Browsing by Author "Wilkins, Jesse L. M."
Now showing 1 - 20 of 38
Results Per Page
Sort Options
- Classroom Assessment in Malawi: Teachers' Perceptions and Practices in MathematicsSusuwele-Banda, William John (Virginia Tech, 2005-01-30)This study investigated teachers' perceptions of classroom assessment in mathematics and their current classroom assessments practices. Specifically, the study sought to gain an understanding of the extent to which teachers use different classroom assessment methods and tools to understand and to support both the learning and teaching processes. The following three questions guided the study: 1) How do primary school teachers perceive classroom assessment in mathematics? 2) What kinds of assessment methods and tools do teachers use to assess their students in mathematics? 3) What is the influence of teachers' perceptions of classroom assessment on their classroom assessment practices? The study used a questionnaire to establish the teachers' perceptions of classroom assessment in mathematics, a lesson observation protocol, and pre-lesson and post-lesson observation interview protocols as main sources of data collection. The data collected through observations and interviews helped to map out patterns between perceptions of classroom assessment and the teachers' classroom assessment practices. Document analysis was used to triangulate the information collected through observations and interviews. In addition, document analysis provided first hand information on the kind of written feedback students get and the nature of activities they do. A total of six teachers (three male and three female) were drawn from two primary schools in Malawi. The data suggest that teachers perceive classroom assessment as tests that teachers give to their students at specified time intervals. What teachers said about their teaching was not reflected during their teaching. Since teachers perceived classroom assessment as tests, they showed limited ability to use different methods and tools to assess their students while teaching. The teachers' perceptions of classroom assessment have influence on their classroom assessment practices. Five of the six teachers perceived assessment as testing, and classroom assessment practices were not clearly embedded in their teaching. Teacher experience and teacher education program did not seem to contribute much to teachers' perceptions of classroom assessment; however, teacher's academic qualification seemed to influence teachers' flexibility to accept new ideas.
- Development of an Instrument to Evidence Knowledge Abstractions in Technological/Engineering Design-Based ActivitiesFigliano, Fred Joseph (Virginia Tech, 2011-05-02)This document outlines the development of a Design Log Instrument (DLI) intended for use in identifying moments of abstraction as evidence of STEM content knowledge transfer. Many theoretical approaches to explaining knowledge transfer are rooted in a belief that transfer occurs through knowledge abstraction (Reed, Ernst, & Banerji, 1974; Gick & Holyoak, 1980, 1983). The DLI prompts participants to be reflective during technological/engineering design activities. During the development of this instrument, a three-phase multiple case: embedded design was used. Three distinct Phases accommodated the collection and analysis of data necessary for this investigation: Phase 1: Pilot Case Study, Phase 2: Establishing Content Validity, and Phase 3: Establishing Construct Validity. During Phase 3, data from the DLI was collected at each of seven work sessions from two design teams each working through different engineering problems. At the end of Phase 3, a comparison of abstractions found in DLI responses and observation data (Audio/Video transcripts) indicated the extent to which the DLI independently reflected those abstractions revealed in observations (Audio/Video transcripts). Results of this comparison showed that the DLI has the potential to be 68% reliable to reveal abstracted knowledge. Further analysis of these findings showed ancillary correlations between the percent abstractions found per DLI reflective prompt and the percent abstractions found per T/E design phase. Specifically, DLI Reflective Prompts 2 and 3 correlate with T/E Design Phases 3 and 4 (58% and 76% respectively of the total abstractions) which deal with design issues related to investigating the problem and developing alternate solutions. DLI Reflective Prompts 4 and 5 correlate with T/E Design Phases 5 and 6 (22% and 24% respectively of total abstractions) which deal with design issues related to choosing a solution and developing a prototype. Findings also indicate that there are highs and lows of abstraction throughout the T/E design process. The implications of these highs and lows are that specific phases of the T/E design process can be targeted for research and instruction. By targeting specific T/E design phases, a researcher or instructor can increase the likelihood of fostering abstractions as evidence of STEM content knowledge transfer.
