Search

In 1998 Marzano proposed a taxonomy of learning that integrated three domains or systems: the self system, which involves student motivation; the metacognitive system, involving goal setting and planning; and the cognitive system, required to complete the task at hand. Although extant for 20 years, a paucity of studies have utilized this taxonomy, even though employing Marzano's taxonomy as a framework is particularly appropriate for studies involving student affect. This study provides an exemplar of the use of Marzano's taxonomy as a framework to investigate the impact of a classroom intervention using active and social strategies to enhance student participation. Further, this paper provides suggestions for employing Marzano's taxonomy in other areas for practising teachers, teacher educators, and educational researchers.

There is a dream, among mathematics educators, for a common direction that would unite teachers, policy makers and researchers, with researchers collaborating to accumulate knowledge and thus contribute to the understanding and improvement of school practice. But this dream becomes less and less realistic in view of the tendency of mathematics education to multiply and diversify its epistemological foundations, theoretical approaches, methodologies and research questions. In response to concerns and challenges, mathematics educators started engaging in "meta-studies" such as attempts at outlining a definition of the domain, more or less comprehensive syntheses of research approaches and theories, comparison of theories, and analyses of articles published in leading journals and conference proceedings over the past 10 or more years. The author decided to take a more empirical approach and have a good look at a more or less "random sample" of papers, in the sense that they were not specially selected to represent this or that issue, theme, theoretical perspective or mathematical topic. In reading the reports, the author had several questions in mind. In this article, the author will initially give an overview of the results from this empirical study, and then elaborate a little more on one particular issue (namely, task problematization). She will give examples of task problematization from the 55 PME papers, but she shall also go beyond the sample and speak more broadly on various approaches to this kind of work, known to her from other publications.

Understanding the intricate quantifier relations in the formal definitions of both convergence and continuity is highly relevant for students to use these definitions for mathematical reasoning. However, there has been limited research about how students relearn previous school mathematics for understanding multiply quantified statements. This issue was investigated in a case study in a 5-week teaching unit, located in a year-long transition course, in which students were engaged in defining and proving sequence convergence and local continuity. The paper reports on four substantial changes in the ways students relearn school mathematics for constructing quantified statements: (1) endorse predicate as formal property by replacing metaphors of epsilon strips with narratives about the objects [epsilon], N[subscript epsilon], and |a[subscript n] - a|; (2) acknowledge that statements have truth values; (3) recognize that multiply quantified statements are deductively ordered and that the order of its quantifications is relevant; and (4) assemble multiply quantified statements from partial statements that can be investigated separately. These four changes highlight how school mathematics enables student to semantically and pragmatically parse multiply quantified statements and how syntactic considerations emerge from such semantic and pragmatic foundations. Future research should further investigate how to design learning activities that facilitate students' syntactical engagement with quantified statements, for instance, in activities of using formal definitions of limits during proving.

In Hong Kong, pupils at Grades 3, 6 and 9 are required to take the Territory-wide System Assessment (TSA). TSA is a low-stakes standardized assessment, tracking pupils' attainment of the basic competence levels in English, Chinese and Mathematics. By design, TSA does not constitute part of individual pupils' school results, but generates assessment information which monitors teaching and learning in school as a quality assurance mechanism. Despite its low-stakes nature, key stakeholders remain skeptical about its rationale, purpose and usefulness, especially when it comes to ranking school performance. Because of this, this paper examines the extent to which the Grade 3 TSA could serve its purpose to enhance teaching and learning and promote constructive alignment of assessment and learning. The paper begins with an overview of the international assessment arena, followed by a discussion on the Hong Kong assessment reform landscape. Then, the paper reviews three issues in TSA, namely pedagogical, societal and political concerns. Next, it addresses whether TSA has become a high-stakes test; whether it brings about drilling; and whether it facilitates positive learning. Lastly, three sustained recommendations as to how the Grade 3 TSA could concurrently serve its evaluative, pedagogical and learning purposes are provided.

In undergraduate college courses, assignments designed to showcase preservice teacher learning traditionally include classroom presentations, papers, projects, and tests. Often, these activities do not translate into permanent products that will be utilized outside the course. The purpose of this pilot study was to investigate the effects of self-created video podcasts on content acquisition among undergraduate preservice special education teachers. Preservice teachers in two undergraduate courses were randomly assigned to one of two conditions: (1) podcast condition (treatment group; n = 25) and (2) traditional live presentation condition (control group; n = 25). The participants in each group were then assigned targeted instructional strategies for teaching mathematics. Pre-post measures included a comprehensive multiple-choice test and an open-ended assessment targeting each assigned strategy. During the last 2 weeks of the semester, participants presented their final product and completed the posttests. Results suggested that the participants in the treatment group (podcasts) acquired a deeper understanding of their assigned strategy than the participants from the comparison group (live presentations). The treatment group also outperformed the control group on the comprehensive multiple-choice test.

