IMPACT OF INQUIRY-BASED TEACHING ON STUDENT MATHEMATICS ACHIEVEMENT AND ATTITUDE. A dissertation submitted to the

Transcription

1

2 IMPACT OF INQUIRY-BASED TEACHING ON STUDENT MATHEMATICS ACHIEVEMENT AND ATTITUDE A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirement for the degree of DOCTOR OF EDUCATION (EdD) In the Division of Teacher Education Of the College of Education, Criminal Justice, and Human Services 2006 by Taik Kim

3 ABSTRACT In 2002, University of Cincinnati faculty members from the College of Engineering and the College of Education, Criminal Justice, and Human Services proposed the Science and Technology Enhancement Program Project (STEP) to improve students learning in the secondary mathematics classroom using modules of inquirybased teaching. The purpose of this study was to determine the impact of the STEP Project on students achievement in and attitude toward mathematics. Hierarchical linear models (HLM) were used to evaluate the impact of the STEP Project. The sample group for the study was 130 ninth grade students enrolled in Integrated Algebra I in a large urban school district. The school was one of eight secondary schools that participated in the STEP Project. The classes in the treatment group were three of five classes ordered in terms of the highest, middle, and lowest mean GPA. The control group consisted of two other middle GPA classes. The classes had an average of 25 students. Teachers who previously had been involved in the STEP Project taught all treatment and control classes. The inquiry-based teaching activities provided by the project were confined to the treatment classes. The mathematics achievement test scores and a survey measuring students attitudes toward mathematics were obtained for both groups of students. An effect on mathematics achievement was significant at p <.08 (ES =.51). The inquiry- based teaching methods had no substantially different effects on students of different genders (p =.42, ES =.07). These results suggest that inquiry-based teaching improved mathematics achievement. Furthermore, the inquiry-based teaching affected students attitudes toward mathematics (p <.07, ES = 3.07). Especially, African ii

4 American students who had preexisting favorable attitudes toward mathematics were significantly affected by treatment (p <.02, ES =.02), while the treatment affected African American students overall at p <.08 (ES =.58). iii

5 Copyright Taik Kim. All Rights Reserved. iv

6 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES Page vii viii CHAPTER 1: Introduction 1 Background of the Study 3 Purpose of the Study 5 Research Questions 6 CHAPTER 2: Literature Review 9 Constructivism and Inquiry-Based Teaching 10 Studies Related to Inquiry-Based Teaching 14 Studies Related to Attitude toward Mathematics 18 Studies Related to the Technological Approach 20 CHAPTER 3: Methodology 25 Setting and Participants 25 Data Collection 26 Procedure 28 Data Analysis 29 CHAPTER 4: Results 32 Missing Data 34 Mathematics Achievement 35 Importance of Gender 41 Effects of GPA 42 Attitude Measures 43 v

7 Page CHAPTER 5: Discussion 48 Summary of Findings 50 Limitations 52 Relationship to Other Research 53 Implications of Future Research 54 Conclusion 55 REFERENCES 56 APPENDICES A. Mathematics Attitude Survey 66 B. Sample of Mathematics Test Items 68 C. Sample of Inquiry-Based Teaching Modules Let s Make a Cozy House 73 vi

8 LIST OF TABLES Page Table 1: Descriptive Statistics 33 Table 2: Mean Z-score for Each Test of the Treatment & Control Groups 34 Table 3: Final Estimation of Fixed Effects for HLM 3 39 Table 4: Mean Scores on the Pre- & Post-Surveys Measuring Attitude 45 Table 5: Final Estimation of Fixed Effects for HLM 2 47 vii

9 LIST OF FIGURES Page Figure 1: Group Differences from the Pre-Test to the Post-Test 41 Figure 2: Gender Difference 42 Figure 3: High and Low GPA Group Difference 43 Figure 4: Group Differences from Pre-Survey to Post-Survey 46 viii

10 Chapter 1 INTRODUCTION Third International Mathematics and Science Study (TIMSS), a study of comparisons among world nations in mathematics and science achievement, reported disappointing results for the United States in mathematics achievement (Stigler, 1995). The average score of 12 th grade students on the 1995 test was 461, which was below the international average of 500. U.S students outperformed only the students in Cyprus and South Africa. Also, the United States average score of eighth-grade students was mediocre. In 2003, the TIMSS report showed little improvement for eighth-grade students from the second set of uniform tests in 1999, despite a massive effort to reform and improve mathematics education. Meanwhile, the average score of 504 for 8 th -grade students of the U.S exceeded the international average score 466, but the US average was still 101 points less than the score for Singapore which had the top average scale score. It had been more than 15 years since a presidential commission acknowledged problems with the educational system and encouraged educators to improve students academic achievement in a report entitled A Nation at Risk (1983). On mathematics and science tests in 1999, students from a dozen of the 38 nations participating in the study did better than students from the United States. Specifically, the average U.S. mathematics score was 104 points lower than the top average national performance score. Klein (2003) implied that the unacceptable performance of American students, especially high school students, was due to the reform movement, and that the way to improve students 1

11 mathematics achievement is to go back to the traditional basic mathematics or direct instruction. Undoubtedly, one of the major factors that affects the learning of mathematics is the instructional method (Eggen & Kauchak, 2001). The National Council of Teachers of Mathematics (NCTM, 2000) suggests that mathematics in academia should be conveyed and taught as a subject that improves one s logical thinking, analytical reasoning, and evaluating skills, all of which are essential to every career path. That such vital skills may be developed by learning mathematics makes the subject valuable to gaining a wellrounded education. There is a consensus that students need to understand mathematics, and that students do construct mathematical knowledge rather than infer it in finished form from the teacher. McNair (2000) concluded that traditional teacher-centered teaching methods, which are reliant on hierarchical transference of knowledge rather than on student-based knowledge construction, do not improve student s logical thinking, analytical, and evaluating skills. If traditional systems are not working, other instructional methods need to replace them. In the traditional form of classroom instruction, a teacher demonstrates procedures, then assigns homework, and later administers a test (Gersten, Taylor, & Graves, 1999). A new type of teaching is needed to stimulate students motivation and adjusts teachers teaching styles to individual students learning styles. Several studies suggest that reformed teaching methods like real life situations, problem-based lessons, and authentic learning can improve students attitude toward mathematics (Martin, Sexton, Wagner, & Gerlovich, 1997; Portal & Sampson, 2001). However, the research on students academic achievement in the field of mathematics with respect to use of inquiry-based teaching in the classroom is limited. Little research 2

12 exists regarding the effect of inquiry methods on students attitudes and learning in mathematics, although there is some research evaluating the teaching method. Since there is a possibility for improvement in achievement and attitude as a result of using inquiry teaching methods, quantitative research is needed to gain a clearer, more complete measure of data. Background of the Study In 2002, ten University of Cincinnati (UC) faculty members from the College of Engineering and the College of Education, Criminal Justice, and Human Services proposed the Science and Technology Enhancement Program (STEP) to improve students learning in the secondary mathematics classroom using modules of inquirybased teaching. The STEP Project was funded by the National Science Foundation. Eight public schools with whom the College of Education, Criminal Justice and Human Services had previously interacted participated in this project. The schools in the STEP Project were four urban and four suburban schools, each having Grades 7 through 12 or 9 through 12. The two primary goals of the STEP Project were to produce scientists, engineers, and secondary school science and mathematics educators; and to design, develop, and implement hands-on activities and inquiry-based modules that relate to issues in the students communities and authentically teach science and mathematics. Proposing inquiry-based teaching as a method to improve students academic achievement and concurrently to boost students attitude toward mathematics, the STEP Project encouraged the development of modules based on real-world situations, and using hands-on activities and technology to teach mathematics and science. 3

13 In addition to the ten UC faculty members, Project STEP employed four undergraduate fellows, 22 secondary school science and mathematics teachers, and nine graduate fellows. One of the key requirements to be selected as a graduate fellow for this program was to have a strong mathematics or science background. Seven graduate fellows were chosen from the College of Engineering and two graduate fellows belonged to the College of Education, Criminal Justice, and Human Services. The primary role of the graduate fellows was to develop inquiry-based learning activities using current instructional technology, as well as to implement these activities in secondary classrooms with teachers. During the first year of the 2-year project, the fellows spent 10 hours per week in classrooms. They worked working directly with teachers of mathematics, science, and technology classes. The fellows were trained to create and implement handson activities and technology-driven, inquiry-based teaching modules by taking a new education course, an advanced course covering instructional technology and the local, state, and national standards in the field of mathematics, science, and technology. After spending two years as a Project STEP graduate fellow, the researcher analyzed the achievement and attitude results of algebra students in an urban school that participated in the project. Promoting inquiry-based teaching as an ideal teaching method in our technology-centered world, the STEP Project encourages the development of modules based on real-world situations, hands-on activities, and the use of technology in teaching mathematics. Inquiry-based teaching in the STEP Project typically has the following characteristics. Lessons begin with collecting data. Then, students organize the information and identify relevant real-world situations. During this process, students are required to use technology and utilize critical thinking and logical reasoning skills to 4

14 analyze the outcomes (Grabinger, 1996; Duffy & Cunningham, 1996). Compared to traditional teaching, students are more actively involved in learning while investigating the problem, designing solution strategies, and finding solutions (Slavin, Madden, Dolan, & Wasik, 1994). This study was not designed to evaluate the outcomes of Project STEP but to examine the relationship between inquiry instruction and students grades in and attitudes toward mathematics. Purpose of the Study The purpose of this study was to determine the effects of inquiry-based teaching on students mathematics achievement and attitudes toward mathematics. This study analyzed students mathematics test scores and examined students attitudes toward mathematics when the inquiry-based teaching was used as the instructional method. One of the main goals in learning mathematics is to improve reasoning and analytical skills and to become mathematical problem solvers (NCTM, 1989). The inquiry-based teaching can be a valuable tool to enhance students learning, and to improve attitudes toward mathematics. If the inquiry-based teaching module were found to be successful, the findings would be of value to educators who seek an effective strategy to improve students test scores in and attitudes toward mathematics. Therefore, this study is significant to educators who seek to know more about how students achievement and attitudes toward mathematics can be improved. This research is also important for helping teachers with their instructional design because it aids in the implementation of a teaching model that could be more effective for secondary students. 5

