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An evaluative study of a remedial program using computer diagnosis of third grade error patterns in subtraction

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An evaluative study of a remedial program using computer diagnosis of third grade error patterns in subtraction
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Bosnick, Janet Nathanson
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vii, 152 leaves : ; 28 cm.

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Arithmetic ( jstor )
Art teachers ( jstor )
Computers in education ( jstor )
Error rates ( jstor )
Mathematics ( jstor )
Mathematics education ( jstor )
Mathematics teachers ( jstor )
Observational research ( jstor )
Students ( jstor )
Teachers ( jstor )
Arithmetic -- Remedial teaching ( lcsh )
Dissertations, Academic -- Educational Leadership -- UF
Educational Leadership thesis Ph. D
Subtraction -- Computer-assisted instruction ( lcsh )
Subtraction -- Study and teaching ( lcsh )
City of Chiefland ( local )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Bibliography: leaves 146-150.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Janet Nathanson Bosnick.

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AN EVALUATIVE STUDY OF A REMEDIAL PROGRAM USING COMPUTER DIAGNOSIS OF THIRD GRADE ERROR PATTERNS IN SUBTRACTION By JANET NATHANSON BOSNICK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987

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ACKNOWLEDGEMENTS Throughout the years spent on this endeavor, I have received help in many ways from many people. My supervisory committee members. Dr. Bill Hedges, chair, Dr. Pat Ashton, Dr. Forrest Parkay, Dr. Robert Drummond, and Dr. Mary Grace Kantowski, were unfailingly interested and supportive. More than once they spent their valuable vacation time reading and responding to my work. Dr. Mary Grimes of the University of North Florida has been my expert advisor in the "art of pedaguese." She is a true educator and there is no doubt that I will be a better educator as a result of her mentorship. I am proud that Mary is my friend. Radie Armstrong, Carolyn Fabal, and I have shared countless hours discussing course concepts, studying for exams, and supporting each other as each of us has taken another step toward completion of this degree. I am grateful for their friendship and support. Ruth and Larry Brigant have consistently encouraged me. When I first thought of starting this program, Ruth said, "Sure, you can do it!" She was right and she was always there to make sure I stood by her words! ii

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Steffi and Alan Hinunelstein, Beth and Pat Sullivan, and Catie and Dick Wallace stuck with me. I do not know how many times they heard, "I can't go, I have to study" or "I'll have more time this semester." I could rarely find the extra time to show them that I cared about their friendship. They believed in me and knowing that they did provided me with the extra impetus to keep working. Betty Holler has been a constant source of encouragement to me. Her optimism, coupled with my pessimism, provided for the humor necessary to complete this endeavor with at least some degree of sanity remaining. Not enough can be said about the patience, tolerance, and support of my mother and sister. My mom spent many lonely hours while I was busy "getting my degree." My sister tackled the New York City Fifth Avenue Library and braved the grounds of Columbia University to get me needed references. I love them both dearly. My husband. Jay, is mentioned last on this page but is first in my heart. Jay gave me support, courage, and a kick in the pants when I needed it. My success is his success. His love is my strength. iii

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TABLE OF CONTENTS Page • • ACKNOWLEDGEMENTS ABSTRACT CHAPTERS I INTRODUCTION ^ Purpose of the Study 3 Research Questions 3 Rationale ^ Limitations ^ Assumptions ^ Definition of Terms 10 Summary Overview of the Remainder of the Study 11 II REVIEW OF RELATED LITERATURE 13 Error Analysis 13 Remediation Studies 24 Diagnostic and Remedial Models 34 Summary III METHODOLOGY 48 Setting 49 Subjects 51 Achieving Entry 52 Procedures 53 Data Collection 61 Instrumentation 66 Limitations of the Diagnostic Instrument 67 Validity 67 Data Analyses 67 IV FINDINGS 71 Research Question 1 71 Research Question 2 88 Research Question 3 96 Research Question 4 101 Research Question 5 116 IV

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Page V DISCUSSION, CONCLUSIONS, IMPLICATIONS, AND SUGGESTIONS FOR RESEARCH 134 Responses to the Research Questions 135 Conclusions 1*3 Implications 1*3 Suggestions for Further Research 144 REFERENCES BIOGRAPHICAL SKETCH 151 V j 1 I

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN EVALUATIVE STUDY OF A REMEDIAL PROGRAM USING COMPUTER DIAGNOSIS OF THIRD GRADE ERROR PATTERNS IN SUBTRACTION by Janet Nathanson Bosnick December, 1987 Chairman: William Hedges Major Department: Educational Leadership The purpose of this study was to ascertain the effects of using computer generated diagnostic data on third grade students' subtraction error types to facilitate effective remediation of computational errors in subtraction. A case study approach which included 4 days of observation of three third grade teachers and their classes, a training component on computational error patterns and the use of a computerized diagnostic instrument, followed by an additional 4 days of observation was used. Preand posttraining data were collected on the teachers' methods of diagnosis and remediation, use of instructional time, and student achievement. Data were also reported on the effects of the training component. Two of the three teachers used Mathematics Assistant I, the computerized instrument. The following changes in behavior were noted for these two teachers: (a) the amount of time spent on large group instruction was reduced in vi

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favor of small group and individual instruction directed at specific errors of individual students, (b) diagnosis of students' error types was more accurate, and (c) remediation was more successful than during the pre-training period. vii

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CHAPTER I INTRODUCTION If teachers are to effect significant improvement in the arithmetic skills of children, they must be trained in the use of feasible and effective remedial strategies. To develop and implement effective remedial strategies requires the collection and analysis of data regarding training problems, error types, implementation difficulties, and remedial activities, as well as cost and time factors for delivery of instruction. Data must be collected which reflect the reality of the instructional and remedial processes. This case study provided such data. Effective diagnosis of students' difficulties as evidenced by error patterns would appear to be a logical first step in the process of skills remediation. A body of literature exists in which the elements of effective diagnosis and the need for this tool are addressed. Lack of teacher training and lack of time are often cited as major reasons for the infrequency of in-depth diagnosis of student work by teachers. Methods of diagnosis such as oral interviews and observation are acknowledged to be desirable, but these require a great deal of instructional time and are therefore relatively low in usability. -1-

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-2An examination of the diagnostic process suggests that alternatives to interviews and observation are available. The types and causes of arithmetic errors are addressed in an extensive body of research. Subtraction errors are among the most prevalent and persistent (Brown & Burton, 1978; Brueckner, 1930; Cox, 1975; Graeber & Wallace, 1977) . Research has been conducted to identify effective methods for remediating subtraction problems. Remedial strategy decisions are often based on specific diagnostic data. Ross and Stanley (1954) listed five questions in the process of educational diagnosis. They are (a) Who are the pupils having trouble? (b) Where are the errors located? (c) Why do the errors occur? (d) What remedies are suggested? and (e) How can errors be prevented? The issues of time, training, or feasibility that are needed to obtain these data were not, however, addressed in previous studies. While computers may not be the panacea sought by educational theorists and practitioners, they may offer one solution to the problems of diagnosis and remediation of computational error patterns in the field of arithmetic. The literature relating the use of microcomputers to diagnosis and remediation is expanding. Computers may also offer one possible effective tool in alleviating the problems of teacher time and teacher training.

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-3Purpose of the Study The purpose of this study was to ascertain the effects of using computer generated diagnostic data on third grade students' subtraction error types to facilitate effective remediation of computational errors in subtraction. This study was designed to yield information which could lead to a more comprehensive and hence more generalizable body of information on diagnosis and remediation of errors for teacher training. Research Questions Several specific research questions were addressed in this study: 1. What are the differences in the teacher's diagnostic procedures before and after training in identifying computational error patterns which includes a computerized diagnostic program? Specifically, (a) What diagnostic methods are used by teachers to identify student subtraction errors? (b) Do selected teachers identify problem areas for intact classes or do they identify specific error types for specific students? (c) How much time is spent on the diagnostic process? and (d) How accurate are teachers' diagnostic processes for identifying individual error patterns?

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-42. What are the differences in the inethod(s) of remediation before and after computational error pattern training which includes a computerized program? Specifically, (a.) Do the teachers remediate individually, in small groups, or in intact class situations? (b) Is the remedial strategy the same as the original teaching technique? (c) Do the teachers use coaching, demonstration, written corrective responses, student interviews, or other techniques to remediate? and (d) What supplementary materials are used, if any, for remediation? 3. What are the differences in the amount of time before and after training spent by selected teachers on intact classes, small group, and individualized instruction? 4. What are the differences in the effectiveness of the remediation as evidenced by the eradication of students' error patterns before and after teacher computational error pattern training which includes a computerized diagnostic program? 5. What are the equipment/materials requirements, changes in teachers' knowledge of computational error patterns, and training problems associated with the 2 -hour training program using a computerized diagnostic instrument on teacher's diagnosis of computational error patterns which has as its three components (a) a general overview of computational error patterns in addition.

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-5subtraction, multiplication, and division as identified by past research; (b) a focused introduction of error patterns specific to subtraction as identified by Janke and Pilkey (198.5) and as addressed by the computerized diagnostic program Mathematics Assistant I (1985); and (c) a modeling/coaching situation designed to teach the use of the computerized program. More specifically, (a) What hardware, software, and other support materials are needed for training teachers using this model? (b) How do each of the training components affect the teacher's knowledge of computational error patterns? and (c) What problems are encountered by the trainer in delivering the instruction? Rationale Educators are today, more than ever before, held publicly accountable for students' progress. This phenomenon has evolved, in part, through increased public awareness of and increased dependence upon standardized testing. Goldberg and Harvey (1983) reported that "the unprecedented attention now being paid to education is evidence of public concern" (p. 15) . In reply to A Nation at Risk , the report compiled by the National Commission on Excellence in Education, Goldberg and Harvey (1983) recommended that "citizens across the U.S. hold educators and elected officials responsible for providing the leadership necessary to achieve . . . reforms" (p. 17) .

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-6Specifically, there has been a demand for increased effectiveness in math instruction. "The concern for improving the mathematics achievement of U.S. students has reached record proportions" (Klingele & Reed, 1984, p. 712) . Remedial programs which both improve students' mathematical competency and are time and cost effective must be developed. Richbart (1980) suggested factors that should be considered in setting up or evaluating a remedial mathematics program. These include knowing where the program is going (planned course) , knowing where to begin (diagnosis) , knowing what to use (methodologies) , knowing what to expect (sufficient time and load) , and letting everyone involved know what is going on in the program (communication) . Time factors, dollar estimates, and outcome worth are important factors to consider in assessing the feasibility of implementing remedial programs. The time needed for computerized diagnosis versus that required for teacher diagnosis, as well as the time involved for teacher training, must be considered. The cost of hardware, software, and other support materials cannot be ignored. And the effect on student achievement (does this program work?) is of major concern. Finally, unanticipated problems of implementation must be addressed. "Decision makers are interested in the consequences of the real alternatives they face, not in

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-7averages of studies with unknown properties conducted over three decades" (Levin, Glass, & Meister, 1986, p. 71). Programs that meet all of these demands must go beyond the mere reporting of scores and instead be focused upon the careful monitoring of students' work. Careful analysis of the computation process is vital if remediation is to be successful. Although computer assisted instruction, tutoring, and remediation programs are available, teachers are not apt to make effective use of them until they are proficient at identifying root problems. Ashlock (1982) suggested that if the written work of a child is to provide useful information for diagnosis, that work must not only be scored, it must be analyzed as well. Teachers need to observe what the children do and do not do, they need to note the computation which has a correct answer and the computation which does not have a correct answer, and they should look for those procedures used by a child which might be called mature and those which are less mature. Usual scoring techniques do not distinguish among procedures used to arrive at correct answers; frequently they do not even distinguish between situations in which the child uses an incorrect procedure and situations in which the child does not know how to proceed at all. Pincus, Coonan, Glasser, Levy, Morgenstein, and Shapiro (1975) agreed that one important aspect of a teacher's job is to determine how a

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-8child goes about getting the wrong answer. Careful examination of the kinds of errors made by children reveals patterns that are quite logical to the child, if not to the teacher. Sovchik and Heddens (1978) stated that "elementary teachers who diagnose and remediate mathematical difficulties are like a vanishing breed of general practitioners in medicine" (p. 47). The amount of time needed to diagnose and the degree of training necessary to become proficient at diagnosis are probable reasons for the limited amount of diagnosis being performed by classroom teachers. "Diagnostic teaching requires that teachers (1) distinguish between conceptual and careless errors, (2) identify the precise nature of the careless errors, (3) infer the conceptual basis (cause) of the conceptual errors, and finally, (4) prescribe appropriate remedial procedures" (West, 1971, p. 467). Traditional paper and pencil tests do not yield sufficient data on types of errors. To obtain a better understanding of a child's procedures, teachers often use oral interviews. Anderson (1981) advocated that proper diagnosis include examination of children's written work and observation of how each child proceeds. Hopkins (1978) noted that " it would appear that a teacher could spend the entire year diagnosing particular strengths and weaknesses of children, at the expense of the introduction of new skills and concepts" (p. 49) . A time-saving

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-9diagnostic technique which increases instructional time is needed . Ronau (1986) advocated use of the microcomputer as a possible diagnostic tool. Software has been developed in which the most commonly committed student errors can be identified and valuable data upon which teachers can base remediation can be provided. Limitations Only the diagnosis and remediation of computational errors in subtraction were addressed in this study. Other error patterns or teacher behaviors related to the remaining three operations are not presented. So, while the research summary includes material broader in scope, this study was more finite in its focus. Assumptions The following assumptions guided the development of the design and data collection. 1. Math skills are interdependent. A math error in one operation could affect the chance of success in another operation. 2. Traditional methods of diagnosis such as interviews and test examination tend to yield insufficient information due to time constraints on the teacher.

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-103. Teachers could do a better job of remediation if they had adequate diagnostic information prior to treatment . 4. Teachers traditionally are given neither the training nor the time to diagnose adequately. Definition of Terms Algorithm . A rule or procedure used for solving a mathematical problem. Computational error . An algorithmic error in the addition, subtraction, multiplication, and/or division of whole numbers. Computerized diagnostic program . Known as Math Assistant I in this study — a computer program capable of diagnosing 20 types of addition errors and 20 types of subtraction errors. Effective remediation . Eradication of an error pattern. Error pattern . Repeated applications of erroneous definitions or incorrect procedures (Ashlock, 1982) . Inversion error . Subtraction of the larger minuend from the smaller subtrahend in a subtraction problem. Summary To be cost effective, remedial instruction should focus on specific errors. Many studies have been

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-11conducted to identify those computational errors most frequently made by students. Subtraction errors, especially when zero or regrouping is involved, are acknowledged as a major problem. Since subtraction is generally a topic in the third grade curriculum, the subtraction strategies of third grade students warrants careful examination. A remedial program that incorporated the use of computerized diagnostic data for the identification of error patterns in subtraction was examined in this study to determine whether a remedial program which used it would minimize the time and training needed while providing detailed diagnostic information to help teachers plan for further instruction. It was expected that teachers, equipped with the skill of interpreting such data, would then be more capable of making sound decisions regarding instruction and remediation. Overview of the Remainder of the Study The literature pertinent to the research problem areas is reviewed and evaluated in Chapter II; the design and methodology that was used in conducting this study are explained in Chapter III; analyses and evaluation of the data gathered in the research are presented in Chapter

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-12IV. A summary, conclusions, and recommendations based on the results obtained from this research study are presented in Chapter V.

