<p> This exploratory study seeks to investigate how a mathematical education game, <i>DragonBox12</i>+, effects students&rsquo; learning about algebra. Data for this research was collected from middle school 7^th grade students in the Northeast region of the United States of America. The interviews and classroom observations were recorded on videotape. The research results showed that the video game <i>DragonBox 12</i>+ affects students&rsquo; attitude of mathematics and learning of mathematics by the help of using game mechanics to teaching algebraic rules. </p>

<p> A Multi-Tier System of Support (MTSS) for academic skills is widely recognized as the best practice framework for supporting all students. Additionally, the recent shift from constructivist pedagogy toward more intentional teaching of mathematics at the preschool level has encouraged more explicit mathematics instruction with younger children. In spite of these advances, there are no published best practice guidelines for implementing MTSS for mathematics at the prekindergarten level. The current study sought to investigate one possible way to implement effective instructional practices for preschool mathematics within a multi-tier system, including the use of validated screening and progress monitoring instruments. A centers-based mathematics curriculum was implemented at the universal level within an inclusive preschool classroom. Universal screening was conducted using curriculum-based measurement (CBM) in order to identify at-risk students in need of additional instruction. A supplemental prekindergarten program was implemented with small instructional groups at the secondary tier of support. Students receiving supplemental instruction were progress-monitored using growth-sensitive CBMs in a multiple baseline across dyads research design. Results and limitations of the study are discussed. Finally, topics for future exploration in preschool mathematics are suggested. </p>

Trots att problemlösning beskrivs som själva hjärtat inom matematiken, och som något önskvärt av både lärare och elever, har problemlösning ofta en liten plats i svensk matematikundervisning. Istället har rutinuppgifter haft en framträdande roll, inom en ofta läroboksorienterad undervisning. Syftet med denna litteraturstudie är att beskriva hur matematiklärare kan genomföra matematikundervisning där problemlösning ingår. Detta syfte nås genom systematiska litteratursökningar för att samla forskning inom det gällande forskningsläget. De framträdande resultaten för denna litteraturstudie visar på hur lärare använder sig av sex olika didaktiska strategier vid matematikundervisning där problemlösning ingår. De sex olika didaktiska strategierna är att läraren skapar en tydlig lektionsstruktur, antar en aktiv lärarroll, gör undervisningen motivationsskapande, använder sig av språket som en påverkande faktor, anpassar kontexten i uppgifterna och är medveten om elevers kognitiva resurser. Fortsatt forskning på området bör undersöka vilket utrymme problemlösning bör få i matematikundervisning, jämfört med till exempel rutinuppgifter.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / "Just In Time Teaching" (JiTT) is a teaching practice that utilizes web based technology to collect information about students' background knowledge prior to attending lecture. Traditionally, students answer either multiple choice, short answer, or brief essay questions outside of class; based on student responses, instructors adjust their lectures "just-in-time." In this study, modified JiTT techniques in the form of online review modules were applied to a first semester calculus course at a large midwestern state university during the spring 2012 term. The review modules covered algebra concepts and skills relevant to the new material presented in calculus lecture (the "just-in-time" adjustment of the calculus lectures was not implemented in this teaching experiment). The reviews were part of the course grade. Instead of being administered purely "just-in-time," the reviews were assigned ahead of time as part of the online homework component of Calculus-I. While previous studies have investigated the use of traditional JiTT techniques in math courses and reported student satisfaction with such teaching tools, these studies have not addressed gains in student achievement with respect to specific calculus topics. The goal of this study was to investigate the latter, and to determine whether timing of the reviews plays a role in bettering student performance. Student progress on weekly Calculus-I online assignments was tracked in spring of 2012 and compared to student scores from weekly Calculus-I online assignments from spring 2011, when modified JiTT instruction was not available. For select Calculus-I online assignments during the spring 2012 term, we discovered that the review modules significantly increased the number of students receiving perfect scores, even when the reviews were not purely administered ``just-in-time." Analysis of performance, success of review assignments, and future implications are also discussed.

Doctor of Philosophy / Department of Physics / N. Sanjay Rebello / Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that students often encounter in math classes. Research in physics education has reported students’ lack of ability to transfer their calculus knowledge to physics problem solving. In the past, studies often focused on what students fail to do with less focus on their underlying cognition. However, when solving physics problems requiring the use of integration, their reasoning about mathematics and physics concepts has not yet been carefully and systematically studied. Hence the main purpose of this qualitative study is to investigate student thinking in-depth and provide deeper insights into student reasoning in physics problem solving from multiple perspectives. I propose a conceptual framework by integrating aspects of several theoretical constructs from the literature to help us understand our observations of student work as they solve physics problems that require the use of integration. I combined elements of three important theoretical constructs: mathematical resources or symbolic forms, which are the small pieces of knowledge elements associated with students’ use of mathematical ideas; conceptual metaphors, which describe the systematic mapping of knowledge across multiple conceptual domains – typically from concrete source domain to abstract target domain; and conceptual blending, which describes the construction of new learning by integrating knowledge in different mental spaces. I collected data from group teaching/learning interviews as students solved physics problems requiring setting up integrals. Participants were recruited from a second-semester calculus-based physics course. I conducted qualitative analysis of the videotaped student conversations and their written work. The main contributions of this research include (1) providing evidence for the existence of symbolic forms in students’ reasoning about differentials and integrals, (2) identifying conceptual metaphors involved in student reasoning about differentials and integrals, (3) categorizing the different ways in which students integrate their mathematics and physics knowledge in the context of solving physics integration problems, (4)exploring the use of hypothetical debate problems in shifting students’ framing of physics problem solving requiring mathematics.

