This study focused on the mathematical content knowledge of prospective teachers in Iceland. The sample was 38 students in the School of Education at the University of Iceland, both graduate and undergraduate students. All of the participants in the study completed a questionnaire survey and 10 were interviewed. The choice of ways to measure the mathematical content knowledge of prospective teachers was grounded in the work of Ball and the research team at the University of Michigan (Delaney, Ball, Hill, Schilling, and Zopf, 2008; Hill, Ball, and Schilling, 2008; Hill, Schilling, and Ball, 2004), and their definition of common content knowledge (knowledge held by people outside the teaching profession) and specialized content knowledge (knowledge used in teaching) (Ball, Thames, and Phelps, 2008). This study employed a mixed methods approach, including both a questionnaire survey and interviews to assess prospective teachers' mathematical knowledge on the mathematical topics numbers and operations and patterns, functions, and algebra. Findings, both from the questionnaire survey and the interviews, indicated that prospective teachers' knowledge was procedural and related to the "standard algorithms" they had learned in elementary school. Also, findings indicated that prospective teachers had difficulties evaluating alternative solution methods, and a common denominator for a difficult topic within both knowledge domains, common content knowledge and specialized content knowledge, was fractions. During the interviews, the most common answer for why a certain way was chosen to solve a problem or a certain step was taken in the solution process, was "because that is the way I learned to do it." Prospective teachers' age did neither significantly influence their test scores, nor their approach to solving problems during the interviews. Supplementary analysis revealed that number of mathematics courses completed prior to entering the teacher education program significantly predicted prospective teachers' outcome on the questionnaire survey.Comparison of the findings from this study to findings from similar studies carried out in the US indicated that there was a wide gap in prospective teachers' ability in mathematics in both countries, and that they struggled with similar topics within mathematics. In general, the results from this study were in line with prior findings, showing, that prospective elementary teachers relied on memory for particular rules in mathematics, their knowledge was procedural and they did not have an underlying understanding of mathematical concepts or procedures (Ball, 1990; Tirosh and Graeber, 1989; Tirosh and Graeber, 1990; Simon, 1993; Mewborn, 2003; Hill, Sleep, Lewis, and Ball, 2007). The findings of this study highlight the need for a more in-depth mathematics education for prospective teachers in the School of Education at the University of Iceland. It is not enough to offer a variety of courses to those specializing in the field of mathematics education. It is also important to offer in-depth mathematics education for those prospective teachers focusing on general education. If those prospective teachers teach mathematics, they will do so in elementary school where students are forming their identity as mathematics students.

Due in part to the growing popularity of the Advanced Placement program, an increasingly large percentage of entering college students are enrolling in calculus courses having already taken calculus in high school. Many students do not score high enough on the AP calculus examination to place out of Calculus I, and many do not take the examination. These students take Calculus I in college having already seen most or all of the material. Students at two colleges were surveyed to determine whether prior calculus experience has an effect on these students' effort levels or motivation. Students who took calculus in high school did not spend as much time on their calculus coursework as those who did not take calculus, but they were just as motivated to do well in the class and they did not miss class any more frequently. Prior calculus experience was not found to have a negative effect on student motivation or effort. Colleges should work to ensure that all students with prior calculus experience receive the best possible placement, and consider making a separate course for these students, if it is practical to do so.