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CHAPTER II REVIEW OF RELATED LITERATURE In this chapter the literature covering error analysis in basic computation, in diagnostic and remedial models of arithmetic skills, and in remedial studies is reviewed. The error analysis section contains a summary of the research in which common errors in arithmetic are identified. Research on the identification of subtraction errors is examined in detail. The diagnostic and remedial model section includes a representative sample of approaches to the diagnosis and remediation of arithmetic difficulties. An overview of the techniques designed for the remediation of specific subtraction errors is presented in the remedial studies section. Error Analvsis As early as 1913, Phelps conducted a study in which he examined speed and accuracy in relation to computation of basic addition facts. Twenty-five 1minute tests were administered to 270 eighth grade students over a 5-day period. Papers were analyzed by direct inspection. Phelps found that there was not a strong positive relationship between speed and accuracy. -13-

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-14More importantly, he found that there were certain types of errors that were common to many of the students in the schools which participated in the research. Phelp's study was an important one as it marked the beginning of a shift of interest from such measures as grade norms and standards to the problems of the individual child. Smith (1916) examined errors of third through eighth grade students in addition, subtraction, multiplication, and division. He grouped errors under the general headings of lack of knowledge of the process, errors in carrying and copying figures, incorrect combinations responses (facts), the interfacing of one process with another, and students' inability to carry on two or more processes simultaneously. Categorization of errors was accomplished through test inspection as well as observation of students as they worked. Lazar (1928) also investigated the errors of sixth graders in addition, subtraction, multiplication, and division. Through observation of students as they worked, oral questioning, and test examination, he constructed an inventory of errors commonly committed. In subtraction, fact errors and errors involving borrowing were the most frequently committed errors of the seven categories. Brueckner (193 0) studied error patterns for elementary students in addition, subtraction, multiplication, and division. The results of subsequent studies paralleled

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-15the results of Brueckner's study. He cited the most common source of difficulty as errors in basic combinations. The most common faulty algorithms involved carrying or borrowing, coupled with difficulties with zero. Roberts (1968) reported the failure strategies of a sample of 766 third grade students in all four operations according to the ability levels of the students in the sample, using a guartile distribution. Of the 2795 errors found, 36% were due to defective algorithms. In fact, he concluded, "weakness in algorithmic technigues accounts for the largest number of errors in all quartile groups except the lowest quartile" (p. 443) . Eighteen percent of the errors were computational, that is, they reflected the incorrect use of addition or multiplication tables. He concluded that "the actual number of errors due purely to careless numerical errors and/or lack of familiarity with the addition and multiplication tables is fairly constant throughout all four ability levels" (p. 443) . He noted, further, that many students who did not know how to work a problem filled in the answer blank anyway. He attributed this fact to teacher pressure to complete the problems and to "pupils' personal predilection to 'fill in' answers to gain a sense of having completed a task, no matter how poorly" (p. 466) . Finally, Roberts called for remedial programs in which more careful analysis of the

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-16student's method or lack of method prior to treatment is given. In a 2-year study, Cox (1975) classified and analyzed the systematic errors made by 700 second through sixth graders in the operations of addition, subtraction, multiplication, and division. Results indicated that 5% to 6% of the children made systematic errors in addition, multiplication, and division, while 13% made errors in subtraction. Cox noted that "in every case, the child's behavior indicated that he realized patterns and structures are necessary for solving computational problems. He simply had not perceived or recorded the correct pattern" (p. 218) . A follow-up study done 1 year later to see whether systematic errors persisted indicated that "almost one-fourth of the children in the follow-up study were making either the identical systematic error or another systematic error on the same algorithm one year later" (p. 220) . In 1977 Graeber and Wallace completed a study analyzing the addition, subtraction, and multiplication errors for elementary students. Samples were taken from pretests on specific skills in the four operations. They reported that the frequency of systematic errors for addition was 12%, for subtraction 40%, and for the two multiplication skills 19%. It is interesting to note that Cox (1975) also found the percent of systematic errors in

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-17multiplication and addition tests to be similar, while the percent of systematic errors in subtraction was considerably higher. It should be noted that errors were classified as either systematic errors, fact errors, three or more random errors, two errors, or one or no errors. Also, "only 32 of the 1,088 tests analyzed (2.9%) included three items that were incorrect due to fact errors" (p. 64) . The authors concluded that teachers do not always use the results of pretests to diagnose for remediation. Engelhardt's (1977) study of the computational errors of 198 third and sixth grade students on an 84-item test was an extension of Robert's previous efforts. He categorized errors into eight types instead of the four studied by Roberts. These types were errors in basic fact, in grouping, inappropriate inversion, incorrect operation, defective algorithm, incomplete algorithm, identity errors, and errors involving the use of zero. When ranked by quartiles, a direct relationship between the number of items attempted, the level of competence, and the number of errors committed was found. The numbers of items attempted increased with competence and the number of errors decreased. Over 40% of the total errors were committed by students in the lowest quartile. Engelhardt concluded that competent performance is distinguished from incompetent performance by an absence

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-18of errors dealing with basic facts, grouping, inappropriate inversion, defective algorithm, incomplete algorithm, and the zero concept. The error type which distinguished highly competent performance was the defective algorithm error type. He concluded that, since basic fact errors were the most common type in all quartiles, teachers should give increased attention to this area. Brown and Burton (1978) completed a study using a computer program, entitled "Buggy," which was designed to diagnose procedural "bugs" in addition, subtraction, multiplication, and division. They examined responses for 1325 fourth, fifth, and sixth graders. They found that nearly 40% of the students consistently used incorrect algorithms. They concluded that most of the difficulties arose when borrowing was required, especially when a zero was involved. The most common bug occurred when borrowing from a column in which the digit was zero. The students changed the zero to nine but did not continue regrouping from the next column to the left. This error occurred alone or in combination with other bugs 153 times in the 1325 student tests. They found that students were remarkably competent procedures followers, but that they often followed the wrong procedures. In a survey by Kilian, Cahill, Ryan, Sutherland, and Taccetta (1980) 97% of the arithmetic errors were found to be either procedural or related to calculation. Again,

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-19these results suggest incorrect use of algorithms and/or knowledge of multiplication tables. The 121 elementary students included in the sample produced 685 errors on 3,294 multiplication examples. The authors concluded that errors that might appear random or careless when the work of an individual child is considered take on discernible patterns when the work of many students is considered. The teacher, in selecting or constructing multiplication examples for practice, in formulating a review lesson for multiplication, or in introducing multiplication algorithm, should be aware that these kinds of mistakes occur most frequently, (p. 24) Janke (1980) studied the performance of mentally retarded students between the ages of 8 and 18. Using the eight error categories established by Engelhardt in 1977, Janke found that 82% of all student errors reflected the categories of basic facts, incorrect operations, and defective algorithms. More recently, McKillip (1981) conducted a significant study of division errors. He measured the skills of students, ages 9 and 13, who completed exercises for the National Assessment of Educational Progress. He reported that at age 9, 74% of the students correctly divided with a 2-digit dividend, but without using the division algorithm. By age 13, about 70% correctly worked exercises with 3-digit dividends, using the division algorithm. He found that the most striking category of errors was that labeled "unclassified." The percent varies from 10 percent to over 40 percent.

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-20and in every case the percent of unclassified errors far exceeds the percent of classified errors. It seems reasonable to assume that errors in basic facts, in multiplication, and in subtraction were responsible for many incorrect results on exercises. . (p. 35) Information on many aspects of mathematics are presented in the National Assessment of Educational Progress (NAEP) mathematics reports. For the purpose of this study, only those data pertaining to subtraction are reported. The results of the first NAEP as reported by Carpenter, Coburn, Reys, and Wilson (1975) indicated that for 9 -year-olds, the most frequent error types were reversals (18%) and subtraction where regrouping was required. The results of the second NAEP mathematics assessment, as reported by Carpenter, Kepner, Corbitt, Lindquist, and Reys (1980) again indicated reversal errors to be the most common error type for 9-year-olds. The The third NAEP mathematics assessment, as reported by Lindquist, Carpenter, Silver, and Matthews (1983), showed subtraction results similar to those found in the first two assessments. In 1985, Janke and Pilkey tested 376 students in grades 2-6 in addition, subtraction, multiplication, and division. The Educational Ability Series test for the appropriate grade level was administered and student errors were diagnosed by computer, by test examination, and by oral questioning if the error could not otherwise be identified. Of the 970 subtraction errors identified,

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-21409 were errors in the basic facts and 277 errors were classified as "subtracting absolute differences" (inversion errors) . A list of the 20 most common subtraction errors identified in this study is as follows: 1. Left digit of subtrahend was subtracted from greater place value. 2. No answer was recorded for any column with a missing subtrahend digit. 3. Made all borrowed or regrouped digits a "1" rather than the place value. 4. Did not rename digit from which borrowing was made. 5. Borrowed or regrouped only furthest left minuend digit. 6. Wrote "0" in the answer instead of regrouping. 7. Made each minuend digit a 2-digit number without regrouping. 8. Wrote "1" in the answer when minuend and subtrahend digit were the same before or after regrouping. 9. Wrote "0" in the answer when minuend and subtrahend digit were the same before regrouping. 10. Basic fact or an unrecognized error. 11. Wrong operation. 12. Subtracted absolute difference. 13. Made the digit that was borrowed from larger instead of smaller.

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-2214. Wrote "0" in the answer when the minuend or siibtrahend digit was 0. 15. Wrote the answer backwards. 16. Multiplied instead of subtracting. 17. All regrouping was done from the left of a 0. 18. All regrouping was done from the left of a 0 with no regrouping needed to make each 0 a 10. 19. Each consecutive ending was made a 9 with no regrouping. 20. Made a digit smaller without regrouping. Identification of error types was only one aspect of Janke and Pilkey's research. Other facets of their work are discussed in the diagnosis section of this chapter. The studies cited in this section do not comprise the entire body of research on errors in computation. However, these studies are representative and they do provide some common characteristics which are relevant to this study. These are as follows: 1. There is an established body of research in which the errors that children make most often are specified. See Table 2.1 for the most common errors found in the studies discussed in this section. 2. Test examinations and/or oral interviews were the primary methods of identifying error types in all of the studies except those done by Brown and Burton and Janke and Pilkey. The time and training necessary to identify the errors were not discussed but can be assumed to be

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-23 «3 U •H +J > C •H >i rH (0 •i •H rH C p -P IS a< -H 0 -p o 0 (1) a rH 0 3 ^ o ft M 4J 0 u P Vh T) -H c c u 0 •H Q) (1) 4J 0) 0 > N H rH X! 3 O O c U 0 12 rH -H Pi H U S Cn 0 -P c +J o <0 o c u c o rH u o C (U 0 :3 4J +J •H tJ>-p +J 0 ft IS 13 C +J 0) cn (0 •H 0) c u O Q) H < ;3 P o QUO < P4 n 2 V4 (V >4 in CO H H H H H en CO in CO H (0 a) o u (0 0) 10 « c o +J c o u m -p -o u (0 45 (U rH CP c w p u (0 Q) rH c w -p T3 U la 0) rH CP c u xi 0) rH c c: o +J c o 03 >1 0) >^ rH -H (U 0) c

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-24extensive and beyond the capabilities of the average elementary school teacher. 3. In subtraction, inversion errors, errors associated with the concept of one and zero, and basic facts errors accounted for a majority of all errors children committed. Remediation Studies Journals such as The Mathematics Teacher and The Arithmetic Teacher and texts such as Ashlock's Error Patterns in Computation contain suggestions on how to remediate computational errors. However, few studies have been conducted which contain specific remedial techniques according to the specific errors committed. Harvey and Kyte (1965) conducted one of the earliest studies directed at the remediation of a specific arithmetic error. A diagnostic test designed to identify errors in multiplication where problems contained zeroes was administered to 517 sixth grade students. Fifteen teachers were then provided with diagnostic information, upon which they based their individual and group remediation over the next 4 months. A posttest was administered, the results of which indicated that 73.1% of all the multiplication errors where zeroes were involved had been successfully remediated. The actual remedial methods used by the teachers in the Harvey and Kyte study were unclear. The remediation

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-25was done in a classroom setting and no record of the amount of time spent in group or individual instruction was documented. Further, it is unclear how teachers accomplished day-to-day diagnosis of student errors. There is no record of the relationship between the type of remediation used and the specific algorithm errors initially diagnosed. Blankenship (1979) studied the acguisition, generalization, and maintenance of borrowing skills among students who demonstrated systematic inversion errors in subtraction where regrouping was required. She administered the Kev Math: Diagnostic Ar ithmetic Test to 105 learning disabled students. Nine students were identified as having regrouping errors on problems not containing a zero in the minuend. However, one student changed his error pattern from inversion to placing a zero in the answer of the column where borrowing was necessary. Blankenship 's research design consisted of three stages: establishing of a baseline, intervention, and maintenance. Nine types of subtraction regrouping problems were included. They were 1. Two digits minus one digit regrouping from tens to units. 2 . Three digits minus one digit regrouping from tens to units. 3. Two digits minus two digits regrouping from tens to units.

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-264. Three digits minus two digits regrouping from tens to units. 5. Three digits minus three digits regouping from tens to units. 6. Three digits minus two digits regrouping from hundreds to tens. 7. Three digits minus three digits regrouping from hundreds to tens. 8. Three digits minus two digits regrouping from hundreds to tens, and from tens to units. 9. Three digits minus three digits regrouping from hundreds to tens, and from tens to units, (p. 34) Each day during this study, students solved five problems from each of the nine designated problem types. So, a total of 45 problems a day per student were attempted . No instruction or feedback was given during the baseline procedure. The intervention process consisted of a demonstration plus feedback. The experimenter provided a verbal and visual demonstration of a row 1 (type 1) problem. The first demonstration took about 40 seconds. The student was then asked to compute a sample problem of the same type. If the problem was correctly worked, the student was assigned the remainder of the worksheet for the day. The worksheet consisted of five examples of each of the nine problem types. Students who answered incorrectly were given additional demonstration and

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-27feedback until they correctly worked the sample problem. Worksheets were returned to the students with feedback on correct and incorrect problems for row 1 problems only. The maintenance procedure consisted of two checks conducted at 2-week intervals. The maintenance procedure was identical to the baseline procedure. All students improved their accuracy during the intervention. Five of the eight students generalized to at least one row of the noninstructed problem types. Three students generalized to between two and seven problem types. Only one student showed no generalization ability at all. Results of the maintenance procedure indicated that six of the eight students made no mistakes on the instructed problems. The other two students reverted to their previous error patterns. Blankenship concluded that "once a teacher has determined that a student is making systematic inversion errors, he/she could easily apply the technique used in this study" (p. 49) . In addition, she stated that "teachers should be aware that generalization to other problem types may not always result from using demonstration plus feedback" (p. 49) . Duncan (1981) compared the effectiveness of using base 10 blocks and expanded notation and the associative property of addition in teaching subtraction with regrouping to third graders. No significant difference

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-28was found between the mean scores of the base 10-block group and the expanded notation group. Hetzel (1981) developed a remediation plan for students who were identified as having consistent errors in column subtraction. One hundred sixty students in grades 3, 4, and 6 were tested and their errors classified in 10 categories of error types. Sixth grade students tutored the errant students on the specific error type. After one tutoring session of 20 minutes' duration, 81% of those tutored showed no recurrence of consistent errors on the posttest administered 2 days after the tutoring. Skrtic, Kvam, and Beals (1983) studied the remediation of subtraction errors requiring regrouping in learning disabled (LD) students. Piaget's cognitive development theory (1975) and Bruner's three levels of instructional presentation (1966) provided the theoretical framework for the development of this remedial approach, which was designed to provide the LD student with "real life" experiences. Skrtic, Kvam, and Beals (1983) accomplished the identification of error patterns through interviews in which the students were asked to verbalize their computational procedures. "After more than 50 interviews with LD adolescents, it was found that, apart from basic and careless errors, virtually all errors are due to a lack of understanding of place value and the faulty execution of an algorithm" (p. 33) .

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-29Their remediation strategy reflected these findings, specifically addressing place value and the subtraction with regrouping algorithm. Concrete and graphic materials were used in conjunction with numerals to illustrate the relationship between direct experience and abstract symbols. The approach matches numerals with concrete and graphic materials, gradually removing the less abstract materials as the student becomes better able to deal with symbols alone. In the end the student is expected to perform exclusively in the symbolic mode of representation. (p. 33) Students worked for a period of 50 minutes during each of the first four days. During the last 3 days they focused on the subtraction algorithm in 30-minute sessions. Materials were used in the following sequence: (a) concrete materials with numerals, (b) concrete/graphic materials with numerals, (c) graphic materials with numerals, (d) graphic worksheet with numerals only, (e) numerals with place value designations, (f) spaced numerals, and (g) numerals on a typical math worksheet. The researchers stated that "not only did the procedures and materials effectively remediate subtraction errors, the junior high school LD students who participated in this program enjoyed using them" (p. 38) . The identification of gaps in understanding and presentation of content using materials and procedures which are appropriate for the students' cognitive levels were emphasized in this approach.

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-30Cebulski (1984) described three studies which were conducted to determine the source and frequency of subtraction difficulties for third and fourth graders. In his first study, students' verbal descriptions and written solutions were used to classify types of errors. Eleven of the 56 third graders missed 20 out of 20 subtraction examples with inversion errors being the most frequent. The error patterns of the 11 children persisted: "after a one month interval, all eleven of these subjects again erred on every problem, with inversion again the main source of error" (p. 5) . Number facts accounted for a majority of the errors for students who solved at least one problem correctly. The causes of subtraction errors were examined in the second study. Cebulski hypothesized that errors were due to failure to recognize when to borrow, or to low levels of motivation. Students were assigned to either an instructions intervention, motivation intervention, or a control group. The instructions intervention required that students borrow on every problem. The motivation intervention provided a reward of a small toy chosen by the student upon successful completion of a specified number of correct solutions. No instruction was provided during any of these conditions. Results showed that there was no significant difference between the number of subtraction problems solved correctly for the instruction or motivation groups

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-31relative to the no-treatment controls. "These results also suggest that in order to be effective, remediation may require an instructional component" (p. 8) . The effectiveness of two remedial programs for subtraction were compared in the third study. Students were assigned to one of three remediation models: training in the component skills required for borrowing, feedback in the form of correctly worked solutions, or a no-treatment control situation. The component skills training model required children to work through five self-instructional booklets which presented problems requiring regrouping and provided immediate feedback. The criterion training model required students to work through four training booklets with 1 to 10 problems on each page. Students checked their solutions against a correctly worked problem. Subjects in the control group received no intervention. Students worked in their groups for one 45minute session per day for three consecutive days. Results indicated that, for both groups, there was a significant increase in the number of problems solved correctly and a decrease in the number of borrowing errors. So both systematic instruction in component skills and feedback in the form of correctly worked solutions proved to be effective in remediating subtraction difficulties. Voran (1985) compared two methods for remediating subtraction difficulties at the fifth grade level. Based

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-32on their performance on a 10-item diagnostic pretest, 44 students who responded correctly on seven or fewer items were assigned to one of three groups, according to their error categories: algorithm errors, reversal errors, and fact errors. Students within each category were randomly assigned to either a control group or an experimental group . The control procedure consisted of a general reteaching model, using an instructional sequence which presented the concepts first in concrete form and gradually shifted to abstract symbolic form. Instructional materials were not specified for the control teacher. The experimental procedure consisted of the utilization of instructional materials designed specifically to address each of the three error types identified. The materials included instructions for the teacher-directed sessions, games, puzzles, and follow-up worksheets . A posttest was administered 1 week after the week of instruction. A second posttest was given following a 3-week maintenance period. A third posttest was given after 4 weeks had lapsed, with no further instruction intervening. No significant difference between experimental group mean scores and control group mean scores was found on any of the three posttests. Voran attributed the lack of

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-33significance to flaws in the instruction of the control group . Voran did propose a diagnostic model, based upon her research. She . proposed that diagnosis include the following steps: 1. A test for the computational skill itself. 2. A test for basic facts. 3. A test for understanding (probably an interview setting) . 4. An error category section ( a list of common error types and incorrect responses which would be catagorized under each type (p. 155) . Several conclusions can be drawn from these studies. They are 1. Many of the studies had small sample sizes and were conducted with learning disabled students. Therefore, for these studies, there is not sufficient evidence to warrant generalizing to normal populations. 2. Lack of controls or flaws in the experimental methods of some of these studies weaken the validity of the results. 3. The paucity of research directed at the remediation of specific errors in arithmetic makes it difficult to suggest procedures which will result in effective remediation. 4 • Techniques which incorporate verbal and/or written feedback specific to the error committed appear to have a

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-34better chance of success than do those which omit this process. Diagnostic and Remedial Models Brueckner (1935a) defined educational diagnosis as "the techniques by which one discovers and evaluates both strengths and weaknesses of the individual as a basis for more effective guidance" (p. 2) . This definition posits an inextricable link between diagnosis and remediation. Tyler (1935) observed that "the essence of educational diagnosis is the identification of some of the causes of learning difficulty and some of the potential educational assets, so that by giving proper attention to these factors more effective learning may result" (p. 96) , reinforcing Brueckner 's position. More recently, Reisman (1972) supported this insistence upon linkage when she observed that "diagnosis, then, is a process of determining the facts which need to be taken into account in making academically oriented decisions" (p. 4) . Academically oriented decisions are those instructional decisions associated with the presentation and remediation of cognitive skills. In a more limited vein, Underhill, Uprichard, and Heddens (1980) described mathematical diagnosis as "a process by which learner problems are identified through an analysis of errors and other pertinent data" (p. 131) .

