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ENE-wheat} » i 4 . 4 Its/K 4.. 3A.... 3.1V g! ) S U ITHE: LIBRARY Michigan State University This is to certify that the dissertation entitled A COMPARATIVE EVALUATION OF PROGRAMMED AND LECTURE INSTRUCTION IN COLLEGE BUSINESS MATHEMATICS presented by Manfred E. Swartz has been accepted towards fulfillment of the requirements for Ph.D. degreein Teacher Education {2&4} fl- 7’7 Mm Major professor Date April 15, 1985 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES “ \— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. ‘FINES will be charged if book is returned after the date stamped below. “WI- A COMPARATIVE EVALUATION OF PROGRAMMED AND LECTURE INSTRUCTION IN COLLEGE BUSINESS MATHEMATICS By Manfred E. Swartz A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Teacher Education 1985 ABSTRACT A COMPARATIVE EVALUATION or PROGRAMMED AND LECTURE INSTRUCTION IN COLLEGE BUSINESS MATHEMATICS By Manfred E. Swartz The evaluative study was conducted during one ll-week term using 235 students enrolled in Business Mathematics l21 at Ferris State College. an open-admissions institution specializing in occupationally oriented business, health. and technical programs. The two instruc- tional treatments employed in a nonequivalent control group design were (l) programmed, self-paced (n = 123. two sections) and (2) lecture, teacher-paced (n = llZ, five sections). Special attention was given to aptitude-treatment interaction. Pretests included the ACT Mathematics test. which was used for four-level blocking (High, Mid-High. Mid-Low. and Low) in the ATI investigation; the three additional ACT tests; the Mathematics Attitude Inventoryi and a background questionnaire. Posttest measures were a comprehensive final examination and the final course grade. ‘The comprehensive final examination was used as a pretest in one section. Representativeness of the two treatment groups was ascertained by traditional methods. A t-test applied to the estimated gain scores produced large t-values for each method (p < IN”). and a comparison of Manfred E. Swartz the mean gains favored the programmed method (p (.05). The analysis of variance F-value for the ATI was significant (p < .05). Scheffeks post hoc comparisons identified the superiority of the programmed method for the High and Mid-Low aptitude groups. Interestingly. the Mid-Low lecture groupfis achievement fell below that of the Low lecture group. Further analysis revealed that preexisting differences in attitudes toward mathematics (motivation) were associated with the achievement of Mid-Low and Low lecture groups. Students who scored 17 and below on the ACT Mathematics test had less than an 80 percent chance of earning a "C" or better. Stepwise multiple regression showed that grade prediction was aided by the use of the Self-Concept in Mathematics attitudinal scale. There were no differences attributed to students' preferences for the method of instruction (Hotelling's T2). Recommendations included (I) the replication of the study. (2) the use of attitudinal assessment and four-level blocking in ATI studies. (3) the continued use of both instructional methodologies for business mathematics. but greater use of the programmed treatment. (4) a prerequisite learning experience for some students. and (5) future placement and sectioning studies using the discriminant analysis method and including measures of cognitive style. ACKNOWLEDGMENTS I am grateful to many people for their assistance and support throughout the course of my doctoral studies. I wish to express sincere appreciation to the following: The Ferris State College administration who initiated a cooperative program with Michigan State University and who provided continued support: Vice-President Donald Priebe. Associate Vice- President Gary Nash. Dean Karl Walker. and Dean Joel Galloway. The doctoral committee: Dr. Ed Moore. Chairman. for his expert guidance; Dr. Rex Ray. for beneficial consultation and valuable instructional experiences; Dr; Ben Bohnhorst. for the many new concepts he imparted; and Dr. Charles Eberly. dissertation director. for his able editing assistance and assuring attitude. The Office Administration department: [ha Mal Lund. Department Head; Professor Vern Benson; Professor Bill Bennett; Professor George Hewitt; Professor Jim Lindsay; and Professor Rose Ann Swartz. My family: Rose Ann. for her enduring confidence; and Mark and Matt. for their unquestioning recognition of the demands. The office staff: Aggie Dora. Linda Burnes. and Kathy Gregorich. for unknown hours of typing and proofreading. TABLE OF CONTENTS Page LIST OF TABLES ......... . ........ . . . . . . . vi LIST or FIGURES . . . . . . . . . . . . . . . . . . . . . . . . viii Chapter I. INTRODUCTION . . . . . ........ . . . . . . . . . 1 Purpose of the Study . . . . . . . . . . . . . . . . . 1 Need for the Study . . . . . . . . . . . . . . . 2 Setting of Ferris State College . . . . . . . . . . 2 Current Context . . . . . . . . . . . . . . . . . . 3 Relationship to Theory . . . . . . . . . . . . . . . 4 Research Questions and Evaluative Criteria .. .. .. 5 Achievement Gain for Two Methodologies . . . . . . . 5 Sectioning by Ability . . . . . . . . . . . . . . . 6 Prerequisite Learning Experience . . . . . . . . . . 6 Attitude Assessment . . . . . . . . . . . . . . . . 7 Student Evaluation of Methodology . . . . . . . . . 7 Definition of Terms . . . . . . . . . . . . . . . . . 8 Limitations and Delimitations . . . . . . . . . . . 9 Limitations . . . . . . . . . . . . . . . . . . . . lO Delimitations . . . . . . . . . . . . . . . . . . . lO Conduct of the Study . . ..... . . . . . . . . . . ll II. RELEVANT LITERATURE . . . . . . . . . . . . . . . . . . 12 Business Mathematics . . . . . . . . . . . . . . . . 13 Programmed and Lecture Methods . . . . . . . . . . . . l8 Aptitude-Treatment Interaction . . . . . . . . . . . . 26 Course Placement . . . . . . . . . . . . . . . . . . . 35 Cognitive-Style Variables . . . . . . . . . . . . . . 39 Summary . . . . . . . . . . . . . . . . . . . . . . . 42 III. PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . 44 Population . . . . . . . . . . . . . . . . . . . . . . 44 Research Design . ....... . . . . . . . . . . . 45 Basic Design . . . . . . . . . . . . . Rival Explanations . . . . . . . . . . Evaluative Criteria Achievement Gain for Two Methodologies Sectioning by Ability O O O O O O O Prerequisite Learning Experience . . Attitude Assessment Student Evaluation of Methodology Measurement Instruments and Variables ACT Assessment Battery . . . . . . . . Self-Reported High School Grades . . Mathematics Attitude Inventory . . . Business Mathematics Questionnaire . . Business Mathematics Final Examination Course Grade . . . . . . . . . Methodology Evaluation Survey Instructional Methodology Traditional Lecture Method . . Programmed. Self-Paced Method Data Collection . . . . . Statistical Processing . . Data Entry . . . . . . . Analysis Programs . . . IV. FINDINGS . . . . . . . . . . Data-Collection Results . Verification of Assumptions 0 O O O O 0 O Pretest Equality . . . . . . . . Assumptions for Analysis of Variance Analysis of Research Questions . . . . Achievement Gain for Two Methodologies Sectioning by Ability Prerequisite Learning Attitude Assessment Summary V. SUMMARY AND RECOMMENDATIONS Summary Literature . . Method . . . . Results Recommendations Student Evaluation of Methodology 0 O O O O O O O O O O O O O O O O O O O C O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0 O O O O O O O O 115 Page APPENDICES O O 0 O O O O 0 O O O O O O 0 O 0 O O O O 0 O O O O O 118 A. BUSINESS MATHEMATICS QUESTIONNAIRE . . . . . . . . . . . 119 B. BUSINESS MATHEMATICS 121 COMPREHENSIVE FINAL EXAMINATION O O O O O O O O O O O O O O O O O O 0 O O 121 C. CWRSE OUT]. INE--LEC1.URE o o o o o o o ooooooooo 129 Do COURSE OUTL IN E--mmRAMMED o o o o o o o o o o o o o o o 131 E. BUSINESS MATHEMATIC$--METHODOLOGY EVALUATION . . . . . . 134 BIBLIOGRAPHY . . . . . ......... . . . ......... 136 LIST OF TABLES Nonequivalent Control Group Design . . . . . . . . . . Description of the ACT Mathematics Usage Test . . . . . Nonspurious Item-to-Scale Correlations for the Mathe- matics Aptitude Inventory . . . . . . . . . . . . . . Cronbach's Alpha Reliability Coefficients for the Six Scales of the Mathematics Attitude Inventory . . . . Content Analysis of the Business Mathematics Final Examination: 33-Item Version . . . . . . . . . . . . Difficulty and Discrimination Indices for the Business Mathematics Final Examination: 34-Item Version . . . Inter-item Correlations for the Methodology Evaluation Questionnaire . . . . . . . . . . . . . . . . . . . . Multivariate Ability and Attitude Comparison of Class Selected for Final Examination Pretesting to Other Classes . . . . . . . . . . . . . . . . . . . . . . . Comparison of Pretest Ability and Attitude Measures for Business Mathematics Teaching Methodologies . . . Comparison of Demographic Data for Business Mathematics Teaching Methodologies . . . . . . . . . . . . . . . Test of Assumptions for Analysis of Variance . . . . . Gain Score t-Tests for Instructional Methods . . . . . Analysis of Variance Components for Final Examination Scores in Business Mathematics . . . . . . . . . . . Post—Hoe Comparisons for Aptitude by Treatment Inter- actions for Final Test Scores in Business Mathematics vi Page 46 50 55 56 57 58 6O 64 7O 72 75 77 78 8O 4.7 4.8 4.9 4.13 4.15 4.16 4.17 4.18 Analysis of Variance Components for Final Grades in Bus‘ness Mathmatics O O O O O O O O O O O O O O O 0 Analysis of Variance Components for Motivation in Mathematics Attitudinal Scale . . . . . . . . . . . . Analysis of Variance for Enjoyment of Mathematics Atti tUdina] sca]e O O O O O O O O O O O O O O O O O 0 Frequency and Proportion of Students Earning C- or Higher Final Grades in Business Mathematics 121 . . . Comparison of Success in Business Mathematics Based on Prior Enrollment in Mathematics 090 for the l to 8 Score Range . . . . . . . . . . . . . . . . . Comparison of Success in Business Mathematics Based on Prior Enrollment in Mathematics 111 for the 9 to l7 Score Range . . . . . . . . . . . . . . . . . Comparison of Success in Business Mathematics Based on Prior Enrollment in Mathematics 121 for the 18 to 22 Score Range . . . . . . . . . . . . . . . . Prediction of Final Grades Using Ability and Attitude Measures for Lecture and Programmed Students combined 0 O O O O O O O O O O O O O O O O O O O 0 O Stepwise Multiple Regression of Ability and Aptitude and Demographic Measures on Grades for the Lower Aptitude Students in the Lecture Method . . . . . . . Stepwise Multiple Regression of Ability. Attitude. and Demographic Measures on Grades for Lower-Ability Students in the Programmed Method . . . . . . . . . . Comparison of Student Evaluation of Teaching Method- ologies Used in Business Mathematics . . . . . . . . Post-Hoe Comparisons for Aptitude Groups on Use of Out-of-Class Time in Business Mathematics . . . . . . vii Page 83 85 86 89 9O 91 93 96 99 100 102 103 Figure 4.1 4.2 4.3 4.4 LIST OF FIGURES Means for the Final Test Scores . . Means for Final Grades . . . . . . . Means for Motivation in Mathematics Means for Enjoyment of Mathematics . viii Page 82 84 85 87 CHAPTER I INTRODUCTION Educators recognize the need for evaluation of teaching and learning activities sponsored within educational institutions. Brinkerhoff et a1. (1983) said that evaluation should be part and parcel of any educational effort. Bohnhorst (1982). in discussing the curriculum planning time-line. stated that an adequate concept of curriculum includes evaluation of the results of instruction. In defining a program as a set of procedures designed to accomplish a particular objective. Ebel (1980) stated that "program evaluation has been generally accepted as an important aspect of the educational enterprise" (p. 281). McKinney (1977) noted the growing demand for efficient and effective education and stated that "the issue is not whether to evaluate. but how" (p. 1). Thus. evaluation is viewed as a very important aspect of educational activity. W To respond to the need for evaluation of an educational program. this study was designed to evaluate selected aspects of the business mathematics course (Business Mathematics 121) at Ferris State College. The principal aim was to judge the worth of two instructional methodologies. lecture and programmed. now in use. A further aim was to determine whether the former practice of sectioning students by ability levels should be reinstated. The study also examined instruc- tional effects that are relevant to a growing body of literature on aptitude and treatment interactions. The study focused on whether or not both lecture and programmed teaching methodologies should be continued. and potential changes that might be made to optimize student learning. Information was developed about the characteristics of students in each instructional treatment. the amount of gain in business mathematics knowledge under each treatment. the advisability of a prerequisite course for certain students. the advisability of ability-level sectioning for different methodologies. and the value of using attitudinal as well as ability measures in sectioning decisions. Imam The need for the study grew out of current educational prac- tices and decisions to be made about the course by the Ferris State College business mathematics faculty. The need was further justified by the studyhs relationship to aptitude-treatment interaction theory. WWW Ferris State College provides occupationally oriented educa- tional programs for students from a variety of academic backgrounds. Students who complete high school with less than a ZJK)grade point average are admitted to a general studies curriculum. Upon earning at least a 2.00 grade point average. they are eligible to select another program of study. In contrast. professional programs such as Optom- etry. Pharmacy. and Computer Information Systems attract students with strong academic backgrounds in the mathematics and science areas. The business mathematics course at Ferris State College serves students enrolled in two- and four-year degree programs. Since many programs are considered to be "open-admissions." students arrive with varying mathematics backgrounds. W In an effort to respond to the perceived needs of students. two teaching methods were adopted. One used the traditional lecture method. and one used a self-paced method with programmed materials. Until 1981. students were pretested at the start of the term and sectioned by ability level into one of the instructional methods. The upper one-fourth and lower one-fourth of the distribution were assigned to the self-paced treatment. The middle one-half was assigned to the lecture treatment. 1his approach permitted the high-ability group to move through the programmed material rapidly and exit from the course early. It also permitted the low-ability group to proceed at their pace with more individual help from the teacher and tutor. In the lecture method. a homogeneous ability group was formed. thus minimizing the problems associated with teaching to a wide ability range. Although this rationale had appeal. no research was performed to ascertain its efficiency. As a result. the practices drifted. and it was suspected that the full range of ability levels was present in both teaching methods. A systematic evaluation of student achievement was needed to lend credibility to the current practice or to aid in decision making for the future. .Belntigflfihin—IQ_Ih§Q£¥ College students differ in many important ways. Students may differ in ability. motivation. attitude. learning style. sex. and socioeconomic status. for example. As a result. teachers are expected to deal with classroom situations where a wide range of characteristics are present. This diversity presents a challenge to the typical class- room teacher whose goals are to maximize student achievement (Peterson. 1982). The traditional approach to instruction--that of lecture. class discussion. and laboratory exercises--assumes that students who work hard are capable of achieving the instructional goals. Yet it is known that all students do not achieve equally well. Therefore. the study of individual differences. as they relate to classroom achievement. has become a matter of theoretical development and research. Since the student learner can be characterized in different ways. it seems that an instructional design that accounts for individ- ual differences has an opportunity to be more effective than one that treats all students similarly. An approach that matches students' characteristics with the properties of instructional methodology is required. There is a need for more knowledge about student character- istics that favor one instructional approach over another. Cronbach (1967) and Gagne (1967) have suggested that no single instructional approach provides optimal learning for all students. More investigation which might identify the interactions between learner characteristics and instructional methodology is needed. Such studies may show which treatments will serve the largest number of students most efficaciously. In discussing the relationship between learner aptitudes and learning environments. Snow (1970) stated that: What is needed is a grand Darwinian matrix of organisms by environ- ments where both can be characterized by many dimensions and parti- tioned to show the particular types of treatments where particular types of learners thrive. (pp. 67-68) Thus. the need for the study of learner Characteristics that will interact positively with instructional methods has been recognized. Studies supported by this rationale have been named aptitude- treatment interaction (ATI) research. Such studies focus on what ‘works. for whom. and under what conditions. This type of study may be considered an evaluative study (Grasso. 