15 Research Questions The purpose of this study was to determine the effects of inquiry-based teaching on students achievement and attitudes toward mathematics. The following questions guided the research effort: 1. Was there significant improvement on the mathematics achievement test scores after students were taught using the inquiry-based methods? a. Was there significant improvement of the test scores in the treatment group after inquiry-based teaching compared to those in the control group? b. Was there significant improvement of the test scores of male and female students within the treatment group after inquiry-based teaching compared to those in the control group? c. Was there significant improvement in the test scores of students in the treatment group after inquiry-based teaching compared to those in the control group, based on different levels of GPA? d. Was there significant improvement in the test scores of different ethnic groups within the treatment group after inquiry-based teaching compared to those in the control group? 2. Was there significant improvement in the students attitudes toward mathematics after inquiry-based teaching? a. Was there significant improvement in the attitudes of the treatment group after inquiry- based teaching compared to those in the control group? 6

16 b. Was there significant improvement in the attitudes of male and female students within the treatment group after inquiry-based teaching compared to those in the control group? c. Was there significant improvement in the attitudes of students in the treatment group after inquiry-based teaching compared to those in the control group, based on different levels of GPA? d. Was there significant improvement in the attitudes of different ethnic groups within the treatment group after inquiry-based teaching compared to those in the control group? To answer the first research question, differences in pre- and post-test scores were analyzed after the treatment group received inquiry-based teaching. The same tests were given to the control group. After pre-tests, the middle and post-test were given after 5-week periods. The content area for teaching was the Coordinates and Functions chapter in the Integrated Mathematics I textbook (Rubenstein, Craine, & Butts, 2002, pp ). An attitude survey was used to investigate students attitudes toward mathematics before and after inquiry-based teaching. Hypothesis Statements The first research hypothesis is that inquiry-based teaching will improve students academic achievement in mathematics. The null hypothesis is that there will be no statistical difference in the test scores after using inquiry-based teaching. The second research hypothesis is that inquiry-based teaching will improve students attitudes toward mathematics. The null hypothesis is that there will be no statistical difference in the students attitude toward mathematics after inquiry-based teaching. 7

17 This study was a quasi-experimental design due to the lack of randomization in the nonequivalent group design used for comparison. This design compared the test scores and attitude survey responded of experimental and control groups before and after inquiry-based teaching. In the study, it was expected that inquiry-based teaching would enhance students learning of mathematics and positively influence students overall achievement and their attitudes toward mathematics. Definitions of Terms For this study, definitions of the following terms were used: Attitude towards mathematics: Positive or negative feelings in situations involving mathematics. It was simply defined as a personal preference for or dislike of mathematics (Papanastasiou, 2000). Treatment group: Students who were presented with the inquiry-based teaching in the classroom during periods two, three, and five. Control group: Students who were presented with traditional teaching methods in the classroom during periods one and four. 8

18 Chapter 2 LITERATURE REVIEW In 2001, the U.S. Congress modified the Elementary and Secondary Education Act (ESEA), which became the No Child Left Behind Act. This Act asserts that a large number of students have not yet gained proficiency with mathematics and that public schools should teach all students to be proficient in reading and mathematics by 2003 and 2004 respectively. However, the remaining question was how the public schools would accomplish this intricate task. A study on international comparisons in mathematics and science achievement (National Center for Educational Statistics [NCES], 1999) provided information about how well the U.S educational system was working and what needed to be done to improve it (National Research Council [NRC], 1999). Part of the TIMSS study indicated that the poor achievement of secondary students was a result of teaching methods (Linn, Kessel, & Slotta, 2000; Schmidt et al., 1999; Stigler & Hiebert, 1999). Schmidt et al.(1999) found that American students were weak in not only the logical and analytical skills used to solve complicated mathematics problems, but also in their understanding of the basic concepts and process of inquiry. NCTM recommended a new type of teaching to improve learners logical and analytical skills by engaging them with today s technology-oriented society and stimulating students motivation (NCTM, 1989, 1991, 2000). The NRC (1989) also maintained that the classroom environment needed to change in order to improve students analytical thinking and logical reasoning skills. In 1991 NCTM (1991, 2000) strongly recommended a change from the traditional lecturebased teaching to methods based on constructivism. NCTM provided two important 9

19 recommendations: (1) Make a stronger connection between mathematics and the students real world, and (2) move the focus of the classroom environment from inactive learning, like rote memorization of formulas and procedures, to interactive lessons that rely heavily on students participation in the thinking and logical reasoning processes of mathematics. Despite these new guidelines and recommendations, classroom mathematics instruction has hardly changed (Hoff, 2003; Silver & Stein, 1996). In their large scale study analyzing 364 science and mathematics lessons taken from observations of and interviews with teachers in schools across the United States, Weiss et al.(2003) reported that the quality of classroom instruction was poor, lacking in lessons that were challenging to students. Constructivism and Inquiry Based Teaching Constructivism is one explanation for how learners gain knowledge of new ideas. Eggen et al. (2001) describe constructivism as a view of learning in which learners use their own experiences to create understanding of knowledge that makes sense to them rather than having understanding delivered to them in already organized forms (p. 246). Novak and Gowin (1986) define the concept of constructivism as people using their experiences or prior knowledge to understand new things. The constructivist framework of Vygotsky (1978) can be a key to teaching and learning mathematics. This theory stresses that students need to understand certain basic concepts and fundamental structures before learning new ones. This is especially important because students who do not understand basic concepts in mathematics are 10

20 unable to learn higher-level mathematical concepts. This theory of learning may accurately describe what failed to happen for some students who struggle in math. If previously taught ideas were not retrievable, then, in effect, they would be lacking the fundamental mathematical concepts needed for further learning. According to Martin et al.(1997), inquiry-based methods are a form of constructivist teaching and learning. They defined inquiry as an attempt to discover new information by the inquirer. Inquiry-based teaching stresses that the inquirer learns by relating new knowledge to the real world. Inquiry-based teaching is designed to give students mathematical practice that relates to how the real world operates. The focus of teachers and curriculum builders should be on enhancing these methods. This strategy has been proposed not only to achieve more meaningful mathematics education for all students, but specifically for the improvement of mathematics performance among high school students. In 1995, the NRC made public the National Science Education Standards to improve science education. The National Science Education Standards stressed that using teaching methods that attempt to increase students curiosity, such as problembased instruction, helped the students develop reasoning skills (National Research Council, 1995). A major aspect of the Standards was inquiry-based teaching. The term inquiry describes students abilities to carry out scientific investigations as part of the course curriculum. It relates to teaching and learning strategies that encourage and enable students exploration. The National Science Education Standards promoted by the NRC (1998) also explained that inquiry is a proper way to learn and teach scientific concepts. 11

21 However, inquiry-based teaching is not a common term in mathematics education. Inquiry-based teaching is another form of problem-based instruction where students become active investigators and the teacher s role is mainly facilitative (Eggen et al., 2001). Problem-based instruction is based on John Dewey s belief that children are active learners who learn by discovering their surroundings (Dewey, 1916). Bruner (1966) stated that students connect newly-learned knowledge to what they already know. The inquiry-based teaching encourages active knowledge construction and connection by selecting and transforming newly-learned information, and then quickly connecting it to old knowledge. Piaget (1970) also indicated that the learner s knowledge is constructed by a process of cognitive interpretation, reevaluation, and connectional reorganization. In response to nationwide calls for changes in the teaching of mathematics, the University of Chicago School Mathematics Program (UCSMP) developed a mathematics curriculum with an emphasis on reading, problem-solving, and applications to real-world problems. In 1989, the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics, a new set of guidelines for effective teaching and learning strategies. According to these standards, problem solving and reasoning as well as basic mathematics fundamentals should be emphasized in teaching to promote more positive attitudes in students. This type of mathematics instruction, known as problem-based instruction, is a collection of integrative teaching strategies that use problems as the focus for the direction of learning (Krajcik, Blumenfeld, Mark, & Soloway, 1994). Problem-based instructional strategies typically have the following characteristics: lessons begin with a problem or question and solving 12

22 the problem or answering the question becomes the focus of the lesson (Grabinger, 1996; Duffy & Cunningham, 1996). As in inquiry-based teaching, students are actively involved in learning while investigating the problem, designing solution strategies, and finding solutions (Slavin, Madden, Dolan, & Wasik, 1994). Similarly, the teacher s role in problem-based instruction is primarily facilitative. To learn mathematics is to learn objective knowledge and to independently develop the necessary thinking, reasoning, evaluating, and logical processing skills that are the foundations of almost all human activities. When people think of what makes up mathematics, people think of learning concepts like addition or multiplication. However, mathematics is a much larger, deeper, and more abstract subject. Most mathematicians would agree with Hoffman (1989) that there could be no description of mathematics that covers its broad scope. Nevertheless, we could use views on various aspects of mathematics to ask, What is mathematics? within specific categories. For example, if mathematics is applied in conjunction with a branch of science, we need the scientific method in order to gain, organize, and apply new knowledge of mathematics. This method begins with exploring a situation, predicting a hypothesis, performing experiments and formulating the simplest general rule. Then the general rule that organizes these main ingredients becomes known as a law or principle. Since mathematics is the science of patterns, we must understand the order of the pattern and be able to interpret other patterns or phenomena. In terms of mathematics as science, school mathematics should be based on an epistemological perspective using experiences from students lives to teach abstract principles. 13

23 There is more to learn about mathematics than skillful computation. NCTM (2000) suggested that students need to understand basic concepts and fundamental structures of mathematics that lie beneath the rules and procedures of simple arithmetic. To understand the structures of mathematics, students need to learn how to relate their conceptual and procedural knowledge to their world, and how to grasp both the interrelations among concepts and operations for higher-level study. An understanding of the mathematical structures underlying the procedures and concepts should be fundamental to meaningful learning. Inquiry-based teaching, as a form of constructivist teaching and learning (Martin et al., 1997), should start with what students know and then apply the next step and allow students to attempt to discover new information, thereby expanding their existing knowledge and connecting it to larger, more complex ideas. Studies Related to Inquiry-Based Teaching The relationship between teachers instruction and students learning has been the subject of much research for many years (Boaler, 1998; Portal & Sampson, 2001; Silver & Stein, 1996; Zahorik, 1996). Portal and Samson (2001) examined whether motivation was a factor in students learning mathematics. The purpose of the study was to find the evidence that students achievement in mathematics was related to motivation by analyzing teacher observations, student surveys, and students grades. The participants were 1,460 high school students and 76 teachers from a suburb school in a middle class community of a large urban area. Portal and Samson suggested that instead of conventional teaching, the instruction should consist of Hands-On Math with real life applications in order to increase students motivation. Zahorik (1996) explored how 14

