Preservice Teachers' Understanding of Perimeter and Area

Fifty-four postgraduate (elementary school) preservice teachers were given four tasks, two to assess their understanding of perimeter and two to assess their understanding of area. The teachers were asked to prepare a question that would assess student understanding of perimeter. Then they were given three problems and asked to decide whether the problems had sufficient information for a solution. The type of question prepared for the first task and the number of preservice teachers who stated that the other three tasks had insufficient information indicate a procedural understanding of perimeter and area, rather than a conceptual and relational understanding     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

Area and Perimeter: Preservice Teachers' Confusion

Measurement topics, including perimeter and area, are basic to the competency domain of elementary teachers. This study examines solution strategies, used by elementary preservice teachers, to find the perimeter and area of a shaded geometric figure. Many of these preservice teachers incorrectly found the perimeter of the figure by the same method they used to find the area. Examination of incorrect strategies by preservice teachers provides an opportunity to investigate inadequacies in the mathematical backgrounds of these students as well as to make curriculum changes in teacher education programs.     

Family Mathematics Nights: An Opportunity to Improve Preservice Teachers' Understanding of Parents' Roles and Expectations

Providing preservice teachers with opportunities to engage with parents and begin to see them as collaborators in their children's education is a persistent challenge in mathematics methods courses and teacher preparation programs more broadly. We describe the use of family mathematics nights as a model for engaging parents and preservice teachers. These events helped preservice teachers feel more comfortable in working with parents, while also giving them a friendly space for learning more about parents, the ways they work with their children, and activities they do with their children. Incorporating surveys about parents' needs and suggestions for preservice teachers allowed us to use the results as a catalyst for fostering discussions around parent engagement later in the methods course.     

Preservice Mathematics and Science Teachers' Case Studies of a Diverse Population of Workers

Sixteen preservice science and mathematics teachers in the last term of a Master of Arts in Teaching (MAT) program completed case studies of workers from a variety of settings including a radio station, department store, manufacturing plant, health care facility, water treatment plant, and engineering firm. These preservice teachers then examined their findings and reflected about the knowledge and skills necessary for high school graduates to be successful in the present day workplace. Finally, the preservice teachers examined how they could personally contribute to these desirable student outcomes within the high school science or mathematics curriculum for which they will eventually be responsible in their future teaching situations.     

The DESTIN: Preservice Teachers' Drawings of the Ideal Elementary Science Teacher

The aim of this study is to report findings from the Drawing‐Elementary‐Science‐Teacher‐Ideal‐Not, or the DESTIN procedure. The study utilizes a simple drawing procedure accompanied by a narrative and discussion for understanding preservice teachers' images of science, science teaching, and the science teacher. Ninety drawings from two sections of an elementary science methods course were analyzed. Looking at the pre‐drawings from the beginning of the semester and post‐drawings done at the end of the semester, the findings relate the value of using drawings in teacher education and the usefulness of this procedure to promote discussions about science teaching, the construction of new images and practices for teaching elementary science, and teacher identity. The DESTIN procedure has potential as a productive activity for teacher education and long‐term professional development by making more explicit teachers' views of science teaching and learning and their past experiences as science learners.     

Preservice Mathematics Teachers' Ambiguous Views of Technology

While the nature of mathematics and the nature of science have received much attention, there is a lack of research on the nature of technology. This study sought to investigate preservice teachers' perceptions about the nature of technology and its role in mathematics education and society. Based on two philosophical theories of technology, the author analyzed 22 preservice teachers' philosophy statements on educational technology and in‐depth interviews with 5 of them. The findings reveal that the teachers had ambiguous notions about the characteristics of technology, which were contingent on contexts of technology use, human–machine relationships, and other factors. However, in most cases, they had an instrumental view of technology and a techno‐centric mindset. Although the participating teachers were encouraged to reflect on broad socio‐cultural issues associated with technology, most of them did not have a linguistic and conceptual framework from which to examine thoroughly how technology might influence human consciousness. As the educational technology and teacher education communities have broadened their understanding of technology and what it takes for an organizational change, preservice teachers need adequate opportunities to explore the pedagogical, epistemological, ontological, and cultural implications of the educational use of technology.     