Comments on six theses about mathematics education are presented. How theories on teaching and learning mathematics are derived or developed, how philosophies are related, and their role in mathematics education are among the points discussed. (MNS)

How the computer as a metaphor affects our understanding of the processes of learning and teaching is explored. After describing reflection and recursion in mathematics and their roles in thinking and learning, models of the mind and teaching effects are discussed, with self-awareness as a theme. (MNS)

The primary goal of this paper is to provide a commentary on the teaching and learning of geometry in the early years of schooling with the set of papers in this issue as a guiding factor. It is structured around issues about geometry education of young learners, such as: what should we teach in geometry and why; representation of geometrical ideas; the teaching and learning of geometry; and assessment of children's learning in geometry. The author outlines his views based on the literature and the papers in this issue and concludes with an outlook on the future teaching and learning of geometry in schools.

This paper presents an intervention study whose aim was to promote teacher beliefs about mathematics and learning mathematics and student competences in mathematical modeling. In the intervention, teachers received written curriculum materials about mathematical modeling. The concept underlying the materials was based on constructivist ideas and findings from mathematics education. Teacher beliefs about mathematics, learning and self-efficacy were expected to have a major impact on their classroom practices. We therefore assessed teacher beliefs about the usefulness of mathematics, learning (constructivist and socio-constructivist beliefs) and teacher self-efficacy when teaching modeling (teacher or class level variable). The student level variables assessed were modeling competence and other individual factors, such as basic mathematical skills and cognitive abilities. The effectiveness of the intervention was measured in a pre-post control group design using multilevel structural equation modeling. The results showed no direct effect of the intervention on student modeling competence. However, they did reveal that the intervention had a significant effect on teacher beliefs about learning (constructivist and socio-constructivist view) and an effect of these teacher beliefs about learning on student modeling competence. Further, the results showed that students' gains in modeling competence is not only mediated by teacher beliefs, but also influenced by individual factors. Implications for teaching as well as limitations of the study are discussed.

This paper reports on combining semiotic and discursive approaches to reification in classroom interactions. It focuses on the discursive characteristics and semiotic processes involved in the teaching and learning of square roots in a ninth grade classroom in Japan. The purpose of this study is to characterize the development of mathematical discourses in a series of mathematics lessons in terms of the "commognitive framework" and the model of semiotic chaining. To achieve this objective, the notion of reification is revisited from cognitive, semiotic, and discursive points of view, and classroom activities are observed and analysed in terms of the combined theoretical framework. The results suggest that the model of semiotic chaining is not incommensurable with a strong discursive approach and the changes of meta-discursive rules--an essential aspect of the reification of new signifiers--that take place in the phases of the learning of addition on square roots.

This paper is centered on creating equitable opportunities for learners in mathematics education. Through observations of teacher practice, the paper seeks to theorize how teachers enact their dispositions toward mathematics instruction. These observed propensities, in relation to teachers' aims for students to "take up their space" in and beyond the mathematics classroom, then inform a model of equitable mathematics instruction. Teachers' dispositions are considered in relation to Mason's discipline of noticing and Bourdieu's notion of the symbolic violence of dominant discourses.

Drawing on ecological psychology, this paper considers how student engagement is an accomplishment of the classroom system. Specifically, this paper presents an analysis of two teachers and their students who were using a project-based unit in their mathematics classes. The two teachers used identical curricular materials, but had dramatically different personal histories of teaching and different instructional practices. Our goal is to investigate the role of the teacher in supporting student engagement by considering the kinds of opportunities to learn that were presented by the teacher, and the relationship between those opportunities to learn and the ways their students engaged. Pragmatically, this paper contributes to our understanding of how teachers' framing of activity significantly impacts the ways that students are likely to engage tasks. Theoretically, this paper highlights the interactional nature of learning, with the goal of clarifying why learning is not simply an individual accomplishment.

The current study considered young students' (7 and 8 years old) experiences and perceptions of mathematics lessons involving challenging (i.e. cognitively demanding) tasks. We used the Constant Comparative Method to analyse the interview responses (n = 73) regarding what work artefacts students were most proud of creating and why. Five themes emerged that characterised student reflections: enjoyment, effort, learning, productivity and meaningful mathematics. Overall, there was evidence that students embraced struggle and persisted when engaged in mathematics lessons involving challenging tasks and, moreover, that many students enjoyed the process of being challenged. In the second section of the paper, the lesson structure preferences of a subset of participants (n = 23) when learning with challenging tasks are considered. Overall, more students preferred the teach-first lesson structure to the task-first lesson structure, primarily because it activated their cognition to prepare them for work on the challenging task. However, a substantial minority of students (42%) instead endorsed the task-first lesson structure, with several students explaining they preferred this structure precisely because it was so cognitively demanding. Other reasons for preferring the task-first structure included that it allowed the focus of the lesson to be on the challenging task and the subsequent discussion of student work. A key implication of these combined findings is that, for many students, work on challenging tasks appeared to remain cognitively demanding irrespective of the structure of the lesson.

