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SPATIAL ABILITY AND COMPUTERENHANCED PRESENTATION AN APTITUDETREATMENT FIELD MODE OF LINEAR INTERACTION IN A INEQUALITIES: STUDY OMER KISER A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE 0O DOCTOR OF PHILOSOPHY Copyright Omer Kiser 1986 ACKNOWLEDGMENTS Many during sion people my years of deep contributed of graduate appreciation advice, study extended services, research. to Dr. support expres Elroy Bolduc, friend major professor, patient guidance, encouragement, wisdom throughout years. Mary Grace Kantowski , my friend, facilitator, member doctoral committee, gave unselfi shly of her time Dr. to offer Robert advice Soar, and e member encouragement doctoral dissertation. committee, gave unselfishly time to offer constructive criti cism advice on the stati stical analyses involved research. Donald . Bernard, member doctoral committee, shared eas offered encouragement complete dissertation. would also like thank Mark Hale member doctoral committee, service interest. Other individuals contributed a support capacity to complete this re search. would like thank William . Grams, Chairman of the Department Mathematics degree. Edith Carolyn Robinson, Mary recognition serving Ruth Meeker as a "panel Ekstrom, deserve special of experts" in rating a series dimensional test items, spatial used ability my dissertation, requirements. I would two like thank Robert King editing manuscript Debbie Breedlove patience in typing final manuscript. also want to thank students, teachers, univer sity personnel made pilot studies main study possible. a more deepest Doris personal gratitude Trigg, evel wife their want , Marcia, constant express my parents, encouragement love throughout educational endeavors over years. TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES S . . . . lx ABSTRACT CHAPTER PURPOSE, THEORY, AND RESEARCH QUESTIONS Introduction Purpose of Procedures Research 0 a . . 1 Study stions . a . 4 Rese arch Hypothe ses Signifi cancer Definition Assumptions Limitations Summary of the Terms Overview Study S. . . 12 REVIEW OF RELATED LI TERATURE . . . 14 Compute Spatial Visual Cogniti Aptitud Summary rAssis Visualizat Learning . ve Style . eTreatment Instruction ion Spatial sual Ability Interactions PROCEDURES Pilot Study . . . . a 44 Introduction Study L aJ a a a a a am 2 Page i.ar CHAPTER PROCEDURES Main Study (Cont.) S . . . Overview Study S . . 66 Instrumentation . . . . 69 Henderson Treatments Taxonomy Teaching Model . . . . 73 ComputerEnhanced oftw are . . 76 Statisti Mode Formal Hypothe ses Further Formal analysis hypothes RESULTS CONCLUSIONS, Conclusions DISCUSSION, AND From RECOMMENDATIONS othe ses scusslon Recommendations Further Research APPENDIX MATE RIALS MATERIALS FOR FOR PILOT PILOT STUDY STUDY MATERIALS USED IN ALL PILOT STUDIES AND MAIN STUDY SCATTERPLOTS REGRESSION ANALYSIS ASSUMPTIONS VALIDITY OF THE EXPERIMENT REFERENCES BIOGRAPHICAL SKETCH Page LIST OF TABLES Table Descriptive Descriptive stati stati stics stics on mean on mean pretest posttest scores scores Dichotomization of spatial ability scores Twoway ANCOVA results on mean posttest scores Partition analy of total sum squares S1S Descriptive outcomes statistics ANCOVA on aptitude Summary Multiple showing of variance aptitudes, regression partial in performance treatment equations regre ssion accounted effects, achievement coefficients each treatment Comparison equations reasoning of be tween group sim spatial ability pretest ability ple regression where are stract constant Summary of variance aptitudes in performance , treatment accounted effects, S. . . .a 100 Summary of variance aptitudes in performance treatment disregarding accounted effects, . . . . 101 Reduced multiple achievement efficient regression showing or each partial treatm equations 1 regress ent CO * *. 102 . 93 Page I k Table Page Comparison equations abstract constant of between field reasoning group mode and pr simple ability etest regre where ability ss5on are Intercorrelations of aptitudes outcomes treatment LIST OF FIGURES Figure Graphs of types interactions Graph of levels sordinal of spatial fraction across fixed ability Graph of levels disordinal eraction across fixed treatment Graph Graph of the of simple CEI group and ability where statistical regression traditional stract analysis equations group reasoning on spatial ability a pretest ability are constant Graph of ence YB spatial Graph CEI mode confidence  YA in ability of simple group and ability ability band predicted regression traditional where pretest abstract ability about achievement equations group on differ for field reasoning are constant Outline Henderson moves mode in teaching a concept Outline Henderson moves teaching model general zation S Examples of scatterplots assumptions regression Regression (AGroup) achievement on spatial ability Page Figure Regression (AGroup) Regression (BGroup) of achievement on field of achievement mode on field mode ability ability Regressio ability achievement on abstract areas oning Group) Regre ssion ability of achi BGroup evement on abstract areas oning Regre ssion of achievement on pretest (AGroup) ress of achievement on pretest (BGroup D.10 Regre ssion ability of abstract reasoning on spatial Group) D.11 Regre ssion abstract reasoning on spatial ability Group) D.12 Regre ssion of abstract reasoning on field mode ability C AGroup) D.13 D.14 Regres mode Regre sion of abstract ability ( ssion BGroup test reasoning ) . on spatial on field ability (AGroup D.15 D.16 Regression (BGroup Regression (AGroup) of pret of prete on spatial on field mode ability ability D.17 Regression (BGroup pret on fi mode ability D.18 Regre ssion of pret est on ab stract reasoning ability (AGroup) D.19 Regression ability of pretest (BGroup) . on abstract reasoning Page Abstract of Dissertation Submitted to the Graduate School of the University of Florida Partial Fulfillment Degree of Doctor Requirements of Philosophy SPATIAL ABILITY FIELD MODE IN A COMPUTERENHANCED PRESENTATION AN APTITUDETREATMENT OF LINEAR INTERACTION INEQUALITIES STUDY Omer December, ser 1986 Chairman Major De Dr. Elroy apartment . Bolduc, Instruction Ed. D. Curriculum This aptitudetreatment interaction ATI) study was designed to investigate effect instructional treatments on the achievement of students having different eve Is of spatialvisual field mode abilities. instructional tional treatments presentations were computerenhanced of the procedures solving tradi linear or absolutevalue inequalities college algebra. Subjects study classes were of coll students algebra enrolled at EmbryRiddle intact Univers Daytona Beach , Florida, fall trimester, 1986. Group class received traditional treatment .  ^jjfc . were used measure students ' two dimensional spatial visual ability The Group Embedded Figures Test Abstract Reasoning Subtest Differential Aptitude Test were used measure students ' field mode and general reasoning abiliti , respective A pretest was admini stered a measure requl site skills prior twoweek study. first cl ass period, ability tests pretest were administered to both groups. Subj ects A Group received a sound traditional sentation topi Subjects B Group rece ived a highly visual computerenhanced presentation topi Two interactions were studied interaction between treatments spatial ability when achievement test score was dependent variable inter action between treatments and eld mode ability when achi evement test score was dependent variable A multiple interactions. G regression general analysis reasoning was ability used and to study pretest s these cores were entered as covariates analyst i Spatial ability field mode were each entered dependent variables in separate regression procedures , while achievement test score was taken endent variable. A significant nr rnii rro A cntin t1a1 nhi 1 i t v treatments Cs. rP o+"t*7o n I nl i L j i i * U  r, was found between field mode treatments. B Group, however, significantly higher posttest achievement scores than A Group across entire range of field mode scores. CHAPTER PURPOSE, THEORY, RESEARCH QUESTIONS Introduction Computerass is ted instruction (CAI) relates aptitudetreatment interaction (ATI) research made contributions toward improved mathematics instruction. In recent years, a variation computeras sted instruc tion emerged in which graphics capability microcomputer to enhance s utilized individualiz in ongoing ed instruction ssroom which interactions learner receives. instruction s variation (CEI), is of CAI, of educational termed computerenhanced interest to ATI researchers have visual an interest difference variables learning because as cognitive theorists. it focuses style res on such field earchers individual mode), spatial ability, learning attitudes, rate learning of vi sual materials. Visual learning theorists have an interest as motion such simulation, computer sound graphics color cueing, capabilities dynamic shading graphing, sequencing pacing visual materials facilitatina student achievement. researchers contend that individual difference variables are primary source of variation evaluation of innovative traditional methods instruction. Whenever instruction is presented groups of students, individual diff erences are bound to affect outcomes entially thrust, relates to visualized on identifying instruction individual visual difference learning, is variables focused which interact with different types of instructional presentation formats, different types of educational ectives different amounts of reali stic detail contained visualization used to illustrate instructional content, different media production variables , and diff erent techniques of organic zing managing media. integration with visual learning strategies S eems to hold great potential investigation in mathematics education. Purpose of Study purpose of this study compare proficiency of subje after receiving a comput ere enhanced treatment a topic received college traditional algebra to treatment subjects wh of the same have topic. Procedures subjects investigation were students from intact classes of coll algebra from EmbryRiddle Univer sity Daytona Beach, Florida. subjects in the traditional procedures treatment solving experienced linear a presentation or absolutevalue of the inequalities recommended the School Mathemati Study Group as a sound presentation. subj ects in the experimental treatment experienced a highly visual computerenhanced presentation procedures inequalities solving on visual linear learning or absolute theories value Dwyer his colleagues Dwyer, 1972 Kress Gropper, 1964a, 1964b; Salomon, 1979). subjects both treatments the same same educational criterion measure objectives e. Salomon experienced s (1972) prefer ential model was used to generate hypotheses .  