1979). W This evaluation of several aspects of student achievement was performed to facilitate decision making about how best to offer the Business Mathematics 121 course. Five research questions were posed with criteria for acceptance. Achisxementjajm Methodologies Should the offering of both lecture and programmed instruc- tional methodologies be continued? An affirmative answer required that both methods produce an equal gain in achievement. The expected gain in student knowledge from pretest to posttest in each methodology was stipulated as statistically significant at the .05 level of confidence. If either method yielded less achievement. that method was to be considered for abandonment. If both methods failed the criterion. a complete reappraisal of the teaching-learning conditions was planned. If both methods satisfied the gain criterion. then the respective gains were to be compared to determine their equality. If one method only was associated with superior achievement. adoption of that method was to be recommended. SectionianLAbfljjx Should sectioning by ability level for different instructional methods be reinstated? This question. dealing with aptitude-treatment interaction. was related to the first question. Assuming the two methods were sup- ported. there was a need to know how best to implement the methodolo- gies for different student ability groupings. The stipulated statisti- cal criterion was the.05 level of significance for the F-test for interaction. If achievement for different ability groupings varied with the instructional methodologies. then sectioning by ability level was advised. WW Should a prerequisite learning experience be established for students with low mathematics ability? Students should have at least an 80 percent chance of earning a "C" or better grade in business mathematics when they enter the course. It was determined that if an ability grouping fell below 80 percent in earning grades of at least "C" level. a prerequisite course in mathe- matical skills and concepts was necessary. The 80 percent criterion. based on admissions test data. is currently used to determine course placement in mathematics and English at Ferris State College. WW Should attitudes toward mathematics be considered with mathematics ability in sectioning decisions? If attitude assessment accounted for a significant increase in the prediction of achievement variance. above that of ability assess- ment. then attitude assessment and a multivariate procedure for sec- tioning were to be prepared for adoption. The decision criterion was thee.05 level of significance for the F-value associated with the attitude measures. WWW Should the collection of student opinions of the course method- ology be implemented? If students differed in their reactions to the course method- ology. the continued collection of these data for course monitoring was to be recommended. The .05 level of confidence was used as the cri- terion for determining need for the recommendation. Wm: The following definitions for terms used in the study provided a common basis for understanding. Ability. The term "ability" was used to describe students' learning ability as measured by the ACT tests and high school grade point average. .Acniexement. The term "achievement" referred to students' knowledge of business mathematics as measured by the final business mathematics test. Amiiiude. The term "aptitude" generally referred to a charac- teristic on which students differ. In this study the ACT mathematics test was used as the aptitude variable. W. Gain in business mathematics achievement was estimated from pretesting one representative class with the final examination. The class mean was used to approximate the pretest busi- ness mathematics score for all students. .Lectune_method. The term "lecture method" was used to describe the traditional. teacher-directed classroom where lecture and discus- sion are the primary modes of instruction and students are tested as a group. W. The term "programmed method" was used to describe a classroom environment that is guided by a programmed text that permits students to proceed at their own pace. When instructional units are completed. students can be tested individually. Although self-pacing is under individual control. the assigned work must be completed by the end of the academic term. .Mathemaiics_attitudes. The term "mathematics attitudes" is used to describe a set of attitudes toward mathematics as measured by the Mathematics Attitude Inventory (Sandman. 1973). Six scales com- prised the instrument. They were: 1. Perception of the Mathematics Teacher--A student's view regarding the teaching characteristics of his/her mathematics teacher. 2. Anxiety Toward Mathematics-—The uneasiness a student feels in situations involving mathematics. 3. Value of Mathematics in Society—A student's view regarding the usefulness of mathematical knowledge. 4. Sel f-Concept in Mathematics--A student's perception of his/her own competence in mathematics. 5. Enjoyment of Mathematics-~The pleasure a student derives from engaging in mathematical activities. 6. Motivation in Mathematics--A student's desire to do work in mathematics beyond the class requirements. W The evaluative study was conducted in an educational environ- ment that imposed several restrictions. ‘The restrictions also included those imposed by the design of the research. 10 Limitations Two limitations included generalizability and nonrandom groups. .Genenalizabilrnp The study was limited in generalizability to students who enrolled in business mathematics at Ferris State College during the Winter Term. 1983-1984. The findings should not be routinely generalized to other academic terms or to other academic settings. N9fl£flflfl9m_gronp§. The nature of class scheduling prevented random assignment of students to instructional treatments. Although randomization was not possible. representativeness was possible and was expected. This matter is discussed in Chapter IV. Delimitations The delimitations were concerned with the scope of the study and the comparability of instructional methods. ‘laniab1e_selegtign. Educational evaluation can include a vast array of topics. too broad for a single study of this type. A focus is required. This study centered on student achievement and factors directly related to achievement. 11m. Comparisons between programmed and conventional instructional methods have suffered on logical grounds because the time factor has seldom been held constant. Programmed instruction usually includes self-pacing. which permits early departure from the treatment. Since self-pacing and early completion of course work are considered to be a part of the motivational strategy of the programmed methodology. 11 the potential for time variations is an integral part of the comparison in this study. W The study report proceeds with a review of relevant literature; then the procedures of the study are described. This material is followed by a description of the findings. The last section discusses the conclusions and implications for instructional decision making. CHAPTER II RELEVANT LITERATURE An evaluative study is designed to assess the effects of instructional strategies and to make recommendations for the improve- ment of instruction. A complex set of factors are relevant to such an undertaking. These factors include the subject matter. the instruc- tional method. the students' ability and attitudinal characteristics. the interaction of instructional methods and student characteristics. the variables used to assess the instructional effects. the statistical procedures used in assessing effects. and the decision making about student placement in optimal learning conditions. All of these factors are intertwined with features of the institutional setting where the study is conducted and. to some degree. the professional responsibili- ties of the researcher. Consequently. the review of literature touches a broad range of relevant educational factors. Comparative studies that used two or more instructional methods in high school or college-level business mathematics were reviewed first. These were followed by studies that compared programmed. indi- vidualized. or personalized educational systems to the lecture method in subject areas such as college-level general mathematics. introduc- tory algebra. developmental mathematics. or what is described in some 13 settings as remedial mathematics. The ability level of a portion of the students in this study makes such literature relevant. The arith- metic/algebraic base of the business mathematics subject matter (Kaliski. 1975) added to the relevance of the studies. Then. studies that compared the traditional lecture method to a variety of methods in the general mathematical subject area were reviewed. Studies that sought to identify aptitude-treatment interactions were reviewed next. These studies attempted to identify student char- acteristics that led to improved achievement in one or another instruc- tional treatment. The intended outcome was a procedure for decision making about course placement. After a section on aptitude-treatment interaction. studies that involved course placement and promising variables not included in this study but relevant to subsequent studies of a similar nature were reviewed. Busjoossflathomofloo Seven studies that compared instructional methodologies in business mathematics were reviewed. Four of the studies reported improved achievement with programmed or individualized approaches. One study reported mixed results. and the other two reported no differences. Harsher (1983) investigated the achievement. achievement retention. and attitudes toward subject matter effects of three methods of teaching secondary business mathematics. The instructional methods were the (l) teacher-directed conventional method. (2) student-directed individualized method. and (3) student-directed competency-based 14 method. The study used random assignment of classes to treatments in a pretest-posttest control group design. Thirty-two classes in 17 high schools participated. Students in the individualized and competency— based groups received self—instructional packages. However. the competency-based packages included goal-related components. The study produced mixed results. An analysis of covariance on achievement measures yielded no significant differences. Analysis of covariance was deemed inappropriate for retention and attitudinal measures; consequently. Johnson-Neyman solutions were used to locate regions of significance on specified ranges of covariables and between pairs of treatment groups. For some students. self-directed instruction that included goal-related information appeared to elicit superior retention results as compared to self-directed instruction without goal-related information. For other students. self-directed instruction appeared to be superior to teacher-directed instruction in eliciting favorable attitudes. For still other students. self-directed instruction without goal-related information appeared to elicit more favorable attitudes than self-directed instruction with goal-related information. Miller (1984) experimented with the use of a remedial mathemat- ics program for adult business mathematics students who scored at or below the 9.9 grade level in the mathematics fundamentals part of the Test of Adult Basic Education (TABE). Eighty students were involved in a posttest-only control group design with random assignment to groups. Forty students received the Individualized Manpower Training System (IMTS) remedial program; the control group did not. The purpose was to 15 determine whether participation would increase achievement in business mathematics and decrease the failure and dropout rate. The MANOVA results indicated that the achievement level of the IMTS group exceeded that of the control group (p < .05). However. no significant differ- ences were observed in the number of students who passed. failed. or dropped the course. Brown (1984) compared programmed instruction to the lecture- demonstration method of instruction in high school business mathematics by using a nonequivalent control group design. Programmed business mathematics review materials were prepared to fit the unique format of the vocational office education class. Units involving numeration and whole numbers. decimals. fractions. and percentages as used in business were developed. Each unit was followed by a review unit and a unit test. The material was designed to be completed in six to eight hours. Eighty students enrolled in six pre-employment vocational office education classes participated in the study. Three classes received the programmed material and three received the traditional approach. Analysis of covariance revealed that the programmed instruction method yielded a higher level of student achievement (p = .076). Wells (1982) compared university student achievement in business mathematics by comparing an individualized approach to the traditional lecture approach on two instructional units. percentage and business applications. The data were statistically treated using a multivariate analysis of covariance in a nonequivalent control group design. The findings indicated that the use of individualized 16 instruction produced higher achievement than the use of traditional instruction. In a nonequivalent control group design study of achievement in business mathematics. Liquori (1973) compared the Personalized System of Instruction (PSI) to the traditional lecture method. Two class sections were assigned to each method. with some variations in tutorial assistance within the PSI classes. The study also included an analysis of the effects of a graded and ungraded comprehensive final examination for each methodology. No differences between PSI and lecture method- ologies were observed. However. students with graded final examina- tions in both methodologies revealed better posttest achievement than did those students assigned the nongraded final. The findings sug— gested that the anticipation of a summative evaluation might aid in the integration of course material and that this integration might not occur in PSI classes without summative evaluation. Williams (1975) used a pretest-posttest control group design to compare a mastery learning instructional approach to a conventional strategy. The mastery strategy used small-group peer instruction. small—group teacher instruction. and programmed instruction. The conventional strategy used a lecture-discussion methodology. Both achievement measures and students' ratings were used in analysis of variance and t-tests for statistical testing. The mastery treatment accounted for significantly better achievement when the subject matter was of moderate difficulty. but not for the most difficult and least difficult units. Student ratings in the mastery strategy revealed a 17 preference for small groups with the instructor and small groups with peer instruction. Independent study with programmed materials was least preferred. Oravetz (1966) used a nonequivalent control group design with testing for effects extended in time to compare the effects of daily drill patterns in business mathematics presented (1) by a tachistoscopic-type device. (2) by an instructor-prepared series of audio-oral rapid mental calculation exercises with each other. and (3) with no drill. The drill groups received 10 to 15 minutes of the respective presentations in each class session. At the conclusion of the course. both drill groups revealed higher achievement than the nondrill control group; this difference held when assessed six weeks later. When student background was considered. the drill effects were not apparent for those with more than two years of high school mathe- matics. The drill patterns appeared to be most effectively suited for students with less than two years' previous mathematics instruction. In another review of comparative studies in business mathemat- ics. Brown (1984) cited three studies (by Myers. Swindle. and Pappin) where higher achievement was experienced by students in programmed methods. However. studies by F012 and Neaville (in Brown) found no differences. In summary. Brown's review and this review located 12 studies of instructional methodology in business mathematics. Eight studies found achievement or other features that favored the programmed approach. one found mixed results. and three reported no difference. In no case did the lecture approach prove to be superior in business mathematics instruction. Only two of the seven studies reviewed directly were of experimental design. and they reported results favoring the alternative to the lecture treatment. In four quasi- experimental studies. three studies reported results that favored the alternative to the lecture. One reported no differences. Thus. the relationship of the type of design used and the studies' outcomes appeared to be consistent. ELQ9Lomm§d_ofld_L§£Iu£§_MoIhQfl§ The lecture method of instruction. in such prevalent use. is at the root of considerable methodological research on student achieve- ment in mathematics. Support for the lecture method can be found readily. Mackenzie (1975) noted that as students hear the lectures. see the lecturer. see the argument unfolding at the chalkboard. and take corresponding notes for themselves. the communication becomes more enriched than with independent reading from a text covering the same material. He added that the lecturer's ability to enhance what is said or explained through sidelights or anecdotes from the rich history of the discipline. or with illustrative examples to reinforce the topic under consideration. makes the lecture method potentially effective in improving the learning and attitudes of students. In contrast. Weisen- glass (1976) believed that the lecturer. no matter how hard he/she tries. cannot present new information in the correct context and at the 19 correct pace for all the students. Some are bored because the pace is too slow; some are confused because the pace is too fast. Others have recognized that the lecture has a place for some types of instruction. For example. Woods (1983) asserted that the linking of a lecturekspurpose to its form and structure helps faculty to organize instruction. He considered the classical model of instruc- tion best to transmit information. the problem—centered model to create interest. and the sequential approach to promote understanding. Mathematics education is an excellent medium for research on the effectiveness of the lecture method. The lecture has frequently been compared to the programmed method. as in this study. and to other teaching methods. J. Adams (1981) compared student achievement in programmed and lecture methods of teaching remedial college algebra. The pretest- posttest control group design included testing for effects extended in time. One hundred sixty-four randomly selected students (82 per group) were pretested to determine levels of algebra achievement and attitudes toward mathematics. One hundred thirty who completed the course were posttested. The course withdrawal rate was 21 percent. Analysis of covariance revealed a significant difference in achievement (p < .001) favoring the programmed method. Mathematics background was more influential than attitude in predicting achievement. Follow-up in a regular algebra course produced no differences in achievement related to the method of prior instruction. 