were not w ell su ited fo r the study; th e re fo re, they were not given the advanced d iag n o stic te s t s. The th ird group was the remainder of the stu d e n ts. They demons tra te d the a b i l i t y to perform some of the tasks but had d i f f ic u l tie s