University of Southern California Dissertations and Theses

Actual and ideal instructional practices in California high school gifted geometry education

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Actual and ideal instructional practices in California high school gifted geometry education

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Content ACTUAL AND IDEAL INSTRUCTIONAL PRACTICES IN CALIFORNIA
HIGH SCHOOL GIFTED GEOMETRY EDUCATION
by

Edit Tanahan
A Dissertation Presented to the
FACULTY OF THE ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION
December 2006
Copyright 2006 Edit Tanahan
ii
Dedication
This dissertation is dedicated to Zaven and Martha Tanahan
iii
Table of Contents
Dedication ii
List of Tables v
Abstract xiii
Chapter 1: Introduction 1
Statement of the Problem 1
Purpose of the Study and Conceptual Framework 3
Significance of the Study 4
Specific Research Questions 5
Overview of Method 5
Assumptions 7
Limitations 7
Delimitations 8
Chapter 2: Literature Review 9
Conceptions of Gifted Education and Identification of 9
Gifted Learners
Giftedness in the Mathematics Classroom: Developing 14
Mathematics Promise
The Status of Classroom Practices in Gifted Education 17
Instructional and Curricular Models for Teaching Gifted Learners 22
Gaps in the Literature: Comparisons Between Content Specific 34
Actual and Ideal Classroom Practices in Gifted Education
Summary and Conclusions 36
Chapter 3: Methodology 39
Introduction 39
Research Questions 39
Nature of the Study 39
Subjects 40
Instrumentation 41
Research Procedure 44
Data Analysis 48
Validity and Reliability of the Survey and the Study 49
Chapter 4: Results 54
Chapter 5: Data Analysis 75
Summary of Key Findings 75
iv
Individual Item Analysis for the Survey’s Twelve Instructional 79
Practices
General Conclusions About the Key Findings Reflecting on the 93
Literature Review
Implications for Practice Gleaned from the Investigation 97
Suggestions for Future Research 100
Bibliography 102
Appendices A-D 110
Appendix A: Instructional Practices--Geometry Teacher Survey 110
Appendix B: Information Sheet for study 118
Appendix C: Cover Letters 1-3 121
Appendix D: Telephone Script 124
v
List of Tables
Table 1.1: Alignment of the Twelve Survey Items with the Instructional 52
Approaches for Teaching Mathematics Suggested in the
California Mathematics Framework (2005)
Table 1.2: Alignment of the Twelve Survey Items with the Conceptual 52
Learning and/or Computational/Procedural Learning
Components of Instruction Suggested by the California
Mathematics Framework (2005)
Table 1.3: Alignment of the Twelve Survey Items with the Four 52
Components of a Differentiated Curriculum for
Gifted Education as Suggested in the California
GATE standards (2001)
Table 2: Definition of Groups 55
Table 3: Mann Whitney U/Wilcoxon Matched Pairs Test: 57
An Examination of Differences in Teachers’ Instructional
Practices with Gifted and/or Typical Students Under
Actual and Ideal Instructional Circumstances
Table 4: Teachers’ Mean Frequencies of Twelve Instructional 62
Practices with Four Student Groups
Table 5.1: Actual Teacher Frequencies with Gifted Learners 62
for Instructional Practice #1
Table 5.2: Actual Teacher Frequencies with Gifted Learners 63
for Instructional Practice #2
Table 5.3: Actual Teacher Frequencies with Gifted Learners 63
for Instructional Practice #3
Table 5.4: Actual Teacher Frequencies with Gifted Learners 64
for Instructional Practice #4
Table 5.5: Actual Teacher Frequencies with Gifted Learners 64
for Instructional Practice #5
Table 5.6: Actual Teacher Frequencies with Gifted Learners 65
for Instructional Practice #6
vi
Table 5.7: Actual Teacher Frequencies with Gifted Learners 65
for Instructional Practice #7
Table 5.8: Actual Teacher Frequencies with Gifted Learners 66
for Instructional Practice #8
Table 5.9: Actual Teacher Frequencies with Gifted Learners 66
for Instructional Practice #9
Table 5.10: Actual Teacher Frequencies with Gifted Learners 67
for Instructional Practice #10
Table 5.11: Actual Teacher Frequencies with Gifted Learners 67
for Instructional Practice #11
Table 5.12: Actual Teacher Frequencies with Gifted Learners 68
for Instructional Practice #12
Table 6.1: Ideal Teacher Frequencies with Gifted Learners 69
for Instructional Practice #1
Table 6.2: Ideal Teacher Frequencies with Gifted Learners 69
for Instructional Practice #2
Table 6.3: Ideal Teacher Frequencies with Gifted Learners 70
for Instructional Practice #3
Table 6.4: Ideal Teacher Frequencies with Gifted Learners 70
for Instructional Practice #4
Table 6.5: Ideal Teacher Frequencies with Gifted Learners 71
for Instructional Practice #5
Table 6.6: Ideal Teacher Frequencies with Gifted Learners 71
for Instructional Practice #6
Table 6.7: Ideal Teacher Frequencies with Gifted Learners 72
for Instructional Practice #7
Table 6.8: Ideal Teacher Frequencies with Gifted Learners 72
for Instructional Practice #8
Table 6.9: Ideal Teacher Frequencies with Gifted Learners 73
for Instructional Practice #9
vii
Table 6.10: Ideal Teacher Frequencies with Gifted Learners 73
for Instructional Practice #10
Table 6.11: Ideal Teacher Frequencies with Gifted Learners 74
for Instructional Practice #11
Table 6.12: Ideal Teacher Frequencies with Gifted Learners 74
for Instructional Practice #12
viii
Abstract
The purpose of this study was to investigate the differences between the
frequencies of teachers’ instructional practices with gifted and with typical learners
under actual and ideal instructional circumstances. One hundred seventy California
high school geometry teachers completed a 12-item survey developed by the
researcher. The Mann Whitney U/Wilcoxon Matched Pairs test, with a p value less
than .05, was used to reveal any significant differences between teachers’ responses
about instruction for gifted and typical students. Subjects responded with their 1)
actual frequency of practices with typical learners vs. gifted learners, 2) ideal
frequency of practices with typical learners vs. gifted learners, 3) actual frequency of
practices vs. the ideal frequency of practices with typical learners, and 4) actual
frequency of practices vs. the ideal frequency of practices with gifted learners. In
actual instruction, there were no significant differences between eight of the twelve
instructional practices for typical and gifted learners. This implies that minimal
differentiation of instruction occurs for gifted learners in the regular geometry
classroom. Therefore, gifted learners receive much the same exposure to subject
matter as do their typical counterparts. In ideal classroom practices as perceived by
the teachers there were no significant differences between the frequencies of only
four of the twelve instructional practices for typical and gifted learners. Those four
instructional approaches include student note-taking during a lecture, students’
memorization of theorems and postulates, cooperative group work, and the use of
visual media to teach geometry. Ideally teachers would double the amount of
ix
differentiation in instruction for gifted learners and typical learners. Teachers would
incorporate practices more frequently that foster conceptualization and less
frequently use those practices featuring memorization and drill. The increased
frequencies stated for instructional practices for gifted learners in ideal educational
settings implies that teachers understand that instruction should be differentiated for
gifted learners, but that in actuality, teachers do not significantly differentiate
instruction between typical and gifted learners in high school geometry.
1
CHAPTER 1
INTRODUCTION
Statement of the Problem
The National Council of Teachers of Mathematics (NCTM) in its
Curriculum and Evaluation Standards for School Mathematics (1989) “propose that
all students be guaranteed equal access to the same curricular topics; it does not
suggest that all students should explore the content to the same depth or at the same
level of formalism” (NCTM, 1989, p. 131). These standards reflect what educational
researchers have come to understand about meeting the academic needs of gifted
learners in the classroom—gifted learners need complex and novel learning
experiences in order to reach their academic potential.
As with gifted students in general, teachers must specifically individualize
classroom practices to create challenging and appropriate learning opportunities for
all learners. It is the case much too often, however, that “America’s school system
keeps bright students in line by forcing them to learn in a lock-step manner with their
classmates. Teachers and principals disregard students’ desires to learn more—much
more—than they are being taught” (Colangelo, et al. 2004, p. 1). As research has
generated more understanding about teaching gifted children, many instructional
methods and models have frequently been recommended for effectively activating
and building upon the academic and intellectual promise of students. Such models
for teaching and learning are meant to reduce and eliminate boredom, challenge
minds and activate untapped potential in this special group of students.
2
The existence of different models for curriculum and instruction for gifted
learners is due to years of research in gifted and talented education. The Classroom
Practices Study (Archambault et al., 1993), The Successful Classroom Practices
Study (Westberg et al., 1997), and A Nation Deceived (Colangelo, et al., 2004), are
only a few of the studies that have revealed the curricular and instructional strategies
that teachers implement with gifted learners do not significantly differ from those
practices for typical learners in spite of widespread recommendations to do so. For
those few teachers who actually do differentiate classroom practices for gifted
learners, research shows that similarities exist amongst them: they receive support
from their schools and work collaboratively with colleagues to create lesson plans
where gifted learners have opportunities to go beyond the standard expectations of
the classroom (Westberg, et al, 1997; Reis & Westberg, 1994). These and similar
studies have mainly focused on the status of gifted education at the elementary and
middle grade levels. Research about giftedness at the high school level has often
focused on acceleration and/or advanced placement classes, with very little content-
specific focus (Colangelo, et al., 2004).
Gifted learners in the mathematics classroom have the potential to think and
problem solve at complex levels if they are challenged by the curriculum and the
instruction in the classroom. For instance, teachers who teach mathematics to gifted
students must be aware of and build on students’ problem solving abilities, logical
reasoning, self-monitoring in terms of analysis of work, and spatial perception.
Johnson (2000) states that gifted learners differ from the general population in
mathematics by the “pace at which they learn,” the “depth of their understanding,”
3
and by the “interests they hold” (p. 1). Hence, as in other subject areas, what
mathematics teachers do in terms of classroom practices becomes a crucial aspect of
a successful and meaningful mathematical experience for gifted learners.
Purpose of the Study and Conceptual Framework
The purpose of this study was to investigate (1) the actual practices and (2)
the knowledge of ideal classroom practices of high school Geometry teachers when
teaching gifted learners. “While popular wisdom may say gifted children can teach
themselves and learn by do-it-yourself trips to the library, experts say the truth is that
academically talented students need qualified, informed teachers” (Colangelo, et al.,
2004, p. 49). Hence, the study focused on the relationship between what teachers
believe should be the methods of teaching gifted learners and how they actually
teach gifted learners.
Actual instructional practices in the geometry classroom were investigated by
asking about how often teachers actually carry through various classroom practices
for both gifted and typical students. Teachers’ beliefs about ideal classroom
practices were investigated in much the same way, where they were asked to report
how frequently they felt the same practices should ideally be carried out in their
hypothetical geometry classroom. As a result of this study, what truly occurs in
geometry classrooms in terms of providing for the different learning needs of gifted
learners, what classroom practices teachers believe should occur, and whether the
frequency of those actual and ideal practices are different when teaching gifted
versus typical learners was revealed.
4
The conceptual framework of the study is based on the work of Goodlad on
the ideal educational system (Goodlad, 1984), the California Department of
Education (2001, 2005), and the recommendations of the National Council of
Teachers of Mathematics (1989). According to Goodlad (1984) school must be a
place that allows students to reach their maximum potentials. This is achieved by
having teachers who will provide access to knowledge in the circumstances in which
they must teach. What happens in the classroom is not ideal because many children
are lost in the process and are given opportunities to activate their knowledge—ideal
teaching and learning will be achieved when teachers and schools realize that
children have special needs and that learning cannot be a standardized phenomenon.
The GATE standards by the California Department of Education (2001) and the
recommendations for exemplary teaching by NCTM (1989) further extend
Goodlad’s work and help form the conceptual framework of this study. They
suggest that learners of mathematics receive opportunities that foster
conceptualization rather than procedure, which is seldom seen in high school
classrooms. For gifted learners, especially, learning and maximizing upon
mathematical abilities may be heightened by teachers’ choice of classroom
techniques (Goodlad, 1984; California Department of Education, 2001; NCTM,
1989).
Significance of the Study
Past studies conducted targeting classroom practices with gifted learners have
mainly focused on the elementary and middle grade levels, and are only briefly
content-specific. This study was significant due to its focus on the teaching of high
5
school geometry. The results of the study are significant to educational leaders,
directors and participants of teacher education programs, and to secondary school
mathematics teachers because the results may be used to inform decisions about
what types of support these individuals need (e.g., professional development or
training for teaching the gifted) in order to appropriately and affectively meet the
academic needs of gifted students in the classroom.
Specific Research Questions
1. What ideal strategies do high school geometry teachers believe meet the
needs of gifted learners in the geometry classroom?
2. What strategies do high school geometry teachers actually use when
teaching gifted learners in the geometry classroom?
3. How do actual and ideal instructional practices of high school geometry
teachers with gifted learners compare to those same instructional
practices with typical learners?
Overview of the Method
The method of data collection for this study was a self-administered survey
that was sent to 330 teachers across California. In order to develop the survey
instrument, semi-structured interviews of eight California high school geometry
teachers were conducted. The interview responses informed the researcher of the
most common replies geometry teachers provided when asked about actual and ideal
classroom practices for typical and gifted learners in geometry. These responses
were translated into survey questions so that the study could be quantified. The
survey instrument contains “how often should you” and “how often do you”
(i.e. ideal and actual) questions about classroom practices for typical and gifted
learners. Therefore, each of the twelve survey items was responded to four times,
6
with a different perspective each time. These perspectives were termed “Groups”.
For example, Group 1 consisted of a teacher’s responses about how many times they
actually carry out each of the twelve instructional approaches with typical learners,
and Group 2 consisted of that same teacher’s responses about how many times they
actually carry out each of the twelve instructional approaches with gifted learners.
Groups 3 and 4 consisted of the same teacher’s responses about how many times
they ideally carry out each of the twelve instructional approaches with typical and
gifted learners, respectively.
The survey underwent two pilot studies with nine high school geometry
teachers each time. Each pilot study helped develop the clarity of questions, the
length of the instrument (so that it would take approximately ten minutes to
complete), and the readability and ease of the directions. Based on the teachers’
suggestions from the first pilot study, the survey’s format was condensed into a
shorter version of the instrument, and another set of nine teachers were asked to
indicate which version they preferred. The second version and final version of the
survey (Appendix A) was unanimously voted as “better than version 1” during the
second pilot study and hence used here.
The final version of the survey instrument was sent to a random sample of
330 geometry teachers across California. Two mailings of the survey took place to
maximize the number of returned surveys. For each mailing, there was included a
slightly different cover letter explaining the significance of the potential subject’s
voluntary participation in the survey (Appendix C). The researcher contacted by
phone those teachers who did not respond with a completed survey after the first
7
mailing, and then sent the second survey packet to those who complied to having a
second packet of survey materials sent to them. For the very few teachers for whom
it was impossible to contact by phone (due to various circumstances such as the lack
of a telephone in the teacher’s room) to ask them for permission to be sent a second
packet, another cover letter was adopted explaining that the teacher was being sent a
second packet without having been contacted via telephone. The two mailings as
well as the telephone contact were meant to maximize the number of participants in
this study (Fink & Kosecoff, 1998).
Assumptions
It was assumed that the participants responded honestly to the questions in
the survey. It was also assumed that the investigation was valid because the survey
instrument is reliable and valid. The survey is reliable and valid due to the measures
taken during its development and due to the two pilot studies that helped ensure
validity and reliability.
Limitations
The study was limited by factors such as teacher attrition rates. In order for
the study to be valid, the 330 names and addresses provided by Quality Educational
Data (QED) had to be current in terms of teachers still teaching at the schools under
which they were listed. To control for this possibility in the weakening of the
validity of the study, it was necessary to contact the schools individually before
mailing out the surveys in order to check if the teacher still worked at the listed
school. Approximately 20% of the teachers no longer taught at the listed school,
and therefore for each teacher who was taken off the list, the school was asked to
8
provide the name of another teacher who fit the participant criteria. In this way,
random selection was preserved. However, it should be noted that during the three
week period between the time the schools were contacted and the time that the
surveys were first mailed, additional teachers from the list may have changed jobs or
schools, making teacher attrition an uncontrollable, if small, factor in the validity of
the study.
Delimitations
The constraints on generalizability are governed by the fact that the study
was conducted with California teachers; therefore, the results, conclusions, and
recommendations may be generalized only to other California geometry teachers or
to those in similar situations.
9
CHAPTER 2
LITERATURE REVIEW
The purpose of this dissertation study was to investigate the actual and
knowledge of ideal instructional practices of mathematics teachers in secondary
school gifted geometry education. The following is a discussion of what the
literature says about gifted education that is relevant to this study. Those areas of
relevance include: (1) a review of the conceptions of giftedness and the methods of
identification of gift learners, and implications for classroom instruction, (2) how
gifted learners may reach their mathematical potentials, (3) the status of gifted
education in terms of teachers’ classroom practices, and (4) what experts agree are
the leading instructional and curricular models for gifted education and their
implications for classroom practices for teachers of the gifted. Also included here
are the conclusions the researcher has drawn from the literature in relation to this
study and the gaps in the literature that may be further explored with research.
Conceptions of Gifted Education and Identification of Gifted Learners
Definition of Giftedness and Gifted Education
Gifted education refers to the education of advanced learners. Gifted
individuals perform in the academic subject areas, the arts, and in leadership roles at
high levels of accomplishment when compared to their peers, and/or they have the
potential to do so if they are challenged and motivated (Ross, 1993; Clark, 1997).
Gifted individuals have historically most often been perceived and identified as those
with high IQ scores (Terman, 1926). Terman’s (1926) claim that the higher a
person’s IQ score, the brighter that person, is a cognitive view of giftedness that
10
lasted over forty years as the leading definition of giftedness. However, it is
accepted today that a person who is gifted may be advanced intellectually, in
creativity, in leadership roles, in knowledge of academic subjects, and/or in the
performing and visual arts (Marland, 1972). Recent definitions of giftedness have
focused more on the ability to develop intellect, engage in higher order thinking
skills, student performance, and on the recognition that gifted students require
supplemental services that are normally not found in schools today (Ross, 1993;
Clark, 1997). The United States Department of Education agrees that gifted children
“exhibit high performance capability in intellectual, creative, and/or artistic areas,
possess an unusual leadership capacity, or excel in specific academic fields. They
require services or activities not ordinarily provided by the schools” (Ross, 1993, p.
26). If this definition is interpreted at the classroom level, then the implication is that
gifted learners need curriculum and instruction that is not always traditional but
varied in depth, complexity, pacing, and content. Gifted learners in the
heterogeneous classroom (which is often representative of an entire school’s
population) must be exposed to lessons, activities, and experiences in both social and
academic settings that build upon their potentials.
A more theory-based conception of giftedness in addition to the above federal
and intelligence-based definitions may be found in Sternberg’s Triarchic Theory of
Intelligence. Sternberg’s conception of giftedness in individuals has three parts:
componential, experiential, and practical workings of the mind. He defines
giftedness as “mental activity directed toward purposive adaptation to, selection and
shaping of, real-world environments relevant to one’s life” (Sternberg, 1985, p. 45).
11
How well an individual deals with problem-solving and decision-making, how well
he carries them out, and how well he obtains new information from which to
problem solve and judge make up the first parts--metacomponents, performance
components, and knowledge-acquisition components--of the triarchic theory
(Sternberg, 1985; Sternberg, 1997). The second part of Sternberg’s conception is the
experiential subtheory, which deals with how well a person carries through a task
depending on the familiarity of the task and how well and quickly the novel task
becomes automated (Sternberg, 1997). The third and final part of Sternberg’s
conceptions of intelligence and giftedness, practical subtheory, states that the more
capable a person is to adapting, shaping, and selecting ideas in context, the more
intelligent a person (Sternberg & Davidson, 1986).
A synthetic implication for instruction may be attained by combining these
widely accepted conceptions of giftedness and gifted education. Gifted education is
the education of high-ability learners to meet their cognitive, practical, creative, and
social capacities. Clearly, there may be many different combinations of gifted
characteristics in any one individual who is gifted, and hence, there are many
academic and social needs to which instruction in the classroom must attend.
Teachers must provide opportunities in the classroom for gifted learners that would
not normally be found in the traditional classroom. They should facilitate in the
“fluency—the production of many ideas, originality—the uniqueness of ideas,
elaboration—the extension of ideas, and flexibility—modifiability of ideas” in gifted
students’ potentials (Torrance, 1979, p. 361). Further, teachers must first believe that
modifying curriculum may help, be willing to try different instructional techniques,
12
be aware of gifted children and their unique personality traits, and be supported by
their administration in their endeavors for trying to make change in their classrooms
(Torrance, 1979).
As the complexity of gifted individuals has been unfolded, educators’ focus
has shifted to tapping student potential through self-realization, self-actualization,
and challenging learning opportunities. Hence, identification and teaching of gifted
learners have evolved into many specific and individualized components, as
summarized in subsequent sections of this literature review.
