REALITY-BASED MATHEMATICS VERSES TRADITIONAL …
REALITY-BASED MATHEMATICS VERSES TRADITIONAL MATHEMATICS
Except where reference is made to the work of others, the work described in this project is my own or was done in collaboration with my Advisor. This project does not include proprietary or classified information.
________________________________________________________________________
Kristen Daniel Brady
Certificate of Approval: Certificate of Approval:
_________________________________ ________________________________
Donald R. Livingston, Ed. D. Sharon M. Livingston, Ph. D.
Associate Professor and Co-Project Advisor Assistant Professor and Co-Project Advisor
Education Department Education Department
REALITY-BASED MATHEMATICS VERSES TRADITIONAL MATHEMATICS
A project submitted
by
Kristen Daniel Brady
to
LaGrange College
in partial fulfillment of
the requirements for the
degree of
SPECIALIST IN EDUCATION
in
Curriculum and Instruction
LaGrange, Georgia
July 21, 2011
Abstract
In today’s schools, students often lack knowledge that enables them to relate material learned in the classroom to tasks encountered in their everyday lives. An action research was conducted implementing reality-based mathematics tasks into daily lesson plans of middle school mathematics teachers. Quantitative and qualitative data were collected for this study. Pretests and post-tests were administered to control and experimental groups. Data were also collected through student surveys and interviews with members of the administration and faculty. Although the test scores of the experimental group did not display a significant gain over the control group, the attitudes of the students toward mathematics became more positive after the action research.
Table of Contents
Abstract…………………………………………………………………………………...iii
Table of Contents………………………………………………………………………....iv
List of Tables ………………………………………………………………....v
Chapter 1: Introduction……………………………………………………………………1
Statement of the Problem…………………………………...………………….1
Significance of the Problem……………………………………...…………….1
Theoretical and Conceptual Frameworks…………………………...…………2
Focus Questions……………………………………………………..………...4
Overview of Methodology………………………………………...…..……....4
Human as Researcher……………………………………………………….....5
Chapter 2: Review of the Literature……………………………………………………..6
Reality-Based Content Around the Globe………….………………………....6
The Benefits of Reality-Based Mathematics Curriculum.…………………….8
Students’ Attitudes Regarding Mathematics………………………………...11
Opposing Views……………………………………………………………...12
Implementing a Reality-based Curriculum..…………………………………13
Chapter 3: Methodology…………………………………………………………………16
Research Design……………………………………………………………...16
Setting…..………………………………………………………………….…17 Sample and Participants………….…………………………………………...17
Procedure and Data Collection Methods…..……..……..……………………18
Validity, Reliability, Dependability, Bias and Equity Measures.…………….20 Analysis of Data……………….……………………………..……………….22
Chapter 4: Results……………………………………………………………………..…25
Chapter 5: Analysis and Discussion of Results……….………………………………....37
Analysis……………………………….……….…………………..………….37
Discussion…..……………………….………...………………………..…….45
Implications………………………….………..………………………………47
Impact on School Improvement.………...……………………………………48
Recommendations for Future Research……....………………………………49
References……………………………………………………………………………….50
Appendixes………………………………………………………………………………54
List of Tables
Table 3.1 Data Shell ...…………………………………………………………….…..…18
Table 4.1 –Dependent t-Test for Pretest and Post-test for Experimental Group………..26
Table 4.2 –Dependent t-Test for Pretest and Post-test for Control Group………………27
Table 4.3–Independent t-Test for Pretest for Experimental and Control Group………..28
Table 4.4 Independent t-Test for Post-test for Experimental and Control Group……….29
Table 4.5 - Chi-Square Statistic for Students Attitudes Towards Math Pre survey ..…..30
Table 4.6 Dependent t-Test for Pre- and Post-survey for Mathematics…………………32
CHAPTER ONE: INTRODUCTION
Statement of the Problem
One of the important aims of teaching mathematics is to prepare students to meet the mathematical requirements of everyday living (Sparrow, 2008). In a vast number of middle school math classrooms, students are unable to translate words into numeric operations in order to accurately solve word problems. This, then, becomes a larger issue when trying to apply traditional methods of mathematic problem solving to use in real world situations. Since one goal of becoming a productive teacher is to assist students to become successful citizens, the current methods of teaching need to be reevaluated to meet the need of today’s vastly changing society. When students are able to build upon prior knowledge they are able to relate to the information being taught. Studies show that they even learn more effectively. So, can the use of a reality-enhanced math program make a significant difference on the problem solving abilities of sixth grade math students?