Affect is an important element of the learning process both in the classroom and with educational technology. This paper presents analyses in relation to the identification of Action Units (AUs) related to affective states and their impact on learning with a tutoring system. To assess affect, a tool was devised to identify AUs on pictures of human faces. Action Units are combinations of individual facial muscles or groups of muscles that create facial expressions in association with affect. Pictures from a population of students were taken while using an intelligent tutoring system for mathematics in a secondary school in a suburban school in Veracruz, Mexico. The students were asked to interact with the tutoring system for 40 minutes and they were photographed with the tool at a rate of 1 picture every 5 seconds acquiring a dataset consisting of 16,800 photos. To achieve identification, the software analyzes individual pictures using Principal Component Analyses (PCA) and Euclidian distance. The tool developed to classify affective states shows 88.88% accuracy in the identification of AUs when matching the recognized AU to the Cohn-Kanade AU-Coded facial expression database. The analyses also elicited the most common AUs for the population and their association with learning with the intelligent tutoring system. These preliminary results shed light on the issues of affect in relation to learning mathematics with tutoring systems and pave the way for the implementation of coping strategies based on the automatic recognition of facial expressions.

In this paper, we engage in a thought experiment about how students might notate their reasoning for composing fractions multiplicatively (taking a fraction of a fraction and determining its size in relation to the whole). In the thought experiment we differentiate between two levels of a fraction composition scheme, which have been identified in prior research, and discuss differences in systems of notation that students might produce, depending on the level of their scheme. We situate this discussion within a framework for learning in order to articulate how developing systems of notation with students can contribute to their learning.

Statistical data can be represented in a number of qualitatively different ways, the choice depending on the following three conditions: the concepts to be investigated; the nature of the data; and the purpose for which they were collected. This paper begins by setting out frameworks that describe the nature of statistical thinking in schools, and suggests how they can be used explicitly by teachers and students to support work on statistical investigations. It goes on to describe some of the ways in which students at school level can interact with different statistical representations using technology, and illustrates the interactions using common examples of statistical diagrams, graphs and tables. The descriptions of these interactions are based on a theory of representational versatility that espouses the importance of engaging with statistical representations in a conceptual as well as a procedural manner. Some implications for the teaching and learning of statistics are also considered. (Contains 3 tables and 4 figures.)

A worldwide problem in Engineering education is the high rates of students' failure and drop out particularly at the beginning of the course. This is related to the process by which students learn Mathematics. An innovative methodology of teaching calculus was developed and it is presented in this paper. The approach, based on both course modularisation and the error theory, is student centred. It fosters student/lecturer commitment and emphasizes continuous assessment of the process. Knowledge, in this work, is regarded as a construction and reconstruction process based on phenomenological and hermeneutic concepts. Survey results demonstrated the effectiveness of the applied methodology. (Contains 3 figures.)

Reflects on the work of Robert B. Davis in mathematics education. Discusses his ideas about theories and practices in mathematics education and instruction. (Contains 25 references.) (Author/ASK)

College students were given two paper-and-pencil conditional reasoning tests, one week apart, in varying order. The three problems on each test were presented either verbally or with line drawings as concrete referents. Performance was significantly poorer on the problems with drawings. (MNS)

The first 12 volumes of "The Mathematics Teacher" were examined to determine the extent to which societal changes regarding women were reflected in mathematics education. Evidence on the extent of participation by women and attitudes concerning differences between boys and girls in learning mathematics are presented. (MNS)

Principles on which the teaching of mathematics is based are discussed. Sections concern the skill principle, the curriculum principle, and learning strategy, with many classroom illustrations. (MNS)

The forum is focused on thinking about and with algorithms as a way of unifying all one's mathematical endeavors. The lead article by Maurer presents examples and discussion of this point. Responses, often disagreeing with his views, are by Douglas, Korte, Hilton, Renz, Smorynski, Hammersley, and Halmos. (MNS)

Advanced is the hypothesis that students organize their beliefs about mathematics to resolve problems that are primarily social rather than mathematical in origin. The contextuality of cognition, meaning-making, and learning in interactive situations are each discussed. (MNS)

Begins with the assumption that by practicing something one often learns something else. A discussion is presented on the historical and social development of knowledge, the cognitive development of students, the role of teachers, and the meaning of learning situations. (PK)

This article is based on the 1987 presidential address to the National Council of Teachers of Mathematics. The work of George Polya is used to draw conclusions about teaching and learning mathematics. (PK)

