This thesis presents two research studies aimed at increasing our knowledge about how games can be used in math education. A number of studies have explored the effects of games in classrooms. However, their effects vary across study contexts and it is not yet clear the best ways to implement them. While prior research shows that numerical linear board games are effective learning activities in preschool classrooms, the first study presented aims to replicate and add to previous research by exploring the effects of numerical linear board games when presented in different formats. Board games are typically studied in a table top version. We introduce a physically active, life-size, walkable version, which is based on an embodied cognition perspective. The second study presented in this thesis is an overview of a professional development workshop where teachers and students played technology-based games and then designed and created their own games. We explore the effects of playing teacher-created games on student learning as well as the implications which creating games can have on their development of computational thinking skills.

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Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / The main objective of the study is to identify the general characteristics of groups within a typical Studio College Algebra class and then adapt aspects of the course to best suit their needs. In a College Algebra class of 1,200 students, like those at most state funded universities, the greatest obstacle to providing personalized, effective education is the anonymity of the students. Data mining provides a method for describing students by making sense of the large amounts of information they generate. Instructors may then take advantage of this expedient analysis to adjust instruction to meet their students’ needs. Using exam problem grades, attendance points, and homework scores from the first four weeks of a Studio College Algebra class, the researchers were able to identify five distinct clusters of students. Interviews of prototypical students from each group revealed their motivations, level of conceptual understanding, and attitudes about mathematics. The student groups where then given the following descriptive names: Overachievers, Underachievers, Employees, Rote Memorizers, and Sisyphean Strivers. In order to improve placement of incoming students, new student services and student advisors across campus have been given profiles of the student clusters and placement suggestions. Preliminary evidence shows that advisors have been able to effectively identify members of these groups during their consultations and suggest the most appropriate math course for those students. In addition to placement suggestions, several targeted interventions are currently being developed to benefit underperforming groups of students. Each student group reacts differently to various elements of the course and assistance strategies. By identifying students who are likely to struggle within the first month of classes, and the recovery strategy that would be most effective, instructors can intercede in time to improve performance.

The math achievement of students is low in a small rural district in Colorado. The purpose of this study was to explore the efficacy of a summer third through fifth grade math program in improving math scores. Piaget's theory of cognitive development was used as the theoretical foundation for the math instructional resource delivered to the struggling students in the program. A quasi-experimental design was used to address whether the math scores improved for the participating students (n = 145) and whether the participating students experienced a smaller summer loss in academic achievement than the students who did not participate (n = 457). Ex post facto data included pre- and post- math assessments. The math instruction and assessments were administered to third through fifth grade students as part of the school district's academic program during the 2012, 2013, and 2014 summer schools. A dependent samples t test was used to analyze the data to determine if the students' achievement scores improved for those attending summer school. The results did not indicate any significant improvement. An ANOVA was then used to determine if the summer math program decreased summer loss of learning in participating students. Participating students experienced significantly less summer loss than did non-participating students. Therefore, recommendations for the summer math program include more instructional time and moving the program closer to the beginning of the school year to avoid any summer loss. This study will have a positive social impact as it influences decisions made by the school district to improve the summer math program and produce students who are better prepared for postsecondary school options.

This study identified variations in the learning trajectories of Tier II students when learning equivalent fraction concepts using physical and virtual manipulatives. The study compared three interventions: physical manipulatives, virtual manipulatives, and a combination of physical and virtual manipulatives. The research used a sequential explanatory mixed-method approach to collect and analyze data and used two types of learning trajectories to compare and synthesize the results. For this study, 43 Tier II fifthgrade students participated in 10 sessions of equivalent fraction intervention. Pre- to postdata analysis indicated significant gains for all three interventions. Cohen d effect size scores were used to compare the effect of the three types of manipulatives—at the total, cluster, and questions levels of the assessments. Daily assessment data were used to develop trajectories comparing mastery and achievement changes over the duration of the intervention. Data were also synthesized into an iceberg learning trajectory containing five clusters and three subcluster concepts of equivalent fraction understanding and variations among interventions were identified. The syntheses favored the use of physical manipulatives for instruction in two clusters, the use of virtual manipulatives for one cluster, and the use of combined manipulatives for two clusters. The qualitative analysis identified variations in students’ resolution of misconceptions and variations in their use of strategies and representations. Variations favored virtual manipulatives for the development of symbolic only representations and physical manipulatives for the development of set model representations. Results also suggested that there is a link between the simultaneous linking of the virtual manipulatives and the development of multiplicative thinking as seen in the tendency of the students using virtual manipulative intervention to have higher gains on questions asking students to develop groups of three or more equivalent fractions. These results demonstrated that the instructional affordances of physical and virtual manipulatives are specific to different equivalent fraction subconcepts and that an understanding of the variations is needed to determine when and how each manipulative should be used in the sequence of instruction.

