The 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.

This 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.

Given 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.

The 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.

Children 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 discussed

The 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.

The purpose of this study was to determine the prevalence of using a “mental abacus” by adults whose mathematics education in Korea included extensive use of the actual abacus as both a teaching and computational aid. One hundred and sixty-nine Korean adults between the ages of 25 to 65 who had abacus training and its uses for a minimum of one year participated in the study. The study had two phases: a quantitative phase and a qualitative phase. The quantitative phase focused on the participants’ perceptions of their training and use of the abacus as well as an assessment of their basic arithmetic competencies. This served as a context for a more in-depth analysis of their perceptions of, and thinking about, the use of the abacus in arithmetic operations obtained in the qualitative phase. All participants were asked and then answered a total of 6 questions regarding basic background information about their abacus training as well as their current use of the abacus for arithmetic computations in order to examine the extent of Korean abacus uses. The questionnaires included an assessment of participants’ arithmetic computation skills. Among them, 59 adults were selected and interviewed to explore the extent of the “mental abacus” influence on their qualitative thoughts and tasks. From this research, it was expected that the study would provide information concerning the power of Stigler’s mental abacus in mathematics and how it relates to Korean adults’ daily life. Apparently, although computation tools such as calculators and computers are widely available and convenient to use, the abacus is still used as one of the arithmetic tools by Korean adults. Considering the fact that the Korean national standard mathematics education curriculum has not included abacus training, although some commercial educational institutions included it, the rate of learning the abacus and the period and frequency of its use tell us that abacus skill could affect the basic mathematics competency of Koreans. The data show that Korean adults who have been educated in abacus use provide self-reported evidence that they have the competency of mental computation and the ability to develop a mental abacus image depending on their period of frequency of abacus use. Further evidence indicates that most Korean abacus users who participated in the study report self-confident and accurate perceptions of their ability and arithmetic accuracy in doing basic arithmetic computations. Moreover, they are more confident and accurate in addition test problems than subtraction, multiplication, and division from the assessment results. It is concluded that mental abacus image occurrence may be associated with mental computation among some Korean adults who had learned to use the abacus in the past. Many of the Korean adult participants in this study who trained on the abacus can automatically conjure the mental abacus image and employ the skill during mental computation. The ease with which the abacus mental representation is activated during mental calculation is related to how frequently or intensively the adults practiced or exercised its use. Among further findings, about the positive aspects of “mental abacus” use, most of the Korean adults in the study expressed opinions that there were positive influences of having learned the abacus, not only increased mathematics competency but also an additional “reward” in greater competency in other academic subjects and activities. This study reveals that intensive training with an abacus and the continuous use of an abacus can promote mental visualization and manipulation of the abacus during arithmetic computations, and result in a sense of positive effects from “mental abacus” use among those who have had sufficient opportunities to use the abacus.

This study investigated the potential application of web-based pedagogy for teachers of elementary statistics, using A Pedagogical Information System (APIS), in order to explore ways in which teachers can enhance instruction via technology in the classroom. This information system would provide new opportunities for practical teacher education, given that the literature on statistics pedagogy and web-based teaching and learning is sparse. The purpose of the study was to examine issues of efficacy, efficiency, and effectiveness of the use of Internet websites in the instruction of introductory statistics at the college level. The design of the study was qualitative and used a phenomenological interpretive method. The participants in this study were four instructors of introductory statistics courses at the undergraduate level of higher education in a northeastern state in the United States. They were interviewed individually for approximately 60 minutes in a semi-structured format in which they discussed their experiences and perceptions about using Internet sources for instruction of students in Introductory Statistics. The results of the study indicated that teachers welcomed incorporating technology into their classrooms. Participants found that having students with different learning styles, allocating time to deal with details, and having difficulty managing the interplay between topics as being key problems that could arise in the organization of a web-based introduction to an elementary statistics course. Although all participants agreed that there is a lack of uniformity among statistical topics, most wanted to emphasize descriptive statistics, distributions, probability, inference, confidence intervals, and regression. They acknowledged that web-based learning would radically alter student and teacher roles, so that the teacher would become a mentor and students would become active learners. Instruction would be geared more to practical applications than to theory. Instructors would have to make decisions about how much web-based information they would use and would have to become knowledgeable about web content.

This study focuses on the history of school mathematics through the discourse surrounding the function concept. The function concept has remained the central theme of school mathematics from the emergence of both obligatory schooling and the science of mathematics education. By understanding the scientific discourse of mathematics education as directly connected to larger issues of governance, technology, and industry, particular visions for students are described to highlight these connections. Descriptions from school mathematics focusing on expert curricular documents, developmental psychology, and district reform strategies, are meant to explain these different visions. Despite continued historical inquiry in mathematics education, few studies have offered connections between the specific style of mathematics idealized in schools, the learning theories that accompanied these, and larger societal and cultural shifts. In exploring new theoretical tools from the history of science and technology this study seeks to connect shifting logic from efforts towards rational organization of capitalist society with the logic of school mathematics across the discursive space. This study seeks to understand this relationship by examining the ideals evinced in the protocols of educational science. In order to explore these architectures, the science of mathematics education and psychology are examined alongside the practices in the New York City public schools--the largest school system in the nation. To do so, the discourse of the function concept was viewed as a set of connections between mathematical content, psychology, and larger district reform projects. Four architectures--the mechanical, thermodynamic, cybernetic, and network models--are examined.

The in-class learning environments of a traditional and hybrid mathematics course were compared. The hybrid course had half the face-to-face meetings as the traditional course; outside of class, the students in the hybrid section completed asynchronous online assignments that involved watching content-delivery videos. Moving the content delivery outside of the classroom for the hybrid format had an impact on the interactions between the students and the professor inside the classroom. Quantitative and qualitative analysis of verbal discourse determined that the hybrid class format reduced the amount of in-class time devoted to direct instruction and increased the level of student discourse. Students assisted other students, had the freedom to make mistakes, and were able to receive personal guidance from the professor. The professor was able to address student misconceptions on formative assessments in class. Previous studies of the hybrid class model had focused on comparing differences in examination scores, GPAs, and pre- and post-test scores between the traditional and hybrid class model rather than comparing what is occurring inside the classroom. Quantifying what effect the shift from the traditional to the hybrid class model had on discourse inside the classroom is a first step towards confirming how the different methods of content delivery affects the in-class learning environment, and provides insight into certain pedagogic advantages the hybrid format may offer.