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The effect of the availability of The Geometer's Sketchpad on the locus-motion problem-solving performance and strategies

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Title:
The effect of the availability of The Geometer's Sketchpad on the locus-motion problem-solving performance and strategies
Creator:
Robinson, Stephanie O
Publication Date:
Language:
English
Physical Description:
vii, 132 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Art sketches ( jstor )
Computers in education ( jstor )
Constructivism ( jstor )
Geometry ( jstor )
High school students ( jstor )
Learning ( jstor )
Mathematics ( jstor )
Mathematics education ( jstor )
Students ( jstor )
Teachers ( jstor )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1994.
Bibliography:
Includes bibliographical references (leaves 123-131).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Stephanie O. Robinson.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AKN2739 ( NOTIS )
33354956 ( OCLC )

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THE
ON


EFFECT OF THE
LOCUS-MOTION


AVAILABILITY OF
PROBLEM-SOLVING


THE GEOMETER'S
PERFORMANCE AND


SKETCHPAD
STRATEGIES


STEPHANIE


ROBINSON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY


OF FLORIDA














ACKNOWLEDGEMENTS


The


completion


document


and


the


expert


ences


that


precede


d it


have


been


possible


only


with


the


help


many


individual


Kantowski


have


SThe


been


words


a constant


wisdom


source


of Dr. Mary


energy


Grace


and


inspiration


throughout


my graduate


studi


As chairperson


committee


, she


has been a wonderful


rol


model


both


personally


and


professionally


The


experiences


that


gained


during


years


of working


and


with


Elroy


Bolduc


have


provided


a judi


cioUS


balance


between


tasks


and


people.


Thanks


is extended


to Dr.


James


Algina


his


patient


assistance


in the


data


analy


S1S


gratitude


ext


ended


other


members


committee


and


those


who


parti


cipated


at the defense


dissertation


John


Gregory,


. Charl


Nelson,


Eugene


Todd.


Throughout


one


doctoral


studi


the


support


Hlow


doctoral


students


, th


ose


who


have


gone


before


those


who


will


follow


, is


a driving


force


in the


success


comply


etion


program


heartfelt


thanks


and


endship


to Dr.


Katheryn


Fouche


. Thomasenia


Adams,


Sebastian


Foti


, Ms. Virginia


Harder


, Mrs


. Juli








professionally.


Without


the assistance and


encouragement


the


teachers


and staff


at Gainesville High


School


who


participated


in my


doctoral


study,


none of


this would have


been written


I


appreciate


the help of Nancy Galloway,


Paige Lado,


Nanette Greene,


Michelle Sanders,


Jeff Kanipe,


and Jeff Rung.


wish


to acknowledge


the encouragement,


support,


and


love of


my family


, who saw me


through


this experience:


parents,


Pauline and Harold


Osgood


, who never


gave


brother

Carmen,


and

who


sister;

thought


especially my


would never


daughter


finish and


Kalena


get


"real


Finally


, but foremost,


wish


thank my husband,


Robert Robinson,


love


, patience,


and


understanding


as I


fulfill my


dream,


now our


dream.















TABLE


ACKNOWLEDGEMENTS


ABSTRACT


OF CONTENTS


* JLJ


. . VI


CHAPTERS


INTRODUCTION


. . 1


Purpose
Rationa
Outline
Research
Definit
Summary


of the Study
le .


o
h i
io]


. .
. .


f the Study . 1
Questions . 1
n of Key Terms . 1
. 1


REVIEW


OF RELATED


LITERATURE


Cognition and Constructivist
Spatial Visualization .
Geometry .
Summary .

METHODOLOGY .


Theory 16
* 24
* 35
. 42


* 44


Overview


Res
Sel
The
Loc
Sel
Pro
Des
Sta


arc
cti
Spa
s-M
cti
edu
rip
ist


the Study


Ques
of
al V
ion
and


S .
uat
liz
lem
cri


S
ion
ati
In
pti'


of the In
Procedure


* .. 44
* S 45
Instruments 46
on Battery .. 46
ventory 50
on of Population Sample 54
* 55
ructional Phase 55
57


Summary


S. . 58


ANALYSIS


OF THE


DATA


. 60


Research


h


Questions . 6


. J15


page


w








SUMMARY


AND


CONCLUSIONS


. a a a 82


Overview
Results
Limitati
Implicat
Summary


of the Study 82
and Discussion. 83
ons . 93
ions and Recommendations 95
. . 99


APPENDICES


LOCUS-MOTION


INVENTORY


PROBLEMS


SCORE


SHEET


LOCUS-MOTION


INVENTORY


CONSENT


FORMS


INSTRUCTIONAL


ACTIVITY


SHEETS


INTERVIEW


QUESTIONS


LIST


OF REFERENCES


BIOGRAPHICAL


SKETCH


a a a a a a a
*















Abstract of


Dissertation Presented to


the Graduate


School


the


University


of Florida


in Partial


Fulfillment of


the


Requirements


for the Degree of


Doctor


of Philosophy


THE EFFECT OF THE AVAILABILITY


OF THE GEOMETER'S SKETCHPAD


ON LOCUS-MOTION PROBLEM-SOLVING


PERFORMANCE AND


STRATEGIES


Stephanie O.

December,


Chairman:


Robinson

1994


Mary Grace Kantowski


Major


Department:


Instruction and


Curriculum


The purpose

problem-solving


this study was


strategies used by


to investigate


geometry


the


students


following


instruction


in a


dynamic technological


environment.


The study


explored


the relationships


among the


students'


spatial


visualization ability


mathematical


ability,


problem-solving


strategies with


without


the


availability


of The Geometer's


Sketchpad.


Solution


strategies were examined


dynamic visualization as


for tendency to


related


use drawings


availability


technology


during


locus


of points


problem-solving


session.


Three measures of


including


spatial


ETS Card Rotations,


visualization


Cube


were


Comparisons,


taken,


and Paper


Folding tests.


A locus-motion problem


inventory was


administered.


Participants


for the


study


consisted


158








students were randomly


assigned


two groups


for the


Locus-


Motion

availab


Inventory:

le. and th


those who


had the computer


lose who did not.


Individual


software

interviews


were held with


two students


from each


class,


one


from each


group.


availability


the computer was


significant


factor


for performance on


LMI.


Covariates


spatial


ability or mathematics

variance. Teaching the


achievement accounted


specific skills


for most


of drawings


motion


on the computer


resulted


those


strategies


being


used similarly


students with and without


the Sketchpad.


Although


this


indicated


in data


analyses


nonsignificant


differences


implications


are


positive.


Results


suggest


that


strategies


learned with


the


technology


are


transferable


paper


and pencil


situations,


and


that


active


important


participation


to successful


instructional


activities


performance and use of


strategies.


Interview data


revealed


that students


were able


to solve


problems


successfully


on the written LMI


that


they


had


previously missed.


Students were able


to go


beyond


the


known


problems


solve


novel


questions


with


some


discussion of


the components of


the question


itself.


Language and


communication were critical


to expanding the


student's


zone of


proximal


development.













CHAPTER


INTRODUCTION


The


technological


society


today


demands


fundamental


restructuring


the educational


environment.


The


impact


technology provides


two main


issues


mathematics

learning of


education:


mathematics


the classroom


the changing perspectives

and the changing role of


(Mathematical


Sciences


on the

technology


Education Board


[MSEB],


1990) .


Technology


can assist


the


introduction,


development,


and reinforcement of


mathematical


concepts.


the curriculum

opportunities

cognition and


different


computers


for new


calculators offer


conceptualizations of


opportunities


sequences,


learn new


with a higher


instruction and


content,


degree


sophistication at every


level.


The overall


goal


of mathematics


education


students


learn and


experience


power


of mathematics


(National


Council


of Teachers of


Mathematics


[NCTM],


1989).


Understanding mathematical


concepts


the key to


success


power


in mathematics.


However,


the results


evaluations


by the National Assessment


Educational


Progress


(NAEP)


indicate


that


performance by


students








foundations


for mathematical


understanding need


to be


strengthened.

conceptually


A curriculum that


oriented is


is active and


recommended to assist students


gaining the


foundations and


understanding


(NCTM,


1989).


environment


skills of


of exploration and discovery will


inquiry


help develop


and problem solving.


Technology may


the key to unlocking the door


revised mathematics curriculum and


learning


to a


environment.


the


influences that shape mathematics education,


technology


stands


out as


the one with


the greatest


potential


revolutionary


impact"


(MSEB,


1990,


22)


The


technological


classroom must


society


adapt


is a reality


the changing


the mathematics


scientific workplace or


fail


to reach


primary


goals of


providing power


and


opportunities


the students of


today


world


tomorrow.


According to


the constructivist


perspective


learning,


students


build


their


own


interpretative


frameworks


for making


world


sense


(Schoenfeld,


the world,

1987a). In


including the


implementation


mathematical


of a dynamic


technological


environment


that


promotes


visual


and


dynamic


problem-solving


strategies may


influence


curriculum


, the


instructional


techniques,


learning perspectives


the mathematics classroom.


Students construct


their


own


knowledge by


engaging


in problem solving


and


actively







students


new


through


experiences


interaction


in the classroom.


of previous


knowledge with


Providing multiple


ways


in which


to approach mathematics


learning and


problem


solving


allows


students


to build knowledge


from their


own


cognitive


level.


Alternative approaches and modes


thinking need


to be examined


for their


effectiveness


facilitating understanding


concept acquisition.


The geometry


inductive


environment


curriculum with


potential


deductive reasoning provides


for problem solving


and spatial


a rich

thinking.


However,


the


traditional


geometry


classroom,


buried


in proof


and structure,


has


been particularly


conducive


to an


inquiry


approach.


A modified geometry


curriculum may


encourage a more


versatile,


integrated


approach


learning.


The


use of


computers and dynamic modeling


has drastically


changed


the nature of


mathematics


education


algebra


(Hershkowitz,


1990).


Such a


change


is needed


in geometry


may


be promoted by


combining


observations


intuition


with multiple


representations


in a


dynamic environment.


Purpose


Study


The


purpose


this


study was


investigate


the


problem-solving


strategies


used by


geometry


students


following


instruction


in a


dynamic technological


environment.


Specifically


, the study


explored


the








strategies with and


Students were


without


instructed with


the availability


presentation


technology.


"hands


experiences using a software


product


called The Geometer's


Sketchpa d


(Jackiw


, 1993) .


Specific


lessons


locus-motion


concepts


provided the basis


exploring problem-solving


performance


strategies.


Solution strategies


were


examined


for tendency to use drawings and


dynamic


visualization as


technology


related to


during the


the availability


problem-solving


the


session.


Rationale


To engage


in mathematics


is to participate


activities


of problem solving.


Learners


have


to construct


their


own knowledge,


through


a process of


reflection


(Davis,


Maher


Noddings,


1990;


Noddings,


1990; von Glaserfeld,


1987)


process


of purposeful


activity


(Krutetskii,


1976;


Sowder,


1989).


Metacognitive aspects of


learning,


such


as self-monitoring,


pertinent

explicit


to efficient


instruction


-regulation,


construction

the curricul


-evaluation,


of knowledge

um (Silver,


are


and require

1985)


Constructivism


emphasizes


role of


construction


process as


well


as the awareness of


that process and how to


modify


(Confrey


, 1990).


This study


attempted


investigate


the effect of


a specific


learning


environment


the content and nature


students'


learning.








future


theories and


practices


(Balacheff


et al.,


1990;


Sowder,


1989).


The Cognitive Flexibility Theory


learning


, knowledge


representation,


and knowledge


transfer


acknowledges


the


ability to


restructure


one's knowledge spontaneously


in many


ways


in response


to changing


situational


demands


(Spiro


Jehng,


1990).


Revisiting the


same material


from multiple


perspectives


the key to


learning


and


transfer


in complex


domains


(Spiro et


al. ,


1991) .


Cognitive


Flexibility


combines


the constructivist


nature of


learning


patterns


learning


conceptual


failure to examine dynamic

perspectives (Spiro et al.,


cases

1991).


from multiple

Use of


dynamic


cases


in this


study


added


to the


knowledge


concerning


students'


response


to multiple


representations


novel


situations


the complex domain


of mathematics.


Application of


the Cognitive


Flexibility Theory util


izes


the


random


access


capability


the computer to


provide


flexibility


and


pluralistic representations


(Spiro


Jehng,


1990) .


Computer


environments


are


an ideal


tool


to provide


multiple


representations


in mathematics,


allowing


one


shift


from one representation


to another to


find


the


form


most


useful


(Dreyfus,


1990;


Fey,


1989;


Kaput,


1987,


1989;


Senechal,


1990;


Sowder,


1989) .


"Interactive


technologies


provide a means of


intertwining multiple


representations


mathematical


concepts and relationships--like graphs and








aspects


computer


environment need


to be examined,


including


how


knowledge


affected by the environment and


how students'


cognitive behavior and


constructs


are modified


(Balacheff


et al.,


1990).


The


interactive computer


revolutionizing the


study


environment


is also


of shape and visualization


within


the


study


of mathematics


(Senechal,


1990;


Tall


Thomas,


1989;


Tillotson,


1984/1985).


"Today the microcomputer


increasing the range of


presence


stimulating


aids


in mathematics


a great deal


visualization


classrooms


enormously


is also


research and development


in this


area"


(Bishop,


1989


, p.


Spatial


ability


often a


controversial


subject


when discussing mathematical


performance and achievement.


The definition


of spatial


ability


the


number


of components


factors


involved,


the


relationship of


spatial


ability to


problem solving


and


mathematical


performance,


ability to


learn


increase


spatial


skills are all


topics


that


require


continued research and


investigation


(Steen,


1990).


Some


researchers


suggest


that


spatial


ability


unitary trait


(Johnson


& Meade,


1987;


Moses,


1984)


Others


distinguish


Tillotson,

distinct c

subject tc


(Conner


1984/1985) ,


components.


instruction


Serbin


or three

Whether s


, 1980;


(Linn


patial


is also an


Tartre,


Petersen,


ability


issue.


Many


1990;


1985)


innate


researchers







1977/1978;


Tillotson,


1984/1985;


Vinner,


1989).


Furthermore,


instruction


is not


only


desirable,


but


necessary to


improve spatial


skills


(Dreyfus,


1990;


Krutetskii,


1976) .


These


topics will


be explored


further


the


literature review.