20 R. Adams (1981) used a nonequivalent control group design with pretesting and posttesting to compare student achievement resulting from the personalized system and lecture method of teaching interme— diate algebra. Although the sample sizes at Yarapai College were small. the personalized system produced significantly higher achieve- ment scores. Attrition from the two methods was equal. Watson (1983) compared an individualized system of instruction with a choice of assessment to the traditional lecture method with an end-of-course examination. The students were enrolled in a mathematics course. Discrete Modelling I. over a three-year period at an Australian university. The design used intact class groups in a nonequivalent control group design with separate samples covering a three-year period. With attitude and achievement as outcome measures. the individualized system produced better attitudes and a higher passing rate. but the lecture group had better long-term retention of concepts. It was suggested that preparation for the end-of-course examination in the lecture treatment aided in long-term retention. Schielack (1983) assessed the relative merits of the Keller Personalized System of Instruction (PSI) and the traditional lecture— discussion method in mathematics achievement and attitudes for elementary education majors. A pretest-posttest control group design was used. Also investigated was the existence of aptitude-treatment interaction using general reasoning ability as an aptitude measure. The sample consisted of 30 PSI and 28 lecture students. PSI students performed significantly higher (p < .001) than lecture students on the 21 final examination and had significantly more positive (p < .05) attitudes toward mathematics. ‘There was no aptitude-treatment interaction. Reinauer (1981) investigated the use of aptitude and attitudinal measures in predicting technical mathematics achievement taught by a computer-assisted. self-paced method and the conventional lecture method. Interaction between student characteristics and instructional methods was considered also. The sample for analysis in a pretest-posttest control group design consisted of 88 students selected from seven lecture sections and 72 students from six self- paced sections. The pretest measures. verbal reasoning. numerical ability. student attitudes. mathematics enjoyment. and mathematics anxiety. accounted for 37 percent of the achievement variance. There was no significant treatment interaction; consequently. placement decisions were not recommended. Overall. the self-paced group scored significantly higher than the lecture group. although this was not an indication of the efficiency of the self-paced instruction format for all students. Shine (1983) compared the effectiveness of programmed instruc- tion to lecture instruction in the teaching of digital computer arith- metic to 41 postsecondary electronic technology students. The analysis used a pretest-posttest. two-group simple randomized design. Students were randomly assigned to sample sizes of 21 and 20 and assigned to the programmed and lecture methods. respectively. .Students were pretested and posttested with specially prepared digital computer arithmetic 22 tests. The conclusions were that both methods produced a gain (p < .05) and that the programmed method was as effective as the»conven- tional lecture method. Schwarze (1980) compared a mastery learning approach to conventional lecture instruction in a remedial mathematics program situated in an urban community college. The mastery learning procedures included short introductory lectures. carefully sequenced examples and problems. frequent formative testing. immediate feedback. and Unmediate corrective follow-up. The two-randomized-groups design was used. Students were pretested and posttested for achievement levels and attitudes toward mathematics. The two groups were equal on the pretest measures. On the posttest achievement measure. the mastery learning group performed significantly better than the conventional instruction group. In attitude assessment. the mastery learning group revealed a slight increase. while the conventional instruction group showed a significant decrease. Truckson (1983) explored the development of junior college students' problem-solving ability in arithmetic by comparing three methods of instruction: (1) the lecture method. (2) the heuristic method with problem solving. and (3) the lecture method with problem solving. The design was a pretest-posttest nonequivalent control group that used analysis of covariance. The instructional period was nine weeks in length. and posttests included both achievement and attitude measures. The methods using problem-solving instruction produced significant evidence that problem-solving skills were used. 23 The processes being taught were being used on the tests. However. in actual arithmetic achievement. the three methods were equal. Walker (1981) compared the traditional lecture-discussion method with the lecture-discussion method supplemented with a pro- grammed text to teach arithmetic concepts to prospective elementary school teachers. The pretest—posttest control group design was used. Effectiveness was examined by achievement and attitudinal measures that were administered as pretests and posttests. Results were based on sample sizes of 23 students in each treatment group. All of the null hypotheses were supported. The addition of programmed support mate- rials did not produce increased attitude or achievement gains over results for the traditional lecture-discussion method. Cope (1980) compared the lecture—discussion method of teaching business calculus to the lecture/small-group-discussion method. The experimental group received lectures two days a week with small-group discussion on the following two days. The nonequivalent control group design with pretest and posttest was used. Students in the lecture- discussion group performed as well as those who experienced the lecture four days per week. The students in the experimental class experienced significantly fewer withdrawals. failures. and absences. They also indicated a strong preference for the experimental method. Bouknight (1984) investigated the effects of two teaching strategies with different emphases on two dimensions of learning outcomes in an introductory college mathematics course. The verbal strategy emphasized the interrelationships of the content. providing 24 students with opportunities to express verbally their understanding of the relationships. The computational strategy stressed computational skills and procedures. providing routine drill-like exercises for the development of computational competency. Ninety-nine students were randomly assigned to one of the two teaching strategies in a posttest- only control group design. For four consecutive class periods students received instruction via video-taped lessons and were assigned approp- riate homework. 0n the fifth session they were posttested and a reten- tion test was given six weeks later. Based on the posttest data. evidence of the two learning outcomes was confirmed. The instructional strategy influenced verbal knowledge outcomes H>‘<.0001) and computa- tional knowledge outcomes (p <.05). However. the retention test supported only the verbal strategy (p < .0001). For short-term goals both strategies were effective; however. for longer-term retention. instruction should emphasize the verbal expression of interrelation- ships in the subject matter. Godia (1982) investigated the student achievement and attitude change for college freshmen enrolled in remedial arithmetic under two different instructional approaches. A posttest-only control group design with random assignment to groups was used. One group of 227 students was assigned to a small-group-instruction approach that used calculators. an instructional support system. and a textbook. Class size ranged from 25 to 30 students. The second approach enrolled 94 students in a large-group-instruction mode using diagnostic remediation and instructor-made materials. Two-factor analysis of variance showed 25 that achievement in the small-group approach was higher (p < .001). although both groups showed substantial gain. Both groups experienced positive improvement in attitude toward mathematics. but the gain was greater on the part of the large-lecture group (p uo_66m c. mo_umsozumx Luzonoh c. be c. mo_uoeocumz ccmxop mu_umeu:umz co_un>_uox ucos>0acu uaoocoU-$_om mo o:_m> >uu_xc< ecu we co_uao0com m.>couco>c_ ou:u_uu< mo_uwsocumz ozu no» mco.un_occou u_mum-0uueou_ m:o_c:amcozuu.m.m o_omh 56 indicate their college class. the year of high school graduation. semesters of high school mathematics. number of previous college mathematics classes. and the college mathematics classes taken. The results aided in the description of the student sample for the study. Table 3.4.-—Cronbach's alpha reliability coefficients for the six scales of the Mathematics Attitude Inventory Groupa Scale Norm Pilot 1. Perception of the Mathematics Teacher .83 .88 2. Anxiety Toward Mathematics .86 .89 3. Value of Mathematics in Society .77 .77 4. Self-Concept in Mathematics .83 .87 5. Enjoyment of Mathematics .85 .88 6. Motivation in Mathematics .76 .74 aNorm data--Sandman. 1973. Pilot data--Swartz. 1982a. Businessflathmatios finaLExamioaiion The primary dependent variable. the Business Mathematics Final Examination. was a 33-item. four-option multiple-choice test (Appendix BL. It was developed from an instructor-made. open-ended-items test that had been administered to both lecture and self-paced classes as a comprehensive final examination (Swartz. 1982bL An analysis of answers by type of instruction revealed some items were sensitive to the teaching methodology. ‘These items were rewritten to eliminate that S7 bias. It was pretested as a 34-item test. and. subsequently. one item was dropped because it lacked instructional relevancy. The test content areas related to certain chapters. or units of instruction used in the two teaching methodologies (Table.3£D. Items were taken from the core of content comnmnlto both methods (Appendices C and 0). Table 3.5.--Content analysis of the Business Mathematics Final Examination: 33-item version. Number of Lecture Programmed Content Area Items Chapter Units Percentages 3 10 l. 7 Bank statement reconciliation 2 3 l. 2 Taxes. tax rates 4 20 l. 14 Interest. principal. rate. time 4 l4 2. 1-2 Discount. proceeds. and maturity value 4 15 2. 4-6 Interest--effective rate 1 l7 2. 7 Markon and selling price 4 l3 2. 9 Trade discounts 4 ll 2. 12 Payroll. gross pay. and FICA 4 18-19 2. 13-15 Depreciation 3 25 Handout Total 33 To obtain options for the multiple-choice format. student errors were recorded and tallied for both methodologies. The most frequently obtained errors were identified and selected as distractors for the draft of the test. The 34-item version of the test was administered to 145 students at the end of the fall term. 1982. 58 The test yielded a mean score of 26.2 and a standard deviation of 6.03. The high score was 34 and the low score was 9. The Kuder— Richardson Formula 20 for reliability was 0.88. and all items were of approximate difficulty and provided positive discrimination (Table 3x». These results seemed sufficiently satisfactory to allow the test to serve as the comprehensive final examination. Table 3.6.--Difficulty and discrimination indices for the Business Mathematics Final Examination: 34-item version. Diffi- Discrimi- Diffi- Discrimi- Item culty nation Item culty nation 1 .84 .40 18 .61 .28 2 .90 .32 19 .93 .33 3 .96 .24 20 .88 .33 4 .74 .40 21 .54 .58 5 .76 .35 22 .49 .69 6 .95 .21 23 .64 .32 7 .84 .47 24 .54 .55 8 .72 .39 25 .63 .35 9 .83 .45 26 .45 .49 10 .89 .45 27 .86 .60 11 .89 .28 28 .86 .57 12 .91 .35 29 .87 .60 13 .94 .39 30 .88 .60 14 .76 .62 31 .76 .41 15 .63 .60 32 .88 .46 16 .82 .60 33 .76 .51 17 .57 .54 34 .66 .50 Counsefinaoo The second dependent variable was the final grade assigned by the instructor. Grades were assigned on a 12-point scale ranging from 59 A to F (1.6.: A = 4.0: A" = 3.7: 8+ = 3.39 B = 3.0: B" = 2.7: C'I' = 2.39 C = 2.0. C- = 1.7. D+ = 1.3. D = 1.0. D- = 0.7. F = 0.0). .Mothodoloox_fixaluotion.§unxex A standardized instrument that permitted students to evaluate the method of instruction without evaluating other instructional features was not located. Consequently. a seven-item questionnaire to which students could respond on a four-point. forced-choice. Likert- type scale was developed. The Methodology Evaluation Survey was administered to students during the sixth week of the term. The administration was timed to reach all students before those in the programmed. sel f-paced course completed the Final Examination and stopped attending class. and to avoid proximity to the administration of a classroom examination in the lecture method. The instrument was given to 225 of the 235 students who completed the course. 'Ten students were absent or did not respond. The instrument (Appendix E) included three items (Items 1. 2. and 7) selected from an instructional-evaluation item bank developed by the Teaching and Learning Center at the University of Michigan. Items 3. 4. 5. and 6 were created specifically for the survey. .Xalidity. The items were reviewed by the teaching staff in the study and were considered to have face validity. Inter-item correla- tions were 0.59 for Items 2 and 3 and Items 4 and 5 (Table 3.7). The highest correlation was 0.85 for Items 3 and 7. The mean inter-item correlation was 0.68. which supported the instrument's construct validity. 60 Table 3.7.--Inter—item correlations for the Methodology Evaluation Questionnaire. Item Item 1 2 3 4 5 6 7 l 1.00 .74 .68 .72 .61 .63 .71 2 1.00 .59 .64 .63 .64 .62 3 1.00 .77 .65 .70 .85 4 1.00 .59 .69 .76 5 1.00 .69 .70 6 1.00 .70 7 1.00 Reliability. Cronbach's alpha. an accepted measure of reliability. was 0.94. Hence. the instrument proved to have a high degree of internal consistency. MW Two instructional methods are used in the business mathematics course. the traditional lecture method and the programmed. self-paced method. The respective methods and staffing patterns are described below. museum Each class was staffed by one professor who was responsible for all instructional activity. including order of topic presentation. student assignments. testing. and grading. A blackboard and an overhead projector were available for use in each classroom. A comnuui 61 text. BusinosLMofliomatioLioLQoJJooes (Rice et a1.. 1983) was used in the lecture course. ELQQEimmodi_§§lizflo§§d_M§Ith Each class was staffed by one professor and two upper-class tutors. This instructional methodology used W .Mathemjtigs. 4th edition (Huffman. 1980). Each unit of materials included a survey test. unit objectives. instructional material. and unit posttest. The survey tests. varying from 8 to 14 items. provided the student with an indicator of the need to do the unit or skip it and do the unitfls posttest. If the student achieved less than 100 percent accuracy on the pretest. he/she was directed to complete the unit. The units! objectives indicated what was to learned within the units. These objectives. similar to performance objectives. informed the student what kind of behavior (ins. list. define. compute. explain. eth was to be applied to the content. although performance standards were not listed with each objective. The instructional material was presented in a sequence of small steps or "frames" that included content and a question or problem. Correct answers. to be "hidden" before attempting the work. were provided on the left side of the page. If used as directed. the answers provided immediate feedback to the learner. The rate of presentation was student Controlled. ‘This format was highly similar to the description of programmed instruction provided by Schalock (1976). At the end of the instructional units. the students answered questions on a posttest. called a "checkpointd' These tests varied in 62 length from 30 to 35 items. and they were to be completed before the work proceeded to the next unit. In this part of the process. instructors corrected (or marked) the checkpoints but did not grade them. A performance standard was introduced at the checkpoint. Students who achieved a lower performance standard redid the missed items until the 75 percent level of accuracy was achieved. The program technique was linear. All students moved through the same material. although at their own pace. After the question was answered. the student moved to the next frame. regardless of whether the answer was right or wrong. In the posttest phase. if the standard was not achieved. only the missed items were redone. Students were not directed to new material to cover the deficiency as is offered in the "branching" or "responsive" system of programmed instruction (Schalock. 1976). Several units were then combined. based on similarity of content. for achievement testing and grading. Regardless of student performance on these tests. they proceeded to the next unit of instruction called for by the course outline. When individuals finished the content requirements. they took the comprehensive final examination and left the class. Some students completed the work in as little as six weeks. but about 70 percent required the entire ten-week period. Those who failed to complete the content in the ten-week term still completed the comprehensive final examination. 63 QatoJoJJootjon The study was conducted winter term of 1983-84 at Ferris State College. ACT test scores were requested from the college computer center when the class rosters were printed. The data were analyzed 'hnmediately to determine the most representative class to receive the Business Mathematics Final Examination as a pretest. The Mathematics Attitude Inventory was the initial instrument administered on the first class meeting. followed by the Business Mathematics Questionnaire. 'The Final Examination was administered to the specially selected section on the second class meeting. Instruc- tors were given a small supply of assessment materials for students who arrived at class for the first time on the second or third class day of the term. The class selected to receive the Final Examination as a pre- test was the 2:00 p.m. section. a large section which was to use the programmed method. A visual inspection of the ACT Mathematics test section averages revealed that the selected class was in the middle of the distribution. The representativeness of this class was verified by performing a multivariate test of significance. comparing its students to the other students on the ability and attitude measures (Table 3.8). The F-value associated with Hotelling's T2 was 1.2438. with a proba- bility level of 0.254. which did not approach significance. Of special interest was the difference in mathematics ability. 