Non‐Traditional Preservice Teachers and Their Mathematics Efficacy Beliefs

In Florida, recent legislative changes have granted community colleges the ability to offer baccalaureate degrees in education, frequently to non‐traditional students. Based on information obtained from the literature covering preservice teachers' math knowledge, teachers' efficacy beliefs about math, and high‐stakes mathematics testing, a study examined a population of preservice teachers in a new Florida teacher preparation program. The research investigated relationships surrounding non‐traditional preservice teachers' characteristics such as: ages, high‐stakes math failures, lower division mathematics history, and math methods course performance, in relation to their efficacy beliefs about mathematics. Results revealed that preservice teachers' ages, lower division mathematics history, and math methods course performance, did have a significant relationship with their math efficacy beliefs, as measured by the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI); the variable of high‐stakes math failures did not. Additionally, a multiple regression model including the aforementioned variables did predict preservice teachers' MTEBI scores, but did not generalize to the greater population. The findings from this study can assist new teacher preparation programs in isolating variables that identify preservice teachers who are at risk for poor mathematical attitudes; can posit avenues for fostering positive math beliefs in preservice teachers; and can recommend further research in this area.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.     

Crossing the Barriers Between Preservice and Inservice Mathematics Teacher Education: An Evaluation of the Grant School Professional Development Program

A 2‐year school‐based mathematics professional development program is described and evaluated after its first year of implementation. Included in this program as its first course was a unique methods course in elementary education involving both preservice students and inservice teachers who cooperatively studied and applied reform pedagogy. The program resulted from the collaborative efforts of two institutions of higher education, a neighboring school district, the principal and teachers of one school within that district, and the state office of education. Evaluation of the first year of the program consisted of assessing the beliefs and perceptions of both preservice students and inservice teachers, along with an assessment of the mathematical achievement of the children within the classes of those teachers. Pre‐ and post‐assessments of the preservice students and inservice teachers' beliefs regarding reform pedagogy were administered using the IMAP [Integrating Mathematics and Pedagogy] Web‐Based Beliefs Survey (2006). Likert scale surveys were used to assess perceptions regarding course climate and participant relationships from both teacher groups. The mathematical achievement of children was assessed in three ways: The Wide Range Achievement Test‐3 ( Stone, Jastak, & Wilkinson, 1995 ), the Utah state criterion‐referenced assessment, and performance assessments developed specifically for use at the school. Data obtained from all sources indicated positive effects upon teachers and children, thus providing substantial evidence in support of both the value of the methods course itself and the overall professional development program. An additional evaluation will be conducted following the second year of the program.     

Science as a Learner and as a Teacher: Measuring Science Self‐Efficacy of Elementary Preservice Teachers

Academic science achievement of U.S. students has raised concerns regarding our ability as a nation to compete in a global economy. Additionally, research has shown that many elementary teachers have weak science content backgrounds and had poor/negative experiences as students of science, resulting in a lack of confidence regarding teaching science. However, efforts to increase science self‐efficacy (SE) in preservice teachers can help to combat these issues. This study looked at a sample of preservice elementary teachers engaged in a semester‐long science content course, using Bandura's concept of SE as a conceptual framework. Our quantitative data showed significant increases in science SE on both subscales (personal efficacy and outcome expectancy). Our qualitative data showed that students communicated an increased sense of confidence with regard to the discipline of science. In addition, students reported learning science pedagogy through the instructor's modeling. Combining our findings resulted in several meta‐inferences, one of which showed students growing as both confident learners of science and teachers of science simultaneously. We created a construct new to the literature to describe this phenomenon: “teacher‐learner,” for students are both learning science and learning to teach science simultaneously through the content course experience, resulting in increased science SE.     

Relationships Between the Process Standards: Process Elicited Through Letter Writing Between Preservice Teachers and High School Mathematics Students

The current body of literature suggests an interactive relationship between several of the process standards advocated by National Council of Teachers of Mathematics. Verbal and written mathematical communication has often been described as an alternative to typical mathematical representations (e.g., charts and graphs). Therefore, the relationship between these two processes has been characterized as interchangeable. The current study examined mathematics preservice teachers’ elicitation of the process standards from high school students during a letter‐writing project. Correlational analysis illustrated two sets of relationships: one between communication, problem solving, and reasoning and proof; and a weaker, parallel relationship between representation, problem solving, and reasoning and proof. Additionally, it was found that written communication and representation were elicited in isolation of each other significantly more often than in conjunction, supporting claims of the literature. Implications of these findings and suggestions of future research are described.     

Issues and Challenges for Middle School Mathematics Teachers in Inclusion Classrooms

The purpose of this qualitative study was to investigate typical middle school general education mathematics teachers' beliefs and knowledge of students with learning disabilities and inclusive instruction and to gain an understanding of the process of inclusion as it is implemented in middle school classrooms. In‐depth interviews, surveys, and classroom observations were conducted with seven teachers. The constant comparative method was used to analyze all interview and observation data. The findings reveal that even teachers who believe that inclusion is being successfully implemented are unclear about their responsibilities towards included students and the learning characteristics and specific mathematics teaching approaches that would be effective. The general educators feel that they were grossly under‐prepared during preservice and inservice for the realities of inclusion teaching. The study provides insights that can be used to enhance preservice and inservice programs for teachers and underscores the necessity for building teamwork and collaboration among general and special education middle school teachers.     