I treatments constructs was of general mode of presentation. reasoning, spatial aptitude ability, cognitive style were investigated because their relevance this study to previous studi Research Questions primary research questions to be answered this study are following. Is a computerenhanced treatment traditional topic coll algebra more effective than traditional treatment? Will difference mean posttest achievement scores treatments traditional topic college algebra change across varying level spatial ability? Will difference mean posttest achievement scores treatments traditional topic in college algebra change across varying levels student field mode? Research Hypotheses es. There spatial student is no significant ability posttest solving There linear between levels performance or absolutevalue no significant levels treatment procedures inequalities. levels between field mode posttest levels performance treatment of the student procedures solving linear or absolutevalue inequalities. There performance traditional no significant of the co treatment difference mputerenhanced groups between the linear solving or absolutevalue inequalities. Significance of the Study A recent report, An Agenda Action (National Council of Teachers introducing imagery of Mathemati , visualization , 1980), . and sp rec 'atia ommends 1 concepts into traditional mathematics classrooms 1980 It further recommends that computers be used in school mathematics programs nontraditional problemsolving strategies simulations. encourages computer use teaching traditional mathematics topics "diverse * __ _ . a . a S  L * i To demonstrate that microcomputer effective instrument to assist enhance instruction. To contribute rese arch involving ATI between levels spatial ability in subjects instructional treatments in mathematics achievement (Cronbach Snow, 1977). To contribute to research involving between levels field mode (cognitive style) in subjects instructional treatments in mathemati achievement (Witkin, Moore, Goodenough, Cox, 1977). To contribute to visual learning research related to line drawings, motion simulation, sound cueing, dynamic shading, black white vs. color v (Dwyer, isuals, 1972a; sequencing Gropper and pacing c Kress, 1964a, >f visuals 1964b). To demonstrate an innovative, effective, timesaving topic strategy in mathematics teaching utilizing traditional microcomputer as recommended NCTM CUPM. laPf n4 +4inn nnn r ^: F P rm o Spatialvisual ability. Spatialvisual ability ability a sequence or 1 ess to mentally of movements. complex manipulate visual The objects twodimensional objects appear stimulus within pattern involving a more (Michael, Guilford, Fruchter, Zimmerman, 1957). Spatialvisual ability was measured as a compos score Form Board Tes t (Vzl) Punched Holes Test from Educational Testing Service (Ekstrom, French, Harman, 1963; French, Ekstrom, Price, 1963). Mathematical procedure. A mathematical procedure task are involving types study: mathematical of mathematical spatialvisual tasks stimulus procedures algebraic patterns discussed tasks There in this (Cooney, Davis, Henderson, 1975; Davis, 1978). Spatialvisual task. A spatialvisual task mathematical procedure involving a simple or complex visual stimulus pattern which requires some measurable degree of spatialvisual ability. this study, visual stimulus patterns included line drawings overlapping shaded plane regions one or more linear or absolute value inequalities. Algebraic task. An algebraic task is a mathematical aa 1  * a a I Y1~1~Y~ I~: Ylll It Cognitive style. Cognitive style refers characteristic selfconsistent modes functioning which individuals show their perceptual intellectual activities. Theoretically, cognitive style determines most effective mode of instruction evaluation given learner (Witkin, 1950). Field mode. Field mode one manife station individual colleagues s broader (Witkin, cognitive style. , Faterson, Witkin Goodenough, Karp 1962) define field mode as a relative continuum of consis tent cognitive functioning defined perceptually ability perce an object as di stinct from surroundings. Field mode was measured as a compos score o Figures n Part Test (GEFT) Part from III Cons Group ulting Embedded Psychologists Press (Oltman, Raskin, Witkin, 1971). Field mode is defined terms degree dependence on the structure of the prevailing visual field, ranging from great dependence one extreme termed field dependenceto greater ability to deal with presented figure field from analytically configuration or to separate in which an embedded occurs other extremetermed field independence (Witkin, Good annuah  Ka rn . l Q;7I Computerassisted instruction (CAI). Computer assisted instruction computer is a method is a primary deliver instruction system. which It implies instructional application computer tradi tional teaching tutorials, demon methods, station, such as drilland simulation, and practice, instructional games (Burke, 1982; Coburn, Kelman, Robert, Snyder, Watt, Weiner, 1982). Computerenhanced instruction (CEI). Computer enhanced instruction that facet of computerassisted instruction in which unique capabilities microcomputer color, sound motion cueing, graphing, shading are utilized teacher student ongoing classroom interactions an instructional to enhance instruction which learner receives. Assumptions Lindquist (1951) indicated that a significant interaction between or more treatments experi ment may only partially explained differences instructional ectiveness treatments. There are three Doss ible causes   treatmentbylevel on a third variable; second, a significant interaction can occur chance third , a signifi cant interaction can occur res influencing one level some uncontrollable an experimental variable treatment others. It is assumed present study that of significance that occurs can be replicated true interaction. Several experimental computer programs design basis of Dwyer s visual learning theory (197 the were Henderson utilized Taxonomy Teaching compute Model renhanced Henderson, treatment 1976 teach proce dures solving linear or absolutevalue inequality es. It is assumed present study that using Apple IIe experimental software in conjunction with Henderson Teaching Model is a sound representation CEI. Group Embedded Figures Test GEFT) s intended a group form Embedd ed Figures Test (EFT most direct criterion measure field mode (Witkin, Oltman, Rask in, Karp, 1971). correlations a number studies between GEFT are reasonably high. Numerous embedded studies context demonstrate in the that can ability be taken overcome as an indicator of relatively differentiated functioning perception Witkin, et al., 1971; 1962). Witkin, It i Goodenough, s assumed Oltman, present 1977b; study Witkin that GEFT, as a research instrument , has high construct validity measurement of field ind epend ence /dependence. Thurstone s Punched Holes Test Form Board Test (Vz1) were adapted Guilford colleagues as criterion measures Cognition of Figural Transforma tions (CFT). definition spatialvisual ability used Punched this Holes study Test assumed, Form Board therefore Test, that as research instruments, have high construct validity measure ment of spatialvi sual ability. is ass umed Differential that the Ab Aptitude Test stract (DAT Reasoning Form Subtest high construct validity measurement of general reasoning ability (Bennett, Seashore, We sman, 1973). Subj ects this investigation were students enrolled sity sections Daytona of coll Beach, algebra Florida. Due at EmbryRiddle to student Univer placement in multiple sections throughout school day, assumed that di f ference s in general reasoning ability, reading were ability, uniformly prior distributed knowl across edge both course treatment content sections. Nonsignifi cant differences between groups on mean Limitations limitation present study that intact classes were used. Because random selection was possible, generalization of the findings to other groups must be done with caution. second limitation that majority subjects in both treatment groups were It is assumed that males Western cultures tend to have higher spatial ability than females that they tend to be field independent (Fennema Sherman, 1977). Male dominance the study therefore will affect general inability of the findings to largely male groups. third limitation of the present study that it is only twoweek treatment. A longer treatment could have yielded more conservative treatment effects could have reduced possibility novelty or Hawthorne effects in the experimental treatment. Summary Overview Chapter provides a rationale this investi gation including a statement purpose, a description research procedures, significance of the study. It then examined th rese arch ques tions to be addressed es. of key terms, stated assumption s limitations study. Chapter is a survey of research literature parallel five components of the present study: computer assi sted instruc tion; spatial visualiza tion spatialvisual ability; visual learning theory; cognitive style; research relates to spatial ability, cognitive style, media attributes mathematics achievement. Chapter describes test pilot teaching studies used methodology to develop computer field software used computerenhance d in structional (CEI) treatment. It also describes techniques, procedures , and criterion measures (ATI) of the used inves main data, study. including tigate aptitudetreatment Chapter scriptive IV presents statistics, interaction an analysis inter correlations , and multiple linear regression analysis Kerlinger findings Pedhazur, discus 1973) . ses Chapter results summarizes implications these results; also makes recommendations future research. CHAPTER REVIEW OF RELATED LITERATURE inconsistent of the relation to individualize styles , specific results of mathematics d instruction, abilities, or of studies learning cognitive social environmental present matical There very conditions a general learning can only specific suggest instructional theory i theories conditions S imposes that r mathe ible. s that apply t of mathematics instruction: descriptions 1979, pp. 36 res ults are or local theories local (Radatz, 362 In this section, research literature presented parallel five components of the present study: computerassi sted instruction; spatial visualiza tion spatialvi sual ability; visual learning theory; cognitive style research relates to spatial ability, cognitive style, media attributes in mathematics achievement. ComputerAss is ted Instruction of the earliest most prominent applications often study refers to that method instruction which uses computer as a primary delivery system. It implies instructional application the computer traditional teaching methods such as drillandpractice tutorial demonstration, simulation, gaming (Burke, 1982; Coburn et al., 1982). There are numerous citations in the literature related to CAI or computerbased instruction (CBI equivalent. present study deals with that facet of CAI termed comput erenhance d instruction (CEI) , in which unique capability microcomputer computation, motion simulation, color, sound, graphing dynamic shading are utili zed by teacher student in ongoing ass room interactions to enhance instruction which learner receives. This enhancement replace traditional ssroom enrich instruction classroom in some cases instruction supplement in other cases. There are a limited number of studi related pedagogical effectiveness of CAI an instructional medium. In a study of factors that inhibited more wide spread utilization of computer technology instruc tional purposes Anastasi o Morgan 1972) found most critical