Students


skill


need


of spatial


explicit


visualization,


instruction not only


also


the


in the ability to


monitor their


own endeavors


in the


learning process


(Dreyfus,


1990;


Silver,


1985;


von Glaserfeld,


1987).


Students


need


to attend to the metacognitive


aspects


learning:


self-monitoring


cognitive activity


(Schoenfeld,


-regulation,


1989;


and


Silver,


-evaluation

1985).


Constructivism


emphasizes


that


students


can


learn


improve


their reasoning


and problem-solving


ability


and become


agents of


their


own


learning


(Paris


Byrnes,


1989;


Zimmerman,


1989)


This


investigation explores


whether


instruction


solving


can


strategies


influence awareness of


in novel


dynamic problem-


problem situations.


Improvement


intertwined with


assessment


the mode of


mathematical


instruction


ability


activity


in progress.


A change


in assessment


that more closely


correlates


becoming


to instruction


clearer that a


representative


process


needed


single


of student


in terms of


product


(NCTM


static lev

learning, e

(Campione,


, 1989).


rel


assessment


dither i

Brown,


.n terms


Connell,








the current knowledge


level


the student,


leading him


or her


into new mathematical


territory


(Campione


et al.,


1989;


Ferrara,


1987/1988;


Lampert,


1991)


SThe difference


between


what a


student


can do alone and what he


or she


can


do with


the assistance of


a teacher


or more capable


peer


(Rohrkemper


zone


, 1989)


of proximal


been


development.


termed by Vygotsky


Placing the emphasis


as the


on the


initial


and


possible


learning


states


student,


the


environment


with


in which


the constructivist


learning


is taking place,


perspective of


cognitive


is consistent

e change


(Linn


Songer,


1991) .


The


variety


tasks


offered


this


study


introduced students to


locus of


points


and


extended


the


student's


knowledge of


problem-solving processes


beyond


the


practiced


level


to the


next


dynamic


level


thinking.


Although


it has


been


proposed


that knowledge


constructed by the


learner


that mathematics


is a complex


domain,


research


needed


to provide descriptions


these


theories


applied


to specific


domains


(Hiebert


Wearne,


1991;


Wearne


Hiebert,


1988)


According to


Kantowski


(1981),


problem-solving proce


sses


depend


as much


on the


type


of problem as on


style of


solver.


Krutetskii


takes


this


idea


one step


further to state


that


specific content


ability


depends on


instruction.


ill-structured


domain


of mathematics


terms.


A well-defined


broad

unit


to be


investigated


in geometry


such


in global


locus of







use


of dynamic


technology


a constructivist


learning


environment.


content


area


locus


of points


was


chosen


because


the


subject


independent


of placement


the


curriculum


(provided


that


sic


concepts


have


been


introduced)


often


left


the


geometry


curriculum


due


time


constraints.


The


topic


locus


of points


enhanced


a dynamic


environment


use


of diagrams.


The


computer


has


the


potential


to provide


fruitful


environment


exploring


diagrams,


conj


ecturing


searching


patterns


and


generaliz


nations


(Bishop,


1989;


Hershkowitz,


1990;


Janvier,


1987;


Lesh,


1987;


Senechal,


1990;


Steen,


1990) .


Two


issues


are


raised


the


use


of calculators


and


computers


in the


classroom:


understanding,


finding


and


a balance


problem-solving


of conventional


ability


skill,


that


appropriate


new


technology


discovering


ways


that


technology


ens


entirely


new


approaches


thinking


about


mathemati


ideas


problems


(Fey,


1984).


Emphasis


geometry


may


shift


to experimentation


with


shapes


relations,


inquiry


building

inherent


strong


geometric ir

mathematics.


tuition


The


and


ins


spirit


tructional


phase


this


study


explored


use


of a relatively


new


teaching


technique


medium


geometry


assroom,


providing


some


insight


into


ways


curriculum


might


change


and


techniques


of designing


structuring


new


learning








Many


researchers


agree


that


computer


environments


provide


multiple


representations


, important


flexibility


thought


problem


solving


(Dreyfus,


1990;


Kaput,


1987;


Sowder


, 1989;


Tall


Thomas,


1989).


However


res


earchers


must


concentrate


environment


on determining


on what


how


effect


students


this


learn


learning


(Balacheff


1990;


Sowder


1989


Although


Sowder


(1989)


referring


study


of algebra,


geometry


the


same


sentiment


holds


true:


"Not


much


research


been


done


evaluate


programs


what


students


or how


learn


programs


interacting


might


be used


with


within


the


the


context


. curriculum"


Outline


Study


Thi


study


was


designed


as a


"cons


tructivi


teaching


experiment"


(Cobb


Wood


Yackel


, 1990)


investigate


problem-solving


following


strategies


instruction


used


in a dynamic


geometry


students


technological


environment


to explore


relation


ships


among


students'


spatial


visualization


ability


, mathematical


ability,


problem-


solving


Sketchpad


strategies


This


with


study


without


addressed


availability


some


the


needs


research


a domain


-specific


content


area


locus


points.


Three


independent


measures


spatial


visualization


al.,








Card Rotations, Cube

locus-motion problem


Participants

students fro


Comparisons,


inventory


for the study


m seven geometr


and Paper


(LMI)


consisted

y classes


Folding.


was also administered.


158 geometry


at one high school.


All


students participated


in the


instruction


using The


Geometer 's


Sketchpad.


The


instructional


phase


included


lessons


that


extended previous


lessons


on the Sketchpad


introduced


the concepts


locus of


points.


Students


were


randomly


assigned


to two groups


for the Locus-Motion


Inventory:


available


sess


those who would have


those who would not.


ion followed a


the computer


The


constructivist approach


software


problem-solving


of creating


"problematic environment


that would


elicit


the


student's


adequate constructive endeavors"


(Fischbein,


1990


, p.


Data


analysis


had both


a quantitative and


a qualitative


component.


Research Questions


The


study


addressed


three questions of


interest:


Are


correct


response


score


(CRS),


tendency to


use drawings


, and/or use of


dynamic


strategies


(DS)


that


students


utilize


to answer


locus-motion


problems


related


students'


spatial


visualization


index


(SV)


mathematics


achievement


(MA)?


Does Sketchpad


availability


(SA)


affect


correct


__








Are


tendency to use drawings


(DR)


and/or


use


of dynamic strategies


(DS)


that


students


utili


when


attempting to solve


locus-motion


problems


related


students


' correct response score


(CRS)?


Definition


of Key Terms


Soatial


visualization,


or spatial


ability,


is defined


this


study


as the ability to recognize


the


relationships


among the elements


manipulate mentally


a given


figural


one or more of


configuration,


those


parts.


Spatial


visualization


index


(SV)


is operationalized


the


sum of


the


z-scores,


or standardized


scores,


the


three


tests


in the


Spatial


Visualization Battery.


Mathematics achievement


(MA)


score measured


on a


geometry


semester


examination given


to all


participants.


Locus-motion


inventory


(LMI)


is a set


problems


relating to


locus of


points.


Nine


problems were


used


the written


portion


LMI,


five


problems were


available


interviews.


Correct


response


score


(CRS)


, or


locus-motion


problem-


solving performance,


is the


number


of correct


responses


written


LMI.


Locus-motion problem-solvinq


strategies,


or methods of


solutions,


are scored


in two ways:


tendency to


use


drawing


(DR)


use of


dynamic problem-solving








Dynamic


software,


or dynamic technological


environment,


refers


to The Geometer's Sketchpad


(Jackiw


, 1993)


implemented


on an


local


area network.


Summary


Mathematics education


in the


process of


change,


change,


indicated within


the Standards


that


recommends


new


curricula


and modes


instruction and provides


mathematical


power to


students.


innovations


are


proposed,


planned,


implemented,


investigation


into


their


feasibility


and


effectiveness


is needed.


Technology


provides


and


one vehicle


the methods employed


This


chapter presented


for variations


in the content addressed


the mathematics


the rationale


classroom.


a study


designed


to examine


the effect


instruction


in a dynamic


technological


environment


on performance and strategies


employed


in attempting


locus-motion


problems.


The


investigation


included


the exploration


of relationships


among the students'


spatial


visualization ability,


mathematical


without


ability,


and problem-solving


the availability


the Sketchpad


strategies with


during problem


solving.


The


research


questions


pertaining to


the


study,


well


as the key


definitions,


have


been


outlined here.


Chapter


the questions


contains a


* .


interest.


review


focusing


literature

on previous


pertaining to


research








procedures


utilized


study


found


Chapter


Chapter


presents


results


data


analy


ses


The


findings


their


implications


instruction


and


research


are


found


Chapter














CHAPTER


REVIEW


OF RELATED


LITERATURE


The


theoretical


premise


that


supports


this


study


that


the


use


of a dynamic


technological


environment,


such


The


Geometer'


Sketchpad


, can


enhance


construction


knowledge


students


influence


problem-solving


strategies


using


dynamic


visualization


skills


The


constructivist


eory


learning


promotes


the


beli


that


students


build


their


own


knowledge


through


activity


and


experience


Dynamic


software


can


provide


experien


ces


visualizing


, conjecturing


, and


building


frameworks


within


specific


content


domain.


Experience


will


not


only


influence


students'


ability


to perform


success


fully


on the


spec


ific


content


problems


activities


, but


use


solving


problems


that


are


nove


and


perhaps


more


diffi


cult


The review


literature


presented


thi


chapter


there


fore


, focuses


on studi


research


pertaining


cognition


cons


tructivist


theory,


spatial


visualization,


and


curriculum


instruction


in geometry


The


connections


between


constructivism


Cognitive


Flexibility


Theory


are


explored.


Within t


sec


tions


on cognition a


d geometry,








CoQnition and


Constructivist


Theory


Mathematics educators generally


agree


with


the basic


tenet


of constructivist


theory that


learners


construct


knowledge within an active environment.


Krutetskii


states


that


"mathematical


abilities


exist


only


a dynamic


state,


development;


they are


formed


developed


mathematical


activity"


(1976,


brain


is not


passive,


active


, and


engages


in processes


selection,


interpretation,


inference


(Orton,


1987).


Noddings


(1990)


summarizes


views of


constructivist


theories


follows:


knowledge


is constructed


Mathematical


knowledge
process of


is constructed,


least


in part,


through


reflective abstraction.


There exist


activated


structures account


explain


the result


cognitive


processes of


structures


that are


construction.


for the construction;
of cognitive activity


These


that


they


in roughly the


way


a computer program accounts


for the output


computer.
3. Cognitive
development. P


structures


urposive


under


activity


continual


induces


transformation


existing


structures.


The environment


presses


the


organism


to adapt.


Acknowledgement


of constructivism


as a cognitive


position


leads


the adoption of


methodological


constructivism.


Methodological


constructivism


in research


develops methods of


study


consonant with


assumption


cognitive


constructivism.


b.
teaching


Pedagogical
consonant w


constructivism suggests methods


ith


cognitive constructivism.


An example of how


students


build knowledge


can


seen


in a study


-


Davis


in a


fifth


parade


class


(Davis


& Maher


I


__







pizza.


two


Detailed


boys evinced


thinking,


investigation


solution


the constructing


layers of


conversation.


solution


the


processes


in layers


two students


interacted with


each


other


and with


various


solution


approaches


, the


successful


solution emerged


from


use of


concrete


representation.


students


broke


problem


down


into parts


, the


tendency was


to look at


the


parts


separately


, not at


the whole


picture.


The


building


concrete representations may


have allowed


the


students


concentrate on


the parts


, but


still


be aware of


entire


problem setting.


"Put


into


simple


terms,


constructivism can be described


as essentially


a theory


about


the


limits of human knowledge,


a belief


that all


knowledge


is necessarily


a product


our


own


cognitive acts"


can be,


and need


(Confrey,


to be,


1990,


planned by the


108) .


learner


Cognitive acts


or by


instructor.

supported by


Following

student i


a five day


interviews,


study with


Confrey


one


develop


teacher,

d a model


teacher'


constructivis


instruction that supports

m. Constructivist models


pedagogical


teaching recommend


learning


environments


that


promote


process


ses


including


acquisition


of basic


concepts,


algorithmic


skills,


problem-


solving


heuristics,


and habits of


reflective


thinking


(Davis,


Maher


Noddings,


1990)


One


le skills


that must


be developed


flexibility


One








representational


systems


that


they


instinctively


switch


the most


convenient representation


to emphasize


any given


point


solution


process"


(Lesh,


Post,


Behr,


1987,


Successful


problem solvers are able


to build


mathematical


representations


of problem


situations


relate


problems together that have a


similar


structure


(Schoenfeld,


1987a).


Examining the skills


that


successful


problem solver


the gifted student


possess


reveals


that


their


thinking


process may


be different


(Dover


Shore,


1991) .


Dover


Shore


(1991)


studied


11-year-old


students


them


school-identified gifted and


11 of


average


ability)


focusing


on set-breaking with


the water


jug problem.


Results


showed


a three way

flexibility


interaction among giftedness,


The gifted


speed,


students exhibited more


and

planning,


more


reflection


, and more


flexibility.


Successful


learners construct


these


metacognitive


representations at


representation


for the


same


time


problem.


that


Davis


they


build


(1986)


conducted


three diverse


studies


that


yielded similar results.


One


study


involved


a mathemati


teacher


solving


a problem not


in his

musical


area


expertise.


theme.


A second study


third study was a


looked at


task-based


interview


a university


calculus


student


working


a quadratic equation.


In each


situation


'J


representations


were built:


one


38) .







metalevel,


an observation of


the method


resolving the


question.


These and


other metacognitive


skills must be addressed


directly


and students


taught


to recognize and apply them in


the


proper situations


(Campione,


Brown,


Connell,


1989;


Hershkowitz,


1990;


Janvier


, 1987;


Lester,


1989;


Schoenfeld,


1987).


Hembree


(1992)


found similar


conclusions


in a


meta-


analysis of


487


reports and studies


in problem


solving.


Results


instructional methods


on problem solving revealed


that


students who received


instruction


in problem solving


and heuristics


explicit


had higher


training.


scores


Effects of


than did


those with no


classroom-related


conditions


were


also examined.


Computer


assisted


problem


instruction


provided better results


than did


paper


and pencil


problem


solving.


Teaching the


specific subskill


of diagram drawing


also


showed


positive


results.


"Taken


together,


these


findings


suggest


that


the dominant


factor


in problem solving


less


IQ than mental


development"


(Hembree,


1992,


268).