Mathematics education in the elementary schools has experienced many changes in recent decades. With the curriculum becoming more complex as a result of each modification, immense pressure has been put on schools to increase student proficiency. The Common Core State Standards is the latest example of this. These revisions to the mathematics curriculum require a comprehensive understanding of mathematics that the typical elementary teacher lacks. Some elementary schools have begun changing the organization of their classrooms from self-contained to departmentalized as a possible solution to this problem. The purpose of this quantitative study was to examine the effects of elementary departmentalization on student mathematics proficiency. This was done by exploring and comparing the background and educational characteristics, teaching practices, assessment methods, beliefs, and influence of departmentalized elementary mathematics teachers. The study also investigated the circumstances under which there are significant differences in mathematics proficiency between departmentalized and non-departmentalized elementary students, and examined if these differences continued into students' eighth-grade years and/or led to higher level eighth-grade mathematics course attainment. Additionally, the study aimed to determine if there was a relationship between elementary departmentalization and mathematics proficiency and also to identify additional factors that could lead to mathematics proficiency. Data came from the U.S. Department of Education's Early Childhood Longitudinal Study, Kindergarten Class of 1998-99 (ECLS-K) data set. The ECLS-K is a national data set that followed the same children from kindergarten to eighth grade focusing on their school experiences from 1998 to 2007. Numerous statistical analyses were conducted on this rich data set, utilizing the statistical software Stata 13 and R. The results of this study indicate that there is a significant difference in the mathematics proficiency of departmentalized and non-departmentalized students when teachers have below-average mathematics backgrounds. The students of the mathematically below-average departmentalized teachers displayed the highest mathematics proficiency as well as the biggest gain in mathematics proficiency, and these higher proficiencies and gains continued into later grade levels. However, when exploring differences in mathematics proficiency among all students, there were no conclusive differences between departmentalized and non-departmentalized students. Regression models yielded inconclusive results as well, even after controlling for factors pertaining to classroom size, student demographics and socioeconomic status, student confidence, parental background, teacher knowledge and instructional practices, and prior student mathematical proficiency. Other findings include self-contained and departmentalized third-grade teachers being very similar in their educational backgrounds and teaching practices, whereas departmentalized and non-departmentalized fifth-grade teachers were found to be fairly different in their educational backgrounds and instructional practices. However, in both grade levels, self-contained teachers appeared to be more reliant on printed materials than departmentalized teachers.

Preschoolers can often rattle off a long sequence of numbers in order, but have problems in reporting the exact number of objects even in a small set, and have trouble in comparing numerical relation of two sets that differ by exactly 1 item. The present study showed that representing and highlighting sets by showing a circular, enclosed diagram around them with or without a grouping gesture helps children to enhance their understanding of cardinality and to improve their overall math competence. Nighty-three preschool students, ages ranging from 3years-10 months to 4 years-9months (M= 51.82 months, SD= 3.56 months), from three public schools in Harlem, New York participated in this study. Children from each school were ranked based on their pre-test score on the Test of Early Mathematics Ability (TEMA-3), and were then assigned randomly to one of the three math comparison groups or the reading control group. Children in diagram-plus-gesture math group, were asked to draw a bubble by making a grouping gesture around each of the two sets on a touch screen device, indicate the number of fish in each bubble, and judge whether there were the same number of fish in each bubble, and in case the number was not the same, indicate which set had more fish. Children in the diagram only condition simply saw bubbles around sets without the need to do a grouping gesture around them. Children in the no diagram- no gesture condition neither saw a bubble nor did a grouping gesture. All participants played on the software for 4 sessions within a two-week time period and the data were examined microgenetically. Results showed that all children in the math comparison groups improved in their math scores during the game-play and improved in their overall math competence from pre- to post-test, unlike the children in the reading control group. More importantly, children who saw the circular diagram (bubbles) around sets with or without the grouping gesture outperformed children who never saw bubbles nor made a grouping gesture in their accuracy, understanding of cardinality, and overall math competence from pre to post. Further, children with lower executive functioning skills benefitted from performing the grouping gesture in addition to seeing the circular diagram. Gestures can have the same form as diagrams, and hence, they may carry information that is redundant with diagrams. Such redundancy reinforces the message by presenting information in two modalities-- a redundancy that may not be necessary for some, but beneficial to others (i.e. children with low executive functioning skills). Finally, over the course of game-play children who did the grouping gesture never counted the two sets together as one set when asked to compare their numerical relation-- a mistake many preschoolers make; children in the other groups made that mistake occasionally. Because gestures are actions and dynamic by nature, they appear to be especially suited for changing actions and promoting early counting skills.

