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The effect of metacognitive strategy instruction on sixth-graders' mathematics problem-solving ability

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The effect of metacognitive strategy instruction on sixth-graders' mathematics problem-solving ability
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Chicola, Nancy Anne
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xiii, 175 leaves : illustrations ; 29 cm

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Mathematics -- Study and teaching (Elementary) ( lcsh )
Metacognition -- Testing ( lcsh )
Problem solving in children -- Testing ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references.
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Submitted in partial fulfillment of the requirements for the degree, Doctor of Philosophy, School of Education and Human Development, Administration, Curriculum and Supervision.
Statement of Responsibility:
by Nancy Anne Chicola.

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Full Text
I
THE EFFECT OF
I METACOGNITVE STRATEGY INSTRUCTION
ON SIXTH-GRADERS MATHEMATICS PROBLEM-SOLVING ABILITY
by
Nancy Anne Chicola
B. A., State University of New York at Buffalo,
College of Education, 1966
M. A., Northeastern Illinois University, 1976
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Administration, Curriculum, and Supervision
1992
!Al


1992 by Nancy Anne Chicola
All rights reserved.


Tli!is thesis for the Doctor of Philosophy
degree by
Nancy Anne Chicola
has been approved for the
School of Education
by


Chicola, Nancy, Anne (Ph.D., Administration, Curriculum, and Supervision)
The Effect of Metacognitive Strategy Instruction on Sixth-Graders Mathematics
Problem-Solving Ability
1
Thesis directed by Adjunct Professor Robert J. Marzano and Professor Michael J.
Murphy''
There has long been interest in the variables that influence mathematics
problem-solving, achievement because ability at problem solving has always been
associated with aptitude and intelligence. Metacognition is hypothesized to have a
strong impact on student performance in mathematics problem-solving tasks, but there
has been considerable confusion as to its components. This study integrates
metacognitive strategy instruction and mathematics problem solving with the intent
of providing educators with the tools that will increase students problem-solving
ability.
This study investigates the relationship of direct teaching of curriculum-
embedded metacognitive strategies with the mathematics problem-solving
performance, self-perception, attitude, and metacognitive strategy use of sixth-grade
students. A multigroup, pretest-treatment-posttest design was used. Subjects were
randomly assigned to four treatment groups, three experimental and one control. All
four treatment groups received problem-solving strategy instruction. In addition,
each experimental group received the following forms of metacognitive strategy


IV
instruction: Experimental 1, Self System; Experimental 2, Task System; and
Experimental 3; a combination of Self and Task System. An Analysis of Covariance
(ANCOVA) was used on the pre- and posttest scores to determine significance among
means.
1
Empirical and theoretical evidence is presented to support the conclusion that
metacognitive strategy instruction is an important aspect of mathematics problem-
solving instruction. The Task System of metacognition, which includes knowledge
and control of goal setting, planning, monitoring, and evaluating, may be the most
important aspect of this instruction. However, the Self System of metacognition,
which includes knowledge and control of attitude, attention, and commitment,
although important, may not be of substantive value in direct instruction. It was
found that subjects instructed in the Task System strategies improved problem-
solving performance and attitude toward problem solving when compared to all other
i
treatment groups.' Results of this study suggest that improvement of attitude emanates
from students continued success with problem-solving performance rather than direct
instruction in strategies to change attitudes.
The form and content of this abstract are approved. I repommgnd ityipubligation.
Signed


!
V
To the memory of my mother, Constance Vera Wastyk, and to my father, Steven
Wastyk. They inspired me to become a life-long learner.
To my husband, Michael J. Chicola, without whom this research would be
incomplete.


VI
ACKNOWLEDGMENTS
I am extremely appreciative to my dissertation directors, Dr. Michael J.
Murphy and Dr. Robert J. Marzano, for their advice and support during the editing
and revising of this thesis. Additionally, for providing patience and persistence to
insure quality research.
To Dr. William Juraschek for his critical editing and persistent scrutiny of my
resources, I am most grateful for the time spent discussing the conceptual idea of
metacognition and its instructional value.
I am especially appreciative to my fourth and fifth readers, Dr. William L.
Goodwin and Mr. Ernest O. Porps, for their critical reading of this thesis.
A special thanks to Dr. Karen Harvey for her inspiration to finish what I had
already started and for her direction with my career in education. Also, to Ms.
Deena Tarleton and the staff, community, and children at Willow Creek Elementary
School, Englewood, Colorado, without whom this research would not have been
possible.
To my family and friends I extend particular recognition for all their loving
support throughput this onerous process. I wish I could thank you all individually,
but I hope it is sufficient to say that I recognize all of you. In particular, Dr. Debra
Pickering gave me that consistent push that I needed to pursue my degree to fruition.


CONTENTS
LIST OF FIGURES......................................................xi
LIST OF TABLES ......................................................xii
CHAPTER
1. INTRODUCTION.....................................................1
Purpose ..................................................4
Significance of Study.....................................5
2. REVIEW OF RELATED LITERATURE................................... 12
Descriptions of Metacognition........................... 12
Metacognition and the Self System ................ 21
Metacognition and the Task System................. 27
Metacognition and Strategy Instruction............ 29
Metacognition in Specific Domains ...................... 31
,1
Metacognition and Reading......................... 31
Metacognition and Mathematics Problem Solving .... 35
A Working Model of Metacognition........................ 37
Self System .......................................38
, Task System..........................................39
Metacognitive Strategy Instruction................ 40
Definition of Terms......................................42
General Hypotheses.......................................45


viii
Concluding Remarks ............
3. RESEARCH DESIGN.......................
'i
Summary of Problem Statement .
Method ........................
: Experimental Design ....
Subjects.................
Instruments..............
Materials................
i Procedures..................
Limiting Factors...............
in
Hypotheses.....................
Summary of Research Design . .
I
4. RESULTS...............................
Problem-Solving Performance . .
Hypothesis 1.............
Hypothesis 2.............
Self System Strategy Performance
Hypothesis 3.............
Task System Strategy Performance
Hypothesis 4.............
Attitude Toward Problem Solving
Hypothesis 5....................
47
49
49
50
50
51
52
58
59
61
62
64
65
67
67
72
74
74
78
78
82
82


ix
Problem-Solving Performance and Attitude................... 86
Correlations Between Pre- and Posttests.................... 90
Conclusion .................................................92
5. DISCUSSION.......................................................94
Hypothesis 1: Metacognition and
Problem-Solving Performance.................................94
Hypothesis 2: The Metacognitive Self and Task System and
Problem Solving Performance...............................96
Hypothesis 3: Metacognitive Self System and Self-Esteem ... 98
Hypothesis 4: Metacognitive Task System and
Strategy Use ..............................................100
Hypothesis 5: Attitude and Problem-Solving
Performance ...............................................101
The General Relationship Between Attitude and Ability.....103
The Contribution of Metacognitive Strategy Use to
Mathematics Problem Solving Performance ...................104
Conclusions................................................106
Implications for Teaching and Learning.....................107
Recommendations for Future Research .......................108
APPENDIX ................................................................110
A. Attitude Toward Problem Solving in Mathematics Scale.......Ill
B. Learning and Study Strategies Inventory (Adapted)...........115
LASSI Test-Retest Data ....................................121
C. Problem Solving Pretest .........................122
D. Problem Solving Posttest....................................128


X
E. Sample Lesson Plan...........................................135
F. Questions or Before, During, and After.......................145
G. Student-Generated Strategies, Self System ...................148
H. Student-Generated Strategies, Task System....................152
I. Self-Perception Profile for Children, Reliabilities..........156
J. Figures for Pre- and Posttest Scores by Treatment and Level . 158
.i,
REFERENCES ...............................................................167
ft
i
i


LIST OF FIGURES
Figure
2.1 Metacognitive Learning Model................................43
3.1 Scoring Point Scale for a Holistic Approach
to Evaluation...............................................55
4.1a Correlations for Problem Solving and Attitude
Control Group and Experimental 1............................ 87
4. lb Correlations for Problem Solving and Attitude
Experimental 2 and Experimental 3 .......................... 88
4.2 Correlations for Problem Solving and Attitude
Achievement Levels (low, middle, high) ..................... 89
A. 1 Problem-Solving Assessment
Pre- and Posttest Scores by Treatment.......................161
A.2 Problem-Solving Assessment
Pre- and Posttest Scores by Level ..........................160
A.3 What I Am Like
Pre- and Posttest Scores by Treatment.......................161
A.4 What I Am Like
Pre- and Posttest Scores by Level ..........................162
A.5 LASSI
Pre- and Posttest Scores by Treatment.......................163
A. 6 LASSI
Pre- and Posttest Scores by Level ..........................164
A.7 Attitude Toward Problem Solving Scale
Pre- and Posttest Scores by Treatment.......................165
A. 8 Attitude Toward Problem Solving Scale
Pre- and Posttest Scores by Level ..........................166


LIST OF TABLES
Table
3.1 Mean Grade Equivalent Scores by Ability Level
Quantitative Section on the Iowa Test
of Basic Skills ..............................................53
4.1 Problem-Solving Assessment
Means and Standard Deviations by Treatment.................. 68
i
4.2 Problem-Solving Assessment
hieans and Standard Deviations................................69
4.3 Problem-Solving Assessment
Summary of Analysis of Covariance.............................70
4.4 What I Am Like
Means and Standard Deviations ................................75
4.5 What I Am Like
Means and Standard Deviations ................................76
4.6 What I Am Like
Summary of Analysis of Covariance........................... 77
4.7 Adapted LASSI
Means and Standard Deviations by Treatment.................. 79
4.8 Adapted LASSI
Means and Standard Deviations by Level...................... 80
4.9 Adapted LASSI
Summary of Analysis of Covariance............................ 81
4.10 Attitude Toward Problem Solving Scale
Means and Standard Deviations by Treatment.................. 83
4.11 Attitude Toward Problem Solving Scale
Means and Standard Deviations by Level...................... 84
4.12 Attitude Toward Problem Solving Scale
Summary of Analysis of Covariance........................... 85


Xlll
4.13 Correlations
Pre- and Posttest by Treatment...............................91
4.14 Correlations
Pre- and Posttest by Achievement Levels...................... 92


CHAPTER 1
INTRODUCTION
This study investigates the effect of the explicit teaching of metacognitive
strategies within a mathematics problem-solving curriculum with sixth-grade students.
It is this researchers contention that the explicit instruction of metacognitive
strategies will result in improved problem-solving performance for students. Further,
the attitudes, strategy use, and self perceptions of these students will be enhanced
after instruction.
Both educational practitioners and researchers have long been interested in the
variables that influence mathematics problem-solving achievement as problem solving
is highly correlated with aptitude and intelligence. For example, Polya (1957) related
problem-solving ability to intelligence and conscious thinking. Further, he noted that
problem solving should be the primary vehicle through which intelligence is
developed. Dewey (Archambault, 1933) considered problem solving and reflective
thinking to be synonymous. He also believed that developing problem-solving ability
would naturally lead to increased knowledge of subject matter.
When most educators discuss problem solving, they are referring to questions
for which there is no automatic path to follow to reach an answer. Charles and
Lester (1982) defined a problem as being a task for which one "wants or needs to


2
find a solution, ...has no readily available procedure for finding the solution, and
...must make ah attempt to find a solution" (p. 5). They emphasized an increasing
demand for people to be able to analyze, plan for, and solve problems as a skill for
the future. The use of problem solving as a valuable instructional tool will better
prepare students to learn things that are presently unknown.
In spite of the perceived importance of problem solving, recent studies (Bums,
1984) have indicated that although students are able to compute adequately, they are
unable to solve problems. Students are especially lacking in the ability to understand
and solve problems which require higher-level cognitive skills. Specifically, it has
been noted that the amount of information our society makes use of is increasing
geometrically. Future workers (Commission on Standards for School Mathematics,
1989) will no longer be expected to rotely perform tasks on an assembly line, rather
their knowledge and problem solving ability will be essential in a new information
society. The twenty-first century will demand workers who are well-educated, and
citizens who are ;good problem solvers.
In an attempt to explain students poor performance on problem-solving tasks,
some researchers (Garofalo & Lester, 1985) have studied the role that metacognition
plays in the performance of cognitive tasks. Although metacognition is hypothesized
to have a strong impact on student performance in mathematics problem-solving