factor to be lack of evidence of CAI A C E AL rrA A S A 5 UA I1A 15 U ,CC, uL L ***A Vt aa nk effectiveness conflicting of CAI inconclusive traditional results. instruction Wing report (1967), example , found that computerbased simulations in economics resulted in mixed findings. Katz (1971) , other hand, found that problemsolving high school algebra to negative results. Other studies, however, generally conclude that an instructional program supplemented with at least as effective frequently more effec tive than, a program utilizing only traditional instructional methods Abramson Weiner, 1971 Magidson, 1978 Scrivens, 1970 uppes Morning star, 1972) Research into area of CAI effectivene typic ally inve stigated one or more four criterion variables student achievement, student attitude toward CAI unit toward comply etion subj ect matter, and/or mastery time savings learning, relative learning retention. Weiner Related (1971) first Suppes crite rion, Morningstar Abramson (1972) found positive grades achievement as compared gains drill traditional andpractice methods. Morgan Richardson (1972) found itive achievement gains tutorials high school algebra as compared traditional methods. Kulik, Kulik, Cohen 1980) found C to have (1983) found positive effects on student attitudes toward learning. programmed Battista instruction S tee le regarding (1984) student compared feelings cognitive development found more positive gains CAI. Kulik and BangertDrowns (1984) found that, whereas programmed and individualized instruction limited success in rai sing achievement improving attitudes, raised achievement, positively affected time needed teaching learning, enhanced attitudes toward computers. Kockler (1973) found that even though tutorials college level do not always result greater achievement than traditional methods, time it takes to learn substantially reduced. Krupp 1972) showed that took adults 510 who hours learned computer to reach same programming level through proficiency students took 30 hours traditional instruc tion. Lunetta Blick (1973) compared computer simulations of high school physics experiments with traditional laboratory experiments found that students learned as much in oneeighth time. Even though students learn more or may learn more quickly through CAI, there some evidence that they retain as much traditionally taught students. Winc at both elementary secondary levels. Proctor (1968) found that tutorials college to equal retention. Carrier (1985) found that drillandpractice in fourth grade class ses to greater gains on some tests no higher retention. Because an abundance of CAI effectiveness studies exists, because individual studies have failed produce conclusive evidence of effectiveness, various researchers literature have attempted order to narratively to formulate conc review lusions research and/or establi a more broadlybased case CAI. These endeavors have also resulted in conflicting inconclusive findings. Vinsonhaler Bass (197 , for example, in their review major computerbased drillandpractice studies argue that elementary mathematics performance gains months uncommon. hand, over children Jamison, a review in traditional Suppes of research Wells on eff classrooms (1974), ectiveness was other of alterna tive media in class rooms including instructional televi sion, programmed instruction, CAI, argue that no significant differences achievement occurred. Studies involving did, however, report savings of student time in learning a significant effect. Edwards, Norton, Taylor, Weiss, Dusseldorp 1975) concluded in their review that normal that , __ normal instruction alone. Computerbased teaching with tutorials or simulation significantly reduced time required for students to 1 earn. Thomas (1979), comprehens ive review of secondary level research, concluded that achievement gains over other, more tradi tional methods are norm. Retention equal to that obtained favorable traditional attitudes instruction, toward fosters computers more subject being taught. students gain mastery status in less time. It should be noted that rese arch prior to 1970 involved personal enhanced a mainframe computer, intervention computer. rese in the With arch advent involved ongoing classroom of the computer instruction that occurs. present study more line with this latter tradition in CBI research. Contradi ctory outcomes of these reviews signaled need synth esis of CAI effe ctiveness literature other than narrative means. subsequent reviews were aimed at a quantitative integration of outcomes individual research efforts utilizing research integra tion methodology 1976). Hartley in elementary of Glass (1977) fo second known caused as metaanalysis on mathematics schools (Glass education reported that ra w , I . .... percentile. Burns metaanalytic Bozeman techniques, (1981), also a continuation integrated findings mathemati education elementary secondary schools. They found that computerbased tutorials raised achievement test results extremely ability groups standard deviations. Comput erbased drill and practice raised gifted ability groups standard deviations. Kulik et al. (1983) used metaanalysis integrate findings from 51 independent evaluations computerbased teaching grades 612. Computerbased teaching It also raised final positive exam effect scores on later .32 standard retention s deviations. cores on attitudes toward computer course study. This metaanalysis study also demonstrated that reduced amount of time needed to learn percent. A few advantage studies of CBI a have ,t higher focused levels educational of education. Jamison et al. (1974) concluded that computerbased teaching college level was "about as effective" traditional teaching when used as a replacement rather than as a supplement. Kulik et al. 1980) used Glass's metaanalytic techniques to integrate findings from independent evaluations of comDuterbased collecre teachina. meta m students toward instruction toward subject matter they amount were time studying needed also substantially instruction. On the reduced average conventional approach required hours instruction time week, while computerbased approach required hours week. Empirical establish research statistically findings that generally use have of graphics failed instruc tion improves learning (Merrill Bunderson, 1981; Moore Nawrocki, 1978). Some studies do show, however, that in specific instances graphics have a positive effect (Moore Nawrocki, 1978). Since 1981 studies have been conducted numerous areas mathemati investigating ways the microcomputer classroom graphics to enhance can ongoing be used interactively mathematics instruction. Frand sen (1981) described enormous potential personal tion computer through to enhance graphical secondary solutions trigonometry triangles instruc graphical displays trigonometric functions. Edwards 1982) described development an interactive problemsolving program linear algebra that uses rotations, dilations, refl sections of lines dimensions. Ignatz Ignatz (198 described an interactive video svstistn 1h ' 1 m11 , aI a 7 wnd 'i nn ml nr ncsart Ttt <3T~ a I 4 III I Ci ^4 i * Rudnytsky (1982) demonstrated that computer graphics could be used interactive presentations teach concepts mathematics more easily manipulating relationships among visual objects. The author argued that such presen stations allow students to operate real mathematicians following upon intuition with testing proof, rather than rote memorization. Rieber (1983) investigated effectiveness systematic thought LOGO s turtle teaching graphics simple g both eometry providing concepts to second grade children. The group did significantly better than traditional group thinking skills geometry. Nygard and Ranganathan (1983) examined computer graphics also display discussed system capabilities applications learning concepts, presentations; an interactive principles, they graphics rules, problemsolving techniques mathematics classrooms. advantages of generating graphics displays ongoing classroom instruction were discussed; advantages included greater spee d in learning and as well as greater flexibility translation, rotation, and scale drawing of visual concepts. Thomas (1984) presented an interactive BASIC program that can used effectively a CEI presentation to graphically demonstrate Central Limit Theorem. 'V~ntl.a..e 4V n (, QOA; ,. AT r * n4ar ,n4 no nr 3 nhl 4 no c nl ooA\ F=TAT T^ planes "explosion" of ellipses into parabolas then into hyperbolas Henderson video (1983, instructional 1985) modules developed serve interactive, an instruc computer tional supplement learning mathematics. Results of field trials showed that modules were effective teaching and/or and had reteaching beneficial mathematical effects on concepts affective secondary as well school as cognitive outcomes. In 1981 Burns and Bozeman stated, no ultimate effectiveness presented, th studies po of learning either CAI, a e in final answer or guarantor analysis t relat success synthe to a significant in instructional supplemented t least mathematics. one CAI, to CAI be many enhancement environments or replaced b curricular area authors conclude that additional res each essential with respect to diff erences in student aptitudes, attitudes toward learning, masterly learning time, retention. present a traditional study topic examined college whether a CEI algebra presentation more effective than a traditional addressed student presentation attitudes of the toward topic. learning It also a CEI While Spatial Visualization SpatialVisual Ability There are many different rese arch emphases which have contributed t psychological o understanding construct spatial nterface visual between ability the and mathematics education. research literature contains many potentially fruitful approaches which mathematics educators can use either classroom or for guidance further research efforts Different approaches spatialvisual ability relationship to mathematical ability to cognitive development have been taken research psychologists mathematics education research ers. Alan shop (1973 a mathematics education researcher, warns that . the lead our them goals research a direction concerns, caution ideas keen approaches psychologist which therefore judgment i which away we must n select will ts m from exercise ng those enable develop our own field . 