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Chapter 5 will address implications of the findings, implications for the literature, implications for practice, and implications for future research. The chapter will culminate with a summary of the discussion.
The purpose of the study was to determine the effect of real-world mathematical applications on students' achievement and classroom learning environments. The study sought to test the following hypotheses.
H-1: There are significant differences on achievement scores by group status (experimental or control), academic status (gifted or non gifted), 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).
As indicated in the discussion that follows, overall, the results supported the hypotheses; however, there were a number of notable exceptions.
Hypothesis 1: Student Achievement
Hypothesis 1 was supported by the results of the statistical analysis. There were significant differences between adjusted mean achievement scores in the main effect group status and the interaction effect of academic status and group status. The real-world mathematical applications curriculum did make a difference. This real-life curriculum should be used in the classroom, as it affected students' performance dramatically.
As a result of these findings, Soaring with Numbers is effective, particularly with the African-American nongifted student, but also with the African-American gifted student, too. As stated earlier in this paper, what is good for the gifted is also good for all students. This statement particularly applies here because it is so true: The gifted and the nongifted achieved significance in the real-world mathematical applications curriculum. Through working in such a curriculum, students understand the usefulness of mathematics as it applies to the real world. Through this real-world curriculum, they also realize the connections to other disciplines and within mathematics itself. Students are better able to cope with society after benefitting from such a real-world curriculum.
Hypothesis 2: Classroom Learning Environment
Perceptions of participants toward the classroom learning environment varied from component to component. In general, those participants exposed and those participants not exposed to the real-world mathematics applications curriculum had similar perceptions toward involvement, affiliation, rule clarity, task orientation, satisfaction, and innovation of the classroom learning environment.
However, gifted participants had more favorable perceptions toward three of the CES subscales. Gifted participants had significantly higher adjusted mean scores than nongifted participants toward involvement, task orientation, and satisfaction components. In the interaction effect, group status x academic status, the experimental gifted had significantly higher adjusted mean scores than other groups in rule clarity, task orientation, satisfaction, and innovation.
A subjective reason for these findings might be that a real-world mathematics application curriculum enhances students' abilities to manipulate and understand their learning environments. Another possibility is that real-world mathematical applications demonstrate students' strengths in those particular subscale components.
Following is a brief discussion of the implications of the findings for each component score of the CES.
Involvement. Hypothesis 2.0 regarding the involvement component was supported with regard to the main effect group status and the interaction effect of academic status and group status. However, it was not supported with respect to main effect academic status. The gifted participants became extremely involved in the topics and problems of Soaring with Number. This curriculum was particularly challenging at all levels. Gifted African-American students become easily bored. The real-world mathematics curriculum motivated students to become involved in the many varied activities, using highly interesting resource books to obtain the facts necessary to solve the problems. When students are involved in the real-world mathematics curriculum, they are understanding the principles and practices of mathematics, making connections in mathematics, and applying mathematics. This knowledge and understanding will lead to better classroom environments, curricula, and teaching approaches.
Affiliation. Hypothesis 2.1 regarding affiliation was supported with respect to the main effects academic status, group status, and the interaction effect of academic status and group status. The similar perceptions on affiliation of participants exposed to real-world mathematical applications and participants not exposed to real-world mathematical applications could be attributed to the fact that these classes (including social studies, reading, language arts, physical education, and science) have just an hour each day of the school week in which to learn the disciplines' concepts for that day, pick up homework, correct papers, and discuss new concepts. Students need the time to communicate and discuss with their peers. These results support the idea that all students need more time in class for cooperative learning in which they can help each other and discuss concepts and problems; thus, they will get to know each other.
Rule clarity. Hypothesis 2.2 regarding rule clarity was supported with regard to main effects academic status and group status. Nevertheless, it was not supported with respect to the interaction effect of academic status and group status. The experimental gifted were so involved in the real-world mathematical applications curriculum that they quite naturally participated in an environment in which the rules were quietly adhered to. These experimental gifted African-American students appreciated and recognized structure and organization. They made good judgments in which rules were necessary to accomplish the task. These very students were learning respect for one another through staying within rule boundaries.
Task orientation. Hypothesis 2.3 regarding task orientation was supported with regard to the main effect group status. However, it was not supported regarding academic status and the interaction effect of academic status and group status. The gifted African-American students were more motivated to stay on task with the real-world mathematical applications curriculum than were the nongifted. The gifted, as educators of the gifted indicate, are persistent in what they are interested in and can concentrate for long periods of time. This persistence and interest was only amplified by real-world mathematical applications in task orientation. Again, these experimental gifted African-American students were so interested and challenged with the multitude of activities and projects in Soaring with Numbers that this became quite evident in their CES task orientation scores.