0.7 points. On this key measure. the t-value was 0.75 with a probability of 0.455. Consequently. it was concluded that there was no evidence to suggest :mN.o u a .NNN .N. u ea .mm:~._ u m:_m>-n meme.m_ u A ..mc.__oooz 6A N oom.o we.o em.m s.o~ o_.m A.o~ .u_oaea;.mz e_ co_um>_.oz w-.o _~._ mm.m m.ON mN.: m._N mu_uaem;pmz do acme>omcu eo~.o -._ e:.m o._~ _:.s m._~ .U_pmem;Uaz c_ uaoocou-c_am Nam.o mm.o m~.m :.m~ me.m o.e~ >CoLUOm c_ .u_om5m;pmz .0 m32, mem.o mm.c- _o.: ~.m_ s:.s m.e_ .U_um2m;oaz ecmzoe >om_xc< ems.o ma.o- mm.m e.MN .e.m N.m~ Loeumme mu_oaem;omz we. to co_oamuLaa momumEv£umz ULQZO... movaumufix mmo.o sm._ mm.o mm.~ Nm.o om.~ um__om oum_cm>_u_:zsu.m.m o_omk 65 that the selected students were significantly different from the others. and it was deemed that they were reasonable candidates to receive the Final Examination as a pretest. The average score on the pretest for the 60 students who completed the class was 8.4 on the 33-item test. The high score was 19; the low score was 4. Since the chance score for a 33-item. four- choice exam is 8.25. the class average was considered a chance score on the test. representing very little knowledge of the subject matter. The posttest consisted only of the Business Mathematics Final Examination. It was administered by the instructors in the lecture classes during the last week of the term. Instructors administered the test in the programmed sections as students completed the course requirements. Those not finished before the end of the term completed the test on the last class day. StatistiooLEmossioo Catalan A computer-based record for each student was developed. 'The data were maintained under the author's ID on the interactive terminal system (MUSIC) used by the IBM 4381 mainframe computer at Ferris State College. In addition. an estimated gain score was computed for all students by using a data-transformation option within the processing program. The name. social security number. ACT scores. and final grade were transferred from the student master file to a user's file at the conclusion of the course. The remainder of the data were entered via 66 the keyboard by Testing Office personnel at Ferris State College. Following data entry. the accuracy was verified by visual inspection of the file contents and the original document. W The data were analyzed by the BMDP statistical package on the Ferris State College mainframe computer. The BMDP programs are a recognized resource for processing social science data (Iverson & Norpath. 1976; Tabachnik & Fidel. 1983). Statistics computed for the purpose of this study were (1) frequency distributions. means. and standard deviations (BMDPZD); (2) Hotelling's T2 and t-tests (BMDPBD); (3) Bartlett's test for homogeneity of variances and equality of cell frequencies (BMDPQD); (5) analysis of variance and covariance (BMDPZV); and (6) correlational analysis and stepwise multiple regression (BMDPZR). CHAPTER IV FINDINGS The study was designed to evaluate the effectiveness of the lecture and programmed. self-paced instructional methodologies in the Business Mathematics 121 course at Ferris State College. Five evalua— tive questions concerned with (l) the continuation of offering two methods. (2) sectioning by ability level. (3) prerequisite learning experiences. (4) attitudinal assessment. and (5) student evaluation of instruction were raised. The methodology and evaluative criteria were presented in the previous chapter. The statistical analysis was guided by the research questions. which are restated as hypotheses in this Chapten. The statistical treatment required that the data meet certain requirements. Data:Collootion.Besults A total of 112 students completed the lecture instruction. and 123 students completed the programmed instruction. The ACT measures. mathematics attitude measures.lnathematics background measures. and the final course grade were collected for all students. However. four lecture method students and ten programmed method students who received failing grades declined to take the final examination. Missing data on an important dependent variable present a dilemma of throwing out cases 67 68 and collected data or introducing contrived scores. A procedure recommended by Tabachnik and Fidell (1983) was used here. The cases with missing data were kept in the data set by using the following procedure. Within each instructional group. the mean final score for the failing students who took the Final was computed. Respectively. these means were inserted in the studentfis record to substitute for the missing data. Then. a stepwise multiple regression was executed for each method using the final examination as the dependent variable. ‘The resultant equations were then used to predict a new final test score. The predicted scores were substituted for the means. Then. the regres- sion process was repeated to assure that the revised final test score was identical to predicted test score. Five students from each instructional method did not complete the course methodology survey. The evaluative survey data did not lend themselves to the same kind of missing-data treatment as the achieve- ment data. Consequently. the student survey analysis was based on 107 lecture and 118 programmed students. iotiiioationJLAssumptions Similarity of students'cmaracteristics in the two instruc- tional methods was a primary assumption requiring verification or manipulation of the data to attain equality. In addition. the analysis of variance statistical procedure required that assumptions about equality of cell frequencies and homogeneity of variance be met before data analysis (Tabachnik & Fidell. 1983). 69 .ELQIQ§1_EQfl§111¥ Pretest data were obtained from student records. ACT test scores. a Mathematics Attitude Survey. and from the Business Mathe- matics Questionnaire. The ACT and attitudinal measures were continuous variables. while the questionnaire included categorical variables. .Continuous_yaniables. There were 112 students who completed the lecture treatment and 123 who completed the programmed treatment (Table‘4J). The withdrawals from the course were proportional with 26 (19 percent) leaving the lecture treatment and 30 (20 percent) leaving the programmed treatment. Pretest equality was analyzed with a multi- variate test. Hotellinghs T2 (Tabachnick & Fidell. 1983). The F-value for the test was 1.211 and had a probability level of 0.276; thus the two groups were acceptably similar. Closer inspection of the proba- bilities for the individual variables revealed that a significant difference was observed on the ACT Social Studies test. Also. the probability associated with the mathematics test was near significance. Although the multivariate test implied that the significant difference may well have been a chance difference. a Type I error. it was decided to consider the social studies test as a covariate in the analysis of variance test. However. the homogeneity of regression requirement for analysis of covariance could not be met. Consequently. two-way analy- sis of variance (aptitude group by method) was used. The mathematics test score was positively correlated with the social studies test (r = (L35) and was used as the blocking variable in the analysis of variance. thus reducing the potential influence of the 70 en~.o u a .NNN .N_ n we .__~._ n e:_m>-a oa.m_ u A ..mci__euo: N m_n.o em.o- e~.m ~.o~ m_.m e.o~ .e_oaso;omz c_ co_om>_ooz NNm.o mo.o o_.: _._N om.e _._N mu_omeoeoaz to ucaesoacw Ame.o .m.o mm.m e._~ mm.q ~._~ .U_pese;umz cm paeueou-c_em .e:.o sn.o- m~.m o.e~ sa.m e.m~ >oe_uom c_ .e_omeo;umz .0 e=_m> e_e.o om.o mo.: w.e_ wm.: _.N_ mu_paee;um: ecmzoe >oeixe< ONN.o MN..- em.m e.m~ :e.m o.m~ Luau... me_umee;.az .0 co_oaoecea mu m u mEmfiumz _u._m30._. mwvnu _ uu< mam.o m_.o :m.o m.~ mm.o m.~ u___om “weapon do :0m_cmaEOUII._.: o_nmk .mo_mo_ovo:uoe 71 social studies variable. Tabachnick and Fidell (1983) stated that blocking is a useful technique for reducing the effect of pretest differences where random assignment of subjects to experimental treatments is not possible. Although the ACT means were below those for Ferris State College students in general. it was probable that the score distributions over- lapped considerably. When compared to Ferris students in mathematics ability. the business mathematics students were underrepresented in the upper part and overrepresented in the lower part of the score distribu- tion. A chi-square goodness-of-fit test (Ferguson. 1966) produced a significant value of 24.16. The critical value for 5 degrees of free- dom at the .01 level was 15.09. No college-wide data were available for mathematics attitudes. .Categonigal_xaniables. The categorical variables from the Background Questionnaire were examined with the use of chi-square tests (Table 4.2). Although not statistically significant. it appears that the programmed method included a slightly higher proportion of freshman students. The students were highly similar in the amount of high school and college mathematics in their backgrounds. Twenty-seven students had completed general mathematics (Mathematics 090) and almost one-half had completed the first course in the algebra sequence (Mathematics 111). Slightly over one-third completed the first course in college algebra (Mathematics 121). and a small group had Completed advanced mathematics. 72 Table 4.2.-—Comparison of demographic data for business mathematics teaching methodologies. Lecture Programmed Combined .Measure Freq Pct Freq Pct Freq Pct College Freshman 63 26.8 76 32.3 139 59.2 Sophomore 27 11.5 37 15.7 64 27.2 Junior 17 7.2 6 2.6 23 9.8 Senior 5 2.1 A 1.7 9 3.8 Total 112 “7.7 123 52.3 235 100.0 Chi-square = 7.65, df = 3, p = 0.05h High School Math Semesters One or two 38 16.2 26 11.1 64 27.2 Three or four 39 16.6 52 22.1 91 38.7 Five or six 19 8.1 29 12.3 A8 20.4 Seven or more 16 6.8 16 6.8 32 13.6 Chi-square = 5.69, df = 3, p = 0.128 College Mathematics Courses Zero 19 8.1 19 8.1 38 16.2 One 63 2 .8 66 28.1 129 54.9 Two 25 10.1 27 11.5 52 22.1 Three 4 1.7 7 0 11 “.7 Four or more 1 O.h A 1 7 5 2.1 Chi-square = 1.89, df = 3, P = 0.596 College Mathematics Enrollment General Mathematics Yes IA 6.0 13 5.5 27 11.5 No 98 h1.7 110 h6.7 208 88.5 Chi-square = 0.25, df = 1, p = 0.6A3 Table 4.2.--Continued. 73 Lecture Programmed Combined Measure -————————— Freq ,Pct Freq. Pct Freq Pct Algebra 1 Yes 55 23.4 57 24.3 112 47.7 No 57 24.3 66 28.1 123 57.3 Chi-square = 0.18, df = 1, p = 0.672 College Algebra Yes 36 15.3 49 20.9 85 36.2 No 76 32.3 74 31.5 150 63.8 Chi-square = 1.50, df = 1, p = 0.22 Above College Algebra Yes 10 4.3 17 7.2 27 11.5 No 102 43.4 106 45.1 208 88.5 Chi-square = 1.38, df = l, p = 0.24 Sex Male 42 17.9 56 23.8 98 41.7 Female 70 29.8 67 28.5 137 58.3 Chi-square = 1.55. df = I, p = 0.213 Field of Study Business-~0A 26 11.0 9 3.8 35 14.9 Business--Non-0A 67 28.5 93 39.6 160 68.1 Nonbusiness 19 8.2 21 8.9 40 17.0 Chi-square = 12.21, df II N U T) 74 The sex ratio was balanced between the two groups. although the pattern was the reverse of that of Ferris students generally. Males make up 60 percent and females 40 percent of the college student popu- lation. The opposite ratio was present here. A distinct difference was observed between the two methods in the students' field of study. Students from the office administration curriculum were underrepresented in the programmed method. and students from other business majors were underrepresented in the lecture method. The differences were significant beyond the 0.01 level. ,Sumnuugb The multivariate test on the continuous variables and the chi-square tests on eight of nine categorical variables demonstrate that the lecture and programmed groups were statistically similar to each other. AssumptionLioLAnastio o_f_!ar_i_ano_e Assumptions for analysis of variance are based on independence. equality of cell frequencies. and homogeneity of variance. Independents. Representativeness of subjects in treatment groups satisfies the independence assumption. Data from the multivari- ate test and chi-square test previously reported (Table 4.1) reveal the independence of the students within the instructional groups. W. Four aptitude groups were formed from the combined score distributions on the ACT Mathematics test. The divisions were selected to represent the ACT Math quartiles as closely as possible. The Low-ability group included ACT Mathematics scores 75 from 01 to 08. the Mid-Low group included scores from 09 to 12. the Mid-High group included scores from 13 to 19. and the High-ability group included scores from 20 to 32 (Table 4.3). The chi-square test of independence for equality of cell frequencies produced a nonsignifi- cant value. thus supporting equality. Table 4.3.--Test of assumptions for analysis of variance. Final Examination Groups N Mean 5.0. High Ability Lecture 25 24.5 7.02 Programmed 34 27.5 4.47 Upper-Mid Ability Lecture 27 23.7 5.64 Programmed 32 25.2 5.32 Lower-Mid Ability Lecture 23 19.7 4.02 Programmed 32 24.0 4.86 Low Ability Lecture 37 21.4 4.50 Programmed 25 19.3 4.30 Chi-square = 6.06. df = 7. p = 0.533 F = 1.84. df = 7.227. p = 0.075 W. Bartlett's test for homogeneity of variance was used to test the homogeneity of the cell variances on the dependent variable. the Final Examination. The probability associated with the F-value of 1.84 was 0.075. near but not reaching significance. 76 The standard deviations for the lower three ability groups appeared quite similar. 'The greatest difference existed within the upper ability group. where the ratio of the standard deviations is less than two to one. The pattern of equality observed in the pretest data was sup- ported by the test for equality of cell frequencies and homogeneity of variance. Consequently. the application of analysis of variance for comparative purposes was valid. AnalysimBosoaLoLOuostions The findings for the research questions are presented below. To support the presentation. the questions are restated as null hypoth- eses to aid the clarity of the discussion. thioxomontjainjoo INLMothodoJooies The evaluative question was: Should the offering of both lecture and programmed instructional methodologies be continued? The question had two aspects. one comparing the estimated achievement gain for each method to no gain. and the other comparing the respective gains to each other. ,flypothesis_i§: The estimated mean gain score for the lecture method is not significantly different from zero (MgL = 0). The mean lecture posttest score was 22.3 and the estimated mean gain was 13.9 points (Table 4.4). The t-value was very high and did not register a probability level in the fourth decimal position. 77 Consequently. the gain score was significantly different from zero and Null Hypothesis la was rejected (Hla:MgL#0). HypothesfiJb: The estimated mean gain score for the programmed method is not significantly different from zero (MgP=0). The mean programmed posttest score was 24.3 and the estimated mean gain was 15.9 points. As with the lecture gain. the t-val us was quite high and statistically significant. Consequently. the gain score was significantly different from zero and Null Hypothesis lb was rejected (Hlb:MgP¥0). Table 4.4.--Gain score t-tests for instructional methods. Posttest Gain Method N t-value Prob. Mean 8.0. Mean 5.0. Lecture 112 22.3 5.58 13.9 5.58 26.3 0.000 Programmed 123 24.3 5.47 15.9 5.47 32.3 0.000 Difference 2.0 1.01a 1.98 0.043 aStandard error. Hypothesis_1g: The difference between the estimated mean gain scores for the two methodologies is not significantly different from zero (Mg(L-P)=0). The difference between the methods was 2.0 points. and the standard error for the difference between the means was 1A”. However. the appropriate test of significance was a two-way ANOVA. which pro- vided control over the aptitude groups as well as method (Table 4J9. 78 The F-value for treatment was 6.51. and the probability level was (L011. Consequently. the null hypothesis was rejected (ch:Mg(L-P)#0). There was a statistically significant difference in pre- to posttest gain for the programmed method compared to the lecture method of instruction. The full implication of this finding required the analysis of the following research question. Table 4.5.--Analysis of variance components for Final Examination scores in business mathematics. Effect SS df MS F Prob. Mean 122689.31 l 122689.31 4839.82 0.000 Aptitude—- ACT Math 1120.19 3 373.40 14.73 0.000 Treatment-- lecture/prog. 165.01 1 165.01 6.51 0.011 Interaction 328.24 3 109.41 4.32 0.006 Residual 5754.44 227 25.35 W The evaluative question was: Should sectioning by ability levels be initiated for the different instructional methodologies? The question implied that the respective aptitude groups should perform equally well to maintain the current practice of not sectioning. Unequal performance for equal ability groups would support a 79 recommendation for sectioning. The question called for an analysis of aptitude-treatment interaction. .HxpQIh§§1§_Iofiiing_fon_AIl. 'The null hypotheses were that the difference between the final test scores would not be significantly different from zero for the aptitude groups. An analysis of variance as performed to test the hypothesis (Table 4.5). : The difference between the mean Final test scores for the High—aptitude lecture and programmed groups is not significantly different from zero (MhL-MhP=0). ,flypothesis_2b: The difference between the mean Final test scores for the Mid-High aptitude lecture and programmed groups is not significantly different from zero (MmhL-Mth=0). .flypothesis_29: The difference between the means for Final test scores for the Mid-Low aptitude lecture and programmed groups is not significantly different from zero (MmlL-MmlP=0). ,flypothesis_2d: The difference between the means for Final test scores for the Low-aptitude lecture and programmed groups is not significantly different from zero (MlL-MlP=0). The F-test for treatment (F = 6.51. p = 0.011) revealed that a significant difference in performance existed (Table 42». The differences between the means and the»95 percent confidence intervals for Scheffe's post hoc comparisons (Glass & Stanley. 1970) were computed (Table 4:». The null hypotheses for the Mid-High and Low aptitude groups were not able to be rejected (H2b: MmhL-Mth=0 and H2d: MlL-M1P=0). However. the null hypotheses for the High- and Mid-Low aptitude groups were rejected (H2a: MhL-MhP#0 and H2c: MmlL-Mm1P#0). In both of the latter comparisons. the programmed method students demonstrated superior achievement. 