Reflective Tutoring: Insights into Preservice Teacher Learning

An undergraduate seminar was designed to help preservice teachers focus on students' learning. Preservice teachers planned and conducted weekly tutoring sessions with fourth graders and discussed their experiences in weekly discussions. The author investigated what preservice teachers learned about teaching mathematics from their focus on students' learning of mathematics. The author examined the tasks that preservice teachers posed to children, the questions they asked of children, and the reflections they wrote about their experiences. The article describes what the preservice teachers learned from their experiences and provides insights into their knowledge and skills for developing children's mathematical power.     

Analyzing Student Work as a Professional Development Activity

When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.     

Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.     

New directions in differential equations: A framework for interpreting students' understandings and difficulties

The purpose of this paper is to offer a framework for interpreting students' understandings of and difficulties with mathematical ideas central to new directions in differential equations. These new directions seek to guide students into a more interpretive mode of thinking and to enhance their ability to graphically and numerically analyze differential equations. The framework reported here is the result of investigating in depth six students' understandings through a series of task-based individual interviews and classroom observations. The two major themes of the framework, the function-as-solution dilemma theme and students' intuitions and images theme, extend previous research on student cognition at the secondary and collegiate level to the domain of differential equations and reflect the increased recognition of situating analyses of student learning within students' learning environment. For new areas of interest such as differential equations, mapping out students' understandings of important mathematical ideas can be an important part of curricular and instructional design that seeks to refine and build on students' ways of thinking.     

Sociocultural Factors Influencing Students' Learning in Science and Mathematics: An Analysis of the Perspectives of African American Students

The voices of African American students reveal sociocultural factors that influence their achievement in science and mathematics classes. Using a sociocultural theoretical framework ( Mercer & Covey, 1980 ), this ethnographic study interpreted the perspectives of five African American students as they discussed their learning experiences in science and mathematics classrooms. This framework acknowledges the vulnerability of the educational system to societal influences that inevitably assert cultural values and norms. The students' discussions provided insight into their beliefs about the varied ways in which sociocultural factors impact their learning in science and mathematics classrooms.     

Writing as a Tool to Demonstrate Mathematical Understanding

In this study, I examine how using a writers' workshop model in mathematics creates a space for students to write about their mathematical thinking and problem solving and how their writing impacts instruction. This case study of one classroom with one teacher spanned 6 weeks and included 18 implementations of an adapted version of the Writers' Workshop (WW) in a fourth‐grade mathematics class. On a biweekly basis, the data were reviewed and changes made to the model. The analysis of the students' writing revealed (a) their understandings and misunderstandings of the mathematical content, (b) their readiness for more challenging tasks, and (c) their connections to prior knowledge. Students used writing to demonstrate their understanding of mathematics and show their mathematical processes. In some cases, examining only the numerical work failed to illuminate the students' understanding, their writing provided deeper insight. Students recognized writing as a tool for learning; this was evident in interview responses.     

A Modeling Perspective on the Teaching and Learning of Mathematical Problem Solving

This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an intervention program consisting of a sequence of six modeling activities. To examine students' modeling abilities, the experimental and control groups completed a modeling abilities test on three occasions. Results showed that students' models improved as they worked through the sequence of problem activities and also revealed a number of factors, such as students' grade, experiences with modeling activities, and modeling abilities that influenced their modeling processes. The study proposes a three-dimensional theoretical model for examining students' modeling behavior, with ubsequent implications for the teaching and learning of mathematical problem solving.     

Assessing Students' Mathematical Communication

Assessment of students' mathematical communication through the use of open-ended tasks and scoring procedures is addressed, as is the use of open-ended tasks to assess students' mathematical communication by providing students opportunities to display their mathematical thinking and reasoning. Also, two scoring procedures (quantitative holistic scoring procedure and qualitative analytic scoring procedure) are described for examining students' communication skills.     

Toward Understanding the Role of Technological Tools in Statistical Learning

This article begins with some context setting on new views of statistics and statistical education. These views are reflected, in particular, in the introduction of exploratory data analysis (EDA) into the statistics curriculum. Then, a detailed example of EDA learning activity in the middle school is introduced, which makes use of the power of the spreadsheet to mediate students' construction of meanings for statistical conceptions. Through this example, I endeavor to illustrate how an attempt at serious integration of computers in teaching and learning statistics brings about a cascade of changes in curriculum materials, classroom praxis, and students' ways of learning. A theoretical discussion follows that underpins the impact of technological tools on teaching and learning statistics by emphasizing how the computer lends itself to supporting cognitive and sociocultural processes. Subsequently, I present a sample of educational technologies, which represents the sorts of software that have typically been used in statistics instruction: statistical packages (tools), microworlds, tutorials, resources (including Internet resources), and teachers' metatools. Finally, certain implications and recommendations for the use of computers in the statistical educational milieu are suggested.     

Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

贯通中学和大学数学教与学 集成造就学生数学素养和创新实践能力

针对如何破解我国从应试教育向素质教育转变步履维艰的难题,本文详细介绍了如何贯通中学和大学数学教与学,集成培养学生数学素养和创新实践能力的具体做法和成效.     

Exploring the Use of New Representations as a Resource for Teacher Learning

An important goal of mathematics education reform is to support teacher learning. Toward this end, researchers and teacher educators have investigated ways in which teachers learn about mathematical content, pedagogical strategies, and student thinking as they implement reform. This study extends such work by examining how one elementary school and one high school teacher learned from students' interpretations of new conceptually based representations contained in instructional materials aligned with the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Results indicated that teaching with new representations provided a rich context for teacher learning at both the elementary and high school level, and three dimensions were identified along which such learning occurred. The results suggest that pedagogical content knowledge with respect to representations is an important facet of teacher cognition that should be studied in greater depth.     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

Students Explaining Solutions in Student-Directed Groups: Cooperative Learning and Reform in Mathematics Education

Cooperative learning experiences can contribute to mathematics education reform by stimulating student communication. Sixth grade student conversations were recorded on four occasions over a four month period when they were working in cooperative groups. The results indicated that routine compliance with the requirement to “explain” superseded authentic dialogues about mathematical ideas. Student conversations were influenced by the model of explanation exchanges emerging from the teacher's visits to groups. Teacher influence was mediated by students' past experiences. The findings suggest that teachers implementing reform should help students develop criteria for judging mathematical arguments and confront student conceptions directly to deepen debates.     

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.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

Teaching and Learning Mathematics With Interactive Spreadsheets

Learning to teach mathematics at the middle and secondary levels should include many opportunities for teachers to learn how to use technology to better understand mathematics themselves and promote students' learning of mathematical concepts with technology-enabled pedagogy. This article highlights work done in a variety of preservice and in-service mathematics teacher education courses to help teachers use commonly available spreadsheets as an interactive exploratory learning tool. Several examples of teachers' subsequent use of spreadsheets in their own teaching are also discussed.