Exploring the Link Between Preservice Teachers' Conception of Proof and the Use of Dynamic Geometry Software

This case study investigated how secondary preservice mathematics teachers perceive the need for and the benefits of formal proof when given geometric tasks in the context of dynamic geometry software. Results indicate that preservice teachers are concerned that after using dynamic software high school students will not see the need for proofs. The participants stated that multiple examples are not equivalent to a proof but, nonetheless, questioned the value of formal proof for high school students. Finally, preservice teachers found the greatest value of geometric software to be in helping students understand key relationships within a problem or theorem. Participants also tended to study a problem more deeply with the software than without it.     

Interpreting Middle School Students' Proportional Reasoning Strategies: Observations From Preservice Teachers

This paper reports the findings of an investigation of 11 preservice secondary school teachers' interpretations of the development of proportional reasoning strategies used by middle school students. The preservice teachers examined samples of solution strategies generated by middle school students in proportional reasoning situations and prepared written responses of their views concerning the developmental levels indicated in the students' work. Each preservice teacher also participated in an hour‐long interview, in which the researchers asked for elaboration and clarification of the written responses and, in some cases, challenged the preservice teachers to consider alternative interpretations for the middle school students' work. The interviews were audiotaped for later analysis by the investigators, and key aspects of both the written and audiotaped responses were entered into a spreadsheet and later tabulated into categories indicating trends in the preservice teachers' interpretations. Some implications for the preparation of preservice middle school science and mathematics teachers are included.     

Mathematical Habits of the Mind for Preservice Teachers

This article provides an overview of mathematical habits of the mind and discusses how the concept relates to Polya's problem solving principles as well as exemplification. Specific problems are discussed as a means to assist preservice elementary school teachers' in their development of mathematical habits of the mind. Without a technique to begin solving these rich problems, preservice teachers may have difficulty getting started. The process of preservice teachers outlining their thinking as they progress through Polya's process is discussed. Students' reflections from this technique are discussed to explore the outcomes that may be expected from establishing an environment where students are encouraged to develop mathematical habits of the mind.     

The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

High school students’ understanding of the function concept

This paper is a study of part of the Algebra Project's program for underrepresented high school students from the lowest quartile of academic achievement, social and economic status. The study focuses on students’ learning the concept of function. The curriculum and pedagogy are part of an innovative, experimental approach designed and implemented by the Algebra Project. The instructional treatment took place over 7 weeks during the Junior Year of 15 students from our target population. Immediately after instruction, a written instrument was administered followed, several weeks later, by in-depth interviews. The results are that many of our participants achieved a level of knowledge and understanding of functions on a par with beginning college students, including preservice teachers, as reported in the literature. Many conceptual difficulties that have been reported in the research literature were not as prevalent for our participants and some of them were capable of solving difficult problems involving composition of functions. We conclude that, with appropriate pedagogy, it is possible for students in the Algebra Project's target population to learn substantial and non-trivial mathematics at the high school level, and that the Algebra Project approach is one example of such a pedagogy.     

Preservice teachers’ mathematics task modification for emergent bilinguals

Implementing mathematically challenging tasks is difficult for teachers when working with emergent bilinguals because cognitively demanding tasks in mathematics commonly have high language demand. Currently, inadequate teacher preparation for teaching emergent bilinguals is becoming a significant concern in the United States as this population of students is rapidly growing. This study investigated how two mathematics preservice teachers (PSTs) support middle school emergent bilinguals to understand cognitively demanding mathematical problems through task modification. Fieldwork with a concurrent intervention was designed for the PSTs to work with emergent bilinguals in a one‐on‐one setting. The PSTs modified cognitively demanding mathematics tasks and designed a lesson for the emergent bilinguals based on the modified tasks. The results revealed that the task modification made by the PSTs tended to shift from reducing cognitive demands in mathematics and language to maintaining the demands through learning strategies of contextual support.     

Science Camp as a Transformative Experience for Students,Parents, and Teachers in the Urban Setting

This paper reports on creative partnerships to create learning communities to benefit students, parents, and teachers in the urban setting. Science Camp is a university day camp offering problem‐based science exploration for urban middle school students and an introduction to college life. For parents it is an opportunity to learn about college opportunities for their children and to gather information about funding sources. For middle school science teachers and preservice teachers it is a learning laboratory for conducting problem‐based learning in the urban setting. The findings describe the effects of the learning communities on the transformation of students, parents, preservice teachers, and in‐service teachers.     

An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals, and percents

Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N = 581).In addition to using geometric figures and manipulatives, teachers used natural language such as the words nanny and house to characterize mathematical procedures or algorithms. Some teachers used the words or phrases bigger, smaller, doubling, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers and students, specifically equations with missing equals signs and not multiply/dividing by one to find equivalent fractions. No evidence though of a relationship between representational forms and degree of correctness of solutions was found on student work. However, when students exhibited misconceptions, those misconceptions were linked to teachers’ use of idiosyncratic representations.     

