Spatial ability and field mode in a computer-enhanced presentation of linear inequalities : an aptitude-treatment intera...

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Title:
Spatial ability and field mode in a computer-enhanced presentation of linear inequalities : an aptitude-treatment interaciton study
Physical Description:
xiii, 241 leaves : ill. ; 28 cm.
Language:
English
Creator:
Kiser, Omer Lee, 1943-
Publication Date:

Subjects

Subjects / Keywords:
Learning, Psychology of   ( lcsh )
Computer-assisted instruction   ( lcsh )
Field dependence (Psychology)   ( lcsh )
Curriculum and Instruction thesis Ph. D
Dissertations, Academic -- Curriculum and Instruction -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 226-239.
Statement of Responsibility:
by Omer Lee Kiser.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000928173
oclc - 16109452
notis - AEN8912
System ID:
AA00002160:00001

Full Text

















SPATIAL ABILITY AND
COMPUTER-ENHANCED PRESENTATION
AN APTITUDE-TREATMENT


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


r-Assis


Visualizat
Learning .
ve Style .
e-Treatment


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


Computer-Enhanced


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


Two-way


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
(A-Group)


achievement


on spatial


ability


Page











Figure


Regression
(A-Group)


Regression
(B-Group)


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
B-Group


evement


on abstract


areas


oning


Regre


ssion


of achievement


on pretest


(A-Group)


ress


of achievement


on pretest


(B-Group


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


A-Group)


D.13


D.14


Regres
mode


Regre


sion


of abstract


ability (


ssion


B-Group


test


reasoning
) .


on spatial


on field


ability


(A-Group


D.15


D.16


Regression
(B-Group


Regression
(A-Group)


of pret


of prete


on spatial


on field mode


ability


ability


D.17


Regression
(B-Group


pret


on fi


mode


ability


D.18


Regre


ssion


of pret


est


on ab


stract


reasoning


ability


(A-Group)


D.19


Regression
ability


of pretest
(B-Group) .


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


COMPUTER-ENHANCED PRESENTATION
AN APTITUDE-TREATMENT


OF LINEAR
INTERACTION


INEQUALITIES
STUDY


Omer


December,


ser


1986


Chairman
Major De


Dr. Elroy


apartment


. Bolduc,


Instruction


Ed. D.


Curriculum


This


aptitude-treatment


interaction


ATI)


study


was


designed


to investigate


effect


instructional


treatments


on the


achievement


of students


having


different


eve


Is of


spatial-visual


field


mode


abilities.


instructional


tional


treatments


presentations


were


computer-enhanced


of the procedures


solving


tradi-

linear


or absolute-value


inequalities


college


algebra.


Subjects


study


classes


were


of coll


students


algebra


enrolled


at Embry-Riddle


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


two-week


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


computer-enhanced


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 rn-ii -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


Computer-ass is ted


instruction


(CAI)


relates


aptitude-treatment


interaction


(ATI)


research


made


contributions


toward


improved


mathematics


instruction.


In recent


years,


a variation


computer-as


sted


instruc-


tion


emerged


in which


graphics


capability


microcomputer


to enhance


s utilized


individualiz


in on-going


ed instruction


ssroom


which


interactions


learner


receives.


instruction


s variation


(CEI), is


of CAI,


of educational


termed


computer-enhanced


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


er-e


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


Embry-Riddle


Univer


sity


Daytona


Beach,


Florida.


subjects


in the


traditional


procedures


treatment


solving


experienced


linear


a presentation


or absolute-value


of the


inequalities


recommended


the School


Mathemati


Study


Group


as a sound


presentation.


subj


ects


in the


experimental


treatment


experienced


a highly


visual


computer-enhanced


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 computer-enhanced


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 absolute-value


no significant


levels


treatment


procedures


inequalities.


levels


between


field


mode


posttest


levels


performance


treatment


of the


student


procedures


solving


linear


or absolute-value


inequalities.