Logical statements are prevalent in mathematics, the sciences, law, and many areas of everyday life. The most common logical statements are conditionals, which have the form “If H..., then C...,” where “H” is a hypothesis (or condition) to be satisfied and “C” is a conclusion to follow. Reasoning about conditionals is a skill that is only superficially understood by most individuals and depends on four main conditional contexts (e.g., intuitive, abstract, symbolic, or counterintuitive). The purpose of this study was to test a theory about the effects of context ordering on reasoning about conditionals. To test the theory, the researcher developed, tested, and revised a virtual manipulative educational mathematics application, called the Learning Logic App. This study employed a convergent parallel mixed methods design to answer an overarching research question and two subquestions. The overarching research question was “How does the order of teaching four conditional contexts influence reasoning about conditionals?” The two subquestions examined this influence on reasoning in terms of performance and perceptions. This study involved two phases. During Phase I, 10 participants interacted with the Learning Logic App in a clinical setting. The researcher used information gathered in Phase I to revise the Learning Logic App for Phase II. During Phase II, 154 participants interacted with the Learning Logic App in a randomly assigned context ordering in an online setting. In both phases, the researcher collected quantitative and qualitative data. After independent analyses, the researcher made meta- inferences from the two data strands. The results of this study suggest that context ordering does influence learners’ reasoning. The most beneficial context ordering for learners’ performance was symbolic-intuitive-abstract-counterintuitive. The most beneficial context ordering for learners’ perceptions was intuitive-abstract-counterintuitive-symbolic. Based on these results, the researcher proposed a new context ordering: symbolic-intuitive-abstract-counterintuitive-symbolic. This progression incorporates a catalyst at the beginning (symbolic context) which aids the learner in reassessing their prior knowledge. Then, the difficulty of the contexts progresses from easiest to hardest (intuitive-abstract-counterintuitive-symbolic). These findings are important because they provide an instructional sequence for teaching and learning to reason about conditionals that is beneficial to both learners’ performance and their perceptions.

The purpose of the current research was to examine the effects of a sequence of classroom activities that integrated mathematics content with music elements aimed at providing teachers an alternative approach for teaching mathematics. Two classes of third grade students (n=56) from an elementary school in Southern California participated in the research. A random assignment pretest-posttest control group design was used to examine students' changes in mathematics content achievement and the disposition between the two groups. The students in the music group received music-mathematics integrated lessons. A quasi-experiment time series design with multiple pretests, mid-tests and posttests was utilized for investigating the effects of music-mathematics integrated lessons on students' mathematics process ability levels. The results demonstrated that the intervention of a series of music-mathematics integrated lessons had positive effects on the music group students. The findings showed that the music group students had statistically significantly higher scores on mathematics achievement, and mathematics dispositions after the intervention. Moreover, the music group students also showed statistically significant improvement on scores in the mathematics process abilities from pretests to posttests. The study results suggested that music, with its unique features, can be used as a resource for students to make these connections and also as a way for students to represent mathematics in alternative ways. The findings suggest that teachers should take advantage of the opportunities that music offers to help all students learn mathematics in challenging and enjoyable ways developing students' mathematics achievement, mathematical process ability, and mathematics dispositions.

Bilingual fourth graders’ math stories were collected to explore their math identities. Students expressed identities of powerfulness and powerlessness and identified the key resources they need in order to feel like successful mathematicians. These resources included collaboration, manipulatives, their native language, and the support of family. Implications and suggestions for educators are discussed. / text

This study examines the impact of the municipality level characteristics on the average Math achievement of students in third year of lower secondary schools in Mexico. Using data from different Mexican and international sources and multi-level regression models the present work shows that municipality characteristics provide additional explanation of the unexplained variability in educational achievement controlling for school-level factors and even without accounting for student characteristics. Although school factors are highly correlated with municipality’s characteristics, the present study finds that unobservable characteristics of the municipality are playing an important role in Mexican students’ achievement which goes beyond the possible impact that school factors have on achievement. / text

So many students enter college without the conceptual knowledge of mathematical principles needed in order to succeed in higher education. Pre-service teachers entering teacher education programs are not exempt from this dilemma. While training to be educators, many pre-service teachers struggle to understand the concepts behind elementary level mathematics. These pre-service teachers will then continue in the education field and teach mathematics to the future generation. Will they teach their students the way they were taught? The purpose of this study is to investigate how pre-service teachers view their past experiences with math during their K-12 education and to compare those views with their perceptions of how they will teach math in the future. Using both quantitative and qualitative methods through surveys, short answer responses, and interviews, this study examines 38 pre-service teachers currently taking math methods courses at Southern Illinois University Carbondale in order to find out (1) how pre-service teachers view their past experiences with mathematics during their K-12 education, (2) what pre-service teachers' visions of how they will teach mathematics to their students in the future are, (3) which appears to have more influence on pre-service teachers' perceptions of their own future math teaching practices: their past learning experiences or their current teacher preparation program, and (4) what pre-service teachers perceive as effective ways of mathematics teaching and learning. Findings revealed that pre-service teachers tend to view their past K-12 math education experiences as mostly consisting of steps and procedures they were taught to memorize, but they have strong feelings about teaching mathematics for conceptual understanding instead of focusing on memorization like they were taught during their math classes in K-12 education. The results from this study also revealed that pre-service teachers feel it will be difficult not to fall back on the way they were taught mathematics when encountering unfamiliar concepts they have to teach. The need for more field experiences and learning how to incorporate project-based learning and presenting diverse ways of problem solving also came out as ways to improve teacher education programs.