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-35There is not a great difference in the definitions of the 1930s and more recent ones. The definitions imply that a set of procedures or steps exists which should be followed in performing a diagnosis. Implicit in all of these definitions is the fact that the purpose of diagnosis transcends the diagnostic process itself. The diagnosis is useful only to the extent that it results in decisions about future instruction and remediation. Tyler (1935) listed 11 characteristics of a satisfactory diagnosis. A diagnosis 1. must concern itself with worthwhile objectives; 2. must provide valid evidence of strengths and weaknesses related to the objectives; 3. must be reasonably objective, so that other competent persons may arrive at similar results; 4. must be reliable, so that additional diagnoses covering other samples of pupil reactions will not give widely different results; 5. must be carried to a satisfactory level of specificity; 6. must provide comparable data; 7. must provide sufficiently exact data; 8. must be comprehensive; 9. must be appropriate to the program of desired education; 10. must be practicable; and

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-sell, should be conducted by persons who are well qualified as educational diagnosticians (pp. 110-111) . Ross and Stanley (1954) divided the process of educational diagnosis into two components: correction and prevention. Their concept of diagnosis consists of five steps or levels: (a) locating individuals needing diagnosis, (b) locating the nature of the difficulty, (c) locating the cause (s) of the errors, (d) remedial procedures , and (e) error prevention (preventive diagnosis) . Reisman (1972) identified five processes in the diagnostic teaching cycle. These are as follows: 1. Identifying the child's weaknesses and strengths in arithmetic. 2. Hypothesizing possible reasons for these weaknesses and strengths. 3. Formulating behavioral objectives to serve as a structure for the remediation of weaknesses or the enrichment of strengths. 4. Creating and trying corrective remedial procedures . 5. Continuing evaluation of all phases of the diagnostic cycle to see if progress is being made in either getting rid of troubled areas or in enriching strong areas (p. 5) . The first four steps in both Reisman 's and Ross' models are necessary in the process of correction. Individuals in need of help can be identified by several

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-37methods, including testing, observation, and interviews. The nature of the difficulty then dictates the specificity with which effective diagnosis must be conducted. Locating causes of errors is a necessary first step, since remedial methods are dependent upon the recognition of error origins. Ross and Stanley's last step, prevention, is a step often overlooked. These researchers pointed out how diagnostic procedures can be applied to other similar situations. Given the knowledge gained from diagnosis, instructional strategies can be developed to prevent the errors from recurring. Pupil performance can be diagnosed in several ways. Observation of the student at work, oral interviews, analysis of written work, and evaluative testing are the most frequently advocated methods (Brueckner, 1935b; Junge, 1972) . Classroom teachers traditionally use a combination of these methods. However, time limitations tend to minimize the frequency, accuracy, and specificity of the classroom teacher's diagnostic procedures. Training in the identification of error patterns is also required if a teacher is to be an accurate and efficient diagnostician. Several researchers have addressed teacher training for diagnosis and remediation. Mettair (1982) conducted a study to determine the effect of inservice on elementary teachers' abilities to diagnose and remediate error patterns. He found that before training, the teachers

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-38could neither diagnose many of the common computational error patterns nor select appropriate remedial techniques. After training, the teachers were better able to identify specific error patterns and select appropriate remedial methods, based on the causes of the errors. Sadowski (1979) compared the accuracy of prescriptions based on test results plus interview with prescriptions based on test results only. She found that "accuracy for content category was very high, accuracy for activity level was moderately high. Consistency was moderately high between test-only prescriptions and testand-interview prescriptions" (p. 1422A) . Diagnostic and remediative models have been developed to assist the teacher in diagnosis. Janke and Pilkey (1985) compiled important features of an effective diagnostic program common to the National Council of Teachers of Mathematics as well as other researchers, such as Bright, Signer, and Hill. These included (a) identifying a comprehensive number of errors, (b) identifying the specific errors which can provide a model of students' misunderstandings, (c) providing an immediate diagnosis of the specific error, (d) identifying errors from the actual answers generated by the student, (e) providing for individualized test construction by being capable of identifying errors for any test item made by the teacher, and (f) scoring tests and providing

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-39diagnostic reports which identify the specific errors made by each student. Diagnostic programs which provide important data for teachers are needed, but they must be constructed so that they can be easily applied by teachers with minimal training. They must also be appropriate for repeated diagnostic application and they must be time efficient. Underhill (1972) presented a model in which diagnosis is an integral component of the continuous instructional process. The major components consist of (a) identification of objectives on a linear hierachy. (addition, subtraction, multiplication, and division are perceived as steps on a continuum) ; (b) preassessment to determine where the student is functioning on the content chain or continuum (a survey test covering those increments appropriate for the grade level of the target group is used to gather this information and understanding is measured at the abstract level at this point) ; (c) the institution of instruction or enrichment activities based on the results of the preassessment, or, where students demonstrate mastery of objectives, advancement to step one of the next increment in the model; (d) concept instruction for students, using a concrete/semi-concrete approach (instruction can be in large groups, small groups, or individualized, based upon teacher discretion) ; (e) diagnosis, using an analytic test

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-40( analytical tests allow the teacher to examine the subconcepts in greater detail than do survey tests and include problems to assess understanding at the semi-concrete or abstract levels within a given increment); (f) direct students into enrichment, practice or further instructional strategies based on the results of the analytical tests; and (g) return of student to step one of the model with a new objective when they have demonstrated mastery of the objective in question. This model reflects the way that many classroom teachers structure their diagnosis and remediation. The model is based upon the principle of mastery learning. The number of problems missed, rather than the type of error committed, is the basis for diagnosis and thus for remediation. Instruction after diagnosis is generally the same as was the initial instruction. A continuum of objectives is required. Mathematics teachers in many school systems have identified objectives required for given grade levels and have arranged them in a sequential manner. The specificity of instruction for those objectives is sometimes left to the individual teacher. Often it is also that teacher's responsibility to develop examples or to find marketed materials which reflect the objectives. Denmark (1976) proposed a diagnostic model which extends beyond the identification of types of computational errors. Denmark's model relates the

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-41identified error to a probable cause. The four steps of the model are as follows: 1. Administration of a brief test. All of the test items are related to one increment of a continuum. The test may be written or it may be a game. If a game is used, a wrong response should not stop the students' continuation of the game. 2. Identification of incorrect answers. 3. Reference to a predetermined table or chart which relates specific errors and/or combinations of errors to a probable cause. 4. Having arrived at a possible cause, selection of appropriate remediation activities from a catalog of tested prescriptions, (p. 61) . Unlike Underhill's model (1972) Denmark's model (1976) links remediation to specific errors. A continuum of objectives and charts of error causes is included in his design. The teacher is still responsibile for identifying the error types on a teacher-made activity. A level of teacher expertise is required. Berman and Friedeni^itzer (1981) presented a fieldtested instructional model which included the diagnosticprescriptive approach for remediating addition and subtraction errors where regrouping is required. Steps of this model are as follows: 1. Diagnosis of error patterns through oral

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-42interviews and the use of manipulatives and pictorial representations of concepts. 2. Identification of the causes of the errors (s). These include the lack of experience in manipulation of concrete objects, inability to relate physical object manipulation, terminology and abstract notation, inappropriate vocabulary for describing material manipulation, presentation of materials without consideration for the child's preferred learning style, premature transition from concrete to abstract, and rote learning of an algorithm without demonstration of understanding of the same. 3. Remediation based on the premise that learning is an active process dependent upon a high level of interaction between the learner and the environment. Manipulative materials are used to model general placevalue concepts. Structured materials such as base-10 blocks are used to assist in the transition from the concrete to the abstract. According to Herman and Friederwitzer (1981), students "should have many experiences with both operations without doing any writing at all. Their manipulation of the materials and their verbal descriptions of the activities provide the teacher with insight into their conceptual development" (p. 112) . 4. Gradual introduction of algorithm. The teacher records the manipulation, using a code of dots and slashes for place value. This activity continues until the

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-43students can explain the procedure fluently. Students then record their own manipulations one step at a time. Eventually the students are expected to complete the entire algorithm without the use of manipulatives. This model is more appropriate for a small group setting than it is for use by a teacher with a class of 30 students. Frequent diagnosis in a larger group setting is both difficult and time consuming. Although these models contain diagnostic methods upon which remediation can be based, they lack elements which reflect the "reality" of the average classroom teacher's experience. Instead, use of these models 1. Rely on the teacher's ability to develop continuums or examples that fit on continuums, thus, forcing the teacher to develop many of the materials used in remediation. 2. Require a great deal of record keeping because of the many possible combinations of increments within existing continuums. 3. Require other ways to address the on-going diagnostic needs of classroom teachers who wish to conduct frequent assessments. The proponents of the above models believe that they can easily be used by classroom teachers. However, it is doubtful whether any one of these models can be used in its entirety, given the present structure of classrooms. Teachers will, instead, select parts of particular models

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-44and adapt them in accordance with what they believe is most effective, manageable, and comfortable (Abbott & McEntire, 1985) . In addition, none of these models meets all the criteria established by Janke and Pilkey (1985) for a good diagnostic program. The scoring of tests and identification of specific errors provide invaluable information for teachers. But, the ease of obtaining these data must be established before a majority of teachers would consider a program for use. On the other hand, an efficient, accurate program might encourage teachers to diagnose more frequently and perhaps use their instructional time more effectively. Teachers habitually use published worksheets; they are a reality in the classroom. A diagnostic program must allow the teacher to gather information using this type of material, or it will not tend to be adopted for regular use. Janke and Pilkey (1985) field tested a computerized diagnostic software program to determine whether the software contributed to the students' learning process. Two schools from the same district participated in the study. One school served as the control and one was the experimental school. Three hundred seventy-six students in grades 2-6 were pretested, using the Educational Ability Series Test for the particular grade level. Each student in the experimental group was tested once every 4 weeks for a period of 12 weeks. The diagnostic results

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-45were then given to the teachers. The diagnostic data consisted of 1. Individual student reports identifying the student's specific errors on designated problems. The frequency of the error was also reported. 2. Class performance reports containing an analysis of each problem. 3. Class performance reports with students grouped by error types. The teachers were free to choose their own remediation method. All students were then posttested. The average gain score for the experimental group, where the teacher received the diagnostic feedback, was significantly higher than that for the control group, where feedback was not provided. Janke and Pilkey (1985) concluded that "the significantly improved achievement in this study may simply reflect increased direct teaching time in the experimental group. The experimental and control groups had an equal number of classroom hours in mathematics instruction" (p. 49) . They also concluded that "perhaps computerized error diagnosis provided specific information which helped the teacher to provide appropriate remedial instruction" (p. 49) . The teaching strategies were not documented, but the teachers did report a general strategy which they felt was quite effective in the remediation process. "The strategy was to verbally describe and then visually demonstrate the

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-46error on the board to an individual student or group of students making the same error" (p. 49) . In addition, they stated that "helping students to understand their own error was judged to be more effective than using alternative remedial methods such as low-stress algorithms (Hutchins, 1976) or physical materials such as Cuisenaire rods, chip trading activities, and place-value boards and sticks" (p. 49) . It seems reasonable to infer that a specific diagnosis, accompanied by a verbal explanation of the error might solve many of the remediation problems facing arithmetic teachers. The key to the process seems to rest in the ease with which teachers can obtain specific diagnostic data. A computerized program may help to solve this problem. These models have in common the following attributes: 1. A plan to help teachers address students' arithmetic difficulties via diagnosis and remediation. 2. The opportunity to select parts of models as teachers perceive them to fit their needs. Summary One can draw the following conclusions regarding the entire body of research summarized herein: 1. Diagnosis is essential to effective remediation. 2. Researchers have identified the errors students can be expected to make.

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-473. Not enough is known about which remedial technique is most effective for addressing specific subtraction errors . 4. There exist both mastery and error analysis approaches to remediating arithmetic difficulties. Teachers do not generally follow one model exclusively. This fact can be confirmed by teacher observation. 5. Diagnosis is a time-consuming task. Diagnosis is frequently not carried out because of lack of teacher training in error analysis and insufficient time. 6. A diagnostic method must not require a high level of expertise on the part of teachers, must be time efficient, and it must yield useful results if we expect teachers to use it. 7. The success of computerized diagnostic programs suggests that this may be a feasible and effective approach to diagnosis. 8. Training teachers for use of a computerized diagnostic program and observation of teachers following the training may provide useful data as to what teachers are currently doing and what they might need to be more effective diagnosticians.

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CHAPTER III METHODOLOGY The purpose of this study was to ascertain the effects of using computer generated diagnostic data on third grade students' subtraction error types to facilitate effective remediation of computational errors in subtraction. Bogdan and Biklen (1982) defined a case study as a "detailed examination of one setting" (p. 58) . They advocated use of the case study method as appropriate for persons seeking to describe and analyze complex experiences. Since the research study required an indepth and realistic representation of the instruction and instructional practices, a case study approach was used. The researcher designed a preand post-training data collecting model, where teachers were observed both before and after being trained in error types and computerassisted diagnostic procedures. The researcher collected benchmark data on the initial remedial and diagnostic methods of the teachers participating in the study. During the training phase the researcher recorded the time and difficulty factors of training which she provided for the teachers. Following the training, teachers were again observed to note the remedial and diagnostic methods used by the teachers at -48-

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-49that point. Data were also collected on the effectiveness of the remedial methods as evidenced by the elimination or persistence of students' error patterns. Observation and direct questioning techniques were selected as the primary means of collecting data because of the qualitative nature of the data. Specifically, the following steps were completed: 1. The researcher observed three third grade teachers and their students to gather data on various factors involved in the diagnosis and remediation of computational errors in subtraction, the effectiveness of the remediation, and the use of instructional time. 2. The researcher provided training in computational errors in subtraction and a computerized diagnostic program for the teachers. 3 . The researcher then completed a second set of observations to identify changes, if any, in various factors related to the diagnosis and remediation of computational errors in subtraction, the effectiveness of the remediation, and the use of instructional time. The teachers were not informed that the researcher would be noting changes in behavior. Setting This study was conducted at Chiefland Elementary School, one of five elementary schools in Levy County, Florida. It is located approximately 35 miles south of

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-50Gainesville, Florida, home of The University of Florida. Fanning was Levy County's chief occupation. Those who were not fanners were generally unemployed. Of the 710 students enrolled at Chief land Elementary during the time of this study, 78% were white, 21% were black, and 1% were classified as "other." An indicator of their SES is the fact that 50% of the population qualified for free or reduced lunch (Chief land Elementary School Annual Report, 1987) . Chiefland was selected as the site for this study for several reasons. First, the October, 1986 Florida State Student Assessment Test scores of the third grade at Chiefland indicated that subtraction was an area in which students demonstrated weakness. Seventy-eight percent of the third grade students met Standard F, subtraction of whole numbers, satisfactorily. However, the district percentage for Standard F was 83% and the state percentage was 91%. So, a study focused on the teaching of subtraction, as this study, could provide relevant data. These data could be used as a basis for instructional and remedial decisions directed at improving students' subtraction abilities. Second, the facilities included an Apple computer, which was required for the training session. Third, the principal was a doctoral candidate at the University of Florida and was extremely supportive of any study which had the potential of impacting positively

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-51on teacher effectiveness. Accordingly, the school facilities were freely available to the researcher. Subi ects The subjects were the three third grade teachers and their students at Chief land Elementary School. As indicated, the three teachers, known to this study as Teacher A, Teacher B, and Teacher C, varied widely in their amount of teaching experience at the elementary school level. Teacher A had taught 17 years. Teacher B had taught 29 years, and Teacher C was a first year teacher and a participant in the Florida Beginning Teacher Program. The composition of the three classes is presented in Table 3.1. Table 3.1 Class Composition Teacher Girls/Boys Black/White Total A 11/10 7/14 21 B 12/10 10/12 22 C 11/09 7/13 20 Seventy-two students were involved in the study. Parents were informed of the study via a letter from the researcher, co-signed by the principal of the Chief land

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-52Elementary School. Signed consent forms were returned to the appropriate teachers for each student. Achieving Entry Earning the trust and confidence of the subjects was a primary concern to the researcher. To facilitate this process, an initial conference between the researcher and the participating teachers was set up by the Chiefland principal. It was necessary that the teachers feel comfortable during the researcher's observations of their classes as well in the follow-up question and answer sessions. The researcher first met with the third grade teachers during their planning period on Wednesday, February 25. In order to minimize the distance between the researcher and the teachers, the researcher dressed for this occasion in a casual skirt and blouse rather than in a suit. The intentions of the researcher and purpose of the study were explained at this time. The following main points were addressed: 1. The researcher was completing a doctoral dissertation. 2. The researcher was concerned with effective arithmethic instruction and the efficient use of teacher time.