Identification of Gifted Learners in the Classroom and Implications for Instruction
Given that each gifted individual is unique in terms of the individual’s
abilities, academic and social needs, and the way he/she portrays his/her skills, the
identification procedures for being gifted are also very complex and individualized.
Identification, evaluation, and assessment of gifted learners have branched away
from single assessments (such as an IQ test) to multiple assessments with multiple
criteria (Robinson, 1994; Terman, 1926). Multiple assessments include teacher
screening and identification, observation, portfolio reviews, tests, family history and
background evaluation, and peer identification (Clark, 1997). As stated by Robinson
(1994), educational leaders “have extended try-out periods for students to display
their talents, and [they] have adopted non-traditional assessments” in all areas of
giftedness so as to allow equal identification for those students that could potentially
be identified as gifted (p. 22).
The identification process should be a positive experience for youth and an
enlightening experience for educators in the sense that identification should reveal
13
both the student’s areas of giftedness and her weaknesses. Instruction may then be
modified for the gifted child. According to the National Report on Identification,
Assessment, and Recommendations for Comprehensive Identification of Gifted and
Talented Youth, identification procedures should advocate the best interests of each
potentially gifted child, be guided by quality teacher recommendations and research
of committees, be inclusive of all student groups (disadvantaged and minority
groups), encompass a broad definition of giftedness so as to not overlook any
individual, and modify resources (such as assessment tests and interviews) to fit
unique individual characteristics as much as possible (Richert, 1985). Identification
procedures inform teachers’ classroom practices for gifted children because the
student’s skills are diagnosed through the identification assessments used. Hence,
identification may inform classroom practices because of individualization, which
refers to educators’ use of all of the information gained about the student to provide
“meaningful learning experiences in the most effective and efficient way” (Fox,
1979, p. 105). Individualized instruction is “instruction developed for the learner,
based on assessment data and the logical next steps in learning” (Swassing, 1985, p.
46). The many instructional and curricular modifications recommended for gifted
learners as a result of identification are differentiated in terms of product,
environment, content, and process (i.e., instruction) (Passow, 1981; Maker, 1982;
Swassing, 1985) as discussed later in this chapter.
The many facets of a gifted student’s learning ability and a gifted program all
call for specific teacher criteria to be in place. Such criteria involve teachers
possessing high degrees of content knowledge, an understanding of pacing, breadth,
14
and classroom layout, an understanding of how to facilitate self-directed learning and
holding students accountable for that learning, using resources beyond the textbook,
creating a classroom based upon inquiry learning and open-ended questions, and
having the confidence and the willingness to try new ideas (hence, providing an
example of learning for students) (Ofsted Report, 2001).
Giftedness in the Mathematics Classroom: Developing Mathematical Promise
Being gifted in mathematics is perceived in large part as potential and
advancement in terms of mathematical ability rather than solely possessing the
ability to score very high on mathematics standardized tests (NCTM, 1995).
Principles of gifted education call for rich and meaningful learning opportunities that
allow for ability, motivation, and prior experiences to be tapped into in the
classroom, especially the mathematics classroom where gifted students’ promise is
often banked due to boring, standardized, redundant, and unchallenging curriculum
and instruction. Mathematical promise, which is a subset of gifted education seen in
mathematics learners, refers to the capabilities upon which gifted learners may
maximize given that they receive fair and equitable education (i.e., challenging
learning experiences for gifted learners) within the general education environment
(Sheffield, 1999; House, 1999; NCTM, 1995). Mathematically promising youth
usually exhibit many of the traits associated with giftedness in general, such
as superior verbal skills, keen powers of observation and perception, deep
curiosity, active imagination, original thinking, quick mastery of new
learning, good memory, analytical thinking, and the ability to concentrate and
work independently. In addition, mathematically gifted individuals manifest
certain characteristics directly related to mathematical performance (House,
1999, pg. 4).
15
Mathematics instructors of gifted learners, similar to all instructors of gifted
students, are presented with the task of unleashing mathematical potentials in
students without constantly dispensing facts and knowledge themselves. Kiesswetter
(1985) summarizes that mathematics teachers must present material in an organized
and logical manner so that students may make the maximum sense out of material.
Teachers should teach through a lens of patterns and rules, teach the same patterns
and rules in different contexts, encourage gifted learners to explore complex
mathematics, require students to participate in reverse processes of solving and
creating meaningful problems that are related to each other (from one mathematics
topic to the next) (Kiesswetter, 1985; California Department of Education, 2005;
NCTM, 1995).
As is the case in other disciplines, the idea of leaning away from traditional
teaching practices towards more differentiated curriculum and instruction can be met
with reluctance from mathematics teachers due to the highly structured and skills-
driven nature of mathematics and its textbooks, and the large number of standards
that must be taught in one school year (NCTM, 1995, Kersh & Reisman, 1985).
Classroom practices are seldom differentiated for gifted learners in mathematics
classrooms across all grade levels, but especially at the secondary school level
(Wenglinsky, 2000; Westberg, et al., 2003). Kersh and Reisman (1985) summarize
that mathematics instruction for the gifted may occur in many settings, but “whatever
the setting, instruction has often been limiting in terms of viewing mathematics as
creative and reflexive experiences” (p. 137).
16
In American schools, mathematics courses are often standardized in
instruction and curriculum (Kersh & Reisman, 1985). Teachers tend to have one
instructional technique in common--they misconstrue the practice of assigning more
work to fill up the extra time gifted learners often have in the regular classroom as
providing differential academic support by enrichment through creative activities.
Some teachers even assign more challenging problems to gifted learners, but do not
realize that simply assigning the problems is not enough—they must support the
learning process of gifted learners with inquiry, discussion, and questioning in order
to get the gifted child to establish ownership of his/her own learning
(Wieczerkowski, et al., 2000; Kersh & Reisman, 1985; Westberg & Archambault;
2004).
Mathematics at the secondary school level may be modified for gifted
learners through several mediums such as Honors programs, Advanced Placement
programs, college level courses, Saturday classes, mathematics academies, and
clubs. But these are mainly programs that accelerate mathematics learning and only
beneficial for a subset of gifted children who are ready to be independent learners.
Gifted children do not all benefit from acceleration, hence they need specific
classroom guidance that will benefit them. Gifted students in the mathematics
classroom will exhibit strong problem solving skills, obtain information quickly,
process mathematical information and make generalizations and different
perspectives in solving problems, retain information such as mathematical rules and
relationships, and “see” mathematics involved even in concepts that are not
mathematically obvious (Krutetskii, 1976). Usiskin (2004) summarizes these traits
17
as “flexibility, curtailment, logical thought, and formalization” (p. 59). These traits
call for teachers to modify instructional practices to empower gifted learners of
mathematics to use mathematics to develop academically, intellectually, and
socially. Teachers must “look like” their learners and establish a relationship of trust
and high expectations in the mathematics classroom (Emerick, 1992; Hansen &
Feldhusen, 1994). According to the California Mathematics Framework (2005),
teachers must teach and re-teach material through different lenses, different
algorithms, and different approaches, and then have students learn from these
experiences by comparing and contrasting these methods. This helps establish a
more conceptual type of knowledge rather than procedural knowledge that is far too
often found in mathematics classrooms and in gifted students’ underdeveloped sense
of mathematics (California Department of Education, 2005; NCTM, 1995).
The Status of Classroom Practices in Gifted Education
Gifted learners are often not different from their general education
counterparts, in that they do not want to be singled out, and they perform to the best
of their abilities usually in spurts (Mead, 1954). Given the fact that many gifted
students truly “act” gifted in school inconsistently and sporadically , experts agree
that “the more diversified, the more complex the activities within which children are
encouraged to play a role, the better chance for the superlatively, discontinuously
gifted child to exercise his or her special talent” (Mead, 1954, p. 214). In a study
done by Shields (2002), gifted students reported that teachers in homogeneous
classrooms harbored higher expectations for them and that teachers in heterogeneous
classrooms did not expect as much from the gifted learners than they would
18
otherwise in a homogeneous classroom. Students (both gifted and typical)
responded with better performance and reported having better self-concepts to high
teacher expectations (Shields, 2002). Hence, the quality of teaching matters
significantly in the success of gifted children in either type of classroom. The quality
of instruction and the methods of instruction used in daily classroom practices have a
large effect on the achievement of students in the regular classroom. Teaching does
matter, and how teachers recognize what gifted students need and what they decide
to do with their knowledge greatly impacts the gifted learner. The content, process,
or product pieces of differentiating curriculum and instruction for gifted learners may
be controlled by the teacher (Kaplan, 2005).
Wenglinsky (2000) explains that how teachers teach matters for the overall
success of school children, as seen through the National Assessment of Educational
Progress Study (NAEP, 1999) (also known as “the nation’s report card” for
teaching). In this NAEP summary entitled How Teaching Matters United States
students’ achievement in mathematics and science, and teacher education and
implementation of differential teaching strategies in those subject areas, are among
the last in a sample of thirteen countries. Also seen in the report is that only 25% of
teachers’ professional development time is spent on special-needs students, which
include gifted learners (Wenglinsky, 2000). More discouraging, however, are the
overwhelming frequency of leading types of activities that eighth grade mathematics
teachers reported using in their classrooms, including teaching out of the textbook,
giving tests, lecturing, and addressing homework problems on the board as a routine.
What they did not report was engaging their students in mathematics conversations,
19
reading and writing about mathematics, collaboration, or assigning challenging
problems (Wenglinsky, 2000). The direct correlation between low achievement in
mathematics and teachers’ low levels of modification of instructional techniques for
gifted learners clearly implies that teaching matters in the development of gifted
learners’ abilities.
The 2003 NAEP progress report and the 2003 Trends in International
Mathematics and Science Study (TIMSS) also showed complimentary results in
terms of student achievement in mathematics at the eighth grade level. Both studies
showed that there was no significant difference in test scores from 1999 to 2003,
which means that mathematics achievement in United States’ eighth graders
remained the same over four years. The United States ranked far below other
countries in mathematical achievement.
A Nation Deceived (Colangelo, et al., 2004) is a study that demonstrates that
American schools ignore student excellence by keeping them bored and
underchallenged in schools. This report shows that there are large numbers of gifted
students in classrooms that go unnoticed—they are not allowed to accelerate, and
when they are, they are accelerated inappropriately. The study revealed two very
important facts about gifted students, and these facts have implications for classroom
instruction. One, acceleration, both grade-based and subject-based is beneficial
academically and socially in the long run. Two, when advanced learners are given
the same curriculum and instruction as their “average” peers, they are often
unchallenged and become bored, they are displeased with their classroom
experiences, and do not want to learn (Colangelo, et al., 2004). These two findings
20
of the study call for classroom approaches that are appropriate for high-ability
learners— techniques sometimes need to be different for the general education
population and the gifted population of students.
A study conducted specifically about the experiences of teachers with both
typical and gifted students in the regular classroom was the Classroom Practices
Study (Archambault, et al., 1993). The results showed that a random sample of 7,000
third and fourth grade teachers in different geographic locations in the United States
who were surveyed and interviewed implemented minimal amounts of differential
instruction and curriculum for gifted learners. In other words, there seemed to be
very little difference between classroom practices for typical and gifted students.
This study was replicated in New South Wales (Whitton, 1997) and with middle
grade teachers in the United States (Robinson, 1998). The findings of these two
studies proved to be almost identical to the original study in 1993. Ten years after
the Classroom Practices Study (Archambault, et al., 1993), Westberg and Daoust
(2003) reported that the replication of the Classroom Practices Study that they
conducted did not show any changes in the classroom practices found in the 1993
study. Even though an emphasis on state standards and different assessments for
mathematics education had been implemented in the last ten years, almost no
differences had been found, again, between how teachers meet the academic needs of
typical and gifted children.
In 1997, the Successful Classroom Practices Survey (Westbert, et al., 1997)
sought to uncover and report those circumstances that contribute to exemplary
instructional techniques of the few teachers who actually did practice differentiated
21
curriculum and instruction as reported in the Classroom Practices Study
(Archambault, et al., 1993). In this multiple site qualitative investigation, the
patterns and themes that emerged were that teachers are willing to modify
curriculum (1) if their efforts are supported by the administration, (2) if they
maintain autonomy, (3) if they are trained and gain advanced knowledge about the
subjects they are teaching and the methods they must use to teach those subjects, (4)
if they harbor a belief that positive change is possible, and (5) if they are given
opportunities during work hours to collaborate with other teachers about teaching
and curricular strategies (Westberg & Archambault, 1997).
Clearly, teachers find it a complex task to integrate appropriate curriculum
and instruction for gifted students and for general education purposes in the same
classroom. Teachers are willing to make changes in their classrooms to better teach
students of all ability levels, but they require support and training to do so
(Westberg, et al., 1997; Archambault, 1993; Reis, et al., 1994).
Another revealing study about teachers’ practices was the Curriculum
Compacting Study (Reis, et al., 1998). Curriculum compacting reduces the amount
of students’ previously learned material and substitutes new content in the
curriculum. Reis, Westberg, Kulikowich, and Purcell (2004), in their research about
teacher training and its effects on the practice of curriculum compacting, found that
teachers are more motivated and less afraid to compact curriculum to meet the needs
of gifted and talented students in their classrooms because “test scores of students
whose curriculum was compacted did not differ significantly from students whose
curriculum was not compacted” (Reis, et al., 2004, p. 106). The study revealed that
22
teachers were excited about modifying curriculum for gifted and talented students,
but only after they had received high levels of training on how to compact
curriculum. Hence, the removal of already learned material with the substitution of
new and rich material can benefit students, and should be a part of teachers’
curriculum planning and instruction.
Starko and Schack (1989), in a study about teacher beliefs regarding
appropriate classroom practices for the gifted, learned that teachers of gifted students
reported five areas of most importance for gifted education: higher level thinking
skills, elimination of previously learned content, preparation for instruction,
independent study opportunities for the gifted, and creativity training for teachers.
The implication of the conclusions of this study when combined with the conclusions
of the previously mentioned studies is that teachers of gifted students apparently
know of appropriate classroom practices for the gifted, but they are clearly not
carrying them through. This may be attributed to lack of training, lack of time
during the school day, lack of support, lack of content knowledge, and/or lack of
knowledge of implementing instructional techniques.
Instructional and Curricular Models for Teaching Gifted Learners
Though the majority of the above studies have generally shown that gifted
learners do not receive fair and challenging instruction in the heterogeneous
classroom, there are in fact specific instructional strategies that have been shown to
increase the chances for gifted learners to meet their potentials. The California
Mathematics Framework (2005) states that any mathematics program must foster
both computational/procedural learning, and conceptual learning. Instructional must
23
involve direct instruction, investigation, classroom practice/drill, small groups,
individualized formats, hands-on practice, projects, and the use of technology in
order to be comprehensive, equitable, and effective for any type of learner
(California Department of Education, 2005).
Research has mainly focused on developing teaching strategies and
frameworks that challenge the gifted learner within the general education classroom;
therefore, the above components of any mathematics program are critical for the
academic and social achievements of gifted learners in heterogeneous classrooms.
The following synopsis of these models shows differentiation as the overarching
pedagogical approach that allows for rich learning and meaningful experiences in the
classroom.
Differentiation refers to the multiple methods of adjusting, modifying, and
customizing curricular and instructional practices such that they best attend to the
learning needs of students. Curriculum that is challenging, complex, encourages
higher-order thinking, in-depth, inter- and intra-disciplinary, and novel, combined
with a variety of instructional approaches such as collaboration, group work,
extended time for projects, open-ended activities, inquiry, discovery, and projects are
the hallmarks of a differentiated program (Kaplan, 1986; Hertzog, 1998; Reis, et al.,
1994).
Differentiation is applicable for both average and gifted students. According
to the California Gifted and Talented Education (GATE) standards, differentiation of
curriculum has four elements: depth, complexity, novelty, and acceleration
(California Department of Education, 2001). Dinnocenti (1998) further explains that
24
the “three components that are most notably associated with differentiation are:
content—what is being taught; process—how it is being taught; and product—
tangible results produced based on students’ interests and abilities” (p. 2).
Differentiated curriculum and instruction, ultimately, provide meaningful
educational opportunities for children where they may learn through multiple
perspectives, big ideas, novel ideas, and acceleration and enrichment. Some or all
of the above tenets of teaching with a differentiated perspective may be found in the
following leading models for the teaching of gifted learners, both in homogeneous
and heterogeneous classrooms and schools. Each model and its implications for the
modification of classroom practices are explained briefly.
Autonomous Learner Model
George Betts’ Autonomous Learner Model (1985) is suitable for teaching
gifted learners in that it fosters “independent learning abilities, creativity, or self-
awareness” that is often lost when the emphasis in schools for the support of gifted
learners is on advanced placement classes, college courses, or acceleration (Maker &
Nielson, 1995, p. 25). This model calls for the teacher to become a facilitator of
learning, rather than a dictator of activities and expectations, or an organizer of
knowledge into conceptual schema. Instructors must teach students how to use
learning strategies on their own and transfer them to novel learning situations (Maker
& Nielson, 1995). The five dimensions of the Autonomous Learner Model include
orientation, individual development, enrichment activities, seminars, and in-depth
study (Betts, 1985). Teachers must possess knowledge and/or training for student
collaboration activities, different discussion and inquiry skills, cross-disciplinary
25
activities, and counseling of students, especially for the gifted at the secondary
school level (Betts, 1985; Udall & Daniels, 1991, Betts & Knapp, 1981).
The Cognitive and Affective Taxonomies Model
The Cognitive and Affective Taxonomies Model (Bloom 1956; Krathwohl, et
al., 1964) is a combination of Bloom’s Taxonomies and Krathwohl’s Affective
Taxonomies. This model calls for instruction that “evokes certain types of thinking
and teachers can be reasonably accurate about the underlying processes that go into a
particular activity” (Maker & Nielson, 1995, p. 55). Essentially, teachers must brake
down learning into smaller steps. Such instruction develops basic skills which then
creates opportunity for critical thinking and analysis. This model implies inquiry,
variety of activities, complexity, discovery, freedom of choice of activities, grouping
and collaboration, and most importantly, the knowledge of specifics that are often
ignored in the general education classroom as appropriate classroom modifications
for gifted learners (Maker & Nielson, 1995; Bloom, 1956; Krathwohl, et al., 1964;
Paul, 1985).
Creative Problem-Solving
The Parnes Creative Problem Solving Model (1975) for differentiating
curriculum and instruction for the gifted places an “emphasis on generating a variety
of alternatives before selecting or implementing a solution” on the part of the student
(Maker & Nielson, 1995, p. 129). In this model, which assumes that gifted learners
may be more creative than the average learner, the teacher must encourage the
deferment of learners’ decisions and judgments until many alternative possibilities
for solving a problem situation have been explored (Parnes, 1975; Parnes, 1988). An
26
environment supported by student collaboration and group work, in either a
homogeneous or heterogeneous group of students, is best for carrying out this model.
Parnes’ model is heavily influenced by Osborn’s (1963) research on fostering
learner imagination. Osborn concluded that teachers need to become directors,
inquirers, and listeners in order to allow students to plan a variety of problem solving
procedures before testing them out. Maker and Nielson (1995) summarize the
process modifications for gifted learners as providing open-ended problems, a
variety of problem situations and assignments, higher and complex levels of
thinking, and freedom of choice in problem solving procedures that are relevant to
students. In order for teaching to best benefit gifted learners, teachers must provide
opportunities for applying previous knowledge to novel, interesting, and realistic
settings. Therefore, the types of activities that teachers choose are critical for
building creativity in problem solving with gifted learners.
The Enrichment Triad Model
Renzulli’s Enrichment Triad Model (1977) is a framework for differentiation
of curriculum and instruction for both gifted and typical students. The model has
three components. General exploratory activities and process thinking are the first
two components and are appropriate for all learners. The third component,
“individual or small-group investigation of real problems, is seen as most appropriate
for gifted learners” (Maker & Nielson, 1995, p. 163). In other words, giftedness may
be developed through the interaction of students with one another in meaningful and
relevant learner settings (Renzulli, 1977; Reis & Schack, 1993; Renzulli & Reis,
1985; Reis & Renzulli, 1985).
27
The Enrichment Triad Model, later on expanded into the Schoolwide
Enrichment Model (SEM) for a larger number of students and teachers, calls for
