Mathematical Dialogue Research Articles

Interactions and tensions between mathematical discourses and schoolwork discourses when solving dynamic geometry tasks: what is internally persuasive for students?

ABSTRACT I explore students’ discourses in small groups working on mathematical problems using GeoGebra, focusing on the Cartesian connection between algebra and geometry. Specifically, the interest lies in what is internally persuasive for students in upper-secondary school (11th grade) with histories of low attainment. Three problem-solving episodes are presented in detail to illuminate different intertwining and interacting discourses that students produce. In all cases, students engaged in visual explorations and expressed joy in the work. Students used discourses of visual appearance and technical symbolism to talk about screen objects and problem solutions. Two cases illustrate how a student makes a conjecture internally before verifying it for themselves and then convincing their peers of the validity of their solution. In a contrasting case, students used visual trial and error and asked an authority to confirm their solution. Discourses concerning the institutional demands of school interfered with students’ mathematical dialogue.

The performance level of Mathematics Teachers at The Secondary Level in Light of NCTM's Practices for Effective Teaching

The study aims to determine the performance level of mathematics teachers at the secondary level in light of NCTM’s effective teaching practices. In the book "From Principles to Procedures" in 2014, it consisted of (32) sub-practices, stemming from (8) main practices. The study concluded: that the level of performance of mathematics teachers at the secondary level in light of NCTM’s effective teaching practices was generally at a low degree, with an arithmetic mean of (1.56), and their performance level appeared in light of the practices of “asking purposeful questions, supporting the productive struggle in learning mathematics, And extracting evidence on students' thinking and use", to a medium degree, and with arithmetic averages (1.84, 1.77, 1.70), respectively. While the level of their performance appeared in the light of the practices of “facilitating meaningful mathematical dialogue, building procedural fluency from conceptual comprehension, setting mathematics objectives to focus learning, implementing tasks that enhance inference and problem solving, and using and linking mathematical representations” to a low degree, with arithmetic averages of (1.55, 1.52, 1.44, 1.36, 1.28) respectively. In light of the results of the study, a set of recommendations were presented.

Teachers’ noticing to promote students’ mathematical dialogue in group work

How can teachers refine their strategies for purposefully engaging students in mathematical discussions when students are working in groups and the teacher enters an ongoing group conversation? In three educational design research cycles, four teachers collaborated with a researcher for one year to analyse, design and evaluate strategies for engaging students in small-group mathematical discussions. The idea of noticing (Mason in Researching your own practice: the discipline of noticing, RoutledgeFalmer, London, 2002; Sherin et al. in Mathematics teacher noticing: seeing through teachers’ eyes, Taylor & Francis, New York, 2011) was used to organize the findings—by paying attention to aspects in the mathematical discussions and interpreting the interactions, teachers could together refine their own actions/responses to better support students’ work. The Inquiry Co-operation Model of Alrø and Skovsmose (Dialogue and learning in mathematics education: intention, reflection, critique, Kluwer Academic Publishers, Dordrecht, 2004) was used as a theoretical base for understanding qualities in mathematical discussions. Ehrenfeld and Horn’s (Educ Stud Math 103(7):251–272, 2020) model of initiation-entry-focus-exit and participation was for interpreting and organizing the findings on teachers’ actions. The results show that teachers became more aware of the importance of explicit instructions and their own role as facilitators of mathematical questions to students, by directing specific mathematical questions to all students within the groups. In this article, by going back and forth between what happened in the teachers’ professional development group and in the classrooms, it was possible to simultaneously follow the teachers’ development processes and what changed in students’ mathematical discussions.

Advancing a Democratic Pedagogy and Supervision Framework: An Illustrative Case of Teacher Questioning in Secondary Mathematics Instruction

This article pushes back against the evalu-centric view of improvement (Hazi, 2018; 2020) in the supervision literature by advocating for a democratic pedagogy and supervision framework developed to support instructional supervision and evaluation dialogue between teachers and leaders. This democratized approach honors and centers the teacher’s expertise and learning as well as the leader’s in the observation, debrief, and reflection process. Through this decentering of expertise in the instructional supervision cycle, our goal is to build leaders’ and teachers’ mutual capacity to develop, implement, and sustain democratic instructional supervision cultures in classrooms and schools. Additionally, we illustrate our framework through a subject/discipline-specific case of instructional supervision in secondary mathematics instruction. Through this illustrative case, we demonstrate how the framework provides school leaders and teachers with specific, shared pedagogical language to engage in standards-based mathematical dialogue during the instructional supervision process. Finally, we discuss the implications of our questioning framework for democratic school leadership, supervisors’ leadership content knowledge, teachers’ discipline-specific work of teaching, and instructional supervision practices, which are often stifled by accountability-driven teacher evaluation education policies that suppress schools’ leadership capacity to apply democratic instructional supervision standards and principles.