80 ._m>coucw oucovmmcoU unnecoa mm ocu ov_mu:o monocommmon m:.o- op mo.: mm.Nn _.N+ om.: m.m_ mN cm.: :._N NM 304 mm.o- o. No._- we.Nh .m.:- ew.: o.SN Nm No.: N.m_ MN zoo-e_z mo.:- on mo._ mm.Nn m._- Nm.m N.mN Nm em.m N.MN NN cm_:-eiz Nm.m- 0. me. - Nm.Nh «o.m- Ne.q m.NN em No.N m.eN mN ;e_: _a>Lauc_ mC_s_o .eemo .o.m can: 2 .o.m cap: 2 mocon_mcou Nmm . . cmoz noEEmLmOLL ocsuuoo we:u_ua< .mo_umEocumE mmo:_m:n cm mocoom umou _mc_d Lo» mco_uomcouc_ ucoeumocu >3 ov:u_uam com mcom_cmaeoo oozlumOmlu.o.: o_nmh 81 These findings provided mixed results with respect to section- ing decisions. The former practice of placing the two Mid-level apti- tude groups in the lecture treatment was not supported. For the Mid- High group the difference was not significant. but it favored the programmed method. For the Mid-Low group. the difference was signifi- cant and it favored the programmed method. The former practice of placing the High- and Low-aptitude groups in the programmed method received only partial support. The High-aptitude group achieved better in the programmed method. upholding past practice. The lowest scoring group was the Low-aptitude pro- grammed method group; hence. the former practice for this subgroup was not supported. The F-test for interaction (F = 4.32. p = 0.006) revealed that a significant interaction existed. The nature of the interaction can be observed by inspecting the Final Test means for the respective aptitude groups (Figure 4.1% It was apparent that the means for the programmed group con- formed to a hierarchical expectation. while the means for the Low lecture and Mid-Low lecture groups were reversed. Koran (1974) described this type of interaction as disordinal. The instructional implication for this type of finding is that the Low students should be placed in the lecture treatment and the Mid-Low students should be placed in the programmed treatment. While it might be possible to find explanations for superior performance for Low students in the lecture method over the programmed method. the superior performance of the Low 82 over the Mid-Low students in the lecture group seemed inexplicable. Hence. further exploration of the data was undertaken. 28— " 27.5 High ./ 27— _/’ all] _/ ./ 26'- /// ,/ ./ , 25.2 Mid-High 25 fl ’ ‘1 21+ _ 24.0 Mid-Low ‘ 23 — 22 A 21 — 20 - 19.3 Low 19 — 18 -V T l I Lecture Programmed Figure 4.1.-—Means for the Final test scores. 83 We: First. the final course grades were put into the same ANOVA procedure as the Final Test Scores (Table 4.7). The F-test for interactions produced a value of 4.84. and the probability level exceeded 0.003. Table 4.7.--Analysis of variance components for Final grades in business mathematics. Effect SS df MS F Prob. Mean 970.15 1 970.15 965.82 0.000 Aptitude-- ACT Math 75.82 3 25.27 25.16 0.000 Treatment—— lecture/prog. 2.17 l 2.17 2.16 0.143 Interaction 14.60 3 4.86 4.84 0.003 Residual 228.02 227 1.00 SoLonoioitous_explonation_oi_attitude§. The lack of adequate explanations for the above findings led to an analysis of the attitudi- nal measures under the assumption that more complete inspection of the aptitude groups was warranted. Each of the six attitudinal measures was assigned as a dependent variable in the two-way analysis of vari- ance. Four of the six variables revealed no interaction to help explain the achievement pattern. However. the Motivation scale pro- duced a significant F-test value for interaction (F = 3.27. p = 0.014) (Table 4.8). The Motivation attitude means for the aptitude groups 84 were graphically similar to those of the posttest achievement means (Figure 43). At the time of pretesting. the Low group in the lecture method reported higher motivation than both the Mid-Low and Mid-High groups. Reasons for this phenomenon are not known. 4.0 _ 3.5 _ 3'0-i /./// ‘ 2 5 _ 2.64 _________ 2.24 Mid-High 2'20 ————— // 2.18 Mid-Low 2.0 _ I, 1.62 1.5 _ 1.39 1.0 _ 1.07 Low .5 _ I I Lecture Programmed Figure 4.2.--Means for Final grades. 85 Table 4.8.--Analysis of variance components for Motivation in Mathematics attitudinal scale. Effect SS df MS F Prob. Mean 96830.00 1 96830.00 9685.93 0.000 Aptitude—- ACT Math 67.22 3 22.41 2.24 0.084 Treatment-- lecture/prog. 1.17 l 1.17 0.12 0.733 Interaction 98.03 3 32.68 3.27 0.022 Residual 2214.77 227 9.80 22- 21.6 -——._._h__ .. ........... __21.2 High 2] ._ ',.——' 21.0 MId’LOW 20-8 ,,,,,, ,/ 20.9 Mid-High v /’ 20.4 " 20- ’1/ 19.4 Low I9- 19.0 / 19" I 1 Lecture Programmed Figure 4.3.--Means for Motivation in Mathematics. 86 Although the interaction was not significant (Table 4.9). the Enjoyment of Mathematics scale produced (Figure 4.4) a pattern highly similar to that of the Motivation scale and the achievement measures. Table 4.9.--Analysis of variance for Enjoyment of Mathematics attitudinal scale. Effect SS df MS F Prob. Mean 101308.06 l 101308.06 6279.00 0.000 Aptitude-- ACT Math 426.77 3 142.26 8.82 0.000 Treatment-- lecture/prog. 3.31 l 3.31 0.21 0.651 Interaction 41.59 3 13.86 0.86 0.460 Residual 3662.51 227 16.13 fiujjtmdes_g§_goyatintes. Pretest differences in attitudes were camouflaged in the total score distribution (Table 4.1). Significant differences were not evident. Also. the correlations with achievement (r = .12 and r = .26) were not high. yet the attitudes appeared to be a pretestable characteristic of Low and Mid-Low students that helped explain achievement outcomes. jaunmgty. It appeared that High-ability students benefit from the programmed instruction. Beyond that clear finding. it appeared that the hypotheses. as stated. failed to account for the complexity represented by the data. Without the serendipitous findings about the 87 attitude patterns of the Mid-Low and Low groups. faculty conclusions about aptitude-treatment interactions could have been reached. 24 23 22 21 20 19 18 —‘ 23.0 ........ _____ .-...._._ 22.6 High ,,,, 22.2 Mid-High 21.7 TTTTTTT 20.4 _ 20.0 Mid-Low 19.6 - 18.8 Low 2/ I I Lecture programmed Figure 4.4.--Means for Enjoyment of Mathematics. What appeared to be a differential effect of instructional method on the lower half of the ability group was a reflection of pretest attitudinal differences. Reasons why students with low ability. but higher motivation and Mid-High ability. but lower motivation were enrolled in the lecture treatment and not enrolled in the programmed treatment were not available. Perhaps advice students 88 received from advisors and fellow students was a factor. or the phenomenon was unique to this study. Only a repetition of this study could suggest which explanation is most plausible. Obviously. future studies should include attitudinal measures and subgroup analysis techniques. Further study of this phenomenon is recommended. Enecequisiioieaonioo The research question was: Should a prerequisite learning experience be established for students with low mathematical ability? The college adopted a practice of recommending mathematics course placement to students who had an 80 percent chance or better of earning "C" or higher grades. However. this criterion has not been applied to business mathematics. The number and proportion of students from each method who earned'TFfl or higher grades based on ACT Mathematics scores were combined into 13 score categories (Table 45K”. Overall. 164 students or 70 percent received at least "C-" grades. The 80 percent criterion for score levels was reached in the score interval for 17 and 18. The criterion implied that 148 students or 63 percent would be candidates for a prerequisite experience. This finding was unexpected. especially since a high proportion had taken college mathematics courses pre- viously. The low probability of success suggested an analysis of student achievement based on prior enrollment in a mathematics course be per- formed. Based on current Ferris State College practice. students with 89 scores from 01 to 08 are recommended to take Mathematics 090. a course that emphasizes arithmetic skills and concepts. Table 4.lO.--Frequency and proportion of students earning C- or higher final grades in Business Mathematics 121. ACT Math Frequency and Proportion Earning C- or Higher Score Interval Lecture Programmed Combined Freq Total Prop Freq Total Prop Freq Total Prop 27-above 3 3 100 l l 100 4 4 100 25-26 2 2 100 4 4 100 6 6 100 I 23-24 3 3 100 10 10 100 13 13 100 21-22 8 10 80 12 12 100 20 22 91 19—20 ll 13 85 8 9 88 19 22 86 17—18 7 8 86 ll 12 92 18 20 90 15-16 5 7 71 8 13 62 13 20 65 13-14 4 6 66 4 5 8O 8 ll 73 11-12 4 8 50 ll 15 73 15 23 65 9-10 5 15 33 l4 17 82 19 32 60 7-8 5 10 50 3 10 3O 8 20 40 5-6 11 16 68 3 ll 27 14 27 52 1-4 5 ll 45 2 4 50 7 15 47 Total 73 112 65 91 123 74 164 235 70 Efiegt_gf_Mathematfigs_QflQ. Student final grades and final test scores for those who took Mathematics 090 compared to those who were not enrolled in another mathematics course suggested that the relationship between enrollment in a recommended prerequisite course and success in business mathematics was random (Table 4J1). In fact. inspection of the data suggested that students without Mathematics 090 were slightly more successful in avoiding "F" grades and low final test scores in the subsequent business mathematics class. Thus. it appeared that 90 Mathematics 090 was not an effective prerequisite course for students whose ACT Mathematics scores were in the 01 to 08 range. Table 4.11.-—Comparison of success in business mathematics based on prior enrollment in Mathenatics 090 for the l to 8 score range. Enrolled Not Enrolled Final Grades Freq Pct Freq Pct A 0 00.0 1 2.4 B 3 14.3 5 12.2 C 5 23.8 15 36.6 D 7 33.3 13 31.7 F 6 28.6 7 17.1 Total 21 100.0 41 100.0 Chi-square = 1.59. df = 3. p = 0.661 r = -0.129 Final Test Scores 28-33 2 9.5 4 9.8 24-27 2 9.5 5 12.2 20-23 5 23.8 18 43.9 7-19 12 57.2 14 34.1 Total 21 100.0 41 100.0 Chi-square = 3.318. df = 2. p = 0.190 r = -O.l30 Effegt of Mathematics 1]]. A similar analysis for students with ACT Mathematics scores in the 09 to 17 range was also performed (Table 4.12). The Mathematics 111 course. Introductory Algebra. is the course to be recommended routinely to students in this ability range. Those who had enrolled versus those who did not were compared on both success 91 measures. final grades and final test scores. No relationship existed between prerequisite course enrollment and success in the business mathematics course. The chi-square values were small. and the corres- ponding probabilities were large. Thus. it appeared that Mathematics 111 was not an effective prerequisite for students whose ACT Mathe- matics scores were in the 09 to 17 range. Table 4.12.--Comparison of success in business mathematics based on prior enrollment in Mathematics 111 for the 9 to 17 score range. Enrolled Not Enrolled Final Grades Freq Pct Freq Pct A 5 7.7 4 12.1 B 16 24.6 9 27.7 C 21 32.3 11 33.3 D 14 21.6 5 15.2 F 9 13.8 4 12.1 Total 65 100.0 33 100.0 Chi-square = 1.045. df = 4. p = 0.903 r = -0.087 Final Test Scores 28-33 15 23.1 7 21.2 20-23 15 23.1 9 27.3 7-14 17 26.1 9 27.3 Total 65 100.0 33 100.0 Chi-square = 0.300. df = 3. p = 0.960 r = 0.035 92 .Ettect_o£_Mathemati;s_121. A third analysis was made for students whose ACT Mathematics scores were in the 18 to 22 range and who enrolled in Mathematics 121. College Algebra. Mathematics 121 experience had a positive effect on business mathematics achievement (Table‘4J3). On the final grades measure. the chi-square value was 4.993 and the corresponding probability of 0.082 approached signifi- cance. The proportions earning "A" or "B" grades differed by 26.8 percent. favoring those who enrolled in Mathematics 121. On the final test. the data produced a significant difference. The chi-square value was 9.483 and the associated probability was 0.024. An inspection of the table shows scores of 20 and above were achieved by a high propor- tion of students who had enrolled in Mathematics 121. and a higher proportion of those who did not enroll scored 19 or fewer points on the Business Mathematics Final Examination. 'Thus. it appeared that enroll- ment in Mathematics 121 was associated with better performance in business mathematics. 93 Table 4.13.--Comparison of success in business mathematics based on prior enrollment in Mathematics 121 for the 18 to 22 score range. Enrolled Not Enrolled Final Grades Freq Pct Freq Pct A 9 27.3 1 5.3 B 12 36.4 6 31.6 C 10 30.3 8 42.1 D 2 6.0 2 10.5 F 0 0.0 2 10.5 Total 33 100.0 19 100.0 Chi-square = 4.993. df = 2. p = 0.0824 r = 0.356 Final Test Scores 28-33 14 42.4 6 31.6 24-27 9 27.3 5 26.3 20-23 8 24.3 1 5.3 7-19 2 6.0 7 36.8 Total 33 100.0 19 100.0 Chi-square = 9.483. df = 3. p = 0.024 r = 0.232 W. The failure of the plausible prerequisite courses to be associated with better achievement in the business mathematics course for students who scored 17 and below on ACT Mathematics is a matter of concern. Several possibilities for this phenomenon could be investigated. 1. Students in the 01 to 17 ability range may have difficulty with transference of learning. 94 2. The mathematics course content may be paced inappropriately for students of this ability level. Perhaps more time is needed. 3. The ACT Mathematics score cut-off points for course place- ment may require upward adjustment. giving students with scores of O9 and 10 the opportunity to take Mathematics 090 before taking the Algebra course. 4. Business mathematics instruction may include compensatory instruction which is sufficient to negate the effects of the mathemat- ics courses on business mathematics achievement. 5. The business mathematics course may be more difficult conceptually than previously perceived. Instead of being a step up from Mathematics 090 and on a level with Mathematics 111. it may be a step up from Mathematics 111 and on a level with Mathematics 121. All of the above may interrelate to explain the results observed. Further study of alternatives is recommended. AttitudLAssossmont The research question was: Should attitudes toward mathematics be considered with ability measures in course-sectioning decisions? The criterion for accepting attitudinal measures was a significant contribution in the explanation of the variance of the dependent variable after the ability measures had been fully used. The Final course grade was selected as the dependent variable because most of the independent variables had slightly higher correlations with the grade than with the final test score. 95 Results. Stepwise multiple regression was applied to the data of the 235 students from the combined treatment groups. The hierarchi- cal regression was controlled to first allow the entry of the ability measures before accepting the attitudinal measures. The F-to-enter level (0.50) was selected to permit the full use of the ACT tests and high school grades before considering attitude measures (Tabachnick & Fidell. 1983). The maximum number of steps in the regression was set at eight; however. five steps proved to be sufficient. The primary predictor in the multiple regression analysis was the ACT Mathematics test. which accounted for 26 percent of the vari- ance in grades (Table 4.14). The high school grade point and two ACT test variables. Social Studies and English. were selected next by the statistical procedure. The F-value associated with the grades was significant at the .01 level. and the F-val ue associated with the Social Studies measure was significant at the .05 level. The con- tribution of the English test was not significant. but its presence exhausted the contribution of the ability measures. Also. the presence of the Social Studies and English tests was a reminder about the verbal aspects of mathematical achievement. The attitudinal measure. Self-Concept in Mathematics. was entered in the fifth step. Its contribution added 4 percent to the explanation of variance in achievement as measured by course grades. The cumulative explanation of variance was .3871 or 39 percent. The F-value associated with Self-Concept in Mathematics was 16.1457. To satisfy the research question. the critical F-val ue at the .01 level of 96 confidence for l and 229 degrees of freedom was 6.76. The observed F-val ue for the attitude measure was well beyond the .01 level; thus the stipulated criterion for accepting the use of attitudinal data in course sectioning was met. Table 4.14.--Prediction of final grades using ability and attitude measures for lecture and programmed students combined. Step Variable Multiple Multiple Increase F-to No. Entered R R2 in R2 Enter 1 ACT Math 0.5064 0.2564 0.2564 80.3493 2 HS GPA 0.5725 0.3277 0.0713 24.6033 3 ACT Soc Stu 0.5848 0.3420 0.0143 5.0040 4 ACT English 0.5864 0.3439 0.0019 0.6690 5 Self-Concept 0.6222 0.3871 0.0432 16.1457 Enhem_yaniables. At the conclusion of the fifth step in the regression. the partial correlations and F-values for the other attitudinal measures and sex were inspected. All were within the chance region. well below the critical F-value of 3.89 at the .05 level. For example. the largest F-value was 1.056 for the motivation measure. The F-value for the demographic variable Sex was 0.37. ,Multiyaniate_pnogedute. If an attitudinal assessment were implemented. the instrument could be administered in the college's new student orientation program. The students' responses could be scanned and passed to the mainframe computer electronically. A scoring pro- gram. now in use. could be used to derive scores for inclusion in the student master file along with ACT test scores and high school grades. 