instruction for the gifted that is modified in the following two ways:
1) teachers should allow students to pursue self-selected interests to their own
desired extent in their own learning styles, and
2) teachers should enrich curriculum by “(1) identifying and structuring
realistic, solvable problems that are consistent with the students’ interests; (2)
acquiring the necessary methodological resources and investigative skills that
are necessary for solving these particular problems; and (3) finding
appropriate outlets for student products” (Renzulli, 1977, p. 10).
The Basic Structure of a Discipline—A Framework for Gifted Education
Bruner’s Basic Structure of a Discipline is a framework for teaching gifted
learners. The underlying idea of this framework is that teaching and curriculum
must be organized and structured; the assumption is that if basic concepts are taught
first and they are taught in a manner that is considered “good teaching,” then every
student will be ready to learn (Bruner, 1960; Maker & Nielson, 1995). In order for
teachers to teach well, their philosophy of teaching should be guided by inquiry
teaching. Bruner’s Basic Structure of a Discipline suggests, hence, that “higher
levels of thinking, discovery, open-endedness” are critical in the planning of
instruction for gifted learners (Maker & Nielson, 1995, p. 94).
The Parallel Curriculum Model
A curricular design for gifted students is the Parallel Curriculum Model
(PCM) (Tomlinson, et al., 2002). PCM includes different formats and approaches in
28
four separate curricular areas, but this “should not be taken to mean that the formats
or approaches must remain separate and distinct in planning or in classroom use”
(Tomlinson, et al., 2002, p. 17).
When designing curriculum for high-ability learners, PCM suggests linking
the Core Curriculum, the Curriculum of Connections, the Curriculum of Practice,
and the Curriculum of Identity together to strengthen the potential to achieve in
gifted and high-ability learners. In the core curriculum, gifted and talented children
learn content first in order to be able to extend their knowledge. An implication for
teachers when developing curriculum with PCM is that they must be thorough in
building basic knowledge so that working memory may be transferred into long-term
memory and retrievable for subsequent learning. The mastery of basic skills is a
critical part of this curriculum.
The purpose of the curriculum of connections, which is similar to
differentiating for depth and complexity, is to enable students to think about and
apply knowledge across many mediums (e.g., disciplines, cultures, perspectives,
locations). The implication for instruction for gifted learners is that teachers must
implement activities that are interdisciplinary and that do not inhibit harboring and
experiencing different perspectives of the content.
The curriculum of practice and the curriculum of identity call for instruction
that encourages students to practice applying their knowledge in a real-world setting
and connecting the discipline to their own experiences. A large part of the education
of gifted learners is to help them develop social skills, and these two tenets of the
PCM are meant to do just that (Purcell, 2002; Clark, 1997). Instruction in the
29
classroom must facilitate problem solving by making connections to prior
experiences.
The Parallel Curriculum Model works for small scale lessons or long term
projects—“with a revised or designed unit ‘in hand’, a teacher can move back and
forth across [a parallel], some, or all parallels in a single unit. Equally attractive, a
teacher might use just one parallel to extend a core unit” (Purcell, et al., 2002, p. 4).
High school teachers may find this model more feasible than most other models
given the many standards they must cover and the flexibility of the model.
Layered Approach Model
The Layered Approach, also developed by Dr. Sandra Kaplan (2005),
emphasizes teachers’ steps towards fostering critical thinking, creativity, and
problem-solving skills in gifted learners after they have defined and set the core
curriculum in the classroom. She explains that changing the content, the process,
and/or the product components of curriculum enables teachers to differentiate the
regular curriculum for advanced learners. It is up to the teacher to decide which of
these three, or all of these three components, of curriculum to differentiate for gifted
students. Successfully and according designing the curriculum and instruction with
altered content, process, and/or product components for gifted learners depends upon
the teacher’s background knowledge of the gifted child and the modifications that
are appropriate for him/her (Kaplan, 2005)
Implications for teachers of the gifted might be that they must go
beyond the textbook to enrichment resources as they cover the content
standards set by their state. In the layered approach, the teacher plays a key
30
role in encouraging a gifted child to step towards thinking creatively and
analytically.
Group Investigations Model
Originally developed for building social relationships between different
ethnicities, the Group Investigations Model (Sharan & Sharan, 1992) is now used for
grouping similar ability students together for collaboration, cooperative group work,
and social interaction in the classroom. Hence, the model, designed for all learners,
calls for the homogeneous grouping of gifted learners. According to Maker and
Nielson (1995)
group investigations is designed to incorporate students’ interests, abilities,
and past experiences in the planning of small-group activities. In this model,
peer collaboration and student choice of projects is emphasized” (p. 199).
According to Group Investigations, teachers must become advisors of the problem-
solving process and of the selection of projects and activities (Sharan & Sharan,
1992; Vygotsky, 1978). The “teacher must create a learning environment that
stimulates interaction, search, and communication while maintaining an indirect,
facilitative style of leadership” (Maker & Nielson, 1995, p. 202). In order for such
an environment to be established, Group Investigations suggests teachers’ classroom
practices should include allowing gifted learners more time for projects, allowing
them to conduct research about self-selected topics, and encouraging the asking of
questions and peer explanations of these questions (Sharan & Sharan, 1992).
Teaching Strategies Approach
The Teaching Strategies Approach (Taba, 1966) is an organization of
teaching strategies meant to help teachers build on student knowledge and potential
31
through discovery and inquiry. “Four strategies have been developed: (a) concept
development, (b) interpretation of data, (c) application of generalizations, and (d)
resolution of conflict (also called interpretation of feelings, attitudes, and values)
(Maker & Nielson, 1995, p. 229). The most important aspect of the teaching
strategies is for teachers to plan lessons with many open-ended questions, teach
through patterns and sequences, debrief with students about ideas and solutions to
reinforce the learning process, and change the pace of teaching depending on student
ability (Taba, 1966).
Taba (1966) believed that all children are capable of learning, but that the
quality of knowledge acquisition is different depending on student ability. Gifted
learners need to develop their intellectual, leadership, and social potentials and hence
should be required to reason and explain their solutions, verbalize their findings,
work in groups often, make independent judgments about problem solving, and learn
through inquiry and discovery. The four strategies of this model require that
classroom practices of teachers be modified for gifted learners by making the teacher
more of a facilitator (rather than a dispenser of content knowledge), a listener, and a
questioner. Especially with gifted learners, it is important the teachers “extend and
lift” discussions to help form patterns and generalizations. This implies that teachers
ask more of gifted students than they do of typical learners. Taba, influenced by
Piaget’s work on cognitive development and developmental stages in children,
explains that teachers must let the learners begin with a concept, then “lift” the level
of thought required to “extend” into the next concept (for example, having the
32
students generate a list about an idea and then having them group the items in the list
into categories) (Taba, 1966; Piaget, 1963).
Other Approaches and Teaching Strategies for the Gifted
Many of these “approaches have all been used by schools in their programs
for the gifted, [but] the emphasis of curriculum intervention has been at the level of
program model, not day-to-day classroom practice” (Van Tassel-Baska, 1997, p.
127). Daily classroom practice, as portrayed by the literature review on the status of
gifted education, is what must be further explored and improved for the best possible
education of gifted learners. Teachers of mathematics may incorporate one or more
of the above models when teaching, and they may be selective with the models
depending on the activity. They may enrich, accelerate, or deepen the lesson by
bringing in other disciplines, student experience, and group collaboration. NCTM
(1999) recommends that teachers let students become owners of their education by
creating problems and explaining them in mathematics, attempting and talking about
challenge problems, working with peers, and working on their own. Curriculum
must “respect the unique characteristics of the learner,” and instruction must reflect
the teachers’ understanding of students’ different learning styles, areas of strength
and weakness, and students’ interests (Tomlinson, et al., 2002, p.4). Essentially,
teachers of gifted learners in the mathematics classroom must “recognize, reinforce,
and reward greatness. As a member of the classroom’s mathematical community,
the teacher must model the same mathematical investigative processes that she
wishes to cultivate in her students” (Greenes & Mode, 1999, p. 121).
33
Stepanek (1999) explains that mathematics teachers may effectively develop
mathematical promise when participating in such activities as those above through
various means. One of those is by using the Problem-Based Learning Model (PBL),
where a problem is presented and students are required to solve the problem
together. In PBL, students are “presented with an ‘ill-structured’ problem’” and they
must seek additional information that is missing and decide on an approach to
solving the problem (Stepanek, 1999, p. 33). The teacher acts as a coach, rather than
the leader of the lesson. Teachers ask students to prove their answers, their methods
of solving, and what they think they could have done better, which gets to the
essence of learning mathematics in a conceptual way.
Another model for learning is Creative Problem Solving (CPS), which has
been a part of education since 1953. Treffinger, Isaksen, and Dorval (2000) explain
that CPS focuses on teaching students to recognize problem situations and the skills
they must have when solving them. Student must first understand the situation, then
generate ideas for potentially generating a solution, and finally prepare to take action
by forming a list of specific action steps. This approach is also appropriate for gifted
learners when learning mathematics because it forces them to conceptualize the
mathematics, rather than memorize steps and algorithms.
Learning centers and seminars have also been suggested to stimulate
students’ interest when learning mathematics (Stepanek, 1999). When a teachers’s
goal is to incorporate advanced material into the curriculum, these two classroom
practices are interesting and more student-based methods of doing so than solely
relying on the textbook (Stepanek, 1999).
34
Williams (1986) provides a list of strategies for differentiating content and
curriculum for gifted learners in the inclusive classroom. Among this list is
evaluating problem situations for consequences and effects on other parts of the
problem, encouraging students to use their intuition and providing classroom
opportunities for students to move forward with their own “hunches”, form examples
of habit and create tolerance for exploration and ambiguity in solutions, ask leading
questions and provide opportunities to seek answers to those questions, teach the use
of paradoxes and discrepancies to test hypotheses, and teach through the use of
connections and analogies between subjects. Clearly implied from the
recommendations of major organizations for the gifted and for mathematics teaching,
the success of gifted students begins with the teacher.
Gaps in the Literature: Comparisons Between Content-Specific Actual and
Ideal Classroom Practices in Gifted Education
Gaps in the literature about classroom practices for gifted students occur in
specific content areas such as mathematics and at the secondary school level.
Studies about curriculum and instruction for the gifted have mostly focused on
elementary and middle school children and teachers of this population. The
literature does not sufficiently cover the gifted and talented high school sector. Due
to the departmentalized structure of most high schools and the nature of the school
day (where students change teachers and classes for different subjects), literature
about classroom practices for the gifted may seldom be generalized across subjects.
Specifically in mathematics at the high school level, alternate and specialized
pedagogical techniques for advanced learners remains a vague concept that is
35
understood by some and carried out by a select few, as revealed by the existing
literature.
The results of the Third International Mathematics and Science Study
(TIMSS, 1999), and the similar results of the Trends in International Mathematics
and Science Study (TIMSS, 2003) showed that eighth graders in the United States
are behind in mathematics and science education when compared with almost half of
the 37 countries in the two studies, and that, in general, challenging learning
opportunities in gifted education are sorely lacking in the United States (NCTM,
1995). Little has been done for secondary school mathematics and for improving the
opportunities for knowledge acquisition of general and gifted education students.
The NCTM Task Force on the Mathematically Promising (1995) has
concluded that measures to improve teaching and learning for gifted and talented
students must occur through evaluation and implementation of multiple types of
teaching. However, it is difficult to find studies or reports where the findings and
recommendations are content specific. There is very little literature, besides
literature on specific lessons or activities for the classroom, on contents such as
geometry as the secondary school level. What happens at the secondary school level
mathematics classroom for gifted learners? Is what happens in the secondary
mathematics classroom similar to what happens at the elementary and middle school
levels in terms of teachers’ practices for gifted and talented students? And
depending on the answer to that question, are secondary school mathematics teachers
aware that there are exemplary practices that they may implement in order to
challenge gifted students and enable them to meet their potentials? If teachers are
36
aware of those practices, how do their beliefs compare with what they actually do?
These are some of the questions that have not sufficiently been answered by past
studies and questions that this study will address for secondary school geometry
education.
Summary and Conclusions
Greenes and Mode (1999) state that math teachers often believe they are
challenging gifted learners because they might be assigning more problems instead
of different problems. “There is a big difference between ‘more’ and ‘different’
problems. Different problems build on the identified strengths of the students. More
problems provide practice and are not designed to respond to individual strengths
and interests” (Greenes & Mode, 1999, p. 125). This suggests a need to investigate
the extent to which the academic needs of gifted learners are met in the secondary
school mathematics classroom.
Recent research suggests critical questions for further studies that are similar
in scope to this dissertation. There is a pressing need to solve the problem of
teachers’ often one-sided beliefs about types of curriculum for diverse student
populations. The fact that many high school teachers teach mathematics as though
every child will benefit from one type of curriculum is a cause for concern this day
in education and evolving technology (Tomlinson, et al., 1996). A question raised
by Greenes and Mode (1999) is whether all types of teachers are capable of teaching
gifted students in mixed-ability classes, and whether their ability is related to the
types and amount of training they receive (if any). Hertzog (1998) asks how
students’ qualities should be assessed and by whom, implying a need to understand
37
what teachers feel is implied on their part by having gifted learners in one’s
classroom. Hertzog (1998) also claims that after her study on open-ended questions
as differentiation through learners’ responses, that it is worthy to explore “how
teacher training or awareness could enhance the ability of teachers to develop and
implement open-ended activities for various instructional purposes” (p. 100). This
question supports the need for this dissertation study--to understand what types of
awareness, acceptance, and resulting implementation teachers have of alternate and
more challenging instructional practices.
As stated eloquently by Hertzog (1998), “we must continue to make
systematic inquiries into curricular strategies that maximize students’ performances”
(p. 101). This study will support these systematic inquiries by exposing the extent to
which secondary school geometry education challenges gifted learners (actual
classroom practices) and the extent to which secondary school geometry education
has potential to challenge gifted learners (ideal classroom practices).
After compiling this literature review, it is clear that the status of gifted
education is not so much different than the status of general education. This almost
lack of difference in treatment between the two groups is even more detrimental to
gifted learners because they are the learners with special needs. The studies and the
reports about gifted education show that teachers seem to know what appropriate and
challenging instruction looks like for gifted students, but that they simply do not or
cannot implement that instruction. The factors that aid in this significant similarity
between the general and gifted education are teachers’ time restraints, training, and
beliefs. Until more studies that are content specific and useful for teachers and
38
teacher educators in their own content areas are conducted and studied during
teacher internships, the status of gifted education will remain the same.
39
CHAPTER 3
RESEARCH METHOD
Introduction
This study investigated the actual and the knowledge of ideal classroom
practices of high school geometry teachers with gifted learners in California. The
purpose of the study was to generate new knowledge about how geometry teachers
teach gifted learners in comparison to how they teach typical learners, and how their
reported actions compare with what kind of instruction they believe should ideally
occur with gifted learners. The significance of the study is that the results help shed
more light on the status of gifted education, specifically at the high school geometry
level, so that educators may continue to create programs and support to improve it.
Research Questions
1. What ideal strategies do high school geometry teachers believe meet the
needs of gifted learners in the geometry classroom?
2. What strategies do high school geometry teachers actually use when
teaching gifted learners in the geometry classroom?
3. How do actual and ideal instructional practices of high school geometry
teachers with gifted learners compare to those same instructional
practices with typical learners?
Nature of the Study
The research design of this study was a quantitative investigation of the state
of instructional practices in gifted geometry education. The study was conducted in
the form of a survey in which the ideal and actual classroom practices of geometry
teachers with gifted learners were compared to each other and to those with typical
learners.
40
Subjects
The population of this study was comprised of approximately 2,250 high
school geometry teachers in the state of California with a sample size of 330
randomly selected teachers from this population (California Department of
Education, 2004; Krejcie, et al., 1970). Simple random sampling was conducted to
obtain the names, school mailing addresses, and phone numbers of 330 potential
participants. This information was obtained from Quality Education Data (QED), an
educational marketing company. QED “is a wholly owned subsidiary of Scholastic
Inc., the global children’s publishing and media company. Since 1981, QED has
been an industry leader in providing marketing, consulting, and list services to the
education marketplace” (QED, 2005).
There were no constraints on participant eligibility other than participants had
to be or had to have been (at some time within twelve months of receiving the
survey) a high school geometry teacher in California. Teachers of all ages, all
ethnicities, both genders, and of all levels of teaching experience were allowed to be
a part of the population of this study. The 170 teachers who chose to participate in
the study had a variety of characteristics as revealed from the demographics section
on the survey. Teachers ranged from having less than one year of teaching
experience to having 39 years of experience, having been trained in gifted education
to not ever having any gifted education experiences, having taught honors and
integrated geometry courses to regular courses, and having education levels from a
bachelor’s degree to a master’s degree and a credential.
41
Instrumentation
The instrument used for data collection for this study was a survey developed
by the investigator (Appendix A). The survey was a self-administered, one-time
instrument designed to take approximately ten minutes to complete. The survey’s
multi-column format, its ordinal scale for the teachers’ responses about their
frequency of classroom techniques, and a few of its items were adapted from the
Classroom Practices Survey (Archambault, et al. 1993); the remainder of the
survey’s items were developed or altered by the researcher for purposes of
applicability to high school geometry teachers’ experiences in the classroom.
Section I of the survey was a demographics section with two fill-in questions
and five multiple-choice questions. These seven questions asked for the number of
years of experience teachers had in teaching mathematics and specifically in
geometry, whether or not teachers believed they had training in instruction for gifted
education, whether or not the teacher’s school or district employed a coordinator for
gifted education, whether or not the teachers collaborate with other educators about
gifted education, the types of geometry classes in which teachers had experience
teaching, and the teachers’ highest level of education. The justification for asking
teachers these seven questions was to get to better know the sample of teachers
providing the responses.
Sections II of the survey contained one question with two choices. Teachers
in this section were asked to identify the type of geometry class they taught: a class
with at least one gifted learner or a class with absolutely no gifted learners and only
typical learners. Based on the choice they made, teachers were given directions
42
directly underneath that choice as to which questions in Section III to answer. If
teachers indicated choice 1 (i.e., they had at least one gifted learner in their geometry
class), then they were asked to respond to all of the items in Section III. If teachers
indicated choice 2 (i.e., they had absolutely no gifted learners in their geometry
class), then they were directed to respond to all items in Section III expect for the
ones about their actual instructional practices with gifted learners, as they did not
have any gifted learners and their responses would be invalid for these questions.
Section III of the survey consisted of 12 instructional approaches, for which
the teacher had to provide the number of times a week or month she implemented the
approach with typical learners and with gifted learners. The teachers also had to
provide the number of times they would implement the instructional approach if they
were teaching under what they consider ideal classroom circumstances, once with
typical learners and once with gifted learners. In summary, in Section III of the
survey, teachers responded to “how often do you” and “how often would you” types
of questions by responding four different times to the twelve instructional
approaches. The teachers revealed the frequency of the same instructional practice
in their classroom four times over, each time with a different group of learners in
mind (i.e., typical or ideal) and a different classroom situation (i.e., actual or ideal).
The four frequencies for each of the twelve instructional approaches were
based on an ordinal scale from 0 to 4. The following are the frequencies associated
with each number: 0 = never, 1 = once a month, or less frequently, 2 = a few times a
month, 3 = a few times a week, and 4 = daily. To simplify matters for purposes of
data entry and data analysis, the set of four responses for the twelve instructional
43
practices were categorized into groups. Group 1 was given the name “Actual-
Typical”, Group 2 was given the name “Actual-Gifted”, Group 3 was given the name
“Ideal-Typical”, and Group 4 was given the name “Ideal-Gifted”. “Actual-Typical”,