3
tasks, there has been considerable confusion as to its components.
The term metacognition first appeared in the literature (Flavell, 1971) in
reference to metamemory. Later, researchers related it to performance on reading
and reading comprehension (Gamer, 1987; Forrest-Pressley & Waller, 1984).
Recently, educators have expanded its meaning in relationship to problem solving in
mathematics (Campione, Brown, & Connell, 1988; Lester, Garofalo, & Kroll, 1989;
Schoenfeld, 1985).
Even with the increased emphasis on metacognition, it continues to be a
"fuzzy" concept. Not until recently has there been a delineation as to the various
components of metacognition. Flavell (1985) identified metacognitive knowledge as
a component containing three variables. Metacognitive knowledge is knowledge
about persons, tasks, and strategies. Person variables include knowledge and beliefs
about humans as, cognitive processors. Task variables encompass comprehension of
i
the nature of information and task demands. Strategy variables include the strategies
known and used in a cognitive task. Others (Brown, 1987; Campione, 1987; Chi,
1987; Gamer, 1987; Wellman, 1985) have identified similar variables.
From this research one can conclude that there are three aspects of
metacognition that are important to study. These three aspects include the Self
I
System, the Task System, and direct strategy instruction. The Self System is the
[
interrelationship between attitude, attention, and commitment, and is influenced by
the beliefs of the'learner. This vital relationship has often been omitted in research


4
on metacognition. The Task System is the interrelationship between goal setting,
planning, monitoring, and evaluating and is influenced by the specific task at hand
and by the Self System of the learner. It is this researchers contention that the
interaction of the Self and Task Systems is an important element of learning in
problem solving. The third, but equally important, component in metacognition is
strategy instruction which emphasizes the control of cognition, or self-regulation, and
focuses on both the Self and Task Systems. It is considered the influencing factor
in actual performance and serves to regulate and oversee learning. Although an
interrelationship'between the Self System, Task System, and strategy instruction has
been acknowledged, there have been few investigations of the specific nature of this
interrelationship! Given its explanatory power for learning in general, metacognition
appears to have, great promise as an explanatory factor in mathematics problem-
l
solving performance.
Purpose
The purpose of this study is to determine the relationship between the
metacognitive components of the Self and Task and the influence of strategy
instruction on performance in mathematics problem solving. Specifically, this study
investigates whether the direct instruction of strategies in the Self and Task Systems
of metacognition improves the mathematics problem-solving performance of sixth-


5
grade students.' This study addresses three primary issues:
i
1. Whether instruction in the Self System strategies for controlling attitude,
commitment, and attention improves students mathematical problem-solving
performance.
2. Whether instruction in Task System strategies for controlling goal setting,
planning, monitoring, and evaluating improves students mathematical problem-
solving performance.
3. Whether instruction in the combination of the Self and Task System
strategies produces a stronger effect on mathematics problem-solving achievement
than instruction in the Self or Task System studied separately.
Additionally, this study attempts to control for affective and ability factors.
Also of interest is the question of whether attitudes and beliefs toward problem
.1
solving and self-icontrol of learning change significantly after treatment.
Significance of Study
This studys significance is based on two elements: (a) the importance of
problem solving to general reasoning and (b) the importance of metacognition to
problem solving. In the context of U.S. students continuous decline in achievement
on recent assessments of mathematical skills, teaching mathematics problem-solving
skills continues to be an area of concern. The nations economic health depends on
the mathematics-related skills contained in problem-solving instruction. It is


6
imperative that students learning of mathematics and problem solving improves. A
successful approach to mathematics problem solving is long overdue.
In April, 1989, the NCTM presented new standards for mathematics
l
performance of;K-12 students. These standards address new societal goals and the
nations movement from an industrial society to an information society. The
numeracy of students is the intent of the new goals. Specifically, the new goals
emphasize the development of each students ability to solve problems as essential to
becoming a productive citizen. NCTM noted that future employees must be able to
set up problems, develop knowledge of a variety of techniques to approach and work
on problems, understand the underlying mathematical features of a problem, work
with others on problems, see the applicability of mathematical ideas to common and
complex problems, and be prepared for open problem situations. This study could
have a major impact on the education community if an approach to mathematics
problem solving: can be established which encourages effective problem solving
among students.
"Metamemory means knowledge or cognitive activity bearing on anything
mnemonic; it is, therefore, metacognition that takes memory enterprises as its object"
(Flavell, 1985, p. 240). Schneider (Forrest-Pressley, MacKinnon, & Waller, 1985a),
'j
in his analysis of the literature on metacognition, concluded that there is a strong
relationship between metamemory and memory behavior, implying that metacognition
has considerable 'utility in teaching effective learning strategies in any curriculum


7
area. Flavell (1985) considered metacognitive thinking as a cognitive tool with broad
applicability. He stated further that metacognition has value in education as a
teaching tool because of the important role that metacognitive skills play in learning.
Metacognition is central in childrens knowledge about cognitive processes.
Students (Borkowski & Kurtz, 1987) often fail to use learning strategies or higher-
order metacognitive processes in problem solving. Further, not only must learners
be aware of specific learning strategies but must be able to effectively use them and
monitor their success. Considering that decisions to call upon certain cognitive
strategies during: mathematics problem solving are metacognitive in nature and greatly
impact the problem solvers chances of success (Silver, 1982), one could conclude
that there is a symbiotic relationship between metacognition and problem solving.
In a review of research on metacognitive strategy use in mathematics problem-
solving activities, Garofalo and Lester (1985) suggested the importance for teachers
of mathematics not only to incorporate metacognitive strategy instruction into
mathematics, but to develop a metacognitive posture toward mathematical
performance with their students. Further, they agreed that mathematics educators
must address the concept of metacognition and the important role it may play in
mathematics problem solving.
Polya (1957), in How to Solve It (first published in 1949) took a giant step in
the area of mathematics problem solving. His goal was to improve the problem-
solving ability of the learner by developing general strategies. He viewed problem


I'l
8
solving as an art that could be learned only by imitation and practice. He outlined
I
a general strategy for problem solving within which each problem solver could plan
a solution unique to the specific problem to be solved. His general strategy had four
steps: understanding the problem, devising a plan, carrying out the plan, and looking
I
back. Within each step, the problem solver must ask several questions to be
successful. Although Polya did not use the term metacognition, this general strategy
to mathematics problem solving prompts the use of many metacognitive strategies.
Wicklegren (1938), another well-known mathematician, warned against a
cookbook approach and espoused a teaching-by-example method in problem solving.
I
He suggested that the levels of skill development influence how well one makes use
of specific thinking strategies; thus, the explicit training of those basic skills is
essential to gain]a level of automaticity.
Although,there have been numerous studies (Krulik & Reys, 1980) involving
mathematics problem solving, most of these studies focus on a particular strategy to
guide problem solvers in their pursuit of an answer; few address metacognition.
Additionally, the Research within metacognition has failed to differentiate between the
unique effect of strategy instruction that relates to the Self System versus strategy
instruction that relates to the Task System.
The Self System encompasses that area of metacognition that is best
described as "knowledge and control of self' (Marzano et al., 1988) which includes
beliefs about self, more specifically described as attitudes, attention, and


9
commitment. However, it is the relationship between these components of self that
can enhance or limit success. Mathematicians (Lester, Garofalo, & Kroll, 1989;
Silver, 1985; Schoenfeld, 1985) assert that belief systems are important in
influencing success in problem-solving performance. Hyde and Bizar (1989)
recognize a direct link between metacognition and effect. Reflective thinking about
feelings and beliefs about chances of success are integral to positive problem-solving
performance. Research in the Self System of metacognition could be important
because ones attitudes about problem solving and the skills involved, ability to focus
attention, and commitment to the task are believed to be important components of
successful problem solving.
"l
The Task; System includes knowledge and control of task, more specifically
goal setting, planning, monitoring, and evaluating. Many math educators suggest
:i
some sort of problem-solving approach that encompasses these components.
However, they fail to show how to use these components effectively with specific
j
problem-solving strategies.
A common complaint about typical step-by-step methods for problem solving
has been the lack of generalizability to other problem-solving situations. An
emphasis of this study is on developing a repertoire of strategies as well as the skills
to access and monitor these strategies as needed. The quest for a universal problem-
I
solving tool has; continued for centuries (Polya, 1981). Metacognitive strategy
instruction can help students develop an awareness of their thinking which leads to


10
this instructional model. Executive control in problem-solving situations could
provide an important alternative to traditional problem-solving instruction.
Finally, the integration of Self and Task System strategies into a
comprehensive instructional program for teaching mathematics problem solving could
have a major impact on the problem-solving performance of students. Task System
strategy instruction has most often been the emphasis of research in metacognition.
The Self System, strategies are often overlooked as an integral part of instruction, yet
those who have studied them have found that they impact problem solving ability
(Garofalo, in press; Harter, 1984; Lester, in press; McCombs, 1988). The key to
the present study is to determine the effect of instruction in metacognitive strategies
for the Task and Self Systems and the interaction of these systems on the mathematics
problem-solving performance of students. Comparisons of each metacognitive system
will help clarify the impact of each system on overall success at problem solving and
define the possible importance of the interaction of both systems. The research of
the last decade i(Forrest-Pressley, MacKinnon, & Waller, 1985a) suggests that
metacognition plays an important role in student learning. The intent of the present
study is to clarify precisely which aspects of metacognition impact student learning
in the area of problem solving.
In summary, the present study focuses on the role of the Self and Task
['
Systems of metacognition and their interaction on the problem-solving performance
of sixth-grade students. Understanding the impact of metacognitive strategy


11
instruction could initiate development of a valuable model for mathematics problem
solving for use, with students.


CHAPTER 2
REVIEW OF RELATED LITERATURE
For the purpose of this study, the review of related research is arranged into
five areas: (a) Descriptions of Metacognition, (b) Metacognition and the Self System,
(c) Metacognition and the Task System, (d) Metacognition and Strategy Instruction,
and (e) Metacognition in Specific Domains. This research provides a foundation for
the development, of a working model of metacognition. The chapter concludes with
(a) a Definition of Terms and (b) the General Hypotheses that will be used in this
study.
Descriptions of Metacognition
Flavell (1971) coined the term metamemory to describe knowledge an
individual is able, to verbalize with respect to information storage and retrieval. His
research into metamemory initiated a decade of research into metacognition and the
construct of metamemory. Flavell (1987) defined metacognition as "cognition about
cognitive objects" (p. 21). He concluded that metacognition is cognition about
anything cognitive, a definition which this author believes contributes to the
"fuzziness" of this topic. Flavell extended his concept of metacognition to include
cognition about anything psychological. This expanded concept encompassed


13
consciousness of emotions and motives or of any kind of monitoring, for example,
when one tries to monitor ones problem-solving activity during a problem-solving
situation.
He divided metacognition into two distinct but interacting constructs:
metacognitive knowledge and metacognitive experience. He stated that metacognitive
knowledge "refers to the part of ones acquired world knowledge that has to do with
cognitive (or perhaps, psychological) matters. As people grow up, an important part
of what they learn or come to believe concerns the mind and other things
psychological" (p. 21).
Flavell (1985) subdivided metacognitive knowledge into three categories:
person, task, arid strategy. The person category refers to beliefs about self and
I
others, and beliefs about cognition in general. The task category can be described
as the recognition that performance is influenced by the nature of the task information
and the nature of the particular demands of the task. When describing the strategy
category, Flavell distinguished cognitive from metacognitive strategies. Cognitive
I
strategies are any strategies that are used to complete a cognitive task, whereas
metacognitive strategies are those that serve to monitor a cognitive task. Further, he
explained, the person, task, and strategy categories interact so that one develops an
intuition about which strategies are better for a particular task.