This dissertation explored how a professional developer, using the Center’s Professional Development Model for Innovating Instruction, supported two teachers’ acquisition of the knowledge needed for their mathematics instruction. Through analysis of detailed field notes and semi-structured interviews of two experienced elementary school teachers working in an urban, high-need school, this dissertation studied how the design and situate components of the Center’s Professional Development Model for Innovating Instruction can lead to multiple ways of supporting teachers’ instruction depending on the teachers’ needs and interests. Findings from these two case studies suggest that there is a need for teacher education mathematics programs—In-service and pre-service—to provide teachers with the knowledge for innovative mathematics instruction needed to create demanding learning experiences in their classrooms. This dissertation elaborates on these results, discusses connections with other research, and ends with implications of these results, in terms of their immediate application and the need for future research.

The role of diagrammatic representations and visual reasoning in mathematics problem solving has been extensively studied. Prior research on visual reasoning and problem solving has provided evidence that the format of a diagram can modulate solvers’ interpretations of the structure and concept of the represented problem information, and influence their problem solving outcomes. In this dissertation, two studies investigated how different types of diagrams influence solvers’ choice of solution strategy and their success rate in solving probability word problems. Participants’ solution strategies suggested that problem solvers tended to construct solutions that reflect the structure of a provided diagram, resulting in different representations of the mathematical structure of the problem. For the present set of problems, a binary tree or a binary table tends to steer solvers to use a sequential-sampling strategy, which defines simple or conditional probabilities for each selection stage and calculates the intersection of these probabilities as the final probability value, using the multiplication rule of probability. This strategy choice is structurally matched with the diagrammatic structure of a binary tree or a binary table, which represents unequally-likely outcomes at the event level. In contrast, an N-by-N (outcome) table steers solvers to use of an outcome-search strategy, which involves searching for the total number of target outcomes and all the possible outcomes at the equally-likely outcome level, and calculates the part-over-the-whole value as the final probability, using the classical definition of probability. This strategy is strongly cued by the N-by-N (outcome) table, because the table structure represents all equally-likely outcomes for a probability problem, and organizes the information so that the target outcomes can be seen as a subset embedded in the whole outcome space. When an N-ary (outcome) tree was provided, choices were split between the two solutions, because the N-ary tree structure not only cues searching for equally-likely outcomes but also organizes the problem information in a sequential-sampling, stage-by-stage way. Furthermore, different diagrams seem to be associated with different patterns of characteristic errors. For example, solving a combinations problem with an N-by-N table tended to elicit erroneous solutions involving miscounting those self-repeated combinations represented by the table’s diagonal cells as valid outcomes. Typical errors associated with the use of a binary tree involved incorrect value definitions of the conditional probability of the outcome of a selection. And the N-ary tree may lead to less successful coordination of all the target outcomes for the studied problems, because the target outcomes were dispersed in the outcome space depicted by the tree, thus not salient. The findings support arguments (e.g., Tversky, Morrison, & Betrancourt, 2002) that in order to promote problem solving success, a diagrammatic representation must be carefully selected or designed so that its structure and content can be well-matched to the problem structure and content. And for computational efficiency, information should be spatially organized so that it can be processed readily and accurately. In addition to the implications for effective diagram design for problem solving activities, the findings also offer important insights for probability education. It is suggested that a variety of diagram types be utilized in the educational activities for novice learners of probability, because they tend to highlight different probability concepts and structures even for the same probability topic.

Quantitative literacy, or numeracy, has been discussed as an essential component of mathematics instruction. In recent years community colleges around the nation introduced a quantitative literacy alternative to the developmental algebra curriculum for students placed into remedial mathematics. The QL curriculum consists of problem situations that are meant to improve numeracy through a combination of collaborative work and a student-centered pedagogy. There is little research that investigates the enactment of that curriculum. Research in K-12 indicates that teachers’ beliefs influence the enactment of curriculum, but studies that connect instructional practices and faculty beliefs are scarce. This study employs a multiple qualitative case study approach to investigate the alignment between four community college instructors’ beliefs about teaching, learning, the nature of mathematics, and curriculum on their enacted practices in two different developmental mathematics courses at a large urban community college (UCC). One is a standard developmental algebra curriculum and the other curriculum is based on quantitative literacy. Data were collected through semi-structured interviews, classroom observations and field notes. The results indicate an alignment between the professed beliefs and enacted practices for all but one instructor in this study. The findings imply that curriculum plays a significant role when its intended design correlates with instructors’ belief systems. The study also discusses the differences in instructional practices across the quantitative literacy and elementary algebra curricula taught by the instructors in this study.