257) There an extensive research tradition concerning spatialvisual ability as it relates to mathematics learning. Only studies pertinent present study will summarized section. Since Galton began stematic investigation imagery 1898, spatial ability become an aptitude Testing Service attempted to collate interpretations factors visualization psychometric that research been domain proposed space Thurstone (1950), workers psyc hological rese arch units Army Air Force (Guilford, 1947; Guilford, Fruchter, Z immerman 1957) force , 1952; French delineated Micha in his three Guilford monograph dimensions, , Fruchter, (1954). or subabilities Z immerman, s task , of spatialvisualization aptitude: spatial relations orientation kinesthetic SR0) , (2) imagery sualization Vzfactor pertinent to the operational definitions present study to the testing instruments used will be examined further. Visuali zation (Vz) as defined task force requires ability to mentally manipulate visual objects involving a specified sequence of movements. objects appear within a more or less complex stimulus pattern. individual finds necessary to mentally rotate, turn, twist, or invert one or more obje or parts configuration relatively constituting explicit a test directions item, to what according nature order French of manipulation (1954 should recommended that (Michael et al., Vzfactor 1957) . be measured performance on the Form Board Test (Vzl) Punched  school of differential psychologists headed Anastasi developed a research interest in documenting, describing, explaining individual different ces students' abilitie 1976), a valuable The research contribution Krutetskii to individual (1969, difference research more , measured or less visual extent ideas to which solving an individual mathematics uses problems and/or performing mathematical procedures. He developed a set degree tasks which spatial include thinking problems which involving make a high valuable connections between spatial abilities mathematical abilities. He documented seven tases pupils good at mathemati who use predominantly spatial ideas problemsolving. Batti (1980, 1981) found that both spatial visual zation cognitive development correlated significantly with geometry achievement. Visual Learning Using visualized transparencies , films, materials televi such sion, as drawings, CAI, slides, programmed instruction to complement classroom instruction become a common Ins tructional strategy at all levels of education. lhwvnrr 1 Q79an tthii 0tm QG ,I nrc con 4 Shth a v i"n i Q I  f Schramm (1962) stated that extensive research needs be conducted on physical character stics of visualized instruction which lead to increased learning attainment of specific indicates educational that objectives. effectiveness Visualization of visualized research instruction in facilitating student achievement is primarily dependent type of visualization used, method by which visualized instruction is presented student, type level educational objectives that are to be achieved, student characteristics, techniques used to focus student attention on essential learning cues visualized materials Kress (Dwyer, Gropper 1972a; , 1964a, Groppe 1964b; Kres Miller, s, 1965; 1969). review of research the field of vis education produced effective numerous ess studies of visual that investigated illustrations relative possessing different amounts of real stic detail being used to complement oral or verbal instruction. A number theoretical orientations include were identified Iconicity Theory in the literature. identified Morris These (1946) Carpenter (1953), Dale (1954) Cone of Experience Theory. convenience, these orientations are referred as realism theories. Realism theory assumes that learning will more situation complete increase. number assumes cues further that in the learning an increase in realism in the existing cues in a learning situation increases probability that learning will be facilitated (Finn, 1953; son, 1954; Knowlton, 1954; Osgood, 1953). According of realis to realism detail theory, which visual they differ sent, amount on a continuum ranging from simple line drawings black white realistic photographs color. Visual learning rese archers contrast, argue that there is a curvilinear relationship between amount of detail a visual evel student achievement (Arno id Dwyer, 1973, 1975, 1976; Broadbent, 1965; Dwyer, 1968a, 1968b; Travers , McCormick, Mondfraus, Williams, 1964). Visuals closely repre senting line drawings containing essential information to be transmitted are more effective more efficient than m 1967a, lore 196 detailed il 7b, 1968a). .lustrations Excesses (Attneave, realism 1954; Dwyer, interfere with effectiveness of visual s (Miller Allen, 1957). Empirical evidence shows that addition of color to media presentations may enhance learning retention (Dwyer, 1971a; white drawings are more effective collegelevel students on comprehension retention tests (Dwyer, 1969b, 1971b). drawings with more detail , more time crucial to attend to all relevant information cues Dwyer, 1969b) . Several findings involving televised instruction are relevan t to the present study. Te levi sion computer presentations both utilize a color monitor in various visual learning strategic es. Televised line drawings complement ed by motion (such as in dynamic shading) facilitate achievement instruction y focusing gaining stud on important ents aspects attention relevant cue s Dwyer, 1969a, 1972b). use questioning to complement simple line drawing an effective technique increa se achievement (Dwyer, 1970) Research investigations have een done on pacing visual pacing materials visual is imposed, adequately (Kress presentations. students Gropper, still 1964a) When be able slower external to perform experimenter control pace can be selected that compatible with high achievement that allows individualization instruction (Kress Gropper, 1964b). Cognitive Style Cognitive style refers characteristic, self consistent their modes perceptual of functioning intell which ectual individuals activities. show Field mode is one manife station an individual s broader cognitive style. There is an extensive research tradition concerning field independence /dependence as a cognitive style. Only selected studies pertinent present study will be summarized this section. Students a sys presented linear with inequalities procedural graphically task are of solving required to diff erentiate a common ove rlapping region (the eas ibility region) from a complex stimulus pattern of shaded half planes. Students have pierce ptual preferences in processing symbolic figural information while performing such spatialvisual tasks. Cognitive style research provides an empirical basis interpretation student pe procedures *rceputal (Witkin, pref erences 1969 ; Witkit in such n, Cox, information Friedman, processing 1976; Witkin Goodenough , 1977). Students who analyze differentiate halfplane regions to obtain feasibility region are termed field independent. Students who fail to analyze differentiate various shaded complex configuration overlapping regions in which embedded (Witkin, 1950; Witkin Goodenough, 1977; Witkin e changing t al points 1967 Salomon of view (1979) relating found compone that nts visually whole visually interacted significantly with field mode learning. In a recent revision of cognitive style theory, Witkin et al. (1977a, 1977b) suggest that cognitiverestructuring ability on which personal field autonomy dependent t are field character independent stics students differ. minimal According structure to the theory, guidance treatments should that provide appropriate field independent students, since they can provide their own structure work autonomous Fielddependent students , however, should excel in a highly structured treatment that provides careful guidance. Cognitive style research as it relates to ATI research will summarized later this chapter. Field mode restructuring a relationship dimensions to certain to spatialvi sual spatial ability. ability stimulus to disembed pattern of shad a feasibility ed halfplane region regions from may a complex involve a res tructuring a vis field, more specifically, involve ability to mentally manipulate the shaded w v 1962). Numerous studies have shown a high positive corre lation between field independence spatial ability (Anglin, Satterly, S chwen, 1976; Anglin, Vaidya 1985; Chansky, DuRapau 1980; Carry, Wachtel, 1981; 1972). In brief, mode related to a variety of perceptual p rob le msolving visualization see tasks Appendix requiring C for skill a discus spatial sion controversy surrounding constructs of field mode spatial ability). AptitudeTreatment Interactions Aptitudes, or individualdifference characteristics, are a complex of personal characteristics which result from a combination of natural ability environmental experlen ces upon which suspected that individual will differ in terms their learning potential. potential number of aptitudes which be related learning is overwhelming , since there are many categories (preferences, intellectual abilities , personality traits, interests, constructs attitudes be derived so forth) (Allport from which Odbert, s cre te 1936 ; Cattell, 1971; Guilford, 1967; Ve rnon, 1969) Aptitude been rapidly he acquires content information from a specific instructional presentation (Dwyer, 1978) . Thur stone (1938) argued that general aptitude or general mental ability consists three cial aptitudes spatial aptitude the ability distances to manipulate sence of verbal shapes , si or numerical zes symbols; numerical aptitude  the ability think with numerical symbols such those used in algebra, chemis try, statistics; verbal aptitud ability think with words. Cronbach (1967 recommended that researchers find aptitudes that interact with variations instructional treatments to design instructional treatments particular ways aptitudes to adapt groups instructional of students. treatments earch to individual differences known as aptitudetreatment interaction (ATI) research As Cronbach (1967) advocated, unless one treatment treatments clearly should the best be differentiated individuals such then a way to maximize their interaction with aptitude variables. If this accomplished sordinal interactions , then learning maximized. Numerous studies conducted since 1968 have sought to determine existence of ATI between subjects ' aptitudes various instructional that potential interactions are likely to reside three classes of aptitude variables specific intellectual abilities like those defined work of Guilford (1967) ; specific personality traits like those defined work of Cattell (1971); aptitudes styles a poorly preferences, defined group learning of cognitive sets, information processing coding strategies, other subtle expe riential variables. Although significant s have been found involving these aptitude variable , they frequently are difficult to replicate. Carry (1968 found an interaction between spatialvisual ability general areas oning ability instructional treatments (graphical analytical) quadratic inequalities. He hypothe sized that spatial visual ability would predict success from graphical treatment success general from reasoning analytical ability would treatment. predict finding was not very considered sound since reliability criterion direction instrument interaction was opposite to theoretical expectation. Webb (1971) made revisions Carry s treatments introduced a new *  *l * C ft r * when factors criterion of general Webb Carry measure reasoning (1975) transfer was spatial replicated regressed visualization. improved 1968 study Carry. tructional treatments criterion test were analyzed terms multiprocess theoretical model 1 of Melton (1967). No significant disordinal interactions were found between aptitude variable combination variables treatments. Eastman Carry studied aptitude tests used Webb study theoretical framework Guilford s Structureof Intellect used model Webb were (1967). Paper The spatial Folding visualization Spatial tests Visual ization These tests were class sified