It is the purpose of this work to evaluate the possible contribution that a semiotic perspective could make to mathematics education and mathematics. This emphasis of this work is on the development of a semiotic perspective, emerging from the work of C. S Peirce and L. Vygotsky, that embodies both the philosophical and psychological notions of the role of the sign. This perspective, termed Developmental Semiotics, evolves through the interplay of the theoretical and empirical aspects of this work. Developmental semiotics is applied in the form of a prototype as an empirical tool to two case studies, one concerning algebra and one concerning co-ordinate representations. The results of these studies serve to inform the evolution of the perspective and provide some conclusions specific to the case studies arising from this way of viewing learning in these contexts. The perspective is refined in the light of the studies and a further case study is carried out, this time in the area of the history of mathematics (the development of Boolean algebra), with the purpose of evaluating the final form of the framework and informing on mathematical development, creativity and motivation. Developmental semiotics has implications at two levels, on the macro semiotic level it brings forward the emphasis on the role of the sign in mathematical meaning-making, it rejects the traditional subject-object dichotomy and through semiotic action accounts for the relativity in mathematical meanings. At the micro semiotic level it describes mathematical development in terms of progressive meaning making instances for increasingly opaque signs and gives specific information about the meanings made by mathematics learners in specific contexts. In some instances it has enabled the identification of meanings that may be useful for mathematical progression. This work hopes to contribute to mathematics education by bringing to the fore the discipline of semiotics through emphasis on the role of the sign as a meaning giving entity, and through provision of a framework and empirical toolkit for the semiotic analysis of specific learning contexts. Furthermore it may also contribute to both mathematics, by exposing the history of mathematics to a semiotic analysis, and to semiotics itself by providing a further context for the application of this cross disciplinary point of view.

<p>This study explored the impact of numbers used in instructional tasks on the construction and generalization of multiplicative reasoning by fourth grade students designated as having learning difficulties or disabilities in mathematics (SLDs). In particular, this study addressed the following research questions: (1) In what ways do SLDs&rsquo; conception of number as a composite unit afford or constrain transition to multiplicative reasoning? (2) Which specific numbers, used in instructional and/or assessment tasks, may support or interfere with SLDs&rsquo; progression from additive reasoning to multiplicative Double Counting (mDC)? Results suggested that in early participatory stages, using numbers with multiples familiar to the students, such as 2 and 5, promoted multiplicative solution paths (e.g., counting by fives while simultaneously keeping track of how many fives they have counted). This use of familiar numbers allowed for students&rsquo; reflection on their multiplicative thinking. Introduction of more difficult numbers&mdash;any number for which the child was yet to master multiples&mdash;tended to limit the multiplicative thinking and move students back to more known (additive) solution paths. In later participatory stages, the introduction of more difficult numbers promoted the progression within mDC. </p>

<p> The purpose of this study was to examine Kuwaiti mathematical elementary teachers&rsquo; perceptions about their ability to integrate M-learning (mobile learning) into their current teaching practices and the major barriers hindering teachers&rsquo; ability to create an M-learning environment. Furthermore, this study sought to understand teachers&rsquo; perceptions about their ability to create a collaborative cloud-computing learning environment that corresponds with the 21st century skills and possibly explain their readiness for future reformation of education in Kuwait. </p><p> Using an Internet-based format to this study quantitative and qualitative data, the Technological Pedagogical Content Knowledge (TPACK) and barriers survey gleaned quantitative information about how mathematics teachers and a head of a mathematics department (n = 562) viewed use of technology as well as the barriers they faced in integrating it into the classroom. Also, qualitative data were collected using a survey of open-ended questions to provide context to survey answers and better understand the barriers and affordance experienced by the participants. Moreover, a 21st century open-ended questionnaire was employed to collect qualitative information from mathematics teachers and head of the departments (n = 21) in regard the their ability to construct a 21st century learning environment based on collaboration and constructivist perspective utilizing a cloud-computing technology. </p><p> Quantitative analysis was utilized to examine elementary mathematics teachers&rsquo; perceptions using the TPACK survey, and the validity and reliability of the TPACK subscales were computed by administering the confirmatory factor analysis. Factors that were elicited were specified as: all seven subscales encompassed in the TPACK survey significantly fit model of factor structures, and the TPACK survey was reliable and valid. In addition, descriptive analysis such as the TPACK subscale means and standard deviations were computed via the SPSS software. </p><p> Qualitative content analysis was used to understand teachers&rsquo; perceptions about their ability to integrate mobile technology, perceptions of the primary barriers and affordance that limited their ability, and their perceptions of their ability to integrate collaborative cloud computing and create a 21st century learning environment based on the constructivist perspective. When analyzed, the self-reported open-ended survey yielded the following specific themes: (a) teachers perceived themselves high in their ability to integrate mobile technology; (b) the primary barriers based on teachers&rsquo; perceptions were budget constraints, IT limitations, time constraints, and administrative support; and (c) teachers perceived themselves high in their ability to integrate collaborative cloud computing to construct a 21st century learning environment based on the constructivist perspective. This study finding could be implemented to create a new modern mathematics elementary curriculum that resolves the current curriculum issues. Future research is recommended in the direction of creating a new mathematical curriculum based on administrators&rsquo;, parents&rsquo;, and students&rsquo; perspectives.</p>

