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THE EFFECT OF REAL-WORLD MATHEMATICAL APPLICATIONS
ON GIFTED AND NONGIFTED STUDENTS' ACHIEVEMENT AND
CLASSROOM LEARNING ENVIRONMENT

A Dissertation Presented to the
Faculty of the College of Education
University of Houston

In Partial Fulfillment
of the Requirements for the Degree
Doctor of Education

by
Ann P. Bevil

Copyright © Ann P. Bevil, May, 2003


§ § §


Acknowledgment

For a lifetime of encouragement and support, I wish to dedicate my dissertation to the memory of my father and to the honor of my mother, Dr. and Mrs. Harold H. Bevil. Throughout the long process of graduate school and research for my study, they were there for me, every step of the way, with their love, wisdom, and sympathetic attention in abundance.

My heartfelt appreciation to Dr. Jacqueline Smith for accepting Soaring with Numbers for the North Forest Independent School District and to Mrs. Annie Crockett for her direction in the most constructive ways to accomplish our goals. Bravo and thanks to the 320 students who participated in the study and to their dedicated teachers: Mrs. Leonetti Chenier, Mrs. Wanda Clarence, Mrs. Brenda Collins, Mrs. Rosetta Davis, Mrs. Annette Harrison, Mrs. Betty Robinson, Mrs. Dorothy Hawkins, Mrs. Verna Henry, Mrs. Delores Johnson, and Mrs. Pam Josey.

Special thanks to my dissertation committee: Dr. Theresa Monaco, Chairperson of my committee; Dr. Hersholt Waxman, my research methodologist; Dr. Lee Mountain, who presented me with the opportunity to develop Soaring with Numbers; and Dr. Yolanda Padron, Dr. Thomas Kellow, and Dr. Jacqueline Smith, consultants to my study. Thank you, Dr. Monaco, for your unflagging guidance through a long and difficult process. I have been so fortunate to have such a kind, helpful group of professionals, always ready with constructive comments. Thank you all for helping me grow professionally.

My thanks, too, to Janice Stensrude, my editor and typist; to Steve and Robin Bambera, who provided the wonderful illustrations for Soaring with Numbers; and to Mary Lee Cook, who read my paper and made so many wonderful suggestions.


§ § §


Abstract

For mathematics programs to be effective, real-world applications need to be taught. In addition, concepts need to be integrated with other disciplines and taught through connections to real-world situations. Two key components need to be addressed. First, through real-world applications, students are able to better comprehend the practical application. Secondly, the importance of integrating the disciplines results in students understanding the connections by relating other disciplines to mathematics and within the mathematical domain — hence, a better understanding of the mathematical concepts.

The purpose of the study was to determine the effect of real-world mathematical applications on students' achievement and classroom learning environments. The present study provided the opportunity to test the researcher's theory, based upon personal classroom experience and the supporting opinion of noted educators (Mitchell, 1990; Carnine & Gerston, 2000), that a real-world mathematics curriculum, which connects mathematics to students' experiences, produces a significant improvement in students' achievement.

A nonequivalent control group design was used to test the hypotheses in the study. In this quasi-experimental design, the experimenter had manipulative control over the independent variable, real-world mathematics applications curriculum, and randomization was not possible. The dependent variables were the posttest performances on the Classroom Environment Scale and the Bevil Mathematics Inventory. The study examined the influence of the treatment effect on the academic performance and classroom environment of the participants. The pretests and posttests were used as the criteria for determining whether posttest differences could be attributed to the treatment.

The sample was drawn from the population of sixth-, seventh-, and eighth-grade students in a suburban school district with a total enrollment of 12,000. Ethnicities represented in this student body were 83% Black, 16% Hispanic, and 1% White. The total sample for the study contained 320 participants (160 in the control group and 160 in the experimental group). Also, within the 320 participants, there were 160 participants in the nongifted group and 160 participants in the gifted group.

The Classroom Environment Scale used in the study was a shortened version of a self-report survey instrument (Fraser, 1982, 1986; Fraser & Fisher, 1983, 1986; Fisher, 1986; Moos, 1979). The instrument was designed "to focus on students' perceptions of their content area classes rather than their general impressions of school as a whole" (Waxman & Huang, 1998, p. 99). The participants' achievement was measured by using the Bevil Mathematics Inventory.

The data were analyzed using the two-way analysis of covariance. The findings revealed that intermediate and middle school students who were exposed to the real-world mathematical applications curriculum performed significantly better academically than their counterparts who were exposed to the traditional curriculum. Additionally, significant differences were found between adjusted mean achievement scores when academic status and group status were combined. Thus, the experimental nongifted participants scored significantly higher than experimental gifted participants.

