Commognitive Framework Research Articles

Proving routines in a lecturer’s mathematical discourse for proof teaching: a case of analysis lecturing

This study focuses on the characterisation of proof teaching in university lectures on mathematical analysis using the commognitive framework. The case is set around a lecturer with experience in teaching this course and a good reputation among university students. Six lectures of the course were analysed using discourse and thematic analysis. The analysis resulted in a range of characteristics of the mathematical discourse for proof teaching of a lecturer. Then, by exploring the proving routines within the characteristics, we produced a typology of teaching characteristics in the lecturer’s discourse; that typology gave us insights into a pattern of proving specific to our case study. The exploration of the proving routines in the lecturer’s discourse offered an in-depth understanding of teaching proof beyond the macro-level descriptions of lectures in the extant literature. We conclude the study with implications for research and education.

When Learner-Centered Teaching and Learning Goes Online: Zooming into Linear Algebra Tutorials During the Pandemic

AbstractThe COVID-19 pandemic shifted higher education online, drawing attention to synchronous learning and instruction on digital communication platforms. Learner-centered teaching practices in the tertiary level, such as mathematical discussions, have been shown to benefit student learning. The interactions involved in online synchronous mathematical discussions have been studied less. Most research taps into these aspects drawing on students’ academic outcomes and reflective interviews. This study explores instructional practices as they unfold online, with a focus on student-instructor and student-student interactions. We zoom-in to online synchronous teaching and learning processes on a popular communication platform by analyzing Linear Algebra tutorials in the first pandemic year. Using the commognitive framework, we characterize instructional interactions with the construct of a learning-teaching agreement. The analysis resulted in three interactional patterns, where in all cases, the tutorial transformed at some point to the one dominated by the instructor.

Teachers' Multiple Representations on Teaching Fractions at Elementary School: A Commognitive Framework

This research explores how teachers utilize verbal, symbolic, pictorial, and real-life representations using a commognitive framework to support student learning. This research uses a qualitative approach with an exploratory descriptive design. The selection of research subjects was carried out through purposive sampling which was modified to achieve the research objectives. The research focuses on elementary school teachers who have education diplomas and more than 15 years of teaching experience. In this research, our data is in the form of learning video recording data. The collected data was analyzed using the cognition framework and the Lesh Model of multiple representations. The results of this research show two representations that teachers always apply to each commognitive component, namely symbolic representation and real-life representation. Symbolic representations are often used by teachers to solve mathematical problems, especially word problems. Likewise, real life representations are often used because they apply events or occurrences in everyday life. The use of diverse representations has the potential to increase students' understanding and ability to apply fraction concepts in real life.

The role of mathematical connections in object-level and meta-level learning: a potential lens for supporting pre-service teacher learning

ABSTRACT The importance of mathematical connections in teaching mathematics is broadly acknowledged in the literature. Nevertheless, more research is needed to clarify how different types of mathematical connections manifest in classroom interactions and how they can support student learning. This article uses the commognitive framework to analyse 15 pre-service secondary teachers’ (PSTs’) discussions on mathematical connections that hypothetically could be made in a pattern generalisation task. The study seeks to explore the types of connections produced by the PSTs and to examine the potential opportunities for object-level and meta-level learning in algebra discourse. The findings are discussed in light of ongoing efforts to refine teachers’ pedagogical knowledge and practice regarding mathematical connections and to develop better pedagogical tools to communicate about mathematical connections.

Unveiling problem-solving strategies of pre-service mathematics teachers: A visual and discursive exploration

This study explores how pre-service mathematics teachers (PMTs) in South Africa use visualization and self-discourse to solve mathematical problems. Visualization is known to enhance mathematics learning, while effective communication skills are critical for teaching and learning mathematics, especially in contexts, where the language of instruction may not be the first language of students or teachers. By understanding the visualization techniques and discursive properties employed by PMTs, insights can be gained into how to improve mathematics learning and teaching. The study is informed by the commognitive framework and uses qualitative data from a purposive sample of 10 PMTs who participated in a performance test and semi-structured interviews. The study found that PMTs rely on mental visualization for simpler mathematical problems but use both symbolic and iconic visual mediators for more complicated problems. The use of language to engage in self-discursive activity during problem-solving was found to be key for successful visualization.