- Evaluation of the Externship Within a Master's Degree Program for Mathematics Specialists at Virginia TechKreye, Bettibel Carson (Virginia Tech, 2009-03-31)The purpose of this evaluation was to determine the effectiveness of an Externship as the capstone experience of a Master's degree program in preparing seventeen teachers to be Mathematics Specialists. This formative evaluation was based on the teacher's perspective and was naturalistic in design utilizing both qualitative and quantitative research methods. Data sources included a teacher survey, teacher interviews, writing prompts, teacher observations, and teacher final project presentations. This evaluation was designed to answer the question: Do the teachers feel that the requirements of this Masters' Degree Externship have prepared them for their role as a Mathematics Specialist? The success and effectiveness of the Externship was tied directly to the teachers chosen school-based experiences which were designed based on a mathematics needs assessment of their schools. Teachers were found to be involved in all five of the essential components of an externship as outlined in the theoretical framework — application, collaboration, reflection, expectations, and the cohort structure. In addition, teachers were found to integrate their leadership and teaching skills through the engagement of their school colleagues in the improvement of educational experiences for all students. Overall, the teachers felt that the experiences throughout the Externship adequately prepared them for their roles of a mathematics specialist; working effectively with stakeholders; working with issues around curriculum and instruction; planning and delivering professional development; and working as leaders within their schools.
- Examining Connections among Instruction, Conceptual Metaphors, and Beliefs of Instructors and StudentsRupnow, Rachel Lynn (Virginia Tech, 2019-07-29)In this study, I will examine the beliefs and conceptual understanding of instructors and students from two abstract algebra classes. This research takes the form of a case study in which I answer four research questions, each addressing a relationship between instruction and beliefs or conceptual understanding. Specifically, these research questions are: 1. What beliefs do the instructors have about math, teaching, and learning and what relationship exists between these beliefs and instructional practice? 2. What is the relationship between instructional practice and students' beliefs about math, teaching, and learning? 3. What conceptual metaphors do the professors use to describe isomorphisms and homomorphisms and what relationship exists between these metaphors and the mathematical content in instruction? 4. What is the relationship between the mathematical content in instruction and conceptual metaphors the students use to describe isomorphisms and homomorphisms? In terms of beliefs, the instructors articulated considered positions on the nature of math, math learning, and math teaching. These beliefs were clearly reflected in their overall approaches to teaching. However, their instruction shifted in practice over the course of the semester. Students' beliefs seemed to shift slightly as a result of the ways their instructors taught. However, their core beliefs about math seemed unchanged and some lessons students took away were similar in the two classes. In terms of conceptual understanding, the instructors provided many conceptual metaphors that related to how they understood isomorphism. They struggled more to provide an image for homomorphism, which requires thinking about a more complicated mathematical object. Their understandings of isomorphism and homomorphism were largely reflected in their instruction with some notable differences. Students took away similar understandings of isomorphism to the instructors, but did not all take away the same level of structural understanding of homomorphism. In short, relationships between instructors' beliefs and instruction and between instructors' conceptual understanding and instruction were evident. However, certain elements were not made as clear as they perhaps intended. Relationships between instruction and students' beliefs and between instruction and students' conceptual understanding were also evident. However, relationships between instruction and beliefs were subtler than between instruction and conceptual understanding.
- Examining the Relationship Between Students' Measurement Schemes for Fractions and Their Quantifications of AngularityMullins, Sara Brooke (Virginia Tech, 2020-06-26)In the basic understanding of measurement, students are expected to be able to subdivide a given whole into a unit and then change the position of that unit along the entire length of the whole. These basic operations of subdivision and change of position are related to the more formal operations of partitioning and iterating. In the context of fractions, partitioning and iterating play a fundamental role in understanding fractions as measures, where students are expected to partition a whole into an iterable unit. In the context of angle measurement, students are expected to measure angles as a fractional amount of a full rotation or a circle, by partitioning the circle into a unit angle and then iterating that unit angle to find the measure of the given angle. Despite this link between measurement, fractions, and angles, research suggests that there is a disconnect between students' concepts of measurement and geometry concepts, including angle and angle measurement. Therefore, one area of study that might help us understand this disconnection would be to investigate the relationship between students' concepts of measurement and their concepts of angle measurement. This current study documents sixth, seventh, and eighth grade students' measurement schemes for fractions and their quantifications of angularity, and then investigates the relationship between them. This research is guided by the following question: What is the relationship between middle school students' measurement schemes for fractions and their quantifications of angularity? Results indicate that the majority of students involved in this study do not possess a measurement concept of fractions nor a measurement concept of angularity. However, these results demonstrate that there is a relationship between students' measurement schemes for fractions and their quantifications of angularity. It is concluded that students who construct more sophisticated fraction schemes tend to construct more sophisticated quantifications of angularity.