Satisfaction. Hypothesis 2.4 regarding satisfaction was supported by the main effect group status. Nonetheless, it was not supported by main effect academic status and the interaction effect of academic status and group status. Students were able to move about in their classrooms to use reference books and the Internet to look for information to help them solve these creative problems. They worked together to solve different problems. These elements enhance a student's feeling of satisfaction.
If the experimental African-American gifted are achieving in the real-world mathematics curriculum, involved in the real-world mathematics application curriculum, easily able to conform to rules because of their involvement in this curriculum, and are task oriented, it follows that this very same group of students will be satisfied with their tasks at hand in this curriculum. They will be happy and responsive to applying their knowledge and skills in the real-world mathematics curriculum and serving as a vital part of the classroom environment.
Innovation. Hypothesis 2.5 regarding innovation was supported by main effects group status and academic status. However, it was not supported by the interaction effect of academic status and group status. Academic status and group status, combined, did produce a significant effect on the participants' perceptions toward innovation. When the teacher tries new ideas and uses a variety of methods in the real-world mathematics classroom, students benefit from learning a variety of ways to approach the curriculum. The teacher is thus a role model in a classroom environment that is conducive to exploring and trying new ways of solving problems and solving these problems in a creative way. Once again, the real-world mathematics curriculum strengthened the experimental gifted African American's creativity, imagination, curiosity, wide variety of interests, high alertness, and attention characteristics. The Soaring with Numbers curriculum served the experimental gifted African American with its multitude of different kinds of problems (for example, Venn diagrams, planning their clothes and necessities for a trip to various parts of the earth, converting a football field into a soccer field, and discovering the countries that are the world's largest oil consumers and largest oil producers).
If students are satisfied, task oriented, involved, and know their boundaries, a more creative environment is provided. Thus, it is not surprising that the experimental gifted African-American participants' perceptions of their classrooms included significantly higher perceptions than the other groups.
Even though many of these subscales are characteristics of the gifted (for example, involvement, rule clarity, task orientation, and innovation), it is apparent that this real-world mathematics curriculum heightened and enhanced these very characteristics as reflected in the experimental gifted African-American students' perceptions of these subscales. Finally, while the experimental nongifted African-American participants' perceptions were not statistically significant, nevertheless, their minds and interests were stretched, too. What benefits the gifted most certainly benefits all students.
One of the most interesting findings of the current study was the significant influence that the real-world mathematics applications had on the mathematics performance of the student participants. To be sure, participants exposed to real-world mathematics applications appeared to perform better academically in mathematics than those participants not exposed to real-world mathematics applications. These findings were consistent with those of Good, Grouws, and Ebmeier (1991), Connell, Peck, and Buxton (1992), Drager-McCoy (1996), Fuson, Carroll, and Drueck (2000), Huntley, Rasmussen, Villarubi, Sangtong, and Fey (2000), and Pascale (1979). All of these researchers found that students exposed to real-world mathematics applications had more advanced levels of conceptual understanding in arithmetics than their traditionally instructed counterparts.
On the other hand, the current findings regarding real-world mathematics applications were not supported by the work of Cobb, Wood, Yackel and Perlwitz (1992). These researchers found that, in the levels of computation performance on familiar textbook tasks, the students exposed to traditional mathematics applications were comparable to those exposed to real-world mathematics applications. However, in the same study, these authors' findings regarding students' conceptual development in arithmetic were similar to those in the present study.
Somewhat of a surprising finding was the lack of influence that group status had on the mathematics performance of the participants. However, Fuson, Carroll, and Drueck (2000) reported on two studies that concluded that real-world mathematics produced positive results for gifted and nongifted students. More specifically, gifted and nongifted students had similar mathematics scores. Additionally, the National Science Foundation, which funded the Core-Plus Mathematics Project, opined that a real-world mathematics program enhanced the academic performance of not only the gifted students, but all students (Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000). These findings did not correspond to those of Van Tassel-Baska (1994) and Gallagher (1994), who found that gifted students performed better in mathematics than their nongifted peers. A reasonable explanation for the high achievement of nongifted participants is that these very participants were linking mathematical formalism to real-world situations. But, in the control groups with the traditional curriculum, mathematics was "a collection of symbol manipulation rules, and some tricks for solving stereotyped story problems" (Charles & Silver, 1988, p. 32). It could be that those same students would find it difficult to solve authentic problem situations in which they must address specific complexities in real-world mathematics.