Twinned Teachers’ Mathematical Discourse Using Problem-Solving

Discussing and debating mathematical ideas through integrating natural and mathematical language is vital for conceptual understanding and ultimately for learner performance. In schools with low performance, it is likely that teaching mathematics follows an approach largely involving rote learning. In rural Limpopo where schools are isolated, professional development takes the form of twinning, where a well-functioning school twins with a school in need. In this study two teachers twinned to teach algebraic word problems to Grade 11 to improve learner performance through a problem-solving approach. A central aspect of the study was attention to discourse informed by the commognitive framework, where both natural language (learners home language and the language of instruction) and language of mathematics spoken by the teachers and encouraged in the learners, was the focus. This quasi-experimental design was implemented to examine the effect of the twinned teachers discourse in a class where English, the language of instruction, was a second language. In the study a pre-test and post-test were administered to the experimental group of 34 learners, and to a comparison group of 40 learners. The study aimed at testing the hypothesis that the twinned mathematics teachers’ discourse using a problem-solving approach informed by the commognitive framework has a significant effect on learner performance in algebraic word problems. In summary, using the commognitive framework in implementing problem-solving approach with the twinned teachers’ mathematical discourse had significant effect in improving learner performance of algebraic word problems in the target group.

Leveraging interdiscursivity to support elementary students in bridging the empirical-deductive gap: the case of parity

Research has recognized deductive reasoning as challenging but not impossible for young mathematics learners. In this paper, we present a learning environment developed to assist elementary-school students to bridge the empirical-deductive gap in the context of parity of numbers. Using the commognitive framework, we construe the empirical-deductive gap as part of a broader divide between two discourses that abide by different rules of a “mathematical game”: a discourse on specific numbers and a discourse on numeric patterns. Interdiscursivity is leveraged as a mechanism for instructional design, where students’ familiar routines with specific numbers are teased out and advanced to make sense in the new discourse. We mobilize this mechanism to create opportunities for students to play an active role in recognizing issues with empirical reasoning and generating deductive arguments to establish the validity of universal statements. The environment is illustrated with a small group of 8-year-olds who learned to justify deductively that “odd + odd = even”.

How narratives about the secondary-tertiary transition shape undergraduate tutors’ sense-making of their teaching

Drawing on the commognitive framework, we construe the secondary-tertiary transition (STT) as a distinctive element in the pedagogical discourses of various communities. Our interest rests with university tutors in light of the emergent recognition of their impact on undergraduates’ mathematics learning in many tertiary contexts worldwide. We aim to understand the roles of STT communication in tutors’ reflections on incidents that took place in their tutorials. Our participants were undergraduate students in the advanced stages of their mathematics degrees in a large New Zealand university and who were enrolled in a mathematics education course. Throughout the semester, the participants led tutorial sessions for first-year students and wrote reflections on classroom incidents that drew their attention. Our data corpus consisted of 58 reflections from 38 tutors collected over four semesters. The analysis revealed that STT communication featured in tutors’ descriptions of classroom incidents, assisted them in making sense of unexpected events, positioned their instructional actions as replications of what was familiar to them from their own STT experience, and contributed toward generating new pedagogical narratives. We situate these findings in the literature concerning undergraduate tutoring and teachers’ perspectives on STT.

의사 인지 이론 관점에서 본 2015 개정 고등학교 수학에서 대학 수학으로의 정적분 담론의 이행

According to the 2015 national curriculum revision, the high school mathematics 2 textbook introduces the definite integral as the difference of the anti-derivatives at the endpoints. However, because most calculus and elementary analysis textbooks at the university level define definite integral as the limit of Riemann sums, students who learned mathematics according to 2015 curriculum revision in high school are forced to use the word “definite integral” differently in university. Sfard views mathematics as a discourse, and she explains the historical development of mathematics and mathematics learning as the change of discourse in the participationist’s perspective. This article looks at the transition problem of the definite integral from school and university in the commognitive framework of Sfard. The comparative analysis of high school and university mathematics discourse about definite integral demonstrates that the two discourses are incommensurable not only in the different word use of definite integral but also the different meta-rule of using definitions and proving. Furthermore, the historical analysis shows that this meta-level change in the transition can be traced to the mathematical revolution from 18th result-oriented calculus to 19th rigorous calculus based on precise definitions and proof. This analysis helps instructors to understand the nature and meaning of the challenges of students in the transition of definite integral between school and university mathematics, to be aware of the commognitive conflict in the communication with students about definite integral, and to design a lesson that helps student change their discourse. The commognitive perspective can contribute to instructors seeing the didactical possibilities in the transition and improve their teaching approach.

