The “Problem” of Experience in Mathematics Teaching

Prior research has established that teachers' use of curriculum materials is affected by a range of factors, such as teachers' conceptions of mathematics teaching, and the nature and extent of their teaching experience. What is less clear, and far less examined, in prior research is the role that the teacher guide (TG) may play in mediating the influence of these and other factors on teachers' decisions and actions. Accordingly, this study examines how two 6th grade teachers use the TG from Connected Mathematics Project as a resource in making planning and enactment decisions, and factors associated with patterns of TG use. Through cross‐case analysis, the author found that these teachers seemed to draw largely from their previous experiences and their own conceptions of mathematics teaching and learning when making planning and enactment decisions related to mathematical tasks, and not particularly from the TG. For example, when faced with certain planning and instructional challenges, such as students struggling with the content, teachers tended to rely on their particular conceptions of mathematics teaching to address these challenges. Despite the fact that the TG provided suggestions for teachers as to how address such challenges, it was not extensively used as a resource by the teachers in this study in their planning and enactment of lessons.     

Teacher Questioning to Promote Justification and Generalization in Mathematics: What Research Practice Has Taught Us

This article explores the teacher's role in classroom environments that center on learning through student exploration, and reinvention, of important mathematics. In such environments, teachers will often work to create situations where students are invited to express their thinking, most especially to peers. How is this done? In the work reported here, both teacher questioning and teacher listening will play important parts, as does the teacher's background understanding of the mathematics and the children. This study focuses especially on teacher questioning in third- and fourth-grade classrooms associated with a longitudinal study now in its eleventh year. Analyses of videotaped data indicate a strong relationship between (1) careful monitoring of students' constructions leading to a problem solution, and (2) the posing of a timely question which can challenge learners to advance their understanding. What a teacher needs to know in order to work well with student explorations has important implications.     

Conceptions of mathematics and student identity: implications for engineering education

Lecturers of first-year mathematics often have reason to believe that students enter university studies with naïve conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students’ conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students’ conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.     

On the theoretical, conceptual, and philosophical foundations for research in mathematics education

The current infatuation in the U.S. with “what works” studies seems to leave education researchers with less latitude to conduct studies to advance theoretical and model-building goals and they are expected to adopt philosophical perspectives that often run counter to their own. Three basic questions are addressed in this article:What is the role of theory in education research? How does one's philosophical stance influence the sort of research one does? And,What should be the goals of mathematics education research? Special attention is paid to the importance of having a conceptual framework to guide one's research and to the value of acknowledging one's philosophical stance in considering what counts as evidence.     

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.     

Research, practice, uncertainty and responsibility

Three issues concerning the relationship between research and practice are addressed. (1) A certain ‘prototype mathematics classroom’ seems to dominate the research field, which in many cases seems selective with respect to what practices to address. I suggest challenging the dominance of the discourse created around the prototype mathematics classroom. (2) I find it important to broaden the school-centred discourse on mathematics education and to address the very different out-of-school practices that include mathematics. Many of these practices are relevant for interpreting what is taking place in a school context. That brings us to (3) socio-political issues of mathematics education. When the different school-sites for learning mathematics as well as the many different practices that include mathematics are related, we enter the socio-political dimension of mathematics education.On the one hand we must consider questions like: Could socio-political discrimination be acted out through mathematics education? Could mathematics education exercise a regimentation and disciplining of students? Could it include discrimination in terms of language? Could it include sexism and racism? On the other hand: Could mathematics education bring about competencies which can be described as empowering, and as supporting the development of mathematical literary or a ‘mathemacy’, important for the development of critical citizenship?However, there is no hope for identifying a one-way route to mathemacy. More generally: There is no simple way of identifying the socio-political functions of mathematics education. Mathematics education has to face uncertainty, and this challenge brings us to the notion of responsibility.     

