Inquiry-Based Learning and the Art of Mathematical Discourse

Abstract

Our particular flavor of inquiry-based learning (IBL) uses mathematical discourse, conversations, and discussions to empower students to deepen their mathematical thinking, building on strengths of students in the humanities. We present an organized catalog of powerful questions, discussion prompts, and talk moves that can help faculty facilitate a classroom focused on mathematical discourse. The paper brings this discourse alive through classroom vignettes and explores various teacher moves and their impacts. The mathematical theme of the classroom investigations, Maypole dance patterns, stems from the learning guide “Discovering the Art of Mathematics: Dance.” Both authors are part of the NSF-funded project “Discovering the Art of Mathematics,” which provides IBL materials for mathematics for liberal arts courses, see www.artofmathematics.org.     

Promoting Mathematical Communication in the Classroom: Two Preservice Teachers' Conceptions and Practices

Recent reforms in mathematics education have encouraged teachers to engage their students in various forms of communication. Scholars have begun to consider questions such as: In what ways do teachers facilitate and guide classroom discourse? How does the quality of students' reflections impact the development of rich mathematical understanding? In order to address these and similar questions, the authors provide a framework of four constructs that can be used to analyze various forms of classroom communication: uni-directional communication, contributive communication, reflective communication, and instructive communication. Throughout the article,the authors both develop and use these constructs as they consider two preservice teachers concepts of communication and their corresponding classroom practices.     

Transfronteriza pre-service teachers managing,resisting, and coping with the demands of mathematical discourse

This article reports on a case study of a college class for pre-service teachers on the US–Mexico border in which students participated in in-depth discussion around mathematical problems every day. This pedagogical approach promotes the socialization of students into and through the specialized discourse of mathematics. The focus of this paper is on the experience of transfronterizo students in that course. Transfronterizos are Mexican residents who periodically cross the border to attend school. For these students, whose educational background in Mexico allowed them to develop proficiency in elementary mathematical discourse in Spanish, their socialization experience includes ways in which they draw on language, and other social and learning experiences in Mexico. The focus of this paper is an assignment called Thinking Logs, a genre that required the use of mathematical discourse for teaching. Drawing on data gathered from participant observation of the course, interviews, analysis of study session discourse, and genre analysis, I highlight agentive ways that each participant used in their own socialization process. I show how participants improvised writing of models, asked for clarification in the first language, and even resisted the discourse. Students who resisted the demands might incur negative effects. Furthermore, I argue that the role of the guidance from an expert (such as a professor) is imperative in a socialization process, and I offer implications for ways that teachers can guide second language writers to develop mathematics discourse.     

When the Rules of Discourse Change,but Nobody Tells You: Making Sense of Mathematics Learning From a Commognitive Standpoint

The interpretive framework for the study of learning introduced in this article and called commognitive is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather than being necessitated by an extradiscursive reality, is spurred by commognitive conflict, that is, by the situation that arises whenever different interlocutors are acting according to differing discursive rules. The framework is applied in 2 studies, one of them featuring a class learning about negative numbers and the other focusing on 2 first graders learning about triangles and quadrilaterals. In both cases, the analysis of data is guided by questions about (a) features of the new mathematical discourse that set it apart from the mathematical discourse in which the students were conversant when the learning began; (b) students' and teachers' efforts toward the necessary discursive transformation; and (c) effects of the learning–teaching process, that is, the extent of discursive change actually resulting from these efforts. One of the claims corroborated by the findings is that school learning requires an active lead of an experienced interlocutor and needs to be fueled by a learning-teaching agreement between the interlocutor and the learners.     

An Ode to Imre Lakatos: Quasi-Thought Experiments to Bridge the Ideal and Actual Mathematics Classrooms

This paper explores the wide range of mathematics content and processes that arise in the secondary classroom via the use of unusual counting problems. A universal pedagogical goal of mathematics teachers is to convey a sense of unity among seemingly diverse topics within mathematics. Such a goal can be accomplished if we could conduct classroom discourse that conveys the Lakatosian (thought-experimental) view of mathematics as that of continual conjecture-proof-refutation which involves rich mathematizing experiences. I present a pathway towards this pedagogical goal by presenting student insights into an unusual counting problem and by using these outcomes to construct ideal mathematical possibilities (content and process) for discourse. In particular, I re-construct the quasi-empirical approaches of six!4-year old students’ attempts to solve this unusual counting problem and present the possibilities for mathematizing during classroom discourse in the imaginative spirit of Imre Lakatos. The pedagogical implications for the teaching and learning of mathematics in the secondary classroom and in mathematics teacher education are discussed.     