The Area Enclosed by a Function Is Not Always the Definite Integral: Relearning Through Collaborative Transitioning Within a Learning-Support Module

Learning-support system is an umbrella term that we use for digital resources that assign students with mathematical questions and give automatic feedback on the inserted answers. Transitioning between questions and feedback is characteristic to students’ work with such systems. We apply the commognitive framework to explore the role of within-system transitions in students’ mathematics learning, with a special interest in what we term as “reroutinization”—a process of repeated development of conventional routines to be implemented in already familiar mathematical tasks. The study revolves around a digital module in integral calculus, which was designed to support undergraduates with finding areas enclosed by functions. The data comes from dyads and triads of first-year university students, who collaboratively interacted with the module. The analyses cast light on how transitioning within the module aided students to review familiar routines, amend them, confirm, and solidify the amendments. The transition process was not always linear and contained instances of students cycling back and forth between the assigned questions and feedback messages. We conclude with the discussion on the module’s design that afforded reroutinization and suggest paths for further research.

Commognition as an approach to studying proof teaching in university mathematics lectures

In this position paper, we propose commognition for the study of proof teaching at university lectures through an integrative literature review. We critically examine studies that focused on proof teaching but did not use the commognitive framework. Through this examination, we gain an understanding of the pedagogical aspects of proof teaching and address gaps in the literature. After we introduce the key principles of the commognitive framework, we review a set of studies that used commognition for the investigation of lecture-type teaching in higher education. In reviewing these studies, we pay special attention to the added value of the use of commognition. We conclude by proposing an approach to the study of proof teaching by using the commognitive framework. This approach offers insights into opportunities provided by lecturers for the de-ritualization of proving routines.

Statistical Reasoning Process of Students in Decision Making Using Commognitive Framework

Background : Statistical reasoning plays an important role in decision making. This is because decision making requires the ability to connect and relate various aspects and ideas. Objectives :This study aimed to describe the student's statistical reasoning process using a commognitive framework based on the task of making decisions. Design : This study used a qualitative approach with a descriptive exploratory design. Setting and Participants: Respondents in this study were students majoring in mathematics in the fourth semester and sixth semester. There were 58 students involved, 3 of them were research subjects. Data collection and analysis : The authors gave a task containing a question to students. The question had to do with making a decision to choose one of the six employees who were most successful at their job. Results : The findings in this study indicate that there were three main aspects when carrying out the statistical reasoning process using a commognitive framework in making decisions, 1) the visual mediator created when representing data is the foundation for being able to analyze and interpret data. Visual mediators that were irrelevant to data displays had an effect on the emergence of ritualized routine, resulting in illogical decision, 2) substantiations narrative, in the form of involving various aspects of decision making and memorization narrative in the form of rules related to mathematical or statistical concepts in decision making, was a tool to minimize the risks arising from the decisions taken.

Learning research in a laboratory classroom: a reflection on complementarity and commensurability among multiple analytical accounts

With the myriad theories generated through research over the years, a continuing challenge for researchers is to navigate the multitude of theories in order to communicate their research, integrate empirical results, and make progress as a field by building upon empirical research. The Social Unit of Learning project was purposefully designed so that researchers from multiple disciplines with different theoretical perspectives could work together to examine the complexity of the mathematics classroom. In this paper, we reflect on the multiple analytical accounts generated from the project, drawing from the notions of complementarity and commensurability. Two parallel analyses, applying the commognitive framework and the theory of representations respectively, are used as illustrative examples for discussion regarding complementarity and commensurability. The paper addresses two focal questions, as follows: in what ways do divergence or contradiction in incommensurable analytical accounts reflect methodological discrepancies or fundamental differences in the underpinning theories? Furthermore, in what ways do the accounts generated by the parallel analyses predicated on different theories lead to differences in instructional advocacy? The answers to these questions provide empirically-grounded insights into the consideration of incommensurability in educational research, and suggest ways in which researchers and practitioners might apply the notion of complementarity to reconcile or exploit incommensurable analytical accounts that have resulted in different instructional advocacies.

Classroom learning as a deritualization process: The case of prospective teachers learning to solve arithmetic questions

This paper conceptualizes classroom learning and operationalizes the proposed conceptualization. We adopt the commognitive framework, according to which learning is thought of as routinization and routines are conceptualized in terms of <task, procedure> pairs. To examine the usefulness of this conceptualization, we applied our operationalization to examples from a class of preservice elementary school teachers learning to solve a specific type of questions. We analyzed their suggested solutions during three lessons that were recorded and transcribed verbatim. Progress in classroom learning was evidenced by changes in the participants' tasks, which became closer to the tasks expected in the discourse into which they were being initiated. Progress was also evidenced by changes in participants' procedures for tackling the tasks. Finally, in examining the deritualization characteristics of flexibility, bondedness, substantiability and students' agentivity, we found changes in students' performance.