There


performance

traditional


no significant


of the co

treatment


difference


mputer-enhanced


groups


between


the

linear


solving


or absolute-value


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


problem-solving


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,


time-saving


topic


strategy


in mathematics


teaching


utilizing


traditional


microcomputer


as recommended


NCTM


CUPM.


laPf n4 +4-inn nnn


r ^: F P rm o











Spatial-visual


ability.


Spatial-visual


ability


ability


a sequence


or 1


ess


to mentally


of movements.


complex


manipulate


visual


The objects


two-dimensional


objects


appear


stimulus


within


pattern


involving


a more


(Michael,


Guilford,


Fruchter,


Zimmerman,


1957).


Spatial-visual


ability


was


measured


as a compos


score


Form


Board


Tes t


(Vz-l)


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


spatial-visual


tasks


stimulus


procedures


algebraic


patterns


discussed


tasks


There


in this


(Cooney,


Davis,


Henderson,


1975;


Davis,


1978).


Spatial-visual


task.


A spatial-visual


task


mathematical


procedure


involving


a simple


or complex


visual


stimulus


pattern


which


requires


some


measurable


degree


of spatial-visual


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


self-consistent


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


dependence--to


greater


ability


to deal


with


presented


figure


field


from


analytically


configuration


or to separate


in which


an embedded


occurs


other


extreme--termed


field


independence


(Witkin,


Good-


annuah -


Ka rn .


l Q;7I










Computer-assisted


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 drill-and


simulation,


and


-practice,

instructional


games


(Burke,


1982;


Coburn,


Kelman,


Robert,


Snyder,


Watt,


Weiner,


1982).


Computer-enhanced


instruction


(CEI).


Computer-


enhanced


instruction


that


facet


of computer-assisted


instruction


in which


unique


capabilities


microcomputer


color,


sound


motion


cueing,


graphing,


shading


are


utilized


teacher


student


on-going


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


- -


treatment-by-level











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


r-enhanced


Henderson,

treatment


1976


teach


proce


dures


solving


linear


or absolute-value


inequality


es.


It is


assumed


present


study


that


using


Apple


II-e


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


(Vz-1)


were


adapted


Guilford


colleagues


as criterion


measures


Cognition


of Figural


Transforma-


tions


(CFT).


definition


spatial-visual


ability


used


Punched


this


Holes


study


Test


assumed,


Form


Board


therefore


Test,


that


as research


instruments,


have


high


construct


validity


measure-


ment


of spatial-vi


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 Embry-Riddle


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


two-week


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


spatial-visual


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


computer-enhance


d in


structional


(CEI)


treatment.


It also


describes


techniques,


procedures


, and


criterion


measures


(ATI)


of the


used


inves


main


data,


study.


including


tigate


aptitude-treatment


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:


computer-assi


sted


instruction;


spatial


visualiza-


tion


spatial-vi


sual


ability;


visual


learning


theory;


cognitive


style


research


relates


to spatial


ability,


cognitive


style,


media


attributes


in mathematics


achievement.


Computer-Ass 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 drill-and-practice


tutorial


demonstration,


simulation,


gaming


(Burke,


1982;


Coburn


et al.,


1982).


There


are


numerous


citations


in the


literature


related


to CAI


or computer-based


instruction


(CBI


equivalent.


present


study


deals


with


that


facet


of CAI


termed


comput


er-enhance


d instruction


(CEI) ,


in which


unique


capability


microcomputer


computation,


motion


simulation,


color,


sound,


graphing


dynamic


shading


are


utili


zed by


teacher


student


in on-going


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


computer-based


simulations


in economics


resulted


in mixed


findings.


Katz


(1971) ,


other


hand,


found


that


problem-solving


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


-and-practice


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 Bangert-Drowns


(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


5-10


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 one-eighth


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


drill-and-practice


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


broadly-based


case


CAI.


These


endeavors


have


also resulted


in conflicting


inconclusive


findings.


Vinsonhaler


Bass


(197


, for


example,


in their


review


major


computer-based


drill-and-practice


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.


Computer-based


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


on-going


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 meta-analysis


on mathematics


schools


(Glass


education


reported


that


--ra w


, I


. ....












percentile.


Burns


meta-analytic


Bozeman


techniques,


(1981),


also


a continuation


integrated


findings


mathemati


education


elementary


secondary


schools.