Significance of the Problem
In the United States, and especially Georgia, students of all ages have fallen behind other nations in mathematics skills and have been unable to complete globally. Data collected from various tests administered to evaluate progress demonstrated their inadequacies. For example, The TIMSS (Trends in International Mathematics and Science Study) tests that are given to fourth and eighth graders in 43 participating countries, discovered that the American students were ranked in the middle of those with comparable economic development (Gonzales et al., 2004). On a daily basis, nearly every individual uses some type of mathematics. It is unfortunate that “for many children, the mathematics of the classroom has no obvious connection to the mathematics of their world” (Sparrow, 2008, p. 4). Students have found it difficult to relate the material learned in class to their daily lives. At the middle school level, students have struggled to integrate their understanding of the world around them and the ‘rules’ being taught in their mathematics classroom. In regards to standardized testing, students have not applied logic to many reality-based problems and are not accurately solving the problems. If students are unable to visualize the situation problems present and make the connections between traditional mathematics and reality-based mathematics, they will likely have issues throughout high school as well as higher education and in society. According to the 2008 -2009 American Human Development Report, “a whopping 22 percent of us – more than one in five Americans – have below basic quantitative skills, making it impossible to balance a checkbook, calculate a tip, or figure out from an advertisement the amount of interest on a loan” (Burd-Sharps, Lewis, & Martins, 2008, p. 82). Although the statistic is alarming, it is even more disturbing to discover that there are strategies and methods that exist in mathematics that could significantly reduce this number.
Theoretical and Conceptual Frameworks
This study closely relates to cognitive constructivist learning, which states that leaning is an active process. When students are applying their previous knowledge to the concepts they are learning in class, they are engaging and becoming active learners. By using a method of adapted problem solving and implementing reality-based mathematics methods students are able to partake in authentic learning. The constructive approach to teaching provides students the skills necessary to problem solve and search for their own understanding of concepts. Marlowe and Page (2005) explain that learning should be "an active process in which learners construct new knowledge and awareness based upon current and past knowledge and experience" (p. 221).
The Conceptual Framework of LaGrange College Education Department (2010) is composed of three tenets that relate to this study. The tenets include: Tenet One “Enthusiastic Engagement in Learning”, Tenet Two “Exemplary Professional teaching Practices”, and Tenet Three “Caring and Supportive Classroom and Learning Communities.” Each of these tenets further contains competency clusters.
Incorporating reality-based mathematics into a middle school classroom most closely relates to Tenet One, which focuses on knowledge of content. Competency Cluster 1.2, states that candidates relate content areas to other subjects to and see connections in everyday life to make subject matter meaningful (LaGrange College Education Department, 2010). By incorporating realistic examples and mathematical tasks into the mathematics classroom, students are able to see how various contents closely related to something they would encounter daily.
This study also closely relates to Competency Cluster 1.3, which states that candidates understand how to provide diverse learning opportunities that students’ intellectual, social, and personal development based on students’ stages of development, multiple intelligent, learning styles, and areas of exceptionally (LaGrange College Education Department, 2010). Because every person frequently experiences situations involving mathematic calculations, teaching basic mathematics problems and tasks can cover an array of intelligence levels as well as leaning styles.
The LaGrange College Education Department (2010) provides the alignments of the Conceptual Framework’s Tenets, the five standards of the National Council for Accreditation of Teacher Education (NCATE) standards and the five National Board of Professional Teaching Standards (NBPTS) core propositions. For reality-based mathematics, the NCATE standard Element 1B: Pedagogical Content Knowledge and Skills for Teacher Candidates most closely relates. This specific standard aligns with Tenet One, enthusiastic engagement in learning under Cluster 1.2, knowledge of curriculum. Of the five NBPTS Core Propositions, reality-based mathematics aligns with Proposition 2. Proposition 2 states teachers know the subjects they teach and how to teach those subjects to students (LaGrange College Education Department, 2010).
Focus Questions
Can the use of a reality-enhanced math program make a significant difference on the problem solving ability of a sixth grade math student? To assist in providing answers to this thesis question, the following focus questions have been formulated. First, will implementing a reality-based math curriculum improve mathematical understanding and increase math test scores? Secondly, how have the feelings and attitudes of the students been altered after reality based mathematics instruction? Finally, how effective was the change process used to convince colleagues to adopt a reality-based curriculum or to better meet the needs of students in mathematics?
