Proof and reasoning in an inquiry-oriented class: The impact of classroom discourse

Over the past decade, mathematics educators and researchers have become increasingly aware of the impact of social interactions on students\u27 learning (NCTM, 2000; Bowers & Nickerson, 2001; Forman, 2003). Current research indicates that the classroom environment, including the activities and discussions that take place, can have a significant effect on the ways students make sense of mathematical concepts (Yackel, 2001). Understanding mathematics involves knowing how to make sense of key concepts through the processes of reasoning and justification. Educators and researchers agree on the importance of providing students with opportunities in class to explore, conjecture, and prove in order to promote mathematical understanding beyond procedural knowledge (Lakatos, 1976; Rasmussen & Marrongelle, 2006). Although there are a number of studies that investigate many different aspects of classroom discourse and students\u27 learning, there remains a need for more understanding (Franke, Kazemi & Battey, 2007). This study is aimed at investigating the nature and impact of social interactions, both teacher-student and student-student, in classroom discourse. In particular, the study seeks to gain understanding of how interactions influence students\u27 engagement in proof and reasoning activities. In addition, the study analyzes students\u27 argumentation schemes as they occurred in classroom discussions and during student group work. Through the perspective that learning is both a social and an individual activity, this research focuses on the social component of the learning process as it occurs in the classroom. Ethnographic techniques of participant observation and interviews provided methods of data collection, and analysis of discourse and argumentation structures provided a way to interpret the data. This study contributes to the existing research by highlighting certain types of interactions that resulted in students contributing to proof construction and collective reasoning

Dialogue as Data in Learning Analytics for Productive Educational Dialogue

This paper provides a novel, conceptually driven stance on the state of the contemporary analytic challenges faced in the treatment of dialogue as a form of data across on- and offline sites of learning. In prior research, preliminary steps have been taken to detect occurrences of such dialogue using automated analysis techniques. Such advances have the potential to foster effective dialogue using learning analytic techniques that scaffold, give feedback on, and provide pedagogic contexts promoting such dialogue. However, the translation of much prior learning science research to online contexts is complex, requiring the operationalization of constructs theorized in different contexts (often face-to-face), and based on different datasets and structures (often spoken dialogue). In this paper, we explore what could constitute the effective analysis of productive online dialogues, arguing that it requires consideration of three key facets of the dialogue: features indicative of productive dialogue; the unit of segmentation; and the interplay of features and segmentation with the temporal underpinning of learning contexts. The paper thus foregrounds key considerations regarding the analysis of dialogue data in emerging learning analytics environments, both for learning-science and for computationally oriented researchers

Investigating 6th Graders' Use of a Tablet-Based App Supporting Synchronous Use of Multiple Tools Designed to Promote Collaborative Knowledge Building in Science.

At this pivotal moment in time, when the proliferation of mobile technologies in our daily lives is influencing the relatively fast integration of these technologies into classrooms, there is little known about the process of student learning, and the role of collaboration, with app-based learning environments on mobile devices. To address this gap, this dissertation, comprised of three manuscripts, investigated three pairs of sixth grade students’ synchronous collaborative use of a tablet-based science app called WeInvestigate. The first paper illustrated the methodological decisions necessary to conduct the study of student synchronous and face-to-face collaboration and knowledge building within the complex WeInvestigate and classroom learning environments. The second paper provided the theory of collaboration that guided the design of supports in WeInvestigate, and described its subsequent development. The third paper detailed the interactions between pairs of students as they engaged collaboratively in model construction and explanation tasks using WeInvestigate, hypothesizing connections between these interactions and the designed supports for collaboration. Together, these manuscripts provide encouraging evidence regarding the potential of teaching and learning with WeInvestigate. Findings demonstrated that the students in this study learned science through WeInvestigate, and were supported by the app - particularly the collabrification - to engage in collaborative modeling of phenomena. The findings also highlight the potential of the multiple methods used in this study to understand students’ face-to-face and technology-based interactions within the “messy” context of an app-based learning environment and a traditional K-12 classroom. However, as the third manuscript most clearly illustrates, there are still a number of modifications to be made to the WeInvestigate technology before it can be optimally used in classrooms to support students’ collaborative science endeavors. The findings presented in this dissertation contribute in theoretical, methodological, and applied ways to the fields of science education, educational technology, and the learning sciences, and point to exciting possibilities for future research on students’ collaborations using future iterations of WeInvestigate with more embedded supports; comparative studies of students’ use of synchronous collaboration; and studies focused on elucidating the role of the teacher using WeInvestigate - and similar mobile platforms - for teaching and learning.PhDEducational StudiesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/116748/1/casher_1.pd