97 The scores could be processed by the formula generated from the mul- tiple regression procedure to attain a predicted course grade. The formula is: Predicted Grade = -1.807 + .049*(ACT Math) + .472*(HS GPA) + .025*(ACT SS) + .023*(ACT ENG) + .063*(Se1f-Concept) The predicted grade could be translated into a placement recom- mendation and printed on the college's placement profile to accompany course placement suggestions in other subject course areas. such as mathematics. English. and reading development. Wis. Prior analysis of aptitude-treatment interactions in the discussion on sectioning decisions revealed that two attitude measures. Motivation in Mathematics and Enjoyment of Mathematics. had a special relationship to achievement in the lecture group. The Mid-Low aptitude group had a low motivation. low enjoyment of mathematics. and low achievement pattern. while the lowest aptitude group had higher motivation. enjoyment. and higher achievement. This reversal was not evident for the programmed group. Also. the analysis of prerequisite learning (Table 4.9) revealed that Low and Mid-Low lecture groups had relatively low probabilities of "C-" or better grades. In comparison. the Mid-Low programmed group had a higher proportion earning grades of "C-" or better. Given those contrasts. it seemed useful to explore the role of h— ~ 98 the attitudinal variables in explaining achievement for these lower aptitude students within the respective treatment groups. For this analysis. stepwise multiple regression was applied to each instructional group. In contrast to the prior regression for the total group. the restriction to enter ability measures was first removed. The attitudinal and demographic measures had equal opportu- nity to be considered. In each case. a stepwise regression was run; the table of partial correlations was inspected at each step. An optimal set of predictor variables was selected for inclusion. The stepwise multiple regression for 60 students from the lower 1 half of the aptitude distribution in the lecture method showed that across the eight steps 30 percent of the variance in grades was explained by all predictor variables (Table 4.15). The contribution of individual measures after the second step was not significant at the .05 level. However. the F-ratio for the overall regression at the eighth step was 2.71. significant at the.05 level for 8 and 51 degrees of freedom. Setting aside the matter of significance. it was interesting to observe the role attitudes played in this analysis. The primary predictor. Enjoyment of Mathematics. accounted for 8 percent of the variance. It was followed in Step 2 by the English Usage measure. not Mathematics or High School Grades. Then the Value of Mathematics measure was entered. followed by Natural Science and two more attitudinal measures. Anxiety and Motivation. In all. the attitudinal measures accounted for 14 percent or almost one-half of the total 99 explained variance. Perhaps the chief value of this analysis of the lecture method is to contrast it with the same aptitude group that experienced the programmed method. Table 4.15.--Stepwise multiple regression of ability and aptitude and demographic measures on grades for the lower aptitude students in the lecture method. Step Variable Multiple Multiple Increase F-to No. Entered R R2 in R2 Enter 1 Enjoyment 0.2878 0.0828 0.0828 5.2384 2 ACT Eng 0.4599 0.2115 0.1287 9.3038 3 Value 0.4896 0.2397 0.0281 2.0718 4 ACT NS 0.5098 0.2599 0.0202 1.5048 5 Anxiety 0.5247 0.2753 0.0154 1.1439 6 Motivation 0.5320 0.2831 0.0078 0.5769 7 ACT SS 0.5387 0.2902 0.0072 0.5262 8 Teacher 0.5462 0.2984 0.0081 0.5895 The multiple R for the effect of ability and attitudes on achievement for the programmed method reached .7467 and accounted for 56 percent of the variance in grades in the tenth step for 57 students in the lower-half aptitude group of the programmed method (Table 4.16). The F-val ue at the tenth step was 10.501. well past the .01 critical value of 3.12 for df = 6.50. The high school grade point average was entered first and accounted for 27 percent of the variance. The ACT Mathematics and ACT Natural Science variables were entered next. They were followed by the Sex variable. which accounted for an increase of 5 percent in explained grade variance. The coefficient for Sex was negative. meaning the 100 lower-aptitude women did not use the programmed methods as well as men in the same aptitude range. The F-value associated with Sex. 5.2702. was significant at the .05 level for df = 1.52. Table 4.16.--Stepwise multiple regression of ability. attitude. and demographic measures on grades for lower-ability students in the programmed method. Step Variable Multiple Multiple Increase F-to No. Entered R R2 in R Enter 1 HS GPA 0.5166 0.2669 0.2669 20.0225 2 ACT Math 0.6043 0.3652 0.0983 8.3610 3 ACT NS 0.6364 0.4051 0.0399 3.5523 4 Sex 0.6781 0.4598 0.0547 5.2702 5 ACT NSa 0.6643 0.4412 -0.0186 1.7865a 6 ACT Math 0.6320 0.3995 -0.4180 3.9632 7 No. HS Math Course 0.6665 0.4442 0.0447 4.2620 8 ACT Math 0.7099 0.5040 0.0598 6.2736 9 Self-Concept 0.7297 0.5325 0.0285 3.1084 10 ACT NS 0.7467 0.5576 0.0251 2.8324 aRemoved. In the next four steps. the two ACT tests were removed. the number of high school courses was included. and ACT Mathematics was re-entered. Then the attitudinal measure Self-Concept was entered. The F-value associated with Number of High School Courses was significant at the.05 level; however. the 3 percent contribution of the attitudinal measure was not statistically significant. Similarly. the re-entry of the Natural Science measure was not significant. The effects of attitudes appeared to be negligible in this analysis. 101 Student_Exaluation_oi_Mothodolog¥ A seven—item. forced-choice questionnaire designed to obtain student opinions about features of the business mathematics course was administered on the sixth week of the term. before student departures from the self-paced. programmed courses. The first two items asked about the usefulness of the course in developing concepts and skills. The remaining five items asked about opinions of different aspects of the teaching methodologies. fiypothesis_testing. The null hypothesis was that the means of the two methodologies were not significantly different (Hypothesis 3: ML=MP). One hundred seven students from the lecture group and 118 from the programmed group responded to the course evaluation survey (Table 4.17). Five students were absent from each group. The means for each of the items were within two-tenths of a point from a value of three. which is described by the instrument as agreeing with a positively worded statement. A multivariate test. Hotelling's T2. produced a value of 9.63 and an associated F-value of 1.338. The probability level for the finding was 0.23. not significant. Thus. the null hypothesis was not able to be rejected (H3: ML=MPL An inspection of the t-values for each item reveals that one item. concerning use of out-of—class time. produced a significant difference. Research procedure would lead to the conclusion that this finding could be due to chance (Type I error). In contrast. however. the instructional faculty believed that high—ability students who had 102 the opportunity to depart from class upon completion of the material would report greater satisfaction on this item. Since potential for a Type II error existed. a follow-up analysis was performed. Table 4.17.--Comparison of student evaluation of teaching methodologies used in business mathematics. Lecture Programmed Method (n=107) (n=118) t-value Prob. Mean 8.0. Mean 8.0. f Understanding of Concepts 3.0 0.62 3.0 0.59 0.01 0.991 Developing Skills 3.1 0.59 3.1 0.63 -0.97 0.336 Like Method 2.9 0.76 3.0 0.88 -l.35 0.179 I Adjusted to Method 3.0 0.70 3.0 0.77 -0.55 0.582 Use of In-Class Time 2.9 0.79 3.1 0.77 -l.46 0.146 Use of Out-of- Class Time 2.8 0.71 3.1 0.71 -2.79 0.006 Recommend to Others 3.0 0.80 3.2 0.89 -l.35 0.177 Hotelling's T'2 = 9.63 F = 1.338. df = 1,217, p = 0-233 Use_gf_out_ot_glass_time. The post—hoc comparisons for the four ability groups from each methodology showed that. as suspected by the faculty. the largest mean difference occurred for the High aptitude group (Table 4.18). In fact. within the lecture method the High 103 aptitude group had the lowest mean on the use of out-of-class time. while in the programmed method the High aptitude group had the highest mean. This finding. when considered with the finding that the High aptitude programmed group achieved better than the High aptitude lecture group. provided support for the special placement of the high- ability students in the programmed method. Table 4.18.--Post—hoc comparisons for aptitude groups on use of out-of- class time in business mathematics. Lecture Programmed Mean . Aptitude Difference N Mean S.D. N Mean S.D. High 23 2.4 0.89 32 3.2 0.55 -0.8 Mid-High 27 2.8 0.74 31 3.0 0.75 -0.2 Mid-Low 20 2.8 0.44 30 3.0 0.79 ~0.2 Low 37 3.1 0.55 25 3.1 0.76 0.0 Summm Student achievement in lecture and programmed classes of Business Mathematics 121 was evaluated in Chapter IV. Pretest data revealed that students enrolled in respective methods had equivalent characteristics and that comparisons between instructional treatments could be pursued. Both methods produced substantial significant gain in knowledge as measured by a pretest administered to a representative group and by a comprehensive final examination. The programmed method produced higher achievement than did the lecture method. particularly for the High aptitude group. The Low 104 lecture group achieved better than the Low programmed group. and conversely. the Mid-Low programmed group achieved better than the Mid- Low lecture group. Aptitude-treatment interaction was present. However. additional exploration with the attitude measures revealed a preexisting condition. an attitude-treatment interaction. that cast doubt about the observed aptitude-treatment interaction for the Low aptitude groups. An analysis of aptitude and course grades revealed that a substantial proportion of the students had less than an 80 percent chance of receiving "C-" or better grades. A number of students had taken potential prerequisite mathematics courses that were considered appropriate for their ability levels. Their performance in Business Mathematics 121 was compared to students of comparable ability levels who had not taken the course. Interestingly. evidence that prior enrollment in a mathematics course had an effect on success in business mathematics was not demonstrated for two of the three groups analyzed. Reasons for this phenomenon were discussed. The utility of attitude assessment to predict student grades was explored. After ability measures had been entered into stepwise multiple regression. the Self—Concept in Mathematics scale score was entered. It explained 4 percent of the total course variance and was statistically significant. Other attitudinal measures were not useful in predicting success for the combined groups. However. a regression analysis of grade prediction for the Low and Mid-Low aptitude lecture groups revealed that attitude measures accounted for 14 percent. or 105 about half. of the explained variance. In the equivalent programmed group. sex was significantly related to achievement. An instrument designed to assess student reaction to the course methodology did not reveal differences between the methods. The multivariate test was not significant; however. the data suggested that High aptitude. programmed method students perceived that the method enabled them to make better use of out-of—class time than did High aptitude lecture students. CHAPTER V SUMMARY AND RECOMMENDATIONS Summanx Evaluation studies of educational practices are useful to the process of judging the value of such practices and making decisions to improve instructional outcomes. Evaluative studies. like descriptive and experimental studies. generate information useful to future inquiries into the same or similar educational problems. Consequently. this study was designed to evaluate selected aspects of the business mathenatics course at Ferris State College. Two instructional methodologies were used: the self-paced. programmed method and the traditional lecture method. Students who enrolled in the course varied widely in mathematical ability upon entry to the course. Before 1981 these differences were recognized and used to develop sections of homogeneously grouped students for special instructional treatment. The middle 50 percent of the mathematics- ability distribution were assigned to the lecture treatment. The upper and lower 25 percent groups were assigned to the programmed. self-paced treatment. No formal studies to confirm the effectiveness of this practice were performed. Since then. the sectioning practice was discontinued. At the time of the study. students of all ability levels were taught by both methods. The existence of alternative treatments 106 107 with students of varying aptitudes for mathematics instruction gave rise to five research questions. These research questions were the basis for the null hypotheses. 1. Should the offering of both lecture and programmed instruc- tional methodologies be continued? 2. Should sectioning by ability level for different instruc- tional methods be reinstated? 3. Should a prerequisite learning experience be established for students with low mathematics ability? 4. Should attitudes toward mathematics be considered with mathematics ability in sectioning decisions? 5. Should the collection of students' opinions of the teaching methodology be implemented? The criterion for judging Questions 1. 2. 4. and 5 was the.05 level of significance on the appropriate statistical test. The cri- terion for judging Question 3 was based on an institutional practice which sought at least an 80 percent chance for students to earn a "C" or better grade in an entry-level mathematics course. LiteLatuLe These questions provided the basis for a literature review that included (1) studies that compared two or more instructional methods in high school or collegiate business mathematics; (2) studies that com- pared programmed. individualized. or personalized systems to the lec- ture method for entry-level college mathematics;(3) studies that sought to identify aptitude-treatment interactions in mathematics 108 instruction; (4) studies on mathematics course placement; and (5) promising variables relevant to future studies. In 12 studies that focused on business mathematics. seven found support for the programmed approach and one found mixed results. Four studies reported no differences. The lecture method was not found to be superior in any of the studies. Quasi-experimental research designs were used in all but two of the studies. Another group of 13 studies that dealt with introductory college-level mathematics were reviewed. The lecture method was compared to the individualized. programmed. self-paced. or other treatments (some involving the small-group discussion methodL Eight of the studies used an experimental design; the remainder used quasi- experimental designs. A relationship between type of design and results was not apparent. In only one case did the lecture method prove superior. In three studies. achievement was equal and in seven studies the alternative method was favored. Thus. in the 25 studies reviewed. the lecture method produced better achievement only once. Such an outcome could be expected by chance alone. In general. the literature provided support for studies that would challenge the effectiveness of the lecture method in mathematics instruction. The literature on aptitude-treatment interactions in mathe- matics education defied a neat summary. The aptitude dimension has included variables such as mathematical ability. attitudes toward mathematics. anxiety. and locus of control. Assignment of students to aptitude groups has been determined by univariate and multivariate 109 procedures. The treatment dimension has included arithmetic. algebra. and calculus; logical sequencing of content and scrambled sequencing of content; units within courses. full courses. and series of courses; and programmed and lecture methods. Sample size per treatment has varied from 2 to 25 to over 100. Outcome measures have included test scores. grades. problem-solving ability. attitudes. and cognitive-style meas- ures. Mediating variables such as time-on-task have been used. Research designs were evenly divided between experimental and quasi- experimental. The complexity of ATI research noted by Cronbach and Snow (1981) was evident in this review. Snow's (1970) statement of need for a grand matrix that identifies learning environments where learners of different characteristics thrive has yet to be realized. The course-placement literature produced evidence that remedia- tion instruction can be effective (Rhodes. 1984) and that proper instructional sequencing and course placement lead to improved achieve- ment (Bone. 1981; J. Smith. 1982). Also. multivariate procedures including ability and attitudes to predict course grades and influence course placement have potential for improved practices (Byrd. 1980; Decker. 1974; Helmick. 1983; Sims. 1980). Measures of cognitive styles may have potential for future studies. Method The study was conducted in all business mathematics classes during the winter academic term of 1983-84 at Ferris State College. One hundred twelve students comprising five classes completed the 110 lecture teaching methodology. One hundred twenty-three students comprising two classes completed the programmed teaching methodology. Students were assigned to one of four ability-level groups based on the ACT Mathematics test score. The quartiles were used for the blocking. Scores of 01 to 08 were assigned to the Low group. 09 to 12 to the Lower Middle group. 13 to 18 to the Upper Middle group. and 19 to 32 to the High group. The design used was the nonequivalent control group since random assignment to instructional treatment was not feasible. Beyond the ACT Mathematics test. the pretests included the other three ACT subtest scores: English. Social Studies. and Natural Sciences. Also. the ACT self-reported high school grades were obtained. In addition. the Mathematics Attitude Inventory and the Business Mathematics Questionnaire were administered on the first day of class. The Business Mathematics Final Examination was administered as a pretest in the most representative class section so that gain scores could be estimated. The posttest measures were the Business Mathematics Final Examination and the final grade in the course. The data were analyzed by using the BMDP statistical package on the mainframe computer at Ferris State College. Results The first research question was concerned with the continuation of offering the two instructional methodologies. The statistical analysis consisted of t-tests for gain scores for students in each methodology. The lecture method estimated mean gain was 13.9; the programmed method estimated mean gain was 15.9. A large t-value 111 (p < .01) for each method was found. which led to the rejection of the hypotheses that the gain scores would be equal to zero. Both methods produced highly significant gains; consequently. both methods were considered productive and useful. A t-test to compare the respective mean gains was performed. The observed difference of two points favored the programmed method (p < .05). The finding was similar to that of Miller (1984). Brown (1984). and Wells (1982). who studied business mathematics. However. the result should not be taken to imply that the programmed method was superior for all students. The second research question provided a more detailed investigation into the comparative worth of the two methodologies. The second research question dealt with sectioning by ability levels for the different methodologies. Aptitude-treatment interac- tions. if observed. would have supported a return to a practice of sectioning by ability level. The study of interaction also clarified the finding that programmed students experienced superior gain. Analy- sis of variance was applied to the Business Mathematics Final Examina— tion score. Significant F-values for the main effects and the interac- tion effect were found (p < .05). Consequently. Scheffeks post hoc comparison procedures were used to construct 95 percent confidence intervals around the difference between the means for each ability level. Where the confidence interval excluded zero. the null hypothe- sis was rejected. 