Another notable finding of the present study (and somewhat surprising) was the lack of significant impact that real-world mathematics applications had on the participants' learning environment components. Specifically, participants exposed to real-world mathematics applications and those not exposed had similar perceptions on all six components of the classroom learning environment. Perhaps these similar perceptions of participants on the CES are consistent with the middle school findings in other studies (Waxman & Huang, 1998). Middle school students in Waxman and Huang's 1998 study scored about 18% below elementary school classes on the CES, whereas these same middle school students scored about 12% below high school students in the classroom learning environment scores. These scores reflect that the middle school years are not positive and encouraging years for our young students according to the CES.
These findings did not parallel those of Good, Grouws, and Ebmeier (1991) and Anderson (1987). The aforementioned researchers opined that the instruction that students received from real-world mathematics application programs had a direct effect on their achievement.
A reasonable explanation for the prevailing findings can be found in the work of Anderson (1987), who concluded that the perceptions that students hold toward their teachers provide the necessary structure for their learning in the pedagogical process, as well as having a consistent influence on their academic achievement.
Finally, the variable academic status had a significant influence on some of the components of the CES. To be sure, 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. The majority of the sample participants were African American (82.8%), and Gay (1978) stated that gifted African American students have a strong ability for concentration. This characteristic relates to the CES components of involvement and task orientation. The experimental gifted African Americans had significantly higher adjusted mean scores than the other three groups in rule clarity, task orientation, satisfaction, and innovation. Innovation for the gifted African American is demonstrated by many displays of creativity (Swenson, 1978). For example, gifted African American students create their own games, use different materials, find new ways to get attention, and think of original ways to avoid work.
The present study has provided the opportunity to test the researcher's theory, based upon personal classroom experience and the supporting opinion of noted educators (Mitchell, 1990; Carnine & Gersten, 2000), that a real-world mathematics curriculum, which connects mathematics to students' experiences, will produce a measurable improvement in students' achievement. The study has borne out this theory in the value of real-world mathematics curricula, thus leading to a number of implications for practice for educators at all levels.
An earlier version of the curriculum employed in the 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 and conducting the study. Through the successful development, implementation, and testing of a real-world mathematics curriculum, 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 materials the researcher had previously used in the classroom, were developed specifically for use in the study and integrated into the mathematics resource book titled Soaring with Numbers. Constructing such a book, which demonstrates precise understandings of concepts with instructional methods, offers the teacher a more thoughtful, effective way to present a mathematics curriculum.
Educators responsible for improving the mathematics performance of students need to be made cognizant of the influence that a real-world mathematics applications curriculum has on the performance of students in mathematics. Because of the significant achievement performance of the participants exposed to the real-world mathematical applications curriculum, educators should be encouraged to develop their own activities. It was the researcher's intention that mathematics teachers and specialists, in particular, should venture to incorporate a curriculum that includes everyday, relevant, real-life, and real-world mathematics problems in the classroom on a daily basis. Nothing is more exciting to students than to have products in their hands that their teacher has created for their unique use. Naturally, the teacher is excited about resource materials that he or she has created themselves, and that enthusiasm is contagious. The teacher is in the best position to know what works with his or her students and what appeals to them.
Hopefully, this study of a teacher-created resource book will lead to more teachers and educators being innovative by writing their own resource books and implementing their original resource books in studies. The addition of this type of research, that identifies what really works, will be invaluable for practitioners in education.
A plausible explanation for the present findings with regard to the significant influence that real-world mathematics applications had on the performance of participants might be that teachers who had provided instruction to the experimental group (real-world mathematics applications), even though they had similar academic backgrounds and years of experience as those in the control groups (traditional instruction methods), probably viewed real-world mathematics applications as more stimulating and intellectually motivating to students than those in the traditional group. Because of their positive view of the curriculum they were teaching, the experimental group teachers were more actively involved in providing students with practical applications to fit their mathematics understanding. Students exposed to real-world mathematics applications scored statistically higher in various categories of mathematics than those students exposed to the traditional mathematics applications. As a result of their research, Good, Grouws, and Ebmeier (1991) concluded that teachers represent a distinct educational resource and that money invested in attracting, educating, and retaining capable teachers can result in increased student achievement.