- Exploring Middle School Students' Heuristic Thinking about ProbabilityMistele, Jean May (Virginia Tech, 2014-05-04)This descriptive qualitative study examines six eighth-grade students' thinking while solving probability problems. This study aimed to gather direct information on students' problem solving processes informed by the heuristics and biases framework. This study used purposive sampling (Patton, 1990) to identify eighth-grade students who were knowledgeable about probability and had reached the formal operational stage of cognitive development. These criterion were necessary to reduce the likelihood of students' merely guessing answers and important so that the researcher could distinguish between reasoning and intuition. The theoretical framework for this study was informed by Kahneman and Fredrick's (2002) recent revision to the heuristics and biases framework grounded in the research of Amos Tversky and Daniel Kahneman. Kahneman and Fredrick (2002) drew on dual process theory to explain systematic and predictable heuristic ways of thinking. Dual process theory hypothesizes that human thinking is divided into two different modes of processing. One mode, called System 1, is fast and linked to intuition, and the other, called System 2, is slow and linked to reasoning (Evans, 2008; Stanovich and West, 2000). Within dual process theory, System 1 thinking provides a credible system for explaining why people use heuristic thinking (Kahneman and Frederick, 2002). The recent revision to the heuristics and biases framework is focused on three heuristics, representativeness, conjunction fallacy, and availability. These three heuristics are believed to share the same mental process identified by Kahneman and Fredrick (2002), as the attribute substitution process. The clinical task based interview method was used in this study. This technique allowed the researcher to better observe and interact with the participants while exploring the students' probability thinking. The researcher also used think-aloud protocols to better reveal the organic thinking patterns of the students in real time (Ericsson and Simon, 1980; Fox, Ericsson, and Bets, 2010; Van Someren, Barnard, and Sandberg, 1994). The data from the interviews were analyzed using the constant comparison method (Glaser, 1965). This analysis revealed three categories that were combined with other analyses to create profiles for various thinking patterns observed by the researcher. The researcher identified patterns of thinking by students that were consistent with System 1 thinking and associated with the attribute substitution process (Kahneman and Fredrick, 2002). There were also situations in which students demonstrated ways of thinking consistent with System 2 thinking. However, unexpected ways of thinking were also identified by the researcher. For example, there were occasions when students substituted their fraction knowledge when solving probability problems and even seemed to equate probability with fractions. This type of thinking was referred to as the content substitution process in this study. This process occurred when students were using System 1 thinking as well as other types of thinking. In addition, the researcher observed students with thinking patterns that contained characteristics of both System 1 and System 2, which is referred to as slow intuition in this study. Slow intuition seemed to affect students' problem solving strategies as they wavered between multiple problem solving strategies that included either of the two substitution processes: attribute substitution and content substitution. This study contributes to the body of knowledge related to probabilistic thinking. In particular, this study informs our understanding of heuristic thinking used by eighth-grade students when solving probability problems. Further, teaching practices that draw on Fischbein's (1975, 1987) general notion of intuition might be developed and used to improve probability reasoning skills. These teaching practices target students that depend on the attribute substitution process and/or the content substitution process. Each of these heuristic ways of thinking may require different instructional techniques to help students develop more sound ways of thinking about probability. Regardless, teachers need to be informed of the extent that some students rely on their fraction knowledge when solving probability problems.
- First mathematics course in college and graduating in engineering: Dispelling the myth that beginning in higher-level mathematics courses is always a good thingWilkins, Jesse L. M.; Bowen, Bradley D.; Mullins, Sara Brooke (ASEE, 2021-07-03)Background: Graduation rates in engineering programs continue to be a concern in higher education. Prior research has documented an association between students' experiences in first-year mathematics courses and graduation rates, but the influences of the mathematics courses completed and the grades earned are not fully understood. Purpose: The purpose of this study was to investigate the relationship among the first undergraduate mathematics course a student completes, the grade they earn in this course, and the likelihood of graduating with a degree in engineering within six years. Method: The study involved 1504 students from five consecutive cohorts of first-year students enrolled in an engineering degree program at a medium-sized Midwestern public university. Logistic regression was used to model the interrelationship between course and grade in predicting the relative likelihood of graduation for students enrolled in 16 different mathematics courses. Results: Overall, students who take Calculus I or a more advanced mathematics course as their first mathematic course and who are more successful in their first mathematics course are more likely to graduate with a degree in engineering. However, considering grade and course together, some groups of students who are more successful in lower-level mathematics courses are as likely to graduate as students who are less successful in upper-level mathematics courses. Conclusions: Evidence from this study helps to dispel the myth that beginning with higher-level mathematics courses is the optimal course-taking strategy when pursuing an engineering degree. Findings have implications for student advising, curriculum and instruction, high school course-taking, and broadening participation in engineering.