Guilford measuring (CFT) aptitude three Cognition dimensions. of Figural An examination Trans formations graphical treatment used Webb study (1971) showed that tended treat content inductively Both tests used Webb, These however, observations loadings suggest on a deduction need factor. an inductively structured tes t of spatial visualization that involved twodimensional aspects. Using Abstract Reasoning Subtest Differential Aptitude Test (DAT ) (Bennett et al., 1973) as a univocal * I  .1 I ~  and more deductively structured analytical treatments. significant t disordinal interaction between spatial visual zation and treatments oCC urred. This confirmed Carry original hypothesis that spatial visualization will predict success in a graphical treatment that general reasoning will predict Success an analytical treatment. further supported findings of King (1969) that Success on inductive testspredic tssuccess learning inductive material that success on deductive tests predicts success learning deductive material. an attempt answer Carry s (1968) question to extend Eastman Carry s (1975) results to another mathematical content area (the study of linear absolutevalue equations Eastman Salhab (1978) inve stigated interaction between instructional aptitude variables treatments spatial (algebraic ability geometric) general reasoning. Evidence was found which supported stence of a disordinal interaction. Hussien (1980) continued investigation disordinal interactions between instructional treatments (figural verbal) of modulus seven and v arithmetic erbal ability ability. measures, interactions were spatial ability disordinal only interaction between treatments verbal A1%11 44u Tn 04rfl .F~nf4 ~o ri.n 1nh147 ~tna a nnfl P4 nn+ a'h 11 *{4t * fn *; *_~ ^V\ i 1 4^ t7v k l TJra T.TS B In a continuation several studies (Carry, 1968; Eastman Carry, 1975; Webb Carry, 1975; dealing with quadratic used inequalities Salomon , Anglin, ferential Schwen, model 1 Anglin studies (1985 (1972 to capitalize on existing capabilities student preferential model, treatments are matched learners ' higher aptitude These re se archers investigated interaction reasoning of spatialvisual ability with ability presentation modes general of quadratic inequalities. Subtest Form They . of found that Abstract a sound measu Reasoning re of general reasoning ability that does correlate well with spatialvisual ability as Eastman Carry conjectured (1975). treatment was verbalpi ctorial numeric other was verbalsymbolicnumeric. No significant interactions occurred. DuRapau Carry 1981) conducted an ATI study clarify nature relationship between general reasoning ability diff erent strategies processing spatial tasks in their effect on the transfer learning. Spatial visualization was viewed as a single ability rather a dichotomy between gestalt analytic processing strategies of spatial tasks this finding was S U a a. a a  * verbalanalyticfiguralnontransformational approach point Salomon line (1972) symmetry preferential in Euclidean model, twospace present Using study searched significant between levels spatial ability levels of treatment (traditional versus computer enhanced) student posttest performance procedures solving linear or absolutevalue inequalities Cronbach Snow (1977) attributed most to general reasoning ability because difficult to separate effects of a specific aptitude from general reason ning ability. difficult with traditional aptitude constructs aser (197 to call research with "new aptitudes, " including dimensions that are related personality variables such as cognitive styles. cognitive style variable, mode more commonly, field independence/dependence), received considerable attention literature (Witkin Goodenough, 1977 Witkin et al., 1967). McLeod, Carpenter, McCornack Skvarcius, 1978) investigated an ATI between field mode treatments expository versus discovery) achievement. Treatments were based two levels of guidance crossed with two level of abstraction; topic was numeration systems. significant supported revised theory * . A q q 1 m L r ( symbolic significant materials used supported an expository revi mode. cognitive style hypothe sis. McLeod Adams (1980) examined inter action between levels of guidance field mode teaching topic of networks. levels guidance, high, cue were salience, chosen active varying amount involvement structure, student. Visual materials were prepared basis preferential model suggested Salomon (1970). high guidance treatment t was designed a compensatory treatment field dependent students. lowguidance treatment was designed a preferential treatment field independent students. None of the interactions were significant. This contradicted earlier support cognitive style theory found Adams McLeod (1979). Andrews (1984) compared effects of discovery expository learning strategies found learning on field that independent a significant strategies were best field occurred dependent that independents students. discovery while expository strategies were best for dependents Cronbach Snow (1977) argued that further inve stigations information process approaches their relationship to cognitive style other aptitudes are needed to build a comprehensive theory f aptitudes interactions. Using w_  w (cognitive versus style) levels computerenhanced of treatment in student (traditional posttest performance of the procedures solving linear or absolutevalue inequalities mathemati thrust, relates to visualized instruction, focuses on identifying individual difference variables which interact with several different media attributes types instructional presentation formats, kinds educational objectives , amounts of realistic detail contained visuals variables used of media to illustrate production, instructional techniques content, of organizing managing ature media. on interactions differences found Clark between studies (1975) , media testing in surveying attributes such liter individual interactions. In general, theoretical measures studies involved results before gross of which 1970 media were were primarily attributes difficult trait to explain. Snow , Tiffin, Siebert (1965) found that numeric aptitude interacted presentations. significantly Kanner with Rosenstein film (1960) versus found line that color versus black white televi sion presentations interacted learning. significantly Frederick, with general Blount, mental Johnson ability (1968) found that figural versus symbolic versus verbal notation CS. stimulus complexity interacted significantly with locus control information seeking. Dwyer (1970) found that various pictorial attributes such amount of detail line drawings) interacted significantly with grade level in learning. Since 1970, both individual difference measures media attributes have been specified more exactly. Salomon (1970) found that motion versus static media presentations interacted significantly with verbal ability, search cue attendance, in hypothesis embedded generation. figures Farley information Grant (1973) found that black white versus color pictures interact with arousal potential/ stimulation seeking arousal delayed effects. Peterson Hancock (1973 found that figural, verbal, significantly with symbolic pretest modes figural, media verbal, interacted symbolic aptitudes immediate delayed retention. Grippin (1973) found that strong versus weak prompt techniques interact significantly with field mode in learning. Summary There been limited research into pedagogical effectiveness computer technology it relates instruction they related to general mental ability; focus on identifying specific traittreatment interactions. In the past fifteen years, however, a rich tradition in ATI res earch emerged to parallel research endeavors visual spatial learning. visualization, research cognitive methodology style, promises to maximize effectiveness differential treatments traditional mathematics topi Clearly, computer offers unique capabilities computation, motion simula tion, color sound cueing, graphing, dynamic shading, of which can contribute to current research involving visual learning. Computerenhanced presentations traditional standing to improve topics in mathematics motivation student to learn attitudes promise those toward to deepen topics, learning under promise mathematics general. Students presented with procedural task of solving a sys of linear inequalities either graphically algebraically mus t separate specific pieces of information from complex stimulus pattern which it is embedded prior to information reach because an impasse character tW 1 % 1 , 1 i  , ew ti. processing. in performing their 4 t*1i e 4.*. rflf Field this cognitive fl w nj% 4. e dependent procedural style nr  students task suggest P  cognitive entirely style algebra preferences, ic tasks. O theirr perform such students, who procedures have high ability to mentally manipulate halfplane regions, decide to perform the procedure entirely as spatial visual tasks. Still other students adopt a combination of strategies utilizing both graphical algebraic cues performing informationpro students solution cess performing procedure. strategies particular used Research into particular mathematical procedures should a new facet understanding to the problem solving process mathemati The to enric learner recent h the on receives tradition bgoing and t in CEI classroom o contribute research promises instruction which e to ATI research both the involving field mode spatialvisual ability. It also promi ses to contribute to visual learning research involving line drawings, motion, sound color cuelng, dynamic shading, sequencing pacing of visuals. CS. CHAPTER PROCEDURES purpose this chapter twofold: describe pilot studies used to develop field test teaching methodology computer software used computerenhance d to describe instructional techniques, (CEI) procedures, treatment criterion measures used to investigate aptitudetreatment interaction (ATI) main study Pilot Study Introduction Visual learning research suggests that using televised line drawings, shading, complemented an effective motion visual such learning as in dynamic g strategy. Such drawings facilitate student achievement more than static chalkboard drawings focusing on important aspects of instruction focusing students ' attention relevant 1s uI rnf 1 1 w Whan nnmtrnl 1 r'no .