14
Flavell (1987) described metacognitive experiences as "conscious experiences
I1 i
that are cognitive and affective" (p. 24). More specifically, metacognitive
experiences are the experiences you have when you are aware of your thinking. If
you know that you do not understand a problem or that a task is difficult, then you
are having a metacognitive experience. Flavell developed a model (cited in Brown,
I
1987) of metacognitive elements in which cognitive goals, metacognitive knowledge,
metacognitive experiences, and strategy use interact.
The working model for metacognition used in the present study has categories
'll
similar to Flavells person, task, and strategy components. However, there are
I
important differences. In the present model, (a) strategy use is explicitly taught, (b)
self and task are two separate and distinct categories for strategy instruction, (c)
metacognitive experiences and metacognitive knowledge are inseparable in the larger
concept of metacognition, and (d) metacognitive strategy control is a vital
i,
instructional component.
Wellmans definition of metacognition is similar to Flavells definition of
metacognitive knowledge. Wellman (1985) called metacognition a "theory of mind"
(p. 2). He defined it as "the persons knowledge of cognitive processes and states
't
such as memory, attention, knowledge, conjecture, illusion" (p. 1). Wellman did not
separate metacognitive experience from metacognitive knowledge. He held a
developmental perspective of metacognition, based in Piagetian theory. He viewed
metacognition as developing in children and existing in adults. He identified five


15
different but Overlapping sets of knowledge that could be part of a persons
metacognition:,! existence, distinct processes, integration, variables, and cognitive
;i
monitoring. Wellman described existence as the difference between a mental state
ij
and an external, behavior. A person knows that thoughts and internal mental states
exist, but they also know that these internal states are not the same as the actual
external acts or 'events; Wellman used the example of knowing something is true,
but pretending its not. Distinct processes has to do with distinguishing between
[i
different mental acts such as lying and pretending. Integration speaks to the
similarities and interactions among cognitive tasks; although there are differences
among cognitive tasks, one also uses the similarities in mental processes. Variables
refers to those faictors that influence ones ability to accomplish a task; for example,
ones level of concentration is affected by the difficulty of the task. Finally,
cognitive monitoring refers to an individuals ability to accurately assess and monitor
the contents of his or her mind. This is part of a ones understanding of cognition,
which relates to the concept of executive control.
I
The working model for metacognition used in the present study separates the
self from the task for the purpose of instruction. Although Wellman identified five
different sets of knowledge, he did not separate self from task within metacognition.
Wellmans categories are reflected in the Self System and Task System from this
authors model. His variables category closely resembles the commitment aspect of
the Self System; one must consider a different level of commitment depending on the


p
16
degree of difficulty of a task. Cognitive monitoring in Wellmans model relates to
the monitoring of strategies in the Task System. The control of metacognitive
strategies is subsumed under cognitive monitoring in Wellmans model; in the present
model, control is explicitly taught.
Brown (1987) does not separate metacognition into self, task, and strategy
components, although she refers to two important concepts underlying metacognition:
knowledge about cognition and regulation of learning. Knowledge about cognition
is thinking about ones own thinking and is often identified through verbal reports of
ones own cognitive processes. Regulation of learning consists of those activities one
engages in to oversee learning.
A number of theorists in metacognition emphasize the concept of executive
control. Executive control refers to the ability to perform an intelligent evaluation
of ones own mental operations (Brown, 1987). The concept of executive control is
borrowed from the information-processing models of cognition. Brown noted that
the influence of task difficulty and familiarity are important executive control factors
when exploring the planning and monitoring of ones own activities. She viewed
metacognition as covering the regulatory functions of error detection and correction
and, thus, considered it a "tool of wide application" (cited in Flavell, 1985, p. 110).
Brown made the distinction, based on Piagetian theory, between self-
regulation during learning and other-regulation, that is, regulation that comes from
an outside source, as with a parent controlling the behavior of a young child. The


17
transition from other-regulation to self-regulation is one that is a normal part of child
development. The most common example of this is from parent (usually the mother)
to child. As the child gains experience, the parent gradually relinquishes that control
to the child until the child is self-regulated. This transition also occurs in the
relationship between a teacher and a student. In the classroom, the teacher initially
assumes the regulatory role. Mediated learning is an important step in the transition
from other-regulation to self-regulation. Through guided practice, students gradually
learn to regulate their own learning and thinking processes.
The development of self-regulation in the student is critical. Cognition-
initiating the thinking processes-opens the possibility for metacognition.
Metacognitionthinking about your own thinkingis an important tool for students
to become independent learners and thinkers.
The working model for metacognition used in the present study resembles
Browns work in, that both knowledge about cognition and regulation of learning are
addressed. Browns view assumed the learning of strategies without differentiating
a relationship to the self and to the task. However, this authors model separates the
Self and Task System for purposes of metacognitive strategy instruction. Mediated
' i
learning becomes crucial when considering any instructional model using
metacognitive strategies. The other- to self-regulation transition Brown mentioned
relates to the explicit strategy instruction component of the present study. This
component has a strong impact on how strategies are taught and how executive


18
control is obtained.
Gamer (1987) also emphasized the concept of executive control. In describing
metacognition, she drew from two bodies of research: (a) developmental psychology,
for a definition of metacognition, and (b) information-processing cognitive
psychology, for a definition of executive control and its relationship to metacognition.
She concurred with Flavells description of metacognition, which identified
metacognitive knowledge, experience, and strategy use as interrelated variables.
However, Gamer concluded that these variables are not only interrelated but build
on one another.! She asserted that there is a sequence of learning, "one in which
metacognitive knowledge is a basis for metacognitive experiences that in turn prompt
the use of cognition and metacognitive strategies" (p. 20). Gamer, like Flavell,
stated that metacognitive knowledge is highly interactive knowledge about ourselves,
tasks, and strategies. Metacognitive experiences are those experiences that have to
do with progress i toward a goal and occur before, during, and after the task. Before
the task, personal strength is considered; during the task, strategy knowledge is
important; after the task, task information is the focus. Strategy use is both cognitive
and metacognitive.
'i
'l
Like Brown, Gamer (1987) compared metacognition and executive control by
first focusing on their origins. Metacognition grew out of Piagets developmental
theory; executive control theory emerged from information-processing research.
Research in metacognition has focused on adult-child clinical interviews, whereas


19
research in executive control has stressed task analysis and computer simulations with
adult intervention. A conceptual overlap occurs between metacognitive research and
executive contrbl research. This overlap occurs in the emphasis of metacognitive
knowledge for detailed, sequential analysis of tasks; strategy use by learners; and
cognitive monitoring of movement toward goals. Educators combine the knowledge
with the control learners bring to a task and use knowledge and control for their
greater area of Concern: instruction.
il
i
The working model for metacognition used in the present study focuses on
instruction. Garner identified the areas of self, task, and strategies and even included
strategy instruction as an important element of classroom instruction. Yet, unlike the
present research, she failed to separate the self and task for the purpose of
instruction. (It is reasonable to state that most well-known researchers have viewed
1;
metacognitive research from the point of view of the Task System only.) There is
I
a relationship between Gamers notion of a students personal strength before the task
and the Self System as described in this study. Gamers concept of strategy
knowledge during the task fits with explicit strategy instruction in this authors
i
model. Further, there are similarities between the Task System and Gamers ideas
about task information.
Marzano et al. (1988) described metacognition in simpler terms as "...being
aware of your own thinking as you perform certain tasks and then using this
awareness to control what you are doing" (p. 9). (Other researchers, especially Paris
I


20
and Winograd (in press), have concurred with this view of metacognition.) Marzano
et al. identified two principal features of metacognition: knowledge and control of
self and knowledge and control of process. A person brings to a task the knowledge
and control of some level of commitment, specific attitudes, and focus of attention.
These are considered the Self System variables in the present study.
When describing process variables, two elements are (a) types of knowledge,
and (b) executive control of behavior. Initially, Marzano et al. identified three types
of knowledge critical to metacognition: declarative, procedural, and conditional.
Later, McCombs and Marzano (1990) omitted conditional and added strategic
knowledge and teleologic knowledge, that is, knowledge that is "dynamically
organized and supported by differentially developed metacognitive and cognitive
processes" (p. 56).
Marzano et al. identified three aspects of executive control of behavior:
evaluation, planning, and regulation. In a later work, Marzano (1989) described
knowledge and control of task, which included goal setting, planning, correcting for
error, and evaluating. Correcting for error corresponds to the earlier concept of
regulation and infers a monitoring mechanism during the task performance. Goal
setting is an additional behavior that students engage in before making a plan. Goal
setting before mathematics problem solving may be as simple as defining and
deciding to solve a problem, whereas planning includes delineating all of the steps
required to achieve that goal.


21
The working model for metacognition used in the present study has many
similarities to the 1988 model developed by Marzano et al. They separated
metacognition into two distinct areas: self and process, which corresponds to this
authors self and task categories. Further, they acknowledged strategy instruction as
an important and integral part of self and process. Based on this model, the present
study integrates ^explicit strategy instruction into the Self and Task Systems.
In summary, it should be noted that there are common threads in these models
of metacognition. Although there is relative agreement that there are at least two
systems, Self and Task, interacting within a metacognitive model, most research has
I
addressed the Task System without considering the possible interaction of both
systems. It is this authors contention that both systems are integral to metacognition
and that strategy instruction is an important overriding factor in any instructional
model. The next section of the literature review will address the Self and Task
Systems and their relationship to a general metacognitive theory.
Metacognition and the Self System
The self is a variable is often ignored in the study of metacognition. The
focus of most metacognitive research has been on ones knowledge and control of the
task with little emphasis on the more affective issues of the self. Flavell (1987) made
reference to person variables in metacognitive knowledge when he spoke of beliefs


1
22
about self, others, and cognition in general. Yet, he continued to emphasize task and
strategy components while omitting the self component in instructional models for
research. Others (Brown, 1987; Cross & Paris, 1988; Dinnel, 1986; Forrest-
Pressley & Waller, 1984; Gamer, 1987; Palincsar, 1986; Wambach-Schmidt, 1987)
who have studied metacognition have also focused primarily on task variables.
McCombs and Marzano (1990) contended that the self generates both the will
and motivation for self-regulated learning. They described the role of self as
"agent"; the self is the overseer of learning and generates "will" through intentional
|:
choices. An awareness of self as the source of willful, goal-directed actions could
be an important1 part of ones knowledge base. Once this awareness occurs, the
individual is able to regulate learning. According to McCombs and Marzano (1990),
the self has distinct functions during regulation. Those functions
(a) provide a sense of continuity via influencing every aspect of the
processing of self-relevant information, (b) regulate affect or defend self
against negative emotional states via self-enhancement strategies, and (c)
motivate individuals via commitments to future self-relevant goals, (p. 62)
One explanation for the lack of attention to the self in studies of metacognition
could be the difficulty of controlling Self System variables and identifying specific
strategies for improvement. The implications for self-regulated learning are that
interventions must be aimed at the learner and learning environment. Educators must
facilitate the development of an awareness of the learners will and must provide
I
strategies that will facilitate accomplishment of self-goals.