24 teachers applied teaching strategies to generate interest in learning. The population in the research consisted of 65 elementary and secondary teachers who registered in a graduate course on constructivist teaching. Four interactive papers, presenting how teachers made students learning interesting, were collected to analyze the data. The findings of the study indicated that hands-on activities were the main method to improve the students interest. However, teachers failed to connect main concepts and content information to the activities in many cases. Boaler (1998) inspected 3-year case studies of two secondary schools to examine student experience and understanding with different teaching approaches: traditional teaching and activity-oriented teaching. Data collection included the use of case studies, questionnaires, interviews, and student assessments. Both activity-related tests in real world situations and traditional test problems were given to students for the assessments. The researcher found that the two different teaching approaches affected students achievement. Students in the traditional teaching were able to develop procedural knowledge, but were lacking in conceptual understanding, while the students in the activity oriented class achieved more at real-life situations. If lecture-oriented systems the traditional teaching method in which a teacher gives a lecture, homework, and a test are not working, they need to be replaced by other instructional methods. However, several educators suggest that direct instruction could be a better teaching method (Gersten & Keating, 1987; Gersten, Keating & Beacker, 1988; Adams, 1995). They supported the traditional teaching method, which is a form of direct instruction. They thought that direct instruction could teach students more effectively than any other instructional method. Yet those studies were limited to elementary students. No equivalent studies for secondary schools were found. 15

25 Although standard-based teaching, the collection of effective teaching and learning strategies outlined in 1989 by NCTM guidelines, has been criticized by educators who support direct instruction, there are several studies that indicate that constructivist teaching did positively affect students academic achievement in mathematics and attitudes toward mathematics (Nichols & Miller, 1994; Silver & Stein, 1996; McCaffrey, Hamilton, et al. 2001; Kramarski, Mevarech, & Arami, 2002). In 1994, Nichols and Miller observed that cooperative group learning in an algebra II class resulted in a significantly greater increase in students motivation and achievement than the traditional lecture group in algebra achievement. Silver and Stein (1996), participating in a project called Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR), tested the hypothesis that there was improved learning when students received enhanced forms of teaching such as lecture combined with active interaction, quality mathematical activities, and tasks to develop understanding of mathematical concepts. They attempted to investigate the hypothesis that new teaching methods produced a positive impact on students learning of mathematical concepts. QUASAR teachers used specially-designed instruction methods that emphasized mathematical understanding, thinking, reasoning, and problem solving in order to help students develop the necessary understanding of math concepts. Analysis of classroom observation data and over 150 teaching samples were collected over three years supported the conclusion that new teaching methods produced a positive impact on students learning of mathematical concepts. The QUASAR schools, six middle schools located in urban school districts, ranged in size from 300 students to 1,500 students. Silver and Stein (1996) found that existing methods of teaching mathematics could not provide the high 16

26 levels of conceptual and meaningful understanding that students require. They suggested that in order to create a new classroom environment, teachers should include many mathematical activities and ideas to increase the engagement of students. McCaffrey, et al, (2001) & Klein, et al. studied the influence of instructional practices and curriculum on students achievement in high school mathematics. They found that standards-based teaching was positively related to students achievement in integrated mathematics courses, which exactly followed NCTM guidelines, while there was no impact on students achievement in the more traditional algebra and geometry classes. In their study of the effects of metacognitive instruction on mathematics achievement, Kramarski, Mevarech, and Arami (2002) discussed the instructional methods used to improve students active involvement in the learning process and the development of apprehension and conception of mathematical ideas. Metacognitive instruction was designed to improve students reasoning and thinking skills using realworld problems. They concluded that cooperative learning with metacognitive instruction significantly affected outcomes positively compared to the cooperative instruction without metacognitive instruction. The test scores of students who were given cooperative-metacognitive instruction were better than those of the students who had the cooperative instruction only. However, test scores alone cannot yield an entirely accurate indication of the effectiveness of any treatment. According to Eggen et al. (2003), inquiry-based teaching as another form of problem-based instruction emphasizes students active involvement in learning. Thus, inquiry-based teaching frequently stresses that the students learn by relating new knowledge to the real world. Inquiry-based teaching methods, designed to give students 17

27 mathematical practice that relates to how the real world operates, are a fundamental and important teaching method in mathematics; the focus teachers and curriculum builders should be on enhancing these methods. Martin et al. (1997) suggested that hands-on activities are a very useful tool in inquiry based learning. The next step in inquiry-based teaching is to use hands-on activities that give students practical learning opportunities through active involvement. With a proper lesson plan and activity, students not only will come to enjoy the mathematics class they previously disliked, but also will improve their mathematics skills and their attitudes toward mathematics (Li, 2003). Studies Related to Attitudes Toward Mathematics Several studies investigated factors that are important to students learning (Portal & Sampson, 2001; Silver and Stein, 1996; Veenman, 1984; Zahorik, 1996). Portal and Sampson (2001) found that motivation was an important factor in students learning. They suggested that instead of conventional teaching, instruction should consist of Hands-On Math with real-life applications in order to increase students motivation. They suggested that a way to improve mathematics teaching was to implement a new curriculum connecting life experiences to mathematics instruction. Veenman (1984) found that lack of student motivation was one of the greatest concerns of the teachers who were concerned with effective classroom management. Zahorik (1996) also noticed that lack of student motivation was great concern of teachers. In this research, he observed positive behaviors in the classroom, and only a slight improvement in students attitude towards mathematics after modifications of the 18

28 curriculum. His research indicated that providing students with real-life examples to illustrate abstract ideas enhanced students interaction and enabled them to better understand mathematic ideas. However, Kulubya & Glencross (1997) reported no significant relationship between attitudes and mathematics achievement. Papanastasiou (2000) also found that even though the students of Cyprus had lower scores on mathematics than American or Japanese students test scores in the TIMSS study, the percentage of students in Cyprus with positive attitudes towards mathematics was higher than in the U.S. or Japan. The number of students in Japan with positive attitudes towards mathematics was also lower than the number of U.S. students, despite higher test scores among Japanese students. Several studies reported ethnicity gaps in attitudes toward mathematics, affecting overall achievement in the mathematics classroom (Walker & McCoy, 1997; Tapia & Marsh, 2000; Stull, 2002). In their 1997 study, Walker et al. (1997) reported that most African American students did not expect to have mathematics-related careers. Tapia et al. (2002) also found that there was a relationship between ethnicity and attitude toward mathematics. However, Stanley & McCoy (2004) found no significant difference in attitude toward mathematics among students of different ethnicities or of different genders. The National Assessment of Educational Progress (NAEP, 2005) data also indicated the gap between White and Black students among 4 th grade in mathematics has narrowed from 32 of the scale score in 1990 to 26 in For 8 th grade, the gap narrowed from 34 to 34 between 2003 and It is fundamental that every educator understands that each student enters school from a different starting point, with a different background, ability, talent, attitude, and 19

29 overall learning style. The major disadvantage of class lectures is that students are forced to keep up with their teachers, regardless of these differences. The vast majority of students who are labeled as slow learners or at-risk students are sitting in the classroom with overwhelming feelings of stress and frustration because they do not understand what is being taught (Hannula, 2002). If students do not understand the lecture, it will be very hard for them to benefit from the education process no matter how well the instruction is planned or implemented. They then become bored and under-prepared, which makes them even more difficult to teach. Particularly, students on the lower end of the socio-economic scale who do not have mathematical skills struggle through the rest of school. Effective teaching greatly enhances the attitude that students have, but another equally important element to understand is that learning requires students to play an active role in their own education. Many students, especially Black students in urban schools, struggle with learning mathematics (Walker et al., 1997). It is very important to know the factors that affect to students learning (Portal & Sampson, 2001; Silver & Stein, 1996; Veenman, 1984; Zahorik, 1996). Studies Related to the Technological Approach of Teaching Mathematics Since the STEP project emphasized using technology, it is important to review literature on how technology affects students learning. The review of literature in this section includes research on computer-based instruction and research on other applications. 20

30 Research on Computer-Based Instruction Over the past two decades, a substantial number of studies have been conducted to assess the effectiveness of computer-based instruction. The experiments reported in the literature were completed on various student populations, using different instructional and experimental approaches. They also provided mixed results. Heid (1997) found that Computer-Assisted Instruction (CAI) increased students scores, compared to traditional classes. Tiwari s study (1999) showed that when students used CAI, their test scores were better than the control group s. However, Keepers (1995) found that there was no significant improvement in academic achievement using CAI in a calculus class. Soeder (2001) also found no difference between the students who used the computer-aided instruction (software from the Computer Curriculum Corporation) and the students who did not use any new technology. In Soeder s study, 391 students were in the experimental group and 497 students in the control group. The study was conducted over three years to determine if computer-aided teaching in mathematics improved students academic achievement as measured by the Pennsylvania System of School Assessment (PSSA) test. Moch (2002) conducted a study on the effects of computer-based instruction on a high school algebra curriculum. A total of 78 students participated in the study. The study found no significant difference between the control and experimental groups; however, the limitation of Moch s study was that it was a case study without generalizability. In Soeder study, there was the problem of group difference: there were size differences and family background differences between the two groups, and classroom teachers were changed during the academic years. 21

31 Other Applications Hembree and Dessart (1986) conducted meta-analyses based on data gathered from a total of 79 studies on the effects of the calculators in mathematics instruction at the kindergarten through 12 th grade levels. The research found that calculator use had a positive effect on students attitudes toward mathematics, a mean effect of.44. However, using calculator for computation and composite skills did not significantly impact the abilities of students with preexisting low and high abilities. Only students who had average ability improved their paper-and-pencil skills after using calculator with a mean effect of.14, except in Grade 4, in which there were negative effects on their achievement with a mean effect of.15. Studies indicate that spreadsheets are a very useful tool in helping students use graphical methods to solve problems (Aramovich, 1995; Abramovich & Nabors, 1998; Clements & Samara, 1997; Dugdale, 1998). According to these studies, the use of spreadsheets provided students with a deeper understanding of the set of concepts in a problem. Spreadsheets can also be used to illustrate graphically the solving of simple and complex equations (Niess, 1998). McCoy (1996) examined the effects of computers (in the area of programming), computer-assisted instruction, and tool software in mathematics learning. While only achievement in geometry knowledge and problem solving were measured, students in the treatment group had higher scores than students in the control group. McCoy also found that treatment groups from third grade to high school who used fraction software, problem-solving software, and estimation software scored noticeably higher than did control groups. Simulation programs were effective in helping students understand geometric notions and the concept of graphs. According to 22