- The History of the Eastern Mennonite High School Touring Choir: 1917-1981Basham, Dwight Cornell (Virginia Tech, 1999-08-02)For more than 35 years, the Eastern Mennonite High School Touring Choir has provided concerts for choral enthusiasts throughout the United States, Canada, and Europe. The present study documents the development, establishment, and growth of Touring Choir from its inception, through June of 1981: identifying (1) the events leading to the development of Touring Choir; (2) the philosophy and purpose of Touring Choir; (3) the directors of Touring Choir and their musical and professional backgrounds; (4) the contributions of each of the directors; and (5) the origin and development of the Senior Parent Weekend Concert, Tour Concert, Mennonite High School Music Festival, and Commencement Concert traditions. The study reports that the history of the Eastern Mennonite High School Touring Choir can be traced to the expressed desire of the school's founders in 1917 that vocal music be an integral part in the education of its students in order to enhance worship and singing in constituent churches. As touring choirs developed, the unique organization of Eastern Mennonite College and High School provided a basis for emulation by the high school when it became a separate organization during the 1960s, culminating in the establishing of Touring Choir in association with the Mennonite High School Music Festival. Marvin Miller's leadership as Touring Choir director established the Senior Parent Weekend Concert, Tour Concert, Mennonite High School Music Festival, and Commencement Concert traditions which became the foundation of Touring Choir's activities during the 1960s and early 1980s. Implications of the study consider the importance of modeling for music education practitioners. The growth of the Eastern Mennonite High School music department and the success of the Eastern Mennonite High School Touring Choir during the combined tenures of Annetta Wenger Miller and Marvin L. Miller were characterized by an enthusiasm for the subject matter of music, high expectations and motivation of students by challenging them to sing music of enduring value in a variety of styles by master composers, placing responsibility for learning on students, a positive approach in teaching, care and concern for students as individuals, and an emphasis on the expressive aspect of music study. In addition, the results of the music program under Marvin Miller's direction may have been due in part to his overall philosophy of music curriculum design that each student would understand the place of music in his or her life experience.
- Homework Journaling in Undergraduate MathematicsJohnston, Alexis Larissa (Virginia Tech, 2012-03-23)Over the past twenty years, journal writing has become more common in mathematics classes at all age levels. However, there has been very little empirical research about journal writing in college mathematics (Speer, Smith, & Horvath, 2010), particularly concerning the relationship between journal writing in college mathematics and college students' motivation towards learning mathematics. The purpose of this dissertation study is to fill that gap by implementing homework journals, which are a journal writing assignment based on Powell and Ramnauth's (1992) "multiple-entry log," in a college mathematics course and studying the relationship between homework journals and students' motivation towards learning mathematics as grounded in self-determination theory (Ryan & Deci, 2000). Self-determination theory predicts intrinsic motivation by focusing on the fundamental needs of competence, autonomy, and relatedness (Ryan & Deci, 2000). In addition, the purpose of this dissertation study is to explore and describe the relationship between homework journals and students' attitudes towards writing in mathematics. A pre-course and post-course survey was distributed to students enrolled in two sections of a college mathematics course and then analyzed using a 2Ã 2 repeated measures ANOVA with time (pre-course and post-course) and treatment (one section engaged with homework journals while the other did not) as the two factors, in order to test whether the change over time was different between the two sections. In addition, student and instructor interviews were conducted and then analyzed using a constant comparative method (Anfara, Brown, & Mangione, 2002) in order to add richness to the description of the relationship between homework journals and students' motivation towards learning mathematics as well as students' attitudes towards writing in mathematics. Based on the quantitative analysis of survey data, no differences in rate of change of competence, autonomy, relatedness, or attitudes towards writing were found. However, based on the qualitative analysis of interview data, homework journals were found to influence students' sense of competence, autonomy, and relatedness under certain conditions. In addition, students' attitudes towards writing in mathematics were strongly influenced by their likes and dislikes of homework journals and the perceived benefits of homework journals.