23
The Self System involves beliefs the learner holds about self. The present
study focuses on providing strategies to the learner for controlling the attitudes,
commitment, arid attention of the Self System. These components are under the
control of the learner, yet often the involuntary view is held in which the learner
assumes no control.
A key impact on the self is ones beliefs. Lester, Garofalo,'and Kroll (1989)
viewed beliefs as ones subjective knowledge about self. They concluded that beliefs
often influence decisions made during problem solving because of their interaction
with attitude. These researchers undertook a study about the role of metacognition
in the mathematics problem-solving behavior of seventh-grade students. They
analyzed students misconceived beliefs about abilities, how the nature of problem
il
solving influences the formation of attitudes toward mathematics, and, consequently,
how students monitor their work and make use of their knowledge.
Garofalo (in press) suggested that students belief systems are fostered by
school practices, curricula, classroom environments, and instructional practices. In
his study, students were instructed in the subtraction of mixed numbers, then
'I
presented with a relatively simple problem of finding 3 minus 2 1/2. The students
could not formulate a procedure for solution. They tried to find an algorithmic
solution rather than viewing the problem in a logical, realistic, or concrete manner.
They made no attempt to verify or negate their answers.
Garofalo believed that these responses were not unexpected since the students


I
school experiences entailed computation exercises that were solved with the
application of step-by-step procedures. This has strong implications for classroom
instruction. A logical approach to problem solving will influence those belief systems
|!i
so that students may become more effective problem solvers. For instance, students
attitudes may be influenced by clear, explicit instruction about how to approach
mathematics problem solving.
When presenting information about the key influences on problem solving,
i
Lester and Garofalo (in press) asserted that beliefs about self, mathematics, and
problem solving play a dominant role in a students problem-solving behavior. They
reported how students misconceptions influenced attitudes, work monitoring, and
knowledge use. Beliefs affected behavior, especially behavior in mathematics and
mathematics problem solving.
McLeod (1988) asserted that beliefs held by students interact with their
emotional states. This interaction is an important consideration in establishing a
theoretical framework for research on affect in mathematics problem solving. In
i
l
metacognition, both an awareness of ones own cognitive processes and the regulation
of cognition are crucial in problem solving and are linked with affective concerns.
The attitudes a student brings to problem solving result from both knowledge and
beliefs about problem solving, which have been reinforced through his or her prior
experiences with problem solving.
li
Reynolds and Shirley (1988) described the role of attention in studying and


.1
It
I
25
learning. They1 asserted that the extent to which students monitor their attention
l',
while studying may be as important as learning the study skills themselves. When
studying text, students use differentiated attention to determine the information upon
I
which to focus attention, the level of attention that must be allotted the particular text,
and the duration jof attention to successfully learn the information. Attention becomes
an important Self System piece of studying and learning.
'i
Silver (1SJ82) suggested that the belief system of a problem solver is an aspect
of metacognition: that affects the encoding and retrieval of information in mathematics
problem solving. Students are more likely to use a strategy if they believe that
strategy can lead to a solution. Perseverance, strategy selection, and satisfaction may
be explained by the belief system held by the student. Belief systems and their effect
on metacognitive systems may be an important direction for future research (Silver,
1982; Schoenfeld, 1985).
Weinert and Kluwe (1987) linked motivation with metacognition by suggesting
that past performance in a task influences ones motivation to think about various
methods of problem solving. Further, he asserted, past performance affects ones
ability to control that thinking in order to complete a task. He reported that in
metacognition and motivation research an overlap of variables appears. Although
there are differences between metacognition and motivation in learning theory, the
likenesses suggest that an integrated research plan would be reasonable.
In a paper by Borkowski, Carr, Rellinger, and Pressley (in press), the


26
interdependence, of metacognition, attributions, and self-esteem was considered.
"Although motivational states often direct and energize human behavior, they also
play more subtle roles in determining actual strength, shape, or functioning of
cognitive processes" (p. 2). These researchers further concluded that students must
be convinced that the knowledge and strategies they learn in school will allow them
to become proficient adults, able to perform lifes many purposes competently.
Further, "If children can be led to believe they are acquiring powerful and important
tools, self-esteem and self-confidence should also increase" (p. 47). This should build
the motivation for learning new skills. Success in strategy use will affect motivation
and self-esteem, which will promote further strategy learning and transfer.
McCombs (1988) considered motivation an important basis for attitude, which
helps to determine whether or not effective learning takes place. Before a learner is
'j
able to engage in' self-management of his or her own learning, he or she must have
appropriate attitudes and orientations toward learning. McCombs reached five
conclusions:
First, the metacognitive skills of self-awareness, self-evaluation, and
self-regulation provide a basic structure for the development of positive self-
control. Second, perceptions of personal control underlie continuing
motivation and are reciprocally influenced by perceptions of personal
competency of self-efficacy such that both contribute to continuing motivation,
perhaps in their effects on feelings of self-worth. Third, there is a set of
general self-development skills related to the development of self-system
structures and processes that appears to be prerequisite to students ability to
assume personal responsibility and control as well as apply specific learning
skills. Fqurth, skill training programs can be effective in changing both
negative self-views, attitudes, and orientations toward learning and specific


27
metacognitive and cognitive skills required for self-regulated learning.
Finally,^ interventions have tended to focus on external educational practices
in bringing about desired changes in internal processes, interpretations, or
expectations; however, this approach should be combined with interventions
that help, students change inappropriate cognitively mediated processes and
belief systems in order to more directly influence their environment and
positively adapt themselves to changing instructional conditions, (pp. 153-
154)
McCombs (1988) constructed an integrative model of those processes that
underlie the intrinsic motivation to learn. In this model the metacognitive system
[\
interacts with both the affective and cognitive systems. She concluded that
motivation interventions should focus on "modifying the learner vs. modifying
educational practices" (p. 157). The continuing motivation to learn and the ability
for self-directed land self-regulated learning depends upon both a metacognitive self-
awareness and metacognitive task-awareness.
In summary, the effective utilization of the Self System variables of attitude,
commitment, arid attention depend upon beliefs, which, in turn, influence ones
I1
motivation to learn. The learners belief system influences the Self System in a way
that may determine success or failure with the task. Intervention with the Self
System variables to facilitate an awareness of self as agent appears to provide
students with a powerful tool to improve learning.
Metacognition and the Task System
I
Flavells (1987) task category relates to the concept of the Task System of


28
metacognition as defined in this study. When describing task, Flavell spoke of
monitoring comprehension as a key to determining how a task is approached and
what the particular task demands are. One must consider task demands when
monitoring a task; a more difficult task may require closer monitoring, whereas a
comparatively easy task may require only minimal monitoring.
In his descriptions of metacognition, Wellman (1985) concentrated solely on
those behaviors that relate to the Task System. He included a list of sets of
knowledge necessary to complete a theory of the mind. Metacognition as it relates
to the Task System includes the learners knowledge and control of cognitive systems
involved with the task, such as goal setting, planning, monitoring, and evaluating.
Goal setting is the initial step in addressing a task if one hopes to reasonably
complete that task. Goal setting includes acknowledging the existence of a problem
and accurately defining the problem (Sternberg, 1988). Redefining goals when the
original goal is seemingly unattainable helps to make the goal more attainable and
realistic and, therefore, more likely to be reached.
Much of!the theorizing about planning came from artificial intelligence
computer models. "Central to the issues of metacognition are- computer planning
models that attempt to model problem-solving behavior" (Brown, 1987, p. 83). Good
planners make many metaplans and executive decisions, exercise deliberate control
I
over planning processes, make use of knowledge information, show greater
flexibility, recognize the importance of global planning, and develop prototypic


29
procedures (Brown, 1987).
Few studies have addressed the issue of monitoring behavior. Yet, there has
been some agreement about why children do not use monitoring effectively.
According to Chi (1987), when children fail to accurately assess the degree of
difficulty of a task, it may be due more to a lack of knowledge than to poor
monitoring of that knowledge. Chi further states that monitoring is also closely
related to and necessary for executive processes.
Evaluation is a complex activity that occurs both during and after a cognitive
task. Most studies of monitoring (e.g., Weinert & Kluwe, 1987) assume that
children evaluate their progress during and after the task. Many factors enter into
this evaluation. First, the time invested in a strategy may help determine the value
of that strategy for the learner. Second, beliefs about ability and effort influence
value. Third, the childs learning style will determine if one strategy is more
valuable than another. Finally, an estimate of the amount of effort required will
influence the learner.
i
In summary, according to the present study, the Task System includes the
activities of goal:setting, planning, monitoring, and evaluating. This is the system
most generally emphasized in research relating to metacognition.
Metacognition and Strategy Instruction
In a review of studies on metamemory and the teaching of memory strategies,


30
Pressley, Borkowski, and OSullivan (1985) suggested that specific strategy
knowledge is necessary to make a strategy useful. A child needs to know strategy-
specific components to be able to use the strategy in an instance outside the training
situation. Explicit metamemory training proved successful in strategy teaching.
Pressley et al. noted that the teaching of Flavells Metamemory Acquisition
Procedures {MAPs) with specific strategy knowledge had positive results. MAPs
allow a learner to form personalized metacognitive experiences and to use those
experiences.
Researchers (e.g., Harris, Graham, & Freeman, 1988) suggested that strategy
training can produce important metacognitive improvement even without the addition
of explicit metacognitive skill training. It was also suggested that metacognitive skills
I
are an important1 component of performance.
,1
Palincsar (1986) characterized metacognitive instruction as teaching students
to plan, implement, and evaluate strategic approaches to learning. The steps that are
important in the teaching of these components include (a) wise strategy selection, (b)
guided instruction in acquisition and application, and (c) promotion of the utility and
consequences of specific strategies. Her conclusions focused on empowering students
through explicit strategy instruction, with gradual relinquishment of control from
teacher to student.
Campione (1987) found that strategy instruction coupled with instruction as
to the effectiveness of these strategies, was an effective way to improve the skills of


31
many types of students, especially problem students. This metacognitive strategy
instruction should have three essential components: (a) knowledge about the domain,
(b) specific procedures for operating within the domain, and (c) general regulatory
processes that aire task dependent.
I
Larson and Gerber (1987) looked at the effects of social metacognitive training
for enhancing overt behavior skills. These researchers inferred that the simultaneous
training of specific metacognitive awareness skills and metacognitive control skills
are a general and powerful intervention method with delinquent youth. An important
finding was that1 cognitive skills obtained in training generalized to overt behavior
outside of training sessions.
In summary, the research indicates that metacognitive strategy instruction
plays an important role in knowledge acquisition and transfer. Strategy instruction
addressing only the task has been a popular focus in recent research. There also
seems to be a link between metacognitive strategy use and ability. The next issue of
concern includes; the role metacognition plays in learning typical school-related,
domain-specific subject matter.
Metacognition in Specific Domains
' i
Metacognition and Reading
From the review of literature on metacognition, numerous studies in reading
and reading comprehension were found. The primary focus of this research has been


I
strategy instruction that relates to the Task System of metacognition. Forrest-
'i
Pressley and Waller (1984) pointed out the reasons for this plethora of research:
i
First, any conceptualization of reading that includes only decoding and
comprehension is inadequate. Why? Simply because skilled readers are able
to read strategically they can adjust what they do to the demands of the
situation] Second, reading involves metacognitive aspects as well as
cognitive;. That is, we dont just decode words, we also know about
decoding. Skilled readers dont just comprehend; they monitor their
comprehension and if something isnt working they do something about it.
Skilled readers dont just read strategically; they know about and execute
control over their strategic reading, (pp. v-vi)
Cross and Paris (1988), in a developmental and instructional analysis of third-
I
and fifth-graders: metacognition and reading performance, found that metacognitive
instruction in specific Task System strategies improved the reading performance of
the experimental' group. However, there were some inconsistencies in the results.
At the fifth-grade level, the instruction had its greatest effect on the poorest readers
and less of an effect on the better readers. At the third-grade level, the training was
not as effective for the poorest readers, suggesting that relatively good reading-
V
awareness skills are necessary for those poorer readers to be successful.
Salomon, Globerson, and Guterman (1989) used computer programming that
provided metacognitive guidance modeled after what "good readers" do through a
computerized reading partner. Seventh-grade students using this reading partner were
asked general self-guiding questions, introduced to specific reading principles, and
were asked to answer questions about self-monitoring and the use of reading
principles. The researchers found that these computer programs helped students in
I,


33
the regulation of metacognition in reading and writing activities. Improvements in
I
reading and writing were found to be directly related to the subjects ability for
metacognitive reconstruction, that is, verbal reporting of thoughts during the process
of reading and \yriting.
In a discussion of the applications of metacognitive research to classroom
instruction, Gamer (1987) suggested that strategy teaching in the classroom must be
embedded within the curriculum, rather than added as another layer of instruction.
She identified two clusters of metacognitive strategies that relate to reading
instruction: text reinspection and text summarization. These strategies all fall within
the realm of the Task System.
Gamer identified six guidelines that teachers should follow in reading
instruction. First, "teachers must care about the processes involved in reading and
studying, and must be willing to devote time to them" (p. 131). Second, "teachers
must do task analyses of strategies to be taught" (p. 133). Third, "teachers must
present strategies as applicable to texts and tasks in more than one content domain"
(p. 134). Fourth, "teachers must teach strategies over an entire year, not in just a
single lesson or unit" (p. 135). Fifth, "teachers must provide students with
opportunities to practice strategies they have been taught" (p. 136). Finally,
"teachers must be prepared to let students teach each other about reading and
studying processes" (p. 137). Gamer believed that these guidelines are central to
effective strategy instruction; to do any less is a disservice to our children.