<p> There exists a need for instructional designers to understand how to incorporate supportive interventions in online developmental pre-algebra/math course designs. College students at the undergraduate level who require remedial assistance and academic supports in mathematics must successfully complete developmental pre-algebra/math courses. The study describes instructional strategies for procedural, active learning, and cognitive constructivist instructional strategies in problem-based learning. The study included six volunteer instructional designers who shared their perspectives for design practices, supportive interventions, and procedures to assist learners. A sample of convenience purposive sampling strategy was used to allow access to the volunteer participants through public social media. The six participants responded to the 16 related guided interview questions and the data was analyzed. Eighteen individual themes emerged related to supportive interventions used in instructional design regarding, instructional strategies, motivation, learning theories, and interaction by students within the courses. The participants shared seven design models and practices for successful learning, seven supports, and 12 developmentally appropriate design practices, used in their instructional designs. The findings of this study support the premise that by combining cognitive constructivism, social constructivism and confidence builders, to effect motivation and self-efficacy for supportive interventions, the learner could potentially successfully complete the requirements for undergraduate online developmental math courses.</p>

<p> Mathematics performance of students in the United States is concerning. When compared to global peers, students in the United States perform at the lower range in areas of mathematics. Even after controlling for variables such as ethnicity, parent educational attainment, and socio-economic status, students in the United States continue to struggle in mathematics. One area that facilitates learning complex mathematics skills is computational fluency. The Mathematics Advisory Panel Report and the Common Core State Standards in Mathematics recommend that students have daily practice to build computational fluency skills. Evidence based interventions that target computational fluency includes incremental rehearsal, cover copy compare, and performance reward. The interventions were implemented with three students who attended a dual language immersion program and analyzed through single-case research design. The results found the interventions effective for two of the three students. Using these interventions as a package can have beneficial results in the computational fluency of students in dual-immersion programs.</p>

Magister Educationis - MEd / In South Africa it is widely known that most learners struggle with mathematics. The results for mathematics are poor. The department of basic education offers a number of intervention programmes to assist learners in mathematics but the problem still persists. Algebra is the most basic and important topic in mathematics as it becomes an element in almost all the other topics in mathematics curriculum. Algebraic fractions in particular are a challenge for most leaners. Research shows that learners commit a number of errors when they work with algebraic fractions. The study investigated the implementation of an intentional teaching model into grade 9 mathematics learners’ ways of working with rational algebraic fractions. An intentional teaching model is a teaching strategy which emphasizes teaching intentions or teaching objectives are brought to the fore during a lesson, the model emphasizes the use of spiral revision and assessment for learning. Ways of working in this study refers to the way in which learners deal with algebraic fractions when they simplify them including the errors they commit from the misconceptions they have about aspects of working with fractions. The study was conducted in a group of grade 9 mathematics learners at Gugulethu High school, which is located in Guguletu, a township in the Western Cape Province of South Africa. The study is premised on a qualitative research paradigm which focuses on studying situations in their natural settings and applying an interpretive perspective. Data was collected by means of observation and video recording of lessons while learners were engaged in working with algebraic fractions. Learners’ written work was analysed as part of the data collection. The results of the study show that leaners commit a number of errors when they manipulate algebraic fractions. Among other errors are: (i) Cancellation errors which had the highest frequency of occurrence (ii) Defractionalisation (iii) No recognition of the common factor and (iv)Exponential laws error. It was found that the learners’ ways of working with algebraic fractions are mostly characterised by their misunderstanding of exponential laws and difficulty in working with fractions needing the use of factorisation to simplify and find the lowest or highest common denominator during addition or subtraction. The results of the study also reveal that learners struggle to articulate extensively or in detail what they are actually doing as they simplify rational algebraic fraction.

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This study looks at the use of integrated mathematics and music lessons at the high school level. Four lessons were taught by the researcher in both a research and a control class to determine how mathematically motivated music instruction affects students understanding of operations of functions, composition of functions, inverse functions and domain and range. A pretest-posttest was used to determine the effect of these lessons and a questionnaire was used to identify differences between groups and to help determine the effect of musical applications of mathematics on students’ mathematical perceptions, self-efficacy and grit. The pretest-posttest included both a standard mathematics section and a section involving non-musical applications. A gain score approach using independent sample t tests was used to determine the impact of the integrated instruction. The research group demonstrated significantly greater gains both overall and on the applications portion of the exam. Additional qualitative analysis was done to determine how the posttests differed between groups. Three major differences were identified: the research group used function notation more frequently than the control group, the control group demonstrated confusion between composition of functions and inverse functions while the research group did not and the research group showed more mathematical work for the applications portion of the exam than the control group. Qualitative analysis was also done to identify trends in the questionnaire data. Among the major differences between groups was the increased willingness to work with mathematical applications in the future by the research group compared to the control group. The integrated instruction led to comparable and in some cases significantly better mathematics outcomes than the control group and led students to an increased willingness to work with mathematical applications both on the posttest and moving forward.