~ IrlrI nr1 01 selected that is compatible with high achievement that allows individualization of instruction. A carefully structured consisting of a teaching model su offers c learning pported onsiderab process computerassisted opportunity for as compared instruction enhancement to traditional systems (CAI) the , which include opportunity computer to enhance support attitudes also students offers toward instruction first toward the pilot study subject was matter conducted they are studying. following purposes assess overall effectiveness teaching methodology computer software used treatment; assess student attitude toward sequencing pacing visuals toward treat ment The general. Study The enrolled subjects this intact investigation classes in college were algebra, students both taught same instructor, at EmbryRiddle University in Daytona Beach, Florida, during fall trimester, 1984. linear inequalities, a pretest over earlier college algebra material was admini steered to both groups prior two week treatment, an independent sample ttest was used test statistically significant differences mean pretest scores groups. Appendix a copy Pretes Results ttest analysis yielded no significant diff erences between two groups at the level .156; crit .00; see Table .1) . This pretest on earlier mate rial was strong support that control experimental groups were essentially equivalent prior treatment. A twoweek treatment procedural skills solving linear absolutevalue inequalities was conducted Group same A and treatments wer Group e use B cla sses. d in this Since pilot essentially study main study, a detailed description treatments will res served discussion as part main study later this chapter assess overall effectiveness teaching methodology computer software used treatment, a posttest over treatment material was admini stered to both treatment groups conclusion of the twoweek presentation. An independent sample ttest was used to test statistically significant diff erences in the mean posttest scores to tne two groups Table Descriptive statistics on mean pretest scores. Treatments N Mean S.D. Variance Statistic Traditional (Group A) 29 68.03 16.041 257.32 tca = .156 .s* cal n.s. Computer 30 70.57 16.506 272.461 Enhanced (Group B) *t crlt = 2.00 .05) between groups at the level .094; t crit .00; see Table 3.2) Sequencing pacing of visual used treatment were based on visual learning theories Dwyer, Salomon, Gropper, Kress Dwyer, 1972a; Kress Gropper, 1964a, 1964b; Salomon, 1979). software was designed to allow based on shading some student cue s structor feedback: to enhance controlled pacing included visual learning. of vis sound, lual motion, Henderson Taxonomy Teaching Model was used both treatment groups primary instructional mode i. This model useful analy zing procedures moves strategies monitoring in teaching facilitating mathematical different levels of understanding students (Henderson, 1976). A detailed description of the Henderson Model computer software used in the treatment is reserved discussion as part main study later in this chapter. assess pacing toward subject of visuals treatment, compatibility assess with student a questionnaire sequencing attitude was administered last twoweek treatment to Group subjects. See Appendix A for a copy questionnaire a summary res ults An analy responses  W , V   m V Table Descriptive statistics on mean posttest scores Treatments N Mean S. D. Variance Statistic Traditional (Group A) 26 80.50 14.45 208.80 cal = .094n.s.* Computer Enhanced 28 79.00 17.22 296.59 (Group B) *tcrit crnt = 2.00 .05) appropriate to their level of understanding cognitive structure would rearrange topics. Of the subjects, said they would delete topic of quadrati inequalities until later course. In relation topics of the subjects Group B found visual s too fast comprehension. In relation topics found visuals too slow in presentation. Of the subjects, found pacing appropriate comprehension topics. On topics , 29% of the subjects Group found press entation boring times. Overall, enjoy ed the presentation topic. In addition, said that would enjoy other topics in coll algebra taught using computer. Finally, said that they would enjoy other topics in mathematics taught using computer. Of the subj ects, felt they would retain procedures solving linear inequalities because they learned them visually computer. Comments on the Study Although results ttest analysis support superiority of either treatment, analy respond ses questionnaire provide feedback concerning student attitudes toward sequencing pacing of visuals used Results on the questionnaire indicated that pacing visuals have been fast comprehension subjects 1964b) argue that in Group pacing Kress a crucial Gropper variable (1964a, high achievement visual learning. this pacing een slower, subjects in Group have comprehended visual cues better achieved significantly higher post test scores. Spatialvisual ability have interacted with levels of treatment to contaminate interpretation posttest results. There were limitations instruments procedures used in this first study. Pretest posttest have reliability as criterion measures. software have allowed suffi cient instructorcontrol pacing or sufficient branching options learning process to be eff ective. Although first pilot study support overall superiority naire of the related treatment, pacing findings sequencing question computer visuals potential a traittreatment interaction motivate software modification as part of the of a second pilo research t study design and spring 1985. Pilot Study Introduction search ways adapting instructional treat ments to individual differences is known research. goal research to select methods of instruc tion that are mos t effe ctive groups students with partic ular kinds of aptitudes. Kerlinger Pedhazur (1973) distingui shed between kinds of interactions ordinal interaction disordinal interaction. This distinction can be explained graphically Figure 3.1. Figure represents no interaction, where regression lines + BIX A2 + B2X that 2 corres pond to treatment treatment (T2) are parallel. There is a constant diff erence between along dimension the aptitude Stated differently, slopes regress ion lines are identical, entirely difference accounted between treatments difference between intercepts of the regression lines. Figure (b), where regression lines intersect a region aptitude that is of no concern researcher, represents an ordinal interaction. In this case, T S  = A1 Figure Graphs of types of interactions Y2 = A2 + B2X Y1 = A1 + BlX Range of Interest Aptitude No interaction Y2 = A2 + B2X Y1 = A1 + BX Range of Interest Aptitude Ordinal = A2 interaction + B2X = A1 levels. intersect Figure in the (c) , range where aptitude that regression lines important researcher, is superior represents at the lowe a disordinal levels interaction. of X while Here superior at the ess upper one levels treatment As Cronbach is clearly (1967 best advocated, individuals, then treatments to maximize their should be interaction differentiated with such aptitude a way variables. If this accomplished disordinal interactions then learning maximi zed. idea can illustrate ed b y Figure In this figure, there a disordinal interaction , and a point on the aptitude dimension regrets corresponds sion point lines. of intersection difference outcomes significant values of X less than P and values X greater than then individuals should divided with into an aptitude (T1) those e groups higher with ase than their P should an aptitude lower aptitude. given than Those treatment P should given treatment treatments assignment potential to maximize groups individual learning outcomes. Snow Salomon (1968) argued that, in highly likely to visual reside in treatments, potential e Vzfactor interactions of spatial are visualization spring of 1985, second pilot study was conducted following purposes assess teaching used findings overall methodology a revise from Pilot effectiveness computer treatment of the software based Study to develop scoring, procedures interpretation adminis traction, results spatial ability tests practice administration to study of these effect tests; instructional treatments on the performance of students different levels of the Vzfactor. second pilot study was designed primarily study effect of two instructional treatments on the achievement of students of different level spatial visual ability Vz) . This was achieved studying interaction between eve ls of treatment three levels of spatialvisual ability. Salomon s (1972) preferential model was used to generate hypothesis this second pilot study There of spatial no significant ability between levels levels treatment I r 1 a According preferential model, treatments were designed to call on subjects ' higher aptitudes. experimental treatment, therefore, was a highly visual, computerenhanced presentation remini scent of the computer treatment students' was used high modified, Pilot spatial based Study ability. on visual and The learning designed computer research to call software findings from Pilot Study , to allow greater teaching flexibility instructorcontrol in pacing visuals. software was re designed allow options returning previous steps the solution proce dures advancing to subsequent steps more slowly reviewing step s in the procedure; pursuing divergent repeating solution entire strategies procedure selecting and visual zing simpler problems, analogous problems, more enriching problems control treatment was identi traditional treatment used in Pilot Study A detailed description treatments reserved discussion as part main study later this The chapter. Study subjects this investigation were freshmen ~~ a  1 I LI . I I University Daytona Beach, Florida, during spring trimester, 1985. Prior twoweek instruc tional presentation, both class ses were administered a unit test on earlier coll algebra material Test , a pret on the material covered course Pretest), Form Board Test 1), Paper Folding Test (Vz2) . Vz1 Vz2 were produced Educational Testing Service were used as aptitude measures Vzfactor spatialvisual ability. (These tests have reliabilities res pectively. French (1954 recommends them as having high construct validity measure factor Appendix C for sample items from Vz1 Subjects were allowed eight minutes each parts Vzl test three minutes each of the two parts Vz2 test. Both tests were admini steered scored according to specific directions provide ed by Educational Testing Service. Initial differences in prerequisite skills knowledge pretest course (Pretest) material earlier were unit controlled (Test entering scores covariates in a twoway analysis covariance (ANCOVA) procedure. treatment Level were of spatial entered ability as categorical (Vz) levels independent variables. a copy of the pretest unit test. Appendix a copy of the postt est.) Composite scores from Vzl Vz2 were used a basis dichotomi zing spatial ability factor into low, medium, high levels (See Table 3.3). Using ANCOVA procedure, with factors spatial ability three levels) treatment levels) , visual a disordinal ability interaction treatments between achievement spatial occurred. Since F cal 699) greater than F crit 29) , was ected. There was a signifi cant between levels spatial ability levels treatment in student posttest performance of the proc edures solving linear absolutevalue means inequalities. ANCOVA Table analysis. presents Table cell partitions total covariate sum effects squares , main ANCOVA effe analy interaction into effects. Because of a significant ATI, effects spatial ability changed across levels treatment. Figure a graphical presentation these results across levels of each factor. A followup simple effe analysis , using Bonferroni procedure family comparisons), was conducted. A significant difference between Table Dichotomization spatial ability scores. Category Range of Standardized Scores Range of Raw Scores Low Spatials Below Deviation .5 Standard from Mean Medium From 0.5 to +0.5 Spatials Standard from Deviation X 20 Mean High Spatials Above +0. Deviation Standard from Mean 20.18 34.00 Descriptive Scores: Statistics on Composite Spatial Ability Range Grand of Raw Scores Mean = 27.75 Scores = 17 Lowest Score = 1. High est core = 29 Standard Deviation Scores 61 (0 1H Cl p'po en 0 N CD en Cr,, I~ I **lv o r r tr I I om I w  SII II 0 0 N N S rS LA 0 ,. 