Furthermore, gifted participants had more favorable perceptions toward three of the CES subscales. Gifted participants had significantly higher adjusted mean scores than nongifted participants toward the involvement, task orientation, and satisfaction components. Moreover, group status and academic status together produced a significant interaction effect on the rule clarity, task orientation, satisfaction, and innovation components. In all four components, experimental gifted students had significantly higher adjusted mean scores than the other three groups.


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Table of Contents

List of Tables

List of Figures

Chapter 1: INTRODUCTION

Chapter 2: REVIEW OF LITERATURE

Chapter 3: METHODS

Chapter 4: RESULTS

Chapter 5: DISCUSSION

REFERENCES

APPENDICES (Appendices are not included here)


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List of Tables
Table
1Summaries of Real-World Mathematics Curricula
2Studies Showing an Increase in Student Achievement Resulting from the Implementation of Real-World Applications in Mathematics Programs
3Alpha Reliability Coefficients Regarding the Subscales of the Classroom Environment Scale
4Time Sequence of Treatments by Group Status
5Null Hypotheses and Pretest and Posttest Instrumentation
6Evaluation of Assumptions by Academic Status and Group Status
7Sex of Study Participants
8 Ethnicity of Study Participants by Grade Level
9 Group Status of Study Participants by Grade Level
10 Academic Status of Study Participants by Grade Level
11 ANCOVA Results for Differences in Achievement Scores by Group Status, Academic Status, and Their Interaction
12 Scheffé Results Regarding Achievement Scores by Group Status and Academic Status
13 ANCOVA Results for Differences on Involvement Scores by Group Status, Academic Status, and Their Interaction
14 ANCOVA Results for Differences on Affiliation Scores by Group Status, Academic Status, and Their Interaction
15 ANCOVA Results for Differences on Rule Clarity Scores by Group Status, Academic Status, and Their Interaction
16 Scheffé Results Regarding Rule Clarity Scores by Group Status and Academic Status
17 ANCOVA Results for Differences on Task Orientation Scores by Group Status, Academic Status, and Their Interaction
18 Scheffé Results Regarding Task Orientation Scores by Group Status and Academic Status
19 ANCOVA Results for Differences on Satisfaction Scores by Group Status, Academic Status, and Their Interaction
20 Scheffé Results Regarding Satisfaction Scores by Group Status and Academic Status
21 ANCOVA Results for Differences on Innovation Scores by Group Status, Academic Status, and Their Interaction
22 Scheffé Results Regarding Innovation Scores by Group Status and Academic Status
23 Summary of ANCOVA Results for CES Component Adjusted Mean Scores
24 Summary of Null Hypotheses Tests
25 Summary of ANCOVA Interaction Results for Classroom Environment Scale Components

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List of Figures
Figure
1Non-equivalent control group design
2 Trend analysis of adjusted mean academic achievement scores by academic status and group status
3 Trend analysis regarding adjusted mean rule clarity scores by academic status and group status
4 Trend analysis regarding adjusted mean task orientation scores by academic status and group status
5 Trend analysis regarding adjusted mean satisfaction scores by academic status and group status
6 Trend analysis regarding adjusted mean innovation scores by academic status and group status

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CHAPTER 1

INTRODUCTION

Students can understand our world's dependence on mathematics through connecting mathematics to other disciplines and to the real world, advocated J. Price (1996), President of the National Council of Teachers of Mathematics (NCTM). Price made this commitment at the 74th Annual Meeting of the NCTM. With Price's position about real-world mathematics and connecting mathematics to other disciplines, the present researcher's position is positively stated. The present researcher proposes that teachers' mathematics resource books integrate disciplines and develop real-world mathematical applications in order to provide students with positive mathematical achievement and classroom learning environments.

Need for the Study

Numerous studies and reports throughout the past decade showed that traditional mathematics was ineffective in the modern classroom (NCTM, 1989, 2000; Carnine & Gersten, 2000; Verschaffel, De Corte, & Lasure, 1994). In traditional teaching of mathematics, concepts are not integrated and taught to relate to the world. There has been apparent agreement among many educators that students need to understand not only the procedural knowledge, but also conceptual knowledge in mathematics (Cangelosi, 1996; Orton, 1994; Resnick & Ford, 1981; Willoughby, 1990).

Other researchers expressed that, in order to align research-based principles with curricula, students need experience in using mathematics to solve real-life problems (NCTM, 1989, 2000; Mitchell, 1990). A conceptual approach enables students to acquire clear and stable concepts by constructing meanings in the context of physical situations. It also allows mathematical abstractions to emerge from empirical experience (NCTM, 1989). These skills can be acquired in ways that make sense to children and in ways that result in effective learning. A strong emphasis on mathematical concepts and understandings also supports the development of real-world mathematics.