Overview of Methodology
This research study was that of an action research design. The study included the implementation of reality-based mathematics tasks in place of traditional assignments. An urban middle school located in west central Georgia served as the setting. Subjects and participants consisted of two heterogeneous groups of grade six mathematics students, mathematics teachers and the principal of the study site. This design focused on students realizing that math concepts learned in class could be frequently applied outside the classroom. This study required collection of both quantitative and qualitative data. The Data methods used in the study included a pretest and post-test, student surveys and interviews. Quantitative data were analyzed using an independent and dependent t-test and a chi square. Qualitative data consisted of interviews that were specifically coded for themes.
Human as Researcher
After teaching grade six mathematics for the past five years, I have noticed a trend in the ability of my students. Many of the students I have taught cannot extend the concepts learned in the classroom to problems they will face in the real world. Many students learn the bare minimum to get by in class and do not attempt to go further with their mathematical understanding of the concepts. Mathematics is something people use daily and it is critical for students to be able to utilize and adapt basic math skills to apply in real world situations. My assumption is that if I implement an adapted curriculum in which reality-based math is a focus, my students will begin to think critically and, in turn, perform better on standardized tests. I am qualified to conduct this action research because I possess a master’s degree in middle grades mathematics education.
CHAPTER TWO: REVIEW OF THE LITERATURE
Reality-Based Content Around the Globe
According to “international comparisons (e.g., TIMSS, 1995, 2003) mathematically, The Netherlands is one of the higher achieving countries in the world” (Hough & Gough, 2007, p.1). Studies show that U.S. students still do not perform as well as their counterparts in many other countries (Son, 2011). Thus, studying and implementing the mathematical teaching techniques of The Netherlands can only serve as beneficial to students in the United States. Within the last thirty years, The Netherlands implemented a new mathematics approach entitled Reality Enhanced Mathematics, or REM. REM is an innovative approach to teaching mathematics that remains relatively unknown by most teachers in the United States classrooms. Educators note that “traditional math education strategies have focused repetition, drills and problem solving far removed from everyday life” (Kerekes, Diglio, & King, 2009, p.10). The reality approach differs from that of traditional math in that with a reality-based mathematics content “learning mathematics means doing mathematics, of which solving everyday life problems (contextual problems) is an essential part” (Fauzan, Plomp, & Slettenhaar, 2002, p.1). A unique aspect of the REM approach is that the curriculum “uses imaginable contexts to help pupils to develop mathematically, with a strong emphasis on pupils 'making sense' of the subject” (Hough, & Gough, 2007, p.1). The reality-based concept of teaching mathematics not only assists in preparing the students to be mathematical problem solvers in the workplace, but also contributes to build functional math skills necessary in everyday life. Students are able to visualize how various concepts that are being taught in the classroom can benefit them in their future career choices. By teaching mathematics with a reality-based approach “the ability to create engaging math learning experiences can have immediate and long-term positive effects on learners” (Kerekes et al., 2009, p.10). One positive lasting effect is that students will possess valuable mathematical skills that they can utilize successfully outside of academia. They will have the knowledge to apply real world concepts in their future careers.
Many professions necessitate using some form of mathematics. For example, carpenters must be adept in geometry, accountants are required to utilize algebra, and nurses must be knowledgeable in basic arithmetic. Therefore, it is imperative for students to understand why they must learn certain mathematical concepts, and understand the need for them to have the ability to build upon their prior knowledge in math. The failure to teach a student the means “to study mathematics can close the doors to vocational-technical schools, college majors, and careers” (Kerekes et al., 2009, p.10). As students begin their high school careers, and embark on their journeys toward the future, they should already possess basic mathematical skills that would enhance their chances of obtaining a position in their proposed career. One of the five goals of the National Council of Teachers of Mathematics [NCTM], (1989) is for students to become proficient problem solvers in mathematics. Students who are adequately prepared for their proposed career are presented with an array of opportunities and various options in terms of their career choices. Students who are ill-prepared or less proficient in math may be much more limited in their options.