32 McCoy, students who used mathematical software for algebra, geometry, and calculus were more motivated and self-confident about those subjects, in addition to possessing a deeper understanding of those subjects. Funkhouser and Dennis (1993) reported that high school geometry students who used problem-solving software did a better job in both problem-solving and general performance areas than did students not using the software. In addition, students attitudes towards and enjoyment of high-level mathematics topics were more positive than other groups. McCoy s studies, however, noted that there was no achievement difference between the treatment group using computers in mathematics and the control group, yet they noticed that the motivation of students using computers was higher than that of the other group. In a report to the President entitled the Use of Technology to Strengthen K-12 Education in the United States (1997), The President s Committee of Advisors on Science and Technology suggested a focus on learning with technology, not just about technology, and an emphasis on content and pedagogy and not just hardware. According to the committee s recommendations, technology needs to be utilized as a tool to improve students achievement in mathematics. According to NCTM (1989), in addition to instruction in problem solving and reasoning skills, students need access to new technology at all times and at all levels, to promote more positive attitudes. In 2000, NCTM suggested that teachers need to be more skillful at using technology: The effective use of technology in the mathematics classroom depends on the teacher. Technology is not a panacea. As with any teaching tool, it can be used well or poorly. Teachers should use technology to enhance their students' learning opportunities by selecting or creating mathematical tasks that take advantage of 23

33 what technology can do efficiently and well graphing, visualizing, and computing. For example, teachers can use simulations to give students experience with problem situations that are difficult to create without technology, or they can use data and resources from the Internet and the World Wide Web to design student tasks. Spreadsheets, dynamic geometry software, and computer micro worlds are also useful tools for posing worthwhile problems. (NCTM, 2000, p. 25) The impact of the computer in the classroom has become one of the most widely-debated issues since this encouragement of the use of technology was recommended. Most research about computers as educational tools reported that they provide evidence of learning benefits to students, yet there still remain generalization issues (Bangert-Drowns, Kulik & Kulik ;1985). There are no learning benefits to be gained from employing any specific technology to deliver instruction. Even though technology can be a useful tool in teaching and learning mathematics, technology itself does not influence learning under all conditions. If there is no relationship between technology and learning, it may be because the technology is not being used properly. In order to properly use technology as an educational tool, it is necessary to understand the relationship between technology and learning. In other words, when technology is integrated into a mathematics curriculum, educators should develop appropriate lesson plans or activities for use in the classroom. If we define learning as an active, constructive, cognitive process rather than a passive response to a delivered lecture, technology can be a vital resource to integrate new information into understanding. 24

34 Chapter 3 METHODOLOGY This study investigated how inquiry modules affected students mathematics achievement and their attitudes toward mathematics. Quantitative research methods were used to evaluate effectiveness. The research hypotheses stated that there were statistical differences in academic achievement and significant improvement in the students attitude after experiencing inquiry-based teaching methods. Setting and Participants Data were collected in ninth grade mathematics classes from one of eight secondary schools in a large urban school district that participated in the Science and Technology Enhancement Program (STEP). The sample group for the study was 130 ninth grade students enrolled in Integrated Algebra I. The participants were chosen because of their teachers involvement in the STEP Project. The criteria for teachers participating in STEP were a willingness to provide support for the graduate fellows, to guide the teaching framework according to school and State standards, and to be positive, motivating, open, and flexible throughout the year. Due to the lack of randomization, this study was a quasi-experimental design that used a nonequivalent group design for comparison. All participants were between ages 14 and 16; the groups included male and female students. Since the students were under 18 years old, students were allowed to participate in the STEP Project only with their parents consent. At the beginning of the 25

35 academic year, the fellows of the STEP Project collected consent forms, which had been approved by the University of Cincinnati s Institutional Review Board. The school then assigned students to classes based on their grade point average (GPA). There were five different periods and the sizes of the groups were not the same. Teachers who had been involved in the STEP Project taught all classes of the treatment and control groups. The students of both the treatment and control groups had five 50-minute classes per week. the inquiry-based teaching activities provided by the researcher were used with the treatment classes. In the control group classes, the teacher used lecture-oriented teaching methods: Asking students to read the textbook aloud, explaining the content, writing notes on the board, inviting students to answer questions, and requesting the completion of worksheets. Compared to the control group, the students of the treatment group had lessons with three main parts: (1) The teacher s review of the previous lesson and introduction of the present lesson, (2) Inquiry-based teaching and group activities, and (3) The completion of a worksheet. Data Collection Instruments Attitude measures. The students attitudes toward mathematics were measured using scores from the Mathematics Attitude Questionnaire (Appendix A). The Mathematics Attitude Questionnaire was a one-page paper-and-pencil questionnaire with 28 items consisting of 14 positive and 14 negative statements. The construct was adapted from separate attitude surveys by a STEP Project evaluation team that was developed from an instrument designed by Fennema and Sherman (1977) for a mathematics survey. 26

36 The internal panel reviewed and finalized the items in order to validate that survey questions related to the research purpose. The scale ranged from highest score of 5, meaning strongly agree, to the lowest score of 1, meaning strongly disagree, with a neutral score of 3, meaning undecided. The values for negative questions were reversed. The researcher and the teacher administered the pre- and post-class surveys to the treatment group and the teacher conducted the pre- and post-class surveys in the control group. The same survey was given to students at the end of the fourth quarter. The reliability of the survey had the Cronbach s α =.98 for the pre-class survey, and.96 for the post-class survey. However, the researcher did not use six items (#2, #8, #9, #11, #22, and #25: see Appendix A) to analyze data, which were not relevant to the research questions. After conducting a factor analysis to measure the construct validity, two items (#10 and #19) were not included in the data analysis because the loadings criteria of the two items were less than.3 (Hair, Anderson, Tatham, & Black, 1998). The range of possible points for the pre- and post-surveys were from the maximum 100 to the minimum 20. After dropping two items, the first component accounted for 86.3% of the total variability for the pre-survey and 69.2 % for the post-survey. The validity and reliability analysis indicated that instruments were appropriate to interpret data. Mathematics Achievement Measures In order to measure mathematics achievement before and after the inquiry-based teaching, three quarter tests were used. The chapter test at the end of the second quarter was considered a pre-test. The chapter test at the end of third quarter was used as a middle test. In the same way, the chapter test of the fourth quarter was regarded as a post 27

37 test. The first and last chapter tests, a standardized test constructed by the Ohio Department of Education (ODE), were paper-pencil tests to evaluate the content knowledge of the participants. The test contained both basic knowledge and more conceptually oriented items. ODE developed these tests as part of a state initiative. The validity of the test was determined by the Content Advisory Committee to ensure that the content of each item relates to the outcome for that particular strand. The school district used these practice tests for ninth graders in preparation for the Ohio Graduation Test (OGT) for the each quarter. All tests developed by ODE were reported as highly reliable with coefficient above.90. Test questions consisted of those mathematics concepts from Ohio Mathematics Academic Content Standards (arithmetic, measurement, data analysis, geometry, and algebra). Meanwhile, the course instructor constructed a middle test that was reviewed by another mathematics teacher. All test items were constructed based on the school district standards and course objectives. The Cronbach alpha of the middle test was.89. Since three different tests were used, standardized test scores (z-scores) were used to analyze the difference in change over the three quarters, instead of raw scores. Procedure At the beginning of the third academic quarter of 2004, the researcher gave a brief explanation to students enrolled in the Integrated Algebra I courses, and then asked the students to have the consent form signed by their parents in order to participate in the study. Only data from students who returned the consent form with their parents signature were used in the study. The total participants were 128. At the beginning of the study, prior to any use of the inquiry-based teaching in the classroom, a survey was given to measure students attitudes toward mathematics. Before the teaching began, the 28

38 teacher checked the content to ensure that the lessons matched the students present skills and followed the guidelines of the district academic content standards for mathematics. After a brief introduction, students were presented with the instructional activities. During the small group activities, the teacher observed and evaluated students progress. After each class, the teacher collected and inspected the students activity books and then evaluated students activities following the rubric evaluation sheet, which was provided by the researcher. At the end of the quarter, a chapter test was given to students in order to assess how well the students learned the content. Data Analysis Besides descriptive statistics, hierarchical linear models (HLM) (Raudenbush & Byrk, 2002) were used to evaluate the effects of the inquiry-based teaching on the students mathematics achievement and attitudes towards mathematics. HLM is a statistical linear model that analyzes hierarchical data like individual level within a classroom or within a school. Because students are grouped within classrooms, the errors from the observations are not independent of one another, and therefore there will be underestimation of standard errors. Traditional multivariate analysis of variance cannot handle this nested data structure. HLM was used as the analytical method in this research to address the problem of lack of independence among observations and the problem of cross-level relationships. The following is the outline of the dependent variables, independent variables, and covariates that were used in the HLM analysis: 29

39 The analyses were performed with one-tailed hypothesis testing because these studies were meant to find whether inquiry-based teaching method improved students mathematics performance and attitude toward mathematics. The HLM analysis of data for this study is divided into two parts: mathematics achievement and attitude measures. In mathematics achievement, repeated observations of test scores in mathematics at level 1 was the dependant variable, while time (pre, middle, and post) was the independent variable. GPA, gender (male and female), and ethnicity (Caucasian, African American, Others) were covariates of level 2. The independent variable for level 3, i.e. the classroom level, was inquiry-based teaching versus control group. Average class GPA was a covariate. For academic achievement Level 1 (Repeated observations): Dependent Variable: Test score in mathematics Independent Variable: Time (pre, middle, and post) Level 2 (Individual level): Covariates: GPA Gender (Male, Female) Ethnicity (Caucasian, African American, Others) Level 3 (Classroom level): Independent variables: Group (Treatment vs. Control) Covariates: Class average GPA 30

40 In attitude measures, the dependent variables of level 1 as individual level were the attitude questionnaire scores from the post-survey. Covariates of the individual level were the attitude questionnaire scores from the pre-survey, GPA, gender (male, female), and ethnicity (Caucasian, African American, Others). The independent variable of level 2, i.e. classroom level, was the inquiry-based teaching versus the control method. Covariates of the level 2 were the average class GPA and class size. However, since there were valid data from only four students data from the others ethnicity category, 3.13 % among the available data, only the data of Caucasian and African American students were used to perform the HLM analysis. For attitude measures Level 1 (Individual level): Dependent Variables: Attitude questionnaire score from the post survey Covariates: Attitude questionnaire score from the pre-survey GPA Gender (male, female) Ethnicity (Caucasian, African American, Others) Level 2 (Classroom level): Independent variables: Group (Treatment vs. Control) Covariate: Class average GPA 31