<p> Providing students with necessary intervention in the instruction of mathematics can be accomplished through the use of digital resources. Schools reaching a one-to-one (one device for every student) status have the ability to implement a digital intervention on a broad scale. For the purpose of this study, mixed methods research afforded an in-depth investigation into the impact of Khan Academy (digital intervention) embedded in a one-to-one program on student achievement, student perseverance with math, and teacher pedagogy within rural schools. The theoretical framework of educational technology integration known as Theoretical, Pedagogical, and Content Knowledge (TPACK) was woven into this study to answer the guiding research questions. While using Khan Academy in a one-to-one setting, participating students were tested twice a year utilizing the Measurement of Academic Progress (MAP) testing procedure provided by the Northwest Evaluation Association (NWEA). The testing data provided quantitative data for the study. In addition, interviews of teachers and administrators were conducted to reveal themes related to teacher pedagogy and student perseverance with mathematical problems. An analysis of MAP scores revealed that 9th- and 10th-grade high school students utilizing Khan Academy in a one-to-one program demonstrated significantly more growth when compared to national norms. The study involved 227 9th-grade and 114 10th-grade students from three rural high schools in the Pacific Northwest. Results from an analysis of means illustrated that each grade level demonstrated significantly greater growth when compared to national norms. When examined at the school level, each of the three sites once again exceeded growth norms. Further analysis of the interviews revealed a pedagogical shift directly related to the use of Khan Academy embedded in a one-to-one program. Themes related to the success of the program include the ability of Khan Academy to fill individual gaps and practice skills to mastery, the importance of student and teacher buy-in, and the ability for students to have daily, individual access to devices. Furthermore, the interviews revealed teachers and principals did not agree on whether Khan Academy impacts student perseverance; however, a theme related to Khan Academy&rsquo;s ability to impact confidence with math did emerge. This study fills gaps in the existing literature regarding the academic and pedagogical impact of Khan Academy on teaching and learning. In addition, this study addresses a gap in the literature regarding rural schools&rsquo; use of a digital mathematical intervention program as a means of blended learning. Implications for educational policy can result from this study given the current political climate surrounding one-to-one and blended learning deployments. This study demonstrates that when one-to-one deployments are paired with the use of Khan Academy, student achievement can be positively impacted.</p>