An exploration of first year university students' constructions of proofs in elementary set theory

A thesis submitted to the Wits School of Education, Faculty of Humanities, University of the Witwatersrand in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015In this thesis I explored the development of first year university students’ proof construction abilities in the context of consultative group sessions. In order to do this I investigated students’ difficulties in proof construction in the area of elementary set theory and the forms of guidance offered as they participated in consultative group sessions. Vygotsky’s (1987) socio-cultural theory is the theoretical framework for the study. His premise that all higher mental functions which include the activity of mathematical proof construction, develop as a result of mediated activity in the context of more knowing others, motivated my exploration. Ten students purposefully chosen from a first year mathematics major class at the University of Limpopo (a historically disadvantaged university) participated in weekly consultative sessions. Students were encouraged to share their thoughts and ideas and critique other students as they attempted proof construction exercises. The lecturer (myself) was present to offer guidance whenever necessary. By establishing the sociomathematical norms pertinent to successful proof construction, my aim was to support students in becoming intellectually autonomous and to empower those with the potential to become more knowing peers to develop their capabilities. With this in mind I investigated the nature of the interactions of the students and lecturer in the consultative sessions. I also traced the journeys of two case study students as they progressed in the first two sessions. Two complementary analytical frameworks incorporating social and cognitive aspects of students’ development enabled me to obtain a holistic picture of the development and scaffolding of proof construction abilities in consultative group sessions. Students’ difficulties were found to be similar to those reported in the literature and included difficulties within meanings of mathematical terms, symbols, signs and definitions, logical reasoning and proof methods and deductive reasoning processes and justification. The most persistent of these difficulties seemed to be the challenge of knowing how to use the knowledge of the definitions of relevant mathematical objects, proof methods, deductive reasoning processes and justification. This is also referred to as strategic knowledge (Weber, 2001). The two case study students showed great improvement in all aspects of their proof construction abilities as they progressed from the first to the second session. This highlighted the effectiveness of the consultative sessions in facilitating access to the observed students’ zones of proximal development and in allowing students to make functional use of the various mathematical objects and processes needed in successful proof construction. This functional use together with the scaffolding received from their peers and the lecturer enabled students to develop and internalise proof construction skills and abilities. Investigation of the nature of the interactions in the consultative sessions examined the lecturer’s use of requests for clarification, reflection on proof construction strategy, critique and justification, while eliciting elaboration of contributions which could drive the proof construction process forward. The importance of the correct interpretation of definitions and their role in providing the logical structure and the justification of each step of the proof construction was emphasized. As the sessions progressed more knowing peers emerged from the group who took over the role and responsibilities of the lecturer and provided most of the scaffolding to their peers. I often called upon these more knowing peers to explain and elaborate on completed proof constructions. Their presentations were observed to be effective learning opportunities for other students

Transactive Discourse during Assessment Conversations on Science Learning

Transactive Discourse During Assessment Conversations on Science Learning by Homer A. Russell III It has been argued that development of science knowledge is the result of social interaction and adoption of shared understandings between teachers and students. A part of understanding that process is determining how student reasoning develops in groups. Transactive discussion is a form of negotiation between group members as they interpret the meaning of their logical statements about a topic. More importantly, it is a form of discourse that often leads to cognitive change as a result of the interaction between group participants as they wrestle with their different perspectives in order to achieve a common understanding. The research reported here was a correlational study designed to investigate the relationship between the various forms of transactive discussion and learning outcome performance seen in an investigation involving 24 students in a middle-SES high school located in southwest Atlanta, Georgia. Pretest and posttest measures of genetics reasoning, as well as curriculum content test data, were used in this study. Group discussion was captured on videotape and analyzed to determine whether transactional discussion was present and whether or not it had an effect on learning outcome measures. Results of this study showed that participant use of transactive discussion played a role in development of reasoning abilities in the area of genetics. It is suggested that teachers should monitor classroom discourse for the presence of transactive discussion as such discourse plays a role in fostering performance outcomes

Classification of Spoken Discourse in Teaching the Construction of Mathematical Proof