4 CO (Q <"3 cU I X 1 4J~~ ro er 1 N CD O Ia o I 4 o (T 02 ( s .) 0 I It o 0 V en cD Cl) e n oo 'UU N CD o ~ ~~ ~ I_________ M U)l CCI ^ E: I > I X H H  0 Cl) o i < ~o r^ ao 02 rj (N C)  Frl I iC I II I0 C) (( 0 L A *l Slrr m uin n F oo i"i cNO O *r4 ar3 a, (1 r' a) 0 U, 1.4 CQ 0 ci 4IW 0 2 U, L 'o in in rN 'V 4 (MrfirI mP Fr (N ul Ln in m un m M l C.I Cd tHl 0 4 *d *O w ,Q QP a U CO H r C' CV r4CNJ N (N 00 Computer Enhanced Traditional Medium Spatial High Ability Figure Graph fixed of disordinal levels interaction of spatial across ability v high o medi o low spatials um spatials spatials 4. Traditional ComputerEnhanced Treatments Figure Graph levels of disordinal of treatment. interaction across fixed spatials at the level (See Appendix B for detailed discussion of the simple effects analyses) observation cell means in Table it is clear that high spatials better than medium or low spatials treatment that medium spatials better than spatials. However, none ese differences were significant using Bonferroni procedure. Comments on the Study Visual learning theorists predict that students having medium spatial ability should do better than students having spatial ability traditional treatment involving transparencies. Simple line drawings present ed with black white overlays provide more visual cues various shaded regions spatial to the subjects. medium The a spatial Igebraic subjects pres than entations used traditional treatment have required an analysis detail which interfered more with informationproce ssing style the m edium perceptual spatials. preferences In like of the manner, high spatials graphical than presen stations a ges used talt in the processing traditional strategy w treatment which interfered have required more with informationprocessing style perceptual preferences . By presentations have adapted well to either processing style. ability of high spatials to learn from relevant sound, motion, shading cues, their ability manipulate shaded regions readily could account their higher scores in a highly visual computerenhanced treatment. inability transformations of low spatials readily to perform have cognitive to their lower figural scores computerenhanced treatment. There were limitations instruments procedures used this second study. Spatialvisual ability was dich otomized this into study. , medium, disordinal high levels interaction purposes medium spatials have been an artifact this artificial dichotomy factor. interaction disappear if the spatial sual factor was treated as a continuous variable in a multiple regression analy recommended Kerlinger Pedhazur 1973) Pretest posttest have reliability as criterion measures. Pretest effects are always poss ible experimental studio which administer prete sts. Main Study lvs rvi sw n I I*~ fllliv to study treatments effect on the of two performance instructional students different leave is of spatialvisual ability (Vz) , to study treatments the effect instructional on the performance students having different levels of field mode (field independence/dependence), assess overall effectiveness teaching methodology used a CEI treatment. Based on findings from Pilot Study , students having different levels of spatialvisual ability along a continuum exhibit differential performance highly visual treatment which uses microcomputer to enhance mathemati instruction received. this reason, spatial visual in the ability main was study chosen Students as a continuous presented independent with variable procedural tasks of solving a system of linear inequalities graphically or algebraically, must , prior to information processing, s ep ar a te pattern pieces in which of cognitive of information is embedded. style from complex Some preferences, students, perform such stimulus because procedures entirely algebraic Other students, have high ability to disembed feasibility region from processing. Cognitive style provides an empirical basi interpretation of student perceptual preferences in such informationpro cessing procedures. eld independence/depen dence (field mode) one manifestation cognitive style. eld independence/dependence continuum represents level student processing was perceptual therefore pref erence chosen in information second continuous independent variable main study. Salomon (1972) preferential model was used to generate ATI null hypotheses There no significant between levels spatial ability and level s treatment in student posttest solving There is performance linear or absolute no significant ATI procedures value between inequalities. level field mode and levels treatment in student posttest solving There performance linear or absolute is no significant performance procedures value diff computer erence enhance inequalities between traditional treatment groups in solving linear absolute value inequaliti es. deci sion rule rejection null hypotheses was .05. students. preferential mode 1, treatments are matched learners ' higher aptitudes present study invest gated interaction spatialvisual ability (Vz) and field independence/dependence with presentation modes linear inequaliti es. computerenhanced (CEI) treat ment was designed as a preferential treatment high spa tial ability students. was also designed as a lowguidance (discovery) students. compensatory preferential traditional treatment treatment treatment spatial field was independent signed ability students. was ment also designed field dependent a highguidance students. compensatory remainder treat Chapter presents a detailed discussion instrumentation, teaching methodology, treatments, computer software, statistical analy ses used main study. Instrumentation Prior solving twoweek linear presentation absolutevalue procedures inequalities, subjects both asses were admini stered a 60minute pretest on earlier college algebra material. (The subjects main intact study classes were freshmen college algebra, students both enrolled taught same instructor. at EmbrvRiddle University Daytona __r .70. (See Appendix C for copy of this pretest. first treatment , all subjects in both classes were administered seven aptitude measures. Group Embedded Figures Test (GEFT) was administered as a measure of field independence/dependence. Form Board Test (Vz1) Paper Folding Test were administered as measures of the Vzfactor Reasoning Subtest of spatial of the Di visual fferential ability. Aptitude Abstract Test (DAT), Form was admini stered a measure of general reasoning ability Appendix C for sample test items from each aptitude measure a discussion of reliability validity of each scored, instrument.) interpreted aptitude as specific measures d in the were testing administered, procedures manuals obtained from Consulting Psychologi Press ca tional Testing Se rvi ce, Psychological Corporation. conclusion twoweek presentation, subj ects both treatment groups were admini steered a 60minute posttest over treatment material. pointbiserial correlation this teacher rmade test was .75. (See Appendix a copy of the posttest. Henderson Taxonomy Teaching Model Mathematics education concerned primarily with used to achieve this understanding. Henderson and his colleagues developed a hierarchical model for teaching concepts, generalizations, and procedures in mathematics based on Bloom's Taxonomy (Cooney, Davis, Henderson, 1975; Henderson, 1976). Using teaching moves (bits of discourse) strategies (sequences of moves) as originally defined by Taba (1966) , teacher can monitor level of under standing learner also adjust pace type of strategies used teach student effectively that level. The Henderson model ideal teaching the procedural skills Using solving this model, linear absolutevalue teacherthrough verbal inequalities. communication can focus students' attention procedural skills involved by describing briefly what procedure entails and by giving an objective skills. Part of teacher' strategy must be to make sure students know how to complete task. teacher communicates directions, called prescriptions, to advise, guide, and direct student action. A stepbystep prescrip tion is one of the easiest and most effective ways teach a procedural teacher skill. clarifies demonstrating a prescription, procedure designated by prescrip * * give verbal visual cues as to when to perform procedure. Henderson model assumes that students have appropriate prerequisite skills prior learning a new procedure. Otherwi when teacher models appropriate steps in a solution procedure, students standing a new will imitate thus situation. teacher will be able (See Appendix without true transfer C for their an outline under learning moves used the Henderson model examples stepbystep prescriptions.) Henderson model assumes levels of understanding, student at level understanding a proce dure, instance, would need examples nonexamples procedure. Analogy moves are appropriate this level to remind procedures. students Level similarity II understanding to previously should learned achieved app ropri ate Practice practice, is necessary analysis in order , and application to perform moves. procedures with speed accuracy to reinforce appropriate behavior. Corrective feedback must accompany such practice to prompt to motivate correct performance. Eliciting questions such "How procedure solving ,1~~~~~pA DIC lAae ji i41.. T^w n 1 A I cn~ understanding of a procedure. Incorrect performance step a procedure could return teacher immediately from level moves to level moves. This where computer to return a use to earlier teaching steps device in a solution allows procedure students or to review an entire procedure. method of justifying prescriptions to students to enable following them t a pres o determine cription, whether their correct. answer, can after done performing graphical justify algebraic solution to themselves solution procedures procedures. algebraically immediately example, that students center after can value  B graphical C C and solution critical values procedure are B + C and correct. C in Then, looking at combination inequalities of the form C student can apply what learned new solution procedures. As another application move, Henderson model can be used to provide stepbystep prescriptions solving linear programming problems. Treatments Henderson Taxonomy Model was primary instruc  B below, followed particular programs used teaching topics. Topic Graphing a linear equation linear inequality equations Solving algebraical a system ly and g of linear raphically ("Inequalityl" program) Topic Solving a system linear inequalities algebraically graphically (" Inequalityl" program) Topic Solving linear programming problems, involving at most four constraints, algebraically graphically ("Linear Programming" program). Topic Solving linear programming problems, involving five or more constraints, algebraically graphically ("Linear Programming" program). Topic Solving absolute value inequalities form algebraically absolutevalue  B C orlAy  BI graphically. inequalities  