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.     

Preservice Elementary Teachers' Conceptual and Procedural Knowledge of Mean, Median, and Mode

This article describes aspects of the statistical content knowledge of 46 preservice elementary school teachers. The preservice teachers responded to a written item designed to assess their knowledge of mean, median, and mode. The data produced in response to the written item were examined in light of the Structure of the Observed Learning Outcome (SOLO) taxonomy (Biggs & Collis, 1982, 1991) and Ma's (1999) conception of Profound Understanding of Fundamental Mathematics (PUFM). The article describes 4 levels of thinking in regard to comparing and contrasting mean, median, and mode. Several different categories of written definitions for each measure of central tendency are also described. Connections to previous statistical thinking literature are discussed, implications for teacher education are given, and directions for further research are suggested.     

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.     

Making Connections: Elementary Teachers' Construction of Division Word Problems and Representations

If teachers make few connections among multiple representations of division, supporting students in using representations to develop operation sense demanded by national standards will not occur. Studies have investigated how prospective and practicing teachers use representations to develop knowledge of fraction division. However, few studies examined primary (K‐3) teachers' learning of contextual division problems, making connections among representations of division, and resolving the ambiguity of representing quotients with remainders. A written post‐course assessment provided evidence that most teachers created partitive division word problems, used a set model without splitting the remainder, and wrote equations with limited success. Post‐course written reflections demonstrated that many teachers developed pedagogical knowledge for helping students make connections among multiple representations, and mathematical knowledge of unit fractions. These findings suggest two areas that have implications for mathematics teacher educators who design professional development courses to facilitate teachers' learning of mathematical content and pedagogical knowledge of division and fraction relationships.     

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.     

Identification of Error Types in Preservice Teachers' Attempts to Create Fraction Story Problems for Specified Operations

The purpose of this research was to determine if recognizable error types exist in the work of preservice teachers required to create story problems for specific fraction operations. Students were given a particular single‐operation fraction expression and asked to do the calculation and then create a story problem that would require the use of both the expression and calculation to answer the story problem. Distinct error types, determined by mathematical and grammatical characteristics, were identified by the researchers. These error types were then used to analyze an additional set of work samples to determine frequency rates of error types. Future research will involve evaluating the efficacy of specific instructional methods with the goal of reducing the frequency of certain error types.     

算术平均值与几何平均值不等式的推广

利用初等对称多项式得出算术平均值与几何平均值不等式的推广形式,并给出[1]中的一个猜想不等式的证明.     

Preservice Elementary Teachers' Conceptions of What Causes Night and Day

Two different procedures were utilized to study 50 preservice elementary teachers' conceptions of what causes night and day on earth. The numbers of responses representing alternative conceptions for the models with written and verbal explanation procedures were 27 and 14, respectively. The alternative conception most frequently expressed for both procedures was that night and day are caused by the earth revolving around the sun. The frequency with which alternative conceptions were expressed by future teachers is a problem which should be addressed with instruction using models.     

平方平均与算术平均的差距估计

关于n个正数的平方平均与算术平均、几何平均的差的上下界,利用最值压缩定理,给出了两个新的双向不等式.     

Expanding Students' Conceptions of the Arithmetic Mean

The arithmetic mean is the most commonly used measure of central tendancy; nevertheless, many students who can add all the elements of a data set and then divide that sum by the number of elements do not truly understand the concept of mean. This article presents four activities designed to help elementary and middle school students develop a concept of mean. To bring about a desirable level of understanding, all computational formulae and algorithms in mathematics should be preceded by experience emphasizing conceptual understanding. Since that is not the normal instructional sequence for the arithmetic mean, the activities presented in the article assume previous exposure to the computational algorithm for the arithmetic mean.     

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.     

求解多属性决策问题的竞赛图法

对一类已知各方案对每个目标的优先次序的多属性决策问题,借鉴竞赛图的排序方法,提出了确定所有方案总排序的竞赛图法。实例表明,该方法可行、有效且易于实现。     

Metaphors Reveal Preservice Elementary Teachers' Views of Mathematics and Science Teaching

Typically, experiences shape one's attitudes toward the “thing” that is being experienced, whether it is a person, pet, movie, etc. Classroom experiences also shape one's attitudes toward a subject, teacher, learning, and so forth. Studies have shown that attitudes become more negative as students move from elementary to secondary schools. Elementary preservice teachers have had many classroom experiences by the time they reach their methods courses. These varied experiences have shaped their attitudes about the learning and teaching of mathematics and science.     

Graphical Representations of Speed: Obstacles Preservice K‐8 Teachers Experience

This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper‐level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstrated widespread misconceptions about the variable speed. This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speed within a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.     

Web‐Based Instruction on Preservice Teachers' Knowledge of Fraction Operations

This study determines whether web‐based instruction (WBI) represents an improved method for helping preservice teachers learn procedural and conceptual knowledge of fractions.. The purpose was to compare the effectiveness of web‐based instruction (WBI) with the traditional lecture in mathematics content and methods for the elementary school course. The results of this study suggest that the use of WBI in learning fractions is more effective. When compared with the traditional instruction, the WBI treatment results were significantly more effective for procedural and conceptual knowledge of fraction operations.     

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 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.