Educators should be aware that active involvement in the curriculum enhances students' performance. A basis for the NCTM Standards' principle is the key word actively: The Standards' principle is "actively building new knowledge from experience and prior knowledge"(NCTM, 2000, p. 11). Through action and interaction in the mathematics classroom, the teacher should be interacting by attempting to confirm, disconfirm, or modify hypotheses (Steffe & Wiegel, 1992). For example, in the present study, students were using reference books, measuring, graphing, constructing, diagraming, examining, discussing, and questioning. Because these real-world problems make sense to children, the children find them of interest. This directly results in students acquiring skills in ways that result in effective learning. These real-world mathematical problems also better prepare students for life, the workplace, and scientific and technical communities, where they additionally contribute to society.
The Soaring with Numbers experiment also demonstrated the success of cooperative groups in a real-world mathematical applications curriculum. It follows, then, that the use of cooperative groups in mathematics should be explored as an integral part of any new real-world mathematics curriculum. Cooperative groups can be a tool for students to develop their mathematical ideas, strategies, and solutions. In these groups, students get the opportunity to experiment with various approaches to solving problems, to construct mathematical arguments, to respond to others' arguments, and to make conjectures. Also, through the medium of cooperative groups, students are socially active, exercising important social skills as they share and explain strategies, learn to respect their peer's ideas, obtain help, and encourage others to contribute to the group. Thus, students in these groups are developing social and academic skills that are necessary for their futures.
It is recommended that educators carefully examine the components of a real-world mathematics curriculum to assess their usefulness with other disciplines, as well as within mathematics. Disciplines should connect, and connections should be made within mathematics for a more meaningful and better understanding of mathematics by students.
Educators need to be aware of the advantages and disadvantages of implementing realistic mathematical modeling into intermediate and middle school classrooms. By understanding the complexities of these types of programs, educators can be assisted in their efforts to improve the learning environments for their students.
Finally, educators need to consider rethinking the classroom environment in the mathematical sciences for the purpose of understanding the roles of teachers and students. Such a new mathematical environment could allow teachers opportunities to better prepare students, regardless of their academic deficiencies, in all aspects of mathematics. Classroom environment does provide a basis for what occurs in the classroom and can be extended to what occurs in a school environment. It is invaluable to teachers to have an opportunity to look through the eyes of the students (by way of students' perceptions of what is happening in the classroom) in order to examine strengths and weaknesses of the students' learning environment. It is the researcher's opinion that teachers do make a difference in the classroom learning of individual students and that the concept of active teaching is useful for studying classroom teaching and learning. Good active and authentic teaching includes the eight components of the classroom learning environment: involvement, affiliation, rule clarity, task orientation, teacher support, order and organization, satisfaction, and innovation. Through this learning, the teacher is able to make critical changes in classroom practices to strengthen the weaknesses found in the students' perceptions of the classroom. In a positive, coherent, student-centered environment, students will meet with success in achievement.
Classroom Learning Environment
The inconclusive nature of the results of the study with regard to classroom environment suggests that the study should be replicated over a longer period of time to determine if more time in using the real-world mathematical applications curriculum would also produce a positive gain in the classroom learning environment. Additionally, use of the longer form of the Classroom Environment Scale would increase the reliability of the scales (Waxman and Huang, 1998). Positive attitudes are a valued aspect of the mathematics classroom, and results of the present study indicate that carefully crafted research on classroom learning environment is needed.
Of interest, too, would be a comparative study of classroom environment perceptions between majority and minority students from varied cultural backgrounds. Such a study would help teachers understand the cultural factors that affect the quality of learning among various cultures and ethnicities.
Real-World Mathematics Curriculum
That this teacher-made mathematics resource book was successfully implemented and used in intermediate and middle school classes, suggests that other teachers, too, have significant contributions to add to the educational research base. Teachers need to become aware of the potential value of their contributions to the field of mathematics education by writing their own resource books and implementing these original resource materials in a research setting. There is a continuing need for more teacher-developed materials and studies in mathematics. Who knows better what meets the needs and piques the interest of mathematics students than the teacher. Many teachers do innovative, constructive, active lessons in mathematics every day, and these teachers need to be encouraged to go one step further and write their own books and conduct their own studies.