112 For the High ability group. the programmed method was favored and the null hypothesis was rejected. This finding was similar to that of T. Smith (1983) and five studies reviewed by Cronbach and Snow (1977L For the Mid-High group. the programmed method was favored slightly but the null hypothesis was accepted. For the Mid—Low group the programmed method was favored and the null hypothesis was rejected. For the Low group the lecture method was favored slightly. but the null hypothesis was accepted. Thus. clear support for the programmed method was found for two of the four ability groups. the High and Mid-Low groups. However. closer inspection of the means revealed an inconsis- tency in the data. For the programmed aptitude groups. the posttest means on the Final Business Mathematics Test were ordered from high to low. in correspondence with the aptitude groupings. The same consistency was not apparent for the lecture group. The Mid-Low group mean of 19.7 was below the Low group mean of 21.4. This within-method reversal was not anticipated or explainable. Therefore. a follow—up analysis on final grades was performed. The same pattern was observed. Further inquiry into the pretest attitudinal measures was conducted. It was discovered that two attitudinal measures provided useful information. The interaction patterns for Motivation for Mathematics and Enjoyment of Mathematics conformed to the patterns for the posttest measures. The Low lecture group means were higher than the Mid-Low lecture group means. This finding confirmed that a preexistent interaction between method. ability. and attitude was present. 113 Identification of reasons for this occurrence goes beyond the scope of this study. However. the findings suggested that (1) the results of the study should not be generalized beyond the term in which the study was conducted and (2) the rejection of the null hypothesis for the Mid-Low groups might be a false rejection (Type I error). The condition observed in this study may not reoccur. However. the find- ings that supported the superiority of the programmed method for the High ability group should stand if a replication of this study were to be carried out. The third research question was concerned with the need for a prerequisite course for students with low mathematics ability. The criterion was at least an 80 percent chance of earning a "C" or better final grade in Business Mathematics 121. The analysis showed that students who scored below 17 on the ACT Mathematics test. n = 148 (63 percent). would be eligible for a prerequisite course. If the criterion had been dropped to a 70 percent chance of "C" or better grades. then those who scored below 13. n = 117 (50 percent). would be eligible for a prerequisite course. In fact. many students had taken courses that would be logical choices for a prerequisite experience. This permitted an analysis. according to institutional course-placement guidelines. to determine whether potential prerequisite courses were demonstrably effective. However. the effects of completion of a prerequisite course could not be shown. Similarly. Whitesitt (1980) found that all of the remedial mathematics courses at Montana State University were ineffective in 114 developing competence in most areas identified as important to success in subsequent courses. Totten (1983) found similar results at Ferris State College. Several potential reasons for this finding were suggested and should be considered in further research. Suggestions by Totten (1983) should be considered also. It was concluded that a prerequisite learning experience for approximately 60 percent of the students was advisable. but the nature of the experience could not be ascertained. The fourth research question dealt with the usefulness of attitudinal measures in making sectioning decision. Stepwise multiple regression controlled the entry of variables so that the contribution of the ability measures in predicting final grades could be exhausted before the attitudinal measures were entered. In the fifth step. the Self-Concept in Mathematics measure was entered. The F-to-enter was 16.1457. significant at the .01 level. and the increase in multiple R was over 4 percent. Consequently. the use of attitudinal measures in a sectioning procedure should be considered. A prediction equation was developed for this purpose. Since analysis of aptitude-treatment interactions suggested that preexisting attitudes toward mathematics influenced achievement for the lower-half ability group. follow-up analysis was performed. For the lecture method. the best predictor was Enjoyment of Mathematics and. overall. attitudes accounted for 14 percent. or almost one-half. of the explainable variance. For the programmed method. the contribu- tion of attitudinal measures was negligible. The result suggested that 115 special attention to attitudes be given the lower half of the ability distribution in the lecture courses. The fifth research question was concerned with the use of student opinions about the teaching methodology. A multivariate test. Hotelling's T2. was used to analyze the results of a seven-item questionnaire. The multivariate F-value was not significant. which suggested that student opinions about the methodology were not highly contributory to the other outcomes. However. the data suggested that high-ability students in the programmed method felt better about the method's contribution to use of out-of—class time than their counter— parts in the lecture methodology. Recommendations Students gained considerable knowledge about business mathemat- ics under each method of instruction. but the former sectioning prac- tice of placing the high and low quartiles in the programmed classes and the middle quartiles in the lecture classes received only partial support. The programmed method was favored for three subgroups (High. Mid-High. and Mid-Low). The differences in performance for the High and Mid-Low groups were statistically significant. 1. Consideration should be given to making greater use of the programmed method. Replication of the study in other terms would provide greater confidence that the programmed instruction is the method of choice for higher aptitude groups. 2. Especially needed is further study of the lower half of the aptitude distribution. where unexpected differences in attitude were 117 5. Although differential student preferences for course methodology were not found. such data should be a part of future studies. 6. Multivariate approaches to sectioning and placement decisions should be studied. The power of discriminant analysis to classify successfully placed students should be a part of any multivariate studies carried out. 7. Other variables that may help explain achievement. such as cognitive style. should be considered for future studies. APPENDICES 118 APPENDIX A BUSINESS MATHEMATICS QUESTIONNAIRE 119 Business Mathematics Questionnaire The purpose of this survey is to help the researcher learn about the mathematics background of Business Math 121 students. Please answer each question. Soc.Sec.# 1. What is your current status? (check one) 2. When did 3. How many have? 4. How many Freshman Sophomore Junior Senior you graduate from high school? (check one) 1983 1982 1981 1980 1979 or before semesters of high school math courses did you (check one) zero one or two three or four five or six seven or more previous college math courses have you taken? (check one) zero one two three four or more 5. If you checked other than "zero" in #4, please indicate courses you took. (check as many as apply) which 6. Sex 7. Major 1 3 4 l 2 Math 090 or equivalent Math 111 or equivalent Math 121 or equivalent Course above college algebra Male Female Field of Study: APPENDIX B BUSINESS MATHEMATICS IZI COMPREHENSIVE FINAL EXAMINATION IZI I22 BUSINESS MATHEMATICS 121 COMPREHENSIVE FINAL EXAMINATION DIRECTIONS: Use a No. 2 pencil to mark answers on the machine scoreable answer sheet. Make good clean erasures if changing an answer. Do your scratch work on the test itself. A calculator may be used. 144 is 36% of what number? a. b c. d 51.84 5184.00 195.84 400.00 What percent of 350 is 28? a. b. c. d. 8% 12.5% 98% 93% What is 202 of $8,000? a. b c. d 5 $160.00 $400.00 $4000.00 $1600 00 Balance shown on the bank statement of the Murphy Chemical Co., June 30, 19-- was $3,650.55. Balance shown by the checkbook, $3874.12. Checks outstanding: No. 336 - $38.50, No. 387 — $28.43, No. 395 -$6.25, No. 396 - $115, No. 397 - $80.75. Charge for imprinting checks - $9.00. Service charges - $3.50. Deposit in transit, $480.00 What is the Bank Balance? a. b. c. d. $4130.55 $3861.62 $3381.62 $4188.05 What is the Checkbook Balance? 3. b. c. d. $4188.05 $4130.55 $3861.62 $3381.62 I23 6 - 9 The following list itemizes the anticipated expenses of a county government for the coming year. Property in the county is assessed at $62,450,000. Education, $3,600,000 Police and fire protection, $2,100,000 Roads, $759,000 Parks, $220,000 Retirement of Debt, $190,500 6. What is the county's budget for next year? a. $619,500.00 b. $69,309,500.00 c. $6,869,500.00 d. $6,186,400.88 7. What is the tax rate, expressed as a percent? a. 11% b. .0112 c. 9% d. 9.1% 8. What is the tax on $1.00? a. $.09 b. 3.011 c. $.11 d. 3.091 9. What are the annual taxes for a piece of property assessed at $42,000.00? a. $1486.90 b. $163.56 c. $1486.00 d. $4620.00 10. Find the principal (P) when the rate (R) is 10%, the time (T) is 180 days and the interest (I) is $100.00. a. $2000.00 b- $5000.00 c. $5.55 d. $1666.67 11. Find the time (r)when the principal is $15,000.00, the rate (R) is 12%, and the interest is $900.00. a. 139 days b. 144 days c. 180 days d. 72 days 124 12 Find the rate (R) when the principal (P) is $4,000.00, the time (T) is 144 days and the interest (I) is $120.00. a. 75% b. 7.5% c. .075% d. 3/4% 13. Find the interest (I) when principal (P) is $560.00, the rate (R) is 8 1/2%, and the time (T) is 3 months. a. $264.00 b. $47.85 c. $142.80 d. $11.90 14 — 17 Discount Problems I and II. Face Time Date Discount Interest Discount Maturity Proceeds Value Date Rate Rate Value 1. $400 90 days June 16 July 18 8% 9% ? ? II. $200 120 days May 11 June 10 9% 10% ? ? 14. For problem I, what is the Maturity Value? 3. $432.00 b. $409.00 c. $408.00 d. $405.42 15. For problem I, what are the Proceeds? a. $402.08 b. $393.12 c. $402.06 d. $425.52 16. For problem II, what is the Maturity Value? 3. $218.00 b. $206.00 c. $204.65 d. $206.15 17. For problem II, what are the Proceeds? a. $199.13 b. $196.20 c. $204.28 d. $200.85 18. 19 — 19. 20. 22. I25 The retail price of an automobile is $8100. Purchased on a credit plan of 36 monthly installments, the price jumps to $10,100. Calcu- late the true annual percentage rate (effective rate of interest) using the formula: Programmed Classes: APR = 24F D (T + 1) Regular Classes: 2M1 R = P (n + 1) a. 16% b. 12.8% c. 52% d. 15.2% 20 What is the markon and selling price of an article costing $235 and has a markon of 40% of the cost? What is the Markon? a. 894.00 b. $156.66 c. $352.50 d. $141.00 What is the Selling Price? a. $391.66 b. $587.50 c. $329.00 d. $141.00 22 Find the selling price and markon for an article costing $55 and has a markon of 25% of the selling price. What is the Selling Price? a. $73.33 b. $128.33 C. $220.00 d. $68.75 What is the Markon? a. $13.75 b. $165.00 o. $73.33 d. $18.33 126 23 - 26 For problems I and II, find the invoice price and cash price of the following invoices. Assume that they are paid within 1. II. 23. 24. 25. 26. the discount period. List Invoice Price Trade Discounts Price Terms $3,000 33 1/3%, 25%, 15% ? 3/10, n/60 $ 865 33 l/3%, 40%, 10% ? 2/10, n/3O For problem I, what is the Invoice Price? a. $1725.00 b. $1275.00 c. $1062.50 d. $1253.75 For problem I, what is the Amount of Payment? a. $1216.14 b. $1188.25 c. $1624.75 d. $1030.62 For problem 11, what is the Invoice Price? a. $553.60 b. $307.08 c. $311.40 d. $417.48 For problem 11, what is the Amount of Payment? a. $300.94 b. $293.17 c. $530.77 d. $293.41 Value of Returned Amount of Goods Payment $50 ? $12 ? 27 - 3O 27. 28. 29. 30. Overtime is paid for over 40 hours worked per week. time rate is 1 1/2 times the regular rate. 127 as 6.13%.) Person Fletcher Enyart problem $179.40 $191.10 $214.50 $167.70 problem $11.18 $11.00 $11.52 $11.71 problem $171.00 $204.00 $193.00 $182.00 problem $11.29 $11.83 $11.16 $11.47 Total Hourly HOurs Rate 46 $3.90 45 1/2 $4.00 I, what is the Gross Pay? I, what is the FICA? II, what is the Gross Pay? 11, what is the FICA? The over- (Use the FICA rate Gross Pay FICA (at 6_]3%) ? ? 7 9 128 3] — 33 Depreciation Problems. Using the Straight Line Method, cost of equipment = $8800, scrap value = $600, and estimated life = 8 years. 31. What is the amount of depreciation for the first year? a. $1025.00 b. $8200.00 c. $1035.00 d. $2050.00 32. What is the Book Value at the end of the first year? 8. $6775.00 b. $7175.00 c. $8200.00 d. $7775.00 33. What would be the amount of depreciation if this equipment was purchased on October 4. 3. $2050.00 b $1025.00 c. $854.17 d $256.25 APPENDIX C COURSE OUTLINE-—LECTURE 129 130 Fuuum STATE COLLEGE UK}RAHD& MKJHGAN To; Winter Quarter Business Math Teachers PROM. . I . . Malcolm E. Lund, Head, Office ministration SUBJECT: ' ~ Business Math Classes DATE: November 9, 1983 After our meeting the other day, a check was made to determine material that should be included. The following chapters are the ones in the Rice book which should be included: Section Coverage 3 Bank Records 5,6 Fractions ' 10 Percentages 11 Cash and Trade Discounts 13 Markup 14 Simple Interest 15, Notes and Interest Variables 16 Borrowing by Business 17 Charges for Credit 18 Payroll Records 19 Payroll Deductions 20 Property Tax 25 Depreciation Optional: 7.8 Decimals 12 Commission 32 Compound Interest and Present Value MEL:smd APPENDIX D COURSE 0UTLINE--PROGRAMMED I31 132 B—121: Business Mathematics TEXT: Huffman, Pro ammed Business Mathematics, Book 1 and Book 2, Fourth Edition, Gregg Division, McGraw-Hill Book Co. BOOK I gflll_ §A§§§ TOPICS CHECKPOINTS 1,2 ONIIT 3 23 — 30 Addition and Subtraction of Fractions 31 4 33 - 38 Multiplication and Division of Fractions 39 5 41 - 50 The Use of Decimals 51 6 OMIT 7 63 - 66 Introduction to Percent 67 TEST 1 (UNITS 3, 4, 5, 7) 8 74 — 80 The Use of Business Formulas 81 9 OMIT 10 93 — 98 The Percentage Formula 99 11 101 - 106 Percentage Problems in Business 107 12 114 - 122 Bank Reconciliation 123 13 OMIT 14 131 — 138 Property Taxes 139 15 141 - 150 Computation of Commission 151 16 153 - 158 Money Management 159 TEST 2 (UNITS 8, 10-12, 14-16) \‘IONKBJ-TWNI-J CO 10 11 13 14 15 16 PAGES 2 - 8 11 - 18 21 - 30 33-38 41 — 46 49 - 54 57—62 70 — 76 79 - 86 89 - 94 97 — 104 107 — 114 122 - 132 135 — 144 147 — 158 133 BOOK II TOPICS Computing Interest Using Interest Tables Negotiable Instruments Introduction to Discount Discounting Noninterest—Bearing Notes Discounting Interest—Bearing Notes Consumer Credit TEST 3 (UNITS 1-7) Pricing Pol icy Markon on Selling Price and Markon on Cost Computing Cash Discount Special Problems in Computing Cash Discount Computing Trade Discount TEST 4 (UNITS 8—12) Payroll Procedures Determining Gross Pay Determining Net Pay Depreciation - No Text — See Instructor for Handout TEST 5 (UNITS 13-15 plus Depreciation) TEST 6 - Comprehensive Examination CHECKPOINTS 19 31 39 47 55 63 133 145 159 APPENDIX E BUSINESS MATHEMATICS--METHODOLOGY EVALUATION 134 135 Business Mathematics Methodology Evaluation Different methods can be used to teach Business Mathematics. Some of the methods are (a) the traditional lecture method, (b) the programmed, self-paced method, (c) the independent, non—classroom method, (d) the television method, etc. This questionnaire asks your opinion about the method used in this class. It is BEE an evaluation of your instructor. Your opinions are provided for research on instructional methods in Business Mathematics. Your name is provided solely for research purposes. The data will not be used to affect your instructor's perception of you or to evaluate your instructor. NAME: Soc.Sec.No: Question A asks which method you experienced. Questions 1-7 ask your opinion of the method's meaning to you. If you Strongly Agree with the statement, circle 4. If you merely Agree with the statement, circle 3. If you merely Disagree with the statement, circle 2. If you Strongly Disagree with the statement, circle 1. A. The main teaching method used in this course was the: (1) Lecture, discussion method. (2) Programmed, self—paced method. 1. I am gaining a good understanding of 1 2 3 4 concepts and principles of Business Mathematics. 2. I am developing skills needed by l 2 3 4 professionals in business. 3. I like the method used to present 1 2 3 4 material in this course. 4. I adjusted easily to the method of 1 2 3 4 presenting material in this course. 5. My in—class time is well-spent in l 2 3 4 this course. 6. The method used in-class helps me make 1 2 3 4 good use of my out—of-class time. 7. I would recommend Business Mathematics 1 2 3 4 taught by this method to my friends. BIBLIOGRAPHY 136 BIBLIOGRAPHY Adams. J. L. (1981). A study of achievement and attitude toward mathe- matics 1n remedial algebra (Doctoral dissertation. University of Nebraska-Lincoln. 1980). Dissertation Abstracts International. 41. 2980A. Adams. R. C. (1981). A study of the effects of PSI and lecture teaching methods upon student achievement and attitude change in college mathematics (Doctoral dissertation. Northern Arizona University. 1981). Ineumujaijon Abstraois International. 42. 519A. American College Testing. (1973). Technical LERQLI 91 the 891 555555: msni prggram. Iowa City: American College Testing. American College Testing. (1984). Content of the test In the ACT assessment. Iowa City: American College Testing. Artz. A. F. (1984). The comparative effects of the student-team method of Instruction and the traditional teacher-centered method of instruction upon student achievement. attitude. and social Inter- action in high school mathematics courses (Doctoral dissertation. New York University. 1983). Dissertation Abstracts Interns; Iflonol. 