which is the name of Group 1, is the set of teachers’ responses about the actual
frequencies of the instructional approaches for their typical learners. Group 4,
“Ideal-Gifted”, consists of the set of teachers’ responses about their ideal frequencies
of the instructional approaches with gifted learners.
The idea of having columns on the left and the right side of the survey items
for teacher responses was inspired from the Classroom Practices Survey
(Archambault, et al., 1993) because the two studies are very similar in the nature of
their purposes, and because several pilot studies conducted for the Classroom
Practices Survey revealed that the three-column set up provided the most clarity for
participants and the highest response rate. Therefore, in this survey, the two
responses for each instructional item that were targeted towards teachers’ actual
frequency of implementation were placed on the left side of the instructional
practice, and the two responses that were targeted towards teachers’ ideal frequency
of implementation were placed on the right side of the instructional practice item. At
the end of the survey was a clearly marked “optional” section for any comments that
teachers chose to write about their classroom practices in geometry. Thirty-seven
teachers provided comments in that section.
The reliability and validity of the instrument were strengthened by eight pre-
survey interviews. Content validity was established before the survey was developed
because eight high school geometry teachers were interviewed for purposes of
44
composing survey items that reflected practices teachers actually can and ideally
could apply in their classrooms. Hence, the survey items became a reflection of
what eight randomly selected geometry teachers expressed when asked about their
classroom practices with typical and gifted learners. The items on the survey
instrument are a quantified version of interview responses given by experts in the
field (i.e., the eight geometry teachers that were interviewed) and what the literature
says about teachers’ classroom practices in the field; hence, the survey has content
validity (Fink & Kosecoff, 1998).
Also, in order to assure clarity and effectiveness of the survey instrument,
two pilot studies were conducted. For the first pilot study, nine teachers were asked
to take the survey. Revisions were made according to their suggestions at the end of
the survey, and the new instrument was given to nine other geometry teachers. None
of the total eighteen teachers were participants for the actual study. The survey was
refined again with the results and suggestions of the second pilot study. The two
pilot studies assisted in eliminating confusing or ambiguous survey items and
directions, and hence helped create an easy to read, concise survey (Fink &
Kosecoff, 1998).
Research Procedure
Once IRB permission to conduct the study was obtained (Appendix B), the
names, mailing addresses, and telephone numbers of the 330 randomly selected
teachers were obtained from QED. Based on the recommendations of Fink and
Kosecoff (1998), the teachers were then assigned a random four-digit code number
to protect their identity. This coding process was computer-based; the coded list was
45
saved and stored in a password-protected file. The purpose of coding the teachers
was to avoid using names on surveys and hence to protect confidentiality and to
maintain the privacy of all participants. The results of the survey remained
confidential but not anonymous, because, as explained later in this section, the
investigator contacted some teachers by telephone in order to solicit their
participation in the study (if that teacher did not mail back a completed survey after
the first attempt to have him/her participate).
After the coding was complete, a survey packet was mailed to all potential
participants. The packet that was sent included a cover letter highlighting the
importance and the purpose of the study (Appendix C), an information sheet with
IRB approval stamped on it (Appendix B), the survey with the teachers
corresponding code number written in the top right corner, a stamped return
envelope, and a complimentary pencil. Once a survey was completed and returned,
the teacher that corresponded with the code number on that particular survey was
removed from the list of 330 geometry teachers, and no more surveys were mailed to
that teacher.
In order to receive as many completed surveys as possible and achieve a high
confidence level for the results of the study, several measures were taken to ensure
that teachers would want to participate in the study. First, “advance preparation, in
the form of careful editing and tryouts, [will] unquestionably help[ed] produce a
clear, readable self-administered questionnaire” and allowed the researcher to
receive as many responses as possible (Fink & Kosecoff, 1998, p. 31). The editing
that occurred as a result of the two pilot studies also helped ensure that the survey
46
was reliable. Because the survey was mailed and participants were isolated from the
researcher during the administration process, there was little support for clarification
of any potentially confusing areas on the survey. As stated, the pilot tests helped to
eliminate survey content that was ambiguous or time-consuming. In addition to the
pilot studies that assisted the researcher in getting the results she desired, a follow-up
mailing was conducted to teachers who did not return a completed survey (Fink &
Kosecoff, 1998). The same information sheet, another copy of the survey, a stamped
return envelope, another complementary pencil, and a slightly different cover letter
that expressed the importance of teachers’ voluntary participation in the study were
sent in the same packet (Fink & Kosecoff, 1998). The researcher waited three weeks
between mailings, since the teachers were asked to complete and mail back the
survey within one week of receiving it.
After the first mailing, a telephone call was made to the teachers who had not
mailed back a completed survey. Fink and Kosecoff (1998) explain that telephoning
a potential participant, explaining the significance of one’s study, and asking for
voluntary participation can increase the response rate of a self-administered survey.
During this telephone conversation, the researcher utilized a pre-developed script
approved by IRB to solicit the teacher’s participation and inform the teacher that
he/she would be receiving another packet shortly after the contact (Appendix D).
Mention of the telephone contact was made in the cover letter in mailing two
(Appendix C). Those teachers for whom it was impossible to contact by telephone
due to various circumstances in their schools, a cover letter explaining that they were
receiving a second packet in the hopes that they would participate in the study was
47
included with the second mailing to the teachers. Mention of a telephone contact
was not included in this particular letter to the few teachers who were attempted to
be contacted by telephone but were eventually not (Appendix C).
Another method for encouraging as many people to respond to the survey is
providing an incentive or something to show the researcher’s appreciation of the
participants; in this study, that appreciation was expressed through the cover letter
and the complimentary pencil mailed with the survey, which read “Thank you math
teacher!” In each cover letter, it was explained that the pencil was a gift for the
teacher whether the teacher decided to participate in the survey or not (Fink &
Kosecoff, 1998).
In summary, the short cover letters describing confidentiality along with “the
survey aims and participants,” the simple survey procedures, the self-addressed
stamped envelope, the clear and succinct survey items, the “thank you” pencil, and
the follow up mailing and telephone contact after the first mailing, all encouraged the
teachers to complete the survey and return it to the researcher (Fink & Kosecoff,
1998, p. 31).
The researcher waited four weeks after the second mailing before beginning
data entry and analysis; 170 teachers had responded with completed surveys, hence
that is the number in the final sample for this study. The data was recorded into the
SPSS statistical analysis software program and data analysis commenced from there,
as described below.
48
Data Analysis
Data collected for this study was analyzed using the SPSS statistical program.
An individual item comparison for each of the twelve instructional practices on the
survey was done between practices for gifted and typical learners, both under ideal
and current teaching circumstances. Therefore, four comparisons were generated
with each of the twelve survey items to help answer research question 3. The
comparisons made revealed whether or not there was a significant difference
between 1) the actual frequency of an instructional practice with typical and gifted
learner, 2) the actual frequency and the ideal frequency of the same instructional
practice with typical learners, 3) the actual frequency and the ideal frequency of the
same instructional practice with gifted learners, and 4) the ideal frequency of the
same instructional practice with typical and gifted learners. Since data for this study
were ordinal and not continuous, the comparisons were generated using a test for
ordinal data analysis, which is analogous to a dependent t-test (Hocevar, 2005;
Hinkle, et al., 2003). That nonparametric test is called the Mann Whitney
U/Wilcoxon Matched Pairs Signed-Rank test. Significance levels were based on a p
value less than or equal to .05 and a confidence level of 95%.
In addition to the nonparametric test, descriptive statistics (i.e., the
frequencies for both the actual and ideal instructional practices with gifted learners)
were generated in order to understand which of the twelve survey items teachers
thought most appropriate for gifted learners in two different environments (their
classrooms and their ideal classrooms). These descriptive statistics helped answer
49
research questions 1 and 2 about what types of activities teachers believe appropriate
for gifted learners in actual and ideal circumstances.
Validity and Reliability of the Survey and the Study
In order to produce a reliable instrument, the survey underwent two pilot
studies with two groups of teachers, making a total of eighteen geometry teachers
that completed the survey and provided feedback about readability. Since the results
and feedback from the second set of nine teachers who took the survey showed that
there were no areas that produced concern in the survey instrument, the survey is
reliable (Fink & Kosecoff, 1998). As such factors as misinterpretation of items and
clarity of words were eliminated with the pilot studies, the data collected during the
actual study were more reliable (Fink & Kosecoff, 1998).
Internal reliability or consistency was established by the fact that the survey
is very similar to the Classroom Practices Survey, which was administered with
elementary school teachers (Archambault, et al., 1993). The Classroom Practices
Survey has internal reliability--it was also reused ten years after it was developed and
administered for the first time and produced the same results. It was also used in
Australia and the United States with middle school teachers (the items having been
altered to accommodate middle school teachers’ experiences), and again produced
very similar results. Because the survey for this dissertation study utilized the same
format and had a similar purpose as that of the Classroom Practices Survey, and was
different only in the number of items on the survey and its target population, the
survey is in fact internally reliable. This study was simply an implementation of the
Classroom Practices Survey for actual instructional practices of high school
50
geometry teachers, in addition to its implementation for purposes of data collection
about ideal instructional practices (which was not a tenet of the Classroom Practices
Study).
The validity of this research study was strongly influenced by the level of
honesty of the survey responses provided by the participants, the level of completion
of the survey items, and the number of surveys completed and mailed back.
Teachers completed all the questions on the survey that were applicable to them,
therefore the data were complete. Thirty-three teachers could not respond to the
twelve survey items for gifted learners because they reported not having or not
having had any gifted learners in their classes within twelve months of filling out the
survey. In summary, n=170 for Groups 1, 3, and 4, and n=137 for Group 2 (i.e., the
group of questions about actual classroom practices with gifted learners).
Data collected from this survey are very similar to data from other well-
known studies such as the Classroom Practices Study and its replication in two states
ten years later with middle grade teachers (Archambault, 1993; Westberg & Daoust,
2003), and the replication of the Classroom Practices Study in New South Wales
(Whitton, 1997). Those showed that there is little difference between classroom
practices for gifted and typical learners, and so does this investigation. Therefore,
this study has high concurrent validity.
As mentioned before, the content validity of the survey was established by
the fact that the development of the survey items was a result of knowledge gained
from the literature review, previous research about instructional approaches and
models for gifted students, and responses from interviews conducted with a random
51
sample of teachers with the same criteria as the sample that was used in this study
(Archambault, et al., 1993, Fink & Kosecoff., 1998). In addition to this, however,
content validity was further established because of the fact that each of the twelve
instructional items on the survey may be aligned either with (1) one or more of the
methods of instruction suggested by the California Mathematics Framework (Table
1.1), (2) the computational/procedural or conceptual learning components for student
education also suggested by the California Mathematics Framework (2005) (see
Table 1.2), and/or (3) the four components of a differentiated curriculum (California
Department of Education, 2001) (Table 1.3). These components are discussed in the
literature review. The alignment of the instructional items with these components
further helps the validity of the survey and also assists in visualizing the areas of
focus and neglect in teachers’ responses on the survey, as discussed in Chapter 5.
The following is a list of the 12 instructional practices from the survey to help in
reading the subsequent three tables:
Instructional Practices from the survey:
1 = Assign open-ended problems
2 = Assign challenge or bonus problems
3 = Have students work in pairs or groups
4 = Have students take notes on a lesson while you lecture
5 = Have students memorize theorems, postulates, and definitions
6 = Have students discover and prove theorems
7 = Have students write proofs
8 = Use basic-skills worksheets
9 = Use visual media
10 = Incorporate other subject areas into geometry
11 = Allow time for students to pursue self-selected interests
12 = Provide logical reasoning and problem solving activities with real-life
applications
52
Table 1.1
Alignment of the 12 Survey Items with the Instructional Approaches for
Teaching Mathematics Suggested in the California Mathematics Framework
(2005)
TWELVE INSTRUCTIONAL PRACTICES FROM
SURVEY
1 2 3 4 5 6 7 8 9 10 11 12
Direct Instruction X
Investigation X X X
Classroom
Discussion/Drill
X X X
Small Groups X
Individual Formats X X X
Hands-on Materials X
Projects X X X X
Technology X
Table 1.2
Alignment of the 12 Survey Items with the Conceptual Learning and/or
Computational/Procedural learning Components of Instruction Suggested by
the California Mathematics Framework (2005)
TWELVE INSTRUCTIONAL PRACTICES FROM
SURVEY
1 2 3 4 5 6 7 8 9 10 11 12
Instruction Fosters
Computational and
Procedural Learning
X X X
Instruction Fosters
Conceptual Learning
X X X X X X X X X
Table 1.3
Alignment of the 12 Survey Items with the Four Components of a Differentiated
Program for Gifted Education as Suggested by the California GATE Standards
(2001).
TWELVE INSTRUCTION PRACTICES FROM
SURVEY
1 2 3 4 5 6 7 8 9 10 11 12
Novelty X X
Complexity X X X X X
Depth X X X X
Acceleration X
53
Construct validity “is usually established after years of experimentation and
experience” and therefore is beyond the scope of this investigation (Fink &
Kosecoff, p. 35, 1998). However, construct validity may be established in part by
comparing what experts believe participants will say about a specific construct (i.e.,
actual and ideal practices with gifted learners) and what participants actually do say
on an instrument meant to measure that very construct. That comparison is made in
chapter five of this dissertation.
54
CHAPTER 4
RESULTS
This chapter contains the raw data collected for the study. The data help answer
the questions of whether gifted learners are treated differentially from typical
learners in the geometry classroom and what instructional practices teachers actually
implement and believe should be implemented with each group of learners. The
instrument used to collect data was a survey with twelve items about instructional
practices in high school geometry class. The 170 participants answered the same
twelve questions for four different classroom situations, referred to as groups. They
were asked to mark the frequency of each instructional practice using an ordinal
scale ranging from 0 to 4 (0 being “never” implementing the practice, and 4 being
“daily” implementing the practice). The four classroom situations, or groups,
included the actual instruction of typical learners that occurs in classrooms, the
actual instruction of gifted learners that occurs in classrooms, the instruction of
typical learners that would occur under teachers’ ideal circumstances, and the
instruction of gifted learners that would occur under teachers’ ideal circumstances.
To clarify matters, the four groups were given names such as “Actual-Typical” or
“Ideal-Typical”. Table 2 is a key of phrases and definitions about the groups that
will be used in chapter five.
55
Table 2. Definition of Groups
Group
Group
Name
Group
Description
Group Sample
Size
1
Actual-
Typical
Teacher responses about
the actual frequency of
implementation of twelve
instructional practices with
typical learners
N=170
(All teachers responded
to Group 1 questions)
2
Actual-
Gifted
Teacher responses about
the actual frequency of
implementation of twelve
instructional practices with
gifted learners
N=137
(Only 137 teachers
responded to Group 2
questions because 33 did
not have experience with
gifted students and opted
not to answer these
questions)
3
Ideal-
Typical
Teacher responses about
the ideal frequency of
implementation of twelve
instructional practices with
typical learners.
N=170
(All teachers responded
to Group 3 questions)
4
Ideal-
Gifted
Teacher responses about
the ideal frequency of
implementation of twelve
instructional practices with
gifted learners.
N=170
(All teachers responded
to Group 4 questions)
56
In addition to the descriptive statistics done for this study, four cross tabulations
were performed with the Mann Whitney U/Wilcoxon Matched Pairs test with 2
independent variables in order to answer research question 3: How do actual and
ideal instructional practices of high school geometry teachers with gifted learners
compare to those same instructional practices with typical learners? The Mann
Whitney U test, which is appropriate for nonparametric data, “is sensitive to both the
central tendency of the scores and the distribution of scores” (Hinkle, et al., p. 574).
The Wilcoxon Matched Pairs signed rank test with 2 independent samples is the
“nonparametric analog of the two-sample case with dependent samples for ordinal
data” (Hinkle, et al., p. 579). The independent variable in this study is each
instructional practice. In the SPSS statistical software program used to analyze data,
the independent variable was entered as “instructional practice”. The dependent
variables are Groups 1-4, or the number of times a week or month that teachers
incorporate each instructional practice depending on the ideal or actual
circumstances with both gifted and typical learners. Significant differences are
based on a p value < .05. Table 3 shows those differences.
57
Table 3. Mann Whitney U/Wilcoxon Matched Pairs Test: An Examination of
Differences in Teachers’ Instructional Practices with Gifted and/or Typical
Students under Actual and Ideal Instructional Circumstances
SD = Significant Difference between two groups
NSD = No Significant Difference between two groups
Instructional Practice
Group 1
(Actual –
Typical)
compared
to
Group 2
(Actual-
Gifted)
Group 3
(Ideal-
Typical)
compared
to
Group 4
(Ideal-
Gifted)
Group 1
(Actual-
Typical)
compared
to
Group 3
(Ideal-
Typical)
Group 2
(Actual-
Gifted)
compared
to
Group 4
(Ideal-
Gifted)
1.Assign open-ended problems
and/or projects (such as having
students build scale models of
their “dream” house and
providing square footage,
volume, ratio of areas and volume
to that of the real life house, and
the floor plans).
SD
z = -2.015
p = .044
SD
z = -3.906
p = .000
SD
z = -8.043
p = .000
SD
z =-8.119
p = .000
2. Assign challenge or bonus
problems such as those from a
higher level textbook, a resource
book, or enrichment worksheets.
SD
z = -4.304
P = .000
SD
z = -5.953
P = .000
SD
z = -6.066
p = .000
SD
z =-5.868
p = .000
3. Have students work in pairs or
groups during class time.
NSD
z = -.055
p = .956
NSD
z = -.619
p = .536
SD
z = -2.476
p = .013
SD
z =-2.971
p = .003
4. Have students take notes on a
lesson while you lecture.
NSD
z = -.161
p = .872
NSD
z = -.091
p = .928
NSD
z = -1.496
p = .135
NSD
z =-1.327
p = .184
58
Table 3 (Continued)
SD = Significant Difference between two groups
NSD = No Significant Difference between two groups
Instructional Practice
Group 1
(Actual –
Typical)
compared
to
Group 2
(Actual-
Gifted)
Group 3
(Ideal-
Typical)
compared
to
Group 4
(Ideal-
Gifted)
Group 1
(Actual-
Typical)
compared
to
Group 3
(Ideal-
Typical)
Group 2
(Actual-
Gifted)
compared
to
Group 4
(Ideal-
Gifted)
5. Have students memorize
theorems, postulates, and
definitions.
NSD
z = -.845
p = .398
NSD
z = -.826
p = .409
NSD
z = -.630
p = .528
NSD
z = -.531
p = .595
6. Have students discover and
prove theorems such as the
Pythagorean Theorem or the
Isosceles Triangle Theorem.
SD
z = -1.985
P = .047
SD
z = -4.509
P = .000
SD
z = -5.122
p = .000
SD
z =-6.271
p = .000
7. Have students write proofs of
geometry problems from the
textbook or from other teacher
sources (including two-column
proofs, paragraph proofs, and/or
indirect proofs).
SD
z = -2.831
P = .005
SD
z = -3.109
P = .002
SD
z = -2.073
p = .038
SD
z =-2.075
p = .038
8. Use basic-skills worksheets as
assignments for drill and practice.
NSD
z = -1.376
p = .169
SD
z = -2.206
p = .027
NSD
z = -.053
p = .958
NSD
z = -.848
p = .396
59
Table 3 (Continued)
SD = Significant Difference between two groups
NSD = No Significant Difference between two groups
Instructional Practice
Group 1
(Actual –
Typical)
compared
to
Group 2
(Actual-
Gifted)
Group 3
(Ideal-
Typical)
compared
to
Group 4
(Ideal-
Gifted)
Group 1
(Actual-
Typical)
compared
to
Group 3
(Ideal-
Typical)
Group 2
(Actual-
Gifted)
compared
to
Group 4
(Ideal-
Gifted)
9. Use visual media including
(but not limited to) advance
organizers, diagrams and charts,
overhead transparencies,
geometric figures and solids,
television media, and/or
Geometer’s Sketchpad to explain
geometry concepts.
NSD
z = .000
p = 1.000
NSD
z = -.535
p = .592
SD
z = -3.998
p = .000
SD
z =-4.157
p = .000
10. Incorporate other subject
areas into geometry such as art,
architecture, science, or language
arts in the form of projects,
homework, class work, and/or
creative assignments.
NSD
z = -.848
p = .397
SD
z = -2.511
p = .012
SD
z= -6.936
p = .000
SD
z =-7.511
p = .000
11. Allow or make time available
for students to pursue self-
selected interests in geometry.
NSD
z = -1.350
p = .177
SD
z = -3.155
p = .002
SD
z = -9.981
p = .000
SD
z =-9.466
p = .000
60
Table 3 (Continued)
SD = Significant Difference between two groups
NSD = No Significant Difference between two groups
Instructional Practice
Group 1
(Actual –
Typical)
compared
to
Group 2
(Actual-
Gifted)
Group 3
(Ideal-
Typical)
compared
to
Group 4
(Ideal-
Gifted)
Group 1
(Actual-
Typical)
compared
to
Group 3
(Ideal-
Typical)
Group 2
(Actual-
Gifted)
compared
to
Group 4
(Ideal-
Gifted)
12. Provide problems that
encourage logical reasoning and
problem solving with real-life
applications (for example, having
students solve how much it would
cost to paint the entire school
given the cost of paint per
gallon).
NSD
z = -1.437
p = .151
SD
z = -2.754
p = .006
SD
z = -6.515
p = .000
SD
z =-6.882
p = .000
Total Number of Significant
Differences
4 8 9 9
Total Number of Insignificant
Differences
8 4 3 3
Table 4 provides insights into research questions 1 and 2: 1) What ideal
strategies do high school geometry teachers believe meet the needs of gifted learners
in the geometry classroom, and 2) What strategies do high school teachers actually
61
use when teaching gifted learners in the geometry classroom? More importantly,
Table 4 also provides a visual representation of the raw data for latter tables in this
chapter and to research question 3.
Even though the data for this study are ordinal, they were run as continuous
data for the following table in order to generate the differences between the means of
the frequencies for each instructional practice on the survey and to see trends in
teacher responses. Though the following bar graph was not used to generate
significant differences or cross tabulations between groups, it is interesting to see the
trends in how teachers responded for each individual instructional item.
Key for Table 4
Instructional Practices:
1 = Assign open-ended problems
2 = Assign challenge or bonus problems
3 = Have students work in pairs or groups