However


, Resnick


(1989)


offers a


caution about metacognitive


training:


This


points


metacognitive


to a


fundamental


training


efforts


problem with


that


focus


certain
attention on


knowledge about


problem


solving rather than


on guided


and
Such
talk


constrained practices


efforts may


about


be more


processes


in doing problem solving


likely to produce


functions


than


ability to


to perform


them.


Several


studies


cited by


Campione et al.


(1989)


that








given explicit


instructions and did not attend


their


own


processes

included a

aimed at p


learning.


learning phase,


promoting transfer


Campione et al.


(1989)


One study


followed by

Dynamic


and Ferrara


Ferrara


dynamic

tests,


(1987/1988


(1987/1988


C


assessment


as described by

), provided


information about


the ability


the student


as well


as the


student'


potential


improvement.


Assessment,

current knowledge


like i

level


instruction,


should begin at


the student and


lead him


the

or her


into


new mathematical


Ferrara,


1987/1988;


territory


Lampert


(Camp


, 1991) .


lone et al., 1

The difference


989;

between


what a


student can do alone and what he or


she can


do with


the


ass


instance of


a teacher


or more capable


peer


(Rohrkemper,


zone


1989)


of proximal


been


development.


termed by Vygotsky


Schoenfeld


(1978)


describes


as the


this


process


learning:


"One acquires


higher


order thinking


skills


exercising those skills


ZPD with


the help


others and


then


internalizing those


skills,


that


mastering


as an


individual


those


skills


for which


one,


once,


needed


support"


(1987b,


210)


In a


case


study


from


"Computer


as Lab


Partner"


project


, Linn


and Songer


(1991


referred


this difference


as the


range of


possible


cognitive changes.


experiments,


Students


simulations,


interacted with


instruction.


computer-based


Results


indicated


that,


with proper guidance and active engagement








previously


able


to do.


Focusing


on the


zone


of proximal


development


Salomon


(1989)


fostered


the speed


conceptual


produced similar results


in a


change.

study c


reading-related metacognitive guidance with seventh


grade


students.


three


Seventy-four


treatment groups w


program with


varying


students were

ith three ver


levels


randomly


sons of


of guidance


assigned


a computer


varying


cognitive


levels


of questions.


Salomon reached


conclusion


that


"computers can


serve as


tools


that


provide


guidance


in a


child'


zone of


proximal


development"


(1989,


626).


The


studies


Linn and Songer


and by


Salomon reflect


similar


cognitive constructs


those


found


the


Cognitive


Flexibility Theory.


The Cognitive


Flexibility Theory


learning,


knowledge


representation,


and knowledge


transfer


provides


for multiple representations of


the same


material


in rearranged


instructional


sequences


from different


conceptual


perspectives


(Spiro


Jehng,


1990).


The goal


to be able


to apply


independently the


learned knowledge and


processes


new


situations.


An application


this


theory


is seen


in a


project


described by


accessed


Spiro and Jehng


videodisc of


(1990


in which


provided


a random


foundational


instruction

scene-based


materials.


explorations


Students engaged

A cognitive fle


theme-based


xibilit


-


Citizen Kane


w








Commentary was


provided


to expound upon


thematic and


symbolic contrasts within


the scene.


The goal


was


only


to explore


themes and


symbols within Citizen Kane,


also


to demonstrate


complex nature of


literature


provide


students with active experiences


with


processing


that


complex knowledge


and building new metacognitive


skills.


Many


principles of


Cognitive


Flexibility Theory


are


also


basis


constructivism


(Noddings,


1990;


Spiro


et al.,


1991) .


Students


interpret knowledge,


absorb


Active


participation and


exploration


by the


student


crucial.


An environment of


exploration allows construction


of knowledge


situations


with


the


transfer


ultimate goal


of knowledge and skills

SThe metacognitive as


new


pects


constructing


activity with


knowledge


learner


important


control


for transfer.


Purposive


self-regulation builds


new


frameworks.


Those


activities must


provide multiple


representations

Constructivist


the concepts


tenets outlined by


within


the domain.


Paris and Byrnes


(1989)


are


also


reflected


research by


Spiro et


(1990,


1991) .


Instructional


innovations


that are


task-oriented


are


needed


to promote


increased metacognition.


Metacognitive


skills


are a


necessary


characteristic


of good


problem


solvers,


flexible


thinking.


According to Cognitive


Flexibility Theory,


instruction


__~_







multiple

However,


representations of


important


the same concept


to avoid


or theme.


compartmentalization


information,


narrowing the


focus of


the


idea.


Revisiting


the


same material


multiple times,


in multiple ways


promotes


flexible


thinking


by the student.


Mathematics and problem solving


are considered by many


researchers


(and many


students!)


to be complex


or ill-


structured


domains


(Davis,


1986;


Polya,


1957,


Resnick,


1989).


As a


complex


domain,


mathematics can


be approached


from the


Cognitive


Flexibility theoretical


basis of


knowledge representation and


transfer.


Linear presentation


not


sufficient


for understanding;


multiple


representations


are


necessary to


provide


a network


interrelated


ideas


(Romberg,


1988;


Spiro


& Jehng,


1990).


single conceptual


perspective


inadequate (Spiro et al.,

representation of dynamic

transformations, movements


is not


1991). "T

situations


incorrect


he area


involving <


is particularly


, just

the

g action,


intriguing


corresponds


to domains where children


encounter


learning


difficulties"


(Dufour-Janvier


, Bednarz,


Belanger


, 1987


, p.


122


"The dynamic and


interactive media


provided


computer


software make gaining


(traditionally the


mathematicia


intuitive


province of


understanding


professional


interrelationships among


graphic








that


the dynamic display


of multiple


representations


valuable not


only to discover the


relationships,


also


learn


skills of


translating


from one


representation


another,


building


1989).


consonant with


, interpreting,


"The computer


the constructive


perspective


relating knowledge


appears able


(Resnick


to offer qualitatively


new thinking tools mainly through


its graphic


facilities


the


possibility


synchronous


representation


different


but related processes


and situations"


(Balacheff


et al.,


1990,


145) .


Spatial


Visualization


Many


studies have


investigated whether


spatial


ability


is a single t

investigating


spatial


rait


or a composite of


gender


ability


differences in d

Johnson and Meade


traits.


While


developmental


(1987)


patterns


considered


spatial


ability to


be a


unitary trait


Moses


(1977/1978)


investigated


the composition


of spatial


ability


and


came


same conclusion.


Moses described


spatial


ability


as the


ability


perceive


the essential


relationships among the


elements


a given


visual


situation,


to mentally


manipulate one or more of


these elements"


(1977/1978


, p.


The


purpose of


study


conducted by Moses


(1977/1978)


spatial


was


ability


-fold:


investigate


to refine


the definition


relationships


between


18) .








instruction


on spatial


ability,


problem-solving performance,


and


degree


visuality.


study


fifth-grade


students


in four


classes were given


same battery


tests measuring


spatial


ability


and problem solving,


prior


to and


following the


instructional


phase.


The


spatial


tests


were


Punched Holes


Test,


Card Rotations Test,


Form Board


Test


, Figure Rotations


Test,


Cube Comparisons


Test.


Problem Solving


Inventory


constructed by Moses


was


the


sixth


test.

lessons


The


instructional


involving two-


perceptions and


some


phase consisted


three-dimensio


problem-solving tasks


of nine weeks of

nal geometric


using visual


solution processes.


Correlation


coefficients,


factor


analyses,


and regression analyses were


performed


investigate


the decomposability


of spatial


ability.


Pearson


product-moment correlation


coefficients were


used


to analyze


relationships


among the


variable


es.


Analyses of


covariance


were


performed


to analyze


the effects


instruction.


Conclusions


were


reported


follows:


Spatial


ability


ability to manipulate


to manipulate


is not
an enti


parts


fi


decomposable int
re figure and th
gure. Moreover,


:o


the


ie ability
some of


tests


which have


been classified as


spatial


tests


can
are


be solved


pure


spatial


in an analytic manner while


other tests


tests.


2. A]
ability,


.though
it is


spatial
a good


ability
predictor


is a


general


cognitive


of problem-solving


performance. .
3. An individual


with high spatial


ability will


frequently not write down


visual


solution


processes


part


of his


solution.


Problem-solving performance


is best


-


edicted by


C .


.. ... w








Spatial


ability


instruction will affect


is a modifiable quantity,


the spatial


ability


1.e.,
an


individual,


affecting males more


Instruction


in certain


than


visual


females.


processes does


significantly
individual.


affect


the degree of


visuality


Instruction aided neither males


of
nor


females on


their degree of


visuality


scores.


Instruction
significantly


performance of


affect,


in a


in certain visual


affect


the general


an individual;


positive manner


however,
success


processes


does


problem-solving


it does
on spatial


problems .
9. Instruction does


significantly


affect


the


problem-solving performance of males


nor


females.


Instruction has


the same amount of


effect


on the


problem-solving performance of both high


and


spatial
10. I
spatial
problems


ability


individuals.


instruction significantly


problems more


(pp.


affects


success on


than success on analytic


144-154)


Other researchers have distinguished


two or three


distinct


components


(Conner


Serbin,


1980;


Linn


Petersen,


1985;


Tartre,


1990;


Tillotson,


1984/1985).


Components


have


been


termed by researchers as spatial


orientation,


spatial


visualization,

relations, and


spatial


I kinesthetic


perception,


imagery


mental

(Conner


rotation,


Serbin


spatial

, 1980;


Linn


Petersen,


1985;


Tartre,


1990


; Tillotson,


1984/198


"Spatial


visualization


is distinguished


from


spatial


orientation


tasks


identifying what


is to be moved;


task


suggests


that all


or part of


a representation


mentally moved


altered,


is considered


a spatial


visualization


task"


(Tartre,


1990


, p.217).


Tartre continued


describing


spatial


orientation as


"those


tasks


that


require


that


subject mentally readjust her


or his


perspective








Tartre


(1990)


used


these definitions as


she explored


role of


spatial


orientation


component


in the


solution


of mathematics


problems with 57


tenth-grade


students.


The


sample of


students was


chosen from those who


scored


the


top or


bottom third


on the Gestalt


Completion


Test.


According to Tartre


this


test


was chosen because


was


best


test


to capture


the essence of


pure


spatial


thought.


holistically


That


the tasks would be


, it appeared unlikely that


solved


verbal


analytic


processes


would


contribute to subjects'


solutions,


and


items directly required the structural


organization


visual


information


in order to make


sense out of


the


partial


pictures"


(1990,


220) .


problem-solving


interview


consisted


10 mathematics


problems,


geometric and


nongeometric,


Students


that


were asked


could be

to solve


solved

the p


in more


problems,


than


talking


one way.


aloud as


they


did.


Interviews were recorded


and


later


coded


according


following


categories:


Correct


answer


, Done


like,


Failure


to break


Mental


movement,


Misunderstood


problem,


Added marks,


Drew picture


, Drew relation,


and


Estimate error.


Tartre


concluded


that


"spatial


orientation


skill


appears


to be used


specific and


identifiable


ways


. accurately


figure,


estimating the


demonstrating the


approximate magnitude


flexibility to change an


unproductive


mind


, adding marks


to show mathematical








without help


to a


problem in


which


a visual


framework was


provided"


(1990,


. 227).


However,


like other researchers


(Fennema


& Tartre


1985


Ferrini-Mundy,


1987;


Kantowski


, 1981;


Krutetskii


1976;


Moses,


1977/1978),


Tartre


(1990)


noted


that spatial


skill


, and


other mathematical


skills,


may


linked


to more


general


thought


patterns when making


sense of


new material


and may


be directly related


the specific mathematical


skill,


test,


or activity


in progress.


To discuss spatial


ability


in terms


way


in which


individual


solves


or thinks through


the


problem


basis of Krutetskii


(1976)


analysis of


mathematical


ability


in general.


Krutetskii


(1976


described


two


different modes of


thought:


analytic-logical


and visual-


pictorial.


In his


analysis,


these modes of


thinking


corresponding


abilities


were


interconnected


with


the


mathematical


activity


in progress.


Other researchers


have


reached


a similar


conclusion


that


type


activity


is an


important


factor


(Fennema


Tartre,


1985


Ferrini-Mundy,


1987;


Kantowski


, 1981;


Moses,


1977


/1978;


Tartre,


1990).


Thus


, the


issue


the definition of


spatial


ability


component


traits may


further


confused


as the activities


thought


processes


in which


students engage


become


more


dynamic with


of abilities,


use


one means


of computer technology.


psychological


speaking


characteristics








person's activity.


We must stress


that


in analyzing


skills and habits as well


as ability


, we are analyzing


activity"


(Krutetskii,


1976,


71).


Tartre summed


up the discussion of


the definition and


components of


spatial


ability.


Attempting to understand


discuss something


like


spatial o
intuitive
The very


disperses
verbalize
ceases to
mental ac


orientation skill,
and nonverbal, i


act of
it.


the


reaching


which
like


is by


definition


trying to grab smoke:


to take hold


It could be argued


processes


be spatial


tivity.


Any


involved


thinking.


that any
in spatial


Spatial


attempt t
thinking


skill


use


evidence about how


manifested must be


indirect,


since we cannot


get


into


people'


heads


see


what


they


see


in their mind'


eye.


Often,


processes


involved


are


not


even


understood by the


people experiencing them.


The


resulting
does set


indirectness of


limits on


research


it but should not


in this


area


curtail


spatial


skills are


important


to mathematics,


then


researchers must


specific roles
mathematics.


find ways to


that spatial


(1990,


identify


skills play


and describe


in doing


229)


Although


solvingsolving


questioned


the relationship of


and mathematical


, many researchers have


spatial


ability


found


skills


also


that


to problem


been


a correlation


does exist


between


visualization and mathematics


(Battista,


Wheat ley,


Talsma,


1989;


Ferrini-Mundy,


1987;


Tillotson,


1984/198


Usiskin,


1987;


Vinner


, 1989).


Those


researchers


who are


mathematical


pursuing the


convinced


the direct


achievement


relationship of


however


spatial


correlation


, acknowledge


skill


with


that


and mathematics


worthwhile


(Fennema


Tartre,


1985;


Lean


Clements,


a~* a m. a t SS


IA Aa


*f *








. did


not determine


extent


of mathematical


giftedness,


(1976,


but did


315)


determine


Lean and


type,


Clements


or cast


expressed


of mind"


need


additional


investigation


"before


relationships


between


spatial


ability


and mathematical


performance can be


clarified"


(1981,


277).