In any replication of the present study, an additional assessment instrument (for example, the Texas Essential Knowledge of Skills) could be used to determine students' mathematics achievement performance. Judging from the comments of teachers participating in the present study about their participating students' improved TAAS scores, the implication has been drawn that a real-world mathematics applications curriculum benefits all mathematics skill levels and problem-solving.
To extend the findings of the study, a follow-up study could be conducted using various real-world mathematical applications curricula in mathematics classes on different educational levels. Of noteworthy value for education would be a study in the primary grades using a real-world mathematical applications curriculum. Such studies would provide additional data on the influence the real-world mathematics curricula have on young students' mathematics performance. At the other end, in high school, a study could be conducted to provide additional data on the influence that real-world mathematics curricula could have on high school students. For example, high school students could perceive real-world mathematics through aeronautics by covering a plane, flight direction, and the air show. These are relevant mathematical experiences for students.
Another area for future research is a true experimental study, where students are randomly assigned to two groups to explore the influence of real-world mathematical applications versus a traditional curriculum on the academic performance of majority, as well as minority, students. Though quasi-experimental models are often necessary and yield relevant results, the authority commanded by a true experimental study is the ideal.
Another interesting and useful study would be to examine the effectiveness of the components of a real-world mathematical applications curriculum on the academic performance of students in other disciplines (for example, social studies). Yet another study could investigate the influence of demographic variables in conjunction with realistic mathematical models on the academic performance of students. The goal of such a study would be to identify information about differences in relationships between environment and achievement. Additionally, a study to determine the real-world mathematics applications curriculum's most appropriate grade level would be of significant value.
Finally, a study to determine the minority group that would benefit the most from this real-world mathematical applications curriculum would yield valuable results for educators. In this study, the minority groups could include African-American students, Hispanic students, Asian-American students, and White students. Perhaps, this would help address at-risk students throughout the groups. The real-world mathematical applications curriculum benefitted the African-American students particularly in the present study. It should benefit all groups as it is based upon everyday experiences in mathematics that all minority groups need to have to function in our society.
Based upon the findings derived from the data analysis of the investigation, the researcher made the following conclusions. When pretest differences were controlled, student participants exposed to the real-world mathematical applications curriculum appeared to perform better in mathematics achievement than participants not exposed to the real-world mathematics applications curriculum. Study participants, as a whole, regardless of their group status, performed similar in their mathematics achievement. The combination of academic status and group status had some influence on the mathematics scores among study participants.
In general, both experimental and control groups had similar perceptions toward involvement, affiliation, rule clarity, task orientation, satisfaction, and innovation. Gifted participants reflected more favorable perceptions than their nongifted counterparts on involvement, task orientation, and satisfaction components of the classroom environment scale. Regardless of academic status, all participants appeared to have similar perceptions toward affiliation, rule clarity, and innovation.
Gifted and nongifted participants in both the experimental group and the control group tended to have similar perceptions regarding the involvement and affiliation components of classroom learning environment scale. Gifted participants exposed to the real-world mathematics applications curriculum appeared to have more favorable perceptions toward the rule clarity, task orientation, satisfaction, and innovation scales than did nongifted participants exposed to the real-world mathematics applications curriculum. Finally, nongifted participants not exposed to the real-world mathematics applications curriculum appeared to possess more favorable perceptions regarding the task orientation and innovation scales than did nongifted participants who were exposed to the real-world mathematics applications curriculum.
Given the opportunity of exposure to the real-world mathematics applications curriculum, both academic groups, nongifted and gifted, demonstrated positive achievement. In the NCTM Standards (2000), equity is emphasized with high expectations and strong support for all students, as it was with this real-world mathematics applications curriculum. Students and teachers, alike, were appreciative and enthusiastic for this opportunity to learn mathematics with a different approach. Teachers commented on how good this real-world mathematics applications curriculum had been for their students. It heightened students' awareness of what they were capable of accomplishing. These findings infer that the study's real-world mathematical applications curriculum was interesting to the participants and helped them see the importance and utility of continued mathematical study for their own futures. Teachers also commented on improvement in behavior, and one young boy who had been a behavior problem, became a group and class leader as a direct result of this real-world mathematics applications curriculum.
Another interesting point from teachers' notes was that the teachers in the experimental schools felt that their students' TAAS scores had improved. One teacher of mathematics reiterated that, out of 22 students in her homeroom, 18 students received Academic Recognition, and never before had she had so many in any of her classes. Overall, the scores improved in that school; in fact, this school's mathematics scores had never been as high.
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