41 Chapter 4 RESULTS The purpose of this study was to determine if inquiry-based teaching affected students mathematics achievement and their attitudes towards mathematics. Besides descriptive statistics, hierarchal linear models (HLM) were used to evaluate the effects of the inquiry-based teaching on the students mathematics achievement and their attitudes towards mathematics. Descriptive Statistics The treatment group was 45 male (62.5%), compared to 36 (64.3 %) in the control group. Thirty-five students (48.6%) in the treatment group were Caucasian, compared to 24 (42.9 %) of the control group (also see Table 1). To test the group equivalence on the demographic, as well as GPA, pre-test score and pre-survey score, an independentsample t test was conducted to determine whether there was a difference in means between the treatment and control groups, and a Chi-square test for categorical variables was used to evaluate group difference (see Table 1). All p values indicated that the difference between the treatment and control groups was not significant (p >.05) for any variables, except Mean GPA. The P value of Mean GPA (p <.05) implied that two distributions differ significantly from each other. It is needed to be controlled in the data analysis. The treatment groups selected were three of the five classes: the high, middle and low GPA classes. The highest GPA, which of group 3, was The lowest GPA, of group 5, was The students who had a mid-level GPA of 2.13 were assigned to group 32

42 2. Meanwhile, the control group consisted of two classes with the following GPAs: 1.98 for group 1 and 1.12 for group 4 (see Table 1.) Table 1. Descriptive Statistics Total Treatment Control Group Equivalence Test Variables (N = 128) Group (N =72) (N = 56) Statistics p Gender χ² (1)= Male 81 (63.3%) 45 (62.5%) 36 (64.3%) Female 47 (36.7%) 27 (37.5%) 20 (35.7%) Race χ²(1) = White 59 (46.1%) 35 (48.6%) 24 (42.9%) Black 64 (50.0%) 33 (45.8%) 31 (55.4%) Others 5 (3.9%) 4 (5.6%) 1 (1.9%) Mean GPA t (126)=.02 (SD = 1.12) (SD =.92) 2.33 Mean Pre t (122)= -.48 survey (SD = 18.12) (SD = 17.14).71 Mean Pre-test t (105) =.11 (SD = 1.10) (SD =.84)

43 Missing Data There were missing data in the achievement model. 4.6 % of the subjects did not take the pre-test, 3.7 % did not take the middle test, and 5.1% did not take the final. However, one of the benefits of using HLM is that it can handle the unbalanced repeated measures resulted from the missing data. In the case of the attitude surveys, 3.9 % of all participants did not take the pre-test and 7.0 % of subjects did not take the final. Because a group mean is the best estimate for the value on a given variable (Mertler & Vannatta, 2001), the mean values were filled in for missing data using the SPSS process. There were two outliers; after checking the raw data of the survey, two outliers were deleted because two subjects deliberately marked the same response from the beginning to the end of the test. Table 2 indicates the mean Z-scores for each test of both the treatment and control group. Both mean Z-scores of the treatment and control group were lower from the pretest to post-test. However, the mean Z-score of the treatment group was.032 lower from the pre test, while mean Z-score of the control group had a larger drop of.251 Table 2. Mean Z-score for Each Test of the Treatment & Control Groups Pre-Test Middle Test Post-Test Group N M SD N M SD N M SD Treatment Control

44 Effect size In order to measure the practical significance of the effect, effect sizes were calculated. Effect size can be defined in terms of the standardized difference in group means (Tymms et al., 1997). The actual effect size of treatment for the mathematics test scores and attitude measures was each.51 and.58, indicating moderate effect size. Mathematics Achievement In order to measure the impact of inquiry-based teaching on mathematics achievement, a 3-level HLM was used to analyze three different levels. At the first level, academic achievement was measured on the repeated observations for three different tests as pre, middle, and post-test. Individual level as the second level was exactly same procedure as the level one of the attitude analysis. The level three as classroom level was same method as the second level of the attitude study. The 3-level HLM model is specified as follows: Level-1 Model Y = π0 + π1* (TIME) + π2* (TIME) ² + E Level-2 Model π0 = β 00+ β 01*(GENDER) + β 02*(GPA) + R0 π1 = β 10 + β 11*(GENDER) + β 12*(GPA) π2 = β 20 + β 21*(GENDER) + β 22*(GPA) Level-3 Model 35

45 β 00 = γ γ 001(MEANGPA) + γ 002(TREATMEN) + U00 β 01 = γ γ 011(MEANGPA) + γ 012(TREATMEN) β 02 = γ γ 021(MEANGPA) + γ 022(TREATMEN) β 10 = γ γ 101(MEANGPA) + γ 102(TREATMEN) β 11 = γ γ 111(MEANGPA) + γ 112(TREATMEN) β 12 = γ γ 121(MEANGPA) + γ 122(TREATMEN) β 20 = γ γ 201(MEANGPA) + γ 202(TREATMEN) β 21 = γ γ 211(MEANGPA) + γ 212(TREATMEN) β 22 = γ 220+ γ 221(MEANGPA) + γ 222(TREATMEN) Where Y = the outcome variable, mathematics standardized test score. π0 = the initial status. π1 = the linear growth rate. TIME: the number of test (pre, middle, post) TIME ²: the second power of TIME E = the level-1 random errors. β s = the level-2 coefficients. R0 = the level-2 random errors. GENDER & GPA : self-explanatory. γ s = the level-3 coefficients. U00 = the level-3 random errors. MEANGPA: the mean score of GPA TREATMEN : the inquiry-based teaching class vs. control group 36

46 Note that some independent variables or covariates, for example, ethnicity, was not included in the HLM model above, because they were not significant predictors for the outcome in a preliminary analysis. In order to know whether HLM should be used, the intraclass correlation coefficient (ICC) was calculated using the unconditional model. The estimated variability between groups was 68% of the variance in mathematic test scores, indicating a significant amount of variation between groups. To write out the equations for the HLM model, at the first level, mathematics standardized test score as the outcome variables was expressed by the initial status (π0) and the linear growth rate( π1, π2) during a fixed unit of time. Y = π0+ π1* (TIME) + π2* (TIME) ² Where π0 = β 00+ β 01*(GENDER) + β 02*(GPA) π1 = β 10 + β 11*(GENDER) + β 12*(GPA) π2 = β 20 + β 21*(GENDER) + β 22*(GPA) Meanwhile, parameters of level-2 computed from level-3 β 00 = (MEANGPA) (TREATMEN) β 01 = (MEANGPA) +.36(TREATMEN) β 02 = (MEANGPA) (TREATMEN) β 10 = (MEANGPA) +.27(TREATMEN) β 11 = (MEANGPA) +.04(TREATMEN) β 12 = (MEANGPA) (TREATMEN) β 20 = (MEANGPA) +.81(TREATMEN) 37

47 β 21 = (MEANGPA) (TREATMEN) β 22 = (MEANGPA) (TREATMEN) The descriptive statistics of Level 1, 2, and 3 can be found in Table 3. The z-score, instead of direct comparison among raw test scores, was used to find the direction and degree of mathematics achievement scores after teaching because of using three different tests (pre, middle, and post). The minimum and maximum z-scores were and 1.79 respectively. The mean GPA score of all students was 1.86 out of a range of 0 to At level 3, MEANGPA, the group mean GPA (SD =.82), was MEANPRESURVEY, as the mean of pre-survey scores from five groups, varied from to (SD = 6.54). PERCENTAGEGENDER showed that the percentage of males in the entire student population was 64.80%. PERCENTAGEETHNICITY showed that the percentage of Black students was 47.98%. TREATMEN indicated that there were three treatment groups among the total of five groups. Table 3 explains the parameter estimates from HLM analysis and shows how students academic achievement was affected by the treatment. Overall, the effect of treatment was significant at α =.10 level (p <.08, ES =.51). Thus, the inquiry-based teaching did have an effect on mathematics achievement (Figure 1). Figure 1 shows how the z-scores of two groups of students varied over time. The z-scores of the treatment group students increased over time, while those of the control group actually decreased. 38

48 Table 3. Final estimation of fixed effects for HLM 3 Fixed Effect Coefficient SE df t p ES For INTRCPT1, π0 For INTRCPT2, β00 INTRCPT3, γ MEANGPA, γ TREATMEN, γ For Female, β01 INTRCPT3, γ MEANGPA, γ * TREATMEN, γ For GPA, β02 INTRCPT3, γ MEANGPA, γ ** TREATMEN, γ * For TIME slope, π1 For INTRCPT2, β10 INTRCPT3, γ MEANGPA, γ * TREATMEN, γ For Female, β11 INTRCPT3, γ MEANGPA, γ TREATMEN, γ

49 For GPA, β12 INTRCPT3, γ * MEANGPA, γ TREATMEN, γ For TIME ² slope, π2 For INTRCPT2, β20 INTRCPT3, γ MEANGPA, γ TREATMEN, γ ** For GENDER, β21 INTRCPT3, γ MEANGPA, γ TREATMEN, γ ** For GPA, β22 INTRCPT3, γ MEANGPA, γ TREATMEN, γ * Significant at p <.05 (One-tailed). ** Significant at p <.01 (One-tailed). Significant at p <.10 (One-tailed). Note that in the HLM model there is a quadratic term is a better fit to the data. The quadratic effect indicated by the square term (Time²), but the interpretation of the quadratic effects is beyond the scope of the present study. 40

50 Figure 1. Group Differences from the Pre-Test to the Post-Test Control Treatment Mean Z-Score Time Importance of Gender The results for the effects of the inquiry-based teaching on students of both genders are presented in Figure 2. As listed in table 4, female students had better mean GPA scores than males at the beginning, (p <.09, ES = 1.06). Females in the lower GPA class achieved better test scores than others, (p <.05, ES =.73). Overall, there is no gender difference in inquiry-based teaching (p =.42, ES =.07). The z-scores of both the females and the males in the treatment group increased over the time period while simultaneously, the z-scores of the males and the females in the control group decreased. (See Figure 2). 41

51 Figure 2. Gender Difference BOYS, CONTROL BOYS, TREATMEN GIRLS, CONTROL GIRLS, TREATMEN ZTESTR TIME0 Effect on GPA Figure 3 displays how inquiry-based teaching affected groups with different mean GPAs. The results indicated that there was some observed effect of the treatment group for lower GPA students at the later stage although it was not statistically significant (p =.22, ES =.14) However, students who had lower GPAs in the treatment received better test scores than others (p <.01, ES =.63). The result also showed that the test scores of students in the lower GPA class increased significantly (p <.02, ES =.55). 42

