Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.     

Learning to Teach Inquiry: A Beginning Science Teacher of English Language Learners

Early career science teachers are often assigned to classrooms with high numbers of English language learners (ELLs). For the underprepared early career science teacher, these circumstances are challenging. This study examines the changes in beliefs and practices of an early career science teacher who taught high numbers of ELLs in an urban setting. Victoria participated in the Alternative Support for Induction Science Teachers (ASIST) program during her initial two years of teaching. Our research team followed her over a three‐year period, and the data collected included classroom observations and interviews about her beliefs and practices. In addition, documents such as teacher evaluations and classroom artifacts were collected periodically for the purpose of triangulation. The analysis of the data revealed that with the support of the ASIST program, Victoria implemented inquiry lessons and utilized instructional materials that promoted language and science competencies for her ELLs. Conversely, standardized testing and her teaching assignment played a role in constraining the implementation of inquiry‐based practices. The results of this study call for collaborative efforts between university science educators and school administrators to provide professional development opportunities and support to build the capacity of early career science teachers of ELLs.     

Assessing the Use of Graphing Calculators in College Algebra: Reflecting on Dimensions of Teaching and Learning

This article considers various aspects of teaching and learning stemming from the integration of graphing calculator use in a semester-long college algebra course. The project examined four class sections, in which two instructors each taught one section using graphing calculators and one section using a traditional approach. Achievement and attitude data showed no significant differences for treatment or instructor; a significant difference in achievement was found for gender. Students in the calculator sections responded to an open-ended questionnaire about their use of the calculator and were generally supportive of the technology. Students were in agreement about specific topics for which the technology was most useful. Overall, findings indicated that the technology use had a positive impact on various dimensions of student learning.     

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.     

基于解释结构模型的学生学习行为投入评价因素研究

学生学习行为投入是教学研究中的热点问题,对改善学习绩效,提升教育质量具有重要意义.通过文献梳理和专家访谈,识别出学生学习行为投入评价影响因素的15个变量,并确定了各因素变量之间的二元关系.在此基础上,采用解释结构模型方法对这些影响因素进行层级建模,得到学生学习行为投入评价影响因素解释结构模型.研究发现:学术拓展、任务独立、寻求挑战、交流讨论、协作学习、克服困难是影响学习行为投入的直接因素,更多的投入、自我检测、积极响应、出勤和参与决策与管理是根本因素,坚持、专注、自我调节和任务管理作为间接因素发挥作用.最后对研究结果进行相关分析,进而为促进学生学习行为投入提供一定的参考.     

Change in Preservice Teachers' Beliefs: An Evaluation of a Mathematics Methods Course

This study investigated and evaluated the potential impact of an elementary mathematics methods course in promoting teacher beliefs and attitudes that are consistent with the underlying philosophy of current reform efforts in mathematics education. Using the Mathematics Beliefs Instrument (MBI; Hart, 2002 ), data from 89 preservice teachers were used to evaluate the course. Findings from the study suggest a positive relationship between participating in the mathematics methods course and change in teacher beliefs and attitudes. This study also provided additional validation of the MBI.     

From the Written to the Enacted Curricula: The Intermediary Role of Middle School Mathematics Teachers in Shaping Students' Opportunity to Learn

In this paper is reported the extent of textbook use by 39 middle school mathematics teachers in six states, 17 utilizing a textbook series developed with funding from the National Science Foundation (NSF‐funded) and 22 using textbooks developed by commercial publishers (publisher‐generated). Results indicate that both sets of teachers placed significantly higher emphasis on Number and Operation, often at the expense of other content strands. Location of topics within a textbook represented an oversimplified explanation of what mathematics gets taught or omitted. Most teachers using an NSF‐funded curriculum taught content intended for students in a different (lower) grade, and both sets of teachers supplemented with skill‐building and “practice” worksheets. Implications for documenting teachers' “fidelity of implementation” ( National Research Council, 2004 ) are offered.     

Community‐Based Service‐Learning as a Source of Personal Self‐Efficacy: Preparing Preservice Elementary Teachers to Teach Science for Diversity

Bandura (1997) contends that when compared to other sources of efficacy, mastery experiences, when presented appropriately, have the most powerful influence on self‐efficacy. The purpose of this study was to investigate the effects of community‐based service learning (CBSL) experiences on preservice elementary teachers' personal self‐efficacy beliefs about equitable science teaching and learning. Data were collected using pretests‐posttests and post‐questionnaires with the study sample. Findings from this study support Bandura's assertion. CBSL experiences were an important source of personal self‐efficacy and significantly influenced preservice elementary teachers' personal self‐efficacy beliefs about equitable science teaching and learning.     