- How Calculus Eligibility and At-Risk Status Relate to Graduation Rate in Engineering Degree ProgramsBowen, Bradley D.; Wilkins, Jesse L. M.; Ernst, Jeremy V. (Springer, 2019)The problematic persistence rates that many colleges and schools of engineering encounter has resulted in ongoing conversations about academic readiness, retention, and degree completion within engineering programs. Although a large research base exists about student preparedness in engineering, many studies report a wide variety of factors that makes it difficult to address specific issues that prohibit students from completing a degree in engineering. Many studies anecdotally address mathematics achievement as a factor associated with success, but few contain empirical data specifically related to success or readiness to take calculus. This study specifically examines engineering degree completion of calculus eligible students compared to non-eligible calculus students upon acceptance into a College of Engineering as a first-semester freshman, and the mediating effects of being at-risk for non-matriculation on this relationship. A 10-year span of engineering student data, including admission and completion data, was accessed and analyzed to investigate student preparedness (as defined by calculus eligibility) and student success (as defined by at-risk status for non-matriculation) as they related to graduation rate. This study documents a partial mediating effect of at-risk status on the relationship between calculus eligibility and graduation rate; however, calculus eligibility remains a significant predictor of graduation rate and together with at-risk status predicts a significant proportion of the variance in graduation rate.
- Instructors' Orientation on Mathematical MeaningChowdhury, Ahsan Habib (Virginia Tech, 2021-06-11)Students often ask "when is this ever going to be useful?", "why are we doing this?", etc. when speaking about mathematics. If we take this as a question about 'meaningfulness', how can instructors respond and how do they even understand the terms 'meaningful' and 'meaning'? My dissertation looked at how college instructors see their instruction as meaningful or not. Drawing on social and cognitive perspectives of learning, I define four ways to think of what's 'meaningful' about mathematics. From a cognitive perspective, instructors can understand 'meaningful' as mathematical understanding versus understanding the significance of mathematics. From a social perspective where meaning is taken as the experiences of everyday life within communities, teachers can understand 'meaningful' as anything that engages students in practices the mathematics community engage in versus practices non-mathematics communities engage in (e.g. pushing computation or critical thinking as a means for maintaining social hierarchies). Using these four conceptions to categorize instructors' goals, this work focuses on how four undergraduate mathematics instructors thought of their instruction as meaningful and contextual and background factors that influenced those views.
- Integrative Perspectives of Academic MotivationChittum, Jessica Rebecca (Virginia Tech, 2015-03-17)My overall objective in this dissertation was to develop more integrative perspectives of several aspects of academic motivation. Rarely have researchers and theorists examined a more comprehensive model of academic motivation that pools multiple constructs that interact in a complex and dynamic fashion (Kaplan, Katz, and Flum, 2012; Turner, Christensen, Kackar-Cam, Trucano, and Fulmer, 2014). The more common trend in motivation research and theory has been to identify and explain only a few motivation constructs and their linear relationships rather than examine complex relationships involving 'continuously emerging systems of dynamically interrelated components' (Kaplan et al., 2014, para. 4). In this dissertation, my co-author and I focused on a more integrative perspective of academic motivation by first reviewing varying characterizations of one motivation construct (Manuscript 1) and then empirically testing dynamic interactions among multiple motivation constructs using a person-centered methodological approach (Manuscript 2). Within the first manuscript (Chapter 2), a theoretical review paper, we summarized multiple perspectives of the need for autonomy and similar constructs in academic motivation, primarily autonomy in self-determination theory, autonomy supports, and choice. We provided an integrative review and extrapolated practical teaching implications. We concluded with recommendations for researchers and instructors, including a call for more integrated perspectives of academic motivation and autonomy that focus on complex and dynamic patterns in individuals' motivational beliefs. Within the second manuscript (Chapter 3), we empirically investigated students' motivation in science class as a complex, dynamic, and context-bound phenomenon that incorporates multiple motivation constructs. Following a person-centered approach, we completed cluster analyses of students' perceptions of 5 well-known motivation constructs (autonomy, utility value, expectancy, interest, and caring) in science class to determine whether or not the students grouped into meaningful 'motivation profiles.' 5 stable profiles emerged: (1) low motivation; (2) low value and high support; (3) somewhat high motivation; (4) somewhat high empowerment and values, and high support; and (5) high motivation. As this study serves as a proof of concept, we concluded by describing the 5 clusters. Together, these studies represent a focus on more integrative and person-centered approaches to studying and understanding academic motivation.