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-533. The researcher had designed a study to increase faculty awareness of teaching methods, problems, and possible solutions. 4. The researcher was not an expert coming in to solve teachers' problems but rather wished to gain an understanding of these problems. 5. The researcher would not be observing to make a judgment about teacher capabilities. 6. The researcher would not violate the confidentiality of the study by reporting to the principal any instance that might influence the principal's judgment of the teacher's performance. The research design required that there be observations for 4 days, followed by one training session of approximately 2 hours, with 4 additional days of observation of each teacher to occur following the training session. Only math lessons were observed. A time schedule was agreed upon by the researcher and the teachers. The observation schedule was as follows: Teacher A: 9:00-9:45 Teacher B: 10:00-10:45 Teacher C: 12:00-12:45 Procedures A pretest was given to the classes by the appropriate teachers on the Friday prior to the first observation. The pretest was a teacher-made test (See Table 3.2).

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-54Table 3.2 Pretest 1. 68 2. 51 3. 84 4. 739 42 26 55 -208 5. 640 6. 787 7. 326 8. 942 513 499 151 686 9. Ill 10. 500 11. 916 12. 840 22 271 272 171 13. 95 44 14. 425 79 15. 841 623

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-55The test consisted of 15 subtraction examples, 3 examples in horizontal form and 12 examples in vertical form. Thirteen of the 15 examples required regrouping. All students received the same pretest. This was normal procedure for the teachers, as they usually worked together and give the same tests. Teachers were instructed to respond to the tests the way they would normally. Over the weekend, the teachers graded the tests, using their normal grading standards and procedures. The teachers then began remediation, based on their interpretation of the results. The tests and scores were given to the researcher on the following Monday, at the time of the first observation. The researcher analyzed the tests for error types, using Math Assistant I, a computerized diagnostic program. The computer generated results were used to identify the errant students and the specific errors they were making. The researcher then compared the areas of students' weaknesses perceived by the teachers based on their interpretation of test scores with the students' areas of weaknesses identified by the researcher based on specific computerized diagnostic results. After these error patterns were identified the researcher then used this information to observe the congruency between student error patterns and teacher responses to perceived student performance. Large and small group instruction was observed to ascertain whether or not the remediation was directed

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-56at the specific errors of the appropriate errant students . The first group of observations was conducted Monday through Thursday at the times established by the schedule. Questions and other concerns were addressed between 10:50 and 11:30 a.m. in the teachers' lounge during the teachers' lunch break. This site was chosen by the researcher because it provided a relaxing environment in which the teachers were not threatened. The teachers administered a posttest 1 (Table 3.3) on the Friday immediately following the first observation to provide feedback on the effectiveness of the remediation. The examples on posttest 1 were the same as the pretest with the exception of order. The teachers graded this test during the day and provided the researcher with these tests that same day. The researcher then analyzed the tests for error types, using Mathematics Assistant I. The researcher used the computer-generated results of the test to determine which students had been successfully remediated. The determination was based on data indicating whether or not the computational error previously identified during the computer analysis of pretest results was eradicated. The training session was conducted on the Friday afternoon immediately following the first 4 days of observation. The session was conducted in an alcove adjacent to both the computer lab and to the teachers' classrooms.

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-57Table 3.3 Posttest 1 51 26 84 55 68 42 326 151 111 22 942 686 640 513 916 272 787 499 10, 500 271 11, 739 208 12, 840 171 13 425 79 14. 841 623 15. 95 44

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-58The session lasted 2 hours. The training session was designed to increase the teachers' awareness of the types of computational errors in subtraction and of a computerized method of diagnosing these errors. The session took place following classes on the Friday following the first 4 days of classroom observation. To encourage participation, the researcher bought the teachers "Cokes" and made several comments about their dedication to work on Friday afternoons . The group convened around a student work table located in an alcove between the teachers' classrooms and the computer lab. The teachers were asked to write a brief description of what they thought they would be teaching on Monday. They were also asked to indicate which students would be in the various groups if they were in fact going to do any grouping. The descriptions were collected by the researcher. The teachers were then asked to list the types of subtraction mistakes that they knew students tend to make. The errors identified by each teacher were compared. The teachers were not surprised that there was considerable consensus among their responses. The researcher distributed a table of common errors to the teachers. The researcher explained that these were some common errors which researchers have found in error type studies dealing in all four operations. The group reviewed the errors.

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-59Since subtraction was the focus of the remediation being observed, a list of common subtraction errors (see p. 21) was provided to the group. The list contained the 20 subtraction errors identified by Janke and Pilkey (1985). This listing also corresponds to the errors detected and analyzed by Math Assistant I. A brief description of Janke and Pilkey 's study was given by the researcher. An introduction to Math Assistant I followed. The researcher explained that Math Assistant I would allow them to 1. Create an addition or subtraction test. 2. Print the test. 3. Enter the student answers to the test and receive diagnostic analyses. The instructor emphasized that the Math Assistant I program provides 1. Statistics on individual students, including the type of error made and the example containing the error. 2. Class statistics by error types, listing the error type, students who made that type of error, and the number of times each students made the error. The researcher demonstrated the start-up procedure for using Math Assistant I. The researcher also provided a brief discussion on diskette handling and care. Then the researcher provided a demonstration of the program to illustrate its use. To accomplish this, a subtraction test of 10 examples was created by the group. The

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-60researcher sat at the keyboard with the teachers observing. The teachers generated the test examples while the researcher keyed them in. The test was then printed, and copies were distributed to the teachers. The teachers were asked to work the problems but play the role of errant students. They were told to make mistakes such as the ones they had just talked about. The Mathematics Assistant I start-up procedure was reviewed. The researcher entered the teachers' answers as a demonstration of what the teachers would do with their students' answers. Both student and class reports were printed and given to each teacher. Each teacher then demonstrated the start-up procedure and created a 5question test. Teachers also entered fictitious student responses and printed results. The Math Assistant I documentation and disks were left in the computer lab for teacher use. Four days of observation followed the training session. Teachers were asked to continue remediation using any techniques they thought appropriate. The researcher stressed the fact that the teachers were not required to use Mathematics Assistant I if they did not feel comfortable with it. They were asked to provide the researcher with any copies of reports generated by Math Assistant I, if they did choose to use it at any time during the week. The researcher continued to ask relevant questions regarding diagnostic methods, selection of

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-61remedial content, grouping procedures, and perceptions of student achievement. These data were used to identify changes, if any, in these areas. The teacher's lounge continued to serve as the meeting place for questions and answers. Posttest 2 (Table 3.4) was administered to the students on the Friday following the 4 days of observation. Posttest 2 was the same as the pretest and posttest 1 with the exception of order. The teachers graded the tests and reported the evaluation procedure used to the researcher. The tests were given to the researcher the Tuesday following the administration of the exam. The researcher analyzed the tests for error types using Mathematics Assistant I. These data were used to assess the success of the remediation as evidenced by the reduction of the number of computational errors committed by the students. Data Collection Observation was the primary method of data collection. The researcher developed a data collection sheet (Table 3.5) that consisted of 30 one-inch squares. One sheet per class per teacher was used to record data each day of the observations. The sheet served as a seating chart and a place for observer notations. The teachers checked to make sure students were in their proper seats each day. A shorthand developed by the

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-62Table 3.4 Postest 2 1. 84 2. 51 3. 68 4. 787 55 26 42 -499 5. 640 6. Ill 7. 739 8. 840 513 22 208 171 9. 500 10. 916 11. 942 12. 326 -271 272 686 151 13. 841 623 14. 95 44 15. 425 79

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-63+> «) Q> 43 CQ o •H P O 0) IT) rH • iH n 0 u 0) iH XI -p (0 (0 Eh Q 0) 4J Q U 4:! o (0 0) C (0 X! 0) xi o Q) O •H c c •H c (0 >1 c c n CQ H C C •H -H H CQ •H C •H CQ C 0) CO (0 0) Q (I4 (U 0) Si o 0) o •H u as o •H 0) u 0) •H in -H 0) c c (0 (i> Q) -H 1:; (0 Q T3 c o o •H n C H n C C OQ H -H If) CQ (0 <1) C X! W (0 -H -H c o E-i x: o (0 o •H cn 0) XI XI o CQ c c H -H ^ CM CQ H C -H g CQ g g r» o H I H H CQ CQ C C •H -H g g CO CM CQ H 0) O PQ C -H g H CQ

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-64researcher was used to enter relevant data in the boxes. If the student was given individual attention at his/her seat, "I" was entered in the box of the appropriate student. "B" was entered in the box when the student went to the board to work a problem. The appropriate error code number which corresponded to the number of error type (see p. 21) was placed after the B if the student demonstrated that error while at the board. For example, if John made a basic fact error while working at the board, B-10 was placed in John's box. The amount of time for each "I" and "B" interaction was indicated next to the appropriate notation in the box. A second collection sheet (Table 3.6) was used to record observations of the following data: 1. Instructional/remedial procedure. 2. Materials/aids/worksheets used in instructional or remedial procedure. 3. Time for class instruction or small group instruction. 4 . Student/teacher comments relevant to research questions. 5. Miscellaneous notes. The researcher's notes of discussions were used as a source of data. Tape recordings were not made of teacher responses to researcher questions because the presence of a recorder might have interfered with the spontaneity of the responses. The training session was recorded.

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-65Table 3.6 Sample Observation Sheet TEACHER DATE TIME LESSON OBJECTIVE; LESSON FORMAT; Seatwork Large Group Boardwork Total Time_ Total Tiine_ Total Time Examples: Small Group #Groups Group I Focus Group II 1. 12. 2. 3. 3. 4. 4. 5. 5. Group III Focus Group IV 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. MATERIALS /AIDS /WORKSHEETS : COMMENTS RELEVANT TO ERROR DIAGNOSIS: OTHER:

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-66Questions were asked when the information was not made clear by the teacher (s) , The researcher made written notes of the information obtained by questioning at the end of the relevant conversations. Instrumentation Math Assistant I (1985) is a computerized diagnostic program developed by Robert Janke and Peter Pilkey and marketed by Scholastic, Inc. The program has the capability of identifying 20 types of addition and 20 types of subtraction errors. The program is a modification of a model used by Janke and Pilkey in their 1985 study. A field test of the instrument was conducted as part of Janke and Pilkey's study (1985). The study involved error analysis in addition, subtraction, multiplication, and division. The pretest answers of 376 students in grades 2-6 were analyzed for errors. A total of 29,392 responses to test items were analyzed. There were 3,069 incorrect responses. Each student's worksheet was examined to verify the error strategies identified by the software. Seventy of the 80 error strategies included in the software were made by the students. In subtraction, there were 8,176 responses and 970 errors. Of these, 87 errors, or 9%, were not correctly categorized. This percentage of reliability was within the accepted parameters for this study.

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-67Limitations of the Diagnostic Instrument Analysis of student responses was accomplished by entering the responses into the computer, using the keyboard. The teachers entered the responses. There was the possibility of human error when entering the responses. While Mathematics Assistant I was capable of identifying 20 types of subtraction errors, the program does not allow for the detection of multiple errors. Validity The validity of this study may have been jeopardized by the fact that the researcher and observer were one and the same. The researcher attempted to minimize any discrepancies that could have occurred by using low inference data recorded on collection sheets. Data Analyses Error analysis data gathered from this study were analyzed by the use of the computerized diagnostic program Mathematics Assistant I . Research Question 1. The differences in pretraining and post-training diagnostic procedures were the focus of question 1. Observation and teacher supplied diagnostic data provided information on the diagnostic practices of the teachers. The time used for diagnosis by

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-68the teachers was recorded according to the methods used. For example, if the teacher used boardwork as a method of diagnosis, the time students spent at the board was recorded. Teachers supplied time estimates for test grading. Accuracy of the teachers' diagnoses was determined by a comparison of teacher perceived student errors and student errors identified by the researcher using Mathematics Assistant I as the diagnostic tool. Research Question 2. This question focused on the methods of remediation during the pre-training and posttraining periods. Observation of the teachers' remedial strategies during the two 4 -day periods provided the data necessary to answer this question. The differences in remedial methodology were determined by a comparison of the strategies used during the pre-training and posttraining periods. Research Question 3. The time in minutes for large group instruction, small group instruction, and individualization was recorded during the two 4 -day observation periods. Averages for the pre-training and posttraining periods in each of the three categories were calculated. The averages were compared to determine the changes in the use of the instructional time during the pre and post training. Research Question 4. The differences in the effectiveness of the remediation as evidenced by the eradication of students' error patterns were the focus of

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-69question 4. Mathematics Assistant I was used to analyze student responses to questions on the pretest, posttest 1, and posttest 2. Comparison of the frequency of students' error . types on the pretest and posttest 1 provided data to assess the effectiveness of the pretraining. Comparison of the frequency of students' error types on posttest 1 and posttest 2 provided data to assess the effectiveness of the post-training. Students were successfully remediated if the frequency of an existing error type (students committed the error with a frequency of two or more) was reduced to a frequency of one or less. Pre-training and post-training were considered successful if a greater percentage of students were successfully remediated than were not. Research Question 5. The equipment/materials requirements, changes in teachers' knowledge of computational error patterns, and training problems associated with the 2-hour training program using a computerized diagnostic instrument were investigated in question 5. Teachers were asked to write a list a computational errors they thought their students often committed. The trainer reviewed, with the teachers, common errors in addition, subtraction, multiplication, and division as identified by past researchers. Also, a list of 20 subtraction errors was also discussed. A comparison between the teacher generated errors and the trainer supplied errors was made to substantiate the need for such training. Training on

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-70the computerized instrximent Mathematics Assistant I was given using demonstration, then coaching techniques. The answers to the practice test first generated by the teachers during the demonstration period were analyzed for error types so as to identify changes in the teachers' awareness of error patterns.

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CHAPTER IV FINDINGS In an effort to learn more about third grade teachers' diagnostic capabilities and problems in the teaching of subtraction, the author investigated the effects of a remedial program which incorporated the use of a computerized diagnostic instrument. Changes in teacher's diagnostic and remedial behaviors, uses of instructional time, in addition to the effectiveness of the program and training issues were examined. In this chapter, the findings of the study are presented. Research Question 1 What are the differences in the teachers' diagnostic procedures before and after training in identifying computational error patterns which includes a computerized diagnostic program? Specifically, (a) What diagnostic methods are used by teachers to identify student subtraction errors? (b) Do selected teachers identify computational problems for intact groups or do they identify specific error types for individual students? (c) How much time is spent on the diagnostic process? and (d) How accurate are teachers' diagnostic processes for identifying individual error patterns? -71-

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-72The following material contains a description, often verbatim, of selected teachers' reported procedures for gathering and analyzing diagnostic information regarding their students' computational error patterns for subtraction. The data reflecting student performance are contained in Tables 4.1, 4.2, and 4.3. Specific analysis of these data will be found in Tables 4.7, 4.8, 4.9, 4.13, 4.14, and 4.15 of research question 4. Teacher A and the Diagnostic Process Teacher A used both pretest grades and boardwork as the primary means of diagnosis during the pre-training period. In a conversation with the researcher on the first day of the pre-training period. Teacher A stated "Test grades tell me a lot. I can also get a feel for what the students know by their boardwork. I get them up to the board whenever I can. The test scores don't look that great, looks like I'll have to start at the very beginning of subtraction and refresh their memories." Statements such as the preceding ones indicate that Teacher A used test grades as diagnostic indicators for the class as a whole. When asked by the researcher in the same discussion whether specific problems of individual students could be identified from the pretest. Teacher A replied, " I need to work with them a bit more, but it looks like regrouping is the problem. Leatha, Keith, and

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-73Beverly have had subtraction problems since we started, and they are still having problems. I don't think they know their basic facts." Therefore, based on the pretest results, Teacher. A could at best speculate about only three students' problem areas. Teacher A reported that the pretest took approximately 45 minutes to grade. It can be concluded that the time for initial diagnosis was the time it took to grade the test. The method (s) and time for diagnosis are meaningful only to the degree that they are accurate. Teacher A's initial diagnosis was congruent with the pretest analysis by the researcher using Mathematics Assistant I. Therefore, Teacher A's diagnostic procedure was deemed accurate. Leatha, Keith, and Beverly were those students reported as having committed basic fact errors. However, the researcher identified 12 other students as having one or more of five error types, including that of basic facts and regrouping. So, while Teacher A was accurate in the diagnosis to the degree that she considered individual students, this analysis was far from representative for the entire class. The researcher's observation during the 4 days of the pre-training period provided further evidence of Teacher A's diagnostic methods. Boardwork was a part of the instruction during 2 of the 4 days. A total of 27 minutes of boardwork was recorded by the researcher. Each student