Indexing Distributions of Data: Preservice Teachers' Notions of Representativeness

The purpose of the study was to identify strategies used by preservice elementary teachers to index distributions of data. A secondary purpose of the study was to investigate whether representational form influenced the type of representative values constructed. Two hundred eighty‐three preservice teachers were administered a selection of tasks requiring them to construct representative values for a variety of distributional shapes. Participants were asked to provide justifications for their chosen representative measures. Data were analyzed from two perspectives: nature and adequacy of representative measure. Results indicated that when presented with skewed data sets, over half of the participants constructed poorly representative values. Symmetric data sets were best indexed with 67% of participants constructing highly representative values. Representational form played a significant role in determining representative values. Data presented graphically were more likely to be represented using measures of variability than were those presented nongraphically. Conversely, measures of central tendency were more frequently used to represent nongraphical data. The information arising from this study indicates that many preservice teachers construct poorly representative values to index data sets and utilize identical measures of representativeness regardless of distributional shape. The findings of the study highlight the need for exploratory data activities at the preservice level, call for provision of experiences affording insights into construction of representative values, and thus, have implications for the design of preservice mathematics education curricula.     

Linking Preservice Teachers' Mathematics Self‐Efficacy and Mathematics Teaching Efficacy to Their Mathematical Performance

This study examined preservice teachers' mathematics self‐efficacy and mathematics teaching efficacy and compared them to their mathematical performance. Participants included 89 early childhood preservice teachers at a Midwestern university. Instruments included the Mathematics Self‐Efficacy Scale (MSES), Mathematics Teaching Efficacy Beliefs Instrument (MTEBI), and the Illinois Certification Testing System (ICTS) Basic Skills Test. The results indicate that preservice teachers' mathematics self‐efficacy is positively correlated to their personal mathematics teaching efficacy. In addition, their mathematical performance is related to their mathematics self‐efficacy and mathematics teaching efficacy. In regard to affecting student outcomes, only those preservice teachers who are very confident in their ability to teach believe they can have an effect on their students. Implications on teacher education programs are discussed.     

Focusing on Units to Support Prospective Elementary Teachers' Understanding of Division in Fractional Contexts

This theoretical paper proposes a way to extend the partitive and measurement interpretations of whole number division to fractional contexts focusing on the issue of units. We define the unit‐changing and unit‐keeping interpretations for division and suggest a stronger and earlier focus on the concept of units in courses for prospective elementary teachers. As we highlight the importance of the concept of unit, we use the proposed interpretations in the analysis of one example that emerged in our methods course, discuss the idea of inverting and multiplying, and suggest implications for teacher education.     

The Impact of Professional Noticing on Teachers' Adaptations of Challenging Tasks

This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding.     

Mathematical Modeling for Preservice Teachers: A Problem from Anesthesiology

The study reported in this article deals with the observed actions of prospective Swedish mathematics teachers as they were working with a modeling situation. These prospective teachers were preparing to teach in Grades 4 to9 or in the gymnasium (Grades 10 to 12) and were students in a course in mathematical modeling. The larger study of which this study was a part focused on these students' understanding of modeling and how they related mathematical models to the real world. This article also attempts to illustrate how mathematics is intertwined with many other subjects, in this case medicine. This revised version was published online in July 2006 with corrections to the Cover Date.     

Preservice Teachers' Analysis of Children's Work to Make Instructional Decisions

Teaching is an interactive process in which teachers gather information, analyze the results, and construct a response based on this diagnosis ( Cooney, 1988 ). Considering alternatives in constructing a response, that is, making an instructional decision, is of great importance in teaching. How might mathematics teacher educators provide experiences for preservice teachers to begin the development of this skill? In an attempt to determine how these experiences might reveal the level of understanding preservice teachers have in regards to children's mathematical thinking, a study was conducted over three semesters. During the mathematics methods course, preservice teachers were involved in analyzing children's work through the review and discussion of several samples. They were required to determine the error pattern, discuss what might have lead to this misconception, and suggest appropriate instructional strategies that might help this student. Although most preservice teachers could correctly identify the computational error patterns, they had difficulty in determining what might have led to the misconceptions and proposing effective instructional strategies.     

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.