Differences in cognitive structure are approached through consideration of cartesian and non-cartesian knowing. Figuration and configuration are described as two layers of cartesian knowing leading to the third layer, the cognitive structure. Knowing in general is then discussed, with comments on learning in another culture, Botswana. (MNS)

This classic reprint was first published in 1955. Fehr discusses how to determine what is taught, and makes specific comments on the roles of planning, meaning, practice, learning, and problem solving. (MNS)

Examined is what radical constructivism in connection with Intuitionism might mean to mathematics instruction. Implications for mathematics education, especially the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge, are suggested. (YP)

This paper describes some of the myths on training lecturers in mathematics, discusses briefly the nature of teaching and learning, and suggests some ways of analyzing the tasks in teaching and learning. (Author/MK)

Concern about certain of the difficulties encountered by undergraduate students in learning mathematics is discussed. Three attitudes of present-day undergraduates which contribute significantly to the difficulties encountered by students are examined. (Author/MK)

This paper discusses some of the factors which have to be considered in constructing an adequate description of the understanding of mathematics. It examines three models of understanding which have appeared during the last three years. (Author/MK)

When the words "problem" and "answers" are used in connection with computational exercises, students think they are solving problems. Distinguishing between computational forms and problems is illustrated with a variety of topics. (MNS)

Teaching mathematics in hard ways, rather than using easier methods or technology, is described. Employing the most efficient means possible to solve a problem is the essence of good mathematics, rather than wasting time in practicing obsolete skills. (MNS)

This discussion centers around the dangers in making too precise a particular concept in educational psychology that distinguishes between two types of understanding, instrumental and relational. (MP)

This paper reports about professional learning related to mathematics instruction in student-teaching triads. The study focuses on how the joint work of triads offers opportunities for the professional learning of student teachers (interns), mentor teachers, and supervisors. Two elementary student-teaching triads reported different areas of and opportunities for professional learning related to mathematics instruction, with more areas and richer opportunities reported by the triad with more extensive collaboration among its members. Findings suggest the need for professional development activities in which mentors and supervisors can examine their stances toward joint work and professional learning in triads. [ABSTRACT FROM AUTHOR]

This article describes key elements for mathematics teachers, through the faculty development process, to increase student engagement and motivation in learning mathematics. It addresses common barriers facing student motivation and achievement in the field of mathematics and provides potential remedies from a faculty development viewpoint to alleviate these barriers. The paper offers practical examples from the authors' own faculty development programs and experiences in teaching mathematics to both foster and grow the intellectual curiosity of students and offers suggestions for successful faculty development programs. [ABSTRACT FROM AUTHOR]

Past research has demonstrated that males outperform females in mathematics (Hyde, J. S., Fennema, E., & Lamon, S. J., Psychol Bull 107:139–155, ). Research has also shown that encouraging mindful learning–learning information in a conditional rather than an absolute way–can increase mathematics performance in females (Ritchhart, R., & Perkins, D. N., J Social Issues 56:27–47, ). This paper examines the moderating role of mindful learning for gender differences, by manipulating mindful learning for females’ and males’ performance on a novel math task. The results from this study show that males performed better than females when mindful learning was not encouraged (absolute instruction), but males and females performed equally well when mindful learning was encouraged (conditional instruction). Thus we find that mindful learning moderates gender differences in math performance. [ABSTRACT FROM AUTHOR]

This article documents teacher learning through participation in lesson study, a form of professional development that originated in Japan and is currently practiced widely in the US. Specifically, the paper shows how teachers in three different lesson study teams 1) expanded their mathematical content knowledge, 2) grew more skillful at eliciting and analyzing student thinking, 3) became more curious about mathematics and about student thinking, 4) emphasized students' autonomous problem-solving, and 5) increasingly used multiple representations for solving mathematics problems. These outcomes were common across three lesson study teams, despite significant differences among teams composition, and content foci. [ABSTRACT FROM AUTHOR]

Summary: Mathematics textbooks sometimes present worked examples as being generated by particular fictitious students (i.e., person‐presentation). However, there are indicators that person‐presentation of worked examples may harm generalization of the presented strategies to new problems. In the context of comparing and discussing worked examples during extended classroom instruction, the current study compared the impact of person‐presentation to strategy labels on students' posttest accuracy and ratings of strategy generalizability. Five algebra teachers and their 168 students used worked examples either presented using fictitious students or with a strategy label during a multiweek unit on equation solving, with teachers randomly assigned to condition. All students compared and discussed the worked examples. In this context, we found no effect of condition on student accuracy at posttest, nor on their ratings of the generalizability of the presented strategies. We discuss why previously found negative effects of person‐presentation may not have extended to this context. [ABSTRACT FROM AUTHOR]