Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation: public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study documented the voicing of mathematical ideas in public discussion and in teacher–student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher–student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons studied in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation and to student learning ourcomes are central to the development of any theory of mathematics instruction and learning.     

The long term impact of a coherence based model for mathematics intervention

This article presents a large-scale longitudinal study of hundreds of students across the state of Kentucky that participated in a dual-focus mathematics intervention initiative when they were in the third grade. Rather than an exclusive focus on intervention, this initiative focused on both (i) high quality pull-out intervention and (ii) coherence between pull-out intervention and classroom instruction. The study found that over half of the third grade intervention students that participated in this initiative were classified as “novice” (the lowest possible performance category) on state standardized mathematics assessments at the end of the third grade. However, over the course of the following four years, the novice reduction rate of these students was significantly (p < .01) greater than other novices in Kentucky that did not participate in the initiative. These findings indicate that when implementing intervention initiatives to help students that are struggling with mathematics, it may be important to establish coherence between pull-out intervention and classroom instruction. The long term impact of this approach among traditionally underrepresented minorities suggest that this publication may provide insight into important equity issues where long-term analyses may sometimes be needed to capture the full impact of intervention initiatives.     

“He was poking holes …” A case study on figuring out metadiscursive rules through primary sources

An important aspect of participation in a new academic discourse pertains to the metadiscursive rules which govern that discourse. Researchers have documented the viability of using primary sources in undergraduate mathematics education for scaffolding students’ recognition of those rules. Our research explores the related question of whether the use of primary sources can support students’ learning of metadiscursive rules in a way that goes beyond mere recognition. We present a case study of one student’s “figuring out” of metadiscursive rules in a university Analysis course as a result of her experience with a Primary Source Project, illustrate evidence for three dimensions of “figuring out” (adoption, acceptance, awareness) that emerged from that case study, and discuss the implications of our findings for classroom instruction and future research.     

Adolescent girls’ construction of moral discourses and appropriation of primary identity in a mathematics classroom

This qualitative study examines the way three American young adolescent girls who come from different class and racial backgrounds construct their social and academic identities in the context of their traditional mathematics classroom. The overall analysis shows an interesting dynamic among each participant’s class and racial background, their social/academic identity and its collective foundation, the types of ideologies they repudiate and subscribe to, the implicit and explicit strategies they adopt in order to support the legitimacy of their own position, and the ways they manifest their position and identity in their use of language referring to their mathematics classroom. Detailed analysis of their use of particular terms, such as “I,” “we,” “they,” and “should/shouldn’t” elucidates that each participant has a unique view of her mathematics classroom, developing a different type of collective identity associated with a particular group of students. Most importantly, this study reveals that the girls actively construct a social and ideological web that helps them articulate their ethical and moral standpoint to support their positions. Throughout the complicated appropriation process of their own identity and ideological standpoint, the three girls made different choices of actions in mathematics learning, which in turn led them to a different math track the following year largely constraining their possibility of access to higher level mathematical knowledge in the subsequent schooling process.     

The relationship of theory and practice in mathematics teacher professional development: an activity theory perspective

This paper describes certain interactions between the activity of “teaching” and the activity of “researching” in which the teachers participate in a 52-h professional development course aiming to introduce them to research in mathematics education as a tool for inquiry in their teaching. The teachers are involved in different research tasks such as: reading and presenting research papers; analyzing classroom dialogues and tasks; and designing, implementing, and evaluating collaboratively a classroom teaching intervention. From an activity theory (AT) perspective, and in particular Engeström’s (2001) third generation AT, distinguish two activity systems, the activity system of research and the activity system of teaching, to identify links that the participating teachers make. These links indicate the development of an inquiring stance to mathematics teaching and learning as a means of professional learning.     

Meta-rules of discursive practice in mathematics classrooms from Seoul, Shanghai and Tokyo

It is widely acknowledged that discursive practice in classrooms plays a key role in student learning of mathematics. This study extends a previous study on spoken mathematics (Clarke and Xu in ZDM Int J Math Educ 40(6):963–981, 2008) and seeks to compare the discursive practices in classrooms from Seoul, Shanghai and Tokyo with a particular focus on meta-discursive rules (Sfard in Educ Stud Math 46:13–57, 2001; Sfard in Thinking as communicating: human development, the growth of discourses, and mathematizing, Cambridge University Press, Cambridge, 2008) that regulate exchanges between the teacher and students. The analysis centres on the events when the topic of “linear equations” was introduced. The similarities and differences of the three classrooms suggest that while the shared macrocultural values and beliefs may frame the social activity of the classrooms in similar ways, the meta-discursive rules of classroom microculture determine the opportunities for student learning in mathematics.     