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-74Table 4 . 1 Test Results of Teacher A^s Class as Re ported by Teacher A Student Pretest Posttest 1 Posttest 2 Laura 76 82 88 Sonji 82 82 82 Shawn 76 88 94 Brandy 76 88 100 Julie 88 88 94 Brian 70 64 82 Leatha 64 76 82 Lynda 88 88 82 Beth 94 100 100 Paul 70 76 94 Burman 70 76 82 Paige 88 94 100 Donald 16 40 76 Casha 34 58 82 Keith 0 28 64 Jeff 82 70 94 Beverly 34 58 52 Summer 76 88 94 Tara 94 100 94 Wally 88 88 100 Levi 94 94 100

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-75Table 4.2 Test Results of Teacher B^s Class as Reported by Teacher B Student Pretest Posttest 1 Posttest 2 Greg 28 40 40 Clinton 52 70 76 Tracy 76 70 64 Jackie 94 94 94 Harvey 76 100 94 Ellen 82 88 88 Scott 100 94 100 Angela 82 82 76 Christina 94 100 94 April 40 70 70 Chiqueta 70 94 88 Todd 16 52 58 Teresa 28 34 52 Faith 76 82 88 Carneice 94 94 100 Termaine 22 46 58 John 34 46 64 Jason 64 88 88 Roshni 94 94 88 Jessica 94 94 100 Banyon 34 52 70 Charlie 76 76 76

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-76Table 4 . 3 Test Results of Teacher C^s Class as Reported by Teacher C Students Pretest Posttest 1 Posttest 2 Cedrick 76 82 84 Nick 82 82 82 Toni 34 58 82 Kelly 88 76 88 Amy 94 100 94 John G. 40 64 76 Adam 100 94 100 Jessica 94 94 100 Quinn 100 100 94 Rhonda 88 94 94 Brian 94 100 100 Michelle M. 94 94 100 Michelle P. 88 100 94 John P. 0 40 64 Kesha 16 70 58 Deana 58 64 82 Daniel 82 88 94 Leanne 94 100 100 Shane 82 94 82 Boe 16 46 82

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-77was given the chance to work at the board at least once during the 2 days. Of the 15 students identified by the researcher using Mathematics Assistant I as committing certain error types, only 4 students made those same errors while at the board. Posttest 1 was administered on the fifth day of the pre-training period. This day was also the day on which training was provided. Before the training began, the three teachers were asked to construct tentative lesson plans for the next 2 days of instruction. If grouping was to be used, the teachers were to indicate which students would be in each group and what the focus of the instruction was to be for each group. Teacher A's tentative plan did include using groups but she would identify group membership only after the posttests were graded over the weekend, as reflected in her statement, "I'll see the grades and put the kids in groups according to how they did." Therefore, there was no change in Teacher A's diagnostic procedure during the 4-day pre-training period. Upon arriving to observe Teacher A on the first morning of the post-training period. Teacher A gave the researcher diagnostic reports of posttest 1 generated by Mathematics Assistant I . Teacher graded test papers were also given to the researcher at this time. Teacher A had used the computerized program to diagnose the error types

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-78of the class, coming in to work on Sunday afternoon to use the computer for this purpose. For the posttest, it was obvious to the researcher that Teacher A diagnosed specific errors for individual students. Her enthusiasm was reflected by statements such as "The training session was wonderful. I never realized that my students could be making such errors. This program is a God-send." Teacher A reported that it took 1 hour and 10 minutes to enter test examples, student responses, and to generate diagnostic results. But she noted that, "As I get used to using the computer, this should take me less time." The researcher analyzed Teacher A's student responses to posttest 1 and compared them to pretest results. There was absolute agreement between the two sets of reports. It can be concluded that Teacher A's accuracy of diagnosis was enhanced by the use of computer generated data. During the post-training period, Teacher A used boardwork for 2 of the 4 days, for a total of 13 minutes. Teacher A stated, "I use boardwork now because the students like it. It also lets me know if the students are still making the errors." Note that the emphasis was not on the use of boardwork for initial diagnosis, but the use of it as a means of feedback on student progress. Posttest 2 was administered on the fifth day of the post-training period. Teacher A analyzed the results of

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-79this test, using Mathematics Assistant I during her planning period. Copies of the diagnostic reports were provided to the researcher at the end of the day. Teachergraded test papers were also given to the researcher at this time. The change in Teacher A's diagnostic methods was evident from a comparison of these two sets of data. Teacher A reported that it took a total of 55 minutes to analyze posttest 2. This required the assistance of a lab assistant, who watched the computer as the reports were printed, since Teacher A was due back in class. This amount is less time than that required for analysis of posttest 1. Researcher analysis of posttest 2 yielded results identical to those of Teacher A. Accuracy of the diagnosis continued to be high, compared to pretest diagnostic procedures. In summary, it can be concluded that there were two differences in Teacher A's diagnostic methods following instruction in the use of the computerized instrument. The computerized diagnostic program was used as the primary source of data as opposed to the earlier reliance upon test scores and observed boardwork. Diagnosis was used to detect specific errors of individual students, as opposed to the previous practice of identifying general areas of weakness for intact classes.

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-80Teacher B and the Diagnostic Process Teacher B used pretest grades as the primary means of initial diagnosis during the pre-training period. VThen asked about diagnostic methods on the first day of the pre-training period. Teacher B replied, "I used the pretest this time and I generally look at test scores to let me know who's having problems." In the same conversation. Teacher B was asked whether specific students and their problems could be identified. Her response was " The pretest scores were low. For all the years I have been teaching, subtraction has always been a trouble spot. Regrouping always has to be gone over and over. This year is probably no exception. For sure, Greg, Teresa, and John haven't learned their facts yet. Todd doesn't regroup; he just subtracts." Statements such as the preceding ones indicate that Teacher B used test grades as a diagnostic indicator for the class as a whole and specifically diagnosed on a few occasions. Also, past experience was evidently relied upon as a source of diagnostic information. Teacher B was able to identify four students and some specific error types. It could be said that Teacher B's main focus for diagnosis was directed at the class as an intact group and a lesser emphasis was directed at selected individual students and their specific error types.

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-81The initial time for formal diagnosis here was the time required for grading the papers. Teacher B reported that the pretest took 56 minutes to grade. Researcher analysis of the pretest indicated that Teacher B was correct in identifying the four students as having basic fact or inversion problems. However, 11 other students were identified by the researcher as having one or more of five error types identified by Mathematics Assistant I. Teacher B was also correct in diagnosing inversion as a problem common to several members of the class. Five students were identified as making inversion errors. However, six other students with the same problem went undetected by Teacher B's analysis of the pretest, using her habitual diagnostic methods. Observation during the 4 days of the pre-training period provided more data on Teacher B's diagnostic procedures than was initially suggested. Boardwork was engaged in on 3 of the 4 days, for a total of 30 minutes. During this time every student was given the opportunity to work at the board. Six students made errors at the board that coincided with errors detected by the researcher. Teacher B also circulated about the room during the time when students were completing seatwork. Incorrect answers were spotted by the teacher during this time. The researcher observed Teacher B correcting students' seatwork answers 18 times for a total of 29 minutes during

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-82the 4 days. Teacher B directed specific comments to specific errors in 8 of 18 instances. Posttest 1 was administered on the fifth day of the pre-training period. Training was given that same afternoon. Teacher B's responses to the questions asked before the actual training began indicated that large group work rather than small group instruction would take place for the first 2 days of the post-training period. Teacher B did identify the nine students whom she perceived to be experiencing difficulty. The perceived difficulties for these nine students were, again, either basic facts or inversion. Teacher B did acknowledge that, "I might be a student or two off because I don't have my grade book with me. These students did poorly on today's test. I'm sure about the errors that five of them are making. They've been making them all week." Researcher analysis of posttest 1 showed that Teacher B correctly diagnosed seven of the nine students identified during the training session. However, six additional students also showed evidence of existing error patterns . Teacher B reported that it took 50 minutes to grade posttest 1. The time used for diagnosis during the pretraining period included, in addition to the time required for grading tests, that required for observing boardwork.

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-83and for individualizing instruction during seatwork. Thus, the total time required for diagnosis was 109 minutes, or just under 2 hours. Teacher B did not deviate from the lessons she proposed during the training session. Large group instruction was used for the majority of time during the posttraining period. Boardwork was used only 1 of the 4 days, for a period of 13 minutes. Individual attention was given for a total of 13 minutes over the 4-day period. Small group work accounted for a majority of class time on the third and fourth days of post-training period, and totalled 1 hour. During small group activites, all groups played games of the drill and practice nature. All groups played the same games, and Teacher B did not circulate or record results during the activities. Therefore, this researcher concluded that there was little evidence of ongoing diagnosis during the post-training period. Posttest 2 was administered on the fifth day of the post-training period. Teacher B manually graded the tests and supplied the graded tests to the researcher. Teacher B remarked, "I think most of the errors at this point are careless. I need to teach them to work more neatly and to check their answers." Researcher analysis of posttest 2 showed that 12 students were committing one or more of seven error types. Teacher B's diagnosis was less accurate than for the pretest, and students' error problems persisted.

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-84Differences in Teacher B's pretest and posttest diagnostic procedures were evident. Teacher B continued to use test scores as a primary indicator. However, less time on boardwork. and less time on individual help during the post-training days indicated a lessening in the teacher's diagnostic attempts. Teacher C and the Diagnostic Process Teacher C used pretest grades and individualized attention as the primary means of diagnosis during the pre-training period. The researcher asked Teacher C about the diagnostic procedures used. Teacher C responded, "I use test grades to give me a direction. The low grades point out the students having trouble. Then I try to get around to them during class to see exactly what they're having trouble in." Statements such as the preceding ones indicate that Teacher C used test grades as a basis for student identification and then used individual work as the primary method of identifying specific student errors. Teacher C identified six students as having subtraction problems. These six students represented the six lowest grades on the pretest. Teacher C, however, was unable to identify the specific errors the students were making. In the same conversation Teacher C said, "I need to get

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-85around to them before I can tell you what they're doing. I don't have time to look at their answers real carefully. I get the scores and go from there." Initial diagnostic time was the time required for scoring of the pretest. Teacher C reported that 43 minutes were required to grade the pretest. Researcher analysis of the pretest showed that Teacher C had correctly identified six students who demonstrated error patterns. Six additional students demonstrated error patterns which were undetected by Teacher C. Teacher C was able to identify some of the students but was unable to determine with any specificity the nature of their troubles and missed others entirely. Observation during the pre-training period confirmed Teacher C's use of individualization for diagnostic purposes. Small group instruction, boardwork, and individualization accounted for about 90 minutes of class time over the 4 -day period. Teacher C met with the groups and checked at least one example for each student in each group. Several times Teacher C questioned students about their methods. Comments such as, "Show me what you did here. How did you get this number?" and "Explain what you did when you regrouped," are evidence of the diagnostic procedure in action. Posttest 1 was administered on the fifth day of the pre-training period. This was the same day that the training was provided. Teacher C's responses to the

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-86questions posed before training commenced were consistent with the pre-training practices. Teacher C commented "I'll look at the test scores to see if anyone I don't already suspect gets a low grade. Then I'll get to them in class." Teacher C's tentative lesson plans included small group work as an integral part of instruction during the post-training period. Teacher C was able to identify seven students as well as the specific problems they were demonstrating. Researcher analysis showed Teacher C to be correct in the diagnosis of five of these students. Only three other students demonstrated error patterns which were undetected by Teacher C. The researcher arrived at the school early on the first day of the post-training period. Teacher C was seated at the computer while diagnostic reports of posttest 1 were printing. Teacher C turned to the researcher and said that her plans for that day would be changed. Teacher C continued this discussion, referring several times to the fact that students can sometimes get "okay grades" but could still be committing consistent errors. Teacher C then provided the researcher with copies of the computerized diagnostic reports and the graded tests. Teacher C reported that it took 1 hour and 20 minutes to create the test, enter student answers, and to print

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-87the results. Manual test grading time was reported to be 40 minutes. Researcher analysis of posttest 1 matched that of Teacher C. Teacher C's diagnosis was deemed as accurate. Observation during the post-test period provided evidence of on-going diagnosis, as Teacher C continued the practice of working with small groups and with individuals. Questions such as those previously recorded were again evident. Therefore, Teacher C continued to focus for diagnosis on individual students and their problems . Posttest 2 was administered on the fifth day of the post-training period. Teacher C did not have time to analyze the test using Mathematics Assistant I that day but assured the researcher that the test results would be analyzed early Monday morning. Early Monday morning the researcher came by to see whether the teachers had any last minute questions or problems. Teacher C gave the graded tests and computerized reports of posttest 2 to the researcher. Teacher C reported that it took 1 hour and 5 minutes to create the test, enter the student responses, and print the reports. It took 40 minutes to manually grade the tests. It can be concluded that there was a major difference in Teacher C's diagnostic method at this point. The computerized program was added to the teacher's diagnostic

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-88repertoire. The computer results rather than test results became the major indicator of student problems. Individualization continued to be the major source of gathering data. Individual help centered on checking the continued existence of a problem rather than initial detection. Summary of Question 1 Findings To summarize, two of the three teachers demonstrated definite changes in their diagnostic procedures. Two of the three teachers adopted the computerized program as their major diagnostic tool. In addition, two of the three teachers focused on-going diagnosis on the specific error patterns of individual students. The accuracy of these two teachers increased substantially with the use of the computer program to assist in diagnosis. Additional time required to use the program was minimal and was not a deterrent to using the program. Research Question 2 What are the differences in the method (s) of remediation before and after computational error pattern training which includes a computerized program? Specifically, (a) Do the teachers remediate individually, in small groups, or in intact class situations? (b) Is the remedial strategy the same as the original teaching

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-89technique? (c) Do the teachers use coaching, demonstration, written corrective responses, student interviews, or other techniques to remediate? and (d) What materials are used, if any, for remediation? Each of the four parts of this question is addressed in the pre-training and post-training sections. Differences in teacher behaviors are discussed as part of the post-training section. Pre-Training The primary delivery system of remediation for all three teachers during the pre-training period was by large group instruction. Teachers A, B, and C spent a majority of class time presenting the same topic to all students in their respective classes. The proportion of time spent for large group, small group, and individualized instruction is presented in Tables 4.4, 4.5, and 4.6 of research question 3 . Each teacher used small group instruction and individualization as secondary methods of remediation. Teachers A, B, and C grouped students on the basis of pretest scores. The number of groups and number of students in each group varied from teacher to teacher. On the single day of the pre-training period during which Teacher A did group, there were two groups. One group consisted of the six students who received the lowest scores on the pretest. Teacher A sat and worked with this

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-90group. The remainder of the class worked individually in their seats with no assistance from the teacher. Teacher B used small group instruction on the first day of the pre-training period. There were six groups. The grouping was based on test scores. Teacher B walked from group to group not spending more than 3 minutes with each of them. Teacher C used small group work for 2 of the 4 days. On the first day, the group composition was by random selection. Students remediated each other as they quizzed each other on subtraction facts. Groups were selected on the basis of test scores on the second day. Low scorers were grouped, and the remainder of the class worked in their seats. Teacher C worked with the small group and worked with individuals at their seats when the group work was completed. All teachers worked individually with students at some point. Teachers worked with individual students during the small group instruction as well as while they traversed the room during seatwork. Therefore, the teachers remediated for large groups, small groups, and individuals during the pre-training period. Large group work dominated the use of class time. In order to determine whether the remedial strategy was the same as each teacher's initial teaching techniques, the researcher posed the following question to each teacher: "What are the differences in the way you are teaching subtraction now as opposed to the way you first

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-91presented it to this class?" The teachers' responses were as follows: TA: I'm pretty much doing the same things I did when I first taught it. I'm going a bit faster now but I'm still . teaching in the same sequence and highlighting the same points. TB: No real differences. I use different worksheets. Sometimes I even use the same ones, the students don't remember what they've done already. I use a lot of explanation — I model the procedure. TC: I haven't come up with any nifty new ways of teaching subtraction. It's basically teach and let them try it, explain the problems and let them try more. The only thing different now is that I'm not using manipulatives to demonstrate regrouping. These comments indicate that the remedial strategy was basically the same as the original teaching strategy during the pre-training period. The teachers used a variety of remedial techniques. Demonstration was the most commonly used of the methods. All three teachers used demonstration as the major part of the large group instruction. Generally, during the large group instruction, the teachers would put a problem on the board, demonstrate the procedure for solving it, and then give the students a problem to work in their seats. Teachers would then review the problem providing the correct written solution at the board, accompanied by a verbal explanation of the procedure. This procedure continued for a majority of the class time. Problems were generally assigned as seatwork after the large group instruction.