Since the 1960s, the advocacy of teaching mathematics so as to be useful is not without hindrance in school curricula, partly due to the lack of appropriate assessment tools. Practical approaches have been accumulating quickly, but researchers showed that they are not satisfactory in testing students’ ability to apply mathematical knowledge, be they “word problems” in school textbooks, national tests, or large-scale international assessments. To understand the causes behind the dissatisfaction, there is a need to reveal (1) the theories that are used in the test designs, and (2) what the actual assessments are in various curricula. This motive leads to the purpose of the current study, which is to identify empirically consistent theories about students’ ability to apply; the results can be organized as a framework to analyze assessment tools such as PISA, as well as various curricular materials. Based on the current theories, a framework of assessment analysis is created in order to study the coverage of modeling steps of public assessment items. This study finds that, though many education systems have claims of introducing modeling and application into their curricula, high-stake assessments mostly involve a small fraction of the steps that are required in a full modeling cycle. It furthers an earlier result that certain textbooks, though claiming the importance of modeling, almost ignored the first and last steps of modeling. It is found in this study that public assessments are even more limited: most test items that are supposed to test students’ knowledge of application involve only one or two steps of modeling. Furthermore, the tool “modeling spectrum” that is used in the analysis does not only reveal how modeling steps are covered, but can also assists educators to improve or create problems with modeling and application.

The purpose of this study was to determine the modes of acquisition of Shanghai elementary mathematics teachers’ pedagogical content knowledge within their communities of practice. This study uses the qualitative multiple-case study with a survey research approach with two teachers in two public elementary schools, one each from an urban and a suburban district of Shanghai. In total, forty-four teachers, four teaching research coordinators in the two districts and city, one university professor, and four school administrators were involved in the study. The study shows that Shanghai elementary mathematics teachers acquire and develop their pedagogical content knowledge through positive mentorship; active participation in Teaching Research Group activities in the schools, districts, and city; and informal and formal communications with their colleagues in their school communities. The teaching research coordinators help teachers to better understand the elementary mathematics curriculum, topics, and teaching materials, and students’ learning of mathematics. School policies encourage, support, and ensure that teachers’ professional learning and development occur through their participation within teacher-supported communities of practice. This study has implications for the teachers’ communities of practice, in that policy makers and school administrators should enable teachers to share their teaching practices to improve their mathematics pedagogical content knowledge, and therefore improve students’ learning of mathematics.

The goal of this study was to investigate whether mathematics teachers in Brazil had implicit gender biases, and if that potential bias was related to students' confidence and interest. The literature shows that there is a significant gender gap in mathematics achievement favoring boys, and Brazil is a special case in that it has one of the largest divergences in the world. This study investigated whether mathematics teachers in Brazil had implicit gender biases, if that bias was related to their students' confidence and interest in mathematics, and in what ways teachers' bias could be observed in conversations about teaching. For this study I surveyed 40 teachers, along with the students in one of each instructor’s mathematics classes. Teachers were asked to respond to a demographic questionnaire and implicit association test (IAT), while students were asked to respond to a questionnaire measuring their self-assessed confidence and interest in mathematics. At a later date, 10 teachers were selected to be interviewed, based on their IAT scores. The results show that mathematics teachers in Brazil had implicit gender biases regarding mathematics, but that their respective biases varied significantly. Male teachers were significantly biased in favor of boys, while female teachers were not. Teachers' implicit biases also varied depending on their educational levels. Students' confidence and interest in mathematics were shown not to be related to their teacher's measure of bias. However, confidence and interest did vary based on whether students attended public or private schools – with private schools having a significantly larger gender gap in both of these factors, and students' grade -- with the gap being wider among older students. Students' interest in mathematics also proved to be related to teachers' educational level, but their confidence in mathematics was not. Teachers in Brazil believe overall that girls and boys behave differently from each other in school; furthermore, they believe that these differences are due to societal and parental pressures and expectations regarding gender. Teachers who associated mathematics with boys did not appear to be aware of that implicit bias, and in conversation often referred to gender differences in a way that indicated they thought girls had advantages in school that boys did not.