They


found


that


computer-based


tutorials


raised


achievement


test


results


extremely


ability


groups


standard


deviations.


Comput


er-based


drill


-and-


practice


raised


gifted


ability


groups


standard


deviations.


Kulik


et al.


(1983)


used


meta-analysis


integrate


findings


from


51 independent


evaluations


computer-based


teaching


grades


6-12.


Computer-based


teaching

It also


raised


final


positive


exam

effect


scores


on later


.32 standard

retention s


deviations.


cores


on attitudes


toward


computer


course


study.


This


meta-analysis


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


computer-based


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 meta-analytic


techniques


to integrate


findings


from


independent


evaluations


of comDuter-based


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


computer-based


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


on-going


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


problem-solving


program


linear


algebra


that


uses


rotations,


dilations,


refl


sections


of lines


dimensions.


Ignatz


Ignatz


(198


described


an interactive


video


svst-istn


1-h '


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,


problem-solving


techniques


mathematics


classrooms.


advantages


of generating


graphics


displays


on-going


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


* n4-ar ,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


Spatial-Visual


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


spatial-visual


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


spatial-visual


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 sub-abilities


Z immerman,


s task


, of


spatial-visualization


aptitude:


spatial


relations


orientation


kinesthetic


SR-0)


, (2)


imagery


sualization


Vz-factor


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.,


Vz-factor


1957) .


be measured


performance


on the


Form


Board


Test


(Vz-l)


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


problem-solving.


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-


Sht-h-


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


college-level


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


spatial-visual


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


half-plane


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


cognitive-restructuring


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


Field-dependent


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 spatial-vi


sual


spatial


ability.


ability


stimulus


to disembed


pattern


of shad


a feasibility

ed half-plane


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


m-solving


visualization


see


tasks


Appendix


requiring


C for


skill


a discus


spatial


sion


controversy


surrounding


constructs


of field


mode


spatial


ability).


Aptitude-Treatment


Interactions


Aptitudes,


or individual-difference


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 aptitude-treatment


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


spatial-visual


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 Structure-of-


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


two-dimensional


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


absolute-value


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 44-u Tn 04rfl .F~-nf4- ~o r-i.n 1nh147 ~tna a nnfl- P4 nn+


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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 spatial-visual


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


spatial-visual


ability


as Eastman


Carry


conjectured


(1975).


treatment


was


verbal-pi


ctorial


-numeric


other


was


verbal-symbolic-numeric.


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


- *












verbal-analytic-figural-nontransformational


approach


point


Salomon


line


(1972)


symmetry


preferential


in Euclidean


model,


two-space


present


Using


study


searched


significant


between


levels


spatial


ability


levels


of treatment


(traditional


versus


computer-


enhanced)


student


posttest


performance


procedures


solving


linear


or absolute-value


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.


low-guidance


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


computer-enhanced


of treatment


in student


(traditional


posttest


performance


of the


procedures


solving


linear


or absolute-value


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


trait-treatment


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.


Computer-enhanced


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


half-plane


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


information-pro


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


b-going

and t


in CEI


classroom


o contribute


research promises

instruction which

e to ATI research


both

the

involving


field


mode


spatial-visual


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


computer-enhance d


to describe


instructional


techniques,


(CEI)


procedures,


treatment


criterion


measures


used


to investigate


aptitude-treatment


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


nnm-trnl 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


computer-assisted

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 Embry-Riddle


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


t-test


was


used


test


statistically


significant


differences


mean


pretest


scores


groups.


Appendix


a copy


Pretes


Results


t-test


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 two-week


treatment


procedural


skills


solving


linear


absolute-value


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


two-week


presentation.


An independent


sample


t-test


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


two-week


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


t-test


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.


Spatial-visual


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


instructor-control


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 trait-treatment


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 Vz-factor


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 Vz-factor.


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 spatial-visual


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,


computer-enhanced


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


instructor-control


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


two-week


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


(Vz-2) .


Vz-1


Vz-2


were


produced


Educational


Testing


Service


were


used


as aptitude


measures


Vz-factor


spatial-visual


ability.