Learning to signal: Analysis of a micro-level reinforcement model

We consider the following signaling game. Nature plays first from the set {1,2}$\{1,2\}$. Player 1 (the Sender) sees this and plays from the set {A,B}$\{A,B\}$. Player 2 (the Receiver) sees only Player 1’s play and plays from the set {1,2}$\{1,2\}$. Both players win if Player 2’s play equals Nature’s play and lose otherwise. Players are told whether they have won or lost, and the game is repeated. An urn scheme for learning coordination in this game is as follows. Each node of the decision tree for Players 1 and 2 contains an urn with balls of two colors for the two possible decisions. Players make decisions by drawing from the appropriate urns. After a win, each ball that was drawn is reinforced by adding another of the same color to the urn. A number of equilibria are possible for this game other than the optimal ones. However, we show that the urn scheme achieves asymptotically optimal coordination.     

Middle School Students' Conceptual Learning from the Implementation of a New NSF Supported Curriculum: Interactions in Physical Science™

A new National Science Foundation supported curriculum, Interactions in Physical Science?, was evaluated on students’ conceptual change in the twelve concept areas of the national physical science content standard (B) for grades 5–8. Eighth grade students (N=66) were evaluated pre and post on a 31‐item multiple‐choice test of conceptual understanding developed by the Harvard ‐Smithsonian Center for Astrophysics. Significant student gains (p<. 05, t‐test, two‐tailed) occurred in all concept areas in the category of properties and changes in properties of matter; for the force concept areas in the category motions and forces; and for the heat transfer and light interactions areas in the category of transfer of energy. Two of the six concept areas in the category of transfer of energy, chemical and nuclear reactions and the sun as a major source of energy, were not addressed in this study. Significant learning gains as item percent correct were typically close to 20%, though effect sizes were small to medium in magnitude (d = 0.3–0.6). Implications of the study for conceptual change curriculum and teaching are discussed.     

Application of a Near-Optimal Reinforcement Learning Controller to a Robotics Problem in Manufacturing: A Hybrid Approach

Optimization theory provides a framework for determining the best decisions or actions with respect to some mathematical model of a process. This paper focuses on learning to act in a near-optimal manner through reinforcement learning for problems that either have no model or the model is too complex. One approach to solving this class of problems is via approximate dynamic programming. The application of these methods are established primarily for the case of discrete state and action spaces. In this paper we develop efficient methods of learning which act in complex systems with continuous state and action spaces. Monte-Carlo approaches are employed to estimate function values in an iterative, incremental procedure. Derivative-free line search methods are used to obtain a near-optimal action in the continuous action space for a discrete subset of the state space. This near-optimal control policy is then extended to the entire continuous state space via a fuzzy additive model. To compensate for approximation errors, a modified procedure for perturbing the generated control policy is developed. Convergence results under moderate assumptions and stopping criteria are established.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

Reflective Pathways: Analysis of an Urban Science Teaching Field Experience on Noyce Scholar‐Science Education Awardees' Decisions to Teach Science in a High‐Need New York City School

Awardees of the National Science Foundation's Noyce Scholars funds are required to teach science in high‐need urban or rural school districts upon graduation. The purpose of this research was to analyze the reflective considerations that distinguish preservice Noyce Scholar science education majors committed to teaching in high‐need New York City (NYC) schools from those committed to high‐need rural careers. Essays designed to expose their (n = 22) considerations of teaching in NYC written before, immediately after, and two weeks following a one‐week cultural and science teaching internship experience were coded to reveal distinguishing reflective thought patterns. The results showed that those (16/22) whose final essay showed commitment to NYC teaching viewed themselves positively as NYC teachers and that increases in positive self‐perceptions were paralleled by increases in affinity toward living in NYC and NYC schools. Those not inclined to NYC teaching (6/22) expressed increasingly negative views of themselves as NYC teachers, NYC life, and urban education generally. The research provides insights into what Noyce Scholar undergraduates in science education think about when considering a teaching career in urban school districts.