Exemplary mathematics instruction and its development in selected education systems in East Asia

What may teachers do in developing and carrying out exemplary or high-quality mathematics classroom instruction? What can we learn from teachers’ instructional practices that are often culturally valued in different education systems? In this article, we aim to highlight relevant issues that have long been interests of mathematics educators worldwide in identifying and examining teachers’ practices in high-quality mathematics classroom instruction, and outline what articles published herein can help further our understanding of such issues with cases of exemplary mathematics instruction valued in the Chinese Mainland, Hong Kong, Japan, Singapore, South Korea, and Taiwan.     

Mathematics teachers learning with video: the role,for the didactician,of a heightened listening

This article addresses two main questions, how do mathematics teachers learn from using video? and, what is the role of the didactician? A common problem is reported in the difficulty of keeping teacher discussion of video away from judgment and evaluation. A review of mathematics education literature revealed four existing models for the use of video with teachers. Drawing on enactivist ideas (Rosch, Concepts: core readings, pp 189–206, 1999), there are reasons why these models are likely to be productive and therefore suggestions for how teachers can learn from video. However, little is known about the role of the didactician in supporting learning. From empirical data, there is evidence that didacticians need to engage in a particular form of attention that I label a ‘heightened listening’ since there are simultaneous foci (on the content of teachers’ contributions and what kind of a comment is being made), in order to establish discussion norms and to support the development of new ways of seeing and acting in the classroom.     

Understandings of Solutions to Differential Equations Through Contexts,Web‐Based Simulations,and Student Discussion

As in the case of elementary mathematics, the instruction of high‐level mathematical concepts can often be sacrificed at the expense of a focus on algorithmic procedures. Computer‐based simulations can expand an undergraduate mathematics instructor's opportunity to explore high‐level mathematical concepts in an applied environment. This study describes one instructor's approach to incorporating simulations and classroom discussions in a differential equations course and the subsequent effects on student learning attitudes and outcomes. Students made modest gains in the area of conceptualizing and applying ideas regarding solutions to differential equations in this learning environment. Implications of the study include the identification of specific gains relative to computer‐mediated learning environments and recommendations for using simulations to support conceptual development.     

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.     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

Beliefs and Perspectives of First‐Year,Alternative Preparation,Elementary Teachers in Urban Classrooms

Results from an earlier study ( Hart, 2002 ) suggested that a group of 14 teachers participating in an alternative preparation program for elementary teachers had developed beliefs that were consistent with current thinking in mathematics education. The current study follows 8 of those teachers into their first year of teaching in an urban classroom. Qualitative data were collected from three sources: reflection logs, mathematics case discussions, and field notes made during classroom observations. This provided a triangulation of perspectives: the teachers' views of themselves, the teachers' views of others, and the university faculty member's view of them. Also, teachers completed the Standards Belief Instrument (Zollman & Mason, 1996) at the end of Phase I and at the end of Phase II. Results from the instrument and the qualitative analysis suggest that the teachers maintained a strong reform perspective in their beliefs, but they were unable to consistently implement pedagogy that was consistent with those beliefs.     

Enactment of school mathematics curriculum in Singapore: whither research!

The official curriculum for mathematics in Singapore schools is based on a framework that has mathematical problem solving as its primary goal. It is detailed and one may say that the gap between the designated curriculum and teacher intended curriculum is often very narrow. This is so as the main source of instructional materials is textbooks which are very closely aligned with the official national curriculum. There is a dearth of research on the enactment of the curriculum in Singapore schools, with the few research studies done so far appearing to cover only a narrow focus. The author’s view is that, even though only a few such studies have been published, schools have always been engaged in small-scale investigations, the findings of which are necessary to guide decisions on matters related to choice of textbooks and pedagogies for improved student learning. Considering all the published research and the investigative work undertaken by educators in Singapore, it may be said that the conceptual model proposed by Remillard and Heck is rigorous. In addition, the issues in this particular issue of ZDM offer educators, both classroom teachers and others, very good perspectives for research on the enactment of the school mathematics curriculum.     