The purpose of this study is to analyze the patterns of classroom discourse when high school students move from performing prescribed algorithms in order to solve problems for which the process and solution are well-defined to spoken proof, in which ideas are discussed and arguments are formulated and formalized. The study uses a modified version of discourse analysis developed by Arno Bellack and refined for usage in a mathematics classroom by James T. Fey. The analysis framework is supplemented by codes borrowed from Maria Blanton, Despina Stylianou, and M. Manuela David (2009), which is in turn a modified version of a coding system developed by Kruger (1993) and Goos, Galbraith and Renshaw (2002). Twelve mathematics lessons involving two mathematics teachers were recorded, transcribed and coded. Eight of the lessons were classified as “proof-related” and four were designated “non-proof-related.” A lesson designated “proof-related” contained more than half activity that was actively concerned with the construction of proof; whereas a lesson in which no proofs were formulated was designated “non-proof.” Using the codes described above and a variety of qualitative and quantitative measures, the transcripts were examined for constructivist behavior on the part of the teachers and modes of participation on the students’ part. The findings suggest a relationship between a teacher’s beliefs in constructivist principles and the way in which that teacher instructs proof vs. non-proof. More specifically, a teacher who views her/himself as informed by constructivist pedagogical principles may not evince a sharp distinction between her/his teaching of proof vs. non-proof; but a teacher who does not attempt to incorporate constructivist principles on a daily basis may exhibit more constructivist tendencies when teaching proof

The Effects of Group Size on Student Learning, Student Contributions, \Mental Effort, and Group Outcomes for Middle-Aged Adults Working in an Ill-Structured Problem-Solving Environment

Group work has become increasingly important within adult education as educators strive to present students with problems and processes that they encounter in their professional lives. In many work environments, individuals are expected to function as a part of a team to solve complex problems. Consequently, there has been a shift towards teaching students how to solve problems as part of a group rather than individually. An important question becomes What size group maximizes students learning? This study compared student learning, student participation levels, and mental effort for middle-aged, professional students in large (six students) and small groups (three students) while working in a collaborative, ill-structured problem solving environment to determine if group size impacted student performance. This study found that there was no significant difference in learning, participation, and mental effort between large and small groups. It also confirmed earlier research demonstrating that group product scores, even when adjusted for student participation, did not predict individual student learning. A multiple regression was used to determine if group size, participation, mental effort or group scores could be used to predict individual student learning. The study showed that for middle-aged professional students, group size, mental effort, participation, or group quality were not effective predictors of student learning

Thinking, Interthinking, and Technological Tools

Language use is widely regarded as an important indicator of high quality learning and reasoning ability. Yet this masks an irony: language is fundamentally a social, collaborative tool, yet despite the widespread recognition of its importance in relation to learning, the role of dialogue is undervalued in learning contexts. In this chapter we argue that to see language as only a tool for individual thought presents a limited view of its transformative power. This power, we argue, lies in the ways in which dialogue is used to interthink – that is, to think together, to build knowledge co-constructively through our shared understanding. Technology can play an important role in resourcing thinking through the provision of information, and support to provide a space to think alone. It can moreover provide significant support for learners to build shared representations together, particularly through giving learners access to a wealth of ‘given’ inter-related texts which resource the co-construction of knowledge

a Review of Instructional Approaches

UIDB/00183/2020 UIDP/00183/2020 DL 57/2016/CP1453/CT0066 PTDC/FER-FIL/28278/2017Over the past 20 years, a broad and diverse research literature has emerged to address how students learn to argue through dialogue in educational contexts. However, the variety of approaches used to study this phenomenon makes it challenging to find coherence in what may otherwise seem to be disparate fields of study. In this integrative review, we propose looking at how learning to argue (LTA) has been operationalized thus far in educational research, focusing on how different scholars have framed and fostered argumentative dialogue, assessed its gains, and applied it in different learning contexts. In total, 143 studies from the broad literature on educational dialogue and argumentation were analysed, including all educational levels (from primary to university). The following patterns for studying how dialogue fosters LTA emerged: whole-class ‘low structure’ framing with a goal of dialogue, small-group ‘high structure’ framing with varied argumentative goals, and studies with one-to-one dialectic framing with a goal of persuasive deliberation. The affordances and limitations of these different instructional approaches to LTA research and practice are discussed. We conclude with a discussion of complementarity of the approaches that emerged from our analysis in terms of the pedagogical methods and conditions that promote productive and/or constructive classroom interactions.publishersversionepub_ahead_of_prin