4 = Have students take notes on a lesson while you lecture
5 = Have students memorize theorems, postulates, and definitions
6 = Have students discover and prove theorems
7 = Have students write proofs
8 = Use basic-skills worksheets
9 = Use visual media
10 = Incorporate other subject areas into geometry
11 = Allow time for students to pursue self-selected interests
12 = Provide logical reasoning and problem solving activities with real-life
applications
Mean Frequencies:
0 = Never
1 = Once a month
2 = A few times a month
3 = A few times a week
4 = Daily
62
Table 4.
Teacher’s Mean Frequencies of Twelve Instructional Practices with Four
Student Groups
Teachers' M eanFrequenciesofTwelveInstructional Practices
withFourStudentG roups
0
0.5
1
1.5
2
2.5
3
3.5
4
1 23 45 67 89 10 11 12
Ins tructional Practice
Mean Frequenc
Actual--T ypical
Actual--Gifted
Ideal--T ypical
Ideal--G ifted
Tables 5.1-5.12 provide the number and the percentage of teachers out of the
total number of participants that responded to each frequency in each survey item in
Group 2 (Actual-Gifted).
Table 5.1
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #1:
Assign open-ended problems and/or projects (such as having students build scale
models of their “dream” house and providing square footage, volume, ratio of areas
and volume to that of the real life house, and the floor plans).
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 33 19.4 24.1 24.1
1.00 62 36.5 45.3 69.3
2.00 33 19.4 24.1 93.4
3.00 8 4.7 5.8 99.3
4.00 1 .6 .7 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
63
Table 5.2
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #2:
Assign challenge or bonus problems such as those from a higher level textbook, a
resource book, or enrichment worksheets.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 6 3.5 4.4 4.4
1.00 38 22.4 27.7 32.1
2.00 49 28.8 35.8 67.9
3.00 33 19.4 24.1 92.0
4.00 11 6.5 8.0 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 5.3
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #3:
Have students work in pairs or groups during class time.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 6 3.5 4.4 4.4
1.00 9 5.3 6.6 10.9
2.00 32 18.8 23.4 34.3
3.00 52 30.6 38.0 72.3
4.00 38 22.4 27.7 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
64
Table 5.4
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #4:
Have students take notes on a lesson while you lecture.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 3 1.8 2.2 2.2
2.00 7 4.1 5.1 7.3
3.00 31 18.2 22.6 29.9
4.00 96 56.5 70.1 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 5.5
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #5:
Have students memorize theorems, postulates, and definitions.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 13 7.6 9.5 9.5
1.00 9 5.3 6.6 16.1
2.00 25 14.7 18.2 34.3
3.00 50 29.4 36.5 70.8
4.00 40 23.5 29.2 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
65
Table 5.6
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #6:
Have students discover and prove theorems such as the Pythagorean Theorem or the
Isosceles Triangle Theorem.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 7 4.1 5.1 5.1
1.00 34 20.0 24.8 29.9
2.00 58 34.1 42.3 72.3
3.00 28 16.5 20.4 92.7
4.00 10 5.9 7.3 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 5.7
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #7:
Have students write proofs of geometry problems from the textbook or from other
teacher resources (including two-column proofs, paragraph proofs, and/or indirect
proofs)
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 1 .6 .7 .7
1.00 4 2.4 2.9 3.6
2.00 35 20.6 25.5 29.2
3.00 65 38.2 47.4 76.6
4.00 32 18.8 23.4 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
66
Table 5.8
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #8:
Use basic-skills worksheets as assignments for drill and practice.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 26 15.3 19.0 19.0
1.00 35 20.6 25.5 44.5
2.00 45 26.5 32.8 77.4
3.00 20 11.8 14.6 92.0
4.00 11 6.5 8.0 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 5.9
Actual Teacher Frequencies with Gifted Learners for Instructional Practice #9:
Use visual media including (but not limited to) advance organizers, diagrams and
charts, overhead transparencies, geometric figures and solids, television media,
and/or Geometer’s Sketchpad to explain geometry concepts.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 3 1.8 2.2 2.2
1.00 19 11.2 13.9 16.1
2.00 31 18.2 22.6 38.7
3.00 32 18.8 23.4 62.0
4.00 52 30.6 38.0 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
67
Table 5.10
Actual Teacher Frequencies with Gifted Learners for Instructional Practice
#10: Incorporate other subject areas into geometry such as art, architecture, science,
or language arts in the form of projects, homework, class work, and/or creative
assignments.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 11 6.5 8.0 8.0
1.00 56 32.9 40.9 48.9
2.00 42 24.7 30.7 79.6
3.00 22 12.9 16.1 95.6
4.00 6 3.5 4.4 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 5.11
Actual Teacher Frequencies with Gifted Learners for Instructional Practice
#11: Allow or make time available for students to pursue self-selected interests in
geometry.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 66 38.8 48.2 48.2
1.00 49 28.8 35.8 83.9
2.00 16 9.4 11.7 95.6
3.00 4 2.4 2.9 98.5
4.00 2 1.2 1.5 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
68
Table 5.12
Actual Teacher Frequencies with Gifted Learners for Instructional Practice
#12: Provide problems that encourage logical reasoning and problem solving with
real-life applications (for example, having students solve how much it would cost to
paint the entire school given the cost of paint per gallon).
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 13 7.6 9.5 9.5
1.00 39 22.9 28.5 38.0
2.00 55 32.4 40.1 78.1
3.00 20 11.8 14.6 92.7
4.00 10 5.9 7.3 100.0
Total 137 80.6 100.0
Miss System 33 19.4
Total 170 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Tables 6.1-6.12 provide descriptive frequency counts for the survey’s twelve
instructional practices with gifted learners under ideal circumstances. They show the
number and the percentage of teachers out of the total number of participants that
responded to each survey item in Group 4 (“Ideal-Gifted”).
69
Table 6.1
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #1:
Assign open-ended problems and/or projects (such as having students build scale
models of their “dream” house and providing square footage, volume, ratio of areas
and volumes to that of the real life house, and the floor plans.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 4 2.4 2.4 2.4
1.00 40 23.5 23.5 25.9
2.00 83 48.8 48.8 74.7
3.00 29 17.1 17.1 91.8
4.00 14 8.2 8.2 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 6.2
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #2:
Assign challenge or bonus problems such as those from a higher level textbook, a
resource book, or enrichment worksheets.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 4 2.4 2.4 2.4
1.00 11 6.5 6.5 8.8
2.00 50 29.4 29.4 38.2
3.00 69 40.6 40.6 78.8
4.00 36 21.2 21.2 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
70
Table 6.3
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #3:
Have students work in pairs or groups during class time.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 2 1.2 1.2 1.2
1.00 4 2.4 2.4 3.5
2.00 27 15.9 15.9 19.4
3.00 72 42.4 42.4 61.8
4.00 65 38.2 38.2 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 6.4
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #4:
Have students take notes on a lesson while you lecture.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 5 2.9 2.9 2.9
1.00 3 1.8 1.8 4.7
2.00 8 4.7 4.7 9.4
3.00 47 27.6 27.6 37.1
4.00 107 62.9 62.9 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
71
Table 6.5
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #5:
Have students memorize theorems, postulates, and definitions.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 15 8.8 8.8 8.8
1.00 7 4.1 4.1 12.9
2.00 35 20.6 20.6 33.5
3.00 58 34.1 34.1 67.6
4.00 55 32.4 32.4 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 6.6
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #6:
Have students discover and prove theorems such as the Pythagorean Theorem or the
Isosceles Triangle Theorem.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 3 1.8 1.8 1.8
1.00 9 5.3 5.3 7.1
2.00 55 32.4 32.4 39.4
3.00 74 43.5 43.5 82.9
4.00 29 17.1 17.1 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
72
Table 6.7
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #7:
Have students write proofs of geometry problems from the textbook or from other
teacher sources (including two-column proofs, paragraph proofs, and/or indirect
proofs).
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 1 .6 .6 .6
1.00 7 4.1 4.1 4.7
2.00 26 15.3 15.3 20.0
3.00 81 47.6 47.6 67.6
4.00 54 31.8 31.8 99.4
5.00 1 .6 .6 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 6.8
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #8:
Use basic-skills worksheets as assignments for drill and practice.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 43 25.3 25.3 25.3
1.00 41 24.1 24.1 49.4
2.00 47 27.6 27.6 77.1
3.00 25 14.7 14.7 91.8
4.00 14 8.2 8.2 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
73
Table 6.9
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #9:
Use visual media including (but not limited to) advance organizers, diagrams and
charts, overhead transparencies, geometric figures and solids, television media,
and/or Geometer’s Sketchpad to explain geometry concepts.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 1 .6 .6 .6
1.00 3 1.8 1.8 2.4
2.00 24 14.1 14.1 16.5
3.00 49 28.8 28.8 45.3
4.00 93 54.7 54.7 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 6.10
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #10:
Incorporate other subject areas into geometry such as art, architecture, science, or
language arts in the form of projects, homework, class work, and/or creative
assignments.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 1 .6 .6 .6
1.00 20 11.8 11.8 12.4
2.00 58 34.1 34.1 46.5
3.00 60 35.3 35.3 81.8
4.00 31 18.2 18.2 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
74
Table 6.11
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #11:
Allow or make time available for students to pursue self-selected interests in
geometry.
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid .00 10 5.9 5.9 5.9
1.00 54 31.8 31.8 37.6
2.00 66 38.8 38.8 76.5
3.00 28 16.5 16.5 92.9
4.00 12 7.1 7.1 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
Table 6.12
Ideal Teacher Frequencies with Gifted Learners for Instructional Practice #12:
Provide problems that encourage logical reasoning and problem solving with real-
life applications (for example, having students solve how much it would cost to paint
the entire school given the cost of pain per gallon).
Frequency Percent
Valid
Percent
Cumulative
Percent
Valid 1.00 14 8.2 8.2 8.2
2.00 65 38.2 38.2 46.5
3.00 60 35.3 35.3 81.8
4.00 31 18.2 18.2 100.0
Total 170 100.0 100.0
(.00 = Never, 1.00 = Once a month, or less frequently, 2.00 = A few times a month,
3.00 = A few times a week, 4.00 = Daily)
75
CHAPTER 5
DATA ANALYSIS
This study investigated the actual and ideal instructional practices of high
school geometry teachers in California with respect to gifted and typical (i.e. non-
gifted) learners. The three research questions guiding the study were:
1. What ideal strategies do high school geometry teachers believe meet the
needs of gifted learners in the geometry classroom?
2. What strategies do high school geometry teachers actually use when
teaching gifted learners in the geometry classroom?
3. How do actual and ideal instructional practices of high school geometry
teachers with gifted learners compare to those same instructional
practices applied with typical learners?
170 teachers out of a total 330 who were solicited to participate responded by
mailing back a completed, self-reported survey about their actual and ideal
instructional practices with typical and gifted learners in the geometry classroom.
Cross tabulations with the Mann Whitney U/Wilcoxon Matched Pairs Test for
ordinal data were run to analyze the data (Hocevar, 2005; Hinkle, et al., 2003).
Summary of Key Findings
The key findings of this study are similar to those of past studies about
teachers’ classroom practices with gifted learners (Archambault, 1993; Westberg
& Daoust, 2003; Westberg & Archambault, 1997). Teachers show only a slight
difference in their actual instructional practices with typical and gifted learners,
meaning that they generally provide the same sorts mathematics activities for all
types of learners, and at the same level (Sheffield, 1999, 2002). In fact, of the
twelve approaches investigated (including memorization of theorems, the use of
76
visual media, and the writing of proofs), there were only four for which teachers
showed a slight difference between the frequency of implementation of the
practice for typical learners and for gifted learners. Those differences occurred
in assigning open-ended problems, assigning bonus or challenge problems,
having students discover and prove theorems, and writing proofs (California
Department of Education, 2005; NCTM, 1989) (Appendix A). Teachers showed
no significant differences in the remaining eight instructional approaches for
typical and gifted learners, which, similar to past studies, include the use of drill
and practice worksheets, memorization, and taking notes on the teacher’s lecture.
When teachers were asked to indicate the frequency of each instructional
approach used for typical and for gifted learners under ideal circumstances, the
number of significant differences increased. There were eight significant
differences in the twelve comparisons for the “Ideal-Typical” and “Ideal-Gifted”
groups, and only four results that were not significantly different between
frequencies of instructional practices. The significant differences between the
ideal frequency of practices for gifted and typical learners occurred again in
assigning open-ended problems, assigning bonus or challenge problems, having
students discover and prove theorems, and writing proofs, but also included the
use of basic skills worksheets, incorporating other subject areas into the study of
geometry, making time available for the pursuit of self-selected interests, and
providing logical reasoning and problem solving activities (Appendix A).
In actuality and ideally there were no significant differences between
teachers’ responses about the frequency of their implementation of instructional
77
practices 4 and 5 for typical and gifted students. This means that even if teachers
taught in classes they consider ideal, they would still require gifted students to
take notes on a lecture and memorize theorems and postulates at the same rate as
they would require typical learners to perform those actions. According to the
California Mathematics Framework (California Department of Education, 2005)
and Sheffield (1999) such classroom practices are necessary but must be
accompanied by other approaches such as investigation, discovery, group
discussion, and problem solving. Due to the infrequent implementation of these
practices reported by the teachers, this particular finding may be interpreted as
classroom instruction not being as effective as possible towards meeting
students’ learning potentials. In summary, under teachers’ ideal instructional
settings, teachers would like to differentiate between typical and gifted learners
for eight of the twelve instructional approaches from the survey, with gifted
students having more exposure to those activities and instructional methods than
their typical counterparts.
For all instructional practices except students taking notes during lecture
and students being assigned basic-skills worksheets, the plots show a steady
increase in the groups from left to right. The groups are listed from left to right
in the following order: the “Actual-Typical” group, the “Actual-Gifted” group,
the “Ideal-Typical” group, and the “Ideal-Gifted” group. Only do survey items 4
and 8 (taking notes on a lecture and using basic skills worksheets for drill and
practice) show a decrease in the mean frequency for all four groups from actual
to ideal environments.
78
Teachers feel that direct instruction and drill of basic concepts, though
important and appropriate at times (Sheffield, 1999; Goodlad, 1984; NCTM,
1989), are two teaching methods that should ideally be utilized less often than
they are currently being utilized. By comparing the means of instructional
practices mentioned earlier that ideally teachers would implement more
frequently and the recommendations for mathematics instruction made by the
California Department of Education (2005) in Table 1.1, it is reasonable to
conclude that teachers know that teaching with investigative activities and
questions, working in small groups, individual formats, hands-on materials,
projects, and technology are teaching approaches that should occur more often in
the classroom for both typical and gifted learners, but more so for gifted learners.
The California GATE standards prescribe that differentiated curriculum
and instruction show novelty, complexity, depth, and acceleration of learning
(California Department of Education, 2001). A differentiated curriculum offers
multiple opportunities for higher-order thinking by providing challenging, novel,
interdisciplinary, and complex content, fostered by instruction that is multi-
faceted and takes into account the learning needs of gifted children (Kaplan,
1986; Reis, et al., 1994). The effects of a differentiated educational experience
help students make connections, see trends and patterns, and develop big ideas
and conceptual knowledge, instead of gaining only procedural and computational
knowledge (Kaplan, 1986; California Department of Education, 2005; NCTM,
1995). In this study, teachers revealed that they do not frequently incorporate the
facets of differentiated instruction for gifted learners in their classroom, which
79
could mean they are severely limiting learners’ scope of mathematical thinking
(Clark, 1997; Goodlad, 1984). However, teachers show an increase in the ideal
frequency of the twelve instructional approaches that were investigated that do
foster the components of a differentiated program, they reveal a decrease in the
ideal frequency of those instructional approaches that do not. This implies that
teachers feel there needs to be a difference made in their classrooms, and that
certain factors are inhibiting teachers from making that difference.
Individual Item Analysis for the Survey’s Twelve Instructional Practices
The following is an item-by-item examination of the instructional
practices that teachers ranked on the survey. These examinations are the basis
for the key findings previously mentioned in this dissertation.
Instructional Practice 1: Assign open-ended problems and/or projects
The results of this survey item show that teachers believe they should
assign open-ended problems/projects more often to gifted learners than they
actually do in their classrooms. Three-fourths of the teachers believe that ideally,
they should assign such problems a few times a month, and they actually are
doing far less than that in their classrooms (see Tables 5.1 and 6.1).
The Mann Whitney/Wilcoxon Matched Pairs Test shows a significant
difference between the “Actual-Typical” and the “Actual-Gifted” groups for this
instructional practice (Table 3). However, when the means of the two groups are
compared, the true nature of the significant difference is discovered. The low
mean of .9 for the “Actual-Typical” groups indicates that teachers almost never
assign open-ended problems to typical learners and a mean of 1.2 for the
80
“Actual-Gifted” group indicates that teachers assign open-ended problems
anywhere from once a month to a few times a month to gifted learners (Table 4).
It is encouraging that there are stronger significant differences in the other three
comparisons, especially between ideal teaching with typical learners and ideal
teaching with gifted learners. Teachers seem to believe that gifted students need
more open-ended activities than their typical counterparts in the regular
classroom if they are to challenge the learners. There is a steady increase in the
frequency of incorporating open-ended problems for both groups of learners
(typical and gifted) from actual to ideal circumstances (Table 4).
Hertzog (1998) explains that student performance on open-ended
questions and activities allows teachers to understand where students need help
and where they are mastering the content. Open-ended questions are a method of
differentiating the curriculum and instruction in the classroom for gifted learners,
and provide them with avenues of expressing their knowledge in multiple ways
so as to foster conceptualization of ideas (Hertzog, 1998). Therefore, the fact
that teachers report rarely incorporating the use of open-ended activities in their
classrooms is discouraging for the mathematical potential of students, especially
those who are advanced learners.
Instructional Practice #2: Assign challenge or bonus problems
Thirty-six percent of teachers actually assign challenge or bonus
problems to gifted learners a few times a month, and 68% assign the same types
of problems a few times a month or less. Ideally, these teachers say they would
implement this instructional practice much more with gifted learners; in fact,
81
close to 40% said that “a few times a week” would be appropriate, and 20% said
“daily” would be sufficient for gifted learners (see Tables 5.2 and 6.2). It seems
that teachers understand that a challenging curriculum is a central part of a strong
gifted educational program (Clark, 1997; California Department of Education,
2005; Renzulli and Reis, 1985).
Similar to instructional practice 1, there are significant differences in all
four cross comparisons for this instructional practice (Table 3). Teachers more
often assign challenge problems to gifted learners than to typical learners; on
average, they require typical learners to do challenge problems anywhere from
once a month to a few times a month, but require gifted learners to do the same at
least a few times a month (Table 4). Twenty-nine percent of teachers said that
they actually require gifted learners to solve challenge problems a couple of
times a month and 19% said they do so a couple of times a week; in ideal
circumstances, 29% would still require gifted learners to do challenge problems a
couple of times a month, but an increased 40% said that they would incorporate
the challenge problems at least a couple of times per week for gifted learners
(Tables 5.2 and 6.2). In ideal circumstances, teachers expressed that there should
still be a difference in typical and gifted instruction (i.e., there should be more
challenging learning tasks for gifted learners). When comparing their actual and
ideal frequencies for this instructional practice with gifted learners only and with
typical learners only, again they show that they would assign more challenge
problems than they actually do to each group of students.
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Instructional Practice #3: Have students work in pairs or groups during class
time.
Teachers believe that having all students work in groups is a good idea.
Almost two-thirds of teachers have students work in groups during class time
weekly, and they believe students should be working together under ideal
circumstances as well (Tables 5.3 and 6.3). There is no significant difference
between the number of times teachers require typical and gifted learners to work
in groups—this is both good and bad. The implication is that gifted and typical
learners are grouped together all the time, instead of classes alternating between
heterogeneous and homogenous groups in order for students to be able to
experience learning and interacting in all types of situations and with different
people (California Department of Education, 2005; Johnson, 2000; Sharan and
Sharan, 1992; Clark, 1997; Tomlinson, et al., 1996).
Teachers show no significant difference between the ideal frequency of
grouping with typical and gifted students, but they do show a significant
difference between the actual practice with typical learners and the ideal practice
with typical learners, and a difference in actual teaching for gifted students and
ideal teaching with gifted students. By looking at the mean plots and the increase
of frequencies from actual to ideal settings for teachers, it is clear that teachers
would increase the number of times a week they require students to work in
groups if they were teaching under their ideal circumstances (Table 4). What
teachers believe those circumstances are could be one of many things such as 1)
they would like gifted learners grouped together in separate classes, and/or 2)
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they would like training on how to conduct such group activities with gifted
learners (Archambault, et al., 1993)
Instructional Practice #4: Have students take notes on a lesson while you lecture
Seventy percent of teachers lecture while students take notes on a daily
basis. Such a large percentage entails that teachers feel that in their classrooms
they must lecture to teach the standards. This is the same across the board—for
typical and for gifted learners. There are no significant differences between any
of the groups investigated in this study (Table 3). Teachers feel that they should
lecture as often with typical learners as with gifted learners, which is as often as
“daily.”
Under ideal circumstances, teachers still show that they would require
gifted students to take lecture notes as often as they would require typical
learners to do so, but they do feel that there should be less note taking overall
during class time. The bar graph for the average frequencies for this instructional