34
In a study on metacognition and reading, Forrest-Pressley and Waller (1984)
were concerned with three aspects of reading: (a) whether or not children respond
to the demands of the reading situation, (b) the role of metacognition in reading, and
(c) cognitive and metacognitive development of reading skills and related processes.
From a pool of 227 third- and sixth-graders, 72 from each grade were randomly
selected for the study. The Gates-MacGinitie Reading Test and the Canadian Lorge-
Thomdike Intelligence Test were used to place 24 students into one of three groups:
poor, average, and good. The general plan for the investigation included the study
of both reading skills and developmental factors. The students were tested on
i
performance in the reading skills of decoding, comprehension, and strategies and the
developmental factors of language, attention, and memory. It was found that
performance on each type of skill and the ability to verbalize about each skill
increased with reading ability and grade level. It is important to note that the ability
to verbalize the strategy was highly correlated with high performance. In general,
the frequency of metacognitive strategy use increased systematically with reading
ability and grade level.
Future researchers may learn more about cognition and metacognition, the
connection between them, and the importance of explicit teaching of metacognitive
strategies to improve reading performance. Addressing the Task System in strategy
instruction is the common mode of research in metacognition and reading. The Self
System is still an untouched area for reading research and metacognition, at least


35
within the studies included in this review of literature.
Metacognition and Mathematics Problem Solving
Typically, mathematics instruction focuses on increasing students knowledge
of concepts and procedures, not on metacognitive knowledge. There is little research
in the area of metacognition and mathematics, but there is much to be learned from
the metacognitive reading research. Garofalo (cited in Callahan, 1987) noted that
l
mathematics instruction should include questions and assignments that focus on
i
student reflection, analysis, and reporting on mathematical knowledge and behavior.
Further, he suggested that strategy instruction on controlling decisions and actions
can aid students in learning how to control and regulate their behavior.
Leon and Pepe (1983) conducted a study with exceptional children that
appraised the effects of self-instructional strategy (SI) training as a procedure for
remediating deficits in arithmetic computation. These researchers suggested that SI
training is a feasible method for remediating computational deficits in mildly
handicapped students. In addition, it seems that students who experience self-
instructional strategy training are able to generalize to the same types of mathematics
problems in the mainstream setting without substantial cues from the teacher. This
strategy may have considerable effects when mainstreaming the mildly handicapped
!
student.
Dinnel (1986) found that there are strategic differences among first-year


36
college mathematics students with respect to mathematical ability. In this study, he
assessed the information-processing differences between competent and less
competent mathematics students. The competent subjects employed a pictorial
'I
i
strategy, whereas the less competent subjects approached each task linguistically.
Results of an iconic memory task suggested that competent mathematics students have
a larger iconic store visual sensory register which allows for more processing
of stimuli accessed from the external environment. The results of a sentence/picture
completion task1 suggested that the ability to construct pictorial images may be a
!,
factor in mathematical ability. Further, the ability to transpose from one symbol
system to another was an important processing skill in mathematics problem solving.
Competent mathematics students were significantly more accurate in completing a
mental rotations task. This indicates that the ability to hold and manipulate
information in working memory is an important information-processing task. This
ability separates the competent from the less competent mathematics student.
i
Dinnel (1986) found that competent mathematics students have the ability to
access information in the external environment, represent the problem accurately, and
plan for solution, all of which can be considered aspects of metacognition. Exposing
students to a multitude of possible strategies and utilizing different problems as
i
examples and assignments may increase students mathematics ability.
i
Wambach-Schmidt (1987) combined the metacognitive components of self and
l-j
task into a single1 working model with lines of interaction between and among all


37
identified behaviors. Her Zero-Space Model emphasizes no sequence; rather, each
metacognitive behavior is accessed as required by the task. In an investigation of
sixth-graders mathematics problem-solving and metacognitive processes, she found
that sixth graders actively monitor their thinking. They do this in several ways: (a)
analysis of their reading of the problem, (b) evaluation of themselves, the task, and
their methods, (c) verbalization of their strengths and weaknesses, (d) knowledge of
!
what to do when they require assistance, (e) expression of knowledge of their
problem-solving| style, and (f) discussion of useful strategies. From the information
gathered during this study, Wambach-Schmidt created a comprehensive model for
mathematics problem solving incorporating metacognition. The model focuses on the
i
behaviors of attention, scanning, probing, activating (making a plan and carrying it
out), communicating, and evaluating.
i
In summary, the research in mathematics problem solving indicates that both
l
the Self and Task Systems should be addressed to develop a comprehensive model
for problem solving. Some researchers (e.g., Garofalo & Lester, 1985) have met
with success in using a metacognitive model that addresses Self System variables,
whereas others (e.g., Leon & Pepe, 1983; Dinnel, 1986) have met with success with
models that emphasize Task System characteristics.
A Working Model of Metacognition
i
This researcher uses a working model for metacognition that combines
I


38
concepts from a variety of research studies and perspectives. The model contains
three components: (a) knowledge and control of self, (b) knowledge and control of
task, and (c) knowledge and control of strategies.
Knowledge and control of self is referred to as the Self System. Knowledge
'i
and control of task is referred to as the Task System. The various components within
the Self or Task are related or connected in such a way as to form a whole which
suggests a system. Each system is also assumed to be linked between the Self and
i
Task. Because of the interrelatedness of the components within the Self or Task this
author will use the term system when referring to the self and task of metacognition
in the present model, that is, Self System and Task System.
Self System
Variables within the Self System include commitment to a task, the attitude
\t
of the learner about a task, and the attention the learner gives to the task.
Commitment refers to the level of devotion one gives to a task. Through strategy
instruction, students become aware that their commitment to a task leads to effective
problem solving. Instructional strategies and practice in various methods to improve
i
commitment are emphasized in this model. Students often believe that commitment
is involuntary. However, this model considers commitment as a variable that is
i
under the control of the learner.
Another Self System variable, attitude, is dependent upon the beliefs of the


39
learner and has to do with feelings one has toward a task. Beliefs about competency
and self-worth, and beliefs about the subject and the self in relation to that subject
determine the attitude of the learner. Strategies that are geared toward changing
those beliefs are presented and reinforced with the intention of creating a positive
attitude toward a task.
Attention relates to strategies one uses to remain focused on a task. Different
tasks require different levels of attention. For example, when one first learns a
specific process (e.g., riding a bike) there must be a conscious effort to think about
each step while performing the task. Once the skill has been practiced a number of
times, the learner is able to perform the task with a very low level of attention.
Strategies and practice in how one can control the level of attention necessary for a
task are an integral part of this learning model.
Explicit regulatory guides in the Self System before, during, and after the task
are intended to guide learners with questions to ask themselves. It is important for
learners to determine when commitment, attitude, and attention must be modified and
how to control them.
Task System
Variables within the Task System include goal setting, planning to reach
goals, monitoring progress toward goals, and evaluating results. Goal setting occurs
before the task arid must occur within the realm of the task. Goal setting helps the


40
learner direct behavior. When goal setting, the learner consciously determines
direction and defines the problem by setting a goal that seems attainable. Planning
includes identifying the steps required to attain a goal. Monitoring feedback becomes
an important aspect of monitoring. The goal is more likely to be attained if the
student assesses progress toward the goal, determines attainability or progress
enroute, and adjusts accordingly with feedback from others. Evaluating includes
asking questions about goal accomplishment, and about what worked and what didnt
work, and considering what strategy to use the next time a similar task is presented.
The strategy category that Flavell (1987) identified as a part of metacognitive
knowledge is a. vehicle to facilitate students executive control over both the Self and
Task Systems.
Metacognitive Strategy Instruction
Knowledge and control of strategies are referred to in this model as
metacognitive strategy instruction. Metacognitive strategy instruction includes
explicit instruction in monitoring all cognitive processes. Metacognitive strategies
serve to monitor cognitive progress, whereas cognitive strategies are used to make
cognitive progress (Flavell, 1985). The distinction between cognitive and
metacognitive strategies is an important one. Cognitive strategies help one achieve
the goal of the cognitive activity, whereas metacognitive strategies help one to gather
information about the activity and monitor progress in it. An example of a


41
metacognitive strategy in problem solving would be to recheck the solution to a
problem. An example of a cognitive strategy would be to employ the algorithm that
would reach a solution for the problem.
The transition from other- to self-regulation is the goal of strategy instruction
in both the Self and Task Systems of metacognition. Using mediated learning,
'i
parents, teachers, other adults, and peers serve as facilitators and models for self-
regulation. As students are guided by others in learning, they practice those self-
regulatory strategies and begin to acquire them as their own. In this study, specific
strategies taught include (a) an awareness of strategies to use, (b) a demonstration of
the use of those strategies, (c) practice in using strategies, and (d) feedback on the
use of strategies; These four levels of strategy instruction are intended to facilitate
the other- to self-regulation of strategies in the problem-solving process. "Other" in
this context is the teacher and "self is the student. The teacher introduces the
metacognitive strategies, demonstrates, gives examples, guides practice, and, finally,
relinquishes the strategies to the student.
Following Gamers (1987) concerns, this author addresses the "lexical
overlap" that occurs within the strategy choice and usage by learners (p. 24).
Explicit instruction in regulatory guides before, during, and after the task is intended
I
to guide the learner with questions. Derry and Murphy (1986) advocated an
instructional model that incorporates a progression of general to domain-specific
strategy instruction. This authors model of metacognition incorporates some aspects


42
of Derry and Murphys model. In Figure 2.1, a graphic description of the
]'i
instructional model to be used in this study portrays the overlapping variables of the
Self and Task Systems. Strategy instruction is the overriding influence in both
systems. It should be noted that strategies become an integral part of the Self System
and Task System variables. The overlap of the Self and Task Systems creates
I
enhanced performance.
. I
Definition of Terms
Metacognition
This author defines metacognition for the purpose of this study as "being
aware of your thinking as you perform specific tasks and then using this awareness
to control what you are doing" (Marzano et al., 1988). Metacognition encompasses
(a) knowledge and control of self, the Self System, and (b) knowledge and control
I
of task, the Task System. Strategy instruction is a vital element of both the Self and
'li
Task Systems. The Self System has three variables: commitment, attitude, and
attention. The Task System consists of four variables: goal setting, planning,
monitoring, and evaluating. Strategy instruction is explicitly taught within the Self
and Task Systems and utilizes the executive control through which the Self and Task
Systems operate.