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-92Sinall group instruction consisted of a combination of demonstration and coaching. The teacher would pose a problem to the group and would coach students as they solved it. Individual attention was given to most students. Teachers would provide a verbal explanation while demonstrating the correct procedures to students doing seatwork. Coaching was the most commonly used method of remediation during individual seatwork. Coaching was also evident during the time that students were at the board. Corrective comments such as "How did you get the number in the tens place," or "What do you have to do if the bottom number is larger than the top number?" were common among the teachers. Pretest and posttest 1 papers were reviewed by the researcher. Teacher A made no written corrective responses on any of the tests. Teacher B made three corrective responses on posttest 1 for two students. All corrections highlighted the error associated with not completing the example. In each case, the teacher wrote "Finish your problems" on each of the papers. Teacher C made no corrective responses on the pretest but made 26 corrections on posttest 1. Posttest corrections consisted of writing the correct answer next to the incorrect answer on the test. The teachers used the same materials, not necessarily on the same day, during the pre-training period. Remedial materials consisted either of worksheets photocopied from a workbook of activities put out by a

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-93variety of publishers or of copies of teacher prepared worksheets with a number of subtraction examples on them. The published worksheets were game type pages where the students had to solve problems to find the punchline to a joke or the answer to a riddle. The teacher prepared practice sheets consisted of 10 to 20 subtraction examples with spaces provided to show work. Teacher C was the only one of the teachers to use additional materials, with student-made flash cards to practice basic facts in evidence in her classroom. Post-Training During the post-training period, two of the three teachers emphasized small group work as the primary means of remediation. Teachers A and C spent the majority of class time remediating specific students in specific errors in small groups. Composition of the small groups was based on the specific error type the student had committed. Teacher A used three groups during one session. The groups consisted of students making inversion errors, basic fact errors, and regrouping errors where a zero is involved. Teacher A met with one group at a time. The remainder of the students did seatwork, which was corrected by a team of students who were not identified as committing error patterns. Teacher C used small group work and also grouped students

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-94according to the computerized diagnostic information. Teacher C worked with one group of students at a time. Group membership was determined by the error type the students had demonstrated. Students worked on practice problems individually at their seats when they were not in a group. Teacher B continued to use the same strategies in the post-training period as were used in the pretraining period. Therefore, Teachers A and C changed their primary delivery system from large group instruction aimed at the class as a whole to small group remediation directed at individual students and their specific errors. Test scores were no longer the single determining factor for group composition. Students' error types as identified by the use of the computerized diagnostic program provided additional data upon which grouping decisions were made. Teacher B showed no evidence of change from the pre-training strategies. When asked about the differences in remedial strategy during the posttraining and the original presentation of subtraction, the teachers responded as follows: TA: My actual methods are the same but I spend my time differently. My explanations are better because I have a better idea of what the students are doing. I'm not doing anything differently. I'm just doing what I did before, but better. TB: I'm still doing what I've done for all my years of teaching. I keep repeating the procedure, and they eventually get it. TC: I believe more than ever in verbal explanations. I'm having the students verbalize more often. Now that I know what to look for, I can judge their comprehension by listening to them.

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-95There were no major changes in the remedial strategies themselves for the three teachers during the post-training period. Teachers continued to use a variety of remedial techniques during the post-training period. Verbal explanation was the most frequently used of these techniques. While Teacher B continued to explain and demonstrate the correct procedures. Teachers A and C added a new dimension to their remediation. Both teachers provided verbal corrective feedback directed at the specific error the student was committing rather than just the demonstration of correct procedures. Teachers A and C could commonly be heard using statements such as, "This is what you did wrong. This is the correct way" or "You forgot to make the tens place smaller. Let's go through an example together." Therefore, while the teachers still used verbal corrective feedback as their main remedial technique during the post-training period, the specificity of the feedback was increased by two of the three teachers. Data supplied by the computerized program enabled the teachers to focus on individual students and their specific errors, thus adding to the specificity of the feedback. None of the teachers put any form of corrective feedback on posttest 2 . This reflected a change in the initial grading behaviors of Teachers A and C. All three teachers continued to use the same types of worksheets previously used in the pre-training period. Teacher A made an interesting comment about the materials

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-96belng used, "They need to develop materials that are specifically designed to remediate specific errors. Do you think any of the publishers will listen to us?" Teacher B was present during this conversation and added "We can be more careful about the problems we make up; I mean whether we're putting in zeros and so forth., but it would be easier if the publishers accounted for error analysis. Surely they would be as motivated about this subject as we are." Teacher A did have the class create basic fact flash cards similar to those used by Teacher C as a result of finding basic facts to be a major problem area for several of her students. Research Question 3 What are the differences in the amount of time before and after training spent by selected teachers on large group, small group, and individualized instruction? Pre-Training Period The time in minutes for large group, small group, and individual instruction for the pre-training period is presented in Table 4.4. Large group instruction accounted for a majority of the instructional time for all three teachers during the pre-training period. For Teacher A, the average time for large group instruction over the 4 -day period was approximately four times that

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-97Table 4 . 4 Time Analysis During Pre-Trainina Dayl Day 2 Day 3 Day4 Avg. Large Group Small Group Individual 22 0 7 Teacher A 32 19 0 25 9 0 21 0 10 23.50 6.25 6.50 Large Group Small Group Individual 24 16 5 Teacher B 33 0 0 29 0 0 14 0 28 25.00 4.00 8.25 Large Group Small Group Individual 15 32 0 Teacher C 21 34 15 0 9 0 16 0 15 21.50 11.75 6.00

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-98for small group and individualized instruction. For Teacher B, the average time for large group instruction was more than six times that for small group instruction and approximately three times the average for individualization. The average time for large group instruction was almost double the average for small group instruction and more than three times the average for individualized instruction for Teacher C. Post-Training Period The time in minutes for large group, small group, and individual instruction during the post-training period is presented in Table 4.5. Teachers A and C used small group instruction for a majority of the instructional time. The average time for small group instruction was slightly higher than the average for individualization. The average time for large group instruction was higher than that for small group instruction or individualization for Teacher B. A summary of the pre-training and post-training averages by category per teacher is presented in Table 4.6. Teacher A decreased the time spent on large group instruction by half and tripled the time spent on small group instruction. The average time for individualization doubled. The large group average for Teacher B remained about the same while the time for small group instruction

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-99Table 4 . 5 Time Analysis During Post-Training Dayl Day 2 Day 3 Day4 Avg. Large Group Small Group Individual 15 10 17 Teacher A 16 14 13 12 21 6 0 28 10 10.75 18.25 11.50 Large Group Small Group Individual 32 0 13 Teacher B 29 0 0 22 25 0 4 21.75 35 15.00 0 3.25 Large Group Small Group Individual 4 35 11 Teacher C 18 8 15 24 20 8 21 12.75 0 18.50 21 15.00

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-100Table 4 . 6 Summary of Average Time fin Minutes) on Large Group. Small Group, and Individualized Instruction Category Pre-Training Post-Training Change Large Group Small Group Individual Teacher A 23.50 10.75 6.25 18.25 6.50 11.50 -12.75 +12.00 +5.00 Large Group Small Group Individual Teacher B 25.00 21.75 4.00 15.00 8.25 3.25 -3.25 +11.00 -5.00 Large Group Small Group Individual Teacher C 21.50 12.75 11.75 18.50 6.00 15.00 -8.75 +6.75 +9.00

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-101tripled. Teacher B's average for individualization during the post-training was one-third that of the pre-training. Teacher C spent a majority of her instruction time on small group instruction and the least amount of time on large group instruction following the training. All three teachers reduced the amount of time spent on large group instruction during the post-training period. Teacher B had the least amount of reduction. All three teachers increased the amount of small group interaction. Teacher C had the smallest increase in small group instruction but spent the most time using small group instruction. Two of the three teachers increased the time for individualization. Teacher B spent the least amount of time on individualization. Research Question 4 What are the differences in the effectiveness of the remediation as evidenced by the eradication of students' error patterns before and after teacher computational error pattern training which includes a computerized diagnostic program? For the purpose of this study, effective remediation was defined as the eradication of error patterns. Students' error patterns were identified by the researcher using the computerized diagnostic program, Mathematics Assistant I.

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-102The pre-training data on error eradication or persistence for Teachers A, B, and C are presented in Tables 4.7, 4.8, and 4.9, respectively. Only those students who committed an error with a frequency greater than one on both the pretest and on posttest 1 were listed. An error occurrence of only one was attributed to chance. Five error types were identified for each of the three teachers. Error 10 (basic facts or unrecognized error) and error 12 (subtracting absolute differences) were the most frequently committed of the student errors in each of the three classes. The summary of results for the pre-training data are presented by teacher in Tables 4.10, 4.11, and 4.12. For Teachers A and C, the percentage of persistent errors during the pre-training period was greater than the percentage of eradicated errors in three of the five instances. The percentage for persistent errors and eradicated errors was the same (50%) for Teacher A in one instance. For Teacher B, the percentage of persistent errors during the pre-training period was greater than the percentage of eradicated errors in all five of the instances. For the most part, the fact that errors persisted in more cases than they were eradicated for all three teachers would lead one to conclude that the pretraining remediation was not successful. The post-training data for error eradication and persistence for Teachers A, B, and C are presented in

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-103Tables 4.13, 4.14, and 4.15, respectively. Error 10 (subtracting absolute differences) continued to be the most common student error for each of the three classes. The summary, results of the post-training period are presented for Teachers A, B, and C in tables 4.16, 4.17, and 4.18, respectively. For Teacher A, the percentage of persistent errors was greater than the percentage of eradicated errors in one of six instances. The percentages were the same in one instance. For Teacher B, percentages for persistent errors were greater than those for eradicated errors in five of the seven instances. For Teacher C, the percentages for persistent errors were greater than those of eradicated errors in one of four cases. So, remediation was successful for Teachers A and C who had higher percentages of errors eliminated than they did persisting errors in a majority of instances. Remediation was unsuccessful for Teacher B, who had a continued high percentage of unremediated errors. Clearly, the post-training remediation was more successful than the pre-training remediation for two of the three teachers. It cannot be ascertained from these statistics that the training session was the cause of the increase in successful remediation. It is possible that the interaction of several factors contributed to the success of the remediation program.

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-104Table 4 . 7 Error Analysis of Pre-Trainina by Researcher of Teacher A^s Class Using Mathematics Assistant I Error Freq Error Freq Error Student Pretest Posttest 1 Amt Chan( Error 10 Laura 2 2 0 P Shawn 3 1 -2 E Brandy 3 1-2 E Julie 2 2 0 P Brian 4 2 -2 P Leatha 3 2 -1 P Paul 3 0 -3 E Burman 3 2 -1 P Donald 8 5 -3 P Casha 8 5 -3 P Keith 12 9 -3 P Jeff 3 3 0 P Beverly 4 5 +1 P Summer 4 2 -2 P Wally 2 2 0 P Error 12 Laura 2 0 —2 E Leatha 3 0 -3 E Burman 2 2 0 P Donald 7 5 -2 P Beverly 3 0 -3 E Error 13 Sonj i 3 2 -1 P Lynda 2 0 -2 E Paul 2 2 0 P Error 18 Keith 2 2 0 P Jeff 2 2 0 P Error 20 Casha 3 2 -1 P Beverly 4 0 -4 E See pp. 21-22 for descriptions of error types. E means that the error was eradicated. P means that the error persisted.

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-105Table 4.8 Error Analysis of Pre-Trainina by Researcher of Teacher B^s Class Using Mathematics Assistant I Error Freq Error Freq Error Student Pretest Posttest 1 Amt Change Error 04 Termaine 2 3 +1 P John 3 3 0 P Error 10 Gregory 10 6 -4 P Clinton 6 4 -2 P Tracy 3 2 -1 P Angela 3 2 -1 P April 6 3 -3 P Chiqueta 4 1 -3 E Todd 4 2 -2 P Teresa 10 8 -2 P Faith 2 2 0 P Termaine 7 4 -3 P John 8 3 -5 P Banyon 2 1 -1 E Charlie 3 2 -1 P Error 12 April 2 0 -2 E Todd 10 5 -5 P Banyon 5 5 0 P Ellen 3 1 -2 E Clinton 2 2 0 P Error 17 Banyon 3 2 -1 P Error 20 Gregory 2 3 +1 P Teresa 2 0 -2 E Jason 4 2 -2 P E means that the error was eradicated. P means that the error persisted.

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-106Table 4.9 Pre-Trainina Error Analysis bv Researche r of Teacher C^s Class Using Mathematics Assistant I Error Student Error Freq Pretest Error Freq Posttest 1 Amt Chanc 2 0 -2 E Error 10 Cedrick 3 2 -1 P Nick 2 0 -2 E Toni 3 4 +1 p Kelly 2 2 0 P .T r\ V» n n 4 2 -2 P Rhonda 2 1 -1 E John P. 11 4 -7 P ±\ \IZ 0 1 1 u 10 5 -5 P Deana 3 1 -2 E Daniel 3 1 -2 E Shane 2 0 -2 E Boe 5 2 -3 P Error 12 Toni 4 3 -1 P John G. 3 3 0 P Kesha 2 0 -2 E Boe 8 7 -1 P John P. 4 2 -2 P Error 17 Kelly 2 2 0 P Deana 2 2 0 P Error 20 Deana 2 3 +1 P E means that the error was eradicated. P means that the error persisted.

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-107Table 4.10 Summary Results of the Effectiveness of Remediation for Pre-Trainina for Teacher A Error type by number 10 12 13 18 20 Total # of students committing error 15 Total # of students for which error persisted 12 Total # of students for which error was eradicated % of students for which error persisted 80% 40% 66.6% 100% 50% % Of students for 20% 60% 33.3% 0% 50% which error was eradicated

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-108Table 4.11 Summary Results of the Effectiveness of Remediation for Pre-Trainino for Teacher B Error type by number 10 12 17 20 Total # of students committing error 13 Total # of students for which error persisted 11 Total # of students for which error was eradicated % of students for which error persisted 100% 84.6% 60% 100% 66.6% % of students for which error was eradicated 0% 15.3% 40% 0% 33.3%

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-109Table 4.12 Summar y Results of the Effectiveness of Remediation for Pre-Trainina for Teacher C Error type by number 4 10 12 17 20 Total # of students 1 12 committing error Total # of students for which error persisted Total # of students for which error was eradicated % of students for 0% 58.3% 80% 100% 0% which error persisted % of students for 100% 41.6% 20% 0% 100% which error was eradicated

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-110Table 4.13 Error Analysis of Post-Trainina by Res earcher of Teacher A^s Class Using Mathematics Assistant I Error Freq Error Freq Error Student Posttest 1 Posttest 2 Amt Change Error 10 Laura 2 1 -1 E Sonja 3 2 -1 P Julie 2 0 — Z ti Brian 2 2 0 P Leatha 2 2 0 P Burman 2 1 — ± ri Donald 5 4 — 1 P Casha 5 2 -3 P Keith 7 3 -4 P Jeff 3 0 -3 E Beverly 5 3 -2 P Summer 2 0 — z ri Wally 2 0 —2 E Error 12 Burman 2 1 -1 E Donald 5 0 -5 E Error 13 Sonj i 2 0 -2 E Brian 3 1 -2 E Paul 2 1 -1 E Error 14 Paul 2 0 -2 E Error 18 Lynda 2 2 0 P Keith 2 1 -1 E Jeff 2 0 -2 E Beverly 2 2 0 P Error 20 Casha 2 0 -2 E E means that the error was eradicated. P means that the error persisted.

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-IllTable 4.14 Error Analysis of . Post-Training bv Researcher of Teacher B^s Class Using Mathematics Assistant I Error Student Error Freq Posttest 1 Error Freq Posttest 2 Amt Change Error 04 Tennaine 3 4 +1 P U w>Ilil 2 0 P 1 n X yj 6 6 0 P Clinton 4 3 -1 P X X Clw jr 2 4 +2 P Angela 2 4 +2 P April 3 5 +2 P Todd 2 3 +1 P Teresa 8 6 -2 P Faith 2 1 -1 E Tennaine 4 2 -2 P John 3 0 -3 E Charlie 2 2 0 P Error 11 Tracy 3 0 -3 E Teresa 3 1 -2 E Error 12 Todd 5 4 -1 P Banyon 5 1 -4 E CLinton 2 0 -2 E Error 17 Banyon 2 2 0 P Error 18 John 3 2 -1 P Error 20 Gregory 3 2 -1 P Jason 2 2 0 P E means that the error was eradicated. P means that the error persisted.

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-112Table 4.15 Post-Training Analysis by Researcher o f Teacher C^s Class Using Mathematics Assistant I Error Student Error Freq Error Freq Posttest 1 Posttest 2 Amt Change Error 10 Cedrick 2 0 -2 E Toni 4 2 -2 P Kelly 2 1 -1 E John G. 2 3 +1 P John P. 4 1 -3 E Kesha 5 5 0 P Boe 2 1 -1 E Error 12 Toni 3 0 -3 E John G. 3 0 -3 E Boe 7 0 -7 E John P. 2 0 -2 E Error 17 Kelly 2 0 -2 E Deana 2 1 -1 E Error 20 Deana 3 2 -1 P E means that the error was eradicated. P means that the error persisted.