Background review


the study


by Tartre


(1990)


previously


discussed


evinced


beliefs


by many


researchers


, such


as Fennema


and Sherman


(1977),


McGee


(1979),


Conner


and Serbin


(1985),


that


spatial


skills


are


related to mathematics


Ferrini-Mundy


differences


based an


learning


achievement.


investigation of


achievement and spatial


ability


gender

y of calculus


students


upon


premise


that


there


"a well-established


finding


well


of male superiority


as correlational


on tests of


logical


spatial


support


ability,


a relationship


between mathematics


performance and


spatial


ability"


(1987


126) .


Primary


questions


interest were gender


differences


the effects


training program on


calculus achievement


and spatial


ability.


sample


students


included


in three


large groups and


each


four


smaller


groups.


Over


an eight-week


period,


treatment


groups


viewed


six slide-tape modules


taped


commentaries


with


a variety


tasks


situations


with


spatial


visualization and


orientation.


Control


groups


participated








achievement


or spatial


visualization ability.


There


were


interaction effects


calculus


achievement and


the


use


visualization


problems and a

visualization


significant


solids


solving


solid-of-revolution


treatment


of revolution"


effect


(Ferrini-Mundy,


1987,


126)


Ferrini-Mundy


(1987)


suggested


that


significant


training


effects might have resulted


a wider variety


spatial


tests


had been


used


to measure


spatial


ability,


since only the


Space Relations


Subtest


, Form T


the


Differential Aptitude Test was


used.


There were


indications


that


training may


be more


successful


for women


than


for men.


Calculus content and


spatial


visualization was


also


topic of


consideration


in a


study


by Vinner


(1989)


college students.


The course was designed


to emphasize


the


visual


aspects of


every


algebraic concept


theorem.


Comparisons


algebraic versus visual


proofs


indicated


that


students chose algebraic proofs


even when drawings


provided


a visual


proof


upon examination.


Vinner


concluded


that


students


believe


that algebraic proofs are more


mathematically


acceptable,


and memorization


formulae and


algorithms more successful


in assessment


situations.


"Thus,


seems


that


there


is no research


evidence


that


visual


thinking


is not


needed


success


in higher mathematics"


(Vinner,


1989


, p.


150)


Presmeg


(1986)


reached


analogous conclusions


to those








learner to excel


in mathematical


performance.


An initial


investigation by


Presmeg


1985


of mathematical


"stars"


chosen by teachers revealed


nonvisualisers.


cognitive modes


students who were


that


Presmeg researched


, attitudes,


they were


predominantly


the effect


actions


"visualisers.


teacher


upon high school


thirteen


teachers were


grouped according to


visuality


their teaching


seniors


scored


for their mathematical


visuality,


students chosen as


visualisers


were


selected


task-based


interviews.


External


factors


, such


as time


constraints


school


testing procedures and


teaching methods


and


textbooks


that


favor the nonvisualiser


contributed


the


"preponderance


of nonvisualisers amongst mathematical


high


achievers" (

hypothesized


Presmeg,


that


1986,


visual


305).


Presmeg


teachers could


teach


also

visual


students


more effectively than


intermediate


or nonvisual


teachers.


Regardless of


the disputable


nature of


spatial


ability


relationship to mathematical


achievement,


increasing


the awareness and


ability


of spatial


visualization and


spatial


thinking may


benefit


students


aiming them


toward


goal


of mathematical


power.


In view


fact


the difficulties


that most


associated


teachers


with


visual


are


unaware of


processing


mathematics,


may


be overcome,


fact


seems


that


likely that


these difficulties


increased


teacher


awareness of


these


issues could aid


visualisers







Although some


studies


have


not reached significant


results on


the


trainableness


of spatial


ability


, Ben-Chaim,


Lappan,


and Houang


(1988)


provided


data


that supported


possibility that


these skills can


taught,


can be


learned.


A sample of


involving


1000 middle

teachers D


school


anticipated


students at


in a


three


study


sites


of gender


differences,


spatial


grade differences,


visualization ability.


effect of


Immediately


instruction


before and after


three week


spatial


visualization


unit,


students were


administered


the Middle Grades Mathematics


Project


(MGMP)


Spatial


different

subsample


Visualization Test,


types of


items.


students


an untimed


A retention


weeks


test


with


test was given


following the


posttest.


to a

The


results of


training period were significant.


"The most


important


result


this


investigation was


that


after the


instruction


intervention,


middle


school


students,


regardless


sex


, gained significantly


from the


training program


spatial


visualization


tasks"


(Ben-Chaim et al.,


1988,


66).


indicated


previous discussion


the


definition


mathematical


of spatial


ability


problem solving


relationship


and performance,


Tillotson


(1984/1985)


described spatial


ability


as having


least


components,


concluded


that


there


is a


significant


relationship


between spatial


visualization and mathematics,








investigating the


nature of


spatial


visualization,


correlation of


spatial


visualization to problem-solving


performance,


the effect


instruction


on spatial


abilities.


the study


sixth grade


students


five


asses


two comparable schools


were given


same


battery


of four tests measuring


spatial


ability


and


problem


solving prior to,


following


instructional


phase.


Control


experimental


groups were designated by


school.


The


spatial


tests


were


the Punched Holes


Test,


Card


Rotations Test,


Cube Comparisons Test.


A Problem


solving


Inventory


constructed by Tillotson


(1984/198


was


the


fourth


test.


The


instructional


phase consisted


ten


weeks of


lessons


, one


45-minute


lesson


per week,


focusing


activities designed to


improve


the student's


perceptual


skills.


During the


first and


last


weeks


four tests


were


given.


Correlation coefficients,


factor


analyses,


and


regression analyses were


performed


investigate


decomposability

correlation coe


of spatial


fficients we


ability.

re used


Pearson

to analyze


product-moment

relationships


among the


variables.


Analyses of


covariance


were


performed


analyze


effects of


Conclusions were


instruction.


reported


as follows:


1. Spatial
ability. .
2. Spatial


visualization


not a


visualization


current mathematics curriculum


single


a skill


taught


Spatial


visualization


is a good


predictor


3.







Instruction affects


performance on analytic


problems differently than spatial
104)


problems.


(pp.


100-


Supporting the


inclusion of


spatial


skills


the


mathematics curriculum,


Tillotson echoed


the conclusions


many


other researchers


(Ben-Chaim


, Lappan


Houang,


1988;


Dreyfus,


1990;


Hembree


, 1992;


Krutetskii,


1976;


Moses,


1977/1978;


Presmeg,


1986;


Vinner


, 1989)


Furthermore


explicit


instruction


is not only


desirable,


necessary to


improve


spatial


skills


(Dreyfus


, 1990;


Krutetskii,


1976).


Creating the


proper


environment


also an


important


aspect.


According to the constructivist


, a goal


the educator


would be


"the creation


of a problematic environment


that


would


elicit


the student's adequate constructive endeavors"


(Fischbein,


1990


, p.


Geometry


The


correspondence


of spatial


ability


geometry


agreed


upon by many researchers


(Battista,


Wheatley


Talsma


, 1989;


Ferrini-Mundy,


1987;


Tillotson,


1984/1985;


Usiskin,


1987;


Vinner


, 1989) .


Preservice


elementary


teachers


were


relationship


subjects of


between


a study that


strategies


used


investigated


in geometric


problem solving


two abilities,


spatial


visualization and


formal


reasoning


(Battista,


Wheatley


, & Talsma,


1989).


The


study


investigated


following


questions:








problems
ability


related to the students'


or formal


reasoning


spatial


visualization


ability?


Is achievement


in a


geometry


course


preservice elementary teachers


related


to either the


selection of


effectiveness of


problem-solving


strategies utilized when attempting to
problems?


solve geometry


3. Do those elementary teachers who are successful
at geometric problem solving utilize different


strategies


than


those who are not successful?


Five sections of


preservice elementary teachers


females,


males)


were administered a modified Purdue


spatial


visualization


test,


a modified


version


the


Longeot


test,


investigator-constructed


geometry


problem-solving test.


effectiveness of


type of


the strategy was


strategy used


investigated.


and


Strategies


included


drawing


, visualization,


nonspatial.


Percent


use


strategy


and percent


effective


use


of strategy


scores


were


also given.


One of


the most


interesting


results


was


that


although


visualizati

nonspatial


effectively.


was


used more


strategies,


The more


frequently than drawing


the drawing


useful


strategy was


strategies are


and


used more


the ones


used


the most.


Recognition


this discrepancy may


useful


to students as


they monitor their problem-solving


processes.


Battista


indicated


that


"the


balance


between


spatial


logical


ability


likely to


important


factor


geometry performance


in general"


(1990


, p.


48)


The








teachers.


Focusing


on different


levels


of geometry


achievement and


on gender differences,


Battista


(1990)


tested


five


intact classes of


high


school


geometry


students. The variables that were

visualization, logical reasoning,


tested were

knowledge of


spatial

geometry,


geometric problem-solving


strategies,


the discrepancy


between a


student's


spatial


score and


logical


reasoning


score,


and


use of


correct


drawings.


Tests


included


Sheehan'


version of


the Longeot


test of


formal


operations,


Modified Purdue


Spatial


Visualization


test,


Cooperative


Mathematics Test Geometry


Part


Form B,


and


the Geometric


Problem Solving/Strategies


test constructed by


Battista.


Strategies


were classified as drawing,


visualization


without


drawing,


nonspatial,


or none of


the above.


Intercorrelations


between variables


indicated


following


results


(Battista,


1990) :


Spatial


visualization and


logical


reasoning were


significantly


related to


both


geometry


achievement


geometric


problem solving


for males


females.


Spatial


visualization was


significantly


correlated


with strategy variables of


drawings and nonspatial


strategies


for males and with drawings and


correct


drawing


females.


Discrepancy


score


was


significantly


correlated with


drawings,


visualization without


drawing s,


and nonspatial








students with a


level


of geometry


achievement correlations were


significantly


higher


between


spatial


visualization and geometric problem solving than


between


logical


reasoning


problem solving.


Gender


differences on different


variables were also


reported.


Battista suggested that


future


research


investigate


interrelationships


problem-solving


between


strategies


"representational


in geometry"


schemes


(1990,


study


also


"suggests that


instructional


variables


may


critical


factors


in understanding


interrelationships


between


variables,


gender differences,


and geometry


learning"


(1990,


59)


An experimental


verification of


a method


system of


exercises


developing


spatial


imagination


was


conducted


by Vladimirskii


(1971)


to determine


the role


the diagram


in mastering


geometric material.


The


premise


was


that


the


basic task


of geometry


is to develop geometric thought,


to apply theoretical knowledge


in problem solving.


Typically,


study


of geometry


is mostly the memorization


and reproduction


of proofs,


the generalization


concepts.


The diagram


is the


most


often


used


visual


aid,


fully used


enough


to promote


learning


concepts.


The experiments were


conducted with sixth


and ninth







with

cube


the diagram.


imaginations


Preliminary


use


exercises


of solid and dotte


included moving

d drawings.


Resulting problem situations developed


from the


fact


that


book diagrams are too constrained,


thus


restricting the


understanding


the general


concept.


Particular


relationships of


the book


diagram were


taken to


be essential


features


the diagram.


Conclusions


from the control


experiments were:


The diagram may


be a hindrance as well


as a help


the


reasoning process.


the concept has


been defined


learned by


the student,


the


properties cannot be


transferred


new


material


or problems.


The


two goals of


imagination and


foster


the

the


exercise w

formation


ere


to develop spatial


of geometric


concepts,


the complexity


exercises


increasing with


complexity


the diagram.


types


tasks


were


recognition and


composition of


diagrams,


explanations


the geometric


relationships.


The


experiment


concentrated


on the notion


transformation:


translation


, rotation,


reflection


, and


on shifting


figures mentally without


use of models.


Results


from the


sixth grade


indicated


that


graphic material


can


develop spatial


imagination.


Conclusions were


that


major goal


should be


to eliminate


flaws


in present methods








Hershkowitz


(1989)


further


examined


acquisition


basic geometry


concepts


in two experimental


situations.


Subjects


were


students


in grades


five,


siX,


and seven


two


schools,


elementary teachers


, both


preservice


inservice.


process


involved


defining


concept


previously unknown


partic


ipants,


and having them


select


or draw


examples of


the concept.


The


number


critical


attributes


influenced


the accuracy


example


choices.


"There


is a negative correlation between


number to concept attributes and


the mean


success


score


the


task"


(1989,


67) .


A single


prototype


shape


was often


envisioned by the


participants.


role of


visualization


is a complex process,


according to Hershkowitz


(1989)


, in


that


one


cannot


form an


image of


a concept


without


visualizing


its elements,


that


visualization made


may


constrict


the correct


image


concept.


The


computer may


be of


benefit.


"Visualization and


visual


processes


have a


very


complex role


in geometrical


processes


. that


a dynamic


interaction with


a geometrical


microworld


S. contributes


to visual


flexibility.


More


work


is needed


to understand better the


positive


negative


contributions


visual


processes"


(Hershkowitz,


1990


Diagrams


themselves do provide obstacles


learner


(Yerushalmy &


Chazan,


1990;


Zykova,


1969)


Yerushalmy


and


94)







to research


previously mentioned


(Presmeg,


1986) ,


study


Yerushalmy


1986


revealed


that students


using


computer


software called the Geometric Supposer used more diagrams


a generalized


of year test


than did


those students


that


did not have Supposer experience.


Experience with


computer


images may


increase


utilization and


usefulness of


diagrams,


and remove


some


the obstacles


that


they


cause.


From


1984


1988


the effect of


the Supposer on


students and


teachers


high school


geometry


classes


was


studied


(Yerushalmy


et al.,


1990)


In an


inquiry


approach


to the


teaching/learning process,


data


was gathered


from six


sources:


classroom observations,


student Supposer work,


minutes


of monthly teacher meetings,


teacher


interviews,


teacher


writing


reflections,


student


clear materials


interviews.


creating


good


Guidelines


inquiry problems


resulted


from the


investigations.


Inquiry teaching


important for mathematics,

difficulties" (Yerushalmy


and presents

et al., 1990,


"challenges


242)


to bring


into


the classroom.


"The


formulation


inquiry problems


will


inquiry


important


approaches


successful


using


development


other tool-based


of guided


software


environments


in geometry


.and


in other


domains"


(Yerushalmy


et al.,


1990


, p.