A real-world mathematics curriculum (for example, the real-world curriculum constructed for the present study) addresses the very basic content areas that teachers and business community leaders have felt were neglected (Mitchell, 1990). These areas of neglect include statistics, probability, logic and reasoning, percentage, measurement and capacity, algebra, and verbal items. The National Assessment of Educational Progress (1988) commented that students lacked the conceptual knowledge that would lead to the successful application of problem-solving skills and reasoning skills.

Mitchell (1990) further reiterated that curriculum must extend beyond students solving formulas to focusing on realistic applications. The metric system is a good example because, even though students study it in class, they are unable to apply it to everyday mathematical situations. The metric system is totally immersed in this real-world curriculum.

The NCTM Standards' (1989, 2000) basic assumptions state that the curriculum should emphasize the application of mathematics in the way it relates to everyday mathematics. Cangelosi (1996) affirmed that students needed to be provided with experiences that led them to connect each topic to previously learned topics, and to apply their understanding of the topic to solve problems they considered real-life. Because the NCTM Standards were based on theory, Carnine and Gersten (2000) advocated more controlled experimental studies in the field of mathematics. Many of the field-research studies of new, innovative mathematics curricula (for example, Hiebert & Wearne, 1993) were experimental designs that failed to include adequate control and specification so that clear inferences could be drawn about effectiveness. Field testing was conducted only after the instructional or Standards-based curriculum was in place. This lack of field testing resulted in cynicism in the teaching profession. For example, the mathematics textbooks of one small publisher in California were designed to align completely with the objectives required by the Curriculum Commission (California State Department of Education, 1985). The program received a score of 96, in which it outperformed the other textbooks by 16 points. As a result, it gained about 60% of all California sales in the first year (Carnine & Gersten, 2000).

Desiring more factual information on this mathematics textbook, administrators at one school requested research related to this textbook's effectiveness. The evaluation provided by the publisher in response to this request stated that the sample consisted of 18 students. Of these 18 students, 7 were excluded from the final data analysis. Among the 11 remaining students, 61% made gains or had no change, whereas 39% had a loss. Therefore, the average gain of the students was 19 percentile points and the average loss was 22 percentile points. From this and other examples, Carnine and Gersten (2000) asserted that there needed to be a direct "linkage between criteria used to evaluate curricula — or approaches to teaching — and learning outcomes" (p. 140). At the National Institute of Child Health and Human Development conference in the spring of 1998, Birch (1998) found that mathematics education researchers asserted, too, the need for more rigorous testing and validation of reform-based instructional programs.

Ball and Cohen (1996) stated that innovative materials in education do not change school practice much, while Ross (1992) posited that curriculum researchers and designers too often overlook the influence of teachers on the curriculum. Through many experiences as a classroom mathematics teacher over many years, I, too, have come to believe that the effect of the teacher on the curriculum and the classroom environment is very pronounced. I have observed that if a teacher was enthusiastic about the curriculum, likewise students caught the enthusiasm for learning. I assert that the teacher's role in establishing the classroom learning environment is defined by selection of topics and objectives to be taught, as well as the teacher's beliefs about what is to be taught. The real-world mathematics program designed for the present study is a curricular approach and not an instructional approach, yet the classroom environment will be important for the successful implementation of the program. Thus, differences in classroom learning environment were tested, as well as differences in participants' achievement.

Statement of the Problem

For mathematics programs to be effective, real-world applications need to be taught. In addition, concepts should be integrated with other disciplines and taught through connections to real-world situations. Two key components need to be addressed. First, through real-world applications, students are able to better comprehend the practical application. Second, the importance of integrating the disciplines results in students understanding the connections by relating other disciplines to mathematics and within the mathematical domain — hence, a better understanding of the mathematical concepts.

The present study provided the opportunity to test the researcher's theory, based upon personal classroom experience and the supporting opinion of noted educators (Mitchell, 1990; Carnine & Gerston, 2000), that a real-world mathematics curriculum, which connects mathematics to students' experiences, will produce a measurable improvement in students' achievement.

Ball and Cohen (1996) attributed the gap between the intentions of curriculum developers and what actually happens in the classroom to a failure to take into account the role of the teacher in practice. An earlier version of the curriculum employed in the present study was developed by the researcher and used to teach mathematics to fifth-grade students for several years prior to developing the parameters of the present study. Through the successful development, implementation, and testing of this curriculum in real-world mathematics, it was the researcher's intention to answer a perceived need for more teacher-developed materials in mathematics.