Figure 2.1
METACOGNITIVE LEARNING MODEL
Metacognitive Strategy Instruction
v
'N
J


44
A Problem
n
Problems come in a variety of forms. For the purpose of this study, "a
problem is a task for which (a) the person confronting it needs to find a solution, (b)
the person has no readily available procedure for finding the solution, and (c) the
person must make an attempt to find a solution" (Charles & Lester, 1982, p. 5).
Problem Solving
In this study, problem solving is restricted to mathematics in the elementary
school. Schoenfeld (1985) stated that the problem with defining problem solving is
that problems are relative to the skills and abilities of the problem solver. What
seems to be a problem to one person is a fairly simple exercise to another. In this
study, problem solving is a process that is multistep, thought provoking, and, at
times, difficult. 1 Problems to be solved are nonroutine, visual, oral, and written.
!j .
Problems selected for this study are ones for which a solution is not immediately
apparent. In this way, the subjects are not able to solve the problems quickly and
must consider problem-solving strategies from a broader point of view. Students are
taught how to think, not what to think, through the problem solving approach.
Curriculum-Embedded Strategy Teaching
Metacognitive strategies for problem solving emphasizing both the Self and
11


45
Task System are taught within the regular mathematics curriculum for the sixth-grade
level. Students are encouraged to practice problem-solving strategies, justify motives
for various steps, share alternative strategies, and monitor their progress. These
curriculum-embedded strategies are meant to be taught as part of the regular
curriculum, in this case as part of the elementary mathematics curriculum. Although
many of the strategies used are applicable to other curriculum areas, such applications
are not the purpose of this study.
General Hypotheses
From the model used in this study and its supporting research and theory, five
general hypotheses can be generated.
Hypothesis 1
Training in metacognitive strategies for the Self, Task, and combination Self
and Task Systems will enhance a young students ability to problem solve. The
experimental groups that receive Self, Task, and combination Self and Task System
strategy training will exhibit significantly higher scores on the measure of problem-
solving achievement than the control group.
Hypothesis 2
Instruction in a combination of both Self and Task System metacognitive


46
strategies will cause more improvement in problem-solving ability than will
instruction in either strategy alone.
Hypothesis 3
Training in specific metacognitive Self System strategies will improve a
students self-perceptions. The experimental groups that receive only Self System
I
strategy training or a combination of Self and Task System strategy training will
demonstrate significantly higher scores on the measure of self-perception than will
the control group or the experimental group that receives only strategy training.
Hypothesis 4
Training students in specific metacognitive Task System strategies will provide
for better strategy use and improved metacognitive processes in young students. The
experimental groups that receive only Task System strategy training or a combination
of Task and Self System strategy training will exhibit significantly higher scores on
the measure of learning strategies than either the control group or the experimental
group that receives only Self System strategy training.
Hypothesis 5
Attitudes and beliefs about problem solving will be enhanced by explicit
metacognitive strategy training. All three experimental groups will score significantly


47
better on the measure of attitude than the control group.
Concluding Remarks
The study of metacognition has captured the interest of many cognitive
psychologists and educators. Most of the focus has been in the area of strategy
instruction within the Task System. There is a plethora of reading research,
especially from such notables as Gamer, Brown, and Palinscar. Flavell (1971, 1979,
1985) is most noted for his research in memory and metacognition (metamemory).
Borkowski studied the relationship of metacognition to differences in cognitive
behavior between exceptional and "normal" children. Lester, Garofalo, and Kroll
(1989) have been interested in metacognition as it relates to mathematics problem
solving. Much of their focus has been on the Self System variable of attitude toward
I
problem solving. Many have searched for the perfect, general problem-solving
heuristic (Bransford & Stein, 1985; Polya, 1957), with mixed results. Many
problems cannot be solved by using a general rule of thumb. Problem solvers must
regulate their progress and initiate metacognitive behavior that has been internalized
in order to solve nonroutine problems.
I
The present study uses an instructional model that focuses on both general and
specific strategies that can be used in the Self and Task Systems to improve
mathematics problem-solving performance. A primary focus of this research is to
compare the effectiveness of strategy instruction to control commitment, attitude, and


48
attention in the Self System and strategy instruction to control goal setting, planning,
i
'l!
monitoring, and. evaluating in the Task System. There is an overlap between the Self
and Task Systems. Many experts in the field of metacognition view strategy
it
instruction as a'; third system interacting with the other two, whereas in this study,
strategy instruction is used as an intervention in both the Self and Task Systems to
improve performance.
I


CHAPTER 3
RESEARCH DESIGN
This study investigates the effectiveness of the direct teaching of curriculum-
embedded metacognitive strategies to sixth-grade students. Effectiveness is
measured by mathematics problem-solving performance, self-perception, attitude, and
metacognitive strategy use. This chapter describes the method used in the study, the
limitation of the study, and specific hypotheses within the study.
Summary of Problem Statement
The purpose of this study is to clarify the "fuzzy" concept of metacognition
by determining the effectiveness of the direct teaching of Self and Task System
metacognitive strategies to sixth-grade students. Self and Task System variables are
compared to ascertain their relative effects. Comparisons of the effects of strategy
teaching on the performance, self-perceptions, attitudes and strategy use by low,
middle, and high achievers are also made to reveal any aptitude-by-treatment
interactions. The basic assumption underlying the study is that sixth-grade students
benefit from the direct instruction of both Self and Task System metacognitive
strategies. This benefit will be demonstrated in improved performance in
mathematics problem solving, enhanced self-perceptions, positive attitudes, and


50
increased use of metacognitive strategies by the subjects in the experimental groups.
! Method
Experimental Design
The design used for all five hypotheses is classified by Campbell and Stanley
(1963) as a true! experimental. The study employed a multigroup pretest-treatment-
posttest design in which students initially were matched for mathematics ability based
on their quantitative grade equivalent scores on the Iowa Test of Basic Skills (TTBS)
and then randomly assigned to four treatment groups: three experimental and one
control group. Matching plus subsequent randomization produces an experimental
design with greater precision than randomization alone.
The four!treatment groups were selected from three sixth-grade classes that
'1 ,
this researcher taught. Within treatment groups, subjects were classified as high,
medium, or low ability based on their performance on the quantitative section of the
ITBS. The dependent variables were mathematics problem-solving performance, self-
perception, attitude, and metacognitive strategy acquisition. Problem-solving
i.
performance was measured with the Five-Question Problem Solving Assessment,
which was generated from problems developed by Stephens, Hoogeboom, &
Goodnow (1988).1 Self-perception was measured with Harters Self Perception Profile
for Children (1985). Metacognitive strategy acquisition was assessed on an adapted


I
version of the Learning and Study Strategies Inventory (Weinstein, Schulte, &
Palmer, 1987). These variables were analyzed to determine the effects of treatment
and ability level and any interactions. A fixed-effects Analysis of Covariance
(ANCOVA) was used to determine if the differences among means, adjusted for
differences on a pretest, are greater than would be expected from sampling error
alone.
Subjects
All subjects were from a predominately white, upper-middle-class public
elementary school in a suburban area. There were 82 sixth-grade students (42 males
and 40 females)] These students were randomly assigned to one control and three
experimental groups. Scores from the quantitative section of the ITBS were used to
identify students as low, middle, or high mathematics achievers.
According to the scores on the quantitative section of the ITBS, Low, middle,
and high achievers were divided as follows: The low group was comprized of
students who had grad equivalency scores of 4-1 to 5-9, middle from 6-0 to 6-9, and
high from 7-0 to 8-8. Based on this partitioning, the control group contained 19
students: 3 low, 7 middle, and 9 high achievers. Experimental 1 contained 20
students: 3 low, 8 middle, and 9 high. Experimental 2 contained 21 students: 5
low, 6 middle, and 10 high. Experimental 3 contained 22 students: 7 low, 7 middle,
and 8 high. Table 3.1 shows the mean scores from the low-, middle-, and high-


52
ability students within the four treatment groups. Also included are the mean scores
for each treatment group. It should be noted that the low ability group is relatively
small when compared to the middle and high levels.
The low:, middle, and high designations were not used to place students in
I
homogeneous groups for instruction. Heterogeneity of grouping was maintained
within all classroom activities. Also, a small number of students were missing from
either the pre- or posttest due to a variety of causes unrelated to the study. These
were neither systematic nor problematic. Any mainstreamed handicapped students
in these classes were excluded from the study.
Instruments
Attitudes and beliefs about problem solving were measured using the Attitude
Toward Problem Solving Scale. It is adapted from the Attitude Toward Mathematics
Scale, reported in the Arithmetic Teacher (1989) and utilizes a Likert scale indicating
the degree of positive attitude toward problem solving. Students could score from
30 120 points, with a score of 120 indicating a totally positive attitude about
problem solving, (See Appendix A for a sample of this scale.) No reliability
coefficients or validity was reported in the original source for this scale. However,
face validity as determined by this author appears strong for this instrument. That
is, each question addresses attitudes and related feelings when the students are
involved in problem solving.


Table 3.1
Mean Grade Equivalent Scores by Ability Level
Quantitative Section on the Iowa Test of Basic Skills
Ability Experimental 1 Experimental 2 Experimental 3 Experimental 4
Low (=18) 5.27 5.48 5.43 4.93
Middle (n=28) 6.41 6.38 6.53 6.41
High (n=36) 7.59 7.62 7.70 7.63
TOTAL 6.77 6.72 6.66 6.70


54
Problemj-solving performance was measured using five non-routine problems
i
taken from the practice problems of The Problem Solver 7 (Stephens, Hoogeboom,
& Goodnow, 1988). (See Appendixes C and D for actual problems.) As shown in
Figure 3.1, a paraphrased version of a holistic approach to evaluation (Charles,
Lester & ODaffer, 1987) of problem solving was utilized in testing research
Hypotheses 1 arid 2. Using the Charles method students can score between 0 and 20.
This approach places emphasis on both the process and solution.
Three teachers, one of whom was this researcher, participated in blind scoring
of the pre- and posttests of problem-solving achievement. An individual students
score then was determined by averaging the three scores. Interrater reliability was
calculated using ia technique by Breland, Camp, Jones, Morris, and Rock (1987) in
a study conducted by the College Board. The pairwise Pearson Product Moment
correlation coefficients were calculated for the three raters and an average correlation
computed. This represents the reliability of a single rater scoring tests independently.
The Spearman-Brown formula was then used to estimate the reliability of three raters
rating each test as was the case in this study. The reliability was computed to be r
I
= .82. Again, no reliability coefficients or validity information was reported in the
original source. :
The pre- and posttests were rated for equivalency by a panel of two experts
in the field of mathematics. The experts determined that these tests were equivalent.


55
Figure 3.1. Scoring Point Scale for a Holistic Approach to Evaluation (Charles,
Lester, & ODaffer, 1987)
Points Characteristics
0 points Paper is blank. Problem is recopied, but no apparent work is done or there is no understanding of the problem. There is an incorrect answer with no other work shown.
1 point There is a start toward finding a solution that reflects some understanding, but the approach used would not lead to a correct solution. An inappropriate strategy is started but not carried out. The student did not try another strategy, but apparently gave up. The student tried unsuccessfully to reach a subgoal.
2 points Students work showed some understanding of the problem, but used an inappropriate strategy and got an incorrect answer. An appropriate strategy was used, but it was either not carried out far enough to reach a solution or it was implemented incorrectly and thus led to no answer or an incorrect answer. The student successfully reached a subgoal, but went no further. The correct answer is shown but either no work of unintelligible work is shown.
3 points A' solution strategy that could have led to a correct solution is used, but part of the problem was misunderstood or a condition was ignored. Appropriate strategies were applied, but the student answered the problem incorrectly or the correct numerical part of the answer was given and the answer was not labeled or was labeled incorrectly or no answer was given. The correct answer is given with some evidence that appropriate solution strategies were selected, but the implementation of the strategies is not completely clear.
4 points The correct answer was given and the appropriate strategies were selected and implemented. The only error seems to be in copying or computation with a clear understanding of the problem and implementation shown.


56
Another indication that pre- and posttest problems were parallel is that they are
represented in The Problem Solver 7 as equivalent problems.
Harters 'Self-Perception Scale for Children, revised (1985) was used as the
pre- and posttest to determine students self-perceptions and the relationship of these
perceptions to general and specific academic ability and beliefs. The revised version
of this instrument contains six separate subscales: (a) Scholastic Competence, (b)
Social Acceptance, (c) Athletic Competence, (d) Physical Appearance, (e) Behavioral
Conduct, and (f) Global Self-Worth. Scores may range from 36 144, and items are
scored either 4,: 3, 2, or 1. Higher scores reflect more positive self-judgment and
lower scores, less positive. The scale can be administered to entire groups of
students. Subscale reliabilities based on coefficient alpha (an index of internal
consistency) range from .71 to .86. (For complete reliabilities see Appendix I. Also
included in Appendix I are subscale means and standard deviations.)
A version of The Learning and Study Skills Inventory (LASSI) (Weinstein,
,1
Schulte, & Palmer, 1987) adapted for the intermediate grades (4, 5, 6) by McCombs
and Pickering (1990) was used to measure metacognitive strategy use. (See Appendix
B). This adapted LASSI has questions reworded to include age-appropriate
vocabulary for this study. The instrument contains five questions in each of the ten
categories of learning and study strategies. Those ten categories are (a) attitude and
interest, (b) motivation, (c) time management, (d) anxiety, (e) concentration and
attention, (f) information processing, (g) recognizing important information, (h) use


57
of support techniques and materials, (i) self-testing, and (j) test strategies. There is
a possible score of 25 points for each category and a possible total score of 250
points. Higher scores are an indication that students believe they use study strategies
and skills reflected in the category. The test-retest reliability data for the version of
the LASSI, from which the version used in this study was adapted, are given in
Appendix B following the actual inventory used. The validity coefficient reported for
the LASSI was a .60 correlation with the Elaborative Processing Scale of Schmecks
Inventory of Learning Processes. In addition, when the LASSI was subjected to
"user validity," potential users reported few, if any, administration problems and
found the inventory to be useful (Weinstein, Zimmermann, & Palmer, 1988). In the
present study the inventory was used to determine which study strategies students had
and which of those strategies they needed to acquire that would promote good
i .
problem-solving performance.
i
In summary, four instruments were used to measure the dependent variables
in the five hypotheses in this study. The five nonroutine problems from the Problem
Solver 7 were used for hypotheses 1 and 2. The Harter Self-Perception Scale for
Children was used for hypothesis 3. The LASSI was used for hypothesis 4 and the
Attitude Toward Problem Solving Scale was used for hypothesis 5.