Inquiry Teaching and Learning: The Best Math Class Study

This research reports on prospective middle school teachers' perceptions of a “best mathematics class” during their involvement in an inquiry‐designed mathematics content course. Grounded in the prestigious Glenn Commission report ( U.S. Department of Education, 2000 ), the study examined the prospective teachers' perceptions of effective mathematics instruction both prior to and after completing the inquiry course. Pre‐essay analysis revealed that students could be grouped into one of two categories: the Watch‐Learn‐Practice view and the Self as Initiator view. Post‐essay analysis indicated that over two thirds of all students involved in the study changed their views of a best math class after the inquiry courses. The Watch‐Learn‐Practice group's changes focused on developing reasoning skills and learning how one “knows” in mathematics. The Self as Initiator group noted expanded roles for the students, particularly emphasizing the importance of going beyond basic requirements to think deeply about the why and how of mathematics and expanded views of the benefits of group learning.     

Active learning: a case study of student engagement in college Calculus

This paper presents a case study for strategic engagement of students in a Calculus course in order to produce increased learning in the classroom. Since it has been shown that active learning can promote greater comprehension for students in science, technology, engineering, and mathematics (STEM) courses, the researcher utilized many types of active learning techniques to enhance classroom instruction. The key components implemented are presented as a model of enhanced learning through developed classroom engagement. This course redesign model entitled, Strategic Engagement for Increased Learning (SEIL), has the potential to (1) contribute to the body of knowledge on ways to improve mathematics skills for college students, (2) identify successful teaching strategies and technologies that will promote the retention of STEM students, (3) increase the success rate of students taking Calculus, and (4) help produce more STEM graduates needed for the STEM workforce in the United States of America.     

How can emphasizing mathematical modeling principles benefit students in a traditionally taught differential equations course?

This paper reports a mixed-methods, causal-comparative study of two instructional approaches in a differential equations course for engineering undergraduates. One classroom emphasized decontextualized techniques to solve canonical differential equations. The other classroom emphasized modeling principles to derive and interpret canonical differential equations as models of real world phenomena. Both classrooms were lecture based. This paper presents a brief review of both texts and compares a lesson on the same topic from each classroom in order to illustrate what it means to emphasize modeling principles. An ANCOVA was used to compare student performance on common final exam items while controlling for prior mathematics achievement. The main effect for instructional approach was statistically significant suggesting that a modeling perspective for teaching differential equations aided student learning. This quasi-experimental study corroborates previously reported recommendations for improving teaching and learning of differential equations.     

How much time do students have to think about teacher questions? An investigation of the quick succession of teacher questions and student responses in the German mathematics classroom

Several studies have shown that the style of the German mathematics classroom at secondary level is mostly based on the so called “fragend-entwickelnde” teaching style which means developing the lesson content by a teacher directed sequence of teacher questions and student responses. In this article we describe a study on the time the students have for thinking about a teacher question in the public classroom interaction. Our investigation is based on a reanalyasis of 22 geometry lessons from grade 8 classes which mainly deal with a challenging proving content. The results show that the average time between a teacher question and a student response is 2.5 seconds. There are no remarkable differences between different phases of the lessons like comparing homework, repetition of content or working on new content. Moreover, for 75% of the teacher questions the first student was called to answer within a three second time interval.     

Learning cycles for mathematics: An investigative approach to middle-school mathematics

Middle-school-aged students are a complex mixture of often conflictive social, intellectual, and physical ingredients while middle-school mathematics content becomes increasingly abstract and procedure driven. Preparation for algebra and higher mathematics often ignores the developmental and social needs of the middle-school student while overemphasizing computational facility with previously learned elementary mathematics content. The need to explore and question will be encouraged by a classroom atmosphere and instructional strategies that provide for active problem solving and cooperative group learning. This article articulates the philosophical and research bases forlearning cycle pedagogy which, when applied to mathematics teaching, is especially appropriate for the special needs of the middle-school student. The components of, and theoretical/research bases for, the learning cycle are described. Learning cycle pedagogy is the embodiment of the vision of mathematics classroom instruction depicted in the Standards (NCTM, 1989).     

Students discussing mathematics in small-group interactions: opportunities for discursive negotiation processes focused on contentious mathematical issues

Small-group discussions involving students and their teacher that focus on meanings constructed during the mathematics lessons or solutions to problems produced in these lessons offer great potential for debate and argument. An analysis of the epistemological nature of knowledge can give deeper insight, to gain a better understanding of the emerging discontinuities in argumentations, negotiations, and clarifications about contentious meaning differences that arise. In most cases mathematical interactions between students and a teacher about contentions are very fragile and seem to be handled more or less directly—by side-stepping to another topic or by resolving via the teacher’s authority, for example. Therefore, the maintenance of such negotiation processes in mathematics teaching is a specific challenge for students and the teacher. The type of closure of these processes seems to be related to the emerging maintenance processes. In this paper, small-group discussions are interpretatively analyzed in the three steps “Initiation—Maintenance—Closing” with the focus on fundamental (dialogical) learning.     

Interaction analysis of mathematical communication in primary teaching: The epistemological perspective

Communication between students and teacher in the mathematics classroom is a form of social interaction which focuses on a specific topic:mathematical knowledge. This knowledge cannot be introduced into classroom interaction “from the outside”, but grows through the communicative process, in the course of interactive exchanges between the participants of discussion. Although mathematical communication must be seen and analysed in the same way as any other form of communication, the particularity of interactive constructions of mathematical knowledge and its specificsocial epistemology within the context of teaching processes has to be taken into consideration. Also, the institutional influences of school institutions and those of teaching (analysed in the frame of general socio-interactive research approaches) must be considered. An epistemology-oriented interaction research approaches the specificity of amathematical classroom and communication culture in its analyses.     