Mathematics engagement in an Australian lower secondary school

The importance of actively engaging in mathematics discourse in order to learn mathematics is well recognized. In this paper, I use Basil Bernstein’s concepts of pedagogic discourse to document and analyse academic learning time of students in Years 8 and 9 at a suburban lower secondary school: in particular, for what proportion of class time students reported being academically engaged, their explanations for this engagement and how they felt about the discourse. It was found that many students had disengaged from mathematical endeavour as a result of the failure of the instructional discourse either to engage students or to serve the purpose of developing discipline-specific content knowledge. The reasons for this relate to the overemphasis on mundane mathematics resulting in some students lacking the cognitive tools to engage with the concepts and having neither the intrinsic nor instrumental motivation to persist with secondary school esoteric mathematics. The implications for mathematics curriculum development are discussed.     

Environnements Informatisés et Mathématiques: quels usages pour quels apprentissages?

Mathematics, seen as a model of pure science, often conveys the image of a science constructing itself in quite poor technological environments; it nevertheless develops by elaborating (and by exploiting) powerful material and symbolic tools. Actually mathematics teaching is closer to this image of mathematics than to mathematical practice: its goal seems to transmit a form of culture rather than efficient computation tools and theoretical means of their control (Kahane, 2002). This situation is viable if the tools can be held at distance, outside the classroom; it is no longer viable when computation tools (essentially calculators) are imported by students themselves inside the classroom and integrated into their mathematical practice. Thus established conflict between the social legitimacy of these tools and their school illegitimacy (Chevallard, 1992) deeply destabilises mathematics teaching itself. We present here a general framework to think about the integration of the tools in the teaching and learning of mathematics. More precisely, we propose:

1. A theoretical approach, which allows us to understand the influence of tools on human activity and in particular on professional and school education processes;
2. An analysis of computerized learning environments, which shows the importance of students' control of their own activity;
3. Some elements that help to think about the temporal and spatial organization of study in such environments and to guide students' activity;
4. A reflection about the conception of pedagogical resources, which is all the more necessary if one wants to facilitate an evolution of teachers' practices.

     

K-12 Mathematics and the Web

Abstract

The Web offers numerous learning resources and opportunities for K-12 mathematics education. This paper discusses those resources and opportunities. Discussion includes (a) asynchronous and synchronous communication tools, (b) the use of data sets to make connections between mathematics concepts and real-world applications, and (c) interactive environments that promote active thinking by allowing students to manipulate mathematical systems, observe patterns, form conjectures, and validate findings.     

Educational forms of initiation in mathematical culture

A review of literature shows that during the history of mathematics education at school the answer of what counts as ‘real mathematics’ varies. An argument will be given here that defines as ‘real mathematics’ any activity of participating in a mathematical practice. The acknowledgement of the discursive nature of school practices requires an in-depth analysis of the notion of classroom discourse. For a further analysis of this problem Bakhtin’s notion of speech genre is used. The genre particularly functions as a means for the interlocutors for evaluating utterances as a legitimate part of an ongoing mathematical discourse. The notion of speech genre brings a cultural historical dimension in the discourse that is supposed to be acted out by the teacher who demonstrates the tools, rules, and norms that are passed on by a mathematical community. This has several consequences for the role of the teacher. His or her mathematical attitude acts out tendencies emerging from the history of the mathematical community (like systemacy, non-contradiction etc.) that subsequently can be imitated and appropriated by pupils in a discourse. Mathematical attitude is the link between the cultural historical dimension of mathematical practices and individual mathematical thinking.     

Practice and conceptions: communicating mathematics in the workplace

The study examined the experience of communication in the workplace for mathematics graduates with a view to enriching university curriculum. I broaden the work of Burton and Morgan (2000), who investigated the discourse practices of academic mathematicians to examine the discourse used by new mathematics graduates in industry and their perceptions of how they acquired these skills. I describe the different levels of perception of discourse needs and of how they gained the necessary skills. At the lowest level, they learnt through trying out different approaches. At the next level, they were assisted by colleagues or outside situations. At the highest level, a small group viewed communication and interpersonal skills as a scientific process and stood back and used their “mathematical” observation skills to model their behaviour. These graduates did not appear to have systematically studied communication as part of their degree and they were unaware of the power of language choices in the workplace. Those who were working as mathematicians had to come to grips with explaining mathematical concepts to a wide range of people with varying mathematical skills but who generally were considerably less skilled in mathematics. The study revealed that these graduates were seriously underprepared in many aspects for joining the workforce. Many found it difficult to adapt to dealing with colleagues and managers, and developing communication skills was often a matter of trial and error.     

Bridging the macro- and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development

This paper is methodologically based, addressing the study of mathematics teaching by linking micro- and macro-perspectives. Considering teaching as activity, it uses Activity Theory and, in particular, the Expanded Mediational Triangle (EMT) to consider the role of the broader social frame in which classroom teaching is situated. Theoretical and methodological approaches are illustrated through episodes from a study of the mathematics teaching and learning in a Year-10 class in a UK secondary school where students were considered as “lower achievers” in their year group. We show how a number of questions about mathematics teaching and learning emerging from microanalysis were investigated by the use of the EMT. This framework provided a way to address complexity in the activity of teaching and its development based on recognition of central social factors in mathematics teaching–learning.     