242)


Another


Geometric


study


Supposer


involving the


indicated


use of


improved


computer


performance








school


honors


students.


Results


indicated


that


there did


exist


an effect


on problem-solving


skills


on geometry


achievement


from


integrating Supposer activities


into


the


curriculum every two weeks.


Bishop


restates one of


these


obstacles


, "One


problem


geometry teaching


generalized

present many


that


diagram.


impossible

is therefore


diagrammatic examples


to draw


necessary


of a geometric


concept


the


learners are


to be restricted


specificity


the diagram"


(1983,


180) .


Bishop


(1989)


Stewart


(1990)


further reinforced


relationship of


spatial


visualization and geometry


, and


suggested


that


the dynamic


visual


images


of computer


graphics may


development


spatial


skills.


The


obstacles,


computer


only


potential


spatial


to resolve many


visualization,


these


also


processes of


conjecturing,


searching


for patterns,


and


generalizations,


that


processes


"doing


mathematics"


constructing


knowledge


(Bishop,


1989;


Hershkowitz,


1990;


Janvier


, 1987;


Lesh,


1987;


Senechal,


1990;


Steen,


1990)


Summary


review


literature


presented


this


chapter


focused


on studies


and research


pertaining to


cognition


and







constructivist


theory


learning,


students


build


new


knowledge


from


prior


knowledge


experiences


within


environment


active


participation.


research


provided


a strong


basis


investigations


into


student


learning


and


computer


environments


within


domain


specific


content


areas.


Spatial


visualization


geometry


ese


nt rich


areas


examination


content,


cognition,


learning


environment.


The


rese


arch


also indicate


a need


further


examination


of how


students


build


knowl


edge,


specifically


within


an environment


of dynamic


technology


available


the


mathematics


classroom


today.














CHAPTER 3
METHODOLOGY


Overview of


the Study


This


chapter


describes


the research


questions,


the


evaluation


instruments,


participants


for the


study.


It outlines


processes and procedures


the design and


implementation,


The


purpose of


the data


this


analyses


study was


that were used.

investigate the


problem-solving


strategies used by


geometry


students


following


instruction


in a


dynamic technological


environment


within a


Yackel


"constructivist


, 1990)


teaching


Specifically


experiment"


study


(Cobb,


explored


Wood


the


relationships


among the students


' spatial


visualization


ability,


mathematics achievement


problem-solving


strategies


with


and without


the availability


software


called


The Geometer's


Sketchpad.


Students


were


instructed


with


teacher presentation and


the Sketchpad.


Specific


"hands on"


lessons on


experience


locus-motion


using


concepts


provided


the basis


exploring problem-solving


strategies.


The


problem-solving


session


followed


the constructivist


approach


creating


"problematic environment


that


would


- I


I I r 1 --








investigated


dynamic


for the


strategies


tendency to use drawings


in relationship to


use


the availability


the


technology


during the


problem-solving


session


with


the


Locus-Motion


Inventory


(LMI).


Research


Questions


The study


addressed three questions of


interest:


Are


correct response


score


(CRS),


tendency


use drawings


, and/or use


of dynamic strategies


that


students


utilize


to answer


locus-motion


problems


related


students'


spatial


visualization


index


(SV)


mathematics achievement


(MA)?


Does Sketchpad


availability


(SA)


affect


correct


response


score


(CRS) ,


tendency to


use drawings


(DR)


or use


of dynamic


strategies


Are


tendency to


use drawings


(DR)


and/or the


use


of dynamic strategies


(DS)


that


students


utilize


when


attempting to solve


locus-motion


problems


related


students'


correct


response score


(CRS)?


Research


questions were


investigated by more detailed


statistical


questions:


the


relationships


or MA with


CRS


, DR,


DS vary


across


the Sketchpad


availability groups?


relationships do


vary


across


Sketchpad


availability


, what


relationship of


CRS,


or DS








Does Sketchpad


availability


affect


CRS,


or DS


on the Locus-Motion


Inventory?


the relationships of


DR or


DS with


CRS


vary


across


the Sketchpad


availability


groups?


the relationships do not


vary


across Sketchpad


availability


what


relationship of


CRS with DR and


Selection of


Evaluation


Instruments


Measures of


spatial


visualization and mathematics


achievement


were


taken


to examine as


predictor variables,


and


to provide


blocking


or matching variables


statistical


procedures,


such as


those


used by Tall


Thomas


(1989)


their


study


teaching


algebra


with


computer.


Three


independent measures


of spatial


visualization


were


taken.


Mathematics


achievement


data


was


taken


from the


first


semester


geometry


examination.


This


curriculum based


test given


to all


regular


geometry


students


was


created by


Glencoe


the Merrill


Geometry text.


locus-motion


problem


inventory was created


administered


specifically


for this


study.


A description


each


instrument


contained


the next section.


The Spatial


Visualization


Battery


The


Spatial


Visualization Battery


consisted


three








factor


of Spatial


Orientation are


Card Rotations


Test


the Cube Comparisons Test.


An additional


test,


Paper


Folding,


was used


from the cognitive


factor


of Visualization


which also


includes


Form Board and


Surface Development


Tests.


scoring


Since each


schemes,


three chosen


the


tests


results were converted


has different


to standardized


scores and summed


a single


factor


called


spatial


visualization


index.


Standardized


scores,


z-scores


are


corrected


for the mean and


scaled by the


standard


deviation


the


variable.


They are


useful


when


comparing


combining


different


scoring


schemes.


Each


these


tests has


been


used alone


conjunction with


other tests


in studies


involving


spatial


ability


by Moses


(1977/1978)


Tillotson


(1984/1985)


indicated


in previous


chapters,


Tartre


(1990)


reported


that


these


tests


have


also


been


cited as evidence


spatial


ability


Conner


Serbin


(1980)


Linn


and Petersen


(1985).


Originally


ETS considered


three


tests


elements


factor


called


spatial


visualization.


They were


chosen


this


study


based


upon research


to represent


multiple


parts of


the dimensions


spatial


ability


under


consideration.


Card Rotations


Test


The


purpose


this


of problems


test


As








recognize


relationships among


parts


figure


in order


identify the


figure


when


orientation


is changed"


(1984/1985,


42) .


test has


been


identified with


the


dimen


sion


called spatial


visualization


or rotation


(Moses,


1977/1978;


Tartre,


1990;


Tillotson,


1984/1985)


The


test


is two


parts


10 problems each.


Each


problem consists


initial


irregular


figure on


left


followed by


eight


various


representations of


figure.


The


students decide whether


each


one of


the eight


figures


the


same as,


or different


from,


one at


left,


mark


a box


each


or D.


A transformation by


translation


or rotation


considered the same.


transformation


reflection


is considered


different.


Students


were given


minutes


each


two


parts


test.


score


was


number


items answered


correctly minus


number


answered


incorrectly.


Cube


Comparisons


Test


This


test


displays cubes


with


three


faces


showing.


purpose of


test


is to


measure


the


student's ability to


recognize


parts of


a given


configuration


(Tillotson,


1984/19


85)


Although


some


analyses


categorize


this


test


as a measure of


spatial


visualization


(Tartre


, 1990;


Tillotson


, 1984/1985)


, other


analyses


suggest


that


it may relate


to a


different


factor








Each


problem contains drawings of


two cubes


that


can be


drawings of


the same cube


in a


different


orientation,


must


be different


cubes altogether.


The


test contains


two


parts


with


twenty-one


pairs of


cubes each.


Students mark


or D


each


pair.


Students were given


minutes


each


the


two


parts


this


test.


The


score


was


number


items answered


correctly minus


the number


items answered


incorrectly.


Paper


Foldinq Test


The


purpose of


this


test,


also


known


as the


Punched


Holes


Test,


to measure


the student's


ability to mentally


manipulate


a given spatial


configuration


into a


different


one.


It also


been


identified


with


the dimension


called


spatial


visualization


or rotation


(Moses,


1977/1978;


Tartre,


1990;


Tillotson,


1984/1095).


In each


problem,


two


to four


figures


represent


a square


piece


paper


being


folded


then


punched with


a hole.


The


student'


task


is to match


that representation


with


the


correct


representation of


unfolded


piece of


paper with


hole(s)


punched


proper


locations.


The


test


contains


two


parts


with


10 problems each.


Five answer


choices


are


provided


each


problem,


only


one


is correct.


Students were given


minutes


each


two


parts


the


test.


The score was


number


items


answered


. a a


i.I -


* _








Locus-Motion Problem


Inventory


Search


problem-solving


literature


inventories,


revealed several


and many


global


specific to geometry


or spatial


orientation


(Krutetskii,


1976;


Moses,


1977/1978;


Tillotson,


1984/1985).


domain-specific


content,


However,


such


the criterion

locus-motion


to provide

problems,


caused difficulty


specific to


finding a


the content and


problem-solving


applicable


inventory


to dynamic problem-


solving


strategies.


problem-solving


inventory


specific


content


researcher,


locus


incorporating


of points


was created by the


locus-motion problems


similar to


those


found


in geometry textbooks.


The content area


locus


of points


was


chosen because


subject


independent


of placement


in the curriculum


(provided


that


basic


concepts


have


been


introduced)


often


left


the geometry


curriculum due


time constraints.


Studies


Ferrara


(1987/1988),


Campione


(1989),


and


Tillotson


84/1985)


initial


prompted


following


30 problems:


considerations


prerequisite geometry


content,


varying


level


of difficulty,


novelty


of problems.


This


of problems


was


reviewed


six


current


preservice


mathematics


educators,


to assist


devising the


final

light


inventory


of problems.


following


Each


problem was


questions:


reviewed


problem


_ ...._







visualization techniques


of motion


or dynamism be


useful


solving the


problem?


Would


the problem be especially


interesting


in an


interview setting?


Based


upon


the


reviewer ratings,


the set was reduced to nine


problems


the


written LMI.


least one


problem


from each


difficulty


level


was


selected.


problems


were able


to be solved


with pencil


and paper


only.


Five problems were also chosen


to be available


students to solve


in an


interview


setting


See


Appendix A


problems


interest


the researcher was


not only the


performance


on the


problems,


but also


the


strategies


employed


in reaching


solutions.


Problem-solving


strategies


identified


and scored were tendency to


use drawings and use


of dynamic


strategies,


scored


through


observation,


written


work


and


computer


drawings.


Another means of


ascertaining


some measure of


problem-solving process


used by


student


of his


was


to ask the


strategy


student


(Battista,


1990;


or her


Battista


own assessment


, Wheatley,


Talsma,


1989)


Each problem on


the written


portion


consisted


parts.


students were


asked


to provide


the


solution


problem,


to describe


solution


thinking process


involved.


Data


each


problem


consisted


written work


, including


figures and


printed


copies


sketches created


on the computer


students


using The


Geometer's


Sketchpad.








Locus-Motion


Inventory.


score


sheet


for the LMI


can be


seen


in Appendix B.


Correct


response


score


was


awarded


from


0 to


increments


of 0.25,


based


on amount


success.


Scores


for tendency to use drawings were


for no drawings,


one,


for two,


for multiple drawings.


Scores


use


of dynamic strategies were


indication


in words,


or 1,


or figures,


depending


use of


on the


motion,


dynamic thought


processes.


Several


scoring


rubrics


were


used


as models


for the


one created


(Battista,


1990;


Battista,


Wheatley,


Talsma,


1989;


Fennema


Tartre,


1985;


Ferrara,


1987/1988;


Ferrini-Mundy


, 1987;


Krutetskii,


1976;


Moses,


1977/1978;


Presmeg,


1986;


Tartre,


1990) .


The


scoring rubric


also


took


into consideration some concerns


addressed by


Lean


and


Clements


(1981)


about


types


of questions


and


strategies


required,


use of


incorrect


solution


attempts,


and


unwritten strategies.


Although


coming


from


very


different


perspectives,


Spiro et al.


(1991)


Senechal


(1990)

images


students


suggested

used. Ye


taught


more


that


data


rushalmy


with


include


Chazan


the Geometric


thinking process,


number


(1990)


of different


noted


Supposer used


including


free-hand


that


diagrams

drawings,


than


those


that


had not


been


taught


using the Supposer.


Sixteen


percent


problems


were coded


additional


rater


interrater reliability verified


(Ferrara,


1987/19


Ferrini-Mundy,


1987;


Wearne


Hiebert,







Task-based


interviews were conducted with


students,


the


total


sample,


randomly


selected within


each


class


from each SA group.


Davis


(1986)


described


the


task-based


interview


as students


solving


specific problems


talking


aloud,


and an


interviewer


observing those solutions with


audio or video


recording.


Interviews have


been


used


in many


studies


and recommended by many researchers


to explore


problem solving


and


thinking processes of


students


(Bishop,


1983;


Confrey,


1990;


Davis,


1983;


Davis


1986;


Davis


& Maher,


1990;

1986;


Ginsburg

Tartre,


et al.,

1990; vo


1983;


Krutetskii


>n Glaserfeld,


, 1976;


1987)


The


Presmeg,


types


interview strategies varied.


Based


upon


previous


research


interview was designed


to observe


students


in three


situations:


explaining


and confirming the solutions


problems


completed


on the LMI


(Lean


Clements,


1981) ,


attempting problems at


next


level


of difficulty


from


written LMI,


attempting problems available


only


interview


(Campione et al.,


1989;


Ferrara,


1987/1988;


Hoffer,


1983;


Lampert,


1991;


Rohrkemper,


1989).


interviewer


began with


student'


work


from the


written


and


proceeded


to problems


specifically


designated


task-based


interviews


, depending upon


level


difficulty


and


success


the student,


exploring the


activity


development


student within his


(Vygotsky


, 1978)


or her


As Krutetsk


zone of

ii (1976


proximal

)and








the aid


of others.


work was


collected


examination,


as was


transcribed audiotape of


interview


(Davis,


1986;


Dover


Shore,


1991;


Resnick,


1989).


Students were


instructed


talk as they worked,


a technique


used


several


studies and


promoted within a


variety


assessment


tools


(Ginsburg


et al.,


1983;


Huinker


, 1993;


Mashbits,


1975


Selection


Description of


Population Sample


Participants


for the


study


consisted


geometry


students


from seven geometry


classes


at one high school.