New, updated, professionally illustrated materials, adapted from the materials the researcher previously used in the classroom, were developed specifically for use in the study and integrated into a workbook titled Soaring with Numbers. Clearly constructing such a book, which demonstrated precise understandings of concepts with instructional methods, offered the teacher a more thoughtful and effective way to present the curriculum. The researcher heartily concurs with Ball and Cohen (1996), who stated that the teachers play an important role in the development and implementation of the curriculum. Thus, the present study investigated the influence of group status and academic status on the achievement and learning environment scores of intermediate and middle school students.

Answers to the following questions were sought:

1. Do the variables group status and academic status have an independent and combined effect on the achievement scores of intermediate and middle school students?
2. Do the variables group status and academic status have an independent and combined effect on the classroom learning environment scores of intermediate and middle school students?

Purpose of the Study

The purpose of the study was to determine the effect of real-world mathematical applications on students' achievement and classroom environments. Specifically, the study examined the effects of group status and academic status on the achievement scores and classroom environment scores of intermediate and middle school students.

Hypotheses

The hypotheses for the study are:

H-1: There are significant differences on achievement scores by group status (experimental or control), academic status (gifted or nongifted), and their interaction (group x academic).

H-2.0: There are significant differences on involvement scores by group status, academic status and their interaction (group x academic).

H-2.1: There are significant differences on affiliation scores by group status, academic status, and their interaction (group x academic).

H-2.2: There are significant differences on rule clarity scores by group status, academic status, and their interaction (group x academic).

H-2.3: There are significant differences on task orientation scores by group status, academic status, and their interaction (group x academic).

H-2.4: There are significant differences on satisfaction scores by group status, academic status, and their interaction (group x academic).

H-2.5: There are significant differences on innovation scores by group status, academic status, and their interaction (group x academic).

Limitations

The two scales of teacher support and order and organization on the CES had low alpha reliability coefficients. Therefore, these two scales were eliminated from data analysis in the present study.

Definition of Terms

Real-world problem: In solving a real-world problem, students relate mathematical skills to real-life problems and use their skills to help solve problems that are of importance to themselves and to the world (Willoughby, 1990).

Mathematical connections: The following phrases describe mathematical connections in the study:

seeing mathematics as an integrated whole; explore problems and describe results using graphical, numerical, physical, algebraic, and verbal mathematical models or representations; use a mathematical idea to further their understanding of other mathematical ideas; apply mathematical thinking and modeling to solve problems that arise in other disciplines, such as art, music, psychology, science, and business; value the role of mathematics in our culture and society. (NCTM, 1989, p. 84)

Traditional mathematics: Resnick and Ford (1981) defined traditional mathematics as emphasizing arithmetic or calculation, content, and "body of computational rules and procedures" (p. 10). NCTM (1989) described traditional mathematics as solving algorithms through the context of word problems or story problems in a classroom that functions as a collection of individuals, with the teacher as sole authority for right answers. In the traditional mathematics environment, as described by NCTM, the emphasis is on memorization of procedures and mechanistic answer-finding. Traditional mathematics is the control group in the present study.

Real-world mathematics: NCTM (1989) defined real-world mathematics as applications in mathematics to real-world problems in settings that are relevant to students. Real-world mathematical problem-solving is the experimental group in the present study.

Operational Definitions of Terms

Gifted: Gifted were those students who had been identified as gifted according to district and Texas Education Agency (TEA) guidelines.

Nongifted: Nongifted were those students who had not been selected nor identified by the district as gifted.

Control group: The control group included 160 students: 80 gifted students and 80 nongifted students. The control group were given the same pretests and posttests on achievement in mathematics and classroom learning environment that were given to the experimental group. Students in the control group did not receive the real-world mathematical applications as instructed through Soaring with Numbers, but received the district's curricula for gifted and nongifted students.

Experimental group: The experimental group included 160 students: 80 gifted students and 80 nongifted students. The experimental group were given pretests and posttests of concepts taught in the curriculum. Experimental group students received the treatment of real-world mathematical problem-solving applications as instructed through Soaring with Numbers.

Classroom learning environment: The classroom learning environment was determined by a pretest and posttest on classroom learning environment as measured by the Classroom Environment Scale (CES).

Achievement: Each student's achievement was determined by a standardized achievement pretest before receiving the treatment and a standardized posttest after receiving the treatment.

Academic status: Academic status refers to each study participant's identification as gifted or nongifted by the participating school district.

Group status: Group status refers to the study participants' participation in either the experimental group or the control group.

Intermediate schools: The participating school district's intermediate schools included fifth and sixth grades.

Middle schools: The participating school district's middle schools included seventh and eighth grades.


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