The Benefits of Reality-Based Mathematics Curriculum
The notion of incorporating real world situations into everyday mathematics classrooms is a modern approach to problem solving. Reality-based mathematics problems and tasks are more meaningful to students than basic word problems and drill work. When presented with reality-based tasks, students are able to visualize and comprehend how they would respond to a particular situation. Problem solving is the students’ experience of the usefulness of mathematics in the world they live in (NCTM, 1989). Mathematical instruction that revolves around rote learning does not produce knowledge that can be applied in reality. If the mathematical process is not useful and students are unable to relate to the information in the problems, they are simply solving the problem by means of repetition or adhering to mathematical rules. Problem solving is a skill that should be addressed, improved, and enhanced on a daily basis because it is central to inquiry and should be interwoven throughout the mathematics curriculum to provide a context for learning and applying mathematical ideas (NCTM, 2000, p.256). The problems utilized should also be examples that provide realistic value in order to be deemed beneficial for the students. A basic premise in the educational system of the United States is that teachers prepare students for participation in the ‘real world.’ Students are expected to possess the ability to function adequately in society to be successful. Seemingly, the skills they acquire during their years of exposure to formal education should have practical application in their daily lives.
Research shows that “as teachers provide real-life, relevant opportunities for problem based learning; they can facilitate students’ thinking development and thereby content comprehension and internalization” (Kerekes et al., 2009, p. 6). Teachers are in a position to mold students and provide them the necessary tools that would allow them to become successful in a chosen career. According to the Center for the Future of Teaching and Learning (2008), teachers must demonstrate further commitment to deepening subject matter content knowledge and pedagogical skills required to refine capabilities in mathematics. When teachers encourage their students to work diligently and visualize how they can apply academic activities to real life situations, they are assisting them in their future success. Reality-based mathematics activities not only assist students in preparing for their future, but they also help them to understand the calculations they might come in contact with on a regular basis. By giving student problems with real world components, they are able to see how mathematics is embedded in many of the tasks they might complete daily. Mohr (2008) states that “it is safe to say that no matter which job or career individuals may pursue, they will inevitably use math skills learned throughout their primary years of education, and many of those learned during high school” (p. 34).
There is a noticeable difference between basic word problems and realistic, or reality-based, word problems. Hypothetical situations presented in today’s textbooks do not provide the necessary content for students to develop the abilities to apply mathematical concepts to real-life situations. The following is a math problem with hypothetical content: Two trains, 100 miles apart are facing each other. Train A is traveling 80 mph. Train B is traveling 60 mph. At which point will they meet? We probably all remember the problems about animals with two legs and animals with four legs with one group being twice as large as the other and trying to decide the number of each animal according to the total number of legs. Although the situation necessitates basic mathematical skills, it demands no practical application in the real world for the average math student. Wubbles, Korthagen, and Brokeman (1997) maintain that the criterion for a problem to be called realistic is that it should be likely that the problem is experienced by the learner as real and personally interesting. Currently, the many problems that students are assigned to solve in the classroom have very little practical application or minute relevance to their real life situations.
There are many instances when it is apparent to instructors that students are failing to comprehend the rationale behind why they are being taught many of the concepts currently integrated into the mathematics curriculum. Typically, students want justification for why they must master a concept in class and they often want to have a clear understanding in regard to how they will benefit from learning the concept or skill. Sparrow (2008) noted that “if children can experience real mathematics that engages them by connecting with their interests of the moment, and also work with purposeful activities that bring together mathematical skills and knowledge that they have, then there may be a better chance that children will become engaged and experience success in mathematics” (p. 8). Once the teacher begins to implement reality-based components, the students gain a better understanding of the benefits of a part of a curriculum. Students can easily see how this concept can be of assistance in familiar situations. Schools should invest in textbooks that support the notion of using real mathematics and that require students to apply mathematical ideas to real-world problems or by setting tasks within a familiar context. Sparrow (2008) noted that “the technique is useful as it allows children to see the connection of school mathematics to situations and contexts met by people outside the classroom” (p. 5).
Students’ Attitudes Regarding Mathematics
Unfortunately, many students have developed negative attitudes toward mathematics by the time they reach the middle school level. Their attitudes are a result of the exposure to an array of sources including impractical mathematical experiences, the inability to find meaning in concepts and/or lack of their teachers’ expertise of the subject matter. In a study by Lucas, et al. (2007), students surveyed mentioned experiences with cruel and mean mathematics teachers that do not ‘make it real’. They noted that some students do not have adequate teachers for mathematics and consequently, had a dislike for the subject. Many students give the impression that they dislike solving word problems in the mathematics classroom. For the most part, student dissatisfaction is a consequence of the inability to see purpose, meaning or point of the word problems that they are given subsequent to a concept that has been taught. When students are exposed to the technique of reality-based mathematics they tend to have more of a positive attitude toward mathematics in general. Examples of how students can use math in the real world are numerous. When students were asked to identify real world applications, they noted making change, calculating discounts on items, measurement and the use of technology (Lucas, et al., 2007) Of course, the list of opportunities that require the application of math skills in daily life is almost endless. People begin to have a realization of how they can use mathematics to their advantage once they are introduced to reality content. Lucas, et al. (2007) discovered an increase of 21% in positive responses when they asked both students and adults if they use mathematics in everyday life. One of the better examples of how students can use math in the real world is when they compare rates at the grocery store to determine the better deal between two products. This practical situation could enable the student to help save the family valuable money by successfully applying math skills learned in the classroom.