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-113Table 4.16 Summary Results of the Effectiveness of Remed iation for Post-Training for Teacher A Error type by number 10 12 13 14 18 20 Total # of students committing error 13 Total # of students for which error persisted Total # of students for which error was eradicated % of students for which error persisted 53.8% 0% 0% 0% 50% 0% % of students for which error was eradicated 46.2% 100% 100% 100% 50% 100%

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-114Table 4.17 Suimarv Results of the Effective ness of Remediation for Post-Training for Teacher B Error type by number 4 10 11 12 17 18 20 Total # of students 2 11 2 3 1 12 committing error Total # of students 2 9 0 1112 for which error persisted Total # of students 0 2 2 2 0 0 0 for which error was eradicated % of students for 100% 81.8% 0% 33.3% 100% 100% 100% which error persisted % of students for 0% 18.2% 100% 66.7% 0% 0% 0% which error was eradicated

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-115Table 4.18 Summary Results of the Effectiven ess of Remediation for Post-Training for Teacher C Error type by number 10 12 17 20 Total # of studentsitting 7 4 2 1 committing error Total # of students for which 3 0 0 1 error persisted Total # of students for which 4 4 2 0 error was eradicated % of students for which 42.9% 0% 0% 100% error persisted % of students for which 57.1% 100% 100% 0% error was eradicated

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-116Research Question 5 What are the effects of a 2 -hour training program using a computerized diagnostic instrument on teachers' diagnosis of computational error patterns which has as its three components (a) a general overview of computational error patterns in addition, subtraction, multiplication, and division, as identified by past research; (b) a focused introduction of error patterns specific to subtraction as identified by Janke and Pilkey, and as addressed by the computerized diagnostic program. Mathematics Assistant I; and (c) a modeling/coaching situation designed to teach the use of the computerized program. Specifically, (a) What hardware, software, and other support materials are needed for training teachers using this model? (b) How do eguipment/materials reguirements , changes in teachers' knowledge of computational error patterns, and training problems associated with each of the training components affect the teachers' knowledge of computational error patterns? and (c) What training problems encountered by the trainer in delivering the instruction? Persons wishing to implement a successful remedial program must consider the time required for training users in the content area and in the proper use of program materials, in addition to the availability of necessary equipment and supplementary materials. Data describing aspects of a successful remedial program are reported in

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-117detail for this study to provide a frame of reference for others considering incorporating computerized error diagnosis in a remediation program. For the current study, hardware and software requirements are the same for both teacher training and program implementation. A 48K Apple computer with either single or dual disk drives is required. This is the most expensive item of the equipment needed. A printer is also necessary, in order to provide for hard copies of the diagnostic reports. Blank diskettes are needed for data storage. An allowance of two diskettes per teacher is sufficient to accomplish the goals of the training program . Mathematics Assistant I, a diagnostic program produced by Scholastic Inc. and written by Janke and Pilkey (1985) , is the only piece of software necessary. This software is packaged for retail sale with the program diskette, a backup diskette, and a blank data diskette. Printer paper is the only incidental expense, with the average diagnostic report requiring three to four pieces of paper. Supplementary materials are needed for the training component. These materials are supplied at the discretion of the trainer but should include information on the types of errors teachers can expect to encounter. The role of the trainer is an important one. The ideal trainer is an individual versed in computational

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-118error patterns. The presence of such a person allows for a high degree of interaction, immediate answering of relevant questions, and modeling of the procedure for using the computerized program. A list of common computational errors must be provided to teachers if a skilled diagnostician is not available. This alternative delivery system will work only if the teachers are able and willing to review the written information. In addition, teachers utilizing this alternative delivery system would be required to read the documentation supplied with the published program. Finally, the computerized program would have to be self-taught through a trial and error method. The advantages of having a skilled trainer cannot be stressed too heavily. Selfinstruction is not an ideal vehicle for accomplishing necessary instruction for this program. The total time required for the training session was 2 hours. The training session consisted of three components. The first component was a brief discussion of the most common computational errors in addition, subtraction, multiplication, and division, as identified in previous research. The errors on Table 4.19 were reviewed one at a time to ensure that all the teachers understood the patterns. This presentation was completed in 15 minutes. Teachers commented on the types of errors reported as they reviewed the table provided them. Sample teacher/trainer interaction included such comments as

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-119Table 4.19 Common Errors Identified in Past Research Error Patterns Multiplying vertically Adding columns, with no attention to place value or regrouping Examples 123 X 42 186 432 X 229 878 58 + 83. 1311 Using an incorrect operation Difficulty with the concept of zero or one 4 2 6 4 1 1 2 0 0 4 0 0 Working from left to right 385 + 667 9116 Borrowing when not needed 285 53 212 Adding carried numbers prior to multiplying 24 X 6 244

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-120Table 4.19 (continued) Error Patterns Examples Not allowing for having borrowed 63 47 26 Failing to borrow, giving zero as the answer 47 28 20 Failing to add the carried number 769 + 878 X 1537 23 8 164 Carrying the wrong number 93 X 1_ 642 Misplacing columns in multiplication 234 X 12 468 234 Omitting zero in quotient 19 28 / 3052 28 252 252

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-121Table 4.19 (continued) Error Patterns Examples Digits of addends are summed, disregarding place value 33 29 (3 + 17 3 + 2 + 9 = 17) Adding single digits addend to both digits of the other addend 46 3 79 Dividing the divisor into each digit without forming any partial products 101 / 608 Poor alignment of digits in columns 318 1241 509 + 13 Subtracting absolute differences 42 19 37

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-122TA: Do they really do all of these mistakes? R: These errors have been identified as the most common, not all students do all the errors. TC: I don't understand number 7. TB: I don't recall any of my students making error number 15, but maybe I never knew what to look for. I certainly never knew about many of these. As evidenced by the teachers' comments, they were not aware of the variety of computational errors identified by previous researchers' efforts. Gradually the teachers began to conceptualize these errors types in terms of their students' problems and of their own diagnostic capabilities as teachers. Component two consisted of an examination of the teacher generated lists of students' subtraction errors followed by the researcher's provision of a list of 20 subtraction errors identified by Janke and Pilkey (see pp. 21-22) The lists of student errors as perceived by the teachers, reported exactly as compiled by the teachers, are found in Table 4.20. Discussion of the teacher-identified error types revealed three errors common to all lists. They were subtracting absolute values, failing to rename the digit from which borrowing was done, and increasing rather than decreasing the digit borrowed from. Following a discussion of the teacher-generated error listings, the researcher distributed a list of 20 common errors as identified by Janke and Pilkey. The teachers

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-123Table 4.20 Student Errors Perceived by Teachers Errors Perceived Teacher A 1. Subtract top number from bottom number. 2. When borrowing — forgetting to decrease the unit borrowed from (i.e., 36 8 = 38). 3. Forgetting to bring down number in hundreds place (any place past one really) when there is not a number subtracted from that place (i.e., 123 12 =11). 4. Increasing the place borrowed from instead of decreasing it. Teacher B 1. Forgetting to regroup. 2. Forget to make the number in the tens place 1 less. 3. Sometime the student makes the number renamed larger. 4. Sometimes after regrouping the error will be basic facts. 5. Sometimes instead of renaming, the student would reverse the numbers to subtract. 6. Sometimes they will add in part of the problem forgetting to watch the sign. Teacher C 1. When regrouping, they add a number instead of subtract. 2. Subtract the top number from the bottom number. 3. They forget to show that they've taken away from one place.

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-124Table 20 (continued) Errors Perceived 4. Basic facts not known. 5. Borrowing when they don't need to borrow. 6. Not finishing the problem (i.e., 163 19 = 44) 7. Skipping across zeros and not marking them out.

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-125appeared to be astonished by the relatively large number of possible error types contained therein. After teachers had an opportunity to review the list, the following dialogue ensued. R: Here is a list of 20 of the coTnmon subtraction errors that Janke and Pilkey found when they did a study of childrens' computational errors. Do you think you would be able to identify the errors your students are making if you had this error list as a reference? TA: I might be able to pick out more than I can now but I don't think I'd get them all. TB: Probably — but it would take me forever. I probably would deal with the easiest to detect. For sure, I wouldn't do this analysis all too often. TC: It would take practice. I'd probably spend the whole weekend just doing the results from today's test. I'm interested in what you're showing us about errors but I don't think teachers would use this information much if it takes so much time to get the errors. This is only for subtraction, we'd have to be sharp on all the operations. A comparison by the researcher of the teacher error type lists and the 20 errors identified by Janke and Pilkey (see pp. 21-22) showed that the teachers identified without prompting only 3 to 6 of Janke and Pilkey 's 20 errors. Of these errros. Teacher A named three (error numbers 4, 12, 13) , Teacher B named 6 (error numbers 4, 10, 11, 12, 13, 20), and Teacher C named 5 errors (error numbers 4,10,12,13,18). This second component served two purposes. First, the teacher's lack of knowledge of error patterns specific to subtraction, the topic they were currently remediating, was evidenced by the fact that they identified less than 30% of the possible 20 error types.

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-126The teachers' awareness of their limitations as diagnosticians was heightened. Second, the teachers realized that, while error diagnosis is an important part of the teaching/remediating process, a great amount of time and training is necessary for a teacher to be proficient at error pattern diagnosis. The third training component consisted of a description and demonstration of the computerized diagnostic instrument. Mathematics Assistant I. The researcher explained that Mathematics Assistant I would allow them to (a) create an addition or subtraction test, (b) print the test, and (c) enter student responses to the test items and receive diagnostic information for each item/student/class. The researcher explained that Mathematics Assistant I reports provide 1. Statistics on individual students, including each student's score, the type(s) of error (s) made, and the number of the example (s) on which the errors were committed. 2. Class statistics by error type, indicating the error number, names of students who committed the error, and the frequency with which each student made the error. Following the delivery of this information, the researcher initiated the following discussion: TA: Do you have to know a lot of programming and computers to do this? R: No, you only have to know how to start the computer and use the program. There is no programming involved.

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-127TC: Go slow. The kids are good at these things, but I'm still a bit shaky. TB: Will this Apple stay here so we can have access to it in private? R: Yes, that has already been arranged with your principal . In order to familiarize the teachers with the use of the equipment, the researcher demonstrated the start-up procedure. The procedure was executed slowly, since the questions previously asked reflected some participants' computer anxiety. A discussion on the care and handling of diskettes was not necessary, however. The teachers indicated that they knew the "do's and dont's" of diskette management from a lecture previously provided by the computer lab director. The group then elected to create a 10 -quest ion subtraction test, since that was the current topic of remediation in all three classes. The researcher entered the problems at the keyboard while the teachers provided the 10 test examples. The teachers seemed to take the error pattern information into consideration while constructing the examples, as one can see in the following dialogue: TB: They have trouble with zeros. Here's one. . . . TC: Two regroupings, that should be a good one. Try this. . . . TA: Let's try to pick examples to test out if they'll make some of these errors we haven't seen a lot. These errors make you think a bit more about test construction.

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-128Three copies of the 10-item subtraction test (Table 4.21) were printed for use by the teachers. The teachers were asked to work individually and provide answers to the test items as if they were errant students. The researcher then entered the teacher answers to demonstrate the diagnostic capabilities of the program. The teachers were surprised by the program's user friendliness and the ease of the data entry procedure. When all data had been entered, two statistical reports were provided. Table 4.22 is the individual report generated by Mathematics Assistant I. Analysis of Table 4.22 indicates that the three teachers committed only five type of errors. Basic fact errors and subtracting absolute differences (inversion) were the most frequently occurring errors committed by the teachers . Table 4.23 contains an analysis of the class report by error type. After receiving both sets of information, each teacher reviewed the reports for accuracy. They confirmed the fact that all errors committed by the teachers were indeed identified and reported by Mathematics Assistant I. Following this demonstration of the computer program, the start-up procedure was once again reviewed. At this point, Teacher A and Teacher B took notes on the procedure, while Teacher C seemed not to need written reminders. Following the review, each teacher was given

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-129Table 4.21 Test Generated bv Teachers Purina Traini ng Session Using Mathematics Assistant I 1) 200 2) 602 3) 800 145 375 261 Answer: 55 Answer: 227 Answer: 539 4) 4900 5) 6301 6) 7803 598 189 z 567 Answer: 4302 Answer: 6112 Answer: 7236 7) 1800 8) 4008 9) 8047 589 617 153 Answer: 1211 Answer: 3391 Answer: 7894 10) 1006 432 Answer: 574

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-130Table 4.22 Error Analysis by Mathematics Ass istant I of Teacher Responses to Training Test Problem Number 1 23456789 10 Name: Teacher A 12 10 C 12 Name: Teacher B 12 10 1 11 Name: Teacher C Test: Training 4 10 C C 12 C Test: Training 10 10 C 10 C C Test: Training Score; 40% Score: 30% Score: 20% 12 10 C 10 10 10 10 C 12 10 Note: This table reflects the form of the printout of results produced by Mathematics Assistant I.

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-131Table 4.23 Summary of Error Types bv Error of Teac hers During Training Session Error Type : Student Frequency Error 1: Teacher B (1) Error 4 : Teacher A (1) Error 10: Teacher A (2) Teacher B (4) Teacher C (6) Error 11: Teacher B (1) Error 12: Teacher A (3) Teacher B (1) Teacher C (2) the opportunity to complete the start-up procedure. Teacher C went first and quickly moved through the process. Teacher B went through the process, referring to her notes on two occasions. Teacher A first seemed hesitant, but quickly became involved in using the computer. She referred to her notes three times. Then she asked whether I would be available on Monday morning to sit with her when she entered her students' test data. In her words, "I'd feel better if you were around for the first time, in case I get stuck." A time was arranged on

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-132Monday for her to enter the scores under the direct observation of the trainer. The third component of the training phase was instrumental to the success of the training session. The modeling segment helped to ease any possible computer anxiety which existed on the part of the teachers. The effectiveness of the diagnostic program was illustrated by the accuracy, usability, and sophistication of the reports generated. The coaching segment further reduced teachers' fears of not being able to use the computer by providing them with hands-on experience. It was apparent that all anxiety was not eliminated since Teacher A still wanted the security of the trainer when she made her first attempt at entering student data. There were no problems during the training session. The lack of difficulties could be attributed to the small number of participants. It was simple for the three teachers to sit around the computer during the demonstrations. Had there been more participants, seeing the screen would have been more difficult. In addition, all questions were addressed immediately and at length if necessary. All participants knew each other and were not hestitant to ask or answer questions. The possible anxiety of not being proficient at using the computer was minimized because fears were openly expressed. The group supported each other during the hands on component. So group size enhanced the possibility that students felt

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-133comfortable, that they could ask clarifying questions, and that fears of computer incompetency could easily be resolved.

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CHAPTER V DISCUSSION, CONCLUSIONS, IMPLICATIONS, AND SUGGESTIONS FOR RESEARCH If teachers are to effect significant improvement in the arithmetic skills of children, they must be well versed in the use of practical and effective remedial strategies. A remedial program which addresses the instructional and remedial processes and the problems associated with these processes must be developed if teachers are to be expected to succeed in this endeavor. The purpose of this study was to ascertain the effects of using computer generated diagnostic data on third grade students' subtraction error types to facilitate remediation of computational errors in subtraction. A case study approach incorporating 4 days of observation, a 2-hour training session on error patterns and the use of a computerized diagnostic instrument, followed by 4 additional days of observation was used to gather relevant data. A major finding of this study was that a remedial program incorporating the use of a computerized diagnostic program precipitated changes in teachers' behaviors which, in turn, resulted in more effective remediation. Two of the three teachers who used Mathematics Assistant I, the -134-

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-135computerized diagnostic program, consistently showed higher percentages of error eradication than they had demonstrated earlier, when the computerized instrument was not used. Responses to the Research Questions Differences in Diagnostic Procedures Observation during the pre-training period showed that test scores were the primary diagnostic method used by the teachers to determine student weaknesses. Boardwork and individualization also provided the teachers with diagnostic data but were not used to the extent that test scores were. It is most likely that the time constraints on teachers did not allow for extensive diagnostic evaluation. In addition, the teachers were not aware of many of the possible error types in subtraction. Thus, the lack of comprehensive diagnostic procedures was due in part to the lack of teacher training in error patterns. As a result of training, two of the three teachers changed their diagnostic methods to include the computerized diagnostic instrument as the primary source of data. Perhaps the sophistication of the computer generated data and the ease of obtaining it accounted for this change. One teacher did not change diagnostic methods. This teacher announced shortly after the completion of this study that she was retiring. It is

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-136possible that this teacher did not feel a need to improve or change because of her pending retirement. The researcher's discovery that the teachers diagnosed problem areas for intact classes as opposed to identifying specific errors of individual students during the pre-training period was consistent with the finding that teachers used test scores as their primary diagnostic indicator. Lack of teacher training in error pattern detection is again the most likely explanation for their behavior. Once given the training on error patterns and the use of the computerized instrument , two of the three teachers began diagnosing specific errors for individual students. The change in their practices during the posttraining period is again attributed to the training. The lack of change on the part of one teacher is accounted for by the previous explanation relating to this teacher's retirement. The time required for diagnosis during the pretraining period consisted predominantly of the time needed for grading test papers. The time used by teachers to diagnose students while at the board or during individual conferences while students did seatwork was also included in the total time needed for diagnosis. The use of the computer for the purposes of diagnosis did not reduce the amount of diagnostic time needed. In fact, the time required to generate computerized data exceeds the time required to grade tests and to observe students' class

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-137performance. Even though more time was required for the new procedure, two teachers did use the computer for diagnostic purposes during the post-training period. Accuracy of the results generated in the relatively modest time needed to use the program is a probable reason for the teachers' change of behavior. In addition, the training highlighted the need for teachers to be excellent diagnosticians. The teachers saw this tool as a feasible way to address their lack of training and ability in what they perceived to be a crucial area in the teaching of arithmetic. It is also possible that the teachers realized that instructional time previously spent on boardwork and individualization for the purposes of diagnosis could be more efficiently used for remediation. Therefore, the increase in time required to use the computerized instrument was offset by the reduction in the time needed for previous in-class diagnostic proceduress. The degree of diagnostic accuracy during the pretraining period was substantially lower than was that of the teachers following training, when they used the computerized program. This increase in accuracy appears to be a direct result of use of the computer program. The one teacher who did not use the program showed no differences in preand postobservation diagnostic accuracy. This lack of improvement reflects the fact that the teacher continued using the same methods employed during the pre-training period.