The


seven


classes were


taught by


four mathematics


teachers


interested


in encouraging the


use


technology


the


mathematics


classroom,


specifically


in using The Geometer's


Sketchpad


the geometry


curriculum.


The subjects


were


mixed


gender


, age,


racial


background


in grades


All


students


participated


instruction


using The


Geometer's


Sketchpad.


The students were


randomly


assigned


two groups.


One group worked


the LMI


traditional


paper


pencil


manner.


Another


group


the computer


software available


as they


solved


problems.


Two


students


from


each


class period were


randomly


chosen


, one


from


each


group,


to participate


in the


interview


sess


ion.


total


students worked


problems


within an


interview


setting.






55

Procedures


Prior to


study the


University


of Florida'


Institutional


Review


Board granted permission for the


investigation


take


place.


Students were


informed


and had


to obtain parental


permission


(see Appendix C)


participate


study.


The


Spatial


Visualization Battery was administered


two occasions,


before any


exposure to the Sketchpad


and


following the


session.


A script explained


the


purpose


and


directions


for the three parts


the


battery.


First


semester


examination scores were provided by the


teachers.


The


problems


from the


Locus-Motion


Inventory were


presented


the


students


to be worked within one


50-minute


class


period.


The classroom teachers administered


the


paper


pencil


session;


researcher


administered


the computer


sessi


the


on.


LMI.


A script was


The


used by


interviews were conducted


administrators


by the


to execute


researcher,


with


Each


audiorecording


interview


interview transcribed


lasted approximately


afterward.


20 minutes.


Description


Instructional


Phase


The effectiveness of


instruction


using technology


can


best be


explored


reality


the context


classroom


in which


(Fischbein


, 1990;


will


Kulik


occur


Kulik,








classes.


Teachers


used the


software


in a


presentation mode


several


months.


Students had


several


lessons


"hands


experience with


software.


The researcher presented


lessons on


locus


of points.


lessons were


extension


previous


lessons


on use of


the Sketchpad


, and


lessons


that


introduced


the concepts of


locus


of points.


There were


six


lessons,


each


lasting


one


50-minute


class


period.


Each


was


a combination of


teacher


demonstration,


student discovery


, and student


experience


with


problems


on the computer.


Initial


lessons presented


the concept


locus and the steps


involved


in solving


locus


problems,


specifically


applied


to perpendicular


bisectors,


angle


bisectors,


triangles.


Points


concurrency


such


incenter


, circumcenter,


centroid,


orthocenter were


explored.


Other


lessons extended


application


locus


to circles.


applied


Finally,


to the


the concept


intersection


locus


figures,


of points


further


was


the


intersections


loci.


The


different


researcher


ces


conducted all


in results due


lessons


to eliminate


teacher variability,


was


aware


possibility that


a threat


validity may


introduced by


researcher


influencing results.


This


was


minimized


by the


use of


specific presentation


plans,


written activities


students


with


the computer.


Seven


orobiem sets


anrd wnrkshpt-


nc rlir inr +


i nct "r 'int- i r nnn 1







Portions of


lessons were


tested


during the


spring


1993.


The


pilot was a


series of


lessons


that


introduced


students


to The Geometer's


Sketchpad


and


explored


intersection


figures.


lessons


varied


from


"very


guided"


with


specific


instructions


to open


investigations


guided


only


by presenting problem


situations and questions.


The


purpose of


pilot was


to make an


initial


evaluation


the clarity


the activities and


instructions.


Two


groups


three


students each worked


on the activities with


no teacher


assistance or


intervention.


Each


student


provided


observations and


journal


entries


that discussed


the


instructions,


activities,


their


involvement and


attitude


with


the computer


and The Geometer's


Sketchpad.


The


results


the


pilot


study were


used


revise


the activities


that


introduced The Geometer


Sketchpad to


the students.


Statistical


Procedures


Data


analysis


consisted


three


parts:


descriptive


statistics,

qualitative


quantitative st

investigations.


:atistical


analyses


Descriptive


statistics


were


obtained


question


Review of st

dealing with


variables


, appropriate


U


dies in

spatial


under


statistical


literature,


abilities


consideration.


procedures


For


were


each


applied.


especially those


and mathematics,


suggested


several


statistical


procedures applicable


to a


similar


study








Tartre,


1990;


Tillotson,


1984/1985).


Statistical


procedures


for this


study


included


correlation coefficients,


analyses


covariance,


statistical


and multiple regression analyses.


approach


The


often problematic when dealing with


educational

1985). In


issues of


addition


instruction and


to quantitative


learning


statistical


(Schoenfeld,

procedures,


a qualitative component


examined


interview


data


describing the


problem-solving processes of


individual


students.


Summary


This study was designed as a


"constructivist


teaching


experiment"


(Cobb,


Wood


Yackel,


1990)


investigate


problem-solving


strategies


used by


geometry


students


following


instruction


a dynamic technological


environment


to explore


visualization ability,


relationships


mathematical


among the


ability,


students'


and


spatial


problem-


solving


strategies with


and without


technology.


Three


independent measures


of spatial


visualization


were


taken.


They


included


ETS Card Rotations,


Cube


Comparisons


, and


Paper


Folding tests.


A locus-motion


problem


inventory was


also administered.


Participants


for the


study


consisted


geometry


students


from seven


geometry


classes


one


high


school.


students participated


instruction


using The Geometer's


Sketchpad.


The


instructional


phase


1- 1


-I


* *


-- L


J i


I


I







assigned


two groups:


those who would have


computer


software available during the LMI,


and those who would not.


Data

component.

statistics


analysis


had both a


Statistical

, correlations


quantitative and a


procedures,

, analyses o


qualitative


including descriptive

f covariance, and


multiple regression analyses


, were


used


to explore


the


questions of


interest


in the


study.


results of


data


analyses


are


presented


Chapter


discussed


Chapter














CHAPTER


ANALYSIS OF THE DATA


This chapter


contains


the descriptive statistics


variables


under


investigation and results of


the data


analyses


pertinent


to the questions


this


study.


It also


includes


results


the qualitative


investigations


observation and


interview.


This


study was designed as a


"constructivist


teaching


experiment"


(Cobb,


Wood


Yackel,


1990)


to investigate


problem-solving


following


state'


instruction


gies

in a


used by geometry


students


dynamic technological


environment


to explore


relationships


among the


students'


spatial


visualization


index


mathematics


ability,


correct response


scores,


tendency to use drawings,


use of


dynamic


strategies

Three


with and without the Sketchpad

independent measures of spatial


available.

visualization


were


taken at


two different


occasions.


spatial


visualization


index


included


Card


Rotations


, Cube


Comparisons,


and Paper


Folding tests.


A locus-motion


problem


inventory


(LMI


was also administered.


Participants


for the


study


consisted


geometry


students


from seven


geometry


classes at


one


high school.


students








Sketchpad.


The


instructional


phase


included six


lessons


that


were an extension of


previous


lessons on use of


the


Sketchpad


lessons


that


introduced


the concepts


locus


of points.


Students were assigned


individually


and at


random to


two groups


for the LMI


problem-solving


session:


those who


would have


the Sketchpad


available,


those who would not.


The


problem-solving


session


lasted


one class


period


minutes.


Individual


interviews were held


on the


following


day with


two students


from each


class,


one


from each


group.


Interviews


lasted


approximately


20 minutes during which


students worked


locus-motion problems


aloud.


Research Questions


The


study


addressed three questions


interest:


Are


correct


response


score


(CRS)


, tendency to


use drawings


, and/or use of


dynamic strategies


(DS)


that


students


utilize


to answer


locus-motion


problems


related


students'


spatial


visualization


index


(SV)


mathematics


achievement


(MA)


Does Sketchpad


availability


(SA)


affect


correct


response


score


(CRS) ,


tendency to use drawings


(DR)


or use


of dynamic strategies


(DS)


Are


tendency to use drawings


(DR)


and/or the


use of


dynamic strategies


(.DS)


-- V *I -I -


students


utilize when


tha^t








Results


of Data Analysis


Descriptive


statistics


are presented


in Table


4-1,


which


includes


the number


means


(M) ,


and standard


deviations


following variable


Sketchpad a

s: correct


availability

response s


(SA)


core


for the

CRS), tendency


use


drawings


(DR)


use of


dynamic


strategies


(DS),


spatial


visualization


index


and mathematics


achievement


(MA).


MA was measured by


first


semester geometry


examination


scores.


total


60 was possible,


the highest


achieved


was


CRS,


DR and DS were


sums


the


measurements


individual


problems


on the


locus-motion


inventory


(LMI)


Problem


scores


for CRS were


awarded


from


increments


of 0.25


based


on amount


success.


Problem


scores


DR were


for no drawings,


one


for multiple drawings.


Problem


scores


were


depending


figures,


use


on the


of motion,


indication


in words,


or dynamic thought


processes.


was


based


on the


composite


z-scores


for the


three


measures


taken


time


the LMI:


Card Rotations,


Cube


Comparisons,


between


and Paper


Folding Tests.


two occurrences


the spatial


The correlations


visualization


tests


were high;


time


therefore,


were


used


the measures


in analysis.


taken


closest


Correlations


between







Table


4-1.


Descriptive Statistics
Sketchpad


With


and


Without


Variable Sketchpad Number Mean SD

CRS With 78 1.82 1.50

Without 81 1.99 1.62


DR With 78 14.08 6.32

Without 81 15.92 6.12


DS With 78 3.45 2.02


Without 81 3.1 2.08


SV With 75 -0.07 2.30


Without 90 0.28 2.37

MA With 75 42.38 7.97


Without 94 42.86 7.71


Although


students


were randomly


assigned


, for the group


without


the Sketchpad


available


for the


LMI,


the means were


somewhat higher


on matching variables


SV and MA.


The


means


were also


higher


for that


group on


Only the mean


DS was


higher


for the group with


Sketchpad


available.


Due


scoring rubric


for CRS


allowing


scores


increments


0.25


the mean score


low.








reported


in the


following paragraphs.


An alpha


level


.05 was used


tests.


the relationships of


SV or MA


with


CRS,


DS vary


across the Sketchpad


availability


groups?


The


slope


relationships of


CRS,


DR and


DS with


SV and MA


were compared


across Sketchpad


availability


groups


by using multiple regression.


results are


reported


Table


indicate no


variation


over


groups


in the


relationships.


Table


4-2.


Results Comparing


Relationships Across


Sketchpad Availability Groups


Dependent Independent
variable variable MSE F p


CRS SV 2.10 2.36 .1265


MA 2.14 0.80 .3712


DR SV 36.52 0.68 .4123


MA 35.21 1.54 .2165


DS SV 3.54 0.76 .3894


MA 3.78 0.00 .9898


the relationships do


vary


across Sketchpad


availability


, what


relationship of


CRS,


or DS


with SV


and MA?


-r L





1








relationship of MA and


CRS,


MA and DR


and MA and


DS did not


vary


across groups.


Consequently the


pooled


relationships


of SV with


each


of CRS,


and DS as well


as the


pooled


relationship


between MA and


each


of CRS


DR and DS were


investigated using multiple regression.


Results are


shown


in Table


4-3


For


each


of CRS,


and DS there


is a


significant


relationship with SV


and MA


.01.


Table


4-3.


Results of


with


Pooled Relationships
CRS, DR, and DS


of SV and MA


Dependent Independent
variable variable r MSE F p


CRS SV .38 2.11 23.74 .0000


MA .37 2.14 22.22 .0000


DR SV .30 36.44 11.17 .0011


MA .34 35.34 15.05 .0002

DS SV .38 3.54 24.02 .0000


MA .28 3.75 12.27 .0006


Does Sketchpad


availability


affect


CRS


, DR,


or DS


the Locus-Motion


Inventory?


investigate


the question


, for


each


of CRS


DR and


an analysis of


covariance


(ANCOVA)


was conducted


with SV


MA as covariates.


Results are reported


Table


and


- I I 9 A -t


LL_ 1_ L -1 11 1


- -* I -t <








Table


4-4.


Summary


ANOVA


Table


for CRS,


and DS


Dependent
variable Source df SS MS F p


CRS SA 1 1.64 1.64 0.82 0.3673


Error 139 279.36 2.01


DR SA 1 109.46 109.46 3.12 0.0797


Error 139 4883.9 35.14


DS SA 1 2.89 2.89 0.85 0.3595


Error 139 474.51 3.41


the relationships of


or DS with


CRS vary


across


the Sketchpad


availability


of groups?


slope of


relationships


of CRS with DR and DS


were


compared


across


Sketchpad


availability


groups


by using


multiple


regression.


results are reported


in Table


indicate


no variation


across


groups


in the


relationships.


Table


4-5.


Results Comparing Relationships
Sketchpad Availability Groups.


Across


Dependent Independent
variable variable MSE F p


CRS DR 1.15 1.60 .2077








the relationships do


vary


across Sketchpad


availability


, what


relationship of


DR and DS?


The analyses relevant


to question


indicated


relationships


of CRS with DR and


CRS with DS did not


vary


across Sketchpad


availability groups.


Consequently the


pooled


relationship


between DR and


as well


as the


pooled


relationship


multiple

CRS there


between DS and


regression.


is a


CRS were


Results are shown


significant


investigated


in Table


relationship with


using


4-6.


DR and


the


level.


Table


4-6.


Results of Relationships of


CRS with DR and DS.


Dependent Independent
variable variable r MSE F p


CRS DR .73 1.15 176.6 .0000


DS .78 0.95 234.6 .0000


Other


Findings


For the group with


the Sketchpad


available


, the


number


of Sketchpad


drawings


(NSD)


was also


scored.


instances


which


figures were


not saved nor mentioned


a score of


zero was


awarded.


Data


indicates


that students


did


use


Sketchpad when


was available.


mean number


figures


S-A--_ I -a


M.


-- A -


A 1


rF A- .*-


J nrt _


1 A


1








regression


analysis with NSD


as dependent variable


indicate


independent


that


variable and


the relationship was


significant,


Further


visualization


F(1,74)


= 2.11,


analysis of


index


.1443.


tests within


offered some additional


spatial


findings.


three


tests


that


comprised


the spatial


visualization


index


were given


on two


occasions,


before


exposure


the


Sketchpad


following the LMI


sess


ion.


Although


there


was


a high


correlation between


the score


there


were


differences


in the means


that


should be


noted.


Mean


scores


the Card Rotations


test


(CR)


increased


from 98.65


115.79


, an improvement of


17.1


points.


the


Cube


Comparison


test


(CC) ,


mean


scores


increased


from


12.55


16.65,


an improvement of


points.