52 Figure 3. High & Low GPA Group Difference LOWER-GPA, CONTROL LOWER-GPA, TREATMEN HIGHER-GPA, CONTROL HIGHER-GPA, TREATMEN ZTESTR TIME0 Attitude Measures Two levels were considered when measuring attitude towards mathematics: level 1 refers to the individual level and level 2 refers to the classroom/group level. The dependent variable for the individual level was the attitude questionnaire score from the post-survey. The other covariates were attitude questionnaire scores from the pre-survey as well as gender and ethnicity. At Level 2, in which the student population was broken up into the five GPA-grouped classes. Inquiry-based teaching versus the control method was the independent variable. A covariate was class average GPA. In order to know whether HLM should be used, the ICC was calculated using the unconditional model. The estimated variability in group was 39.1 % of the variance in survey scores, indicating a significant amount of variation between groups. The HLM model is specified as follows: 43

53 Level-1 Model Y = β0 + β 1*(BLACK) + β 2* (PRESURVEY) + R Level-2 Model β0 = γ00 + γ 01*(MEANPRESURVEY) + γ 02*(TREATMEN) + U0 β 1 = γ 10 + γ 11*( MEANPRESURVEY) + γ 12*(TREATMEN) β 2= γ 20 + γ 21*( MEANPRESURVEY) + γ 22*(TREATMEN) Where Y = the outcome variable, POSTSURVEY. β0 = the initial status. β s = the level-1 intercept coefficients. BLACK: African American students R = the level-1 random error. γ s = the level-2 coefficients. U0 = the level-2 random errors. PRESURVEY: the score of pre-survey. MEANPRESURVEY: the mean score of pre-survey. TREATMEN : the inquiry-based teaching class vs control class. Note that some independent variables or covariates, for example, GPA, were not included in the HLM model above, because they were not significant predictors for the outcome in a preliminary analysis. To write out the equation for the HLM model, at the first level, mathematics attitude post-survey scores as the outcome variables was expressed by the initial status (β0) and the linear growth rate(β 1, β 2). 44

54 Y = β0+ β 1 * (TIME) + β 2* (TIME) ² β0 = *(MEANPRESURVEY) *(TREATMEN) β 1 = *( MEANPRESURVEY) *(TREATMEN) β 2= *( MEANPRESURVEY) *(TREATMEN) The outcome variable in the HLM analyses was attitude post-survey scores. Mean scores of the pre-survey of attitude were (SD = 16.84), (SD = 20.74), (SD = 18.90), (SD = 24.20) and (SD = 8.27) for each class group (see Table 4). The average score on the pre-survey for the treatment group (groups 2, 3, and 5) was 41.96, compared to for the control group (groups 1 and 4). Also, mean scores on the post survey were (SD = 16.86), (SD = 16.07), (SD = 13.11), (SD = 17.99), and (SD = 6.33) for groups 1 through 5 respectively. The average score on the post-survey for the treatment group was 53.73, compared to for the control group. Table 4. Mean Scores on the Pre- & Post-Surveys Measuring Attitude Size MEAN Postsurvey Group (N) GPA PRESURVEY SD SD Treatment/control

55 Figure 4 illustrates the group differences in the pre- and post-surveys. In the figure, the mean survey scores of the treatment group increased from to 53.73, while the scores of the control group decreased from to Figure 4. Group Difference from Pre-Survey to Post-Survey 60 Group Control Treatment 50 Survey Test Pre Time Post Table 5 shows the parameter estimates from HLM analysis for the attitudes towards mathematics after the treatment at the individual level and at the classroom level. There was a significant effect of treatment at α=.10 level (p <.07, ES = 3.07) at the individual level. The average score of Black students lowered on the post survey (p <.01, ES = 3.99). The treatment also affected Black students attitudes toward mathematics overall at p <.08 (ES =.58). Especially, Black students who expressed highly positive attitudes toward mathematics on the attitude pre-survey scored significantly higher (p < 46

56 .01, ES =.08) on the post-attitude survey than those who had low attitudes toward mathematics on the pre-survey. The data also suggest that students who had lower presurvey scores in the treatment group significantly improved their post survey score (p <.02, ES =.02). Table 5. Final Estimation of Fixed Effects for HLM 2 Fixed Effect Coefficient SE df t p ES For INTRCPT1, β 0 INTRCPT2, γ MEANPRESURVEY, γ TREATMEN, γ For BLACK slope, β 1 INTRCPT2, γ ** MEANPRESURVEY, γ * TREATMEN, γ For PRESURVEY slope, β 2 INTRCPT2, γ MEANPRESURVEY, γ ** TREATMEN, γ * * Significance level at p <.05 (One-tailed). ** Significance level at p <.01 (One-tailed). Significant at p <.10 (One-tailed). 47

57 Chapter 5 DISCUSSION Project STEP, the Science and Technology Enhancement Program, proposed using modules of inquiry-based teaching instead of the traditional methods of delivering formal lectures, using practice exercises, and emphasizing procedures to improve students learning in the secondary mathematics classroom. The purpose of this study was to determine the impact of the STEP lessons on students achievement in and attitudes toward mathematics. In addition, the results of the STEP Project were examined to find evidence of difference between the performance of male and female students, of student groups with varying GPAs, and students from different ethnic backgrounds. Achievement test scores and a survey for measuring students attitudes toward mathematics were obtained from both groups of students. Research question 1 focused on improvement in mathematics achievement as indicated by test scores after students were taught using the inquiry-based methods. The effects on the academic achievement between the treatment and control groups were observed, indicating a possible positive effect of inquiry-based teaching. The result showed that the test score of students in the lower GPA class of the treatment group increased significantly. Particularly, female students who had lower GPAs in the treatment received better test scores than others, even though there were not statistically significant effects on any specific GPA groups nor on students of different genders. Interestingly, there was some observed effect of the treatment group for lower GPA students at the later stage although it was not statistically significant. This result implied 48

58 that longer inquiry-based teaching may generate more positive impact on students mathematics achievement. Urban students, who normally have two or more years below grade level, need to have extra help in addition to the classroom teaching (Balfanz, Ruby, & Mac Iver, 2002). The major disadvantage of class lectures is that students are forced to keep up with their teachers. Careless students make it hard for educators to teach, and make it difficult for other students to concentrate and listen. The vast majority of students who are labeled as slow learners or at-risk students are sitting in classrooms with overwhelming feelings of stress and frustration because they do not understand what is being taught. (Tsseldyke, Spicuzza, Kosciolek, Teelucksingh, Boys, & Lemkuil, 2003) If students do not understand the lecture, it will be very hard for them to benefit from the education process, no matter how well the instruction is planned or implemented. They are then easily bored or under-prepared, which makes them very difficult to teach. This study showed evidence that inquiry-based teaching can be a vital option to improve students academic achievement in mathematics in the form of constructivism. Research question 2 focused on improvement in the students attitudes toward mathematics after inquiry-based teaching. The finding of this research showed that inquiry-based teaching improved students attitude toward mathematics. The effect of treatment was significant at the individual level. One of the important observations was that students who had lower pre-survey scores in the treatment group significantly improved their post survey score, indicating noticeable improvement of students attitudes toward mathematics. This finding can be significant to educators who seek a proper teaching method to improve students attitude toward mathematics. Indeed, this 49

59 result implied that researcher-developed inquiry-based teaching activities were effective teaching tools to improve students attitude toward mathematics. The outcomes of this study support that inquiry-based teaching could be an effective teaching method with appropriate activities or teaching modules. There was also improvement in the attitudes of black students in the treatment group after inquiry-based teaching, compared to those in the control group. Specifically, black students who had positive attitudes toward mathematics on the attitude pre-survey scored significantly higher on the post-survey than those who had less positive attitudes toward mathematics on the attitude pre-survey. This revealed inquiry-based teaching, making connections to real-life situation and using more technology oriented activities, made learning mathematics more interesting to Black students in urban areas. However, there were no significantly different impacts on students of different genders. This result mirrored the national trend that there is no gender gap anymore (NAEP, 2005). Without any doubt, inquiry-based teaching provided not only an important element which students play an active role in their own education, but also increased students attitude toward mathematics. Summary of Findings Research Question 1 Research question 1 focused on improvement in mathematics achievement as indicated by test scores after students were taught using the inquiry-based methods. The effects on the academic achievement between the treatment and control groups were significant at p <.08 (ES =.51), indicating a possible positive effect of inquiry-based teaching. The treatment did not affect any specific GPA groups at p <.22 nor did the 50

60 inquiry-based teaching methods have a significantly different impact on students of different genders (P <.42, ES =.07). The results indicated that there was some observed effect of the treatment group for lower GPA students at the later stage although it was not statistically significant (p =.22, ES =.14) However, students who had lower GPAs in the treatment received better test scores than others (p <.01, ES =.63). The result also showed that the test scores of students in the lower GPA class increased significantly (p <.02, ES =.55). Female students had better mean GPA scores than males at the beginning, (p <.09, ES = 1.06). Females in the lower GPA class achieved better test scores than others, (p <.05, ES =.73). Overall, there is no gender difference in inquiry-based teaching (p =.42, ES =.07). As mentioned earlier, ethnicity, was not included in the HLM model above, because they were not significant predictors for the outcome in a preliminary analysis. Research Question 2 Research question 2 focused on improvement in the students attitudes toward mathematics after inquiry-based teaching. The effect of treatment was significant at p <.07 (ES = 3.07), at the individual level. There was also improvement at p <.08 (ES =.58) in the attitudes of Black students in the treatment group after inquiry-based teaching, compared to those in the control group. Black students who had positive attitudes toward mathematics on the attitude pre-survey scored significantly higher (p <.01, ES =.08) on the post-survey than those who had less positive attitudes toward mathematics on the attitude pre-survey. White students attitudes were not affected by the treatment. 51

61 The data also suggest that students who had lower pre-survey scores in the treatment group significantly improved their post survey score (p <.02, ES =.02). There were not statistically significant effects on any specific GPA groups nor on students of different genders. Thus, GPA and genders were not included in the HLM model above, because they were not significant predictors for the outcome in a preliminary analysis. Limitations In the research, there were limitations. First, the measurements of students achievement were limited to one topic in the second semester. Examination of multiple topics studied over longer periods may result in different outcomes. A second limitation was the number of variables between treatment group and the control group that could affect the outcomes. A third limitation was the measurement of learning correctly. The tests used in this study, typical written mathematics questions, were developed by the school district. However, students can learn mathematics many different ways. Indeed, rubric evaluations to measure students learning in the inquiry-based teaching could not be included in both the treatment and control group students. The fourth limiting factor was students consistent attendance in their classes. One of the most significant problems in the school and districtwide is student absences, especially in the early morning class. Absences from several mathematics classes could severely affect students learning in and attitudes toward mathematics. The student absence rate and its effect on academic achievement were unknown and beyond the scope of the research. Finally, there is the teachers participation factor: Since all teachers would not participate equally as voluntarily, 52