58
Materials
A series of twenty-four 45-minute lessons were developed for use during the
I
experimental period of five weeks. Suggestions for homework and practice were
included for all three experimental groups and the control group. Problems from The
Problem Solver 7 (Stephens, Hoogeboom, & Goodnow, 1988) were incorporated into
these lessons. Any necessary worksheets were also provided. (A sample plan is
shown in Appendix E for each type of lesson presented to the different treatment
groups.)
'|
Group-specific questions for before, during, and after problem solving were
given to the discussion leaders for each experimental group. Questions for
Experimental 1 were developed to elicit Self System strategy use while questions for
Experimental 2 were developed to elicit Task System strategy use. The combination
group, Experimental 3, received questions for both the Self and Task System
ii
strategies. These questions were placed on laminated cards so that they could be
reused during each day of this research. Every student in each of the experimental
groups maintained a copy in his or her problem-solving folder. (Samples of the
question cards for each experimental group appear in Appendix F.)
In order to assure student acceptance of the Self and Task System strategies
l
'I
that were used, students in the experimental groups were queried as to the behaviors
that constitute each. Their responses were duplicated and placed in the folders of the


59
appropriate experimental groups based on the specific metacognitive strategies learned
I
by that group. (Copies of these student responses may be found in Appendix G.)
Procedures
All four treatment groups worked in cooperative learning groups to solve
problems. This researcher utilized cooperative learning groups in order to facilitate
metacognitive strategy use in the three experimental groups. Cooperative learning
provides a vehicle for student interaction and vocalization of thoughts during problem
solving. Cooperation among students may produce higher achievement, more
motivation to learn, and higher self-esteem than traditional instructional methods
(Johnson, Johnson & Holubec, 1987). So that the advantages of using cooperative
learning procedures would not confound this study, these procedures were also used
with the control group.
The control group did not receive any metacognitive strategy instruction;
however they were taught the same problem-solving curriculum, which included
specific problem-solving strategies, as the experimental groups. These problem-
strategies were, taken from The Problem Solver 7 (Stephens, Hoogeboom, &
Goodnow, 1988) and included teaching students to (a) use or make a table, (b) make
an organized list, (c) use or look for a pattern, (d) make a picture or diagram, and
(e) guess and check. The control group received instruction for the same amount of
time as the experimental groups forty-five minutes of instruction including daily


60
practice for twenty-four consecutive days. The pre- and posttesting occurred outside
of this time frame. The control group was presented the lessons without benefit of
the explicit instruction of metacognitive strategies. Only the domain-specific
strategies to be used, objectives to be learned, and problems to be solved were
provided by the teacher. The students were given only the correct answers as
feedback. However, no reflection on the strategies used or predictions for further use
of strategies was presented. Students in the three experimental groups and in the
control group were taught the problem-solving curriculum by this researcher, who
was, at the time, their regular teacher. Instruction occurred in a whole-class setting.
Observation of all three experimental groups and the control group instruction was
completed by this researcher and two other impartial observers.
Experimental 1 was taught the problem-solving curriculum based on the
Metacognitive Learning Model presented in Chapter 2. From this model, only Self
System strategy instruction was presented to this treatment group. A reflection
period that focused on attitudes, attention, and commitment during problem solving
was encouraged. Students were also asked to predict other situations in which they
would use the same Self System strategies.
Experimental 2 was taught the problem-solving curriculum with practice and
reflection on only the Task System variables in the Metacognitive Learning Model.
l1
A reflection period that focused on goal setting, planning, monitoring, and evaluation
during problem solving was encouraged. Students also were asked to predict other


61
situations in which they would use the same Task System strategies.
Experimental 3 students were taught the full problem-solving curriculum based
on the Metacognitive Learning Model (Figure 2.1). These students were given time
to reflect on both Self and Task System strategies used for problem solving. In
addition, they predicted other circumstances for which these strategies would be
applied. j
Limiting Factors
The present study focuses on mathematics problem solving and metacognitive
strategies because of the close relationship between the two and because there has
been relatively little substantive research on the effect of combining mathematics and
metacognitive strategy teaching. The choice of using only sixth-grade elementary
students was an attempt to keep the task within reasonable limits for the author.
These parameters created certain limitations in the study. Specifically, students in
all three experimental groups and in the control group came from families ranging
i
in socio-economic status (SES) from middle-class to upper-middle-class. These
students attended public school in a moderately wealthy suburban district. The
district was involved in a three-year program in the area of thinking skills, with one
elementary school serving as a pilot school for the program developed. The sixth-
grade students from this school were used for the experimental study, with three
classrooms of sixth-grade students participating as subjects. This author delivered


I
62
instruction in all four control and experimental groups. The control group consisted
jl
of one group of sixth-grade students randomly selected from the same pool of
students as the experimental groups. It is assumed that these students from all four
treatment groups represent the population that has been described and that the results
generalize only to other like populations.
Hypotheses
Hypothesis 1
Training in metacognitive strategies enhances a young students ability to
problem solve. Experimental groups 1, 2, and 3, which receive metacognitive Self,
Task, and a combination Self and Task System strategy training, will exhibit
significantly higher scores on the measure of problem-solving achievement than the
control group. The measure of problem-solving achievement will be derived from
l
the Five-Question Problem Solving Assessment.
Hypothesis 2
Compared to instruction in Self or in Task System strategies, the combination
of general metacognitive Self System strategies and Task System strategies more
powerfully effects the mathematics problem-solving ability of students.
Consequently, Experimental 3 will score significantly higher on the Five-Question


63
Problem-Solving Assessment than either Experimental 1 or Experimental 2.
Hypothesis 3
Training in specific metacognitive Self System strategies improves a students
self-perceptions. Experimental 1, which receives training in only the Self System
strategies, and Experimental 3, which receives training in a combination of Self and
Task System strategies, will demonstrate significantly higher scores on the Self-
Perception Profile for Children than either Experimental 2 or the control group.
Hypothesis 4
I
'I
Training students in specific metacognitive Task System strategies provides
for better strategy use and improved metacognitive processes in young students.
Experimental 2, which receives training in only Task System strategies, and
Experimental 3, which receives training in a combination of Task and Self System
strategies, will exhibit significantly higher scores on the Learning and Study
Strategies Inventory than either the control group or Experimental 1.
Hypothesis 5
Attitudes and beliefs about problem solving are enhanced by explicit
metacognitive strategy training. Experimental groups 1, 2, and 3 will score
significantly better on the measure of attitude than the control group. Specifically,


64
the scores derived from the Attitude Toward Problem Solving Scale, will be higher
for those students in the three experimental groups than for those students in the
control group.
Summary of Research Design
This study uses a true experimental design with a multigroup, pretest-
treatment-posttest design. The subjects are primarily white, upper-middle-class,
r
sixth-grade students assigned to this researcher. Materials used in the study consist
of both teacher-made tests and a standardized test, problems from a variety of
sources, a self-perception scale, and an inventory of study strategies. The five
hypotheses are designed to determine the impact of the direct teaching of
differentiated metacognitive strategies on the problem-solving abilities, self-
perceptions, and metacognitive strategy use of sixth-grade students. Aptitude-by-
treatment interactions are also assessed. Comparisons between the effect of the direct
teaching of Self System, Task System, and combination Self and Task System
l!,
metacognitive strategies are made.


CHAPTER 4
RESULTS
The purpose of this study is to investigate the effectivenes of the direct
1
teaching of curriculum-embedded metacognitive strategies as measured by the
mathematics problem-solving performance, self-perceptions, attitudes, and
metacognitive strategy use of sixth-grade students. Identified low-, middle-, and
high-achieving students are compared to determine the impact of teaching
'l
metacognitive Sjelf and Task System strategies.
I
This chapter addresses results in terms of the effectiveness of teaching Self
System metacognitive strategies (Experimental 1), Task System strategies
(Experimental 2), and a combination of both Self and Task System strategies
l
(Experimental 3). A control group received specific problem-solving strategy
instruction only. Comparisons were made to the control group and between each
group. Hypotheses with an explanation of results are addressed, and feasible
conclusions are jsupported.
j
The experimental design for all hypotheses is a multigroup, pretest-treatment-
I
,1
posttest design |(Campbell & Stanley, 1963). Subjects were identified as low-,
j
middle-, and high-achievers as measured by the quantitative section of the Iowa Test
j i
of Basic Skills (ITBS). These levels remain the same for all experiments conducted


66
within this study. Students were then randomly assigned to one of four treatment
groups. The dependent variables are problem-solving performance, self-perception,
!|
metacognitive strategy acquisition, and attitude. These variables were analyzed to
determine the effects of treatment and ability level.
For all hypotheses, a fixed effects ANCOVA was employed to compare the
performance of j the four treatment groups using the pretest scores as the covariate.
"The ANCOVA model treats both between-group and regression variance as
systematic (nonerror) components" (Huitema, 1980, p. 53). As explained by
Huitema, Analysis of Variance (ANOVA) is a statistical technique used to determine
the differences among two or more means. Simply stated the ANCOVA model adds
I
a regression term to the ANOVA model. A smaller error term will then occur,
which accounts for an increase in power. This allows the possibility of data yielding
significant results in ANCOVA when ANOVA yields nonsignificant results.
Each analysis utilized four ireament groups: (a) Self System strategy
instruction, (b) iTask System strategy instruction, (c) combination Self and Task
System strategy; instruction, and (d) control. The levels within treatment groups
consisted of low, middle, and high achievement as identified by grade equivalent
scores on the ITBS quantitative section.


;l
I
! 67
I Problem-Solving Performance
Hypothesis 1 .i
Lj
Training; in metacognitive strategies enhances a sixth-grade students ability
to problem solve. Experimental groups 1, 2, and 3, which receive metacognitive
Self, Task, and ia combination Self and Task System strategy training, respectively,
'j
will exhibit higher scores on the Five-Question Problem-Solving Assessment.
1
The meins, standard deviations, and adjusted posttest means from the Five-
Question Problem-Solving Assessment are displayed in Tables 4.1 and 4.2. Table
t!
4.3 shows the results of a 3 x 4 (level x treatment) ANCOVA using pretest scores
as the covariate.
'i
The analysis revealed that there was a significant difference between the
adjusted posttest means of the four treatment groups. The effect of treatment was
:i
statistically significant at the p < .05 level, F(3,61) = 3.325. The null hypothesis
that all four adjusted means were equal was therefore rejected.
r
i,
A planned pairwise comparison using the Dunnett Multiple Comparison (MC)
method to test the null hypothesis that the combined adjusted means of the three
experimental groups came from the same population as the adjusted means of the
control group. / The Dunnett MC provides a pairwise contrast to determine which
l
means or groups of means are significantly different when compared to one
h
predesignated mean, usually the mean of the control group. The Dunnett MC is a


Table 4.1
Problem Solving Assessment
Means and Standard Deviations
Treatment Pretest Posttest Adj. Posttest
M SD M SD M
Control 6.00 2.54 11.11 3.58 11.01
Experimental 1 5.95 2.86 11.85 3.60 12.27
Experimental 2 5.86 2.29 13.05 4.24 13.55
Experimental 3 5.81 2.18 11.86 2.35 12.31


Table 4.2
Problem-Solving Assessment
Means and Standard Deviations by Level
Level Pretest Posttest Adj. Posttest
M SD M SD M
Low 4.17 2.04 9.28 3.01 11.11
Middle 5.59 1.80 10.96 3.08 11.19
High 7.03 2.48 14.20 2.66 13.61
ON
NO