In a study completed by Fauzan et al. (2002), it was stated that once students in the Netherlands were taught mathematics based on the REM methods, “interviews with the pupils showed that the pupils liked the new approach. The students realized that there were some positive changes on themselves, especially in reasoning and being more active and creative” (p. 2). Students tend to become more engaged and excited about working in groups and as individuals once they are exposed to the reality-based activities they will be completing verses with same work book pages they are completing for repetition and mastery with each unit taught. After reviewing several studies on reality-based mathematics, it was difficult to report anything negative about the students’ attitudes mentioned. With reality-based mathematics, “not only do teachers and students learn the math content better (internalize), but they also enjoy the experience and can see connections to other applications” (Kerekes et al., 2009, p.10).
Opposing views
Not everyone is a firm believer that Reality Enhanced Mathematics is the solution to mathematical success for every child. There are several skeptics of the REM approach. Studies have been conducted demonstrating the fact that the “activities tended to 'run out of steam' after a couple of lessons and then teachers resorted to 'page 35, exercise 2.1a' — a dull, repetitive task that would soon be forgotten by the pupils” (Hough & Gough, 2007, p.1). A significant effort has to be made on the teachers’ part to ensure each lesson and problem is one that would be relevant to the pupils in their classroom. The task of doing so would most likely increase the workload with no evidence of it being a successful teaching method. If additional time and effort are required, many teachers may resort to utilizing the typical textbook problems and enhancing them by adding words. The procedure would not be one that would be considered successful. Teachers would have to be equipped with ample training on amending the concept for the curriculum to be taught correctly and mastery of the content to be adequately met.
Other studies by Heck and van Gastel (2006) discuss the impact that the REM program has had on students once they reach the college level and have a “disjointed body of knowledge and skills” (p. 928). They disclose that the students do in fact have more of a heterogeneous knowledge base, but that they are not accustomed to the abstract mathematics that they would face as they pursue a higher education. The students included in the study possessed mathematical knowledge, but many of the basic skills and mathematical formulas were lost.
Sparrow (2008) discussed the actuality that just because a word problem has a reality component it does not necessarily make it a reality-based problem. The context of the problem has to be meaningful to the student. For example, many area problems will discuss finding the area of a room prior to purchasing carpet to install. As Sparrow reasons, the context is situated in a reality setting, but students in elementary or middle school would not find carpeting a room meaningful to them. Even though basic math problems can be filled with reality based content, it does not necessarily mean that the students will be able to understand and relate to the material.
Implementing a Reality-Based curriculum
When attempting to implement a new curriculum, there are many complications that may arise. The idea of changing or putting into place an adaptation to a current curriculum is often problematic because people, in general, tend to resist change and perceive change as an additional burden. Teachers “are faced with a number of challenges when asked to implement a new curriculum. They are often mastering new content as well as adapting to new methods of teaching, assessing students in new ways, and learning how to use new curriculum materials” (Obara & Sloan, 2009, p. 1). Many teachers are inclined to rely upon what they already know and incorporate the methodologies that their own teachers used in school. Teachers could observe an organizational change process as something being added to their already demanding workload and reject the idea of being open-minded or objective about an innovative technique or method of instruction. Fauzan et al. (2002) discuss that with the implementation of a new curriculum the change process is only possible if there is support given by both the government and key mathematics educators. Educators play an important role in helping to prepare their peers to ensure that they are capable of teaching and disseminating the new mathematical component.