62 the results from this study would apply only to teachers who had credentials comparable to those selected for the STEP project. Relationship to Other Research The results of this study supported those of Silver and Stein (1996), who investigated the theory that a new teaching method produced a positive impact on students math learning. Silver and Stein noticed that QUASAR provided the high levels of conceptual and meaningful understanding that students required. The findings of Kramarski, Mevarech, and Arami (2002), who concluded that metacognitive instruction positively affected research outcomes, also were similar to the results of this research. The current study indicates that inquiry-based teaching affected students learning p < 0.77 and their attitudes towards mathematics (p <.07, ES = 3.07), at the individual level. The results of this study contrasted with those of Keepers (1995), Soeder (2001), and Moch (2002), who found no significant improvement in academic achievement using technology-based instruction in the mathematics class. Also, this research found inquiry-based teaching positively affected the attitude of African American students, especially those who had pre-existing positive attitudes toward mathematics (p <.02, ES =.02). Inquiry-based teaching may have affected the attitude towards mathematics of all Black students at p <.08. The findings of Veenman (1984) and Zahorik (1996) were similar to the outcomes of this study: modifications of the curriculum produced a slight improvement in students attitude towards mathematics. However, the research indicated that simply providing students with real-life examples to illustrate abstract ideas enhanced students interaction and enabled them to better understand mathematic ideas without lowering achievement. This complemented the 53

63 findings of Portal and Sampson (2001) who suggested that a way to improve mathematics teaching was to implement a new curriculum connecting life experiences to mathematics instruction. It is widely believed (Martin et al.,1997; NCTM, 1989, 1991, 2000) that with a proper lesson plan and activity, students can enjoy the mathematics class and at the same time improve their mathematics skills and their attitude toward mathematics. The result of this research supported the suggestions of Portal and Sampson. There was clear evidence of improvement on standardized student achievement exams and attitudes of certain groups in this study. Previous studies indicated that inquiry-based teaching might be a good teaching strategy to improve the attitude of Black students who had a poor attitude toward mathematics (Walker & McCoy, 1997; Tapia & Marsh, 2000; Stull, 2002). In short, inquiry-based teaching may improve students mathematics scores overall and the attitude of Black students who had pre-existing positive attitudes toward mathematics. Implications of Future Research The future study of inquiry-based teaching or other instructional strategies in mathematics could include a variety teaching methodologies which should benefit any group of learners. Based on the findings in this study regarding the improvement of student attitudes toward mathematics, more study is needed to see how educators could improve the attitude of some groups of students who have high GPAs but negative attitudes toward mathematics. There is more to learn in the field of mathematics than skillful computation: Students need to understand basic concepts and fundamental structures of mathematics that underpin the rules and procedures of simple arithmetic. To understand the structures of mathematics, students need to learn how to relate their 54

64 conceptual and procedural knowledge to their world, and how to grasp both the interrelations among concepts and operations for higher-level study. An understanding of mathematical structures underlying the procedures and concepts should be fundamental to meaningful learning. Therefore, there is a continual need to study what factors prevent students learning. In other words, investigating to know where students stand is one of the most important steps in using any teaching method. More research should be conducted on a larger scale, over longer periods, of as many mathematics topics as possible. Conclusion The purpose of this study was to determine the effects of inquiry-based teaching on students mathematics achievement and their attitudes toward mathematics. The results indicated that inquiry-based teaching may have a positive affect on attitudes toward mathematics and students achievement. The study indicated that there was a possibility of improving the achievement of students in urban schools. By changing students attitude towards mathematics, there may be increased numbers of students who choose to pursue science-related degrees, which was one of the goals of the STEP Project. This research found that inquiry-based teaching may be an effective teaching method to improve students academic achievement. With proper instructional activities, inquirybased teaching might be a valuable tool not only to enhance students achievement, but also to improve their attitudes toward mathematics. The findings in the study also indicated that the inquiry-based teaching might be a suitable teaching tool to students in an urban school. 55

65 References Abramovich, S. (1995). Technology-motivated teaching of advanced topics in discrete mathematics. Journal of Computers in Mathematics and Science Teaching 14(3), Abramovich, S., & Nabors, W. (1998). Enactive approach to word problems in a computer environment enhances mathematical learning for teachers. Journal of Computers in Mathematics and Science Teaching 17(2/3), Balfanz, R., Ruby, A., & Mac Iver, D. (2002). Essential components and next steps for comprehensive whole-school reform in high poverty middle schools. In S. Stringfield & D. Land (Eds.), Educating at-risk students (pp ). Chicago: University of Chicago Press. Bangert-Drowns, R. L. (1985). Meta-analysis of findings on computer-based education with precollege students. Chicago: American Educational Research Association. (ERIC Document Reproduction Service No. ED263905) Bangert-Drowns, R. L., Kulik, J. A., & Kulik, C. C. (1985). Effectiveness of computer-based instruction in secondary schools. Journal of Computer-based Instruction, 12(3), Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics, 29(1),

66 Bruner, J. S. (1960). The process of education. Cambridge, MA: Harvard University Press. Clements. D., & Sarama, J. (1997). Computers support algebraic thinking. Teaching Children Mathematics, 3 (6), Duffy, T., & Cunningham, D. (1996). Constructivism: Implication for the design and delivery of instruction. In D. Jonassen (Ed.), Handbook of research for educational communications and technology (pp ). New York: Macmillan. Dugdale, S. (1998). A spreadsheet investigation of sequence and series for middle grades through precalculus. Journal of Computers in Mathematics and Science Teaching 17(2/3), Eggen, P., & Kauchak, D. (2001). Educational psychology: Windows on classrooms (5 th ed.). Upper Saddle River, NJ: Prentice Hall. Fennema, E. & Sherman, J. (1977). Sex-related differences in mathematics achievement, spatial visualization, and affective factors. American Education Research Journal, 14, Funkhouser, C., & & Dennis, J. R. (1992). The effects of problem-solving software on problem-solving ability. Journal of Research on Computing in Education, 24(3), Gersten, R., & Keating, T (1987). Long-term benefits from direct instruction. Educational Leadership, 44(6),

67 Gersten, R., Keating, T., & Becker, W. (1988). The continued impact of the direct instruction model: Longitudinal studies of follow through students. Education and Treatment of Children, 11(4), Gersten, R., Taylor, R., & Graves, A. (1999). Direct instruction and diversity. Upper Saddle River, NJ: Merrill/Prentice Hall. Grabinger, R. (1996). Rich environment for active learning. In D. Jonassen (Ed.), Handbook of research for educational communications and technology (pp ). New York: Macmillan. Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1998). Multivariate data analysis (5th ed.). Upper Saddle River, NJ: Merrill/Prentice Hall. Hannula, M. S. (2002). Attitude towards mathematics: Emotions, expectations and values. Educational Studies in Mathematics, 49, Heid, K. (1997). The technological revolution and the reform of school mathematics. American Journal of Education, 106(1), Hembree, R. & Dessart, D. J. (1986). Effects of hand-held calculators in precollege mathematics education: A meta-analysis. Journal for Research in Mathematics Education, 17(2), Hoff, D. J. (2003). Large-scale study finds poor math, science instruction. Education Week, 23(1), 8. Hoffman, K. P. (1989). The science of patterns: A practical philosophy of mathematics education. Paper presented to the Special Interest Group for 58

68 Research in Mathematics Education at the annual meeting of the American Educational Research Association, San Francisco, CA. Keepers, J. K. (1995). The effect of a CAS on the development of student understanding of the function concept in a precalculus course. Dissertation Abstracts International, 56, Klein, D. (2003). A brief history of American K-12 mathematics education in the 20 th century. Retrieved October 23, 2004 from Krajcik, J., Blumenfeld, P., Mark, R., & Soloway, E. (1994). A collaborative model for helping middle grade science teachers learn project-based instruction. Elementary School Journal, 94(5), Kramarski, B., Mevarech, Z., Arami, M. (2002). The effect of metacognitive instruction on solving mathematical authentic tasks. Educational Studies in Mathematics, 49, Li, Q. (2003). Would we teach without technology? A professor s experience of teaching mathematics education incorporating the internet. Educational Research, 45(1), Linn, M. C., Kessel, C., & Slotta, J. D. (2000, Oct. 31). Teaching and learning high school mathematics through inquiry: program reviews and recommendations. Retrieved Month Day, Year, from 59

69 Martin, R., Sexton, C., Wagner, K., & Gerlovich, J. (1997). Teaching science for all children. Boston: Allyn and Bacon. McCaffrey, D. F., Hamilton, L. S., Stecher, B. M., Klein, S. P. et al. (2001). Interactions among instructional practices, curriculum, and student achievement: The case of standards-based high school mathematics. Journal for Research in Mathematics Education, 32(5), McCoy, L. P. (1996). Computer-based mathematics learning. Journal of Research on Computing in Education, 28(4), McNair, R. E. (2000). Life outside the mathematics classroom: Implications for mathematics teaching reform. Urban Education, 34(5), Mertler, C. A. & Vannatta, R. A. (2001). Advanced and multivariate statistical methods. Los Angeles: Pyrczak Publishing. Nichols, J. D. & Miller, R. B. (1994). Cooperative learning and student motivation. Contemporary Educational Psychology, 19, Moch, P. (2002). Using technology with high school algebra students to enhance attitudes and academic performance. Unpublished doctoral dissertation, University of Central Florida. National Assessment of Educational Progress. (2005). The nation s report card. Retrieved September 1, 2005, from National Center for Education Statistics. Third international mathematics and science study. (1999). Retrieved November 8, 2001 from 60

70 National Center for Education Statistics. Third international mathematics and science study. (2003). Retrieved January 27, 2006 from National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (1991). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Retrieved September 20, 2005, Year, from National Research Council (1989). National science education standards. Washington, DC: National Academy Press. National Research Council (1999). National science education standards. Washington, DC: National Academy Press. Nichols, J. D. & Miller, R. B. (1994). Cooperative learning and student motivation. Contemporary Educational Psychology, 19, Niess, M. L. (1998). Using computer spreadsheets to solve equations. Learning and Leading with Technology, 26, 22-24, No Child Left Behind Act (2001): A Description of State Responsibilities published by the Council of Chief State School Officers. Retrieved August 8, 2005, from 61