BI Solving form > C algebraically graphically "Abs oluteValuel" program Topic Solving a system absolutevalue  B Subjects traditional treatment experienced highly structured, expository press entation procedures solving linear or absolutevalue inequalities recommended (Beckenbach School Bellman, Mathematics 1961). Subj Study ects Grou this (SMSG) highly structured treatment were provided self qui zzes (with answers) at the of each class meeting to reinforce correct student performance questioning with procedures. teacher Opportunity corrective feedback for was provided used at each to clarify subsequent algebraic s meeting. solution chalkboard procedures with was frequent reference to the real line. An overhead projector was used conjunction effective visual with presentation chalkboard of the to provide graphical most solution procedures Henderson pos s Model ible was without used using teach computer. stepby step The prescriptions the correct solution procedures. (See Appendix C for examples stepby step prescriptions used traditional treatment.) Subjects highlyvisual, experimental computerenhanced treatment experienced presentation pro cedures solving linear or ab solutevalue inequalities, based on visual learning theories Dwyer (1972a), Salomon (1979 , Gropper Kress (1964a, 1964b). Unique teacher to verify algebraically correctness graphical solution procedures. For example, after s student visualized center value = B/A critical values = B/A + C/A = B/A  C/A xaxis inequality Ax BI teacher then verified the procedure going through algebraic steps at the chalkboard. A dis cover approach was used designing treatment. In this lowguidance (discovery) treatment, students were expected to generate test con lectures performing solution procedures. ComputerEnhanced Software software package which accompanied treatment was prepared according to teaching principles Henderson Teaching Model recent findings from visual learning theory. software package was written in Applesoft Basic Apple IIe microcomputer consisted three programs Inequalityl; Linear Programming; Absolute Valuel. "Inequalityl" program allowed students to scale x and yaxes to enter number P of linear inequalities. program then accepted numerical coefficients C for each inequality in the form Ax + By < C. An externally _  *t *1 , . *  , I I * _____ ___ each student inequality conjectures to allow as to the teacher possible questioning location moves inter cepts shaded half plan program then drew in the correct lines appropriate black halfplane white regions dynamically on a large computer shaded screen. teacher student could then visually verify their conj lectures. After coefficients were entered, another externally controlled pause was built into program allow student conjectures corner points boundary of the feasibility region. program was then designed to isolate edded solution region erasing of the overlapping regions which were part solution region. teacher student could visually verify their conjectures. A builtin option allowed students see an algebraic representation system inequalities under consideration desired. Followup moves teacher using Henderson model could involve solution algebraically "Linear corresponding to obtain system correct Programming" program of linear corner equations points. provided a menu builtin branching options provide instructions in the solution of a linear programming problem, visual ize word word problems problems with with five four constraints, or more constraints. visualize Option 1 a linear programming solution linear problem, programming visualize problem, graphical visualize an example (demonstration) , return to the main menu; (5) ' exit program. Option ' of submenu allowed student solution see a demonstration procedure with of the a particular correct example. steps Option of the submenu allowed student to return main menu to solve problems involving four or five constraints. Option main menu served essentially as a "page turner" to quickly present linear programming word problems on a large computer screen. student a choice of four Option four exercises exercises involving involving four five constraints or more under constraints under Option Teachercontrolled pauses were built into this program to allow teacher questioning moves , such "What or "Is Pacing are constraints?" or "What active of vi suals function could proceed objective to be maximi faster function?" or minimized or slower depending on student feedback. Options to return main menu to exit program were always available order run "Inequalityl" visual solution region. soluteValuel" program provided a menu branching options to solve  Bj  Bl C c; AX BI  B Option began ,  , w w SI  I steps to solve Ax B( followed a specific example. Analogy moves were used to show similarity in these procedures. entering coefficients together with choice or "y" or "> " the student an opportunity see graphical solution pro cedure absolute value inequalities on a large computer screen. Options offered same sequence of learning experiences each particular type inequality. graphical solution procedures employed several graphics capabilities to capitalize on visual learning theory. To graph program drew vertical lines = B/A screen. + C/A center = B/A point  C/A solution computer region (B/A, was made a relevant visual cue learner flashing beeping student a few to conjecture seconds. center This allowed point time critical points before they were eled. program then continued dynamically region shading labeling between center vertical point lines critical solution points verify student conj lectures. Builtin program pauses at this stage allowed teacher clarification student ques tion Divergent solution strategies algebraic verification at the chalkboard were also possible. an additional  B three programs software package certain common elements. Teachercontrol visual displays student allowed conjectures a slower were pace corrective incorrect. "What eedback . .?" was an appropriate teacher move point to enhance under standing immediate of the concepts. execution Program of smaller flexibility or larger allowed teams inequalities as a reinforcement strategy. Statistical Model A multiple linear regression model following form was used to represent data Model + 83X3 + B4X4 B5X5 where Y is dependent variable, that is the observed score on the achievement sttest 1 1 + B2X2 predicted + B3X value + BX 4 4 (mean + B5X value) dependent variable Y at cific values A is a constant; are regress coe cient score; x, is a covariate is a covariate representing representing tne score pretest on general = oc + 6 score ability under consideration (spatial ability or field mode); a dummy variable representing group membership, where = 1 if score subject treatment group score a subject traditional treatment group; a variable representing interaction between where = X3 * X4; variable represents error or res idual term, i.e. , difference between observed and predicted value Therefore, when that group, when predicted group under value consideration takes form + BIX1 I11 + B2X . (1) when = 0, that when group under consideration traditional group, predict value takes form = A + B X1 Equation + B2X + B3X 3 3 represents . . . (2) relationship between score ability under consideration, predicted value, treatment group where pretest general reasoning ability are held constant. Equation ability represents under the relationship consideration, r between , and score predicted = A + B4 + B5 l Formal Hypotheses In order test existence between two treatments and each level spatial ability field mode following set of formal hypotheses were tested: The proportion total variability population accounted for complete model (that includes is not , X3 zero. The above hypoth esis null form : R2 or H1 The slopes and regression lines (for fixed values of general reasoning ability pretest) which correspond traditional groups are equal population. The above hypothes null form : 5 In other levels words, aptitude there no significant in question and ATI levels between treatment student posttest performance the procedures solving linear or absolutevalue inequalities. Rejection this hypothesis parallel means and that a significant regression ordinal lines are or disordinal between 85 1 + B5 Further analysis Whether ordinal or disordinal interaction should occur, following questions were investigated: there edicted a difference group between achievement) group at specific achievement values spatial ability score and field mode score, are these different ces significant? Regions ability scale which differences between group achievement are significant are known regions significance. The null hypothesis associated with this question B5X3 population, specific value = X3 section this hypoth esis means that there a significant achievement some difference values predicted ability group question. that case, regions of significance will investigated using a JohnsonNeymann technique (Borich, Godbout, Wunderlich, 1976). no significant hence, ATI " term occurs, dropped then and, from complete model. Formal hypothesis The intercepts A + B, and A of regression _ _ which correspond and traditional treatment groups. above hypothesi phrased in null form s as follows That particular aptitude in question , the intercepts regression lines which correspond traditional groups are equal population. In other words , there is no significant difference between performance traditional treatment groups in solving linear or absolutevalue inequalities Rejection this hypothesis means that there significant differ ence between performance traditional treatment groups Rejection then implies that there a main effect treatment. there corre lation between field mode scores spatial ability scores A correlational analysis was used examine correlations between aptitude scores Figure summarizes hypotheses which were Invest tigated main study. this figure, treatment traditional group group is represented is represent B Group A Group. Figure Graph of the analysis statistical regions of significance treatment effect Overall test Interaction test : 85 or H2: Significant interaction AR (X5/X1' interaction J.N. technique signifi not cant signif icant BGroup AGroup signifi cant Graph of regions significance Graph effect treatment Treatment effect test AR (X4/X1, Test 85X3 CHAPTER I RESULTS This chapter r presents a detailed analysis data which were required hypothesis testing and interpretation res ults. each subj in both computerenhanced ( CEI) traditional treatment groups raw scores were obtained score on the Group Embe dded Figures Test GEFT), as a measure of field mode ability; compos score on the Form Board Test Vz and Paper Folding Test Vz as measures of spatial visual ability score stract Reasoning Subtest Differ ential Aptitude Test DAT) , Form a measure neral reasoning ability score pretest score achievement posttest. Table .1 presents means, standard deviations range , and number subj ects per treatment group aptitudes outcome measures. insure that Group (traditional treatment) Group (computer enhanced treatment) were essentially anri' rr31 1 n4 S   nrl/i /n ai r* 4. i n a C. Er  I~ *Cc ^ ^ n rn r~in i/" Y ~YI 