Training teachers in maintaining equity in the micro-moments of a mathematical dialogue

This article presents the analysis of a large body of video data of trainee teachers who were learning mathematical dialogue techniques. We explore some of the ways in which trainee teachers can make better use of children's own contributions to foster learning and increase pupil motivation – taking a number of analytic principles from conversation analysis. While the importance of dialogue in mathematics education has drawn increased attention in recent years, there is still relatively little work on how teachers may learn to make better use of dialogue in the classroom. This paper aims to address that lacunae.

The integral as accumulation function approach: A proposal of a learning sequence for collaborative reasoning

Learning mathematical thinking and reasoning is a main goal in mathematical education. Instructional tasks have an important role in fostering this learning. We introduce a learning sequence to approach the topic of integrals in secondary education to support students mathematical reasoning while participating in collaborative dialogue about the integral-as-accumulation-function. This is based on the notion of accumulation in general and the notion of accumulative distance function in particular. Through a case-study methodology we investigate how this approach elicits 11th grade students’ mathematical thinking and reasoning. The results show that the integral-as-accumulation-function has potential, since the notions of accumulation and accumulative function can provide a strong intuition for mathematical reasoning and engage students in mathematical dialogue. Implications of these results for task design and further research are discussed.

Speaking or not speaking as a cultural practice: analysis of mathematics classroom discourse in Shanghai, Seoul, and Melbourne

The benefits of engaging students in mathematics classroom dialogue have been highlighted (and advocated) in a large body of research studies, most of which were conducted in Western cultural contexts. Whether such research advocacy can be extended legitimately to encompass practice in classrooms situated in other cultural contexts, such as East Asian classrooms, remains a matter for empirical investigation. In this paper, we problematize the assumptions underpinning this advocacy of “mathematical dialogue in classrooms” by revisiting several conceptualizations of the role of language in mathematics classrooms and by considering particular forms of communication in classrooms often undervalued in the research literature. To support such problematization, we draw upon an analysis of classroom discourse in nine classrooms situated in Shanghai, Seoul, and Melbourne from the Learner’s Perspective Study. The results of the analysis demonstrate some interesting variations in the extent to which the discursive practice of each classroom afforded or constrained the opportunities for students to speak mathematically, and its consequences for student learning as indicated by student mathematical fluency demonstrated in the post-lesson interviews. Such variations provide insight into the differences in inspirations and cultural values attached to particular types of mathematical performance. We argue that our theories should anticipate application in culturally differentiated settings and be sensitive to the constraints and affordance that culture might place on classroom discourse practices.

A calculus course in knowledge feedback format

This paper presents a design of an undergraduate calculus course which attempts to achieve higher student activity and teacher-student exchange, allowing the teacher to adapt more to the present student group and to increase students work time. This is done by switching between examinations in small format and examination-free, open mathematical dialogue occasions – two learning-enhancing opposites that reinforce each other during the course. In the examination-free dialogue, students solved problems during lectures every week, where the quality of the student-teacher dialogue is essential both for dissection of properties of mathematics and for students’ willingness to continue participating. Thus, students regularly practiced verbal presentation and reflection in mathematics.

Making a case for exact language as an aspect of rigour in initial teacher education mathematics programmes

