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.     

Influence of a Mathematics Teachers' Circle on Elementary Teachers' Use of Problem Solving

Math teachers' circles are a form of professional development that is recommended by the Conference Board of the Mathematical Sciences in their publication Mathematical Education of Teachers II (2012). However, little research has been published on how effective math teachers' circles are in advancing the mathematical knowledge of teachers and influencing their actual teaching practice. This study reports how 25 elementary teachers from schools with minority and economically disadvantaged populations responded to a math circle experience. A pretest and posttest showed a significant increase in problem‐solving ability. Through analysis of journals, all but 4 of the 25 teachers indicated that participation in the math circle increased their understanding of problem solving and advanced their use of problem solving in the classroom.     

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.     

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     

Preservice Elementary Teachers' Use of Mathematics in a Project‐Based Science Approach

The use of a project‐based science (PBS) approach to teaching encourages students to integrate mathematics and science in meaningful ways as they create projects. As a beginning study of how students use mathematics in such an approach, an analysis of 23 projects developed by preservice elementary teachers enrolled in an elementary science course was conducted. Findings showed that students made a number of different types of mathematical errors and underutilized data representation and summary forms. Implications included the importance of developing methods for supporting the use of mathematical tools in utilizing a project‐based approach and considering ways that such tools mediate scientific thinking.     

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.     

Algebra Teachers' Utilization of Problems Requiring Transfer Between Algebraic,Numeric, and Graphic Representations

For students to develop an understanding of functions, they must have opportunities to solve problems that require them to transfer between algebraic, numeric, and graphic representations (transfer problems). Research has confirmed student difficulties with certain types of transfer problems and has suggested instructional factors as a possible cause. Algebra teachers (n= 28) were surveyed to determine the amount of class time they devote to different types of transfer problems and how many times these problems appear on their teacher‐made assessments. Results suggest that teachers dedicate less class time to graphic to numeric transfer problems than to any other type of transfer problem and that these problems appear less frequently on assessments. These are exactly the types of transfer problems that pose the most difficulty for students. It is conjectured that teachers' familiarity with these problems, combined with assumed student mastery, contribute to this mismatch.     

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.     

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.     

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.