How inconvenient assumptions affect preservice teachers’ uptake of new interactional patterns in mathematics: analysis and aspiration through a bifocal lens

In this paper, I highlight the inadequacies of contemporary theoretical and philosophical orthodoxies to fully address pedagogic change. The required change is in mathematics education, and it has to do with enabling preservice teachers, upon graduation, to rework extant power relations in implementing new interactional patterns that centre the mathematics and the learner in dynamic, productive interaction. I interpret data from published research and my own teaching using psychological, overlaid with poststructuralist, constructs of the relationship between knowledge and action. In data interpretation I read through the words for a psychological interpretation (meaning), and I look at the words for poststructuralist indications of subjectification and identity formation (related to how well students recognise themselves as full participants in the combined discourses of mathematics and education). I have my cake and eat it; contradictory notions of the learner and learning to teach in innovative ways are held together to demonstrate (a) how it can happen that a teacher educator’s aspirations can be held ransom to constituted assumptions that inconveniently work against change, and (b) how the recognition that humanist assumptions, mathematical proficiency and agency are discursively constituted (Davies, 1990) can suggest avenues for change in teacher education.     

Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.

Designing a Bridging Discourse: Re-Mediation of a Mathematical Learning Disability

Students with disabilities present a unique instructional design challenge. These students often have qualitatively different ways of processing information, meaning that standard instructional approaches may not be effective. In this study I present a case study of a student with a mathematical learning disability for whom standard instruction on fractions had been ineffective. With regard to theory, I draw on Lev Vygotsky’s framing of disability and then use Anna Sfard’s conceptualization of mathematics as a discourse to design a fraction re-mediation that provided a bridge from the student’s discourse to the canonical mathematics discourse. This bridging discourse was used in 5 videotaped re-mediation sessions with the case study student. A fine-grained analysis of the re-mediation sessions traced the ways in which the student’s discourse shifted over time, which enabled her to solve problems she had previously been unable to solve. This study provides a proof of concept for reconceptualizing remediation and illustrates the potential utility of a bridging discourse to help students who have a history of failure gain access to the canonical mathematics discourse and content.     

Tacit Beginnings Towards a Model of Scientific Thinking

The purpose of this paper is to provide an examination of the role tacit knowledge plays in understanding, and to provide a model to make such knowledge identifiable. To do this I first consider the needs of society, the ubiquity of information in our world and the future demands of the science classroom. I propose the use of more implicit or tacit understandings as foundational elements for the development of student knowledge. To justify this proposition I consider a wide range of philosophical and psychological perspectives on knowledge. Then develop a Model of Scientific Knowledge, based in large part on a similar model created by Paul Ernest (Social constructivism as a philosophy of mathematics, SUNY Press, Albany, NY, 1998a; Situated cognition and the learning of mathematics, University of Oxford Department of Educational Studies, Oxford, 1998b). Finally, I consider the work that has been done by those in fields beyond education and the ways in which tacit knowledge can be used as a starting point for knowledge building.     

Big Math for Little Kids

Big Math for Little Kids, a comprehensive program for 4- and 5-year-olds, develops and expands on the mathematics that children know and are capable of doing. The program uses activities and stories to develop ideas about number, shape, pattern, logical reasoning, measurement, operations on numbers, and space. The activities introduce the mathematical ideas in a coherent, carefully sequenced fashion, and are designed to promote curiosity and excitement about learning and doing mathematics. The program produces playful but purposeful learning of deep mathematical ideas, and encourages children to think about and express their mathematical thinking. Throughout the program, great emphasis is placed on the development of mathematical and mathematics-related language. Our observations suggest two broad questions for future research: What kinds of competence can children develop in the context of a rich mathematics environment? In what ways can mathematics learning promote language and literacy?     

Talking About Teaching Episodes

This paper examines two types of discourse in which teachers engage when discussing case studies based on classroom episodes, and the ways in which the availability of video data of these episodes may motivate a shift in the mode of discourse used. We interviewed two pairs of secondary school mathematics teachers after they had read a case study based on a 16-minute mathematics classroom episode taped in a secondary school in the United States. During each interview, a multimedia version of the case study, including video of the original episode, was available to the participants. We identify two modes of discourse engaged in by the teachers during the interviews: Grounded Narrative and Evaluative Discourse. We examine and identify the characteristics of the two discourse forms, drawn from both video and textual analysis. These characteristics are self-reflective talk, perspective, ethics, and linguistic patterns. The identification of two modes of discourse is relevant for researchers and teacher educators using case studies or video recordings. In addition, the findings provide insight into how teachers are “seeing” classroom events in a video case study.