practice is one of the few that decreases from actual to ideal situations (Table 4).
However, by looking at tables 5.4 and 6.4, it is clear that a large percentage of
teachers give lecture notes daily and would ideally give lecture notes and a daily
basis (70% and 62% respectively). The implication is that though teachers
realize they should focus less on direct instruction, they nevertheless feel
lecturing and students taking notes would be a large part of their ideal
instructional methodology, no matter the types of students in the class. The
California Department of Education (2005) and Greenes and Mode (1999) agree
that a direct instructional approach such as this one should in fact be a part of
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mathematics teaching and learning, but also that too much of it can turn students
off from taking ownership of their education.
Instructional Practice #5: Have students memorize theorems, postulates, and
definitions.
Approximately 37% of teachers ask gifted students to memorize theorems
and postulates as part of their geometry instruction a few times a week; 30% of
teachers ask that students memorize them daily (Table 5.5). Only 34% of the
total teachers have gifted students memorize facts a few times a month or less.
Under ideal situations the percentages are only slightly different, indicating that
teachers feel for the most part having gifted students memorize theorems and
postulates, which is a procedural and computational sort of knowledge
acquisition (Table 1.2) is a good idea for the better part of the week (California
Department of Education, 2005).
There are no significant differences between teachers’ responses for the
“Actual-Typical” and “Actual-Gifted” groups, the “Ideal-Typical” and “Ideal-
Gifted” groups, nor the “Actual-Gifted” and “Typical-Gifted” groups (Table 3).
This means that what teachers are doing now in terms of teaching the
fundamentals of geometry would not change significantly for gifted learners if
teaching situations were ideally different for teachers. Teachers would still have
students memorize facts such as theorems and postulates. House (1999) and
Sheffield (1999) explain that in order for any student, especially one who shows
mathematical promise, to become a mathematical thinker he/she needs
mathematical experiences that encourage discovery, analysis, and justification of
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concepts (not the memorization and regurgitation of facts). They further state
that mathematically promising students may tap into their abilities to become
advanced mathematical thinkers only when they see the purpose of learning the
material. Memorization of theorems, postulates, and definitions, if
unaccompanied by discovery and proof, can limit what students learn in
geometry.
Instructional Practice #6: Have students discover and prove theorems.
There is a significant difference between how teachers approach this
instructional practice with typical and gifted students. Teachers require gifted
students to discover and prove theorems more frequently than they require
typical learners to do the same (Table 4). They also believe that ideally there
should be an increase in the number of times both types of students of students
are exposed to the discovery and proving of theorems, but that gifted learners
should still have more frequent exposure to such learning situations. For gifted
learners, discovery and proving of the theorems should be a weekly part of their
geometry education. This sentiment is a positive point for gifted education in
that teachers realize building conceptual knowledge through justification and
discovery is important in maximizing learning potentials and that proving
theorems is a large part of the content standards for high school geometry
(Sheffield, 1999; California Department of Education, 2005).
Forty-two percent of teachers implement this practice a few times a
month, and only 20% of teachers implement it a few times a week. Ideally, 43%
would incorporate the discovery and the writing of proofs of theorems a few
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times a week (which is a 23% increase from the actual classrooms), but 17%
would like to see it happen on a daily basis for gifted learners (which is 10%
more than in the actual classrooms) (Tables 5.6 and 6.6).
Instructional Practice #7: Have students write proofs.
In teachers’ actual classroom experiences, 47% of teachers reported
having gifted students write proofs of geometry concepts a few times a week and
23% do so on a daily basis (Table 5.7). This indicates that a large portion (70%)
of the teachers believe writing proofs of conjectures is a fundamental process of
learning geometry, which is a positive finding since writing proofs is
incorporated into many California content standards for high school geometry
(California Department of Education, 2005). One third of the teachers require
that gifted students only write proofs a few times a month or less. Under ideal
circumstances, those percentages go up to 48% and 32% respectively, leaving
fewer teachers who still would not require the writing of proofs more than a few
times a month (Table 6.7).
In all teaching scenarios (i.e. actual and ideal circumstances with only
gifted, only typical, or a mix of both gifted and typical learners), significant
differences were found between how often students are asked to write proofs
(Table 3). By looking at the mean plots for this instructional practice, there is a
clear increase from the frequency of the actual practice for both typical and gifted
learners to the frequency of the ideal practice for both types of learners (Table 4).
According to NCTM (1989), students who attempt to write proofs frequently
prepare themselves for spacial thought and make connections among
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mathematical topics that are often dismissed as unrelated to each other. The
majority of teachers seem to believe writing proofs frequently is an appropriate
and important part of a geometry learner’s mathematical experience.
Instructional Practice #8: Use basic-skills worksheets as assignments for drill
and practice.
A particularly interesting finding is seen in the results for instructional
practice 8. Even though more than half the teachers (55%) reported using basic
skills worksheets anywhere from a few times a month to a few times a week for
drill and practice, there is no significant difference between their actual practices
for gifted and typical learners (Tables 3, 5.8, and 5.9). This indicates that gifted
learners are required to participate in the same drill and practice activities as
typical learners, as frequently as typical learners. The assigning of drill and
practice activities are appropriate instructional methods for learning the basic
concepts of mathematics, but not sufficient for helping gifted learners tap into
their mathematical abilities if those activities are the dominant forms of
instruction gifted students receive (California Department of Education, 2005;
NCTM, 1995; Emerick, 1992).
Under ideal teaching situations, however, teachers show a significant
difference in the frequency of this instructional practice with typical and gifted
learners (Table 3). Teachers reported ideally having gifted learners do less drill
and practice basic-skills worksheets than typical learners. In fact, 77% of
teachers believe they would require advanced learners to complete drill and
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practice worksheets only a few times a month or less, and one third of those
would never require their advanced students to spend time on such activities.
Instructional Practice #9: Use visual media.
Teachers felt that using visual media is an appropriate instructional
practice for all geometry students. In their classrooms, 38% reported using
visual media on a daily basis, 24% use it a few times a week, and 39% a few
times a month or less (Table 5.9). There is no significant difference found
between the actual frequency of this instructional practice with typical and gifted
learners; in fact, on average the frequency is almost identical for both groups of
learners at a “few times a week” (Tables 3 and 4).
Teachers would ideally incorporate more visual media with advanced and
typical learners, but the frequency would still be the same for both groups. There
is no significant difference between the how often teachers would use visual
media for “Ideal-Typical” and “Ideal-Gifted” groups. According to the
California Mathematics Framework (California Department of Education, 2005),
geometry standards may be better mastered if instruction is accompanied by
hands-on materials and visual media. Students should have the opportunity to
see how geometric concepts appear in objects and how they are concrete
representations of algebraic abstracts. Visual media significantly increases
students’ understanding of spacial logical (California Department of Education,
2005). Teachers are aware of the positive affects of using visual media in the
classroom because they do not feel there should be a difference in how many
times typical or gifted students are exposed to visual media while learning
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geometry, and they feel that exposure should be frequent for both groups (Tables
3 and 4). In teachers’ perceived ideal circumstances, 55% would incorporate
visual media daily, and almost 30% would use it a few times a week. These
increased percentages of teachers who would utilize different forms of visual
media more often in their daily classroom practices for both types of learners
results in a significant difference between actual and ideal teaching for typical
learners and a significant difference between actual and ideal instruction for
gifted learners (see Table 3).
Instructional Practice #10: Incorporate other subject areas into geometry
An overwhelming majority of teachers that were surveyed rarely (or
never) blend geometry with other subject areas. Hence, they do not help students
make connections and see patterns in the mathematics being learned with either
group of children. It is disheartening to find that there is no real difference
between the results for actual practices for typical and gifted students for this
instructional practice due to the fact that the average number of times students
are exposed to interdisciplinary activities for both groups of students are so low
(in the 1.6-1.8 mean range) (Table 4). This range entails that teachers only
incorporate geometry with other subject areas approximately once a month,
regardless of the students in the class. Students are not receiving ample
opportunities in the core curriculum to apply their knowledge of geometry
outside of the textbook, which may limit high-ability learners from building
knowledge that is conceptual and overarching and maximizing upon what their
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previous knowledge (California Department of Education, 2005; NCTM, 1995;
California Department of Education, 2001; Johnsen, et al., 2002).
It is encouraging to see that teachers believe they would incorporate other
subject areas such as art, science, language arts, and/or architecture into the study
of geometry under ideal circumstances. Thirty-five percent of teachers would
implement this instructional practice with gifted learners weekly with gifted
learners, and almost 20% would do so daily (Table 6.10); for typical learners,
teacher would do so only a few times a month on average (Table 3). These
percentages compared to the much lower 16% and 4% respectively under actual
circumstances implies that teachers want gifted students to learn through multiple
perspectives in geometry.
Instructional Practice #11: Allow or make time available for students to pursue
self-selected interests in geometry.
A startling 48% of teachers reported never allowing or making time
available for their gifted students to pursue self-selected interests in geometry,
which is something that the literature shows is fundamental in the development
of gifted learners’ intellectual potential (Clark, 1997; NCTM, 1995). Also, 36%
said they allow gifted students to pursue self-selected interests only once a month
or less frequently (Table 5.11). Together, this makes 84% of the teachers
surveyed who allow for the self-selection of activities once a month or less
frequently for gifted learners. There is no significant difference between the
frequency of this practice for gifted learners and that for typical learners.
Making time available for self-selected interests is not a common practice in the
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sample of geometry teachers in this study, even though the National Council of
Teachers of Mathematics recommends that such a practice is critical for fostering
students’ increased interest in mathematics and resulting mathematical
knowledge development (NCTM, 1995). As learners are required to select topics
to explore along with the proper instructional guidance in selecting the topic
(even if they do not initially harbor any personal interests in mathematics), they
are more inclined to take ownership of their learning and more likely to continue
selecting topics to study beyond the regular curriculum (NCTM, 1995; Swassing,
1985).
The ideal frequency of this instructional practice is only slightly more
encouraging than the actual frequency of the practice in that almost 40% of
teachers reported that they would allow for self-selected activities a few times a
month instead of the “once a month or less” they actually do for gifted learners
(Table 6.11). Also, teachers would ideally require typical learners to choose
topics of interest to explore more frequently, but that would still be less often
than how many times they would allow the advanced learners to do so (Table 4).
Only 6% of teachers reported ideally “never” allowing gifted learners to pursue
self-selected interests (Table 6.11).
There is a significant difference in all cross tabulations done with the
Mann Whitney U/Wilcoxon Matched Pairs Test except for the “Actual-Typical”
versus “Actual-Gifted” comparison (Table 3). The three other comparisons are
significantly different, especially the “Ideal-Typical” versus the “Ideal-Gifted”
comparison. This could imply that teachers understand the importance of
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allowing for the pursuit of self-selected activities in knowledge acquisition and in
the student’s interest level in the study of geometry.
Instructional Practice #12: Provide problems that encourage logical reasoning
and problem solving with real-life applications.
The trends observed with this instructional practice are much the same as
those seen above. There is no significant difference between how often teachers
actually assign problem solving with real-life applications for gifted and typical
learners. These same teachers, however, reported that they would increase the
frequency of logical reasoning and problem solving for gifted learners by a larger
margin than they would for typical learners in ideal circumstances; 78% actually
implement this instructional practice a few times a month with gifted learner and
81% would ideally implement it a few times a week with gifted learners, whereas
for typical learners, the shift from “once a month” would change to “a few times
a month” on average (Tables 5.12, 6.12, and 3).
Only 7% of teachers reported actually providing logical reasoning
problems with real-life application daily for gifted learners. That percent went
up to almost 20% in ideal circumstances. The significant difference between the
“Actual-Gifted” and “Ideal-Gifted” groups shows that teachers believe that this
instructional approach is appropriate for gifted learners but that they do not
actually carry it out often. In teachers’ ideal worlds, they would differentiate
their instruction for gifted learners from that of typical learners, at least in this
respect (logical reasoning with real-life application). Logical reasoning and
problem solving with real-life applications is a fundamental part of learning
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mathematics—such activities foster conceptual, novel, and complex learning
(California Department of Education, 2001; California Department of Education,
2005). Teacher guidelines in California’s framework for teaching mathematics
specify that teachers must allow students to see mathematics in settings that are
applicable to students’ lives in order to encourage them to want to learn
mathematics. Gifted learners, especially, need such opportunities if they are to
academically excel, not become bored, and be challenged in the regular,
standardized mathematics classroom (NCTM, 1995; Westberg, et al, 1997;
California Department of Education, 2005).
General conclusions about the key findings
The California Mathematics Framework states that instruction should include
direct instruction, investigation, classroom discussion and drill, small groups,
individual formats, hands-on materials, projects, and technology (California
Department of Education, 2005). All of these instructional approaches are designed
to foster either computational or conceptual learning of mathematics, both of which
are important to the study of math for any learner. The California GATE standards
state that curriculum and instruction for gifted learners should reflect all of the above
teaching standards, but should be differentiated also with regards to novelty,
complexity, depth, and acceleration (California Department of Education, 2001).
This study revealed that teachers implement most frequently those instructional
practices for gifted learners that foster computation and procedures instead of
conceptualization, and that the frequency of those instructional practices are, in the
majority, not differentiated from the frequencies for typical learners in the same
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classroom. (California Department of Education, 2005) (Table 1.2 and 3). The study
further revealed that instructional practices that enable conceptualization and
problem solving for both gifted and typical learners would be more often
implemented if teachers taught under classroom settings that they consider ideal,
suggesting apathy for different learning types and exemplary practices that would
benefit the status of mathematical education in current actual times (Goodlad, 1984).
The literature targeting gifted education shows that there are few differences
between classroom practices for typical and gifted learners (Archambault, et al.,
1993; Wenglinsky, 2000, Westberg & Doust, 2003). House (1999) refers to the little
differentiation of curriculum and instruction occurring for advanced learners in the
regular classroom as an unfulfilled design that should be tapping into students’
maximum capabilities. This particular investigation at the high school geometry
level shows similar results.
Though there were significant differences in one-third of the instructional
practices in actual classes, these significant differences occurred between practices
with low frequencies such as “once a month” and “a few times a month.” This could
be categorized as little differentiation for gifted students. For example, there is a
significant difference between typical and gifted groups for instructional practice 1
(i.e., assigning open-ended problems and projects). Though the nonparametric test
applied to the data shows a significant difference between all four groups with this
instructional practice, the mean plots (Table 4) show average frequencies as low as
“once a month” to as high as only “a few times a month” under ideal circumstances
with gifted learners. Another example of the minimal differentiation occurring in
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classes is the significant differences found in instructional practice 7 (i.e., having
students write proofs in geometry). Writing proofs is one of the California content
standards for geometry, and expected to be a part of most lessons (California
Department of Education, 2005). Though there is a significant difference between
the actual frequency of this instructional practice with typical and gifted learners, the
difference is seen in an instructional practice with means of 2.6 and 2.9 (typical and
gifted respectively). On the survey’s frequency ranking chart, “2” represents a
couple of times a month, and “3” represents a couple of times a week. Therefore,
there is a significant difference between the means, but the significant difference is
not so encouraging when looking at the number of times the practice is actually
implemented. Gifted learners are required to prove geometry concepts somewhere
between a few times a month and a few times a week, instead of doing so daily as is
the basic recommendation (Sheffield, 1999; NCTM, 1995). The fact that four of the
twelve instructional practices produced significant differences between instruction
for typical and gifted learners implies that gifted learners may, on occasion, get
treated differentially from typical learners in the regular geometry classroom to their
advantage; however, the infrequency of the treatment is discouraging. In spite of the
more frequent implementation of four instructional practices that foster
conceptualization in mathematics with gifted learners compared to that with typical
learners, the true nature of the finding is still discouraging given the actual average
frequencies and the implication that promising instruction is possible but just not
taking place (House, 1999; Reis, 1994; Sheffied, 1999; Wenglinsky, 2000).
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Literature targeting gifted education also focuses on exemplary approaches
for teaching mathematics to gifted learners (Fox, 1979; Maker & Nielson, 1995,
California Department of Education, 2005; NCTM, 1995). Such teaching involves
grouping of students based on ability, assigning investigative activities, allowing for
discovery learning, enabling students to apply learned material to real-life settings,
and the teacher as coach rather than the leader of instruction (the lecturer) (Clark,
1997; Richert, 1985; Wenglinsky, 2000; Westberg & Archambault, 1997; Betts,
1985). In this study, teachers’ more frequent use of ideal instructional practices that
encourage conceptualization and may be categorized as differentiation shows that the
teachers surveyed in this study have some knowledge of exemplary practices for
gifted students in their classrooms (Sheffield, 1999; Goodlad, 1984; NCTM, 1989).
Those instructional practices being referred to from the survey are assigning open-
ended and challenge problems, asking students to work in groups, asking students
discover and prove theorems, having students write proofs, using visual media,
incorporating other subject areas into geometry, allowing time for self-selected
interests, and providing logical reasoning activities with real-life applications
(Appendix A). Due to these findings and links to the literature review, it may be
concluded that teachers do not actually differentiate classroom practices for typical
and gifted learners, but that their actions may be categorized as a heightened desire
to differentiate in ideal settings (Feldhusen, 1997, Greenes & Mode, 1999). This, at
least, is an encouraging finding.
In geometry instruction, teachers should focus on visual and logical
reasoning. Concepts should be explored via projects and investigations and visual
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media. The study shows that most of the time in geometry classrooms is spent on
note-taking and procedural practice. There is a clear discord between what is
actually happening in geometry classrooms in California to what should be
happening (according to the teachers’ opinions and to studies about exemplary
practices). The ideal classroom for both typical and gifted learners exists in theory,
but cannot be accessed if educational reform in the form of professional
development, pre- and in-service teacher training, administrative support, peer
support, teacher autonomy, and state and district support of differentiation strategies
does not quickly change towards putting learner needs first (Goodlad, 1984).
Implications for practice
A practical implication gleaned from this investigation is that teachers have a
strong sense of what they should be doing in their classrooms for typical and gifted
learners even though they are not implementing most of them (Goodlad, 1984). This
could mean that teachers do not feel they are teaching under their ideal
circumstances and that they either do not have the resources necessary to
differentiate instruction for gifted learners and/or that they simply do not know how
to use them (Westberg & Archambault, 1997; Reis & Westberg, 1994; Feldhusen,
1997; Goodlad, 1984). The following comments made by teachers on the survey
seem to support this implication:
I teach in a school where GATE students are in the same classrooms as all of
the other students, and so I’m having a very hard time imagining
differentiating my instruction for GATE vs. non-GATE kids.
Teacher 7223
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If a district is not willing to pay a teacher for the extra time it takes to prepare
these extra assignments, why should a teacher do it? In most cases, we if it is
important to the district, state, or you—show the funds for teachers.
Teacher 7239
1. Hard to do “ideal” with the load of students I’m given (36-38 per class)
2. Need more training to get ideas on good projects that feel relevant to the
students, & don’t kill me off with the burden of assessing.
3. We’ve got Geo Sketchpad, but the state of our “technology is old & tired-
less reliable every year, so I gave up.
Teacher 8075
Of course, further investigations need to be conducted in order to confirm or negate
these speculations.
Another implication gleaned from this study is that the teaching of the