- Learning Mathematics in Appalachia: Life Histories of Beginning TeachersWatson, Donna Hardy (Virginia Tech, 2005-09-07)Life stories were constructed for three young women from Appalachia to explore their mathematics experiences as students in public schools of the region. Data sources included interviews, school records, and a self-drawn chart of estimated mathematics ability for each year, from kindergarten through twelfth grade. A cross-case analysis revealed similar characteristics among the three women including shyness, difficulty with middle school mathematics and with high school geometry, the choice not to take a mathematics course in the last year of high school, and an awareness of a negative Appalachian stereotype. The mathematics education received by all the women was inadequate as demonstrated by their self-created graphs, their life story accounts, and their initial difficulties in making the minimum required score on the Praxis I Mathematics test. Their subsequent successes in graduating from college can be attributed to their own motivation and tenacity in addition to the encouragement of their families and some teachers. Connections to Standards-based reform in mathematics education include questions about the teaching and learning of geometry and about opportunities for students, especially females, to participate in mathematical discourse throughout their school mathematics experiences, a situation impacted by their expressed shyness and by overt and subtle incidences of gender and racial biases. Appalachian cultural connections seem to be an aspect of fatalism which influences attribution of natural ability versus effort and, in some instances, a climate of male dominance. Connections to the problems of education in rural poverty included a number of ineffective teachers, a situation exacerbated by a sense of social stratification within the Appalachian culture and a reluctance to challenge school or teacher practices. As for learning preferences, the women tended to favor teachers who offered good explanations and who demonstrated caring, which highlights an emphasis placed on relationships within the Appalachia culture. Determining the degree of influence of the Appalachian culture on the education, especially in mathematics, of these three young women was difficult to ascertain. The factors of culture, socioeconomic levels, and rural isolation combined with the effects of race, gender and ethnicity in the individual to impact the opportunities to a quality education.
- Learning progression toward a measurement concept of fractionsWilkins, Jesse L. M.; Norton, Anderson H. III (2018-06-27)Background Fractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of part-whole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students’ development of more sophisticated concepts of fractions, beyond the part-whole concept, lays the groundwork for the later study of important mathematical topics, such as algebra, ratios, and proportions, which are foundational understandings for most STEM-related fields. In particular, the Common Core State Standards for Mathematics call for students to develop measurement concepts of fractions. In order to support such concepts, it is important to understand the underlying mental actions that undergird them so that teachers can design appropriate instructional opportunities. In this study, we propose a learning progression for the measurement concept of fractions—one that focuses on students’ mental actions and informs instructional design. Results A hierarchy of fraction schemes is charted outlining a progression from part-whole concepts to measurement concepts of fractions: (a) part-whole scheme (PWS), (b) measurement scheme for unit fractions (MSUF), (c) measurement scheme for proper fractions (MSPF), and (d) generalized measurement scheme for fractions (GMSF). These schemes describe concepts with explicit attention to the mental actions that undergird them. A synthesis of previous studies provides empirical evidence to support this learning progression. Conclusions Evidence from the synthesis of a series of research studies suggests that children’s measurement concept of fractions develops through several distinct developmental stages characterized by the construction of distinct schemes. The mental actions associated with these schemes provide a guide for teachers to design instructional opportunities for children to advance their construction of a measurement concept of fractions. Specifically, the collection of quantitative studies suggest that students need opportunities to engage in activities that support two kinds of coordinations—the coordination of partitioning and iterating, and the coordination of three levels of units inherent in fractions. Instructional implications are discussed with example tasks and activities designed to provoke these coordinations.
- Mathematical Discussion and Self-Determination TheoryKosko, Karl Wesley (Virginia Tech, 2010-03-26)This dissertation focuses on the development and testing of a conceptual framework for student motivation in mathematical discussion. Specifically, this document integrates Yackel and Cobb's (1996) framework with aspects of Self-Determination Theory (SDT), as described by Ryan and Deci (2000). Yackel and Cobb articulated the development of students' mathematical dispositions through discussion by facilitating student autonomy, incorporating appropriate social norms and co-constructing sociomathematical norms. SDT mirrors these factors and describes a similar process of self-regulation through fulfillment of the individual needs of autonomy, social relatedness, and competence. Given the conceptual overlap, this dissertation examines the connection of SDT with mathematical discussion with two studies. The first study examined the effect of student frequency of explaining mathematics on their perceived autonomy, competence and relatedness. Results of HLM analyses found that more frequent explanation of mathematics had a positive effect on students' perceived mathematics autonomy, mathematics competence, and relatedness. The second study used a triangulation mixed methods approach to examine high school geometry students' classroom discourse actions in combination with their perceived autonomy, competence, and relatedness. Results of the second study suggest a higher perceived sense of autonomy is indicative of more engagement in mathematical talk, but a measure of competence and relatedness are needed for such engagement to be fully indicative of mathematical discourse. Rather, students who lacked a measure of perceived competence or relatedness would cease participation in mathematical discussion when challenged by peers. While these results need further investigation, the results of the second study provide evidence that indicates the necessity of fulfilling all three SDT needs for engagement in mathematical discussion. Evidence from both the first and second studies presented in this dissertation provides support for the conceptual framework presented.