In the United States, and especially Georgia, there is a major push to increase high stakes assessment scores. Most teachers are willing to do whatever is necessary for their students to meet their high expectations and achieve their maximum score considering their capabilities. In turn, there would probably be resistance from teachers who were unwilling to take risks with their test scores (Sacks, 1999, p. 237). With many teachers focused on test scores and mastering the Georgia Performance Standards, or GPS, they are apt to reject the idea committing to the time that is necessary to launch an experimental project. Realistically, teachers of students who score well on the state’s measures are unwilling to examine if there are techniques that will produce even better results (Sacks, 1999). For an organizational change process to truly become successful, “all parties affected by the change must work together to develop a clear vision of where the process is proceeding, recognizing that formulation of a plan is a developmental process” (Zins & Illback, 1995, p. 114).
CHAPTER THREE: METHODOLOGY
Research Design
The focus of this research study is to determine if the implementations of reality-based mathematics tasks and problems have an effect on student learning and, in turn, increase test scores. This study contains a combination of action and evaluation research. Action research design, also known as classroom or teacher research, “encourages school personnel to systematically develop a question, gather data, and then analyze that data to improve their practice” (Gilles, Wilson, & Elias, 2010, p. 91). The evaluation type of research is “typically done by school districts to determine the effectiveness of given products, procedures, programs, or curricula” (Charles & Mertler, p. 299). Treatment and control groups were used in this study. Results from each group are displayed and the data collected is used for comparison with those who did not participate.
Mixed research methods were used throughout this study to collect data. The quantitative methods used in the study were assessments and surveys. Drew, Hardman, and Hosp (2008) state that “quantitative research involves studies in which the data that are analyzed are in the form of numbers” (p.69). Assessment data collected in this study included pretest and post-test that were analyzed using an independent and dependent t-test to determine statistical significance. Surveys were given to both students and colleagues and analyzed statistical means of a Chi Square test. The qualitative method used was the teacher, student, and administrative interviews. Drew et al.(2008) describes qualitative research as a type of “research that involves collecting data in the form of words or a narrative that describes the topic under study and emphasizes collecting data in natural settings (p. 26).
Setting
The research study took place at a middle school in Coweta County, Georgia. The study site consisted of the 925 students. The student body was multi-racial with Caucasians comprising 73% of the enrollment. African Americans (17%), Hispanics (5%), Multi-Racial (3%), and other (2%) accounted for the remaining 27%.
The middle school had 27% of students receiving free and reduced lunch services and nine per cent of the student body had been identified as receiving special education services.
The middle school in which the study took place was chosen because it is my current place of employment. An IRB was completed and accepted. Permission to perform the study was granted by the Coweta County School System and the supervising principal at the study location.
Subjects and Participants
The subjects used in this research were all sixth grade students. There were a total of 52 students studied including 24 males and 28 females. There were a total of 40 Caucasian students, 7 African American students, 4 Hispanic, and 1 multi-racial student. A total of 35 subjects were labeled as gifted learners. The students used in this study were randomly placed in my math class without any input from me. Those scheduling the students did so with no knowledge of my proposed study. The students were easily accessible and the research was easily conducted within the parameters of my classroom. Other participants involved in the action research include the school administration and fellow mathematics teachers.
Procedures and Data Collection
In this research study both qualitative and quantitative methods were used to collect data and answer the focus questions (see Table 3.1).
Table 3.1 Data Shell
|Focus Question |Literature Sources |Type: Method, Data , and|How are data analyzed |Rationale |
| | |Validity | | |
|Will implementing a |Kerekes, J., |Method: |Quantitative: |Quantitative: |
|reality-based math |Diglio, M., & |Pre/post testing |Independent and |determine if there are |
|curriculum improve |King, K. | |dependent T-Test |significant differences |
|mathematical understanding|(2009) |Data: | |in pretest and post-test |
|and increase math test | |Interval | | |
|scores? |Hough, S., & Gough, S. | | | |
| |(2007) |Type of Validity: | | |
| | |Content | | |
| |Sparrow, L. (2008) | | | |
|How have the feelings and |Sparrow, L. (2008) |Method: |Quantitative: |Quantitative: |
|attitudes of the students | |Survey |Descriptive and |determine if there are |
|altered after being taught|Kerekes, J., | |inferential Statistics |significant differences |
|reality based mathematics?|Diglio, M., & |Data: |Chi Square | |
| |King, K. (2009) |Ordinal | | |
| | | | | |
| |Lucas, D. M., Fugitt, J., |Type of Validity: | | |
| |& Ohio Univ., A. (2007) |Construct | | |
|How effective was the |Sacks, P. (1999) |Method: Interview |Qualitative: |Qualitative: |
|change process used to | | |Coded for themes |look for categorical and |
|convince my colleagues to |Fauznn, A., Slettenhaar, |Data: | |repeating data |
|adopt a reality-based |D., & Plomp, T. (2002) |Qualitative | | |
|curriculum in mathematics?| | | | |
| | |Type of Validity: | | |
| | |Construct | | |
To answer focus question one pertaining to student’s achievement, a quantitative assessment was given in the form of a pretest/post-test to the subjects in the study. There were two control classes as well as two experimental classes used in this study. All subjects were given a pretest and a post-test for statistical purposes. After the pretest was evaluated all students were taught using traditional mathematics teaching methods, including whole group instruction, grouping and individual practice. The treatment classes were given reality-based mathematics tasks during the unit. The tasks for this particular unit on probability, included weather forecasting, picking lottery numbers, calculating baseball batting averages and games of chance that the student might come in contact with in everyday life. The task allowed students to see how the mathematical concepts covered in class could be used in the real world. The tasks were completed throughout the entire unit. When all of the concepts were covered in class a post-test was given to gage the increase in knowledge of the content by the students. The survey used in the study was modified from a published document (Moldavan & Tapia, 1996).
