Number 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.

I 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.

The 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.

This 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.

The 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.

The 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.

While 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.

The dissertation is devoted to the history and practice of extracurricular activities in mathematics in Russia. It investigates both the views expressed by mathematics educators concerning the aims and objectives of extracurricular activities, and the daily organization of such activities, including the pedagogical formats and the mathematical assignments and questions to which extracurricular activities have given rise. Thus the dissertation provides an overview of the history of extracurricular activities over the course of a century, as part of the general development of education (including mathematics education) in Russia. The study called for a multifaceted investigation of surviving sources, which include practically all available textbooks and teaching manuals, scholarly articles on conducting extracurricular activities, magazine and newspaper articles on conducting extracurricular activities, surviving memoirs of participants and organizers of extracurricular activities, and much else, including methodological materials preserved in archives, which have been located by the author. Summing up the results of the study, it may be said that two major goals have always been important in extracurricular activities in Russia: the first goal is motivating students; the second goal is preparing the mathematically strongest students and providing them with an opportunity to deepen and enrich their mathematical education. Of course, extracurricular activities have not been aimed exclusively at these two goals, and at different stages of development additional goals (such as ideological preparation) were also formulated. Broadly speaking, it may be said that the history of the Russian system of mathematical extracurricular activities in general has been strongly aligned with the history of the development of the system of Russian school education. The study analyzes the specific character of extracurricular activities at each of the historical stages of Russia's development, in particular, it lists and described the basic forms of extracurricular activities, paying special attention to the indissoluble connection between the so-called mass-scale forms, in which millions of schoolchildren participate, and forms and activities that are engaged in only by a very few. Also provided is a survey of the changes that have occurred in the mathematical problems that are offered to students. The author believes that familiarity with the longstanding tradition of extracurricular activities in mathematics in Russia may be useful also to the international sphere of mathematics educators, since the issue of motivating students is becoming increasingly important. The study concludes with a discussion of the possibilities and the expediency of putting such experience to use.

In 2 experiments, I tested the effects of a treatment package for teaching 4th graders to edit peers’ written algorithms for solving math problems such that an adult naïve reader could solve the problem. In Experiment 1, the editors were the target participants and the writers were the confederates. Participants were placed in a dyad that consisted of a writer and an editor. The writer and editor repeatedly interacted in writing until the writers produced an algorithm that resulted in adult naïve readers solving the problem. The editor was supplied with a checklist as a prompt for the editing process. Each dyad competed against a second pair of students, using a peer-yoked contingency game board as a motivating operation. Experiment 1 demonstrated that the treatment package increased participants’ accuracy of writing math algorithms, so that a naïve reader could solve the math problems. The target participants acquired the verbally governed responses through peer editing alone, and as a result the participants produced written math algorithms. Experiment 2 measured the behaviors of the editor and writer using a multiple probe design across participants with two groups of 4 writers and 4 editors. The dependent variables were: 1) production of previously mastered math problems, such that a naïve reader could read and solve the math problem without ever seeing the problem, 2) the emergence of explanations of “why” (function) from learning “how” to solve a multi-step math problem, 3) production of novel written math algorithms (i.e., find the perimeter and extended multiplication), and 4) cumulative number of untaught math problems attempted. The independent variable was the same as Experiment 1 except a) the editors did not have access to a checklist and b) the peer-yoked contingency game board was removed. The results demonstrated that all participants produced written math algorithms such that both the writers and editors affected the behavior of naïve readers. I discuss the emergence of explanations of the function (“why”) of math that occurred as a result of being able to explain “how” to solve problems. Moreover, the participants attempted more untaught math problems, demonstrating the resistance to extinction for attempting untaught math problems. Findings suggest that as a function of the intervention, reinforcement for solving math problems was enhanced.

The limit concept in calculus has received a lot of attention from mathematics education researchers, partly due to its position in mathematics curricula as an entry point to calculus and partly due to its complexities that students often struggle to understand. Most of this research focuses on students who had previously studied calculus or were enrolled in a calculus course at the time of the study. In this study, I aimed to gain insights into how students with no prior experience with precalculus or calculus might think about limits via the concept of limit at infinity, with the goal of designing instructional tasks based on these students’ intuitive strategies and ways of reasoning. In particular, I designed a sequence of instructional tasks that starts with an experientially realistic starting point that involves describing the behavior of changing quantities in real-world physical situations. From there, the instructional tasks build on the students’ ways of reasoning through tasks involving making predictions about the values of the quantity and identifying characteristics associated with making good predictions. These instructional tasks were developed through three iterations of design research experimentation. Each iteration included a teaching experiment in which a pair of students engaged in the instructional tasks under my guidance. Through ongoing and reflective analysis, the instructional tasks were refined to evoke the students’ intuitive strategies and ways of thinking and to leverage these toward developing a definition for the concept of limit at infinity. The final, refined sequence of instructional tasks together with my rationale for each task and expected student responses provides insights into how students can come to understand the concept of limit at infinity in a way that is consistent with the formal definition prior to receiving formal instruction on limits. The results presented in this dissertation come from the third and final teaching experiment, in which Jon and Lexi engaged in the sequence of instructional tasks. Although Jon and Lexi did not construct a definition of limit at infinity consistent with a formal definition, they demonstrated many strategies and ways of reasoning that anticipate the formal definition of limit at infinity. These include identifying a limit candidate, defining the notion of closeness, describing the notion of sufficiently large, and coordinating the notion of closeness in the range with the notion of sufficiently large in the domain. On the other hand, Jon and Lexi demonstrated some strategies and ways of reasoning that potentially inhibited their development of a definition consistent with the formal definition. Pedagogical implications on instruction in calculus and its prerequisites are discussed as well as contributions to the field and potential directions for future research.