Preservice mathematics teachers' conceptions and enactments of modeling standards

Mathematical modeling has been highlighted recently as Common Core State Standards for Mathematics (CCSSM) included Model with Mathematics as one of the Standards for Mathematical Practices (SMP) and a modeling strand in the high school standards. This common aspect of standards across most states in the United States intended by CCSSM authors and policy makers seems to mitigate the diverse notions of mathematical modeling. When we observed secondary mathematics preservice teachers (M‐PSTs) who learned about the SMP and used CCSSM modeling standards to plan and enact lessons, however, we noted differences in their interpretations and enactments of the standards, despite their attendance in the same course sections during a teacher preparation program. This result led us to investigate the ways the M‐PSTs understood modeling standards, which could provide insights into better preparing teachers to teach mathematical modeling. We present the contrasting ways in which M‐PSTs presented modeling related to their conceptions of mathematical modeling, choices of problems, and enactments over an academic year, connecting their practices to extant research. We consider this teaching and research experience as an opportunity to make significant changes in our instruction that may result in our students enhanced implementation of mathematical modeling.     

Preparing K‐8 Preservice Teachers in a Content Course for Standards‐Based Mathematics Pedagogy

Many K–8 preservice teachers have not experienced learning mathematics in a standards‐based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards‐based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Furthermore, findings are presented from a study on this course that describe students' pre‐ and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards‐based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points.     

Service Learning Within a Secondary Math and Science Teacher Education Program: Preservice MAT Teachers’ Perspectives

Previous literature suggests that service learning may offer new opportunities to support the development of preservice science and math teachers, but few studies examine service learning beyond isolated teaching events. In this qualitative study, we attempt to improve upon this literature by following Master of Arts in Teaching (MAT) students’ views of their service learning experiences throughout their MAT program and first two years teaching. Data sources included audiotaped individual interviews, focus group field notes, and surveys with seven preservice teachers over a three‐year period. Three major findings emerged from the data analysis. First, participants identified characteristics of service learning teaching events that made them particularly useful, and these included the timing of events, targeted grade level, exposure to high‐needs contexts, and opportunities to practice pedagogical skills. Second, participation in the service learning events improved preservice teachers’ confidence and comfort teaching in high‐needs contexts, but several concerns and deficit perspectives about teaching in high‐needs contexts remained. Third, participants indicated specific ways that the service learning teaching events impacted their readiness to teach in high‐needs contexts. These findings may inform other science and math teacher educators seeking to embed service learning opportunities into their programs.     

Transforming the mathematical practices of learners and teachers through digital technology*

ABSTRACT

This article argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: (1) dynamic and graphical tools; (2) tools that outsource processing power; (3) tools that offer new representational infrastructures for mathematics; (4) tools that help to bridge the gap between school mathematics and the students’ world; (5) tools that exploit high-bandwidth connectivity to support mathematics learning; and (6) tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies. Following exemplification of each category, the article ends with some reflections on the progress of research in this area and identifies some remaining challenges.     

Science Talk: Preservice Teachers Facilitating Science Learning in Diverse Afterschool Environments

The purpose of this study was to assess the impact a community‐based service learning program might have on preservice teachers' science instruction during student teaching. Designed to promote science inquiry, preservice teachers learned how to offer students more opportunities to develop their own ways of thinking through utilization of an afterschool science program that provided them extended opportunities to practice their science teaching skills. Three preservice teachers were followed to examine and evaluate the transfer of this experience to their student teaching classroom. Investigation methods included field observations and semi‐structured, individual interviews. Findings indicate that preservice teachers expanded their ideas of science inquiry instruction to include multiple modes of formative assessment, while also struggling with the desire to give students the correct answer. While the participants' experiences are few in number, the potential of afterschool teaching experience serving as an effective learning experience in preservice teacher preparation is significant. With the constraints of high‐stakes testing, community‐based service learning teaching opportunities for elementary and middle‐school preservice teachers can support both the development and refinement of inquiry instruction skills.     

Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function

Using traditional educational research methods, it is difficult to assess students’ understanding of mathematical concepts, even though qualitative methods such as task observation and interviews provide some useful information. It has now become possible to use functional magnetic resonance imaging (fMRI) to observe brain activity whilst students think about mathematics, although much of this work has concentrated on number. In this study, we used fMRI to examine brain activity whilst ten university students translated between graphical and algebraic formats of both linear and quadratic mathematical functions. Consistent with previous studies on the representation of number, this task elicited activity in the intra-parietal sulcus, as well as in the inferior frontal gyrus. We also analysed qualitative data on participants’ introspection of strategies employed when reasoning about function. Expert participants focused more on key properties of functions when translating between formats than did novices. Implications for the teaching and learning of functions are discussed, including the relationship of function properties to difficulties in conversion from algebraic to graphical representation systems and vice versa, the desirability of teachers focusing attention on function properties, and the importance of integrating graphical and algebraic function instruction.