The means


scores


the


Paper


Folding


test


improved


points,


from


10.46.


Additional


analyses examined


the relationship of


individual


tests,


measured at


time of


the


LMI,


the


scores


to determine


any test had


a more


significant


relationship


to CRS,


or DS


Since


there


was


influence of


multiple


SA with


regressions with


on either


CRS,


these


and DS


variables


respectively were


implemented,


each


time


with


CR-post,


CC-post,


PF-post


independent


variables.


Results are


presented


Table


4-7.








Table


Effect


of Spatial


Visualization


Subtests on


CRS,


and DS.


Dependent
variable Source r MSE F p


CRS CR-post .19 2.45 4.28 .0402


CC-post .37 2.18 22.94 .0000


PF-post .32 2.21 17.59 .0000


DR CR-post .19 39.58 2.69 .1034


CC-post .26 38.24 8.03 .0052


PF-post .31 36.09 13.24 .0004


DS CR-post .23 4'.14 7.10 .0085


CC-post .35 3.83 19.47 .0000


PF-post .30 3.88 14.51 .0002


Results


the multiple


regression analyses


indicate


that


CC-post had


the greatest


influence


on CRS


and DS


both


.01.


PF-post had the greatest


influence on DR also


.01.


CR-post had


least


influence on


CRS,


Interview


Results


This


study


also


had a qualitative


component.


Two


I r


qm 1








the written LMI


had been determined


prior to


the


interview,


and


three or


four problems


had been selected


discussion


during the


interview.


In most


cases


two of


problems


were


from the written LMI


one chosen at


next


difficulty


level


from a


of problems


not seen by the


student


before.


Students


solved


problems aloud,


with


varying assistance


Results


from the


analyses of


researcher.

variance comparing the means of


interviewees


on CRS,


, and DS


with


those


students


that


took


only the written LMI


indicated


that


there were no


significant


differences


interviewed were


in the means.


not different


from those


students


not


being


interviewed


for the research


questions.


A main


purpose of


interview


component


the


study


was


to explore


the ability


the student within his


or her


zone


of proximal


discover what


development


student has


Vygotsky,

learned,


1978)


what he


to

is


capable


learning with


the aid


researcher


(Krutetskii,


1976).


Students worked


problems


from the


written


again with


cases,


students


varying


were able


degrees


to solve


assistance.


problems


successfully


on the written LMI


that


they


had


previously


missed.


Ten


interview


students were


also able


work


least


one


problem


that


they


had not


seen


before


which had


a difficulty


level higher than


one


missed


on the







Patterns,


or causes,


failure were also of


interest,


since


they


& Jehng,


19


suggest ways

90). In many


to expand a


cases,


student's


students


"zone"


indicated


(Spiro


that


understanding the


problem caused


biggest


obstacle.


. when you read the


problem--it


hard


understand


, but when


you went


over


way


easy to


comprehend.
R: So the words were


first


stumbling


block?


Yes.


didn't know really what


it was asking


for.


didn't know how to do


number three.


What


I was
instead
R: So


did


trying to make
d of sixty fee


these


aren't


S. see


them


read
feet


t apart.
terrible are


the
fro


problem wrong


m


race mark


they?


out


. I just have a hard time getting the question
the words.


knew what


I had to draw


didn't know


how to


explain


the


locus.


didn't


understand.


. Is


that asking what


will make


it consist


one


point?


didn't


understand


this


one


very well


at all.


S10:


really


didn't


understand


the question.


I'm not sure


times


the words


what they

within the


are


asking.


problems caused


difficulty.


Meanings


terms


, such as


plane M,


tangent,


isosceles


a circle


that


contains


two given


points


, were


often not known.


Students would


ask specifically


, "what


does


this mean?"


Plane M


confused me.


haven't


learned much


about


planes.


R: What


was


hard about


the question?


What


gave


you


: I








Another


such

lack


obstacle was


as distance r,


of real


or radius


use


variable


measurements,


Students commented


numbers or specific directions


to make


on the

the


question understandable.


Can
You


solve


this


don't have any numbers


don't know the radius?


so you


don't know what


to do.


This


hard


to understand


'cause


doesn't say


where

S14:


they

Point


are.

P is anywhere?


Although


one


problem-solving


heuristic


think


similar problem,


students often related problems


exactly to


those


they


had previously


experienced and


did not


take


note


of differences


in the question or


conditions.


This


student


made


use of


heuristic


in a


positive manner:


Matter


fact


very


same


figure as


first


one with


isos


celes


triangle.


Remembering


a similar problem


exactly


like


one on

written


LMI


test,


was a hindrance


student


to another


indicated


student.


on two occasions


that


problem was


like one


already


solved,


or that


remembered


the solution.


In both


cases,


there were distinct


conditions


that


changed


the question and


solution.


It was


interesting to note


the extent


to which students


were


aware


of what


caused


obstacles with


problems.


Students


were


often aware


that


they


did not


understand


question.


As can be


seen


the quotations above,


they








were


less aware


that


they


had not noted differences


problem situations when similar to


those already


experienced.


The need


for understandable materials


problems was


obvious

did not


from the

interpret


investigations.


In many


problem correctly.


cases,

When t


students


:he


conditions


were clarified,


or the correct


figure drawn,


the


student


was


able


to proceed


to a


correct


solution.


The


wording


the


problems was critical


the correct


figure


and


the correct


solution.


three


instances


students made


valid


interpretations of


the question different


from


interpretation


instances


the researcher.


the students produced a


two of


correct


the


three


solution,


but


way


was expressed was


not clear to


researcher.


Communication


was


critical


from the students'


researcher'


perspective.


An emphasis


the mathematics


classroom on communication


important


to understanding,


instruction,


assessment


(NCTM,


1989) .


That


was


clearly


evident


these


interviews.


Another purpose


interview was


to explore


the


use


the computer,


potential


for use.


Students were


asked a


series


survey


questions at


beginning


problem-solving


twofold:


session


to put


see


student at


Appendix E).


ease


The


with


purpose was


interviewer,


to explore


student


attitude


and beliefs about








more mechanical


and boring.


instances,


students


noted


the technical


aspects


technology


obstacle.


In all


instances


, students


indicated


that


everyone


should


learn something


about


computers.


Stated benefits of


the computer


included


accuracy


figures


faster


and measurements,


drawings,


precision,


easier


and ability to move


calculations,


figures


while


seeing


changes


simultaneously.


available


figures


Students


for the written


and measurements


that did not have


interview sessi


the Sketchpad

ons indicated


that


they would have


used


the computer


been


available,


had


the compute


perhaps would have done

r available appreciated


better.


Those


capability


that


ies.


Well,


took the


test


in the


classroom.


could


have done better


it had been


on the computer


Because


would have


do much better than my


visualized


drawings.


it better.
could have


can


put a


point


more


locus,


sca


probably


a midpoint


than mine.
could have


'cause


used


it would


them


knew how to


read


the question


correctly.


you had had


computer


would


have


used


Yeah.
Would


it have helped?


can move stuff


around


like


circle.


think


it helped


learn


think.


It may


have


helped since


it's


more accurate


easier to


just a
kind o


line


see.


can


see


you draw yourself


a bother to


say this


an angle


bisector


that helps.


is what


It's


want and


highlight everything.
R: So the technical


part may not


be worth th


ie effort.








you


think


When we were


the compute
learning,


r helped yo
it did make


in any way?
things clearer


see,


cause sometimes


you sketched


it out


then


you started seeing things and
make sense.


that made


other things


change


the way you might solve?


S9:
your
you


You


It might have.


own


then


think


can


it yourself


used


it did.


things


it makes


the computer


it a


on almost


you


it and


learn


ask.


easier.


every problem,


right?
S13:


Yeah.


Could


you have done


these


you hadn't had


computer?


S13:


It would have


been a


longer.


just


can't


draw.


assessment should be,


interview sessions


themselves were a


learning


experience.


Students


were


able


to solve


problems with


ass


instance.


Students were able


share

the i


their


opinions


instruction,


and knowledge about


the content area,


technology.


I
Did


actually


am glad


that


you went


over


learn something?


Yeah.


just


very well.


never


really
try to


have


listen,


read


been able


then


can


learn math
understand


it myself.


You knew


some of


the answers,


said


you


were


afraid.


had


ideas.


That always


happens


to me


though


won't


put


it down.


won't


want


to embarrass myself


something.


was making


it harder


than


was.


Are


these


hard?


S10:


Yeah.


Why


are


they


hard?


They make


you


think.


Attitudes and


beliefs,


as well


and knowledge and


success


on the content.


were


i mnrnvPd with


rnmmlln I i n tnn





76

Summary


This


used


study


geometry


investigated


students


problem


following


solving


instruction


strategies


a dynamic


technological


environment


explored


the


relationships


between


students


spatial


visualization


index,


mathematics


ability,


correct


res


ponse


scores,


tendency


use


drawings,


use


of dynamic


strategy


ies


with


without


Sketchpad


available


The


study


addressed


three


questions


of interest


Are


the


correct


response


score


(CRS),


tendency


use


drawings


DR) ,


and/or


use


of dynamic


strategies


that


students


utilize


students


answer


' spatial


locus-motion


visual


zation


problems


index


related


or mathemati


achievement


(MA)?


Does


Sketchpad


availability


(SA)


affect


correct


res


ponse


score


CRS),


tendency


use


drawings


(DR) ,


or use


of dynamic


strategies


(DS)


Are


the


tendency


use


drawings


and/


use


of dynamic


attempting


strategic


to solve


that


locus-motion


students


problems


utili


related


when


to the


students


' correct


response


score


(CRS)?


Research


questions


were


inves


tigated


more


detail


statistical


stions


outlined


below.


Do the


relationships


of SV


or MA with


CRS,


, or DS


vary


across


Sketchpad


availability


groups





relationships


not


vary


across


Sketchpad


availability,


what


the


relationship


CRS,


or DS with


SV and


Does


Sketchpad


availability


affect


CRS,


or DS


Locus-Motion


Inventory?


Do the


relationships


of DR


or DS with


vary


across


the


Sketchpad


availability


of groups?


If the


relationships


do not


vary


across


Sketchpad


availability,


what


s the


relation


ship


of CRS


DR and


Results


of both


the


quantitative


and


qualitative


components


of the


study


have


been


described


detail


chapter


and


are


summarized


this


sec


tion.


Initial


analy


ses


indicated


that


none


of the


relationships


of spatial


visualization


index


with


correct


response


score


, tendency


use


drawings


, or use


of dynamic


strategies


varied


across


Sketchpad


availability.


Similarly


none


of the


relationships


of mathemati


achievement


with


correct

dynamic


response score,

strategies varie


tendency


across


use


groups


drawings,

Results


or use or

of multiple


session


analyses


indicated


that


each


of the


variabi


correct


res


ponse


score


, tendency


use


drawings


, and


use


dynamic

spatial


strategies


visual


had


zation


a significant


index


relationship


and mathematics


with


achievement


To investigate


whether


Sketchpad


availability


ects


correct


response


score,


tendency


use


drawings,


and


use


dynamic


strategies


, an analy


S1S


covariance


(ANCOVA)


was






conducted


each


variable


with


spatial


visual


nation


index


and mathemati


achievement


as covariates.


Results


indicated


that


correct


Sketchpad


response


availability


score


does


not


, tendency


significantly


use


drawings


affect


use


dynamic


strategies.


analy


ses


concerning


relationships


correct


res


pons e


score


with


tendency


use


drawings


with


use


dynamic c


strategies


indicated


that


none


of the


relationships


varied


across


Sketchpad


availability


groups


Results


multiple


regression


analy


ses


indicate


ed that


correct


response


score


there


is a significant


relationship


with


both


tendency


use


drawings


and


use


of dynamic


strategies.


For


students


with


the


Sketchpad


available


, th


ere


was


no significant


relationship


between


the


number


computer-


created


use


drawings


correct


Sketchpad when


was


response


score.


available


Subj


Students


ects


did


that


computer


available


during


the


problem


solving


session


created


a mean


of 4


figures


on the


comput


er.


students

computer


that


had


drawings


the

for


computer


one


or more


available


or 94%


create


problems


Further

visualization


anal


index


of the


offered


tests

some a


within


additional


spatial


findings.


three

were


tests

given


exposure

written


that

on two


to the


comprised

occasions


Sketch


session.


!pad and

Although


the


spatial


before


the


immediately


there


was


visualization


students


following


index

any

the


a high








correlation


between


scores,


there were differences


the means.


Additional


analyses examined the


relationships


the


individual


tests


the LMI


scores to determine


any test


had


a more


significant relationship


to correct


response


score


, tendency to use drawings,


or use of


dynamic


strategies.


Results of


the multiple regression analyses


indicate


that


Cube Comparisons


test had


the greatest


influence on


strategies.


correct response


Paper


score


Folding test had


use


of dynamic


the greatest


influence


on tendency to


use drawings.


Card Rotations


test had


least


influence on


correct response


score,


tendency to use


drawings,


use of


dynamic


strategies.


Analysis of


interview data revealed


that


cases


students were


able


to solve


problems


successfully


on the written LMI


that


they


previously missed.


Sketchpad

solutions


availability


did not make a


to interview problems.


difference


In many


cases


the


students


indicated


that


understanding the


problem caused


the


biggest


obstacle.


times


the words within


problems


caused


difficulty.


Another


obstacle


was


use of


variable


measurements,


such


as distance r,


or radius


Students


commented


on the


lack


of real


numbers


or specific directions


to make


the question


understandable.


Although


one


problem-solving


heuristic


think








those


they


had previously


experienced


take


note


of differences


Students


in the question or


conditions.


were aware of what caused


obstacles with


problems.


Students were often aware


that


they


did


not


understand


the question


They


commented


on the


fact


that


they


did not


understand


the question


or did


not know the


definitions of


terms.


They were


less


aware


that


they


not n

those


oted differences


already


in problem situations when similar


experienced.


The need


for understandable materials


inquiry


problems

cases, s

When the


drawn,


was obvious


students did not


conditions


student


from the


investigations.


interpret


were clarified,


was able to proceed


In many


problem correctly.


or the correct


to a


figure


correct


solution.


The wording


problems was critical


correct


figure


the correct


solution.


Another purpose of


interview was


to explore


use


the computer


potential


for use.