The state-of-art in mathematical creativity

Creativity is a topic which is often neglected within mathematics teaching. Usually teachers think that it is logic that is needed in mathematics in the first place, and that creativity is not important and learning mathematics. On the other hand, if we consider a mathematician who develops new results in mathematics. we cannot overlook his/her use of the creative potential. Thus, the main questions are as follows: What methods could be used to foster mathematical creativity within school situations? What scientific knowledge, i.e. research results, do we have on the meaning of mathematical creativity?     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

Teaching Mathematics with a New Curriculum: Changes to Classroom Organization and Interactions

This report describes a high school mathematics teacher's decisions about classroom organization and interactions during his first two years using a new curriculum intended to support teachers' development of student-centered, contributive classroom discourse. In year one, the teacher conducted class and interacted with students primarily in small groups. In year two, he conducted more whole-class instruction. In both years, teacher-student interactions contained univocal and contributive discourse, but in year two the teacher sustained contributive discourse with students for longer periods. The teacher facilitated the most significant changes to classroom discourse in the instructional format with which he had the greatest experience (whole-class instruction). Over the period of this study, two key factors appeared to affect the teacher's decisions about classroom organization and interactions: his perception of students' expectations about mathematics classroom roles and activity, and his own discomfort associated with using a new curriculum. These areas are important candidates for future research about teachers' use of innovative mathematics curricula.     

Modelling in Mathematics Classrooms: reflections on past developments and the future

This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments. After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future—for what remains to be done to enable modelling to make its essential contribution to the «functional mathematics», the mathematical literacy, of future citizens and professionals. What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them? These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging.     

Mathematics Learning and Participation as Racialized Forms of Experience: African American Parents Speak on the Struggle for Mathematics Literacy

This article draws on 3 ethnographic and participant observation studies of African American parents and adults from 3 northern California communities. Although studies have shown that African American parents hold the same folk theories about mathematics as other parents, stressing it as an important school subject, few studies have sought to directly examine their beliefs about constraints and opportunities associated with mathematics learning for both themselves and their children. I argue that, as they situate the struggle for mathematical literacy within the larger contexts of African American, political, socioeconomic, and educational struggle, these parents help reveal that mathematics learning and participation can be conceptualized as racialized forms of experience. As they attempt to become doers of mathematics and advocates for their children's mathematics learning, discriminatory experiences have continued to subjugate some of these parents, whereas others—as demonstrated in their oppositional voices and behaviors—resisted their continued subjugation based on a belief that mathematics knowledge, beyond its role in schools, can be used to change the conditions of their lives. The characterization of mathematics learning as racialized experience put forth in this article contrasts with culture-free and situated perspectives of mathematics learning often found in the literature. As a result of their experiences with oppression in this society, the concept of race has historically played a major role in the lives of African Americans. Although race has dubious value as a scientific classification system, it has had real consequences for the life experiences and life opportunities of African Americans in the United States. Race is a socially constructed concept which is [a] defining characteristic for African American group membership. (Sellers, Smith, Shelton, Rowley, & Chavous, 1998, p. 18)     

Humans-with-media and continuing education for mathematics teachers in online environments

This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25?years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that ??blends?? aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers?? classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.     

What is the Mathematics in Mathematics Education?

In this paper I tackle the question What is the mathematics in mathematics education? By providing three different frames for the word mathematics.

• 1.Frame 1: Mathematics as an abstract body of knowledge/ideas, the organization of that into systems and structures, and a set of methods for reaching conclusions.

• 2.Frame 2: Mathematics as contextual, ever present, as a lens or language to make sense of the world.

• 3.Frame 3: Mathematics as a verb (not a noun), a human activity, part of one’s identity.

After introducing the frames and examining their distinction and their overlap, I discuss their implication with respect to student-centered classroom, context, and culture.