- Measuring the motivational climate in an online course: A case study using an online survey tool to promote data-driven decisionsJones, Brett D.; Fenerci-Soysal, Hande; Wilkins, Jesse L. M. (Elsevier, 2022-12)Creating a positive motivational climate in an online course can engage students in their learning. Instructors may be able to better manage their courses and create a positive motivational climate if they implement online survey tools that allow them to assess the motivational climate of their courses. Teachers and researchers have documented that five student perceptions—empowerment, usefulness, success, interest, and caring—are particularly important for creating a positive motivational climate and are associated with students’ engagement and evaluations of teaching. In this paper, we describe a case study of an instructor who used an online survey tool to assess the motivational climate in his online asynchronous course over time. He then used the feedback to consider improvements that he could make to his course in the future. In addition, we describe how this process of using the online survey tool could be used by instructors to transform education.
- A Mixed Methods Study of Chinese Students' Construction of Fraction Schemes: Extending the Written Test with Follow-Up Clinical InterviewsXu, Cong Ze (Virginia Tech, 2023-01-12)Understanding fractions is fundamental for expanding number knowledge from the whole number system to the rational number system. According to the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (2000), learning fractions is an important mathematical goal for students in grades three through five in the U.S. Moreover, the NCTM suggests that fraction instruction start in Pre-K and continue through 8th grade. At the same time, the Common Core State Standards for Mathematics (CCSSM) suggests that fraction instruction should occur from Grade 3 to 7. In contrast to the time spent on learning fractions in the U.S., students in China spend a relatively short time learning fractions (Zhang and Siegler, 2022). According to the Chinese national curriculum standards, the Chinese National Mathematics Curriculum Standards (CNMCS) for five-four system, the fundamental fraction concepts are taught in grades 3 and 5 only. However, Chinese students continue to have higher performance on fraction items in international assessments when compared with American students (Fan and Zhu, 2004). Consequently, over the last several years, researchers have investigated subject content knowledge and pedagogical content knowledge of Chinese in-service teachers and pre-service teachers via fraction division (e.g., Li and Huang, 2008; Ma, 1999). There are also studies exploring Chinese written curricula of fraction division (e. g., Li, Zhang, and Ma, 2009). Recently, a quantitative study from Norton, Wilkins, and Xu (2018) investigated the process of Chinese students' construction of fraction knowledge through the lens of fraction schemes, a model established by western scholars Steffe (2002) and his colleague Olive (Steffe and Olive, 2010). However, there is a lack of qualitative research that attempts to use fraction schemes as an explanatory framework to interpret the process of Chinese students' construction of fraction knowledge. The main purpose of this study was to investigate Chinese students' understanding of the fundamental fraction knowledge in terms of their understanding of the "fraction unit," referred to as a "unit fraction" in the U.S., using Steffe and Olive's (2010) fraction schemes as the conceptual framework. A sequential mixed methods design was used in this study. The design included two consecutive phases, namely a quantitative phase followed by a qualitative phase (Creswell and Plano Clark, 2011). During the quantitative phase, five hundred and thirty-four Chinese fourth and fifth grade students were administered an assessment. The quantitative data was first analyzed using a Cochran's Q test to determine if the Chinese participants in this study follow the same progression of fraction schemes as their American peers. Results indicate that the development of fractional schemes among Chinese 4th and 5th grade participants in this study is similar to their U.S. counterparts and the Chinese participants in Norton et al.'s (2018) study regardless of the curricula differences across countries or areas in the same country, the textbook differences, and the language differences. Next, two different analysis of variances (ANOVA), a three-way mixed ANOVA and a two-way repeated measures ANOVA were conducted. The three-way mixed ANOVA was used to inform the researcher as to the fraction schemes these students had constructed before the concept of fraction unit is formally introduced and after the concept of fraction unit is formally introduced. The results showed that the fraction knowledge of the students in this study developed from 4th grade to 5th grade. The analysis of clinical interview data confirmed this conclusion. The two-way repeated measures ANOVA was used to determine which model (i.e., linear, circular, or rectangular) is more or less problematic for Chinese students when solving fraction tasks. The results suggest that generally students' performance on linear model tasks was better than their performance on circular model tasks, but there was no statistically significant difference between performance on circular model and its corresponding rectangular model tasks. The results from the quantitative analyses were also used to screen students to form groups based on their highest available fraction scheme for a clinical interview in the second phase, the qualitative phase. In the qualitative phase, a clinical interview using a think-aloud method was used to gain insight into the role of students' conceptual understanding of the fraction unit in their construction of fraction knowledge. In this phase, students were asked to solve the tasks in the clinical interview protocol using the think aloud method. Two main findings were revealed analyzing the clinical interview data. First, a conceptual understanding of fraction units as well as a conceptual understanding of a unit whole play a critical role in the construction of Chinese students' fraction knowledge. Second, the lack of the understanding of a fraction unit as an iterable unit may be one of the reasons that obstructs students move from part-whole concept of fractions to the measurement concept of fractions. This study also demonstrates that a conceptual understanding of fraction units and the unit whole are a necessary condition for constructing of a conceptual understanding of fraction knowledge. Thus, implications of this study suggest that teachers not only should help students build a conceptual understanding of fraction units, but also need to confirm that students have constructed the concept of what the unit whole is before asking students to identify the fraction units for the referent whole. On the other hand, the tasks used in the present study only include continuous but not discrete wholes. Therefore, future research may focus on investigating how students identify fraction units and in what way the iterating operation could be used when students encounter a discrete whole.
- Modeling Students' Units Coordinating ActivityBoyce, Steven James (Virginia Tech, 2014-08-29)Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed.
- Number Sequences as Explanatory Models for Middle-Grades Students' Algebraic ReasoningZwanch, Karen Virginia (Virginia Tech, 2019-04-23)Early algebraic reasoning can be viewed as developing a bridge between arithmetic and algebra. Accordingly, this research examines how middle-grades students' arithmetic reasoning, classified by their number sequences, can be used to model their algebraic reasoning as it pertains to generalizing, writing, and solving linear equations and systems of equations. In the quantitative phase of research, 326 students in grades six through nine completed a survey to assess their number sequence construction. In the qualitative phase, 18 students participated in clinical interviews, the purpose of which was to elicit their algebraic reasoning. Results show that the numbers of students who had constructed the two least sophisticated number sequences did not change significantly across grades six through nine. In contrast, the numbers of students who had constructed the three most sophisticated number sequences did change significantly from grades six and seven to grades eight and nine. Furthermore, students did not consistently reason algebraically unless they had constructed at least the fourth number sequence. Thus, it is concluded that students with the two least sophisticated number sequences are no more prepared to reason algebraically in ninth grade than they were in sixth.
- Predictors of Positive Change in Teaching Practices: A Quantitative StudySanchez Robayo, Brigitte Johana (Virginia Tech, 2023-03-21)Change in educational settings is a complex and multifaceted process that commonly implies change in teaching practices. Different initiatives have shown the significance of teachers and their perceptions when change in teaching practices is intended. Additionally, various factors may influence change in teaching practices at three different moments: before it happens, during, and after its implementation. Considering teachers' perceptions, I studied different factors that may be related to positive change in teaching practices. I studied the relationship between three groups of factors and positive change in teaching practices: motivational factors, including teachers' self-efficacy and autonomy; learning opportunities that include professional development, feedback, and leadership; and the academic and community domains as part of the school climate factor. In particular, I answered the following research question: To what extent do learning opportunities, teacher motivational factors, and school climate predict positive change in teaching practices? In this study I posited that teacher factors such as self-efficacy and school factors such as leadership influence positive change in teaching practices. I also posited that school factors influence the relationship between teacher factors and positive change in teaching practices. To study these relationships, I analyzed data from the Teaching and Learning International Survey (TALIS). This survey provides clustered data: teachers are clustered by schools and schools by countries. I used multilevel modeling statistical methods (i.e., a two-level hierarchical linear model) to examine the Colombian and United Stated datasets. Before estimating the hierarchical linear models, I conducted an exploratory factor analysis (EFA) to identify the teacher-level variables. One follow-up EFA focused on teacher self-efficacy yielded three variables that allowed me to focus on three specific teaching tasks: managing student behavior, motivating students, and varying instructional strategies. I found that learning opportunities, motivational factors, and school climate predict positive change in teaching practices. Learning opportunities, such as feedback from the principal has a stronger effect than feedback from colleagues. The impact of feedback from the principal has significant unnoticeable variability across schools, and it is negatively influenced by the feedback received by the teachers at the same school. Additionally, teachers' self-efficacy in different teaching tasks predicts positive change, however, these relationships differ by country. Finally, distributed leadership as part of school climate is a significant predictor of positive change that also affects it by influencing teacher interactions positively. Implications of these findings are also discussed as it relates to the existing literature and the educational system in each of the two countries.