71 Novak, J., & Gowin, B. (1984). Learning how to learn. New York: Cambridge University Press. Papanastasiou, C. (2000). Effects of attitudes and beliefs on mathematics achievement. Studies in Educational Evaluation, 26, Piaget, J. (1970). Piaget s theory. In P. Mussen (Ed.), Carmichael s manual of child psychology: Vol. 1. New York:Wiley. Portal, J., & Sampson, L. (2001). Improving high school students mathematics achievement through the use of motivational strategies. Chicago, Saint Xavier University. (ERIC Document Reproduction Service No. ED460854) President s Committee of Advisors on Science and Technology. (1997). Statement of principles. Retrieved June 9, 2003, from Raudenbush, S. W. & Bryk, A. (2002). Hierarchical linear models: Applications and data analysis methods (2 nd ed.). Newbury Park, CA: Sage Publications. Rubenstein, R. N., Craine, T. V., & Butts, T. R. (2002). Integrated Mathematics I. Evanston, Illinois: McDougal, Littell. Schmidt, W. H., McKnight, C. C., Cogan, L. C., Jakwerth, P. M., & Houang, R. T. (1999). Facing the consequences: Using TIMSS for a closer look at U.S. mathematics and science education. Boston: Kluwer. 62

72 Silver, E. A., & Stein, M. K. (1996). The QUASAR project: The revolution of the possible in mathematics instructional reform in urban middle schools. Urban Education, 30, Slavin, R., Madden, N., Dolan, L., & Wasik, B. (1994). Roots and wings: Inspiring academic excellence. Educational Leadership, 52, Soeder, K. (2001). The effect of computer-aided instruction on mathematics achievement. Dissertation Abstracts International, 62 (04), (UMI No ) Stigler, J. W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world s best teachers for improving education in the classroom. New York: Free Press. Stigler, J. W., Gonzales, P. Kawanaka, T., Knoll, S., & Serrano, A. (1995). The TIMSS 1995 Videotape Classroom Study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States. NCES ( ), U.S. Department of Education. Washington, DC: National Center for Education Statistics. Tapia, M., & Marsh, G. E., II. (2000). Effect of gender, achievement in mathematics, and ethnicity on attitudes toward mathematics. Paper presented at the Annual Meeting of the Mid-South Educational Research Association. Bowling Green, KY. (ERIC Document Reproduction Service No. ED ) 63

73 Tiwari, T. K. (1999). Integrating computer algebra systems as an instructional aid in an introductory differential calculus course (Doctoral dissertation, Mississippi State University, 1999). Dissertation Abstracts International, 60, Tsseldyke, J., Spicuzza, R., Kosciolek, S., Teelucksingh, E., Boys, C., & Lemkuil, A. (2003). Using a curriculum-based instructional management system to enhance math achievement in urban schools. Journal of Education for Students Placed at Risk, 8(2), Tymms, P., Merrell, C. & Henderson, B. (1997). The first year at school: A quantitative investigation of the attainment and progress of pupils. Educational Research and Evaluation, 3 (2), United States Presidential Commission report: A nation at risk. Retrieved December 2, 2001 from Veenman, S. (1984). Perceived problems of beginning teachers. Review of Educational Research, 54(2), Verderber, N. L. (1992). Tin cans revisited. Mathematics Teacher, 85, Vygotsky, L. (1978). Mind in society (M. Cole, V. John-Steiner, S. Scribner & E. Souberman, Eds.). Cambridge, MA: Harvard University Press. Walker, E. N. & McCoy, L. P. (1997). Students voices: African Americans and mathematics. In J. Trentacosta & M. Kenney (Eds.), Multicultural and gender equity in the mathematics classroom: The gift of diversity (pp ). Reston, VA: National Council of Teachers of Mathematics. 64

74 Weiss, I.R., Pasley, J. D., Smith, P. S., Banilower, E. R., Heck, D. J. (2003). Looking inside the classroom: A study of K-12 mathematics and science education in the United States. Retrieved September 21, 2005, from Zahorik, J. (1996). Elementary and secondary teachers reports of how they make learning interesting. The Elementary School Journal, 96(5),

75 APPENDIX A Mathematics Attitude Survey Description of Attitude Survey At the beginning of the third quarter of 2004, the researcher gave a brief explanation to students enrolled in the Integrated Algebra I courses, and then asked the students to have the consent form signed by their parents in order to participate in the study. Only data from students who returned the consent form with their parents signature was used in the study. At the beginning of the study, prior to any use of the inquiry-based teaching in the classroom, a survey was given to measure students attitudes toward mathematics. The length of survey time varied from 30 to 45 minutes. The Mathematics Attitude Questionnaire-Students was a one-page paper-and-pencil questionnaire with 28 items consisting of 14 positive and 14 negative questions. The construct was adapted from separate attitude surveys by a STEP Project evaluation team which was developed from an instrument designed by Fennema and Sherman (1977) for the mathematics survey. The scale ranged from highest score of 5, meaning strongly agree, to the lowest score of 1, meaning strongly disagree, with a neutral score of 3, meaning undecided. The same survey was given to students at the end of the fourth quarter. 66

76 67

77 APPENDIX B A Sample of Mathematics Test 68

78 69

79 70

80 71

81 72

82 APPENDIX C Samples of Inquiry-based Teaching Module Title: Let s Make a Cozy House Objectives: Students will be able to 1. Explain how heat is transferred from one area to another. 2. Use technology to predict outcomes. 3. Investigate the effectiveness of the insulator. 4. Apply their knowledge to real world problems. Description: First, in this lesson, students will investigate which cup is most efficient in keeping heat out and cold in. Second, students will determine which kind of insulator is the most effective for a house. Third, using Microsoft Excel, students will make a graph representing three data groups and then put them into a regression line and equation. Next, students will investigate how to decrease energy cost in their homes, find sources of escaping heat in their homes, and make a list of insulation and weatherization tips. Finally, students will calculate which insulator is a better buy according to its price and efficiency. Grade Level: 9 Subject Area: Earth Science Duration: 200 minutes (50 min x 4) Standard: Physical Science 73

83 F. Explain how energy may change form or be redistributed though the total quantity of energy is conserved. 17. Demonstrate that thermal energy can be transferred by conduction, convection or radiation (e.g., through materials by the collision of particles, moving air masses or across empty space by forms of electromagnetic radiation). Scientific Inquiry A. Participate in and apply the processes of scientific investigation to create models and to design, conduct, evaluate and communicate the results of these investigations. Scientific Ways of Knowing A. Explain that scientific knowledge must be based on evidence, be predictive, logical, subject to modification and limited to the natural world. B. Explain how scientific inquiry is guided by knowledge, observations, ideas and questions. Mathematics Patterns, Functions and Algebra Standard 2. Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations. 3. Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations. 6. Write and use equivalent forms of equations and inequalities in problem situations; e.g., changing a linear equation to the slope-intercept form. 74

84 8. Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point. 15. Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs. Data Analysis and Probability Standard 2. Create a scatterplot for a set of bivariate data, sketch the line of best fit, and interpret the slope of the line of best fit. Learning Activities Activity I. (50 min) a. Procedure: 1. Divide the class into groups of around four students. Each group fills a plastic cup, a paper cup, and a Styrofoam cup with ice. The respective groups will measure and record the temperature inside each cup every five minutes. Then they will decide which cup is the most efficient at keeping the ice cool. They will develop a conclusion on which cup would be the most useful for containing a cold drink on a hot summer day and why. Activity II (50 min) a. Procedure: 75

85 1. Each student will use Microsoft Excel to make the XY (Scatter) graph for the plastic cup, for the paper cup, and for the Styrofoam cup. 2. Next, students will add a regression line. By adding a regression line, students are able to compare the efficiency in keeping the ice cool. 3. Lastly, the students will find the equation of the line, so that they are able to predict the temperature beyond what was already recorded. Activity III. (50 min.) a. Procedure: 1. Divide the class into groups of approximately four students. Each group will have one cardboard box, a thermometer, an insulating material (newspaper, cardboard box, cotton, form sealer, poly panel, and Aluminum Foil), glue, tape, and a hair drier as a heat source. 2. Each group will start building a simulated house with the box and an insulating material; they must make sure that it has a window with a flap on one of the side walls to observe the temperature changes. Using an insulator, tape and glue, cover the whole house except the window. 3. Using the hair dryer, blow hot air into the house through the window for fifteen seconds. Put in the thermometer, seal the window, wait one minute and record the temperature. 4. After fifteen minutes, check the thermometers again and record the temperature. The whole class will compare their data all together. 76

86 Activity IV: Real-world Applications (50 min.) a. Saving on energy costs A typical American family s home utility bill is about $1,300 a year. Unfortunately, much of that energy is wasted. Energy can be wasted through poorly insulated windows, doors, walls and attics. Improvements in energy efficiency can make your house more comfortable and save a lot of money in the long run. Checking your home s insulation system is a fast, cheap way to reduce energy waste and save money. A good insulation system provides your home with thermal performance, protects it against air infiltration and controls moisture. You can increase the comfort of your home while decreasing your heating and cooling bills by up to 30% from spending just several hundred dollars in proper insulation products. b. Insulation According to the U.S. Department of Energy or DOE ( about a third of the air flows through openings in ceilings, walls, and floors. To decrease this flow, check the insulation in your attic, ceilings, walls, floors and other spaces to see if it meets the recommended level for your area. The measurement for insulation is known as the R-value. The higher the R- value, the better the walls will resist the transfer of heat. The recommendation from the DOE depends on the different costs of heating and cooling and climate conditions around the nation. c. Insulation Tips Students will develop a list of insulation tips derived from any source (books, periodicals, websites, etc.). 77

87 d. Sources of Air Leaks in Your Home Students will list five sources of air leaks in their houses. e. Weatherization Students will define the term weatherization and find tips from any source (books, periodicals, websites, etc.) about how to protect their house from it. f. Challenge: A family s monthly energy bill is $100 per month. The family decides to replace their original insulator with a new insulator A which saves them 20% of their whole energy bill. However, insulator A costs $ What will the yearly energy bill be after the new insulator is installed? 2. How long will it take them to recover the money they spent on the insulator? 3. Insulator B is very cheap only $400. However, it ll only save them 5% off their whole energy bill. Which one would be a better buy if they only stay three years at the same house? Materials Needed: For Activity I: Plastic cup, paper cup, Styrofoam cup, thermometer, 100 watt bulb (heat source), Activity I sheet. For Activity II: Microsoft Excel, Activity II sheet. For Activity III: 12`` x 12`` x 12`` cardboard box, thermometer, utility knife, tape, glue, four or five types of insulator, Activity III sheet. 78

88 For Activity IV: Internet, periodicals, books, Activity IV sheet. Evaluation: Each activity will be worth fifty points, adding up to a total of two hundred points. Rubrics will be used. In addition, a post-test will be given to the students that is worth one hundred points. Safety: Only use the activity materials as directed by the instructor and by the written directions. For example, do NOT eat the ice, touch the hot bulb, play with the knife or with any insulator. Follow all directions carefully. 79

89 80