(These


tests


have


reliabilities


res


pectively.


French


(1954


recommends


them


as having


high


construct


validity


measure


factor


Appendix


C for


sample


items


from


Vz-1


Subjects


were


allowed


eight


minutes


each


parts


Vz-l


test


three


minutes


each


of the


two


parts


Vz-2


test.


Both


tests


were


admini


steered


scored


according


to specific


directions


provide


ed by


Educational


Testing


Service.


Initial


differences


in pre-requisite


skills


knowledge


pretest


course


(Pretest)


material


earlier


were


unit


controlled


(Test


entering


scores


covariates


in a two-way


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


Vz-l


Vz-2


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


absolute-value


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 follow-up


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







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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


Computer-Enhanced


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


information-proce


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


information-processing


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


computer-enhanced


treatment.


inability


transformations


of low


spatials


readily


to perform


have


cognitive


to their


lower


figural


scores


computer-enhanced


treatment.


There


were


limitations


instruments


procedures


used


this


second


study.


Spatial-visual


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*~


fl-lliv












to study


treatments


effect


on the


of two


performance


instructional


students


different


leave is


of spatial-visual


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 spatial-visual


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


information-pro


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


spatial-visual


ability


(Vz)


and


field


independence/dependence


with


presentation


modes


linear


inequaliti


es.


computer-enhanced


(CEI)


treat-


ment


was


designed


as a preferential


treatment


high


spa-


tial


ability


students.


was


also


designed


as a low-guidance


(discovery)


students.


compensatory


preferential


traditional


treatment


treatment


treatment


spatial


field


was


independent


signed


ability


students.


was


ment


also


designed


field


dependent


a high-guidance


students.


compensatory


remainder


treat-


Chapter


presents


a detailed


discussion


instrumentation,


teaching


methodology,


treatments,


computer


software,


statistical


analy


ses


used


main


study.


Instrumentation


Prior


solving


two-week


linear


presentation


absolute-value


procedures


inequalities,


subjects


both


asses


were


admini


stered


a 60-minute


pretest


on earlier


college


algebra


material.


(The


subjects


main


intact


study


classes


were


freshmen


college


algebra,


students


both


enrolled


taught


same


instructor.


at Embrv-Riddle


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


(Vz-1)


Paper


Folding


Test


were


administered


as measures


of the Vz-factor

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


two-week


presentation,


subj


ects


both


treatment


groups


were


admini


steered


a 60-minute


posttest


over


treatment


material.


point-biserial


correlation


this


teacher r-made


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


absolute-value


teacher--through


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 step-by-step 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


pre-requisite


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


step-by-step


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


step-by-step


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


absolute-value


- B


C orlAy


- BI


graphically.


inequalities


- BI


Solving


form


> C algebraically


graphically


"Abs


oluteValuel"


program


Topic


Solving


a system


absolute-value


- B











Subjects


traditional


treatment


experienced


highly


structured,


expository


press


entation


procedures


solving


linear


or absolute-value


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.


step-by


-step


The

prescriptions


the correct


solution


procedures.


(See


Appendix


C for


examples


step-by-


step


prescriptions


used


traditional


treatment.)


Subjects


highly-visual,


experimental


computer-enhanced


treatment


experienced


presentation


pro-


cedures


solving


linear


or ab


solute-value


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


x-axis


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


low-guidance


(discovery)


treatment,


students


were


expected


to generate


test


con-


lectures


performing


solution


procedures.


Computer-Enhanced


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


II-e


microcomputer


consisted


three


programs


Inequalityl;


Linear


Programming;


Absolute-


Valuel.


"Inequalityl"


program


allowed


students


to scale


x- and


y-axes


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


half-plane


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 built-in


option


allowed


students


see


an algebraic


representation


system


inequalities


under


consideration


desired.


Follow-up


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


built-in


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


Teacher-controlled


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.


Built-in


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.


Teacher-control


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 absolute-value


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 Johnson-Neymann


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 absolute-value


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


B-Group


A-Group


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


computer-enhanced (


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 -


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I~ *Cc


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n rn


r~in i/"


Y


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