4A. 3619A. ASCD Committee on Research and Theory. (l980). Msasurjng and of . Alexandria. VA: Association for Super- vision and Curriculum Development. Barger. P. A. (1975). The relationship of the general aptitude test battery and other factors to the successful completion of business mathematics in selected vocational-technical schools In Georgia (Doctoral dissertation. Georgia State University. l975). Dissertation Absiraois International. 35. 669A. Bennett. E. C. (l983). The relationship between selected demographic. academic. and aptitude variables and student grade achievement in a first course on computer science (Doctoral dissertat1on. Mississippi State University. 1982). Dissertation International. 43. 2168A. Berston. H. M.. & Fisher. P. (1978). 9911591515 Du5ifl555 EAIDQEAIISS (Rev. ed.). Homewood. IL: Richard D. Irwin. Inc. 137 I38 Bohnhorst. B. (l982. November I). Lecture presented to EAC 8lOC. Curriculum construction course at Michigan State University. East Lansing. Bone. M. A. (l98l). A comparison of three methods of mathematics placement for college freshmen (Doctoral dissertation. Michigan State University: 1981). LDssertation Abstraots International. AZ. 565A. Bouknight. M. L. (1984). The effects of instructional method on types of learning outcomes as evidenced by differential performance (Doctoral dissertation. North Carolina State University at Raleigh. 1983) Dissertation Abstraots International. AA. 3002A. Brinkerhoff. R. 0.. Brethower. D. J.. HluchyJ. T.. & Nowakowski. J. R. exaluationi A i ior trainers (l983). Erogram are eoooators. Boston: Kluwer-Nijhoff Publishing. Brown. V. L. (l984). An analysis of an individual study instructional approach of teaching mathematical concepts to high school voca- tional office education students (Doctoral dissertation. North Texas State. 1983). Dissertation Abstraots International. AA. 2333A. Byrd. D. L. (l980). A study to determine the significance of certain high school and college data in predicting students' success in college algebra. pre-college algebra. and basic general mathemat- ics at Enterprise State Junior College (Doctoral dissertation. Auburn University. 1979) Dissertation Abstracts International. AD. 5346A. Campbell. D. T.. & Stanley. J. C. (l963). Exoerimeotal are quasi- exoerimental designs ior research. Chicago: Rand McNally & Co. Carstens. J. C. (1973). Relationship between business career objec- tives and the supporting course business mathematics (Doctoral dissertation. Colorado State University. 1971). Dissertation Abstraots International. 32. 4499A. Center for Research on Learning and Teaching. (l976. October). Student reactions to instruction. Memo to the anulty Not 55. Ann Arbor: The University of Michigan. Clithero. D. L. (1984). A comparison and evaluation of two Instruc- tional programs for the teaching of probability and statistics to eighth grade students (Doctoral dissertation. University of Missouri-Columbia. l983). Dissertation Abstraots International. AA. 3620A. '00 SUIHSIand JeLpueuo :oosioueJd ues -§:Urodnsrx Kjvjadmetuas swag ‘('p3) Lefieis '1 '5 UI 'BULUJEGL ;o suoiiipuoo pue uoiionuisul '(L96L) 'N '3 teuBeg 'oo x008 lLIH-MPJ93W =x4°x Men '(‘pe puz> UUII =§5fi56 UUP' UT §I§KTFUF '(996l) 'v '9 ‘UOSDBJGJ ’oo BUIHSIand LLIJJBW =Ho 'anwnLoo 'UUII =F5fi55 TPUUTIF36F :6 §U6TIPDUWUJ '(8L6l) 'l '3 ‘JJeH ? "N 'a ‘sueAa 'VLLS ‘ZF ‘T5U5TIPU35IUT . '(L86L ‘K1ISJGAIU0 SLOUILLI UJeqldoN ‘uoiieiaessip [9401300) siouiLLI uaeuiJou u; sefielloo Aiiunwwoo peioeles 19 soiieweuiew LeiuewdoieAep u; eJnLLP} pue sseoons o; uoiieaededd soiieweuiew pup ‘suoiideoaed iuepnis ‘eouepuedepui -eouepuedep PleI} }0 diqsuoilPlaJ aul ’(L86L) 'r ‘d 'pLeASJepLa 'VQSVL ‘37 ‘ =35¢UI §ISFII§GV U6I15135§§Td '(Z96L ‘109L139UU00 ;0 KlISJaAIUn SUI luoiieiaessip [9401300) LaAal eBeLLoo out 19 siuepnis eaquLe eieipewleiui 30 epniiiie pup iueweAeiuoe eui uo Jeiueo Suiaoini SOLlQWGq19W a $0 esn 9H1 $0 8138;;6 eul '(296L) '5 '3 ‘LEU1U9AL90 (976 L60 03 '51uewnooa OIaa) 'eedfiep uoiieonpg Io Joiooq JO} siueweainbeu io iuewLLiiin; Leiiaed u; AlisaeAiun EAON o; peiiimqns wnDLlDEJd ' JUTUNT pUFf§I Wivmmmmmmutm I6 WMGIWEVIGW'WM) '3 'a uexoea 'VVVL ‘TV ‘T3U5TIPU35IUT '(0951 ‘siiesnuoessew io KaisaeAiun ‘uoii —eiaess;p [9401300) iueweAeiuoe satieweuiew JLSH1 J0 iuewneeai [Quotienaisui lo sioeiie eu; pue siuepnis efielloo Kiiunwwoo .0 seLfiis 9AI1IU503 sq: OIUI U0I195I159AUL uv '(086l) 'v 'w ‘ueLLno 'ouI ‘SJeUSLand uoibuiAJI :xJoA MeN '('pe puz) U5T¢5WTT§UT DUE §5FDTT¢UV ’(L86L) '3 '3 'MOUS ? "F 'l 'UDPQUOJO °3uI ‘SJSUSLand U015ULAJI :XJOA MeN '355U15m U5T15WJI§UT 5U? §§UDITIUV '(LL6L) '3 '8 ‘MOUS ? ‘°P 'l ‘UDPQUOJO 'sxoog [[LJJew =HO 'anwnlog '35505351 =IIU IvfipTWTFUI UHF 5UTUJF§T ‘('p3) uefiea 'w '3 UI ESGOUSJGJJLP LPnPIAIPUI 01 paidepe 9Q U0L13nJlSUI U93 MoH '(L96L) 'f 'l 'HOPQUOJO 'VEL6 ‘TV ‘TFUUTIPU35IUT 515FJT§UV'U5TI§135§§TU '(096L ‘K115J9Aiun 91913 BLSJOGQ 'uoiieilessip Leuoioog) esanoo soiieueuiew uoisioep e u; iemJo; 1591 elnioei Leuoiiipeai e UliM uoissnosip dnoafi [Laws Bu; -Jniee; lewlo; eqnioeL eBJEL e lo uosiledmoo V '(086l) 'Jf "1 '3 tedog 6S1 lbO Glass. G. V.. & Stanley. J. C. (1970). jfigujstioal metnoos in eonoa; tion and osyonology. Englewood Cliffs. NJ: Prentice-Hall. Godia. G. I. (1982). A comparative study of the effects of achievement. changes in attitude toward mathematics and attrition rate of students enrolled in a freshman remedial arithmetic course under two different instructional approaches (Doctoral dissertation. Ohio University. 1981). Dissertation Abstraots International: AZ. 3412A. Hall. A. L. (1972). Business mathematics achievement and career objec- tives at the community college level (Doctoral dissertation. Colorado State University. 1972). Dissertation Abstracts Inter: national. 33. 2565A. Hall. G. (1974. January 8). A oomoaratixe stony of soeoiiio skill oiseleetedemnlmreesanooleritaloourseoontentin a eommnnity oistriot. Practicum submitted to Nova Uni- versity in partial fulfillment of the requirements for Doctor of Education. (ERIC document. ED 097 090) Harkins. R. J. (1980). The effect of mode of presentation on attitudes toward and achievement in mathematics (Doctoral dissertation. Indiana University. 1980). Dissertation Abstraots International. Al. 3930A. Harsher. S. L. (1983). An investigation of the affective and cognitive effects of the teacher—directed conventional method. the student- directed individualized method. and the student-directed compe— tency based method of instruction in secondary business mathenat- ics classes (Doctoral dissertation. University of Maryland. 1982). l Ab_5_tI‘_aQ_t5 9 $3.! 2203A. Helmick. F. I. (1983). Evaluation of the placement test for first-year mathematics at the University of Akron (Doctoral dissertation. The University of Akron. 1983). Dissertation Aostraots lnterna: 1.12%]. 43.: 401A. Hickey. P. S. (1980). A long range test of aptitude treatment inter- action hypothesis in college mathematics (Doctoral dissertation. The University of Texas at Austin. 1980). Dissertation International. Al. 145A. Higab. A. K. A. (1983). The effects of aptitudes and structurally different methods of teaching mathematics on achievement and satisfaction (Doctoral dissertation. University of Southern California. 1983). Dissertation Abstracts International. Al. 3787A. lhl Hinton. J. R. (1980). An analysis of selected cognitive style dimen- sions related to mathenatics achievement. aptitude. and attitudes of two-year college students (Doctoral dissertation. Ohio State University: 1980). Dissertation Anstraus International. 4.1. 576A. Huffman. H. (1980). Erogrammeo business mathematies (3 vols.). New York: McGraw-Hill Book Co. Iverson. G.. & Norpath. H. (1976). Analysis oi xarianoe. Series: Quantitative applications in the social sciences. 07-01. Beverly Hills & London: Sage Publications. Johnson. R. S. (1974). The business mathematics competencies needed by career business students in two-year colleges (Doctoral dis- sertation. State University of New York at Albany. 1973). Disser; tation Anstraets International. 3.4.. 4507A Jorgenson. P. R. (l981).A survey of remedial/developmental mathenatics programs at two-year public colleges (Doctoral dissertation. Wayne State University. 1981). Dissertation Anstraots International. AZ. 973A. Kaliski. B. S. (1975). Algebra or arithmetic? lonrnal of Easiness HUM! 5Q: 289-90 . Kaufman. S. (1980). The comparison between the utilization of behavioral objectives in the teaching of mathematics of merchan- dising in the distributive education curriculum and traditional teaching (Doctoral dissertation. Temple University. 1980). Lussertation Ah5traot5 International. AZ. 676A. Khan. A. A. (1983). A comparison of conventional lecture method of instruction with programmed instruction and the lecture-laboratory approach in teaching introductory physical geography at the uni- versity level (Doctoral dissertation. The Catholic University of America. 1983). Dissertation Anstraots International. AA. 454A. Koran. M. (1974L Aptitude. achievement. and the study of aptitude- treatment interaction. In D. R. Green (Ed. ). lhe aptitude: aeniexement oistinotion. Monterey: CTB/McGraw-Hill. Liguori. R. A. (1974). A controlled study of P.S.I. procedures applied to a college business mathematics course (Doctoral dis- sertation. The University of New Mexico. 1973). Anstraots International. 35. 1551A. Mackenzie. D. (1975). Large group lecturing in mathematics. Eonea; tional Stodies in Mathematics. o. 293-309. 1&2 McComb. J. A. (1984). Mathematics placement procedures and psycho- metric decision theory (Doctoral dissertation. Michigan State University. 1983). Dissertation Abstracts International. 4.4.. 2742A. McLeod. D.. & Adams. V. M. (1980). Aptitude-treatment interaction in mathematics instruction using expository and discovery method. Journal .for Research in Mathematics Education. ll. 225-234. Miller. K. N. (1984). The effect of the individualized manpower training systems instruction program in basic math skills on the achievement level and dropout and failure rate of mathematics of business students at Daytona Beach Community College (Doctoral dissertation. Florida Atlantic University. 1983). Dissertation Abstracts International. AA. 3254A. Muzik. S. T. (1984). A single-subject experimental study of time on task for selected classroom aptitude. treatment and achievement interaction variables (Doctoral dissertation. Washington State University: 1983). Dissertation Abstracts International. AA. 3271A. Newman. J. S. (1983). A comparison of traditional classroom. computer. and programmed instruction (Doctoral dissertation. University of South Carolina. 1983). Dissertation Abstracts International. AA. 976A. Oravetz. R. F. (1967). A study of the effectiveness of different experimental drill patterns in business mathematics (Doctoral dissertation. University of Pittsburgh. 1966). Dissertation Abstracts International. 2].. 3373A. Payne. H. E. (1984). The effects of three instructional techniques on the problem solving ability of general education mathematics students at the Junior college level (Doctoral dissertation. University of South Florida. 1983). .Dissertation.Abstracts International. 4.4.. 2670A. Peterson. P. L. (1982). Individual differences. In Enoxolooeoia_of eflooational.researoh (Vol. 2). New York: Macmillan. Reinauer. C. D. (1981). An investigation of selected intellective and nonintellective variables and their use in predicting success in two Instructional formats of a college algebra course (Doctoral dissertation. University of Houston. 1979). Dissertation Abstracts International. 41. 3932A. 1&3 Rhodes. T. M. (1984). A study to assess and compare the effects and attitude of two remediation efforts in mathematics by the Ohio State University (Doctoral dissertation. The Ohio State Univer- SltY. l983). Dissertation Abstracts International. AA. 3312A. Rice. L. A.. Mayne. F. B.. Deity. J. E.. & Southam. J. L. (1983). Business mathematicsior colleges (8th ed ) Cincinnati: Southwestern Publishing Co. Robertson. D. F. (1980). An investigation of aptitude treatment inter- actions (ATI) with respect to programmed and lecture treatments in a college course in basic mathematics (Doctoral dissertation. University of Minnesota. 1979. Dissertation International. All. 4940A. Sandman. R. s. (1973). lbs dearelobmenti ialidationi' and abolication oi a multidimensional mathematics attitude instrument. Unpublished doctoral dissertation. University of Minnesota. Sandman. R. S. (1980). The mathematics attitude inventory: Instrument and user's manual. Journal .for Beseamh in Mathematics Educa: m: .11: 148-149. Schalock. H. D. (1976). Structuring process to improve student out- comes. In 0. T. Lenning (Ed.). Imoroxing educational outcomes. San Francisco: Jossey-Bass. Schielack. V. P. (1983). A personalized system of instruction versus a conventional method in a mathenatics course for elementary educa- tion majors (Doctoral dissertation. The University of Texas at Austin. 1982). Dissertation Abstracts International. 43. 2267A. Schwartz. H. I. (1980). A mastery learning study in remedial mathemat- ics in an urban community college (Doctoral dissertation. Columbia University. 1980). Dissertation Abstracts International. Al. 577A. Scrittorale. L. (1973). The business mathematics needs of community/ junior college business majors (Doctoral dissertation. Colorado State University: 1972). Dissertation Abstracts International. 33. 6656A. Shine. S. S. (1983). A comparison of programmed instruction versus lecture-demonstration as a method for teaching digital computer arithmetic at the post—secondary school level (Doctoral dissertation. Texas A a M University. 1982). Abstracts International. A3. 3529A. lhh Sims. G. L. (1980). Prediction of success in college algebra at Richland College in Dallas. Texas (Doctoral dissertation. The Florida State University. 1979). Dissertation Abstracts International. an. 5351A. Smith. J. L. (1982). A study of the relationships of sequences of enrollment in college remedial mathematics to grades in a subse- quent college algebra course and to persistence in college (Doctoral dissertation. The University of Alabama. 1982). Dissertation Abstracts International. A3. 1457A. Smith. T. H. (1983). The effects of a self-paced college algebra program on mathematics achievement and mathematics anxiety (Doctoral dissertation. Temple University. 1983). ‘Dissertation Abstracts International. AA. 96A. Snow. R. E. (1970). Research on media and aptitudes. .Dommentaries onresearchininstructionalmeoiainnexaminationofconcebtual schemes xienooints. In G. Salomon & R. E. Snow (Eds.). Bulletin of the Indiana University School of Education. 4o. 63-91. stnoents. Unpublished paper presented in Education 822F at Michigan State University. East Lansing. Swartz. F. (l982b. August). Deuelooment of. a business mathematics .rinal examination. Unpublished paper presented in Education 984. Field Experience in Vocational Education. Michigan State University. East Lansing. Swartz. F. (1983). Business mathematics in Michigan community and two-year colleges. .MBEA.Ioday. AD. 4-5. Swartz. E.. & Swartz. R. A. (1981). College business mathematics: Pretesting for course sectioning. .Business Education.Eorm. 8-9. Tabachnick. B. G.. & Fidell. L. S. (l983)..Dsino.multiuariate statistics. New York: Harper & Row Publishers. Tobias. 8. (1981). Adaptations to individual differences. In F. H. Farley & N. J. Gordon (Eds.)..Esxcnoloo¥.and.eoucationI.Ine state of the onion. Berkeley: McCutchan. helolotachieiingstuoentsmaintaintheiroradesiniutnre ‘matnematies.oonrses1 Unpublished paper submitted in EAC 883. Readings and Independent Study in Administration and Curriculum. Michigan State University. East Lansing. Truckson. E. B. (1983). The effects of heuristic teaching and instruction in problem-solving on the problem-solving perform- ance. mathematics achievement. and attitudes of junior college arithmetic students (Doctoral dissertation. University of Cincinnati. 1982). Dissertation Abstracts International. A3. 3532A. Urbatsch. T. D. (1980). An interaction study: Locus of control. mathe- matical attitude. mathematical anxiety. and mathematical achievement with three modes of instruction--personalIzed system of instruction. traditional lecture and five day lecture (Doctoral dissertation. The University of Iowa. 1979). .Dissertation Abstracts International. AD. 6180A. Walker. 8. W. (1981). An experiment to determine the effects of mathe- matical achievement and attitude toward mathematics of prospec- tive female elementary school teachers by the use of supplementary programmed instruction (Doctoral dissertation. Memphis State University. 1981). .Dissertation.Abstracts International. A2) 2478A. Watson. J. M. (1983). Individualized mathematics instruction in an Australian university (Doctoral dissertation. Kansas State Uni- versity: 1982). Dissertation Abstracts International. .43. 2268A. Weiner. N. C. (1984). Cognitive aptitudes. personality variables and gender difference effects on mathematical achievement for mathe— matically gifted students (Doctoral dissertation. Arizona State University. 1983). Dissertation Abstracts International. AA. 3621A. Weissglass. J. (1976). Small groups: An alternative to the lecture method. lumear college Mathematics Journal. 1. 15-20. Wells. P. L. (1982). A comparison of the achievement of university students taught by an individualized Instructional approach versus the traditional instructional approach in a combined business math/business machines course (Doctoral dissertation. University of Houston. 1981). .Dissertation.Abstraots.Interna; 3:19.011: AZ. 4697A. 1A6 Whitesitt. J. D. (1980). Entrance level competencies needed for begin- ning mathematics courses at Montana State University (Doctoral dissertation. Montana State University. 1980). .Dissertation Abstracts International. Al. 1455A. Williams. J. W. (1976). Mastery learning in business mathematics (Doctoral dissertation. The University of Nebraska—Lincoln. 1975). Dissertation Abstracts International. 3.6.. 4978A. Wilson. B. U. (1982). The effects of pacing and cognitive style upon student achievement and attitude in basic college mathematics (Doctoral dissertation. The College of William and Mary. 1984). Dissertation Abstracts International. A2. 4331A. Witkowski. J. C. (1982). Cognitive-oriented supplementary material and students' cognitive process and performance in college remedial algebra (Doctoral dissertation. Illinois State Uni- versity. 1982). Dissertation Abstracts International. A3. 1458A. Woods. J. D. (1983). Lecturing: Linking purpose and organization. Improrino college and Dnirersiin leaching. 3.1. 61-64. IIIIIII TATE UNIVERS III/1111111III/11111111III/IIIII/I/IIIIIIiIII/Il 3 1293 10719 7869