geometry standards and the standardized testing that are occurring in high school
geometry classrooms take up most of the time that teachers have in the classroom,
therefore, teachers do not have time to be more creative in terms of instruction for
their gifted students (Wenglinsky, 2000). The 37 teachers who wrote optional
comments on the survey almost unanimously made the comment that the reason they
do not treat gifted learners differently from their typical learners is because they
simply do not have the time to lesson plan for the few gifted learners in their
heterogeneous classrooms after having to cover all of the California geometry
standards and preparing their students for standardized testing. The following are a
few examples of the type of comments made by the teachers:
Ideally- the standards wouldn’t restrict us and this all could be fun. But I
must admit this new standardized testing emphasis puts a damper on my
creativity.
Teacher 7034
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During my time teaching geometry (and Algebra!) I’ve felt the state is
pushing my teaching in the wrong direction, and it makes me sad. At my high
school, we used to use the CPM program, but dropped it for a more
traditional approach, and many of us miss it! It made it much easier to teach
in a style that had the kids thinking and creating, and discovering. The
emphasis on the teaching to the standards for purpose of good test scores saps
the fun out of it. I’d be interested to see the results of your survey!
Teacher 7104
Too much of what we have to do is controlled by the state standards. It has
caused me to cut out some of projects I used to do.
Teacher 7202
Since they are in same class, with same standards I do not/cannot treat gifted
learners any differently from typical ones. Since we are so bound by
standards and test results, there is little/no time to be creative anymore.
Teacher 7356
Extras are good in theory but with 22 standards not always possible.
Teacher 7164
With so much emphasis on scores from the Geometry California Standards
Test, it’s pretty hard to spend too much time on projects and discovery.
Teacher 7354
These comments support the implication that teachers would provide gifted learners
with challenges and support that would allow them to access higher levels of
mathematical thinking in addition to the regular curriculum if they had the time to
plan and seek resources for doing just that (Johnsen, et al., 2002, Johnson, 2000).
Yet another implication gleaned from this investigation is that teachers are
not adequately prepared to implement exemplary practices with different learners
(Wenglinsky, 2000; Goodlad, 1984; Sheffield, 1999). The fact that teachers know
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they should ideally increase the frequency of exemplary practices in geometry for
both gifted and typical learners implies that there has been some exposure to such
strategies during teacher preparation and in-service, but that perhaps teachers do not
feel confident to actually carry them out in their classrooms (Wenglinsky, 2000).
This speaks of the inadequacy of support and teaching in teacher in-service programs
and the often lack of courses that prepare for future teachers to work with a variety
of students. Specialized teaching strategies for gifted students are often neglected
during teacher training; therefore, studies in the past (Archambault, 1993 and 1997)
and this study reveal a disheartening low frequency of exemplary practices for the
benefit of gifted learners in mathematics.
Suggestions for Future Research
Suggestions to improve this study in the future would be to add an
observational data collection element to its methodology. Observation of teachers
would contribute to the reliability of the self-reported data of the teachers.
Further questions that may be asked after having conducted this study are:
What are the reasons behind teachers’ instructional choices under actual and ideal
circumstances? What do teachers believe are ideal teaching circumstances for typical
and gifted learners? What resources do teachers believe they need in order to
achieve ideal teaching strategies under current educational law? Do factors such as
training in gifted education, collaboration among colleagues, the presence of a gifted
education coordinator in the district or the school, level of teacher education, and/or
number of years of experience teaching geometry make a difference in the classroom
practices of teachers (Reis, et al., 2004)? How does the implementation of
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exemplary practices impact the students’ learning, attitude, and self-actualization
(from both the teacher’s standpoint and the student’s)? How can teachers teach the
gifted and challenge typical learners at the same time (Gutteridge, 1982)? What
types of courses would benefit teachers during in-service programs to help with
teaching special learners? What do students believe should ideally occur in
classrooms? What do students believe are exemplary teaching practices and how
often do they believe those practices should occur?
Further questions to ask in future gifted education research are content
specific to geometry. Possible inquiries would be to ask what sorts of activities
teachers provide to cover each of the geometry standards. What activities in
geometry do typical and gifted students find most interesting, most challenging, and
even most confusing, and why? What do typical and gifted students consider to be
ideal learning environments when studying geometry? What do students (both
typical and gifted) believe the “ideal” teacher should know and do in geometry class?
Do students’ ideal learning circumstances differ from teachers’ ideal teaching
circumstances in geometry? What do gifted students feel their teachers do when
teaching geometry that challenges them? Teachers’ and students’ responses to these
questions could help develop a practical manual for teaching geometry for gifted
learners. Such a how-to manual would be a content-specific resource that teachers
could use as support while teaching.
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BIBLIOGRAPHY
Archambault, F.X, Jr., Westberg, K.L., Brown, S., Hallmark, B.W., Zhang, W.,
Emmons, C. (1993). Regular classroom practices with gifted students: Findings
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Archambault, F.X., Westberg, K.L., Brown, S.W., Hallmark, B.W., Emmons, C.L.,
Zhang, W. (1993). Regular Classroom Practices With Gifted Students: Results of a
National Survey of Classroom Teachers. Storrs: Connecticut: The National
Research Center on the Gifted and Talented.
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Appendix A
INSTRUCTIONAL PRACTICES--GEOMETRY TEACHER
SURVEY (PAGE 1)
SECTION 1
Directions: Please answer the following seven questions to the best of your ability. Darken the
ovals completely and make any erasures complete. Use a pencil.
1. How many years have you taught high school
geometry? __________
2. How many years have you taught high
school mathematics? _________
3. Have you had training for teaching
gifted/talented learners (such as graduate courses,
credential courses, professional development,
seminars, conferences, etc…)?
O Yes O No
4. Does your school (or your school district)
employ a coordinator for gifted and talented
education?
O Yes O No O I don’t know
5. Do you collaborate with other teachers about lesson planning for gifted/talented students?
O Yes O No
6. What type of geometry class(es) do you teach or have you ever taught? (Mark all that apply)
O Regular Geometry O Seniors-only Geometry
O Three-semester Geometry O Honors Geometry
O Other______________________ O Summer Geometry
7. What is the highest level of education that you have completed? (Mark one)
O High school graduate with some college O Bachelor’s Degree w/ credential
O Bachelor’s Degree O Master’s Degree
O Bachelor’s Degree with some graduate work O Master’s Degree with credential
O Doctorate Degree O Other_________________
111
(PAGE 2)
SECTION 2—Your Actual and Ideal Instructional Practices
Directions: Choose and mark ONE of the two options below.
Proceed as directed beneath that choice.
O To the best of my knowledge, I have (or had sometime in the
last 12months) at least one gifted student in my geometry
class(es).
**If you choose this first option, please complete Columns
A, B, C, and D in the table on pages 3-5.
Note: It is important that your responses in Column A (Typical Learners)
and Column B (Gifted Learners) reflect your actual instructional
practices. It is important that your responses in Column C (Typical
Learners) and Column D (Gifted Learners) reflect what you would ideally
do when teaching under your ideal circumstances.
OR
O To the best of my knowledge, I have (or had sometime in the
last 12 months) absolutely no gifted students in my geometry
class(es).
**If you choose this second option, please complete only
Columns A, C, and D in the table on pages 3-5. Note: It is
important that your responses in Column A (Typical Learners) reflect
your actual instructional practices. It is important that your responses in
Column C (Typical Learners) and Column D (Gifted Learners) reflect
what you would ideally do when teaching under your ideal
circumstances. Please complete Columns C and D regardless of the fact
that you do not or did not teach gifted learners.
112
PAGE 3
Response Scale:
0 = Never 1 = Once a month, or less frequently 2 = A few times a
month 3 = A few times a week 4 = Daily
Indicate how often you
ACTUALLY incorporate
each instructional practice
with:
Indicate how often you
would IDEALLY
incorporate each
instructional practice with:
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
INSTRUCTIONAL
PRACTICES
Typical
Learners
COLUMN
C
Gifted
Learners
COLUMN
D
0 1 2 3 4 0 1 2 3 4
1.
Assign open-ended
problems and/or
projects (such as having
students build scale
models of their “dream”
house and providing
square footage, volume,
ratio of areas and
volumes to that of the
real life house, and the
floor plans).
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
2.
Assign challenge or
bonus problems such as
those from a higher
level textbook, a
resource book, or
enrichment worksheets.
0 1 2 3 4 0 1 2 3 4
113
PAGE 4 (Table Continued from page 3)
Response Scale:
0 = Never 1 = Once a month, or less frequently 2 = A few times a
month 3 = A few times a week 4 = Daily
Indicate how often you
ACTUALLY incorporate
each instructional practice
with:
Indicate how often you
would IDEALLY
incorporate each
instructional practice with:
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
INSTRUCTIONAL
PRACTICES
Typical
Learners
COLUMN
C
Gifted
Learners
COLUMN
D
0 1 2 3 4 0 1 2 3 4
3.
Have students work in
pairs or groups during
class time.
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
4.
Have students take notes
on a lesson while you
lecture.
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
5.
Have students memorize
theorems, postulates, and
definitions.
0 1 2 3 4 0 1 2 3 4
114
PAGE 5 (Table continued from page 4)
Response Scale:
0 = Never 1 = Once a month, or less frequently 2 = A few times a
month 3 = A few times a week 4 = Daily
Indicate how often you
ACTUALLY incorporate
each instructional practice
with:
Indicate how often you
would IDEALLY
incorporate each
instructional practice with:
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
INSTRUCTIONAL
PRACTICES
Typical
Learners
COLUMN
C
Gifted
Learners
COLUMN
D
0 1 2 3 4 0 1 2 3 4
6.
Have students discover
and prove theorems such
as the Pythagorean
Theorem or the Isosceles
Triangle Theorem.
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
7.
Have students write
proofs of geometry
problems from the
textbook or from other
teacher sources (including
two-column proofs,
paragraph proofs, and/or
indirect proofs)
0 1 2 3 4 0 1 2 3 4
115
PAGE 6 (Table Continued from Page 5)
Response Scale:
0 = Never 1 = Once a month, or less frequently 2 = A few times a
month 3 = A few times a week 4 = Daily
Indicate how often you
ACTUALLY incorporate
each instructional practice
with:
Indicate how often you
would IDEALLY
incorporate each
instructional practice with:
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
INSTRUCTIONAL
PRACTICES
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
0 1 2 3 4 0 1 2 3 4
8.
Use basic-skills
worksheets as
assignments for drill and
practice
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
9.
Use visual media
including (but not limited
to) advance organizers,
diagrams and charts,
overhead transparencies,
geometric figures and
solids, television media,
and/or Geometer’s
Sketchpad to explain
geometry concepts.
0 1 2 3 4 0 1 2 3 4
116
PAGE 7 (Table Continued from Page 6)
Response Scale:
0 = Never 1 = Once a month, or less frequently 2 = A few times a month
3 = A few times a week 4 = Daily
Indicate how often you
ACTUALLY incorporate each
instructional practice with:
Indicate how often you
would IDEALLY
incorporate each
instructional practice with:
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
INSTRUCTIONAL
PRACTICES
Typical
Learners
COLUMN
A
Gifted
Learners
COLUMN
B
0 1 2 3 4 0 1 2 3 4
10.
Incorporate other subject
areas into geometry such
as art, architecture,
science, or language arts
in the form of projects,
homework, class work,
and/or creative
assignments.
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
11.
Allow or make time
available for students to
pursue self-selected
interests in geometry.
0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4
12.
Provide problems that
encourage logical
reasoning with real-life
applications (for example,
having students solve
how much it costs to
paint the school).
0 1 2 3 4
0 1 2 3
4
117
PAGE 8
Comments (Optional): Please provide any comments you believe will help us in
understanding your thoughts about classroom practices within the Geometry
classroom.
118
Appendix B
University of Southern California
Rossier School of Education
INFORMATION SHEET FOR NON-MEDICAL RESEARCH
Actual and Ideal Instructional Practices in High School Gifted
Geometry Education
I am asking you to participate in a research study conducted by Edit Tanahan, M.Ed.
from the Rossier School of Education at the University of Southern California. The
results of this study will be contributed to a dissertation. You were selected as a
possible participant in this study because you are or were a high school geometry
teacher in California within the last twelve months. A total of 350 participants were
randomly selected from all high school geometry teachers in California. Your
participation is voluntary.
PURPOSE OF THE STUDY
You are being asked to take part in a research study in order to generate knowledge
about the actual and ideal instructional practices of high school Geometry teachers
with gifted learners.
Completion and return of the survey will constitute consent to participate in this
research project.
PROCEDURES
You will be asked to complete a survey questionnaire. The survey questionnaire will
take approximately ten minutes to complete, and may be completed at a time that is
convenient to you within one week of receiving it. Should you choose to participate,
a self-addressed, stamped envelope has been provided in this packet for you to return
the completed survey. The types of questions you will encounter on the survey
questionnaire are about your background in teaching mathematics, about your actual
classroom practices, and your beliefs about ideal instructional practices with
geometry learners. Almost all of the questions will be multiple choice questions, and
an optional “Comments” section will be available at the end of the survey. Section I
of the survey has 7 questions, and Sections II has 12 questions.
USC University Park
Institutional Review Board
UP-05-00096
119
POTENTIAL RISKS AND DISCOMFORTS
There are no foreseeable risks involved in participating in this research study. All
surveys will be kept confidential.
POTENTIAL BENEFITS TO SUBJECTS AND/OR TO SOCIETY
There will be no direct benefit to you. However, this study may contribute to the
knowledge base about classroom practices of secondary school Geometry teachers
with gifted learners.
PAYMENT/COMPENSATION FOR PARTICIPATION
You will receive a complimentary pencil whether you decide to participate in the
study or not. There is no incentive for participating in this study.
CONFIDENTIALITY
Any information that is obtained in connection with this study will remain
anonymous. Any personal information and data collected for the survey will be
coded during data analysis to ensure confidentiality. For example, your name will
not appear on your survey; rather, each teacher will randomly be assigned a four-
digit ID number for coding purposes. The data will be stored and password
protected in a computer file in the office of the principal investigator and only the
principal investigator will have access to the data. All data will be destroyed three
years after this research study is completed. When the results of the research are
published or discussed in conferences, no information will be included that would
reveal your identity.
PARTICIPATION AND WITHDRAWAL
You can choose whether to be in this study or not. If you volunteer to be in this
study, you may withdraw at any time without consequences of any kind. You may
also refuse to answer any questions you do not want to answer and still remain in the
study. The investigator may withdraw you from this research if circumstances arise
that warrant doing so.
IDENTIFICATION OF INVESTIGATORS
If you have any questions or concerns about the research, please feel free to contact
Miss Edit Tanahan at the following: Edit Tanahan, M.Ed.: tanahan@usc.edu,
(818) 957-1691.
RIGHTS OF RESEARCH SUBJECTS
120
You may withdraw your consent at any time and discontinue participation without
penalty. You are not waiving any legal claims, rights or remedies because of your
participation in this research study. If you have questions regarding your rights as a
research subject, contact the University Park IRB, Office of the Vice Provost for
Research, Grace Ford Salvatori Building, Room 306, Los Angeles, CA 90089-1695,
(213) 821-5272 or upirb@usc.edu.
USC University Park
Institutional Review Board
Approval Date: 8/25/2005
UP-05-00096
121
Appendix C
Dear Math Teaching Colleague,
My name is Edit Tanahan. I am a fellow California high school mathematics
teacher, and also a doctoral candidate in the Rossier School of Education at USC
where I am just about to begin my dissertation.
As you and I know, there is still much that needs to be understood about
mathematics education in California. Therefore, I am asking for your help in
conducting a survey of high school geometry teachers to generate more knowledge
about teachers’ instructional practices in geometry education.
You were randomly selected from a pool of mathematics teachers in California to
participate in this dissertation study because you are, or have been at some time
within the last twelve months, a high school geometry teacher. Because you are one
of 350 randomly selected teachers, your voluntary participation in this one-time
and confidential survey is crucial in the validity and ultimate value of the study.
I sincerely hope that I can count on your participation.
In this packet you will find (1) an information sheet about the study and your rights
as a participant, (2) a blank survey questionnaire, (3) a self-addressed stamped
envelope with which to return the survey, and (4) a complimentary pencil. Please
keep the pencil regardless of your decision to complete the survey. If you are not or
never were a geometry teacher in California, please check this line ____ and return
this letter and the blank survey in the stamped, self-addressed envelope.
If you choose to complete the survey, please mail it back in the provided envelope
within 7 days of receiving this packet. Please contact me if you have a concern or
question, by phone at (818) 957-1691 or by email at tanahan@usc.edu.
Thank you for your time and for all that you accomplish with our learners.
Many thanks,
Edit Tanahan, M.Ed.
USC University Park
Institutional Review Board
Approval Date: 8/25/2005
UP-05-00096
122
Dear Math Teaching Colleague,
Greetings! About a week ago, I, Edit Tanahan, contacted you about my dissertation
study about instructional practices in geometry education. We spoke about your
potential participation in my study by completing the short survey that you will find
in this packet. Thank you for agreeing to receive another copy of the survey and
considering filling out and mailing back the survey.
As a high school teacher myself I can understand how busy you are, but I would very
much appreciate your participation in my dissertation study of the instructional
practices of geometry teachers. Because you are one of 350 randomly selected
teachers, your voluntary participation in this one-time, confidential survey is
crucial in the validity and end value of this study. It is very important that you
please complete and return this survey. Your responses will help in the
completion of this study and in the knowledge base that we as educators have about
teaching math.
In this packet, you will find (1) an information sheet about the study and your rights
as a participant, (2) a blank survey questionnaire, (3) a self-addressed stamped
envelope with which to return the survey, and (4) a complimentary pencil. Please
keep the pencil regardless of your decision to complete the survey. If you have
already completed and returned a copy of the survey, please disregard this letter.
If you hopefully choose to complete the survey, please mail it back in the provided
envelope within 7 days of receiving this packet. Please contact me anytime you have
a concern or question, by phone at (818) 957-1691 or by email at tanahan@usc.edu.
Thank you for your time and for all that you accomplish with our learners.
Many thanks,
Edit Tanahan, M.Ed.
USC University Park
Institutional Review Board
Approval Date: 8/25/2005
UP-05-00096
123
Dear Math Teaching Colleague,
Greetings! Three weeks ago, I, Edit Tanahan, sent you a similar packet to this one. I
have not heard back from you and therefore would like to ask once more for your
participation in my study. Please consider filling out the enclosed survey. It is
designed to take less than ten minutes to complete and is a one-time self-
administered instrument.
As a high school teacher myself I can understand how busy you are, but I would very
much appreciate your participation in my dissertation study of the instructional
practices of geometry teachers.
Because you are one of 350 randomly selected teachers, your voluntary
participation in this one-time, confidential survey is crucial in the validity and
end value of this study. It is very important that you please complete and
return this survey. Your responses will help in the completion of this study and in
the knowledge base that we as educators have about teaching math.
In this packet, you will find (1) an information sheet about the study and your rights
as a participant, (2) a blank survey questionnaire, (3) a self-addressed stamped
envelope with which to return the survey, and (4) a complimentary pencil. Please
keep the pencil regardless of your decision to complete the survey. If you have
already completed and returned a copy of the survey, please disregard this letter.
If you hopefully choose to complete the survey, please mail it back in the provided
envelope within 7 days of receiving this packet. Please contact me anytime you have
a concern or question, by phone at (818) 957-1691 or by email at tanahan@usc.edu.
Thank you for your time and for all that you accomplish with our learners.
Many thanks,
Edit Tanahan, M.Ed.
124
Appendix D
Script for Telephone Contact
Tanahan: Hello. My name is Edit Tanahan. I am a teacher from the Glendale
Unified School District in California and a student at USC. In the last month, I sent
you a survey about classroom practices that I am conducting for my dissertation at
USC. I was wondering if you have a few minutes to talk to me (Wait for response).
Tanahan: The reason I am calling you is because I have not heard from you (I have
not received a survey back from you). I wanted to personally contact you to briefly
talk about the study and to ask you, afterwards, if you would please strongly consider
filling out my survey. I understand that your time is precious and already filled with
many important priorities. That is why the survey is meant to take only ten minutes
to complete and the results compiled from across California will benefit the
education of our high school students. The study will help me figure out what types
of activities teachers engage in their classrooms with both gifted and typical learners,
and in which areas there are similarities and differences. I am hoping to shed more
light upon the status of geometry education.
Tanahan: Your participation would help me immensely. Let me stress that should
you choose to allow me to send you a second survey and should you decide that you
would like to complete it when you receive it, that all information is confidential and
it will remain anonymous in the dissertation. Do you have any questions about what
my research is about? (Answer any questions about the research that the teacher
might have)
Tanahan: If, after this conversation, you are willing to reconsider participating in
my study, I would be thrilled to send you another copy of the survey if you do not
have a copy of it anymore. May I send you another copy so that you may look over it
and decide if you want to participate? (Wait for response).
Tanahan (If the teacher says “yes, send me a copy again”): I will send you a copy
immediately. Thank you so much for your time and I look forward to getting a
survey back from you. Have a nice day.
Tanahan (If the teacher says “no, do not send me a copy again”): Thank you for
your time anyway. Have a nice day.
USC University Park
Institutional Review Board
Approval Date: 8/25/2005
UP-05-00096