Stated


benefits


the


computer


included


accuracy


figures


measurements,


precision,


easier


calculations


faster


drawings,


ability to move


figures


while


seeing


changes


in the


figures


and measurements


simultaneously.


Students


that did not have


the Sketchpad


available


for the


written


interview


sessions


indicated


that


they would have


used


the computer


it been available,


perhaps


would







The


interview


sessions


themselves were a


learning


experience.


Students


were able


to solve


problems


with


assistance.


knowledge


Students were able


about


technology.


to share


the content area,


Attitudes


and beliefs,


their


opinions and


instruction,


as well


the


as performance


were


improved


with


communication between


learner


and


instructor.














CHAPTER


SUMMARY


AND CONCLUSIONS


Overview of


Study


The


purpose of


problem-solving


this study was


strategies


used


investigate


geometry


the


students


following


instruction


in a


dynamic technological


environment.


Specifically


study


explored


relationships among the students'


spatial


visualization


ability,


mathematical


ability,


and problem-solving


strategies with


and without


technology.


Students


were


instructed with


teacher presentation and


"hands


experience

Sketchpad.


using


a software


Specific


product called


lessons on


locus-motion


he Geometer's

concepts


provided


Methods


basis


exploring problem-solving


solutions were examined


level


strategies.


drawing


dynamic visualization


used


relationship


availability


technology


during the


problem-solving


session.


Three


independent measures of


spatial


visualization


were


taken


two different


occasions.


spatial


visualization


index


included ETS Card Rotations,


Cube








for the


study


consisted


geometry


students


from seven


geometry


classes at one high school.


All


students


participated


in the


instruction using The Geometer's


Sketchpad.


The


instructional


phase


included


lessons


that


were an extension


previous


lessons on


use of


the


Sketchpad


lessons


that


introduced


concepts


locus


of points.


Students


were


ass


signed


individually


at random


two groups


for the


problem-solving


session:


those


who did


have


the Sketchpad available,


those who did not.


The


problem-solving


session


session


lasted


one class period


50 minutes.


Individual


interviews were held


on the


following


day with


two students


from each


class


, one


from


each


group.


Interviews


lasted approximately


20 minutes,


which


students worked


problem-solving


session


locus-motion problems


followed


aloud.


the constructivist


The

approach


of creating


"problematic environment that


student's adequate constructive


endeavors"


would


elicit


(Fischbein,


1990,


Data


analysis had both a


quantitative and


qualitative component.


Results


Discussion


The


theoretical


premise


that supports


this


study


that


use of


a dynamic technological


environment,


such


The Geometer's


Sketchpad,


can enhance construction








constructivist


theory


learning promotes


the belief


that


students


build knowledge


through


activity


experience.


Dynamic


software can provide experiences


conjecturing,


building


in visual


frameworks within a


zing,


specific


content domain.


The


study


addressed


three questions of


interest:


Are


use drawings


the correct


(DR),


response


and/or use of


score

dynamic


(CRS) ,


tendency to


strategies


(DS)


that


students


utilize


to answer


locus-motion


problems


related


the students'


spatial


visualization


index


(SV)


mathematics


achievement


(MA)?


Does Sketchpad


availability


(SA)


affect


correct


response


score


(CRS),


tendency to


use drawings


(DR)


, or use


of dynamic


strategies


(DS)


Are


tendency to use drawings


(DR)


and/or the


use of


dynamic


strategies


(DS)


that students


utilize


when


attempting to


solve


locus-motion


problems


related


students'


correct


Research


response


questions


were


score


(CRS)?


investigated by more


detailed


statistical


questions


outlined below.


relationships


or MA with


CRS,


DS vary


across


the Sketchpad


availability


groups?


relationships do


vary


across Sketchpad


availability,


what


relationship of


CRS,


or DS


with SV


and MA?








the relationships


of DR or


DS with


CRS


vary


across


the Sketchpad


availability


of groups?


the relationships do


not vary across Sketchpad


availability


Initial


what


is the


analyses


relationship of


indicated


that


CRS and DR and DS?


none


relationships of


spatial


visualization


index with


correct


response


score,


tendency to use drawings,


or use of


dynamic


strategies


varied across Sketchpad


availability.


Similarly


none


the


relationships of


mathematics


achievement


with


correct response


score,


tendency to


use drawings,


or use


dynamic


strategies


varied across groups


Results of


multiple regression analyses


indicated


that each


the


variables

drawings,


of correct

and use of


response score, te

dynamic strategies


ndency to


use


significant


relationship with spatial


visualization


index


mathematics achievement.


To investigate whether


Sketchpad


availability


affects


correct response


score


, tendency to use drawings,


use of


dynamic


strategies,


conducted


each


an analysis of


variable with


covariance


spatial


(ANCOVA)


visualization


was


index


and mathematics


achievement


as covariates.


Results


indicated


affect


that Sketchpad


correct response


availability


score


does


, tendency to


significantly


use drawings,


use


of dynamic strategies.


The availability


computer


during the


problem-


II_


_C _








the


covariate


accounted


spatial


for most of


ability or

variance.


mathematicc

Although


achievement


specific to


the geometric content


locus of


points,


these


findings


supplement


those


by Moses


(1977/1978)


and Presmeg


(1986)


that


spatial


ability


is a


good predictor


of problem-solving


performance.


Moses


(1977/1978)


found


that


students with high


spatial


ability


However


often do


, in this


write down


study,


visual


solution processes.


students were encouraged


to write


down


solutions,


make drawings,


and describe


visual


solution


processes.


Both


spatial


visualization and mathematics


achievement were


significantly related


tendency to


use


drawings.


Additional


findings


included


trends


seen


in the


subtests


spatial


visualization


index.


Although


study


did not


specifically


address


the components


of spatial


ability


nor whether


is a modifiable quantity


, descriptive


statistics

warrants c


provide some


comment here an


interesting

d, perhaps,


information


that


further research.


three


tests


that


comprised


spatial


visualization


index were


given


on two occasions,


before any


exposure


the Sketchpad


and


following the


session.


Although


there


was


a high


correlation between


scores,


there


were differences


means


that should


be noted.


Mean


scores


on the


Card


Rotations


test


(CR)


increased


from 98.65


to 115.79,








improvement


4.1


points.


The means


scores


on the


Paper


Folding


(PF)


test


improved


points,


from


7.99


10.46.


Although


there were differences


in the means of


the


subtests on


two occasions,


without a


control


group


that


did not


use


the Sketchpad


at all,


there can be


no direct


comparison.


However


, the


fact


that


the mean


scores


on all


subtests


overall


index


score


increased


suggests


that


spatial


ability


a modifiable quantity


, as


indicated


the s

1988;


earch


the


Hembree,


literature


1992;


Moses,


(Ben-Chaim

1977/1978;


Lappan


Tillotson,


Houang,

1984/1985;


Vinner,


1989)


Additional


analyses examined


relationship


the


individual


tests,


measured


time of


the


LMI,


scores


to determine


any test had a more


significant


relationship to correct


response


scores


, tendency to


use


drawings,

multiple


or use of


dynamic strategies.


regression analyses


indicated


Results of


that


the


Cube Comparisons


test


the greatest


influence


on correct


response


score


.37)


use


of dynamic


strategies


.35)


Paper


Folding test had


the greatest


influence


on tendency


to use


drawings


.31) .


Card Rotations


test had


the


least


influence


on correct


response


score,


tendency to


use


drawings,


tests,


use


Paper


dynamic


Folding


strategies.


Cube Comparisons


involve


three

three


dimensional


figures


or motion


Whereas,


Card


Rotation








Spatial


Orientation,


the Paper


Folding test


as one


the measures of Visualization.


The relationships


among the


variables


were


also


investigated.


The analyses concerning the


relationships of


correct response


score with


tendency to


use


drawings,


and with


use of


dynamic


strategies


indicated


that


neither


the relationships varied across Sketchpad


availability groups.


Results


of multiple


regression


analyses


indicated


that


correct


response


score


there


a significant relationship with both


tendency to


use


drawings


use of


dynamic


strategies.


Students


extensively used


drawings and


dynamic strategies.


Both


variables


were highly


correlated


to correct


response


scores,


indicating that


they were effective


strategies


locus-


motion


problem


solving.


As reflected

to successful pro


in research,


blem solving


flexible thinking

(Davis,1986; Dover


related

Shore,


1991;


Schoenfeld,


1987;


Spiro


Jehng,


1990) .


Flexibility


indicated


by the


tendency to


use drawings


Based


on the


scoring rubric,


on the average each student


drew more


than


one


figure


each


problem.


Research and


experience


indicate


that


lack


of multiple drawings


is often a


limitation


Results


to success


suggest


in mathematics


that experience with


(Vladimirskii,


computer


1971).


images may


increase


utilization and


usefulness of


diagrams.


The







of rigidity


fixedness associated with


images


(Yerushalmy


& Chazan,


1990;


Zykova,


1969).


Students did


use


the Sketchpad when


was available.


Subjects


solving


that had


session


computer.


the computer


created a mean of


the


students


available during the


4.3


that had


problem-


figures on


the computer


available,


or 94%


created


computer


drawings


one or


more


problems.


Having


learned


content with


the Sketchpad


available,


they


students


did not have


imitated


it available


its capabilities


for the


even when


problem-solving


session


Students


used


such


language as,


traced


the


locus


I" "I


imagined


the motion,


" etc.


that


indicated


that


they


envisioned motion as


they would have on


computer.


Students also commented


that


they would have been more


accurate with


solutions


their drawings,


they


and more correct


had had the Sketchpad


their


available.


Since


the results


indicated


that


both


groups


had


similar


correct response scores,


it appears


that


the


drawings


dynamic strategies


produced


similar results.


In an


environment


in which


assessment may


be different


from


instruction,


these


results


suggest


that


problem-solving


strategies


learned


in one environment may transfer to


others.


computer


this


carried


study


, the strategies


over to


paper


employed


and pencil


with


the


problems.


The








levels


problems.


Although


the difficulty


levels


the


problems


chosen


for the LMI


were agreed


upon by


reviewer


ability


students may


have


been


overestimated.


Although


there was


a correlation between


spatial


visualization and


correct


response score


.38),


the


correlation between


strategies


utili


and


correct


response


score was


significantly


higher


DS).


The


findings agree with


those


of other researchers


that


instruction and


practice


in a


specific content area


are


more dominant


factor


for problem solving than


prior


knowledge and


skill


(Hembree,


1992


Kantowski,


1981;


Krutetskii,


1976).


Interview


concerning the


sessions


use


provided additional


of drawings


findings


the availability


computer that


supports


prior


research


conclusions


that


computer


potential


conjecturing,


searching


to resolve many


for patterns,


obstacles


generalizations


(Bishop,

Senechal,


1989;

1990;


Hershkowitz


Steen,


, 1990;


1990) .


Janvier,


Students


1987;


that


Lesh,


did not


1987;


have


the Sketchpad


available


for the written


interview


sessions


indicated


that


they would have


used


computer


had

Thos


it been available,


e


that had


the computer


perhaps would have done

available appreciated


better.

its


capabilities.


Benefits c


the computer


included


accuracy







changes


figures and measurements simultaneously.


providing


students with


"hands on"


experiences with


these


characteristics,


the development of


spatial


orientation


skill


as described by Tartre


(1990)


may


be accelerated.


Tartre concluded


that


"spatial


orientation


skill


appears


be used


in specific and


identifiable ways


. accurately


estimating the approximate magnitude of


a figure,


demonstrating the


flexibility to change an


unproductive mind


set,


adding marks


to show mathematical


relationships,


mentally moving


or assessing the


size


shape of


part


figure,


and


getting the correct


answer without help


to a


problem


in which a


visual


framework was provided"


(1990


, p.


A main


purpose


interview component


the


study


was


to explore


the ability


student


within his or


zone


of proximal


development


(Vygotsky


, 1978)


and


discover what


student has


learned


what he


capable


learning with


the aid


the researcher


(Krutetskii,


1976) .


the


interview session,


students


were


successful


problems with


varying


degrees


assistance


from


investigator.


Students


were


able


to go


beyond


the known


problems


solve


those


novel


questions


with


some discussion


components


the question


itself.


Once


students


understood


the question,


they


demonstrated


ability to change an


unproductive mind








solutions.


Sketchpad


availability


make a


difference


solutions


interview problems.


Patterns,


or causes,


failure were also of


interest,


since


they


suggest ways to expand a


student'


zone.


Student


comments


indeed


indicated


the most


that


first phase of


important,


the most critical


problem solving


(Polya,


1957).


Understanding the


problem was


the biggest hindrance


to successful


that


solutions.


understanding the


In many


cases,


problem caused


the


students

biggest


indicated

obstacle.


Understanding


problem may have


been


curtailed


lack


an initial


were able

within th


to draw the


eir grasps.


figure.


Once


figure, t

Assisting


students


understood and


he solution was often


students


well


in building that


initial


visual


framework


indeed a


critical


step


in the


problem-solving process


communication were most


(Polya,


important


1957).


Language


to expanding the


student's


zone


of proximal


development.


Students


problems.


were aware of what


caused


Students were often aware


that


obstacles


they


with


did


understand


the question.


As can be


seen


in the quotations


in Chapter


students commented


on the


fact


that


they


understand the question


or did not know the definitions


terms.


Anecdotal


evidence


found


in students'


descriptions


of how they


solved


problems


that


indicates


that


students







previous


problems and


the new


ones


was often a


hindrance.


Students are accustomed


to repeating


on assessments


the same


kind


of problems


experienced


during


instruction and


practice.


The need


comprehensible materials


problems


was


obvious


from the


investigations.


In many


cases,


students


did


interpret


problem correctly.


When


the


conditions were


clarified,


or the correct


figure drawn


student


was able


to proceed


to a


correct


solution.


The


wording


the correct


problems w

solution.


ras critical


Communication


the correct figure

was critical from


the


students'


and


researcher's perspective.


An emphasis


mathematics classroom on communication


important


to understanding,


instruction,


and assessment


(NCTM,


1989)


Limitations


This


study


contained


limitations


that affect


generalizability


results.


Participants consisted


high


school


geometry


students


in one high


school


with


four


teachers.


The domain was


content-specific to


locus


points.


limitation


of scoring the


LMI


lies


the


use


self-report documentation


of how


students


solved


problems,


drawing


conclusions


from the


figures


and written


solutions


provided.


Since


there were


no common


standardized


tests


that


could


be used


for matching,


common




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