Pre-service secondary mathematics teachers have a poor command of the exact language of mathematics as evidenced in assignments, micro-lessons and practicums. The unrelenting notorious annual South African National Senior Certificate outcomes in mathematics and the recognition by the Department of Basic Education (DBE) that the correct use of mathematical language in classrooms is problematic is reported in the National Senior Certificate Diagnostic Reports on learner performance (DBE, 2008-2013). The reports further recognise that learners do not engage successfully with mathematical problems that require conceptual understanding. This paper therefore highlights a need for teachers to be taught and master an exact mathematical language that for example, calls an 'expression' an 'expression' and not an 'equation'. It must support the call of the DBE to use correct mathematical language that will support and improve conceptual understanding rather than perpetuate rote procedural skills, which are often devoid of thought and reason. The authentic language of mathematics can initiate and promote meaningful mathematical dialogue. Initial teacher education programmes, as in the subject methodologies, affords lecturers this opportunity. The language notions of Vygotskian thought and language, Freirian emancipatory critical consciousness and Habermasian ethical and moral communicative action frame the paper theoretically. Using a grounded approach, after examining examples of student language in a practice based research intervention, the design and development of a repertoire of language categories, literal, algebraic, graphical (Cartesian) and procedural (algorithmic) emerged from three one-year cycles of an action research methodology. The development of these repertoires of language was to assist teachers in communicating about mathematical objects through providing a structured framework within which to think and teach. A course model encompassing small group discussions, an oral examination and a self-study action research project, that helped sustain the teaching of an exact mathematical language, is presented. This is supported by student reflections on the usefulness of implementing them.

Embedding Mathematical Dialogue in Parent–Child Shared Book Reading: A Preliminary Investigation

Research Findings: Shared book reading provides a meaningful context for rich conversations to occur between a child and an adult and offers opportunities for children to be exposed to a range of vocabulary and concepts that often extend beyond their everyday experiences. Few studies have examined parent–child shared book reading as a context for embedding mathematical discussion. The purpose of this study was to examine systematically the effect of training parents to focus on mathematical concepts and vocabulary during shared book reading. Specific research questions were as follows: (a) Did parents increase their use of math talk during shared storybook reading following training? (b) Did parents generalize intervention strategies? And (c) did children increase their use of math talk during shared storybook reading? Results from a yoked multiple-baseline design with 6 dyads indicated variability across the dyads with 2 general patterns. Math talk increased following training for 3 of the dyads, whereas verbal mathematical behavior did not show consistent change for the other 3 dyads. Practice or Policy: Results are discussed in the context of home support for early mathematical development.

The Structure of Student Dialogue in Web-Assisted Mathematics Courses

Spring term of 2005, three Web-assisted undergraduate mathematics courses were taught at the University of Idaho: Math 235 Mathematics for Elementary Teachers 1; Math 236 Mathematics for Elementary Teachers II; and Math 391 Modern Geometry. While the content of these courses differ, they share common goals: to foster a deep understanding of critical mathematical content; to train students in the use of computer-based modeling and analysis technologies; and to promote the development of mathematical communication and collaboration concepts, skills, and dispositions. Outside of regular class periods, students participated in an ongoing asynchronous mathematical dialogue using the Idaho Virtual Campus Discussion Tool. The structure of this dialogue was analyzed using graph theoretic methods associated with social network analysis. These findings were compared to student achievement data and the results used to answer the question, “In Web-assisted undergraduate mathematics courses, how is the structure of asynchronous communication related to student achievement?”

Journal of Mathematics and Music (review)

Journal of Mathematics and Music. Edited by Thomas Noll and Robert Peck. Taylor and Francis. Triannual. Vol. 1, no. 1 (March 2007). ISSN 1745-9737 (print); ISSN 1745-9745 (online). Print and online (PDF and HTML) format. Access: http://www.informaworld.com/JMM. Subscription or inquiries originating from North America: Taylor and Francis Group Journals, 325 Chestnut Street, Suite 800, Philadelphia, PA 19106. For other countries see: http://www.tandf.co.uk/journals/ contact.asp. E-mail: katie.chandler@tandf.co.uk. $75 individual; $198 institution (print and online), $188 (online only). The relationship between mathematics and music has fascinated philosophers and musicians for millennia. Pythagoras's mythical discovery that music was sounding number, and thus a sensible manifestation of the underlying numeric reality of the Pythagorean universe, cast a shadow over centuries of musical thought. Neo-Platonic philosophers like Boethius, operating within the Pythagorean tradition, considered musica nothing less than a branch of mathematics itself (that branch concerned with compared quantities, i.e., wholenumber ratios). The math-music relationship has persisted in various forms since then, waxing and waning with the intellectual tenor of the age. We currently appear to be in a waxing phase: since World War II, many prominent composers and music theorists have turned to mathematics as a means of stimulating musical creation and modeling musical thought. While mathematics may not play as central a role in musical composition today as it did in the postwar heyday of Darmstadt, mathematical music theory has rarely been more active. In the United States, the visionary work of David Lewin has captured the imaginations of many scholars, spawning the musictheoretic subfield of transformational theory. In Europe, the work of Guerino Mazzola has had a similarly galvanizing effect, stimulating a growing number of European scholars to pursue connections between mathematical category theory and an astonishingly diverse range of musical phenomena. The present journal arrives perched atop the crest of this wave of burgeoning musicmathematical thought. As the editors make clear in their introduction to this inaugural issue, the journal seeks not only to ride that wave but to channel and focus its considerable energy, funneling the many disparate strands of mathematical music theory into a shared discursive space, one in which connections can be drawn between sub-fields that have previously remained separate. Most notably, the journal aims to integrate American and European traditions of mathematical music theory, areas of inquiry that have thus far remained largely independent from one another due not only to geography, but also, one suspects, to differences in intellectual style. This commitment to trans-Atlantic dialogue is made clear by the international pair of co-editors: the German Thomas Noll and the American Robert Peck, both eminent scholars in their own right. Noll and Peck stress that they wish the journal to be not only a meeting place for American and European music theorists, but also a site for mathematical dialogue between theorists, composers, performers, cognitive scientists, and even those outside of the ivory tower-for example, working musicians involved in such fields as computer music and sound processing. This emphasis on boundary crossing, as the editors put it, is one of the most attractive and exciting aspects of the journal. For musicians not involved in any of these fields, and especially for those with some degree of math phobia, the idea of mathematical thought about music likely conjures up images of an arid, mechanical, hard science approach to the subject, one unresponsive-or even anathema-to the poetic and expressive immediacies of musical experience. This impression, though understandable, is in fact far from the truth. Much work in mathematical music theory is highly poetic and interpretive, ranging from issues of musical semiosis and hermeneutics, to mappings of embodied musical experience, to models of musical phenomenology. …