Focus question two refers to the attitudes of the participants used in the study. A survey was distributed to the treatment group before and after the mathematics unit. The initial survey was be used as a baseline of the attitudes the participants possess pertaining to mathematics. The survey given at the end of the unit was the same as the initial survey. It was given twice to gage any significance in the data. The survey (see Appendix A) was modified from current literature and contained twelve questions. The responses to the survey were in a Likert format.
Focus question three discussed teachers’ attitudes and the change process of implementing a reality-based component to the current standards of mathematics. To answer this question, interviews (see Appendix B) were conducted with various mathematics teachers participating in the study. An administrator was also interviewed regarding the change process.
Validity, Reliability, Dependability, Bias and Equity Measures
Data are gathered through teacher created pretest and post-test that align to the current performance standards for focus question one. An interval scale of measurement was used for this focus question. According to Salkind (2010), and interval scale measurement “is where a test or an assessment tool is based on some underlying continuum such that we can talk about how much more a higher performance is than a lesser one” (p. 140). For focus question one content-validity was accomplished through the use of standard related material used by the mathematics teachers in the county. Because the same pretest and post-test was given, test –retest reliability was used for focus question one. Test-retest reliability according to Salkind (2010) “is used when you want to examine whether a test is reliable over time” (p. 145). In order to ensure dependability, the data collection and treatment were kept consistent including the fact that the setting for the data collection was controlled. I am aware that bias may exist and precautions have been made ensure that the study is not unfair or offensive to any students. Unfair, offensive and disparate impact bias was addressed by using subjects randomly placed in classes for the control and experiment group. Due to the lack of minority students in the school population, ethnic groups were not equally represented. Bias was minimized by teaching the control and experimental groups the same material, in compliance with the required Georgia Performance Standards. Each group was given identical pretests and post-tests.
Data were gathered for focus question two by administering the same survey twice. It was administered prior to and after the unit was taught. An ordinal scale of measurement was used for focus question two. Salkind (2010) states that ordinal level of measurement “stipulates data can be, and are ranked” (p. 382). For focus question two construct-validity was accomplished by comparing student survey results to results in documented research. Salkind (2010) states that construct-validity is a “type of validity that examines how well a test reflects an underlying construct” (p. 378). Reliability was reached using a Cronbach’s alpha correlation and chi square of the survey results. Parallel forms of reliability were also used to compare the survey answers of students in the control group to students in the experimental group. Biases were eliminated with the use of anonymous student surveys.
To gather opinions about the proposed change process or implementing a reality-based mathematics curriculum, data were collected by means of an administrative interview (see Appendix C) as well as an interview of mathematics teachers for focus question three. Ordinal data were used to analyze the teacher surveys. Construct validity is used for this focus question. Reliability was reached by interviewing a random sampling of mathematics teachers from different grade levels the same survey. The administrator who was interviewed checked the transcript for accuracy to ensure dependability of the study. The interview scripts and questions were peer reviewed to eliminate bias.
Many efforts were made to ensure equity within this study. Equity was achieved in this study by ensuring teacher quality. According to Skrla, McKenzie, and Scheurich, (2009), “access to high quality teachers is one of the key factors at the school level that influences student achievement” (p. 31). The quality of the researcher in this study was supported by years of experience, certification levels and advanced degrees. Achievement equity was addressed with lessons that were relevant to the student’s daily life and also covered state standard for testing. The pretest and posttest were reviewed by other mathematics teachers to ensure validity and eliminate bias. Efforts were made to make certain interview questions and surveys were fair and further addressed the purpose of the action research.