Table 4.3
Problem Solving Assessment
Summary of Analysis of Covariance
Source Sum of Squares df Mean Square F Significance of F
Level 91.761 2 45.881 7.980 .001
Treatment 57.355 3 19.118 3.325 .025
Level x Treatment 34.457 6 5.743 .999 .434
Cov: Pretest 261.963 1 261.963 45.565 .000
Residual 350.698 61 5.749
Total 808.216 73 11.071


71
|
more powerful method of planned contrasts when comparing sets of means to one
predesignated mean (Glass & Hopkins, 1984). The MSe for this comparison was the
I
MSe from the ANCOVA. The contrast of the combination of Experimental 1,
Experimental 2, and Experimental 3 with the control group exceeded the critical
value of f and the null hypothesis was rejected at p < .05 level.
To further explore how the treatment groups differed, planned comparisons
using the Dunnett MC, were employed. Again, the MSe and adjusted means derived
from the ANCOVA were used to compute the planned comparison. Each of the
experimental group means was compared individually to the control group mean.
When the combination Self and Task System group (Experimental 3) was compared
to the control group, the f-ratio of 1.63 fell just below the critical f-ratio of 1.67;
therefore, the null hypothesis cannot be rejected. That is, results from Experimental
3 did not differ significantly from the control group. When the Task System group
(Experimental 2) was compared to the control group, the f-ratio of 3.18 was well
above the criticlal f-ratio 1.67; therefore, the null hypothesis is rejected. Those
students receiving instruction in the Task System strategies performed significantly
better on the Five-Question Problem-Solving Assessment than those subjects in the
control group. When the Self System group (Experimental 1) was compared to the
control group, the f-ratio of 1.54 was just below the critical f-ratio of 1.67; therefore,
the null hypothesis cannot be rejected. Results from Experimental 1 did not differ
significantly from the control group.


72
Hypothesis 2
Compared to students receiving only Self or Task System strategy training,
the combination of general metacognitive Self System and Task System strategies
enhances the mathematics problem-solving skills of students. Students in
Experimental 3 will score significantly higher on the Five-Question Problem-Solving
Assessment than students in either Experimental 1 or Experimental 2.
To test the null hypothesis that combined adjusted means from Experimental
1 and 2 came from the same population as Experimental 3, a Dunnett planned
pairwise comparison using the MSt term from table 4.3 was employed. The /-ratio
of .79 was below the critical /-ratio of 1.67; therefore, the null hypothesis cannot be
rejected. Instruction in the combination of metacognitive Self and Task System
strategies did not improve problem-solving performance on the Five-Question
Problem-Solving Assessment when compared to subjects receiving only Self or Task
System strategy instruction. Training in a combination of Self and Task System
strategies did not have a significant effect on subjects problem-solving performance
when compared ;to Experimental 1 and Experimental 2.
One comparison of special interest is the adjusted mean of the Task System
group (Experimental 2) compared to the combined adjusted mean of the Self System
group (Experimental 1) and the combination Self and Task System group
(Experimental 3). The need for this comparison is illustrated by Table 4.1 which
reports the pre-';and posttest scores by treatment. From this table it appears that


73
Experimental 2 outperformed the other two experimental groups. Adjusted means
by treatment by level are reported in Tables 4.1 and 4.2. The r-ratio of 1.66 derived
from this comparison is just below the critical r-ratio of 1.67; therefore, it cannot be
said that Experimental 2 performed significantly better than Experimental 1 and 3,
but there is a strong tendency toward significance, which could indicate further
research especially in light of the findings from hypothesis 1 that Experimental 2 was
significantly different from the control group.
To further explore how the achievement levels differed, planned comparisons
were made for high versus low, high versus middle, and middle versus low using the
Dunnett MC method. The comparisons demonstrated that there was no significant
difference, p > .05, between treatment groups at the low versus middle achievement
levels. It should be noted that there was statistical significance at the middle versus
high and low versus high achievement levels at p < .01, with high achievers
outperforming middle and low achievers. In all four treatment groups high achievers
performed better, yet there was no statistical evidence that middle versus low
achievers differed. Figures A.l and A.2 in Appendix J report pre- and posttest
scores by treatment and level respectively for hypotheses 1 and 2.


74
Self System Strategy Performance
Hypothesis 3
Training in specific metacognitive Self System strategies improves a students
self-perception. Students who receive training in only the Self System strategies
(Experimental 1), and students who receive training in a combination of Self and
Task System strategies (Experimental 3), will demonstrate significantly higher scores
on the Self-Perception Profile for Children than students in either Experimental 2 or
the control group.
The means, standard deviations, and adjusted means for What I Am Like
(Harter, 1985), a profile of self-perception for children, are displayed in Tables 4.4
and 4.5. A 3 x 4 (level x treatment) ANCOVA was performed on the profile scores
using the pretest scores as a covariate. The results of this ANCOVA are shown in
Table 4.6.
The analysis revealed no treatment, level, or interaction effect among the
independent variables. This suggests that there were no significant differences in
self-perceptions among the four treatment groups. Given the lack of significance, no
further analyses were conducted. Figures A.3 and A.4 in Appendix J report pre- and
posttest scores by treatment and level respectively for this hypothesis.


Table 4.4
What I Am Like_
Means and Standard Deviations
Treatment Pretest Posttest Adj. Posttest
M SD M SD M
Control 114.00 13.02 114.54 14.63 114.55
Experimental 1 107.31 17.11 110.06 19.73 115.16
Experimental 2 111.18 18.26 108.29 17.91 109.52
Experimental 3 120.59 12.44 117.12 13.62 111.08


Table 4.5
What I Am Like
Means and Standard Deviations by Level
Level Pretest Posttest Adj. Posttest
M SD M SD M
Low 114.00 16.80 112.76 17.02 109.99
Middle 113.22 14.35 111.26 16.64 110.44
High 111.31 16.50 112.45 16.01 115.10
-o
On


Table 4.6
What I Am Like
Summary of Analysis of Covariance
Source Sum of Squares df Mean Square F Significance of F
Level 344.453 2 172.227 1.963 .151
Treatment 337.488 3 112.496 1.282 .291
Level x Treatment 576.199 6 96.033 1.094 .379
Cov: Pretest 11715.326 1 11715.326 113.514 .000
Residual 4387.292 50 87.746
Total 17317.270 62 279.311


78
Task System Strategy Performance
Hypothesis 4
I
Training students in specific metacognitive Task System strategies provides
for better strategy use and improved metacognitive processes in young students.
Students who receive training in only Task System strategies (Experimental 2) and
students who receive training in a combination of Task and Self System strategies
(Experimental 3) will exhibit significantly higher scores on the Learning and Study
Strategies Inventory than the students in either the control group or Experimental 1.
The means and standard deviations for the Learning and Study Strategy
Inventory Adapted (LASSI) are displayed in Tables 4.7 and 4.8. The results of
i
the 3 x 4 (level x treatment) analysis of covariance using the pretest as a covariate
are shown in Table 4.9.
The analysis indicated no treatment, level, or interaction effect among the
independent variariables. This suggests that there were no significant differences on
the Adapted LASSI among the four treatment groups. Given the lack of significance
no further analyses were conducted. Figures A.5 and A.6 in Appendix J report pre-
and posttest scores by treatment and level respectively for this hypothesis.


Table 4.7
Adapted LASSI
Means and Standard Deviations by Treatment
Treatment Pretest Posttest Adj. Posttest
M SD M SD M
Control 161.88 23.65 154.41 32.10 161.41
Experimental 1 175.75 26.78 174.00 25.12 170.26
Experimental 2 172.13 24.48 164.75 27.61 162.59
Experimental 3 174.95 21.19 171.00 20.86 169.71


Table 4.8
Adapted LASSI
Means and Standard Deviations bv Level
Level Pretest Posttest Adj. Posttest
M SD M SD M
Low 164.19 23.49 155.19 23.95 158.85
Middle 167.70 27.93 159.74 25.74 162.47
High 173.50 21.83 171.82 27.62 171.36
OO
o


Table 4.9
LASSI
Summary of Analysis of Covariance
" Source Sum of Squares df Mean Square F Significance of F
Level 1785.628 2 892.814 2.665 .079
Treatment 1037.327 3 345.776 1.032 .386
Level x Treatment 3461.989 6 576.998 1.722 .133
Cov: Pretest 24580.925 1 24580.925 73.372 .000
Residual 18426.091 55 335.020
Total 49031.471 67 731.813


82
Attitude Toward Problem Solving
Hypothesis 5
Attitudes and beliefs about problem solving, and self-control of learning are
enhanced by explicit metacognitive strategy training. Students in Experimental
groups 1, Experimental 2, and Experimental 3 will score significantly better on the
I
l
measure of attitude, the Attitude Toward Problem Solving Scale, than students in the
control group.
The means, standard deviations, and adjusted means for the Attitude Toward
Problem Solving Inventory are displayed in Table 4.10 and 4.11. A 3 x 4 (level x
treatment) ANCOVA was performed on the inventory scores from the Attitude
Toward Problem Solving Scale using the pretest as a covariate. Results of this
ANCOVA are shown in Table 4.12.
This analysis revealed that there was a statistically significant effect for level
(low, middle, high), p < .05, F(2,51) = 3.298. The Dunnett MC technique was
used to further explore the relationship between high versus middle, high versus low,
and middle versus low. Adjusted means and the MSt from the ANCOVA were used
for these analyses. As Table 4.11 indicates, high achievers had significantly more
i
positive attitudes, ,p < .05, than their middle counterparts. In the Task System group
(Experimental 2), the low achievers had more positive scores than the middle or high
achievers in that group and had more positive scores than the students in any other


Table 4.10
Attitude Toward Problem Solving Scale
Means and.Standard Deviations bv Treatment
Treatment Pretest Posttest Adj. Posttest
M SD M SD M
Control 84.00 13.56 78.73 19.75 79.84
Experimental 1 87.63 16.57 83.81 20.48 81.34
Experimental 2 86.06 16.71 83.53 21.17 81.07
Experimental 3 79.38 15.25 80.19 15.14 84.24


Table 4.11
Attitude Toward Problem Solving Scale
Means and Standard Deviations by Level
Level Pretest Posttest Adj. Posttest
M SD M SD M
Low 74.59 11.68 73.76 13.16 81.72
Middle 87.41 14.79 78.41 20.49 75.21
High 85.42 16.29 86.65 18.68 85.89


Table 4.12
Attitude Toward Problem Solving Scale
Summary of Analysis of Covariance
Source Sum of Squares df Mean Square F Significance of F
Level 1331.326 2 665.663 3.298 .045
Treatment 157.187 3 52.386 .260 .854
Level x Treatment 1303.545 6 217.257 1.077 .389
Cov: Pretest 9515.447 1 9515.447 47.149 .000
Residual 10292.625 51 201.816
Total 22654.734 63 359.599


86
treatment group. There was no significant difference between the middle versus low
or the high versus low achievers. Given that the treatment effect was not significant
no further analyses were conducted. Figures A.7 and A. 8 in Appendix J report pre-
and posttest scores by treatment and level respectively.
Problem-Solving Performance and Attitude
Figure 4.1 displays scatterplots of the intercorrelations within treatment groups
for the posttest scores on the Five-Question Problem-Solving Assessment and the
Attitude Toward Problem Solving Scale. The correlation for all treatment groups
combined (r = .42) demonstrates a moderate, but significant relationship (p < .05)
between variables. Upon further examination the control group (r = .42) and
!
combination treatment group (r = .48) exhibited moderate but significant
relationships between attitude and problem solving (p < .05). The Self System
group revealed the strongest correlation between attitude and problem solving
performance (r = .71, p < .01). The Task System group disclosed no significant
correlation (r = .39, p > .05).
Figure 4.2 presents the correlations between the Five-Question Problem-
Solving Assessment and Attitude Toward Problem Solving Scale within achievement
levels. No significant correlation, r = .09 was found for low achievers. A
significant correlation was found for middle achievers, r = .58, p < .05. The
correlation, r = .44, for performance and attitude was weaker for the high group,