Quality of Instruction: Examining Discourse in Middle Schoo Mathematics Instruction

According to the NCTM reform suggestions, when teachers are orchestrators of student interactions, students adopt a more active role in explaining and learning mathematics. This research, which mapped the nature and role of meaningful mathematical discourse, provides insights into discursive practices that lead to rich mathematical interactions. We observed, coded, and analyzed middle school algebra, number, and data lessons using a grounded theory approach. We organized the observed paths that emerged into a map depicting actual paths for mathematics discourse. The results indicated that communication pathways between the teacher and students occur in many ways, and certain student-initiated questions may trigger predictable teaching patterns. Conversation that originates with the teacher often results in dialogue that is one-dimensional, mostly provides factual information, and rarely results in rich, meaningful mathematical dialogue. However, when students engage in the teacher's conversation or they are persistent in their own questioning, teachers tended to provide more detailed explanations, and teachers often embellished with new examples and representations using nuanced solution methods. Although results seem to indicate that teaching children to be persistent with their questioning will enhance understanding, this behavior may be interpreted as threatening to some teachers. Therefore, caution is warranted when attempting to turn these findings into action. It is important that, before instructing students about being persistent with questions, teachers understand the students' intentions. Although participants did not have negative reactions to persistent student questioning, some children might experience negative responses without proper professional development for teachers.

The "Ladies' Diary": Gender, Mathematics, and Civil Society in Early-Eighteenth-Century England

The influential Ladies' Diary, or Woman's Almanack (1704-1840) owed its long affiliation with mathematics to the contributions of an active readership. This article describes how the Ladies' Diary's first correspondents-in addition to maintaining contemporary hierarchies of class and gender-adhered to rules of polite conversation in generating the periodical's scientific content. The Diary's early mathematical dialogue is analyzed in terms of a dualistic conception of civil society, one that encompasses both the activity of a print-based public sphere and the conventions of civility. This dualistic civil society, it is argued, was particularly suited to women's participation, and its degeneration in the second quarter of the eighteenth century served to masculinize the magazine's mathematical forum. The centrality of readers in forming-and in dismantling-the Ladies' Diary's civil society illustrates the danger of relying on top-down models of scientific dissemination in the study of polite science.

A History of Mathematical Dialogue in Textbooks and Classrooms

Mathematical communication is an important goal of recent educational reform. The NCTM's Principles and Standards for School Mathematics (2000), continuing an emphasis on mathematical discourse from the 1991 Professional Standards for Teaching Mathematics, has a Communication Standard at each grade level. This article examines textbooks and classrooms from antiquity through the nineteenth century in search of historical precedents for mathematical communication in the form of dialogue between teacher and student. Although we have no way of knowing how prevalent this mode of teaching has been, interest in dialogue as a tool for helping students learn mathematics has been ongoing.