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This case study described the experiences of secondary students and preservice mathematics teachers in the teaching and learning of social justice mathematics (SJM). Specifically, participants’ experiences in making connections among the mathematics curricula and the real world, perceptions about mathematics, and responses to an integrated curriculum approach were described. Students participated in SJM activities designed by preservice teacher participants: one component of a pre-existing extracurricular Social Issues Club at a high school in Southeastern Ontario. Mathematics activities, led by the researcher or one of the preservice teacher participants, were designed to complement the social justice issues that were being explored by the members of the Social Issues Club. Data were obtained through observations, questionnaires, focus group, individual interviews, written reflections, and artifacts. Results demonstrated that preservice teacher participants had unique professional and educational encounters prior to SJM that they connected to their SJM experience. Subsequent to this experience, preservice teachers suggested limited ideas about integrating curriculum into their future teaching practice beyond the content and contexts made familiar to them through SJM. With limited exposure to examples of curriculum integration identified by preservice teachers as a barrier, results suggest that preservice teachers need more opportunities to engage in mathematics curriculum integration. Students showed an expanded view of connections between mathematics and the real world through their descriptions of the various ways in which SJM had helped them to apply mathematics concepts and understand the issues they were exploring. They enjoyed SJM’s collaborative mathematics learning approach and valued the opportunity to discuss the social issues about which they were concerned. Although the preservice teachers were confident about what they thought to be topics of interest for secondary students, there was a disconnect between students’ choices of contexts for mathematics learning and the beginning teachers’ assumptions about students’ interest. This finding suggests that there is a need to support preservice teachers to understand students’ interests in mathematics learning and that students’ opinion needs to be solicited. In addition, participants’ visions about enhancing mathematics teaching and learning through collaboration and providing students with autonomy allowed suggestions for the practice of mathematics teaching. / Thesis (Master, Education) -- Queen's University, 2012-10-14 16:18:44.922

<p> Throughout the history of teacher education, the final fieldwork experience has often been called the single most influential experience in teacher preparation programs (Burns, Jacobs, &amp; Yendol-Hoppey, 2016; Feiman-Nemser &amp; Buchmann, 1986; Parker-Katz &amp; Bay, 2008). Though this experience has been expanded to include fieldwork experiences throughout many teacher education programs (Guyton &amp; McIntyre, 1990), the final fieldwork experience remains the closing activity and the lasting image of teacher preparation (Feiman-Nemser &amp; Buchmann, 1986; Rosaen &amp; Florio-Ruane, 2008). Given its importance, though, researchers know relatively little about it. &ldquo;The knowledge thus produced is akin to the quantum theory of physics; we know what goes in . . . and what comes out . . . but not what occurs in the interim&rdquo; (Guyton &amp; McIntyre, 1990, p. 524). Given the current reforms in mathematics education and mathematics teacher education (National Council for the Accreditation of Teacher Education, 2010; National Governors Association Center for Best Practices &amp; Council of Chief State School Officers, 2010), Guyton and McIntyre&rsquo;s observation is still relevant today.</p><p> During the final fieldwork experience, university-based and school-based mathematics educators must work together on behalf of the novice to marry university-promoted theory (especially reform-oriented theory) with the practical classroom expectations of day-to-day teaching life. Though there is much research on how this kind of work should be done and the dilemmas that have arisen during fieldwork (e.g., Knight, 2009; Loughran, 2006; Nolan &amp; Hoover, 2004; Sergiovanni &amp; Starratt, 2006; Sullivan &amp; Glanz, 2013), we have little information about the experiences of the mathematics educators who collaborate during final fieldwork. Furthermore, we have very little information on how these educators navigate mathematics reforms to prepare teachers of mathematics. </p><p> This multi-case study was designed to investigate three novices, their school-based mentors, and their university-based mentor (me) who collaborated during a year-long final fieldwork experience at the close of a middle school mathematics teacher preparation program. To write single case reports that illuminated our collaborative experiences, I wrote the &ldquo;stories&rdquo; of each triad. To collect these stories, I used individual and group interviews, paired conversations, asynchronous text interviews, conference observations, collaborative fieldwork artifacts, my own practitioner-researcher journal, and three cycles of participant member checks. After verifying the veracity of the stories of each triad, I engaged in cross-case analysis to make assertions about the commonalities and unique circumstances that defined these fieldwork cases. This study adds to teacher preparation fieldwork literature by evoking a response from educators working in the field and providing them with examples of open dialogue that created more empathetic collaborative experiences. The study also provides evidence that the empathy generated by sharing stories can create more productive and effective learning experiences for the novices involved. In particular, open dialogue provided the collaborators in these cases with a platform for acknowledging pedagogical differences, negotiating fieldwork expectations, and setting and meeting novices&rsquo; professional goals. For future investigations of teacher preparation fieldwork collaboration, this study provides evidence that a practitioner approach to research affords the researcher exceptional access to the stories of novices and mentors and establishes empathetic bonds that can make the telling of those stories both illuminating and respectful of the voices they represent.</p>