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-138Differences in the Methods of Remediation It was found that during the pre-training period the three teachers remediated their classes primarily as intact units. The same topic was presented the same way to all students in the class, irrespective of individual mastery level. Small group work and individual seatwork accounted for a relatively small proportion of instructional time. Two of the three teachers changed their instructional emphasis to small group work and individualization during the post-training period. Detailed diagnostic data obtained by use of the computerized program alerted the teachers to the specific error patterns of individual students. Remediation of the specific errors of individual students is most effectively accomplished by the use of small groups or by individual instruction. Changes in the criteria for group formation from ranking on test scores to common error types reflect a major change in remedial method. No such change was identified for one of the teachers. This teacher did not use the computerized program and thus did not have the advantage of the detailed diagnostic infoirmation regarding individual students' performance. There were no changes identified in the remedial strategies themselves between the pre-training and the post-training periods. The teachers continued to teach subtraction using the same instructional approaches as

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-139when they originally taught the topic. Perhaps if special remedial strategies appropriate for use with specific error types had been presented as part of the training, the teachers might have tried new strategies, allowing additional changes to be noted. Demonstration to the intact classes was the primary remedial technique used by the three teachers during the pre-training period. The shift in instructional time from large group to small group instruction and to individualization is the most likely reason for the teachers' change to coaching as the primary remedial technique. Detailed diagnostic data enabled the teachers to directly address the students' problems. No changes in supplementary materials were noted between the pre-training and post-training period. The lack of change in the remedial strategies obviated the need for new materials. To this researcher's knowledge, materials which directly address the remediation of error patterns are either unavailable or inaccessible to the average classroom teacher. In any case, materials of this nature were not available to these teachers. Differences in the Amount of Instructional Time Differences in the amount of time spent on large group, small group, and individualized instruction were noted for two of the three teachers. A reduction in the

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-140large group time and an increase in the small group individualization and time characterized the post-training behaviors of two of the teachers. Small group and individualized instruction became the most frequently used type of instruction during the post-training period. The differences in the use of time reflect the differences in the teachers' remedial strategies. No differences in the use of time by Teacher B were noted due to the fact that this teacher did not change any remedial strategies from one observation period to the next. Differences in the Effectivenes s of the Remediation Differences in the effectiveness of remediation during preand post-observation were evident for two of the three teachers. All three teachers had a higher percentage of students errors which persisted than they did errors which were eradicated during the pre-training period. Major differences in the effectiveness of remediation were noted for two of the three teachers during the post-training period. For the two teachers demonstrating differences, a higher percentage of student errors were eradicated and a lower percentage of errors persisted. The detailed diagnostic data, coupled with the emphasis on small group and individualized instruction aimed specifically at individual students and their errors, are a

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-141plausible explanation for the increase in error eradication. No differences were evident for Teacher B. Absence of detailed diagnostic information and the continuation of remediation not specifically directed at individual students and their particular error types are likely reasons why no change was noted. Implementation of the 2 -Ho ur Training Program All three teachers were receptive to the training program. Training did not involve the use of elaborate materials, excessive time, or expensive computer programs and equipment. The fact that the training was directed at the teachers' immediate needs helped establish the positive attitude with which the trainer was met. The teachers were able to see that implementation of the remedial program was possible. The computer, printer, and software program were available in the school. Training on computational error patterns and on how to use the diagnostic software were the only elements necessary for program implementation. The training program had three components. The first component was an introduction to computational error patterns in addition, subtraction, multiplication, and division. This component aroused the interest of the teachers and provided a foundation from which to build. The second component narrowed the topic of error patterns to deal specifically with subtraction. The teachers'

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-142interest was piqued, since subtraction was the topic they were teaching at the time. It was important that the training be relevant to the teachers' needs. The information on error types and their diagnosis alerted the teachers to the fact that they were not competent diagnosticians. In addition, the researcher believes the choice of a timely and pertinent program to have been crucial. Also, the teachers probably received the training well because they felt that the training was for "them." The trainer was able to refer to instances based on observations in their classrooms. The provision for teachers of this personalized attention added to the success of the training. The third component was a modeling/coaching simulation where the power of the diagnostic program was demonstrated. The teachers were presented with a solution to the newly conceived (for them) problem of diagnosis. The training program equipped the teachers with the tool needed to promote more effective diagnosis, and ultimately, as this study has shown, more effective remediation. No training problems were encountered. This is probably because the teachers were willing to learn. Computer anxiety did not frighten them off. The researcher believes that the value of the diagnostic instrument increased the teachers' willingness to learn how to use the computer. The researcher also believes that the size of the group was an important variable in

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-143the success of the training. The training group was small and personal. Each teacher was able to have hands-on practice, with personal attention from the trainer. Conclusions The following conclusions emerge from this study: A remedial program which incorporates the use of a computerized diagnostic program influences teacher behaviors so as to promote more effective remediation of computational error patterns of third grade students' subtraction. Given training in computational error patterns and a computerized diagnostic program, teachers will more efficiently diagnose students' errors. Detailed diagnostic data provide sound basis upon which teachers' remedial decisions can be based. Small group and individualized instruction directed at specific error types and at the students committing these errors is more effective for remediation than is large group instruction. Implications Implications derived from the results of this study are the following: Teachers can use instructional time much more efficiently than they presently do. Detailed diagnostic

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-144data can save valuable instructional and remediation time. This time can be spent in areas that are typically neglected due to lack of time. Diagnostic data can be used to design activities for problem solving or other application level activities. Teacher education programs should include training on computational error patterns and the use of diagnostic instruments capable of detecting such patterns if teachers are to be successful in identifying and remediating error patterns . It may be necessary to re-think how educational leaders decide inservice training priorities. Often teachers perceive inservice training as irrelevant, repetitious, and not directed at teachers' needs. It may be that inservice should be developed as a result of observation of teachers' classes. Suggestions for Further Research 1. This study should be replicated to enhance the validity of results obtained. 2. Additional research is needed to determine the most effective remedial strategies for specific types of errors . 3. Research is needed to ascertain the effectiveness of verbal feedback directed at specific error types versus programmed materials designed to address specific error types .

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-1454. Research is needed to verify whether the results of this study are generalizable to addition, multiplication, and division and to other grade levels. 5. Computerized diagnostic programs need to be developed for other topics in mathematics. Research must be conducted to determine the effects of such programs on teacher behaviors and remediation success rate. 6. More case studies must be conducted to enable teacher educators and researchers to gain a better understanding of the problems confronting classroom teachers .

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REFERENCES Abbott, G.A., & McEntire, E. (1985). Effective remediation strategies in mathemat ics: Characteristics of an effective remedial mathematics teacher; effecti ve remedial math teacher checklist; ma th remediation methods questionnaire . Washington, DC. National Institute of Education. (ERIC Document Reproduction Service No. ED 262 459) Anderson, A. (1981) . The need for fully trained mathematics diagnosticians in the elementary schools. The Arithmetic Teacher . 28, 32-33. Ashlock, R.B. (1982). Error patterns in computation; A semi-programmed approach . Columbus, OH: Charles E. Merrill. Herman, B. , & Friederwitzer , F.J. (1981). A diagnostic prescriptive approach to remediation of regrouping errors. The Elementary School Journal . 82, 109-115. Blankenship, C.S. (1979). Reduction of systematic inversion errors in the subtraction algorithm (Doctoral dissertation. University of Washington, 1979) . Dissertation Abstracts International . 37, 4273A-4274A. Bogdan, R.C., & Biklen, S.K. (1982). Qualitative research for education; An introduction to theory and methods. Boston: Allyn and Bacon, Inc. Bright, G. W. (1984) . Computer diagnosis of errors. School Science and Mathematics . 84, 208-219. Brown, J.S., & Burton, R.R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science . 2, 155-192. Brueckner, L.J. (1930) . Diagnostic and remedial teaching in arithmetic . Chicago: John C. Winston. Brueckner, L.J. (1935a) . Introduction. In G.M. Whipple (Ed.), Educational diagnosis; The thirtv-fourth yearbook of the National Societv for the Study of Education (pp. 1-4) . Bloomington, IL; Public School Publishing Company. -146-

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-147Brueckner, L.J. (1935b). Techniques of Diagnosis. In G.M. Whipple (Ed.), Education diagnosis; The-Thirtyfourth yearbook of the National Society for the Study of Education (pp. 131-153) . Bloomington, XL: Public School Publishing Co. Carpenter, T.P., Coburn, T.G. , Reys, R.E., & Wilson, J.W. (1975). Results and implications of the NAEP mathematics assessment: Elementary school. The Arithmetic Teacher . 22, 438-450. Carpenter, T.P., Kepner, H., et al. (1980). Results and implications of the second NAEP mathematics assessments: elementary schools. The Arithmetic Teacher . 27, 10-12, 44-47. Cebulski, L.A. (1984). Children's errors in subtraction: An inyestioation into causes and remediation. Ottawa , Canada. (ERIC Document Reproduction Service No. ED243684) Cox, L. S. (1975). Systematic errors in the four yertical algorithms in normal and handicapped populations. Journal for Research in Mathematics Edu cation. 6, 202220. Dallenback, L. , Schoenberger, R. , & Wehrs, W. (1977). An eyaluation of the cognitiye and affectiye performance of an integrated set of CAI mate rials in the principles of macroeconomics . LaCrosse, WI: Uniyersity of Wisconsin, 1977. (ERIC Document Reproduction Service No. ED 150 057) Denmark, T. (1976). Reaction paper: Classroom diagnosis. In J.L. Higgins & J.W. Heddens (Eds.), Remedial mathematics; Diagnostic and prescriptiye approaches, (pp. 55-62). Columbus: Ohio State Uniyersity, ERIC Center for Science, Mathematics, and Enyironmental Education. Denmark, T. & Munoz, R. (1980). A diagnostic model for use by classroom teachers. In T.A. Romberg (Ed.), Research reports from the Research Council on Diagnostic and Prescriptiye Mathematics (pp. 130-144) . Gainesyille, FL: Renaissance Printing. Duncan, R.K. (1981) . The effects of two instructional treatments on increasing computational skill in subtraction with regrouping of grade three students (Doctoral dissertation. Indiana State Uniyersity, 1980) . Dissertation Abstracts International . 41 . 3876A.

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1 * -148Edwards, J., Norton, S., Taylor, S., Weiss, M. , & Dusseldorp, R. (1975). How effective is CAI? A review of the research. Educational Leadership , 33/ 147-151. Engelhardt, J. M. (1977). Analysis of children's computational errors: A qualitative approach. British Journal of Educational Psychology . 47, 149-154 Goldberg, M. , & Harvey, J. (1983). A nation at risk: The report of the National Commission on Excellence in Education. Phi Delta Kappan . 65, 14-18. Graeber, A. O. & Wallace, L. (1977). Identification of systematic errors: Final report . Philadelphia: Research for Better Schools, Inc. (ERIC Document Reproduction Service No. ED 139662) Harvey, L.F., & Kyte, G.C. (1965). Zero difficulties in multiplication. The Arithmetic Teacher . 45-50. Hetzel, M. C. (1981). A system for the remediation of student subtraction errors (Doctoral dissertation, Brigham Young University, 1981) . Dissertation Abstracts International , 42 , 983A. Hill, S.A. (1983). The microcomputer in the instructional program. The Arithmetic Teacher . 10, 14-15, 54-55. Hopkins, M. H. (1978). The diagnosis of learning styles in arithmetic. The Arithmetic Teacher . 26, 47-50. Hutchings, B. (1976). Low-stress algorithms. In D. Nelson & R. Reys (Eds.), Measurement in school mathematics (pp. 218-239) . Reston, VA: National Council of Teachers of Mathematics. Janke, R.W. (1980) . Computational errors of mentally retarded students. Psychology in the Schools . 17, 30-32. Janke, R.W. , & Pilkey, P. (1985). Microcomputer diagnosis of whole number computational errors. Journal of Computers in Mathematics and Science . 14, 45-51. Junge, C.W. (1972). Adjustment of instruction (elementary school). Chap. 5. In W.C. Lowry (Ed.), The slow learner in mathematics. 35th yearbook (pp. 129-162) . Reston, VA: National Council of Teachers of Mathematics. Kilian, L. , Cahill, E. , Ryan, C, Sutherland, D. , & Taccetta, D. (1980) . Errors that are common in multiplication. The Arithmetic Teacher . 27, 34-37.

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-149Klingele, W. E.,& Reed, B. W. (1984). An examination of an incremental approach to mathematics. Phi Delta Kappan, 65, 712-713. Lazar, M. (1928). Diagnostic and re medial work in arithmetic for intermediate grades . New York: Bureau of Education of New York City. Levin, H. M., Glass, G. V. , & Meister, G. (1986). The political arithmetic of cost effectiveness analysis, Phi Delta Kappan . 68 . 69-72. Lindquist, M.M. , Carpenter, T.P., Silver, E.A., & Matthews, W. (1983). The third national mathematics assessment: Results and implications for elementary and middle schools. The Arithmetic Teacher . 31, 14-19. Lysiak, F. , Wallace, S., & Evans, C. (1976). Computerassisted instruction 1975-76 evalu ation report 495. Fort Worth, TX: Fort Worth Independent School District, 1976. (ERIC Document Reproduction Service No. 140) McKillip, W. D. (1981). Computational skill in division: Results and implications from national assessment. The Arithmetic Teacher . 28/ 34-37. Mettair, J. E. M. (1982). The use of error pattern analysis in the diagnosis and remediation of whole number computational difficulties (Doctoral dissertation, Texas A & M University, 1981) . Disserta tion Abstracts International . 42., 4343A. Morgan, C, & Richardson, W.M. (1972). The computer as a classroom tool. Educational Technology . 12., 71-72. Piaget, J. (1975) . The eauilibrium of cognitive st ructures. Chicago: University of Chicago Press. Phelps, C.L. (1913). A study of errors in tasks of adding ability. Elementarv School Teacher . 14, 29-39. Pincus, M., Coonan, M. , Glasser, H., Levy, L. , Morgenstein, F., & Shapiro, H. (1975). If you don't know how children think, how can you help them? The Arithmetic Teacher . 22, 580-585. Reisman, F.K. (1972). A guide to the diagnostic teach ing of arithmetic . Columbus, OH: Charles E. Merrill. Richbart, L.A. (1980) . Remedial mathematics program considerations. The Arithmetic Teacher . 28 . 22-23.

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-150Roberts, G. H. (1968). The failure strategies of third grade arithmetic pupils. The Arithmetic T eacher. 15, 442-446. Ronau, R. N. (1986) . Mathematical diagnosis with the microcomputer: Tapping a wealth of potential. The Arithmetic Teacher . 79, 205-208. Ross, C.C., & Stanley, J.C. (1954). Measurement in today ^s schools . New York: Prentice Hall. Sadowski, B.R. (1979). An investigation of the validity, accuracy and consistency of prescriptions derived from a domain-referenced diagnostic arithmetic test and interview protocols (Doctoral disseration. University of Maryland, 1987) . Disseration Abst racts International . 40 . 1422A. Skrtic, T.M., Kvam, N.E., & Beals, V.L. (1983). Identifying and remediating the subtraction errors of learning disabled adolescents. Pointer . 2T_, 32-38. Smith, J.H. (1916). Individual variations in arithmetic. Elementary School Journal . 17, 195-200. Sovchik, R., & Heddens, J.W. (1978). Classroom diagnosis and remediation. The Arithmetic Teacher . 25, 47-49. Tyler, R.W. (1935) . Characteristics of a satisfactory diagnosis. In G.M. Whipple (ed.). Educational diagnosis, the thirty-fourth yearbook of the National Society for the Study of Education (pp. 113-129) . Bloomington, IL: Public School Publishing Co. Underhill, R.G. (1972). Teaching element ary school mathematics . Columbus, OH: Charles E. Merrill. Underhill. R.G., Uprichard, A.E., & Heddens, J.W. (1980). Diagnosing mathematical difficulties . Columbus, OH: Charles E. Merrill. Voran, J.G. (1985) . Comparison of two methods for remediating subtraction difficulties at the fifth grade level (Doctoral dissertation. The Florida State University, 1985) . Dissertation Abstracts International , 46, 1549A. West, T.A. (1971). Diagnosing pupil errors: Looking for patterns. The Arithmetic Teacher , 17, 467-469.

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BIOGRAPHICAL SKETCH Janet Bosnick (nee Nathanson) was born and raised in Hicksville, New York. She graduated from Hicksville High in 1969 with an academic diploma. Janet attended Queens College of the City University of New York where she graduated in 1973 with a Bachelor of Arts degree in mathematics. Mrs. Bosnick began her teaching career in 1973 at Rachel Carson Intermediate School in Flushing, Queens, where she taught math, English, and science in addition to sponsoring the student government. During this time she pursued and earned a Master of Science degree in foundations of education from Hofstra University. In July, 1976, Janet married Jay, a long time friend and tennis partner. In search of a change in life-style, Janet and Jay moved to Jacksonville, Florida in 1979. Jay continued his career as a computer programmer and Janet taught mathematics at duPont Junior High under the mentorship of Dr. Olive Luten. Mrs. Bosnick entered the University of Florida's doctoral program in curriculum and instruction in 1983. Her specialty area was instructional computing. She is currently the director of the Learning Lab of the College of Education and Human Services at the University of North -151-

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1 -152Florida. In addition, to her lab responsibilities, Janet teaches secondary math methods on the graduate and undergraduate levels.

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. William Hedges, Chairman Professor of Educational Leadership I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Patricia Ashton Professor of Foundations of Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Forrest Parkay Associate Professor of Educational Leadership

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. This dissertation was submitted to the Graduate Faculty of the College of Education and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1987 Mary Grace Kantowski Professor of Instruction and Curriculum Robert Drummond Professor of Counselor Education Dean, College of Education Dean, Graduate School