Abstract (if available)

Abstract The purpose of this study was to investigate the differences between the frequencies of teachers' instructional practices with gifted and with typical learners under actual and ideal instructional circumstances. One hundred seventy California high school geometry teachers completed a 12-item survey developed by the researcher. The Mann Whitney U/Wilcoxon Matched Pairs test, with a p value less than .05, was used to reveal any significant differences between teachers' responses about instruction for gifted and typical students. Subjects responded with their 1) actual frequency of practices with typical learners vs. gifted learners, 2) ideal frequency of practices with typical learners vs. gifted learners, 3) actual frequency of practices vs. the ideal frequency of practices with typical learners, and 4) actual frequency of practices vs. the ideal frequency of practices with gifted learners. In actual instruction, there were no significant differences between eight of the twelve instructional practices for typical and gifted learners. This implies that minimal differentiation of instruction occurs for gifted learners in the regular geometry classroom. Therefore, gifted learners receive much the same exposure to subject matter as do their typical counterparts. In ideal classroom practices as perceived by the teachers there were no significant differences between the frequencies of only four of the twelve instructional practices for typical and gifted learners. Those four instructional approaches include student note-taking during a lecture, students' memorization of theorems and postulates, cooperative group work, and the use of visual media to teach geometry. Ideally teachers would double the amount of differentiation in instruction for gifted learners and typical learners. Teachers would incorporate practices more frequently that foster conceptualization and less frequently use those practices featuring memorization and drill. The increased frequencies

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Asset Metadata

Creator Tanahan, Edit (author)

Core Title Actual and ideal instructional practices in California high school gifted geometry education

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Curriculum

Publication Date 10/25/2006

Defense Date 02/14/2006

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Gifted Education,instructional practices,OAI-PMH Harvest

Language English

Advisor McComas, William F. (committee chair), Kaplan, Sandra N. (committee member)

Creator Email edittanahan@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m106

Unique identifier UC1233861

Identifier etd-Tanahan-20061025 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-26879 (legacy record id),usctheses-m106 (legacy record id)

Legacy Identifier etd-Tanahan-20061025.pdf

Dmrecord 26879

Document Type Dissertation

Rights Tanahan, Edit

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu