Academic Commons Search Results
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Academic Commons Search Resultsen-usWhat Makes a Good Problem? Perspectives of Students, Teachers, and Mathematicians
http://academiccommons.columbia.edu/catalog/ac:188061
DeGraaf, Elizabeth Brennanhttp://dx.doi.org/10.7916/D8NV9HC3Thu, 04 Jun 2015 00:00:00 +0000While mathematical problem solving and problem posing are central to good mathematics teaching and mathematical learning, no criteria exist for what makes a good mathematics problem. This grounded theory study focused on defining attributes of good mathematics problems as determined by students, teachers, and mathematicians. The research questions explored the similarities and differences of the responses of these three populations. The data were analyzed using the grounded theory approach of the constant comparative method. Fifty eight students from an urban private school, 15 teachers of mathematics, and 7 mathematicians were given two sets of problems, one with 10 algebra problems and one with 10 number theory problems, and were asked choose which problems they felt were the “best” and the “least best”. Once their choices were made, they were asked to list the attributes of the problems that lead to their choices. Responses were coded and the results were compared within each population between the two different problem sets and between populations. The results of the study show that while teachers and mathematicians agree, for the most part, about what attributes make a good mathematics problem, neither of those populations agreed with the students. The results from this study may be useful for teachers as they write or evaluate problems to use in their classes.Mathematics educationecb2154Mathematics, Science, and Technology, Mathematics EducationDissertationsA Historical Analysis of the Mathematics Major Requirements at Six Colleges in the United States from 1905 to 2005
http://academiccommons.columbia.edu/catalog/ac:187818
Huntington, Heather Leehttp://dx.doi.org/10.7916/D88S4P1PTue, 12 May 2015 18:27:05 +0000This study attempts to document and explore the history of the undergraduate mathematics major at six United States colleges during the twentieth century. The six colleges were chosen based on their geographical diversity and their success in producing mathematics Ph.D. students. Three of the colleges are private, and three are public colleges.
There are five key findings in this paper. Regarding specific courses, in 1955, courses in linear algebra, discrete mathematics, and computer science became widely available. This probably occurred due to the close relationship between discrete mathematics, linear algebra, and computing. Computer programming became easier and more popular during the 1950s, and computer science courses at most colleges migrated from the Mathematics Department to their own department. Yale was an exception; there were computer science courses available in the Mathematics Department at Yale until 2005.
Advanced applied courses (Category 9) became more prevalent in some cases and disappeared in others. These courses may have migrated to other academic departments at the schools where they disappeared. This migration may have occurred at CCNY, Colorado College, Stanford University, and Yale. These findings are consistent with Garfunkel and Young's (1990) research on mathematics courses outside of mathematics departments. At the University of Texas, Austin, the advanced applied courses dramatically increased between 1945 and 2005. This is most likely due to the merger of the Pure Mathematics Department and Applied Mathematics Department between 1945 and 1955.
Between 1975 and 1995, three of the four colleges from middle America and the west had many courses with unspecified content (Category 13). These courses included undergraduate colloquia, seminars, history of mathematics, problem-solving courses, tutorial courses, independent study, and experimental courses. Perhaps the schools outside the east coast experimented more with their upper division mathematics during this time.
The three colleges that produced a significant number of undergraduates who eventually earned a doctoral degree in a STEM field between 1997 and 2006 did not credit general education mathematics courses (Category 2) during the entire study. Furthermore, these top future Ph.D.-producing colleges state that their undergraduates can take graduate courses 23 times in this study, whereas the other three colleges only mention it 9 times. Stanford University encouraged their undergraduates to take on graduate courses the most, then Yale and the University of California, Berkeley.
After 1975, the percentage of mathematics courses needed to obtain the undergraduate degree converged at the colleges in this study to be in the range of 30% to 38%. The percentage never exceeded 40% at any of the schools in this study.Mathematics education, History of education, Higher educationMathematics EducationDissertationsClassification of Spoken Discourse in Teaching the Construction of Mathematical Proof
http://academiccommons.columbia.edu/catalog/ac:187896
Reich, Heidihttp://dx.doi.org/10.7916/D87S7MVFTue, 12 May 2015 00:00:00 +0000The purpose of this study is to analyze the patterns of classroom discourse when high school students move from performing prescribed algorithms in order to solve problems for which the process and solution are well-defined to spoken proof, in which ideas are discussed and arguments are formulated and formalized. The study uses a modified version of discourse analysis developed by Arno Bellack and refined for usage in a mathematics classroom by James T. Fey. The analysis framework is supplemented by codes borrowed from Maria Blanton, Despina Stylianou, and M. Manuela David (2009), which is in turn a modified version of a coding system developed by Kruger (1993) and Goos, Galbraith and Renshaw (2002). Twelve mathematics lessons involving two mathematics teachers were recorded, transcribed and coded. Eight of the lessons were classified as “proof-related” and four were designated “non-proof-related.” A lesson designated “proof-related” contained more than half activity that was actively concerned with the construction of proof; whereas a lesson in which no proofs were formulated was designated “non-proof.” Using the codes described above and a variety of qualitative and quantitative measures, the transcripts were examined for constructivist behavior on the part of the teachers and modes of participation on the students’ part. The findings suggest a relationship between a teacher’s beliefs in constructivist principles and the way in which that teacher instructs proof vs. non-proof. More specifically, a teacher who views her/himself as informed by constructivist pedagogical principles may not evince a sharp distinction between her/his teaching of proof vs. non-proof; but a teacher who does not attempt to incorporate constructivist principles on a daily basis may exhibit more constructivist tendencies when teaching proof.Mathematics educationMathematics EducationDissertationsRedefining Professional Development for Supporting Elementary Teachers Mathematics Knowledge: A Case Study Approach
http://academiccommons.columbia.edu/catalog/ac:187932
Sanchez, Ritahttp://dx.doi.org/10.7916/D84T6HG9Tue, 12 May 2015 00:00:00 +0000This dissertation explored how a professional developer, using the Center’s Professional Development Model for Innovating Instruction, supported two teachers’ acquisition of the knowledge needed for their mathematics instruction. Through analysis of detailed field notes and semi-structured interviews of two experienced elementary school teachers working in an urban, high-need school, this dissertation studied how the design and situate components of the Center’s Professional Development Model for Innovating Instruction can lead to multiple ways of supporting teachers’ instruction depending on the teachers’ needs and interests. Findings from these two case studies suggest that there is a need for teacher education mathematics programs—In-service and pre-service—to provide teachers with the knowledge for innovative mathematics instruction needed to create demanding learning experiences in their classrooms. This dissertation elaborates on these results, discusses connections with other research, and ends with implications of these results, in terms of their immediate application and the need for future research.Mathematics educationMathematics EducationDissertationsQuaternions: A History of Complex Noncommutative Rotation Groups in Theoretical Physics
http://academiccommons.columbia.edu/catalog/ac:187914
Familton, Johannes C.http://dx.doi.org/10.7916/D8FB521PTue, 12 May 2015 00:00:00 +0000The purpose of this dissertation is to clarify the emergence of quaternions in order to make the history of quaternions less opaque to teachers and students in mathematics and physics. ‘Quaternion type Rotation Groups’ are important in modern physics. They are usually encountered by students in the form of: Pauli matrices, and SU(2) & SO(4) rotation groups. These objects did not originally appear in the neat form presented to students in modern mathematics or physics courses. What is presented to students by instructors is usually polished and complete due to many years of reworking. Often neither students of physics, mathematics or their instructors have an understanding about how these objects came into existence, or became incorporated into their respected subject in the first place. This study was done to bridge the gaps between the history of quaternions and their associated rotation groups, and the subject matter that students encounter in their course work.Mathematics educationMathematics, Science, and Technology, Mathematics EducationDissertationsThe Effects of Digital Tools on Third Graders’ Understanding of Concepts and Development of Skills in Multiplication
http://academiccommons.columbia.edu/catalog/ac:187938
Yoon, Esther Jiyounghttp://dx.doi.org/10.7916/D8CC0ZRPTue, 12 May 2015 00:00:00 +0000The purpose of this research study was to examine the effectiveness of two digital tools: a virtual number line (Jumper Tool); and a dynamic hundreds chart (Morphing Chart) in improving children’s understanding of multiplication and number sense. One hundred twenty-two third grade students (69 girls), ages ranging from 8 years-0 months to 10 years-3 months (M = 8.88 years, SD = 0.44) from three New York City public elementary schools, were recruited to participate in the study. Participants were randomly assigned to one of two math treatment groups or a reading control group. Students in the Jumper group used a number line tool, while those in the Morphing group used a morphing hundreds chart. Children’s number sense ability and understanding of multiplication were tested at pre- and posttest to examine group differences. Researchers recorded children’s strategy use and a back-end logging system collected data on accuracy during treatment sessions. No group differences across the Jumper, Morphing, or Control groups were found at posttests when controlling for pretest performance. However, the presence of a tool (Jumper or Morphing) during treatment sessions resulted in better performance than the absence of a tool (No Tool). Strategy use had a significant effect on session performance as well. Fast and Tool Use responses performed better than Delayed responses. Additionally, Fast responses were more likely to be correct than those who used an Advanced strategy. Finally, the results indicated that Fast responses were predictive of children’s performance on multiplication facts and number sense tests and Tool use was predictive of performance on multiplication facts. These findings suggest that having a tool, Jumper or Morphing, helped children solve multiplication problems and that tool use is related to superior mastery of multiplication facts.Cognitive psychology, Mathematics education, Educational technologyejy2109Cognitive Studies in EducationDissertationsDevelopment and Evaluation of a Computer Program to Teach Symmetry to Young Children
http://academiccommons.columbia.edu/catalog/ac:187968
Fletcher, Nicolehttp://dx.doi.org/10.7916/D8RF5T4ZTue, 12 May 2015 00:00:00 +0000Children develop the ability to perceive symmetry very early in life; symmetry is abundant in the world around us, and it is a naturally occurring theme in children’s play and creative endeavors. Symmetry is a type of pattern structure and organization of visual information that has been found by psychologists to aid adults in the processing and recall of visual information. Symmetry plays an important role across branches of mathematics and at all levels, and it provides a link between mathematics and a variety of fields and areas of study. Despite this, symmetry does not figure prominently in early childhood mathematics curriculum in the United States. The purpose of this study is to develop, implement, and evaluate a computer program that expands young children’s innate perception and understanding of symmetry and its subtopics—reflection, translation, and rotation. Eighty-six first and second grade children were randomly assigned to one of two conditions: nine sessions using the symmetry computer program designed for this study, or nine sessions using a non-geometry-related computer program. Results showed that children assigned to the experimental condition were better able to identify symmetry subtypes, accurately complete translation tasks and symmetry tasks overall, and explain symmetric transformations. These findings suggest that children are capable of learning about symmetry and its subtypes, and the symmetry software program designed for this study has the potential to improve children’s understanding of symmetry beyond what is currently taught in the early elementary mathematics curriculum. Recommendations for other researchers, educators, and future research are discussedMathematics educationnsf2109Human Development, Mathematics EducationDissertationsStatistical Models of Identity and Self-Efficacy in Mathematics on a National Sample of Black Adolescents from HSLS:09
http://academiccommons.columbia.edu/catalog/ac:187103
Alexander, Nathan Napoleonhttp://dx.doi.org/10.7916/D8WS8S9MMon, 11 May 2015 15:33:43 +0000The research reported in this study examined statistical relations in black adolescents’ identity and self-efficacy beliefs in mathematics. Data for this research study were drawn from the High School Longitudinal Study of 2009 (HSLS:09; Ingels, Dalton, Holder, Lauff, & Burns, 2011) and the study’s first follow-up (Ingels & Dalton, 2013); additional measures were taken from the National Center for Education Statistics’ Common Core of Data (CCD). Data were analyzed using quantitative methods on a nationally representative sample of secondary school students (N = 1,362) across 944 schools in the United States. Although there has been an increase in qualitative research on mathematics identity and mathematics identity development, few researchers have utilized quantitative methods to empirically examine the relationships existing between identity and self-efficacy. Fewer researchers have used panel (longitudinal) data in their investigations. Findings from this study confirmed the literature in that mathematics identity development pathways are informed by students’ mathematics self-efficacy beliefs. Sex differences were also noted. Specifically, males and females experienced divergence in their mathematics identity and mathematics self-efficacy beliefs during high school; however, the returns of these beliefs on a measure of Algebraic proficiency for females were significantly greater than they were for males, although females maintained less positive beliefs over the course of the study. School belonging and engagement significantly predicted shifts in students’ mathematics identity development pathways and were moderated by self-efficacy beliefs, supporting theories that measures of perceived differentiation (e.g., belongingness) are key factors in student motivation and subsequent outcomes. Additional findings underscored the ongoing need for empirical research on students’ peer networks and mathematics teacher’s classroom practices. Overall, results of this study indicated that variations in identity development and self-efficacy beliefs among adolescents extend beyond many theoretical considerations in both their complexity and measured effects when accounting for a host of contextual and psychosocial factors.Mathematics education, Education, Educational psychologyMathematics EducationDissertationsMathematics Identities of Non-STEM Major Female Students
http://academiccommons.columbia.edu/catalog/ac:186989
Guzman, Anahuhttp://dx.doi.org/10.7916/D8NS0SZXThu, 07 May 2015 00:22:16 +0000The mathematics education literature has documented gender differences in the learning of mathematics, interventions that promote female and minority students to pursue STEM majors, and the persistence of the gender, achievement, and opportunity gaps. However, there is a significantly lower number of studies that address the mathematics identities of students not majoring in science, technology, engineering, and mathematics (STEM). Even more elusive or non-existent are studies that focus on the factors that shaped the mathematics identities of female students not pursuing STEM majors (non-STEM female students). Because the literature has shown the importance of understanding students' mathematics identities given its correlation with student achievement, motivation, engagement, and attitudes toward mathematics, it is vital to understand the factors that influence the construction of mathematics identities in particular of those students that have been historically marginalized.
To address this issue, I explored the mathematics identities held by 12 non-STEM major students (six taking a remedial mathematics course and six others taking a non-remedial mathematics course) in one urban business college in a metropolitan area of the Northeastern United States. This study used Martin's (2000) definition of mathematics identity as the framework to explore the factors that have influenced the mathematics identities of non-STEM female students. The data for this qualitative study were drawn from mathematics autobiographies, one questionnaire, two interviews, and three class observations.
I found that the mathematics identities of non-STEM major female students' in remedial and non-remedial mathematics courses were influenced by the same factors but in different ways. Significant differences indicated how successful and non-successful students perceive, interpret, and react to those factors. One of those factors was non-successful students believe some people are born with the ability to do mathematics; consequently, they attributed their lack of success to not having this natural ability. Most of the successful students in remedial mathematics attribute their success to effort and most successful students in non-remedial mathematics attribute their success to having a natural ability to do mathematics. Another factor was successful students expressed having an emotional connection to mathematics. This was evident in cases where mathematics was an emotional bond between father and daughter and those in which mathematics was a family trait.
Moreover, the mathematics activities in both classrooms were scripted and orchestrated with limited room for improvisation. However, the non-remedial students experienced moments in which their academic curiosity contributed to opportunities to exercise conceptual agency and author some of their mathematics knowledge. Further, successful students in remedial mathematics did not have the ability to continue the development of positive mathematics identities given rigid classroom activities that contributed to a limited sense of community to support mathematics learning.Mathematics educationMathematics Education, Mathematics, Science, and TechnologyDissertationsKeeping up with the Times: How are Teacher Preparation Programs Preparing Aspiring Teachers to Teach Mathematics Under the New Standards of Today?
http://academiccommons.columbia.edu/catalog/ac:184196
Sudarsanan, Shalini Thirumulpadhttp://dx.doi.org/10.7916/D8MC8XVTThu, 12 Feb 2015 00:00:00 +0000The purpose of this study is to determine how well traditional elementary preparation programs and alternative elementary certification programs are perceived by teacher candidates, to prepare them to teach mathematics to the standards required today. Using a qualitative design, this study examines how various ranked elementary teacher preparation programs and alternative certification programs are preparing elementary teachers to teach the mathematics envisioned in the CCSSM. Participants in this study all attended an undergraduate elementary teacher preparation program or an alternate route certification program. To paint a holistic picture of elementary teacher preparation in mathematics, each undergraduate elementary teacher preparation programs was ranked in a different level in the NCTQ Teacher Preparation Review. Qualitative surveys and interviews were used to gather data on participants' perceptions of their preparation in mathematics. Twenty-five participants agreed to take part in this study. Participants filled out a two-part survey. The first part of the survey asked background questions on their coursework and field experiences in mathematics along with a survey on their beliefs about their ability to teach mathematics. The second part was a pedagogy survey that asked participants how they would teach particular mathematics concepts that now require a conceptual understanding in the CCSSM. Seven participants agreed to participate in a follow up interview to further investigate their experiences in mathematics preparation in their teacher education programs. The data showed that there is little consistency in the mathematics education of elementary teachers within a teacher preparation program and across different teacher preparation programs. There is little standardization in the coursework for participants from different preparation programs and participants within the same program. The interviews revealed that the degree to which participants were able to teach mathematics in their field experiences also varied within and across teacher preparation programs. Furthermore, the interviews also unveiled that the CCSSM were also studied and utilized in different capacities within and across institutions. Lastly, the data from the surveys disclosed that the majority of participants feel that they have the ability to be effective teachers of mathematics yet the majority of participants teach mathematics for a procedural understanding.Mathematics educationsts2112Mathematics EducationDissertationsThe Effects Of Elementary Departmentalization On Mathematics Proficiency
http://academiccommons.columbia.edu/catalog/ac:175720
Taylor-Buckner, Nicolehttp://dx.doi.org/10.7916/D8D50K49Mon, 07 Jul 2014 00:00:00 +0000Mathematics education in the elementary schools has experienced many changes in recent decades. With the curriculum becoming more complex as a result of each modification, immense pressure has been put on schools to increase student proficiency. The Common Core State Standards is the latest example of this. These revisions to the mathematics curriculum require a comprehensive understanding of mathematics that the typical elementary teacher lacks. Some elementary schools have begun changing the organization of their classrooms from self-contained to departmentalized as a possible solution to this problem. The purpose of this quantitative study was to examine the effects of elementary departmentalization on student mathematics proficiency. This was done by exploring and comparing the background and educational characteristics, teaching practices, assessment methods, beliefs, and influence of departmentalized elementary mathematics teachers. The study also investigated the circumstances under which there are significant differences in mathematics proficiency between departmentalized and non-departmentalized elementary students, and examined if these differences continued into students' eighth-grade years and/or led to higher level eighth-grade mathematics course attainment. Additionally, the study aimed to determine if there was a relationship between elementary departmentalization and mathematics proficiency and also to identify additional factors that could lead to mathematics proficiency. Data came from the U.S. Department of Education's Early Childhood Longitudinal Study, Kindergarten Class of 1998-99 (ECLS-K) data set. The ECLS-K is a national data set that followed the same children from kindergarten to eighth grade focusing on their school experiences from 1998 to 2007. Numerous statistical analyses were conducted on this rich data set, utilizing the statistical software Stata 13 and R. The results of this study indicate that there is a significant difference in the mathematics proficiency of departmentalized and non-departmentalized students when teachers have below-average mathematics backgrounds. The students of the mathematically below-average departmentalized teachers displayed the highest mathematics proficiency as well as the biggest gain in mathematics proficiency, and these higher proficiencies and gains continued into later grade levels. However, when exploring differences in mathematics proficiency among all students, there were no conclusive differences between departmentalized and non-departmentalized students. Regression models yielded inconclusive results as well, even after controlling for factors pertaining to classroom size, student demographics and socioeconomic status, student confidence, parental background, teacher knowledge and instructional practices, and prior student mathematical proficiency. Other findings include self-contained and departmentalized third-grade teachers being very similar in their educational backgrounds and teaching practices, whereas departmentalized and non-departmentalized fifth-grade teachers were found to be fairly different in their educational backgrounds and instructional practices. However, in both grade levels, self-contained teachers appeared to be more reliant on printed materials than departmentalized teachers.Mathematics education, Education policy, Elementary educationMathematics, Science, and Technology, Mathematics EducationDissertationsInvestigating the Effects of the MathemAntics Number Line Activity on Children's Number Sense
http://academiccommons.columbia.edu/catalog/ac:176163
Creighan, Samanthahttp://dx.doi.org/10.7916/D8ZG6QDWMon, 07 Jul 2014 00:00:00 +0000Number sense, which can broadly thought of as the ability to quickly understand, approximate, and manipulate numerical quantities, can be a difficult construct for researchers to operationally define for empirical study. Regardless, many researchers agree it plays an important role in the development of the symbolic number system, which requires children to master many tasks such as counting, indentifying numerals, comparing magnitudes, transforming numbers and performing operations, estimating, and detecting number patterns, skills which are predictive of later math achievement. The number line is a powerful model of symbolic number consistent with researchers' hypotheses concerning the mental representation of number. The MathemAntics Number Line Activity (MANL) transforms the number line into a virtual manipulative, encourages estimation, provides multiple attempts, feedback, and scaffolding, and introduces a novel features where the user can define his own level of risk on the number line. The aim of the present study was to examine how these key features of MANL are best implemented to promote number sense in low-income second-graders. Sixty-six students from three schools were randomly assigned to one of three conditions; MANL User-Defined Range (UDR), and MANL Fixed Range (FR), and a Reading comparison condition and underwent a pretest session, four computer sessions, and a posttest session. During the computer sessions, researchers coded a child's observed strategy in placing targets on the number line. The results showed that children with higher number sense ability at pretest performed better on a posttest number line estimation measure when they were in the UDR condition than in the FR condition. Conversely, children with low number sense ability at pretest performed better on the number line estimation posttest measure when they were in the FR condition than UDR. Although in general, all children improved over time, children with low number sense ability at pretest were more likely to use the UDR tool ineffectively, thus negatively impacting performance. When children were not coded as responding quickly, target number significantly impacted performance in the computer sessions. Finally, children in the UDR condition utilized better expressed strategies on the number line estimation posttest than children in the Reading comparison group. These findings indicate that prior number sense ability plays a role in how children engage with MANL, which in turn affects the learning benefits the child receives. Implications for researchers, software designers, and math educators, as well as limitations are discussed.Cognitive psychology, Educational technology, Mathematics educationsrc2129Cognitive Studies in Education, Human DevelopmentDissertationsConceptions of Creativity in Elementary School Mathematical Problem Posing
http://academiccommons.columbia.edu/catalog/ac:176065
Dickman, Benjaminhttp://dx.doi.org/10.7916/D8MC8X69Mon, 07 Jul 2014 00:00:00 +0000Mathematical problem posing and creativity are important areas within mathematics education, and have been connected by mathematicians, mathematics educators, and creativity theorists. However, the relationship between the two remains unclear, which is complicated by the absence of a formal definition of creativity. For this study, the Consensual Assessment Technique (CAT) was used to investigate different raters' views of posed mathematical problems. The principal investigator recruited judges from three different groups: elementary school mathematics teachers, mathematicians who are professors or professors emeriti of mathematics, and psychologists who have conducted research in mathematics education. These judges were then asked to rate the creativity of mathematical problems posed by the principal investigator, all of which were based on the multiplication table. By using Cronbach's coefficient alpha and the intraclass correlation method, the investigator measured both within-group and among-group agreement for judges' ratings of creativity for the posed problems. Previous studies using CAT to measure judges' ratings of creativity in areas other than mathematics or mathematics education have generally found high levels of agreement; however, the main finding of this study is that agreement was high only when measured within-group for the psychologists. The study begins with a review of the literature on creativity and on mathematical problem posing, describes the procedure and results, provides points for further consideration, and concludes with implications of the study along with suggested avenues for future research.Mathematics educationbmd2118Human Development, Mathematics EducationDissertationsThe Teacher as Mathematician: Problem Solving for Today's Social Context
http://academiccommons.columbia.edu/catalog/ac:176071
Brewster, Hollyhttp://dx.doi.org/10.7916/D8736P17Mon, 07 Jul 2014 00:00:00 +0000A current trend in social justice oriented education research is the promotion of certain intellectual virtues that support epistemic responsibility, or differently put, the dispositions necessary to be a good knower. On the surface, the proposition of epistemically responsible teaching, or teaching students to be responsible knowers is innocuous, even banal. In the mathematics classroom, however, it is patently at odds with current practice and with the stated goals of mathematics education. This dissertation begins by detailing the extant paradigm in mathematics education, which characterizes mathematics as a body of skills to be mastered, and which rewards ways of thinking that are highly procedural and mechanistic. It then argues, relying on a wide range of educational thinkers including John Dewey, Maxine Greene, Miranda Fricker, and a collection of scholars of white privilege, that an important element in social justice education is the eradication of such process-oriented thinking, and the promotion of such intellectual virtues as courage and humility. Because the dominant paradigm is supported by an ideology and mythology of mathematics, however, changing that paradigm necessitates engaging with the underlying conceptions of mathematics that support it. The dissertation turns to naturalist philosophers of education make clear that the nature of mathematics practice and the growth of mathematical knowledge are not characterized by mechanistic and procedural thinking at all. In these accounts, we can see that good mathematical thinking relies on many of the same habits and dispositions that the social justice educators recommend. In articulating an isomorphism between good mathematical thinking and socially responsive thinking, the dissertation aims to offer a framework for thinking about mathematics education in and for a democratic society. It aims to cast the goals of mathematically rigorous education and socially responsible teaching not only as not in conflict, but also overlapping in meaningful ways.Philosophy of education, Mathematics educationArts and Humanities, Philosophy and EducationDissertationsThe Effects of Number Theory Study on High School Students' Metacognition and Mathematics Attitudes
http://academiccommons.columbia.edu/catalog/ac:176092
Miele, Anthonyhttp://dx.doi.org/10.7916/D8XP733JMon, 07 Jul 2014 00:00:00 +0000The purpose of this study was to determine how the study of number theory might affect high school students' metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics. The study utilized questionnaire and/or interview responses of seven high school students from New York City and 33 high school students from Dalian, China. The questionnaire components served to measure and compare the students' metacognitive functioning, mathematical curiosity, and mathematics attitudes before and after they worked on a number theory problem set included with the questionnaire. Interviews with 13 of these students also helped to reveal any changes in their metacognitive tendencies and/or mathematics attitudes or curiosity levels after the students had worked on said number theory problems. The investigator sought to involve very motivated as well as less motivated mathematics students in the study. The participation of a large group of Chinese students enabled the investigator to obtain a diverse set of data elements, and also added an international flavor to the research. All but one of the 40 participating students described or presented some evidence of metacognitive enhancement, greater mathematical curiosity, and/or improved attitudes towards mathematics after the students had worked on the assigned number theory problems. The results of the study thus have important implications for the value of number theory coursework by high school students, with respect to the students' metacognitive processes as well as their feelings about mathematics as an academic discipline.Mathematics educationMathematics, Science, and Technology, Mathematics EducationDissertationsUtilizing the National Research Council's (NRC) Conceptual Framework for the Next Generation Science Standards (NGSS): A Self-Study in my Science, Engineering, and Mathematics Classroom
http://academiccommons.columbia.edu/catalog/ac:176083
Corvo, Arthurhttp://dx.doi.org/10.7916/D8KD1W2GMon, 07 Jul 2014 00:00:00 +0000Given the reality that active and competitive participation in the 21st century requires American students to deepen their scientific and mathematical knowledge base, the National Research Council (NRC) proposed a new conceptual framework for K-12 science education. The framework consists of an integration of what the NRC report refers to as the three dimensions: scientific and engineering practices, crosscutting concepts, and core ideas in four disciplinary areas (physical, life and earth/spaces sciences, and engineering/technology). The Next Generation Science Standards (NGSS), which are derived from this new framework, were released in April 2013 and have implications on teacher learning and development in Science, Technology, Engineering, and Mathematics (STEM). Given the NGSS's recent introduction, there is little research on how teachers can prepare for its release. To meet this research need, I implemented a self-study aimed at examining my teaching practices and classroom outcomes through the lens of the NRC's conceptual framework and the NGSS. The self-study employed design-based research (DBR) methods to investigate what happened in my secondary classroom when I designed, enacted, and reflected on units of study for my science, engineering, and mathematics classes. I utilized various best practices including Learning for Use (LfU) and Understanding by Design (UbD) models for instructional design, talk moves as a tool for promoting discourse, and modeling instruction for these designed units of study. The DBR strategy was chosen to promote reflective cycles, which are consistent with and in support of the self-study framework. A multiple case, mixed-methods approach was used for data collection and analysis. The findings in the study are reported by study phase in terms of unit planning, unit enactment, and unit reflection. The findings have implications for science teaching, teacher professional development, and teacher education.Science education, Engineering, Mathematics educationScience EducationDissertationsMathematical Modeling in the People's Republic of China ---Indicators of Participation and Performance on COMAP's modeling contest
http://academiccommons.columbia.edu/catalog/ac:176110
Tian, Xiaoxihttp://dx.doi.org/10.7916/D8WQ01Z9Mon, 07 Jul 2014 00:00:00 +0000In recent years, Mainland Chinese teams have been the dominant participants in the two COMAP-sponsored mathematical modeling competitions: the Mathematical Contest in Modeling (MCM) and the Interdisciplinary Contest in Modeling (ICM). This study examines five factors that lead to the Chinese teams' dramatic increase in participation rate and performance in the MCM and ICM: the Chinese government's support, pertinent organizations' efforts, support from initiators of Chinese mathematical modeling education and local resources, Chinese teams' preferences in selecting competition problems to solve, and influence from the Chinese National College Entrance Examination (NCEE). The data made clear that (1) the policy support provided by the Chinese government laid a solid foundation in popularizing mathematical modeling activities in China, especially in initial stages of the development of mathematical modeling activities. (2) Relevant organizations have been the main driving force behind the development of mathematical modeling activities in China. (3) Initiators of mathematical modeling education were the masterminds of Chinese mathematical modeling development; support from other local resources served as the foundation of mathematical modeling popularity in China. (4) Chinese teams have revealed a preference for discrete over continuous mathematical problems in the Mathematical Contest in Modeling. However, in general, the winning rates of these two problem types have been shown to be inversely related to their popularity — while discrete problems have traditionally had higher attempt rates, continuous problems enjoyed higher winning rates. (5) The NCEE mathematics examination seems to include mathematical application problems rather than actual mathematical modeling problems. Although the extent of NCEE influence on students' mathematical modeling ability is unclear, the content coverage suggests that students completing a high school mathematics curriculum should be able to apply what they learned to simplified real-world situations, and pose solutions to the simple models built in these situations. This focus laid a solid mathematics foundation for students' future study and application of mathematics.Mathematics educationMathematics, Science, and Technology, Mathematics EducationDissertationsThe Effects of Mastery of Writing Mathematical Algorithms on the Emergence of Complex Problem Solving
http://academiccommons.columbia.edu/catalog/ac:176864
Fas, Tsambikahttp://dx.doi.org/10.7916/D88K7772Mon, 07 Jul 2014 00:00:00 +0000I tested the effects of mastery of writing mathematical algorithms on the emergence of complex problem solving using a time-lagged multiple probe design across matched pairs of participants. In Experiment 1, 6 participants enrolled in third grade, ranging in age from 8 to 9 years, were selected because they were unable to write mathematical algorithms despite mathematical proficiency. The dependent variables were pre and post algorithm instruction probes consisting of verbally governing algorithm probes and abstraction to complex problems. Abstraction to complex problems was defined as solving untaught complex problems by applying taught algorithms. Verbally governing responses were defined as a functional algorithm on how to complete the mathematical problem. The independent variable was algorithm instruction which consisted of two teacher antecedent models for less complex problems, using an algorithm to complete the problem, then writing the algorithm, followed by learn units to the participants who served as writers. A peer-yoked contingency was implemented to teach the functionality of writing algorithms by providing an establishing operation for participants. The writer solved a mathematical problem and then wrote the algorithm on how to solve the problem. If, after one attempt, the reader solved the problem correctly, both participants moved up on the game board, however, if the reader was unable to solve the problem correctly, the experimenter moved up a space on the game board. In Experiment 2, the effects of the algorithm procedure were further tested with 4 new participants enrolled in second grade and ranging in age from 7 to 8 years. The differences between Experiment 1 and 2 were the age and grade level of the participants as well as the mathematical content taught. The mathematical content taught in Experiment 1 was fractions and multiplication and addition and fractions in Experiment 2. Results of the study show all participants acquired the capability to abstract more complex mathematical skills and write functional algorithms for mathematical problems solved. Participants' overall mathematical skills increased from skill levels prior to algorithm instruction. After serving as a writer, participants were able to abstract two more complex mathematical problems without receiving additional instruction.Mathematics education, Behavioral sciences, Special educationtmf2107Health and Behavior Studies, Applied Behavior AnalysisDissertations'Value Creation' Through Mathematical Modeling: Students' Mathematics Dispositions and Identities Developed in a Learning Community
http://academiccommons.columbia.edu/catalog/ac:176803
Park, Joo younghttp://dx.doi.org/10.7916/D87S7KXXMon, 07 Jul 2014 00:00:00 +0000This study examines how mathematical modeling activities within a collaborative group impact students' `value creation' through mathematics. Creating `value' in this study means to apply one's knowledge in a way that benefits the individual and society, and the notion of `value' was adopted from Makiguchi's theory of `value creation' (1930/1989). With a unified framework of Makiguchi's theory of `value', mathematical disposition, and identity, the study identified three aspects of value-beauty, gains, and social good-using observable evidence of mathematical disposition, identity, and sense of community. Sixty students who enrolled in a college algebra course participated in the study. The results showed significant changes in students' mathematics dispositions after engaging in the modeling activities. Analyses of students' written responses and interview data demonstrated that the modeling tasks associated with students' personal data and social interactions within a group contributed to students' developing their identity as doers of mathematics and creating social value. The instructional model aimed to balance the cognitive aspect and the affective skills of learning mathematics in a way that would allow students to connect mathematical concepts to their personal lives and social lives. As a result of the analysis of this study, there emerged a holistic view of the classroom as it reflects the Makiguchi's educational philosophy. Lastly, implications of this study for research and teaching are discussed.Mathematics education, MathematicsMathematics, Science, and Technology, Mathematics EducationDissertationsResources and Reform: Thinking Through the Costs of a Developmental Math Redesign
http://academiccommons.columbia.edu/catalog/ac:175131
Edgecombe, Nicole Diane; Bickerstaff, Susan E.http://dx.doi.org/10.7916/D85D8Q0FFri, 27 Jun 2014 00:00:00 +0000What college resources are required to implement a statewide redesign of developmental mathematics? This presentation looks at the most significant costs for colleges implementing a math redesign in Virginia, drawing on interviews with faculty, administrators, and staff. It also highlights several considerations for policymakers planning for large-scale instructional reform.Community college education, Mathematics educationnde8, seb2188Institute on Education and the Economy, Community College Research CenterPresentations[A Report on the Future of Statistics]: Comment
http://academiccommons.columbia.edu/catalog/ac:173850
Madigan, David B.; Stuetzle, Wernerhttp://dx.doi.org/10.7916/D8D50K3VTue, 13 May 2014 00:00:00 +0000"Extraordinary opportunities for statistical ideas and for statisticians now present themselves. However, to take advantage of the opportunities, statistics has to change the way in which it recruits and trains students. Statistics has primarily focused on squeezing the maximum amount of information out of limited data. This paradigm is rapidly diminishing in importance and statistics education finds itself out of step with reality. The problems begin at the high school and undergraduate levels, where the standard course includes a narrow set of pre-computing-era topics. At the graduate level, the typical statistics program suffers from the same problem..." -- page 408Mathematics education, Higher educationdm2418StatisticsArticlesMotivation and Study Habits of College Calculus Students: Does Studying Calculus in High School Make a Difference?
http://academiccommons.columbia.edu/catalog/ac:172257
Gibson, Megan E.http://dx.doi.org/10.7916/D8W37TCNTue, 01 Apr 2014 00:00:00 +0000Due in part to the growing popularity of the Advanced Placement program, an increasingly large percentage of entering college students are enrolling in calculus courses having already taken calculus in high school. Many students do not score high enough on the AP calculus examination to place out of Calculus I, and many do not take the examination. These students take Calculus I in college having already seen most or all of the material. Students at two colleges were surveyed to determine whether prior calculus experience has an effect on these students' effort levels or motivation. Students who took calculus in high school did not spend as much time on their calculus coursework as those who did not take calculus, but they were just as motivated to do well in the class and they did not miss class any more frequently. Prior calculus experience was not found to have a negative effect on student motivation or effort. Colleges should work to ensure that all students with prior calculus experience receive the best possible placement, and consider making a separate course for these students, if it is practical to do so.Mathematics educationmeg2154Mathematics, Science, and Technology, Mathematics EducationDissertationsA Pre-Programming Approach to Algorithmic Thinking in High School Mathematics
http://academiccommons.columbia.edu/catalog/ac:174473
Nasar, Audrey Augustahttp://dx.doi.org/10.7916/D8BG2M1MThu, 20 Mar 2014 00:00:00 +0000Given the impact of computers and computing on almost every aspect of society, the ability to develop, analyze, and implement algorithms is gaining more focus. Algorithms are increasingly important in theoretical mathematics, in applications of mathematics, in computer science, as well as in many areas outside of mathematics. In high school, however, algorithms are usually restricted to computer science courses and as a result, the important relationship between mathematics and computer science is often overlooked (Henderson, 1997). The mathematical ideas behind the design, construction and analysis of algorithms, are important for students' mathematical education. In addition, exploring algorithms can help students see mathematics as a meaningful and creative subject. This study provides a review of the history of algorithms and algorithmic complexity, as well as a technical monograph that illustrates the mathematical aspects of algorithmic complexity in a form that is accessible to mathematics instructors at the high school level. The historical component of this study is broken down into two parts. The first part covers the history of algorithms with an emphasis on how the concept has evolved from 3000 BC through the Middle Ages to the present day. The second part focuses on the history of algorithmic complexity, dating back to the text of Ibn al-majdi, a fourteenth century Egyptian astronomer, through the 20th century. In particular, it highlights the contributions of a group of mathematicians including Alan Turing, Michael Rabin, Juris Hartmanis, Richard Stearns and Alan Cobham, whose work in computability theory and complexity measures was critical to the development of the field of algorithmic complexity. The technical monograph which follows describes how the complexity of an algorithm can be measured and analyzes different types of algorithms. It includes divide-and-conquer algorithms, search and sort algorithms, greedy algorithms, algorithms for matching, and geometric algorithms. The methods used to analyze the complexity of these algorithms is done without the use of a programming language in order to focus on the mathematical aspects of the algorithms, and to provide knowledge and skills of value that are independent of specific computers or programming languages. In addition, the study assesses the appropriateness of these topics for use by high school teachers by submitting it for independent review to a panel of experts. The panel, which consists of mathematics and computer science faculty in high school and colleges around the United States, found the material to be interesting and felt that using a pre-programming approach to teaching algorithmic complexity has a great deal of merit. There was some concern, however, that portions of the material may be too advanced for high school mathematics instructors. Additionally, they thought that the material would only appeal to the strongest students. As per the reviewers' suggestions, the monograph was revised to its current form.Mathematics educationaan2112Mathematics, Science, and Technology, Mathematics EducationDissertationsImproving Students’ College Math Readiness: A Review of the Evidence on Postsecondary Interventions and Reforms
http://academiccommons.columbia.edu/catalog/ac:170383
Hodara, Michellehttp://dx.doi.org/10.7916/D8M32SS7Wed, 12 Feb 2014 00:00:00 +0000This paper reviews current research on the effectiveness of interventions and reforms that seek to improve the math preparedness and success of high school students entering college. Based on gaps in the research knowledge, it also provides recommendations for further inquiry in particular areas. The studies reviewed here are selected from research conducted by the Community College Research Center (CCRC) and the National Center for Postsecondary Research (NCPR), from searches of the Education Full Text database for peer-reviewed articles, and from searches of Google Scholar for high-quality reports. The two key criteria for inclusion in the review are that (1) the study in question focuses on (a) an early assessment program in math; (b) a math bridge, boot camp, or brush-up; (c) a reform of developmental math; or (d) improvements to math instruction; and that (2) at least one of the study’s outcomes is related to changes in math or college performance. To evaluate the evidence, I report on each study’s design and findings. I also calculate each intervention’s effect size and categorize the effect size to compare impacts across the studies under review. Overall, the evidence is limited, but some of the interventions and reforms appear promising. The evidence on early assessment is minimal. The evidence on bridges, boot camps, and brush-ups suggests that short-term programs may only have short-term impacts. The evidence on different models of developmental reform varies depending on the reform model. For dominant models, it is positive (for compression models), insignificant (for learning communities), or negative (for modularization). For less prevalent models, it is positive (for mainstreaming) or needs further research (for statistics pathways). In terms of innovations that are strictly pedagogical, the strongest positive evidence is found for using structured forms of student collaboration and for building conceptual understanding through the use of multiple representations when teaching and solving problems. The evidence on computer-mediated instruction in the developmental math classroom is very mixed, with some studies finding positive effects and others finding negative effects.Community college education, Higher education, Mathematics educationCommunity College Research CenterWorking papersCross National Comparisons of Excellence in University Mathematics Instructors - An Analysis of Key Characteristics of Excellent Mathematics Instructors based on Teacher Evaluation Forms
http://academiccommons.columbia.edu/catalog/ac:168517
Grant, Frida Kristinhttp://dx.doi.org/10.7916/D8DJ5CKBMon, 06 Jan 2014 00:00:00 +0000Mathematicians have, historically, not been overly successful in their approach to teaching and much research has looked in to why this is so. Teaching mathematics is based on a solid understanding of the subject; however, instructors also need to be able to efficiently communicate the subject to their students. The purpose of this study was to establish common characteristics of excellent university lecturers in mathematics by applying Marsh's ten evaluation categories. This thesis sought to identify which of these areas were most consistently demonstrated by those university lecturers receiving the highest student ratings and whether there are any areas in which excellent lecturers received inconsistent ratings. The dissertation further used these observations to provide evidence of particular characteristics that are more important than others in the development of excellent university mathematics instructors. This study collected quantitative data in the shape of teacher evaluation forms from both Swedish and US mathematics institutions. The data suggests that instructors acknowledged to be excellent receive high ratings in areas concerning subject matter knowledge, explanatory ability, the fairness of examinations, and enthusiasm and commitment to students. Overall, items that explain a lecturer's persona, character and personality are generally more highly correlated with ratings for the instructor himself whereas categories which describe the preparation, organization and structure of the course, are generally more highly correlated with a student's overall learning experience and Overall Course rating.Mathematics educationfka2107Mathematics EducationDissertationsExploring Algebra-based Problem Solving and Strategies of Spanish-speaking High School Students
http://academiccommons.columbia.edu/catalog/ac:165177
Hernandez-Duhon, Andreahttp://hdl.handle.net/10022/AC:P:21628Fri, 13 Sep 2013 00:00:00 +0000This dissertation analyzes differences found in Spanish-speaking middle school and high school students in algebra-based problem solving. It identifies the accuracy differences between word problems presented in English, Spanish and numerically based problems. The study also explores accuracy differences between each subgroup of Spanish-speaking students in each category. It identifies specific strategies used by successful students when solving algebra problems. The study also sought to identify factors that could serve to predict Spanish-speaking students' ability to accurately solve algebra word problems presented in English and Spanish. A heterogeneous urban sample composed of one hundred and fifty two middle school and high school students were given an assessment composed of pre-approved algebra-based problems and a biographical information sheet. Specific students were then chosen for individual interviews in which researcher sought to gain more in depth information about student's reaction to assessment. The study found that the average accuracy rate for Hispanics non-ELL and non-Hispanic students was significantly higher for numerically based problems than Spanish word problems. Similarly, the average accuracy rate for Hispanics non-ELL and non-Hispanic students was significantly higher in English word problems that in Spanish word problems. Results showed that there was a significant difference in the overall performance of the assessment between Hispanic ELL and Hispanic non-ELL students. On one particular set, set C (Spanish word problems), findings showed that Hispanic ELL students performed better than Hispanic non-ELL students and non-Hispanic students. All other subgroup comparisons did not show a significant difference. The study found that students who were most successful in the assessment: (a) used previous linguistics knowledge and memory of previously seen mathematical problems properly; (b) highlighted the question being asked; (c) used key words to identify mathematical principles and to aid in the translation process; (d) used diagrams, tables and graphs to organize data; (e) showed work and had all processes laid out clearly; and (f) displayed a clear verification process for their answer as strategies for successfully answering the problems. As it was evident through the study, the diversity in the Spanish speaking population and their needs exposes the need for teaching methods, which are inclusive of all populations. Schools must be sensitive to the diversity in which students learn and aim to individualize the teaching for every student. As Hispanics become the largest minority in the United States, understanding the diverse needs of Spanish speaking students in the classroom will be necessary for the development of a better educated society.Mathematics education, Hispanic American studiesach2125Mathematics, Science, and Technology, Mathematics EducationDissertationsWhich Approaches Do Students Prefer? Analyzing the Mathematical Problem Solving Behavior of Mathematically Gifted Students
http://academiccommons.columbia.edu/catalog/ac:161908
Tjoe, Hartono Hardihttp://hdl.handle.net/10022/AC:P:20598Thu, 06 Jun 2013 00:00:00 +0000This study analyzed the mathematical problem solving behavior of mathematically gifted students. It focused on a specific fourth step of Polya's (1945) problem solving process, namely, looking back to find alternative approaches to solve the same problem. Specifically, this study explored problem solving using many different approaches. It examined the relationships between students' past mathematical experiences and the number of approaches and the kind of mathematics topics they used to solve three non-standard mathematics problems. It also analyzed the aesthetic of students' approaches from the perspective of expert mathematicians and the aesthetic of these experts' preferred approaches from the perspective of the students. Fifty-four students from a specialized high school were selected to participate in this study that began with the analysis of their past mathematical experiences by means of a preliminary survey. Nine of the 54 students took a test requiring them to solve three non-standard mathematics problems using many different approaches. A panel of three research mathematicians was consulted to evaluate the mathematical aesthetic of those approaches. Then, these nine students were interviewed. Also, all 54 students took a second survey to support inferences made while observing the problem solving behavior of the nine students. This study showed that students generally were not familiar with the practice of looking back. Indeed, students generally chose to supply only one workable, yet mechanistic approach as long as they obtained a correct answer to the problem. The findings of this study suggested that, to some extent, students' past mathematical experiences were connected with the number of approaches they used when solving non-standard mathematics problems. In particular, the findings revealed that students' most recent exposure of their then-AP Calculus course played an important role in their decisions on selecting approaches for solution. In addition, the findings showed that students' problem solving approaches were considered to be the least "beautiful" by the panel of experts and were often associated with standard approaches taught by secondary school mathematics teachers. The findings confirmed the results of previous studies that there is no direct connection between the experts' and students' views of "beauty" in mathematics.Mathematics educationhht2105Mathematics, Science, and Technology, Mathematics EducationDissertationsApplication of ordered latent class regression model in educational assessment
http://academiccommons.columbia.edu/catalog/ac:161911
Cha, Jisunghttp://hdl.handle.net/10022/AC:P:20599Thu, 06 Jun 2013 00:00:00 +0000Latent class analysis is a useful tool to deal with discrete multivariate response data. Croon (1990) proposed the ordered latent class model where latent classes are ordered by imposing inequality constraints on the cumulative conditional response probabilities. Taking stochastic ordering of latent classes into account in the analysis of data gives a meaningful interpretation, since the primary purpose of a test is to order students on the latent trait continuum. This study extends Croon's model to ordered latent class regression that regresses latent class membership on covariates (e.g., gender, country) and demonstrates the utilities of an ordered latent class regression model in educational assessment using data from Trends in International Mathematics and Science Study (TIMSS). The benefit of this model is that item analysis and group comparisons can be done simultaneously in one model. The model is fitted by maximum likelihood estimation method with an EM algorithm. It is found that the proposed model is a useful tool for exploratory purposes as a special case of nonparametric item response models and cross-country difference can be modeled as different composition of discrete classes. Simulations is done to evaluate the performance of information criteria (AIC and BIC) in selecting the appropriate number of latent classes in the model. From the simulation results, AIC outperforms BIC for the model with the order-restricted maximum likelihood estimator.Educational tests and measurements, Statistics, Mathematics educationjc2320Human Development, Measurement and EvaluationDissertationsSocial Capital and Adolescents Mathematics Achievement: A Comparative Analysis of Eight European Cities
http://academiccommons.columbia.edu/catalog/ac:161622
Gisladottir, Berglindhttp://hdl.handle.net/10022/AC:P:20481Thu, 30 May 2013 00:00:00 +0000This study examines the impact of social capital on mathematics achievement in eight European cities. The study draws on data from the 2008 Youth in Europe survey, carried out by the Icelandic Center for Social Research and Analysis. The sample contains responses from 17,312 students in 9th and 10th grade of local secondary schools in the following cities: Bucharest in Romania, Kaunas, Klaípéda and Vilnius in Lithuania, Reykjavík in Iceland, Riga and Jurmala in Latvia and Sofia in Bulgaria. The study builds on social capital theory presented in 1988 by the American sociologist James Coleman. He argued that social capital in both family and community is a key factor in the creation of human capital, meaning that children that possess more social capital in their lives will do better in school. Several prior studies have empirically supported the theory, although most of those studies were carried out in the United States. The current study tests whether the theory of social capital holds across different cultures. The findings partly support the theory, showing that the key measures of social capital are positively correlated with mathematics achievement in all of the cities. The impact however was less in many of the cities than expected. Additionally, Coleman's key social capital variable did not positively associate with mathematics achievement in cities around Europe. The implications of that finding are discussed in the thesis.Mathematics education, Sociologybg2347Mathematics, Science, and Technology, Mathematics EducationDissertationsThe Mathematical Content Knowledge of Prospective Teachers in Iceland
http://academiccommons.columbia.edu/catalog/ac:161616
Johannsdottir, Bjorghttp://hdl.handle.net/10022/AC:P:20479Thu, 30 May 2013 00:00:00 +0000This study focused on the mathematical content knowledge of prospective teachers in Iceland. The sample was 38 students in the School of Education at the University of Iceland, both graduate and undergraduate students. All of the participants in the study completed a questionnaire survey and 10 were interviewed. The choice of ways to measure the mathematical content knowledge of prospective teachers was grounded in the work of Ball and the research team at the University of Michigan (Delaney, Ball, Hill, Schilling, and Zopf, 2008; Hill, Ball, and Schilling, 2008; Hill, Schilling, and Ball, 2004), and their definition of common content knowledge (knowledge held by people outside the teaching profession) and specialized content knowledge (knowledge used in teaching) (Ball, Thames, and Phelps, 2008). This study employed a mixed methods approach, including both a questionnaire survey and interviews to assess prospective teachers' mathematical knowledge on the mathematical topics numbers and operations and patterns, functions, and algebra. Findings, both from the questionnaire survey and the interviews, indicated that prospective teachers' knowledge was procedural and related to the "standard algorithms" they had learned in elementary school. Also, findings indicated that prospective teachers had difficulties evaluating alternative solution methods, and a common denominator for a difficult topic within both knowledge domains, common content knowledge and specialized content knowledge, was fractions. During the interviews, the most common answer for why a certain way was chosen to solve a problem or a certain step was taken in the solution process, was "because that is the way I learned to do it." Prospective teachers' age did neither significantly influence their test scores, nor their approach to solving problems during the interviews. Supplementary analysis revealed that number of mathematics courses completed prior to entering the teacher education program significantly predicted prospective teachers' outcome on the questionnaire survey.Comparison of the findings from this study to findings from similar studies carried out in the US indicated that there was a wide gap in prospective teachers' ability in mathematics in both countries, and that they struggled with similar topics within mathematics. In general, the results from this study were in line with prior findings, showing, that prospective elementary teachers relied on memory for particular rules in mathematics, their knowledge was procedural and they did not have an underlying understanding of mathematical concepts or procedures (Ball, 1990; Tirosh and Graeber, 1989; Tirosh and Graeber, 1990; Simon, 1993; Mewborn, 2003; Hill, Sleep, Lewis, and Ball, 2007). The findings of this study highlight the need for a more in-depth mathematics education for prospective teachers in the School of Education at the University of Iceland. It is not enough to offer a variety of courses to those specializing in the field of mathematics education. It is also important to offer in-depth mathematics education for those prospective teachers focusing on general education. If those prospective teachers teach mathematics, they will do so in elementary school where students are forming their identity as mathematics students.Mathematics education, Teacher educationbj2231Mathematics, Science, and Technology, Mathematics EducationDissertationsMathematical Word Problem Solving of Students with Autism Spectrum Disorders and Students with Typical Development
http://academiccommons.columbia.edu/catalog/ac:174825
Bae, Young Sehhttp://hdl.handle.net/10022/AC:P:20457Fri, 24 May 2013 00:00:00 +0000Mathematical Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development - Young Seh Bae - This study investigated mathematical word problem solving and the factors associated with the solution paths adopted by two groups of participants (N=40), students with autism spectrum disorders (ASDs) and typically developing students in fourth and fifth grade, who were comparable on age and IQ (greater than 80). The factors examined in the study were: word problem solving accuracy; word reading/decoding; sentence comprehension; math vocabulary; arithmetic computation; everyday math knowledge; attitude toward math; identification of problem type schemas; and visual representation. Results indicated that the students with typical development significantly outperformed the students with ASDs on word problem solving and everyday math knowledge. Correlation analysis showed that word problem solving performance of the students with ASDs was significantly associated with sentence comprehension, math vocabulary, computation and everyday math knowledge, but that these relationships were strongest and most consistent in the students with ASDs. No significant associations were found between word problem solving and attitude toward math, identification of schema knowledge, or visual representation for either diagnostic group. Additional analyses suggested that everyday math knowledge may account for the differences in word problem solving performance between the two diagnostic groups. Furthermore, the students with ASDs had qualitatively and quantitatively weaker structure of everyday math knowledge compared to the typical students. The theoretical models of the linguistic approach and the schema approach offered some possible explanations for the word problem solving difficulties of the students with ASDs in light of the current findings. That is, if a student does not have an adequate level of everyday math knowledge about the situation described in the word problem, he or she may have difficulties in constructing a situation model as a basis for problem comprehension and solutions. It was suggested that the observed difficulties in math word problem solving may have been strongly associated with the quantity and quality of everyday math knowledge as well as difficulties with integrating specific math-related everyday knowledge with the global text of word problems. Implications for this study include a need to develop mathematics instructional approaches that can teach students to integrate and extend their everyday knowledge from real-life contexts into their math problem-solving process. Further research is needed to confirm the relationships found in this study, and to examine other areas that may affect the word problem solving processes of students with ASDs.Special education, Mathematics educationysb2102Health and Behavior Studies, Intellectual Disabilities-AutismDissertationsTeachers' Conceptions of Mathematical Modeling
http://academiccommons.columbia.edu/catalog/ac:161497
Gould, Heather Tianahttp://hdl.handle.net/10022/AC:P:20442Thu, 23 May 2013 00:00:00 +0000The release of the Common Core State Standards for Mathematics in 2010 resulted in a new focus on mathematical modeling in United States curricula. Mathematical modeling represents a way of doing and understanding mathematics new to most teachers. The purpose of this study was to determine the conceptions and misconceptions held by teachers about mathematical models and modeling in order to aid in the development of teacher education and professional development programs. The study used a mixed methods approach. Quantitative data were collected through an online survey of a large sample of practicing and prospective secondary teachers of mathematics in the United States. The purpose of this was to gain an understanding of the conceptions held by the general population of United States secondary mathematics teachers. In particular, basic concepts of mathematical models, mathematical modeling, and mathematical modeling in education were analyzed. Qualitative data were obtained from case studies of a small group of mathematics teachers who had enrolled in professional development which had mathematical models or modeling as a focus. The purpose of these case studies was to give an illustrative view of teachers regarding modeling, as well as to gain some understanding of how participating in professional development affects teachers' conceptions. The data showed that US secondary mathematics teachers hold several misconceptions about models and modeling, particularly regarding aspects of the mathematical modeling process. Specifically, the majority of teachers do not understand that the mathematical modeling process always requires making choices and assumptions, and that mathematical modeling situations must come from real-world scenarios. A large minority of teachers have misconceptions about various other characteristics of mathematical models and the mathematical modeling process.Mathematics educationhtg2103Mathematics, Science, and Technology, Mathematics EducationDissertationsStrategy Instruction in Early Childhood Math Software: Detecting and Teaching Single-digit Addition Strategies
http://academiccommons.columbia.edu/catalog/ac:160522
Carpenter, Kara Kilmartinhttp://hdl.handle.net/10022/AC:P:20091Wed, 01 May 2013 00:00:00 +0000In early childhood mathematics, strategy-use is an important indicator of children's conceptual understanding and is a strong predictor of later math performance. Strategy instruction is common in many national curricula, yet is virtually absent from most math software. The current study describes the design of one software activity teaching single-digit addition strategies. The study explores the effectiveness of the software in detecting the strategies first-graders use and teaching them to use more efficient strategies. Instead of a business-as-usual control group, the study explores the effects of one aspect of the software: the pedagogical agent, investigating whether multiple agents are more effective than a single agent when teaching about multiple strategies. The study finds that while children do not accurately report their own strategies, the software log is able to detect the strategies that children use and is particularly adept at detecting the effective use of an advanced strategy with a model that performs 67% better than chance. Overall, children improve in their accuracy, speed, and use of advanced strategies. Of the three teaching tools available to the children, the count on tool was most effective in encouraging use of an advanced strategy, highlighting a need to revise the other tools. Low-performers correctly used advanced strategies more frequently across the six sessions, while mid-performers improved after just one session and high-performers' correct use of an advanced strategy was consistent across the sessions. Whether a student saw lessons featuring a single agent or multiple agents did not have strong effects on performance. More research is needed to improve the strategy detection models, refine the tools and lessons, and explore other features of the software.Cognitive psychology, Educational technology, Mathematics educationkkc2123Cognitive Studies in Education, Human DevelopmentDissertationsExamining the Effects of Gender, Poverty, Attendance, and Ethnicity on Algebra, Geometry, and Trigonometry Performance in a Public High School
http://academiccommons.columbia.edu/catalog/ac:160486
Shafiq, Hasanhttp://hdl.handle.net/10022/AC:P:20075Wed, 01 May 2013 00:00:00 +0000Over the last few decades school accountability for student performance has become an issue at the forefront of education. The federal No Child Left Behind Act of 2001 (NCLB) and various regulations by individual states have set standards for student performance at both the district and individual public and charter school levels, and certain consequences apply if the performance of students in an institution is deemed unsatisfactory. Conversely, rewards come to districts or schools that perform especially well or make a certain degree of improvement over their earlier results. Albeit with certain conditions, the federal government makes additional education money available to the states under NCLB. While testing is nothing new in American public education, the concept of district/school accountability for performance is at least relatively so. In New York City, where New York State Regents Examinations (NYSRE) have been a measure of student performance for many years, scores on these tests are low, often preventing students from receiving course credit, which in turn results in failure to graduate on schedule. In addition, rates of graduation from public high schools are low. The city and state have kept data on student performance broken out by a number of factors including socioeconomic status, ethnicity, attendance, and gender which point to an achievement gap among different groups. This study investigates a series of those factors associated with the mastery of high school Algebra, Geometry, and Trigonometry. This study concerns itself specifically with the effect that gender, socioeconomic status, attendance, and ethnicity may have on student achievement in a mathematics course and on standardized tests, specifically the NYSRE, an annual rite of passage for students in grades 9 through 11. This research considered and ran tests on data gathered from a single large New York City high school. In this study, a 12 two-way (between-groups) univariate analyses of variance (ANOVAs) were conducted to assess whether there were differences in students' mathematics achievement scores by gender, ethnicity, attendance, and family socio-economic status (SES). In addition, three Pearson correlation analyses were conducted to determine whether there was a correlation among Integrated Algebra, Geometry, and Algebra II/Trigonometry unit examination scores and Regents scores. Nine Pearson correlation analyses were conducted to determine whether there was a correlation between Regents scores and mathematics achievement unit examination scores. A correlation was run between each mathematics achievement score with the Regents score from each subject. Six two-way (between-groups) ANOVA were also conducted to assess whether there were difference in students' mathematics achievement among Black males, Black females, Hispanic males, and Hispanic females. Data were gathered, merged, and transferred into a Statistical Package for the Social Sciences (SPSS) 19.0 (IBM, 2010) for analysis. The findings indicate that attendance and family SES have a meaningful relationship to mathematics achievement in the New York City public high school which was the subject of this investigation. On the other hand, gender and ethnicity showed no relationship to students' mathematics achievement. As an implication of this research, school policies must focus more on the achievement gap of students from low-SES families and must encourage students to maintain good attendance. Students should have access to different forms of academic interventions that go beyond after-school or Saturday tutoring; academic intervention services; community counseling or mediation; or peer intervention or peer counseling through which students learn basic mathematics skills from each other to achieve college readiness.Mathematics education, Mathematicsmhs2143Mathematics, Science, and Technology, Mathematics EducationDissertationsTetrahedra and Their Nets: Mathematical and Pedagogical Implications
http://academiccommons.columbia.edu/catalog/ac:160274
Mussa, Deregehttp://hdl.handle.net/10022/AC:P:20037Tue, 30 Apr 2013 00:00:00 +0000If one has three sticks (lengths), when can you make a triangle with the sticks? As long as any two of the lengths sum to a value strictly larger than the third length one can make a triangle. Perhaps surprisingly, if one is given 6 sticks (lengths) there is no simple way of telling if one can build a tetrahedron with the sticks. In fact, even though one can make a triangle with any triple of three lengths selected from the six, one still may not be able to build a tetrahedron. At the other extreme, if one can make a tetrahedron with the six lengths, there may be as many 30 different (incongruent) tetrahedra with the six lengths. Although tetrahedra have been studied in many cultures (Greece, India, China, etc.) Over thousands of years, there are surprisingly many simple questions about them that still have not been answered. This thesis answers some new questions about tetrahedra, as well raising many more new questions for researchers, teachers, and students. It also shows in an appendix how tetrahedra can be used to illustrate ideas about arithmetic, algebra, number theory, geometry, and combinatorics that appear in the Common Cores State Standards for Mathematics (CCSS -M). In particular it addresses representing three-dimensional polyhedra in the plane. Specific topics addressed are a new classification system for tetrahedra based on partitions of an integer n, existence of tetrahedra with different edge lengths, unfolding tetrahedra by cutting edges of tetrahedra, and other combinatorial aspects of tetrahedra.Mathematics education, Mathematicsdhm2114Mathematics, Science, and Technology, Mathematics EducationDissertationsMathematics Self-Efficacy and Its Relation to Profiency-Promoting Behavior and Performance
http://academiccommons.columbia.edu/catalog/ac:160627
Causapin, Mark Gabrielhttp://hdl.handle.net/10022/AC:P:19428Mon, 25 Mar 2013 00:00:00 +0000The purpose of this study was to verify Bandura's theory on the relationship of self-efficacy and performance particularly in mathematics among high school students. A rural school in the Philippines was selected for its homogenous student population, effectively reducing the effects of confounding variables such as race, ethnic and cultural backgrounds, socioeconomic status, and language. It was shown that self-efficacy was a positive but minor predictor of future performance only for male students who previously had higher mathematics grades. The effects were different between genders. It was not a strong predictor for women regardless of previous grades, and men with weaker mathematics skills. On the other hand, mathematics self-efficacy was predicted by previous mathematics achievement for women; and also the number of siblings and parental education for the higher performing women. The use of a second language in the mathematics classroom negatively affected confidence and performance. It was also found that there were differences in terms of academic behavior, peers, and family life between students with high and low self-efficacy. Positive behaviors were found for all female students regardless of self-efficacy levels and fewer were found among men. Negative behaviors were only found among low self-efficacy students. No differences were found in terms of the lives and families of the participants, but the interviews revealed that family members and their experiences of poverty affected educational goals and ambitions. In terms of other dispositional factors, students expressed classroom and test anxieties, concerns of being embarrassed in front of their classmates, and beliefs that mathematics was naturally difficult and not enjoyable. The students who did not talk about any of these themes were better performing and had higher self-efficacy scores.Mathematics education, Educational psychologyMathematics, Science, and Technology, Mathematics EducationDissertationsGood Mathematics Teaching: Perspectives of Beginning Secondary Teachers
http://academiccommons.columbia.edu/catalog/ac:159494
Leong, Kwan Euhttp://hdl.handle.net/10022/AC:P:19322Mon, 11 Mar 2013 00:00:00 +0000What is good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own view on what they consider is good mathematics teaching and what is not. The purpose of this study was to determine how beginning teachers define good mathematics teaching and what they report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. In addition, factors that influence the understanding of good mathematics teaching were explored. A mixed methodology was used to gather information from the research participants regarding their beliefs and classroom practices of good mathematics teaching. The two research instruments used in this study were the survey questionnaire and a semi-structured interview. Thirty-three respondents who had one to two years of classroom experience comprised the study sample. They had graduated from a school of education in an eastern state and had obtained their teacher certification upon completing their studies. The beginning mathematics teachers selected these four definitions of good teaching as their top choices: 1) have High Expectations that all students are capable of learning; 2) have strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to classroom. The three most important attributes in good teaching were: Classroom Management, Motivation, and Strong in Content Knowledge. One interesting finding was the discovery of four groups of beginning teachers and how they were associated with specific attributes of good mathematics teaching according to their demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey analysis that influenced their understanding of good mathematics teaching. The study's results have implications for informing the types of mathematical knowledge required for pre-service teachers that can be incorporated into teacher education programs and define important attributes of good mathematics teaching during practicum.Mathematics education, Mathematics, Teacher educationMathematics, Science, and Technology, Mathematics EducationDissertationsWhen beginning mathematics teachers report acquiring successful attributes: Reflections on teacher education
http://academiccommons.columbia.edu/catalog/ac:157016
Wasserman, Nicholashttp://hdl.handle.net/10022/AC:P:19103Mon, 18 Feb 2013 00:00:00 +0000Education plays a vital role in any society; so much so, that countries strive to have not only adequate, but excellent educators in their classrooms. The aim of this study was to understand how beginning secondary mathematics teachers define success and to what experiences they attribute that success. Specifically, the central research question addressed was, "To what degree were significant attributes or experiences, important to the success of the first year teaching, learned pre-teacher education program, during a program, or post-program?" The practical goal of filling classrooms with great educators needs to be informed by research on how best to recruit highly qualified candidates into the field of mathematics education and how best to facilitate the teacher preparation process. This study employed a mixed methodology, using a sample of beginning secondary mathematics teachers to gather both quantitative and qualitative data on when they reported gaining influential knowledge or experiences. In particular, input from those who have had some success as beginning mathematics teachers was desired. The interview protocol designed for these participants added depth to the survey responses. Emphasis was placed on the relative importance of the three stages, pre-, during, and post-program, in developing common attributes associated with good teaching. Two characteristics were generally discussed as developing pre-program: being a self-starting and hard-working individual, and holding a belief that every student can learn. Beginning teachers viewed these traits as important for their success. Participants also felt that they acquired both practical classroom tools and educational theory from their teacher education program; having program instructors model pedagogy and mathematical instruction, and having opportunities to practice incorporating theory into their teaching were also seen as important. These aspects distinguished particularly prominent roles that the teacher education program played in shaping its graduates. Classroom management and being flexible and adaptive to different contexts were the most notable qualities frequently reported as being learned post-program. The study's results have implications for informing the types of students a mathematics education program should try to attract or recruit and defining areas where practicum or internship components might be incorporated into the teacher education process.Mathematics education, Teacher education, Secondary educationMathematics, Science, and Technology, Mathematics EducationDissertationsProof and Reasoning in Secondary School Algebra Textbooks
http://academiccommons.columbia.edu/catalog/ac:156775
Dituri, Philip Charleshttp://hdl.handle.net/10022/AC:P:19092Fri, 15 Feb 2013 00:00:00 +0000The purpose of this study was to determine the extent to which the modeling of deductive reasoning and proof-type thinking occurs in a mathematics course in which students are not explicitly preparing to write formal mathematical proofs. Algebra was chosen because it is the course that typically directly precedes a student's first formal introduction to proof in geometry in the United States. The lens through which this study aimed to examine the intended curriculum was by identifying and reviewing the modeling of proof and deductive reasoning in the most popular and widely circulated algebra textbooks throughout the United States. Textbooks have a major impact on mathematics classrooms, playing a significant role in determining a teacher's classroom practices as well as student activities. A rubric was developed to analyze the presence of reasoning and proof in algebra textbooks, and an analysis of the coverage of various topics was performed. The findings indicate that, roughly speaking, students are only exposed to justification of mathematical claims and proof-type thinking in 38% of all sections analyzed. Furthermore, only 6% of coded sections contained an actual proof or justification that offered the same ideas or reasoning as a proof. It was found that when there was some justification or proof present, the most prevalent means of convincing the reader of the truth of a concept, theorem, or procedure was through the use of specific examples. Textbooks attempting to give a series of examples to justify or convince the reader of the truth of a concept, theorem, or procedure often fell short of offering a mathematical proof because they lacked generality and/or, in some cases, the inductive step. While many textbooks stated a general rule at some point, most only used deductive reasoning within a specific example if at all. Textbooks rarely expose students to the kinds of reasoning required by mathematical proof in that they rarely expose students to reasoning about mathematics with generality. This study found a lack of sufficient evidence of instruction or modeling of proof and reasoning in secondary school algebra textbooks. This could indicate that, overall, algebra textbooks may not fulfill the proof and reasoning guidelines set forth by the NCTM Principles and Standards and the Common Core State Standards. Thus, the enacted curriculum in mathematics classrooms may also fail to address the recommendations of these influential and policy defining organizations.Mathematics education, Mathematics, Educationpcd2102Mathematics, Science, and Technology, Mathematics EducationDissertationsThe History of Hebrew Secondary Mathematics Education in Palestine During the First Half of the Twentieth Century
http://academiccommons.columbia.edu/catalog/ac:156161
Aricha-Metzer, Inbarhttp://hdl.handle.net/10022/AC:P:18923Mon, 04 Feb 2013 00:00:00 +0000This dissertation traces the history of mathematics education in Palestine Hebrew secondary schools from the foundation of the first Hebrew secondary school in 1905 until the establishment of the State of Israel in 1948. The study draws on primary sources from archives in Israel and analyzes curricula, textbooks, student notebooks, and examinations from the first half of the 20th century as well as reviews in contemporary periodicals and secondary sources. Hebrew secondary mathematics education was developed as part of the establishment of a new nation with a new educational system and a new language. The Hebrew educational system was generated from scratch in the early 20th century; mathematical terms in Hebrew were invented at the time, the first Hebrew secondary schools were founded, and the first Hebrew mathematics textbooks were created. The newly created educational system encountered several dilemmas and obstacles: the struggle to maintain an independent yet acknowledged Hebrew educational system under the British Mandate; the difficulties of constructing the first Hebrew secondary school curriculum; the issue of graduation examinations; the fight to teach all subjects in the Hebrew language; and the struggle to teach without textbooks or sufficient Hebrew mathematical terms. This dissertation follows the path of the development of Hebrew mathematics education and the first Hebrew secondary schools in Palestine, providing insight into daily school life and the turbulent history of Hebrew mathematics education in Palestine.Mathematics education, History of education, Secondary educationia2213Mathematics EducationDissertationsAnalysis of Mathematical Fiction with Geometric Themes
http://academiccommons.columbia.edu/catalog/ac:153198
Shloming, Jennifer Rebeccahttp://hdl.handle.net/10022/AC:P:14870Wed, 10 Oct 2012 00:00:00 +0000Analysis of mathematical fiction with geometric themes is a study that connects the genre of mathematical fiction with informal learning. This study provides an analysis of 26 sources that include novels and short stories of mathematical fiction with regard to plot, geometric theme, cultural theme, and presentation. The authors' mathematical backgrounds are presented as they relate to both geometric and cultural themes. These backgrounds range from having little mathematical training to advance graduate work culminating in a Ph.D. in mathematics. This thesis demonstrated that regardless of background, the authors could write a mathematical fiction novel or short story with a dominant geometric theme. The authors' pedagogical approaches to delivering the geometric themes are also discussed. Applications from this study involve a pedagogical component that can be used in a classroom setting. All the sources analyzed in this study are fictional, but the geometric content is factual. Six categories of geometric topics were analyzed: plane geometry, solid geometry, projective geometry, axiomatics, topology, and the historical foundations of geometry. Geometry textbooks aligned with these categories were discussed with regard to mathematical fiction and formal learning. Cultural patterns were also analyzed for each source of mathematical fiction. There were also an analysis of the integration of cultural and geometric themes in the 26 sources of mathematical fiction; some of the cultural patterns discussed are gender bias, art, music, academia, mysticism, and social issues. On the basis of this discussion, recommendations for future studies involving the use of mathematical fiction were made.Mathematics education, Mathematicsjrs2137Mathematics, Science, and Technology, Mathematics EducationDissertationsThe Use of Cartoons as Teaching a Tool in Middle School Mathematics
http://academiccommons.columbia.edu/catalog/ac:149393
Cho, Hoyunhttp://hdl.handle.net/10022/AC:P:13930Mon, 09 Jul 2012 00:00:00 +0000This dissertation focuses on examining the use of mathematical cartoons as a teaching tool in middle school mathematics classroom. A mixed methods research design was used to answer how the use of cartoon activities affects teacher and student perceptions of teaching and learning and student intrinsic motivation, interest, and mathematics anxiety in middle school mathematics. 17 students in 7th grade pre-algebra class and one teacher participated in this study. Eight cartoon activities were provided over a 10-week period, but no more than one cartoon activity per class period was given to them. Student surveys were analyzed using quantitative method, such as mean score, frequency, and percentage, and student mathematics journal and teacher journal were analyzed using descriptive analysis. The results of this study revealed that both students and teacher reported positive results from using cartoons in the mathematics classroom. Students became more open as time went on and it was possible to see their mathematical insights as the study progressed. They did not enjoy easy cartoon activities, but relished challenging ones. Their frustration at difficult-to-understand activities shows the importance of carefully matching cartoon activities to student abilities. When cartoon activities have appropriate levels of difficulty and are clearly understandable, students' intrinsic motivation and interest increased, and mathematics anxiety decreased. The teacher reported that students gave up less easily, participated more readily, and were more focused in classes with cartoon activities. Mathematics instruction with cartoon activities has shown the students that they can enjoy learning mathematics, mathematics can be fun, and they do have the ability to be successful in mathematics. The use of cartoon activity proved to be a valuable instructional tool for improving the quality of mathematics instruction in a 7th grade classroom.Mathematics education, Teacher education, Educationhc2483Mathematics, Science, and Technology, Mathematics EducationDissertationsReforming Mathematics Classroom Pedagogy: Evidence-Based Findings and Recommendations for the Developmental Math Classroom
http://academiccommons.columbia.edu/catalog/ac:146949
Hodara, Michellehttp://hdl.handle.net/10022/AC:P:13232Thu, 17 May 2012 00:00:00 +0000For developmental education students, rates of developmental math course completion and persistence into required college-level math courses are particularly low. This Brief examines the evidence base on reforming mathematics classroom pedagogy, which may be a potential means for improving the course completion and learning outcomes of developmental mathematics students. Each study examined was classified into one of six sets: student collaboration, metacognition, problem representation, application, understanding student thinking, and computer-based learning. Because most of the studies across the sets did not employ rigorous methods, the evidence regarding the impact of these instructional practices on student outcomes is inconclusive. Nevertheless, analysis of the studies that did employ rigorous designs suggests that structured forms of student collaboration and instructional approaches that focus on problem representation may improve math learning and understanding. This Brief concludes by making a number of methodological recommendations, proposing several needed areas of research, and suggesting instructional practices that may improve the outcomes of developmental math students.Mathematics educationmeh70Economics and Education, Institute on Education and the Economy, Community College Research CenterReportsReforming Mathematics Classroom Pedagogy: Evidence-Based Findings and Recommendations for the Developmental Math Classroom
http://academiccommons.columbia.edu/catalog/ac:146653
Hodara, Michellehttp://hdl.handle.net/10022/AC:P:13144Fri, 04 May 2012 00:00:00 +0000For developmental education students, rates of developmental math course completion and persistence into required college-level math courses are particularly low. This literature review examines the evidence base on reforming mathematics classroom pedagogy, which may be a potential means for improving the course completion and learning outcomes of developmental mathematics students. Each study examined for this review was classified into one of six sets: student collaboration, metacognition, problem representation, application, understanding student thinking, and computer-based learning. Because most of the studies across the sets did not employ rigorous methods, the evidence regarding the impact of these instructional practices on student outcomes is inconclusive. Nevertheless, analysis of the studies that did employ rigorous designs suggests that structured forms of student collaboration and instructional approaches that focus on problem representation may improve math learning and understanding. This paper concludes by making a number of methodological recommendations, proposing several needed areas of research, and suggesting instructional practices that may improve the outcomes of developmental math students.Mathematics educationmeh70Economics and Education, Institute on Education and the Economy, Community College Research CenterWorking papersPromoting the Development of an Integrated Numerical Representation through the Coordination of Physical Materials
http://academiccommons.columbia.edu/catalog/ac:146450
Vitale, Jonathan Michaelhttp://hdl.handle.net/10022/AC:P:13078Tue, 01 May 2012 00:00:00 +0000How do children use physical and virtual tools to develop new numerical knowledge? While concrete instructional materials may support the delivery of novel information to learners, they may also over-simplify the task, unintentionally reducing learners' performance in recall and transfer tasks. This reduction in testing performance may be mitigated by embedding physical incongruencies in the design of instructional materials. The effort of resolving this incongruency can foster a richer understanding of the underlying concept. In two experiments children were trained on a computerized number line estimation task, with a novel scale (0-180), and then asked to perform a series of posttest number line estimation tasks that varied spatial features of the training number line. In experiment 1, during training with feedback, children either received a ruler depicting endpoint and quartile magnitudes (i.e., 0, 45, 90, 135, 180) that physically matched the on-screen number line (congruent ruler), a proportionally-similar ruler scaled 33% larger than the on-screen number line (incongruent ruler), or no ruler. Children were trained to criterion before proceeding to posttest. Results indicated that while children who used the congruent ruler performed well during training, their performance at posttest was less accurate than the other two conditions. On the other hand, by increasing the difficulty of the learning task, while providing relevant landmark information, children in the incongruent ruler condition produced the highest accuracy at posttest. In experiment 2, controlling for learning task duration, the incongruent ruler and congruent ruler conditions were compared directly. Posttest results confirmed an advantage for children in the more complex, incongruent ruler condition. These results are interpreted to suggest that landmarks representations are an important and accessible means of developing a mature numerical representation of the number line. Furthermore, the results confirm that desirable difficulties are an essential component of the learning process. Potential implications for the design of learning activities that balance instructional support with conceptual challenge are discussed.Cognitive psychology, Mathematics education, Developmental psychologyjmv2125Cognitive Studies in Education, Human DevelopmentDissertationsA Cabinet of Mathematical Curiosities at Teachers College: David Eugene Smith's Collection
http://academiccommons.columbia.edu/catalog/ac:185885
Murray, Diane Rosehttp://dx.doi.org/10.7916/D8RV0MPWMon, 30 Apr 2012 00:00:00 +0000This dissertation is a history of David Eugene Smith's collection of historical books, manuscripts, portraits, and instruments related to mathematics. The study analyzes surviving documents, images, objects, college announcements and catalogs, and secondary sources related to Smith's collection. David Eugene Smith (1860 - 1944) travelled the world in search of rare and interesting pieces of mathematics history. He enjoyed sharing these experiences and objects with his family, friends, colleagues, and students. Smith's collection had a remarkable journey itself. It was once part of the Educational Museum of Teachers College. This museum existed from 1899 - 1914 and was quite popular among educators and students. Smith was director of the museum beginning in 1909, although, he had a major influence on the museum from the moment he began his professorship at Teachers College in 1901. After the Educational Museum of Teachers College disbanded, the collection was exhibited in numerous venues. George A. Plimpton (1855 - 1936) created the Permanent Educational Exhibit that housed both modern educational items, as well as, historical pieces for display. Since Smith and Plimpton were great friends and fellow collectors, Smith's collection was included in the historical section of Plimpton's establishment. Unfortunately, due to the hard times of the world at this moment, the Permanent Educational Exhibit closed in 1917. Smith continued to exhibit his collection of mathematical artifacts through the Museums of the Peaceful Arts, founded by George F. Kunz (1856 - 1932), the New York Museum of Science and Industry, Teachers College, and Columbia University. Smith's research, teaching, and publications were directly influenced by his collection. Throughout most of his published works are images and photographs of items in his collection. He also believed in the importance of having primary sources included in mathematics education. This view he followed in his own teaching, which included research in his collection. David Eugene Smith's collection could never be replicated and thus is quite unique and valuable. Smith donated his collection to Columbia University's Libraries in the 1930s. Various exhibits of his collection have occurred since then, the most recent concluded in 2003. The history of Smith's mathematical collection is important to the history of mathematics education as it displays the importance of preserving mathematical books, manuscripts, portraits, and instruments for future generations to research.Mathematics education, History of education, Museum studiesdrm2132Mathematics, Science, and Technology, Mathematics EducationDissertationsAsian American college students' mathematics success and the model minority stereotype
http://academiccommons.columbia.edu/catalog/ac:146308
Jo, Lydia Hyeryunghttp://hdl.handle.net/10022/AC:P:13045Thu, 19 Apr 2012 00:00:00 +0000The often aggregated reports of academic excellence of Asian American students as a whole, compared to students from other ethnic groups offers compelling evidence that Asian Americans are more academically successful than their ethnic counterparts, particularly in the area of mathematics. These comparative data have generated many topics of discussion including the model minority stereotype: a misconception that all Asian Americans are high academic achievers. Research has shown that this seemingly positive stereotype produces negative effects in Asian students. The aim of this study is to examine differences in mathematics success levels and beliefs about the model minority stereotype among different generations of Asian American college students. This study focuses on comparing three different generations of Asian American students with respect to: (1) their success and confidence in mathematics, (2) their personal views on the factors that contribute to their success, (3) their perceptions of the model minority stereotype and (4) how they believe the stereotype affects them. In this mixed methods study, a sample of n = 117 Asian American college students participated in an online survey to collect quantitative data and a subsample of n = 9 students were able to participate in a semi-structured interview. The results of the study indicated that there were almost no differences in either the mathematics success and confidence level, or in the perceptions and perceived effects of the model minority stereotype across generations. Quantitative results showed that all generations of Asian Americans generally are confident in their mathematics abilities. Qualitative analysis showed that the students felt that there were three reasons for their level of success: parental influence, differences in the education system between the U.S. and their home country, and using mathematics and science to get ahead academically as their native English speaking peers tend to be ahead of them in the liberal arts due to language barriers. Though there were mixed feelings among the sample subjects about the validity of the model minority stereotype, all three generations of Asian American students felt peer pressure from the stereotype to excel in mathematics, more frequently in high school than in college.Mathematics educationlhj2107Mathematics, Science, and Technology, Mathematics EducationDissertationsStatistics for Learning Genetics
http://academiccommons.columbia.edu/catalog/ac:146201
Charles, Abigail Sheenahttp://hdl.handle.net/10022/AC:P:13015Tue, 17 Apr 2012 00:00:00 +0000This study investigated the knowledge and skills that biology students may need to help them understand statistics/mathematics as it applies to genetics. The data are based on analyses of current representative genetics texts, practicing genetics professors' perspectives, and more directly, students' perceptions of, and performance in, doing statistically-based genetics problems. This issue is at the emerging edge of modern college-level genetics instruction, and this study attempts to identify key theoretical components for creating a specialized biological statistics curriculum. The goal of this curriculum will be to prepare biology students with the skills for assimilating quantitatively-based genetic processes, increasingly at the forefront of modern genetics. To fulfill this, two college level classes at two universities were surveyed. One university was located in the northeastern US and the other in the West Indies. There was a sample size of 42 students and a supplementary interview was administered to a select 9 students. Interviews were also administered to professors in the field in order to gain insight into the teaching of statistics in genetics. Key findings indicated that students had very little to no background in statistics (55%). Although students did perform well on exams with 60% of the population receiving an A or B grade, 77% of them did not offer good explanations on a probability question associated with the normal distribution provided in the survey. The scope and presentation of the applicable statistics/mathematics in some of the most used textbooks in genetics teaching, as well as genetics syllabi used by instructors do not help the issue. It was found that the text books, often times, either did not give effective explanations for students, or completely left out certain topics. The omission of certain statistical/mathematical oriented topics was seen to be also true with the genetics syllabi reviewed for this study. Nonetheless, although the necessity for infusing these quantitative subjects with genetics and, overall, the biological sciences is growing (topics including synthetic biology, molecular systems biology and phylogenetics) there remains little time in the semester to be dedicated to the consolidation of learning and understanding.Mathematics education, Statistics, Geneticsasc2119Mathematics, Science, and Technology, Mathematics EducationDissertationsKnowledge-as-Theory-and-Elements
http://academiccommons.columbia.edu/catalog/ac:174305
Munson, Alexander Anhttp://hdl.handle.net/10022/AC:P:12408Tue, 31 Jan 2012 00:00:00 +0000This dissertation will examine the Knowledge-as-Theory-and-Elements perspective on knowledge structure. The dissertation creates a set of theoretical criteria given within a template by which lesson plans can be designed to teach mathematics and the physical sciences. The dissertation also will test the Knowledge-as-Theory and-Elements theoretical perspective by designing lesson plans to teach a branch of mathematics, graph theory, by using the new template. The dissertation will include a comparative study investigating the effectiveness of the lesson plans conforming to the new template and the lesson plans designed by the traditional theoretical perspective Knowledge-as-Elements.Mathematics educationaam2173Mathematics, Science, and Technology, Mathematics EducationDissertationsDiagrammatic Reasoning Skills of Pre-Service Mathematics Teachers
http://academiccommons.columbia.edu/catalog/ac:143877
Karrass, Margarethttp://hdl.handle.net/10022/AC:P:12378Fri, 27 Jan 2012 00:00:00 +0000This study attempted to explore a possible relationship between diagrammatic reasoning and geometric knowledge of pre-service mathematics teachers. Diagrammatic reasoning skills, as a sequence of steps from visualization, to interpretation, to formalisms, are at the core of teachers' content knowledge for teaching. However, there is no course in the mathematics curriculum that systematically develops diagrammatic reasoning skills, except Geometry. In the course of this study, a group of volunteers in the last semester of their teacher preparation program were presented with "visual proofs" of certain theorems from high school mathematics curriculum and asked to prove/explain these theorems by reasoning from the diagrams. The results of the interviews were analyzed with respect to the participants' attained van Hiele levels. The study found that participants who attained higher van Hiele levels were more skilled at recognizing visual theorems and "proving" them. Moreover, the study found a correspondence between participants' diagrammatic reasoning skills and certain behaviors attributed to van Hiele levels. However, the van Hiele levels attained by the participants were consistently higher than their diagrammatic reasoning skills would indicate.Mathematics educationrp2141Mathematics, Science, and Technology, Mathematics EducationDissertationsDiagrammatic Reasoning Skills of Pre-Service Mathematics Teachers
http://academiccommons.columbia.edu/catalog/ac:143601
Karrass, Margarethttp://hdl.handle.net/10022/AC:P:12335Wed, 25 Jan 2012 00:00:00 +0000This study attempted to explore a possible relationship between diagrammatic reasoning and geometric knowledge of pre-service mathematics teachers. Diagrammatic reasoning skills, as a sequence of steps from visualization, to interpretation, to formalisms, are at the core of teachers’ content knowledge for teaching. However, there is no course in the mathematics curriculum that systematically develops diagrammatic reasoning skills, except Geometry. In the course of this study, a group of volunteers in the last semester of their teacher preparation program were presented with “visual proofs” of certain theorems from high school mathematics curriculum and asked to prove/explain these theorems by reasoning from the diagrams. The results of the interviews were analyzed with respect to the participants’ attained van Hiele levels. The study found that participants who attained higher van Hiele levels were more skilled at recognizing visual theorems and “proving” them. Moreover, the study found a correspondence between participants’ diagrammatic reasoning skills and certain behaviors attributed to van Hiele levels. However, the van Hiele levels attained by the participants were consistently higher than their diagrammatic reasoning skills would indicate.Mathematics educationrp2141Mathematics, Science, and Technology, Mathematics EducationDissertationsFostering Confidence and Competence in Early Childhood Mathematics Teachers
http://academiccommons.columbia.edu/catalog/ac:143049
Rosenfeld, Deborahhttp://hdl.handle.net/10022/AC:P:12157Tue, 10 Jan 2012 00:00:00 +0000This study aimed to increase efficacy for teaching mathematics in pre-service early childhood teachers through the presentation of five video lessons on topics in early childhood mathematics. Each lesson entailed reading a short essay on a topic related to children's mathematical thinking and then watching a short video of a child engaged in a relevant task and a clinical interview with an adult. This study also examined pre-service teachers' knowledge of mathematical development (KMD) and intellectual modesty, or the awareness of the limits of one's knowledge, as possible mediators of change in efficacy. Results showed that the video lessons did significantly increase efficacy for teaching mathematics, but that KMD and intellectual modesty were not significant mediators of the change in efficacy. In effect, confidence appeared disconnected from competence. Follow-up analyses revealed the importance of rich mathematical content within the videos in producing increased confidence and competence.Educational psychology, Early childhood education, Mathematics educationder2129Cognitive Studies in Education, Human DevelopmentDissertationsDo Gestural Interfaces Promote Thinking? Embodied Interaction: Congruent Gestures and Direct-Touch Promote Performance in Math
http://academiccommons.columbia.edu/catalog/ac:132260
Segal, Ayelethttp://hdl.handle.net/10022/AC:P:10390Tue, 17 May 2011 00:00:00 +0000Can action support cognition? Can direct touch support performance? Embodied interaction involving digital devices is based on the theory of grounded cognition. Embodied interaction with gestural interfaces involves more of our senses than traditional (mouse-based) interfaces, and in particular includes direct touch and physical movement, which are believed to help retain the knowledge that is being acquired. There is growing evidence that spontaneous gestures affect thought and possibly learning. The author was interested to explore whether designed gestures (for gestural interfaces) affect thought. It was hypothesized that the use of congruent gestures helps construct better mental representations and mental operations to solve problems (Gestural Conceptual Mapping). There is also evidence that physical manipulation of objects can benefit cognition and learning; it was therefore also hypothesized that manipulating objects through direct touch on the screen supports performance. These hypotheses were addressed by observing children's performance in arithmetic and numerical estimation. Arithmetic is a discrete task, and should be supported by discrete rather than continuous actions. Estimation is a continuous task, and should be supported by continuous rather than discrete actions. Children used either a gestural interface (multi-touch, e. g., iPad) or a traditional mouse interface. The actions either mapped congruently to the cognition (continuous action for estimation and discrete action for arithmetic), or not. If action supports cognition, children who use continuous actions for estimation or discrete actions for addition should perform better than children for whom the action-cognition mapping is less congruent. In addition, if manipulating the objects by touching them directly on the screen could yield a better performance, children who use a touch interface should perform better than children who use a mouse interface. The results confirmed the predictions.Cognitive psychology, Educational technology, Mathematics educationan2136Cognitive Studies in Education, Human DevelopmentDissertationsA History of Trigonometry Education in the United States: 1776-1900
http://academiccommons.columbia.edu/catalog/ac:132221
Van Sickle, Jennahttp://hdl.handle.net/10022/AC:P:10377Mon, 16 May 2011 00:00:00 +0000This dissertation traces the history of the teaching of elementary trigonometry in United States colleges and universities from 1776 to 1900. This study analyzes textbooks from the eighteenth and nineteenth centuries, reviews in contemporary periodicals, course catalogs, and secondary sources. Elementary trigonometry was a topic of study in colleges throughout this time period, but the way in which trigonometry was taught and defined changed drastically, as did the scope and focus of the subject. Because of advances in analytic trigonometry by Leonhard Euler and others in the seventeenth and eighteenth centuries, the trigonometric functions came to be defined as ratios, rather than as line segments. This change came to elementary trigonometry textbooks beginning in antebellum America and the ratios came to define trigonometric functions in elementary trigonometry textbooks by the end of the nineteenth century. During this time period, elementary trigonometry textbooks grew to have a much more comprehensive treatment of the subject and considered trigonometric functions in many different ways. In the late eighteenth century, trigonometry was taught as a topic in a larger mathematics course and was used only to solve triangles for applications in surveying and navigation. Textbooks contained few pedagogical tools and only the most basic of trigonometric formulas. By the end of the nineteenth century, trigonometry was taught as its own course that covered the topic extensively with many applications to real life. Textbooks were full of pedagogical tools. The path that the teaching of trigonometry took through the late eighteenth and nineteenth centuries did not always move in a linear fashion. Sometimes trigonometry education stayed the same for a long time and then was suddenly changed, but other times changes happened more gradually. There were many international influences, and there were many influential Americans and influential American institutions that changed the course of trigonometry instruction in this country. This dissertation follows the path of those changes from 1776 to 1900. After 1900, trigonometry instruction became a topic of secondary education rather than higher education.Mathematics education, Higher education, History of educationjrv2107Mathematics, Science, and Technology, Mathematics EducationDissertationsBeginning mathematics teachers from alternative certification programs : their success in the classroom and how they achieved it
http://academiccommons.columbia.edu/catalog/ac:129624
Ham, Edwardhttp://hdl.handle.net/10022/AC:P:9834Fri, 25 Feb 2011 00:00:00 +0000This dissertation focuses on beginning mathematics teachers from alternative certification programs and their perceptions of what is required to be successful. A mixed - methods research study was completed with several goals in mind: (1) identifying how beginning mathematics teachers define success in the classroom during their earliest years, (2) identifying what important factors, attributes, or experiences helped them achieve this success, and (3) determining where these beginning mathematics teachers learned the necessary attributes, or experiences to become successful in the classroom. A sample of beginning mathematics teachers (n = 28) was selected from an alternative certification program in California for a quantitative survey. A subsample of teachers (n = 7) was then selected to participate further in a qualitative semi-structured interview. The results of the study revealed that beginning teachers defined success in their beginning years by their classroom learning environment, creating and implementing engaging lessons, and a belief in their own ability to grow professionally as educators. Mathematics content knowledge, classroom management, collaboration with colleagues and coaches, reflection, a belief in one's ability to grow professionally as a teacher, a belief in the ability to have a positive impact on students, personality, and previous leadership experiences were several of the factors, attributes, or experiences identified as most important by the participating teachers. The participating teachers also felt that before and after, but not during, their teacher preparation program were the stages of teacher development that best instilled the necessary factors, attributes, or experiences to become successful in a mathematics classroom.Mathematics educationeh2351Mathematics, Science, and Technology, Mathematics EducationDissertations