Analysis of Data
The pretest and post-test scores were analyzed using both an Independent and dependent t-test. An independent t-test was run to determine if there was a difference that was significant between the control and treatment group’s pretests. Another Independent t-test was run to determine the significance between the experimental groups and control group’s post-test scores. The results that were obtained were verified at a 0.05 level of statistical significance. A Dependent t-test was run once with the control groups’ pretest and post-test and a second time with the experimental groups’ pretest and post-test. The results that were obtained from both the pretest and post-test were verified at a 0.05 level of statistical significance. The effect size r and Pearson’s r for reliability correlation was calculated for the dependent t-test. Cohen’s d was calculated for the independent t-test. Effect size is used to measure the magnitude of the treatment effect. Salkind (2010).
Student and teacher survey results were measured using quantitative descriptive statistics. Each of the student survey questions were analyzed using a Chi Square and Cronbach’s Alpha. Each question’s significance was determined through the data results of the Chi Square with alpha levels of .05, .01 and .001.
The administrative interview transcript was reviewed by the interviewee and then coded for recurring, dominant and emerging themes.
The research study was analyzed holistically through validation, credibility, and transferability and transformational. Consensual validation was achieved by the study being approved by the faculty of the site where the study took place as well as the faculty of LaGrange College Education Department. An IRB was completed and approved for the study. Eisner (1991) calls the faculty review process ‘Consensual Validation,’ an agreement among competent others that the description, interpretation, evaluation and thematic are right. Epistemological validation described by Denzin and Lincoln (1998) is the cycling back to your literature review as ‘Epistemological Validation,’ a place where you convince the reader that you have remained consistent with the theoretical perspectives you used in the review of the literature. Epistemological validation was achieved by comparing the resulting data to that of documented literature.
Credibility was achieved through the use of multiple data sources within the study. Eisner (1991) calls this process ‘structural corroboration,’ where a confluence of evidence comes together to form a compelling whole. Within Eisner’s definition are embedded the concepts of fairness and precision. To be fair, one must state that he plans to present alternative (opposing) perspectives with which he may not particularly agree. This is done by presenting alternative perspectives in the literature review as well as selecting participants in the data collection process who have opposing views. To ensure that the research and opinions of reality-based mathematics are fair, research representing opposing views was included in the review of literature.
The descriptive nature of this study allowed for transferability. Eisner (1991) calls this process ‘referential adequacy’ where perception and understanding by others will increase because of your research. This study will allow other mathematics teachers to see research that might improve not only their teaching, but student achievement and understanding of mathematical concepts. If the study is not proven to be successful at increasing test scores, mathematics teacher will know which method of teaching will assist the students in becoming successful.
Making the study transformational was achieved through it being useful in many mathematical situations. Catalytic validity is the degree to which you anticipate your study to shape and transform your participants, subjects or school (Lather as cited by Kincheloe & McLaren, 1998). If this study encourages teachers to engage their students in higher order thinking by using reality-based mathematics, then both, teachers and students, will reap the benefits.
CHAPTER FOUR: RESULTS
The results displayed in Chapter Four are listed by focus question. The quantitative data results of the t-tests, chi-square, Cronbach’s Alpha, and independent t-test and dependent t-test for means are displayed in the form of embedded tables within the chapter.
To answer the first focus question, will implementing a reality-based math curriculum improve mathematical understanding and increase math test scores, a pretest and post-test was given to both the control group and experimental group. The scores were analyzed by quantitative means through a dependent t-test to establish significance. The purpose of a dependent t-test was to evaluate the means of the pretest and post-test scores to establish the similarity or difference between the scores, it was also to determine any significance within the scores (Salkind, 2010). The data were then analyzed with an independent t-test to determine if there was any significance between the scores of the experimental group and the control group. The calculations for the dependent and independent t-tests for both the experimental and control group pretest and post-test are provided in the Tables 4.1 and 4.2.
Table 4.1 –Dependent t-Test for Pretest and Post-test for Experimental Group
| |Pretest |Post-test |
|Mean |60.058 |67.154 |
|Variance |294.565 |306.603 |
|Observations |52 |52 |
|Pearson Correlation |0.61 | |
|Hypothesized Mean Difference |0 | |
|Df |51 | |
|t Stat |-3.369 | |
|P(T ................
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