Elementary mathematics specialists in “departmentalized” teaching assignments: Affordances and constraints

In this article, we describe the experiences of three Elementary Mathematics Specialists (EMS) who were part of a larger project investigating the impact of EMS certification and assignment (self-contained or “departmentalized”) on teaching practices and student achievement outcomes. All three of the teachers were “departmentalized,” in the sense that each was responsible for teaching mathematics to at least two groups of students, and accordingly, did not teach all subjects as would a typical self-contained elementary teacher. Each teacher had recently earned an Elementary Mathematics Specialist certificate through completion of a 24-credit, graduate-level program designed to build pedagogical content knowledge and leadership capacity in mathematics. Through a series of observations and interviews over the course of one school year, we examined how the teachers described and navigated specific affordances and constraints they encountered in their particular contexts. Common affordances included opportunities to revise and learn from instruction, and constraints included reduced flexibility introduced by the need to schedule multiple classes of mathematics. Despite these common features, we found important differences between the three models of departmentalization, which we describe as team approach, class swap, and grade-level mathematics teacher. For example, some of the models provided more opportunities for collaboration while others made it difficult for teachers to address potential inequities in learning opportunities across sections. Despite the constraints of their respective models, we found evidence of the EMS-certified teachers drawing on professional expertise in mathematics to meet student needs.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

Student enrichment in mathematics: a case study with first year university students

ABSTRACT

This paper presents an enrichment case study to showcase a possible avenue for attending to the needs of academically strong mathematics students. We report on a group of university students who were presented with the opportunity of exploring a specific first year mathematics topic deeper, using an inquiry-based learning approach as part of an enrichment programme. Following the intervention, students completed a questionnaire and a few were interviewed to establish their experiences of the enrichment programme. We discuss the successes and pitfalls of the intervention and report on the impact it had on the participants.     

Design and research potential of interactive textbooks: the case of fractions

The transition from classical to electronic textbooks seems to be a logical step in the digitization advancing worldwide. However, developing an e-book ought to be more than digitizing text: key features of computer-based learning environments such as interactive exercises, adaptive demands, or automatic feedback should be integrated to take advantage of the digitization. The “ALICE:fractions” project aims at designing and evaluating an interactive mathematics textbook for introducing fractions in the classroom. It is based on research in mathematics education and includes the elements just mentioned. This paper describes the electronic textbook’s implementation and its theoretical background. Moreover, it introduces aspects that allow further information on learning processes to be gleaned. As an example, time on task is regarded in a study with 6th graders who used the electronic textbook in the classroom. Linear mixed models revealed a negative effect of time on task on task success. The effect was moderated by exercise difficulty and slightly by student competence: the effect was less pronounced in difficult exercises and for low-achieving students, whereas for high-achieving students or in easier exercises, the effect intensified.     

Student Attitudes and Calculus Reform

This paper compares the attitudes about mathematics of students from traditionally taught calculus classes and those from a “reformed” calculus course. The paper is based on three studies, which together present a consistent picture of student attitudes about calculus reform. The reformed course appeared to violate students' deeply held beliefs about the nature of mathematics and how it should be learned. Although during their first months in the reformed course most students disliked it, their attitudes gradually changed. One and 2 years after, reform students felt significantly more than the traditionally taught students that they better understood how math was used and that they had been required to understand math rather than memorize formulas.     

Exploring Problem‐based Learning in the Context of High School Science: Design and Implementation Issues

This paper reports on the experiences of a small collaborative inquiry group consisting of a high school science teacher, Deidre, and two university researchers, the authors of this paper, as they explored an active, inquiry‐based approach to teaching and learning referred to as Problem‐Based Learning or PBL ( Barrows, 1994 ; Barrows & Tamblyn, 1980 ). Although PBL is not new and has an established tradition in medical education and other professional schools, the use and scholarship of PBL at the secondary level is only starting to emerge. This small‐scale collaboration allowed the co‐inquirers to delve into the complexities of PBL and to examine its feasibility as a curriculum and instructional approach in the context of high school science teaching and learning. The three collaborators adopted an action‐based inquiry method referred to as Collaborative Inquiry (CI), a “process consisting of repeated episodes of reflection and action through which a group of peers strives to answer a question of importance to them” ( Bray, Lee, Smith, & Yorks, 2000 , p. 6). Data collection methods and sources included student‐generated work, classroom observation, student interviews, and audio‐taped planning meetings. The outcomes of the study focus on the issues that arose during PBL design and implementation, such as selecting a PBL topic, determining the level of structure to be incorporated into the PBL experience, selecting appropriate assessment approaches, facilitating groups, and providing optimal student feed‐back. In addition, outcomes related to student perceptions of PBL indicated that the majority liked learning through PBL because it promoted active learning, made science relevant, provided variety in learning, and supported group work. The authors discuss implications for the adoption of PBL in K‐12 settings.