Pre-university students square-root from squared things: A commognitive account of apparent conflicts within learners' mathematical discourses

This study explores the grasp of square roots among eleven students in a remedial mathematics course, with a special focus on where the same student generated apparently conflicting responses. Building on the commognitive framework, the analysis distinguished between routines that individual students consistently implemented in situations where roots “stood alone” and where they were incorporated in more compound exercises, where roots were extracted from square numbers and from squared radicands, where roots were applied to monomials and binomials, and where parameters named with different letters were involved. Differences were found in routines’ degree of objectification, procedures, and tasks. These differences are explained with a theoretical account, suggesting that what may seem as a conflict within a student’s discourse could be a sensible difference of actions taken in situations that this student construed as different. The contribution of this study to the body of knowledge on teaching and learning of roots and to commognitive research is discussed.

Exploring the Argumentation Skills of Prospective Teachers based on Commognitive Approach using Moodle LMS

While a pandemic is currently sweeping the world, technology becomes a necessity in almost the entire aspects of life, particularly in learning activities. One of conventional learning-based platforms that adopt online system is Moodle Learning Management System. This study explored the argumentation skills of prospective teachers based on commognitive approach using the LMS. This qualitative research used online-based semi-structured questionnaire and interview during a discussion group via LMS platform. The subjects of this study were 38 prospective teachers. The research instrument was a questionnaire with eight items. It was initially assessed with Cronbach’s alpha test in which the value indicated the questionnaire was reliable (0.852). The findings of this study indicate the argumentation skills of prospective teachers correlate with the commognitive framework, including claim with endorsed narratives; data with visual mediators; warrant with routine, word use, and endorsed narratives; and backing with endorsed narratives.

Characterizing how and when a way of proving develops in a primary mathematics classroom: a commognitive approach

In this study we aim to characterize a way of proving which can be produced in a primary mathematics classroom and explore the factors that influence these processes and lead to changes in the way of proving. Assuming proving as a socially embedded activity, we conceptualize it as the interplay between ‘construction’ and ‘substantiation’ based on a well-established theoretical framework in mathematics education: the commognitive framework. A tangible proving task was designed, based on the idea of operative proofs, and implemented in a fifth-grade classroom in England. We analysed the construction and substantiation which fairly associated with discursive features (word use, visual mediator, narrative, and routine) during the proving process. The results show that the interplay between construction and substantiation developed progressively rather than in a straightforward manner, in which previously constructed narratives are reconstructed or substantiated in a more general context. This finding is relevant to understand the factors that might influence primary students to change the use of examples and the way of proving when used as a communicational means in proving. Our study has implications for possible continuity to proving activities in secondary schools, and thus contributes to advancing the research on proving in primary schools.

Minding mathematicians\u2019 discourses in investigations of their feedback on students\u2019 proofs: a case study

This article presents a research apparatus for investigating and making sense of stories that emerge from feedback that mathematicians provide on students’ proofs. Using the commognitive framework, the notion of didactical discourse on proof is developed as a lens for conceptualizing mathematicians’ practice of feedback provision. The notion is accompanied by a tentative organizational frame, within which didacticians and mathematicians can operate as partners. The methodological affordances of this apparatus are illustrated with a case study of a research topologist, who taught a small, graduate course in topology. The emerged characteristics of her feedback and discourse are situated in the literature and used to sketch future research avenues.

Running to keep up with the lecturer or gradual de-ritualization? Biology students’ engagement with construction and data interpretation graphing routines in mathematical modelling tasks

Through a commognitive lens, we examine twelve first-semester biology students’. engagement with graphing routines as they work in groups, during four sessions of Mathematical Modelling (MM). We trace the students’ meta-level learning, particularly as they fluctuate between deploying graphs for mere illustration of data and as sense-making tools. We account for student activity in relation to precedent events in their experiences of graphing and as fluid, if not always productive, interplay between ritualised and exploratory engagement with graph construction and interpretation routines. The students’ construal of the task situations is marked by efforts to keep up with lecturer expectations which allow for changing degrees of student agency but do not factor in the influence of precedent events. Our analysis has pedagogical implications for the way MM problems are formulated and also foregrounds the capacity of the commognitive framework to trace de-ritualization and meta-level learning in students’ MM activity.

Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks

We explore the conjecture that engaging teachers with activities which feature mathematical practices from the past (history-focused tasks) and in today’s mathematics classrooms (mathtasks) can promote teachers’ problematizing of mathematics and its pedagogy. Here, we sample evidence of discursive shifts observed as twelve mathematics teachers engage with a set of problematizing activities (PA) – three rounds of history-focused and mathtask combinations – during a four–month postgraduate course. We trace how the commognitive conflicts orchestrated in the PA triggered changes in the teachers’ narratives about: mathematical objects (such as what a function is); how mathematical objects come to be (such as what led to the emergence of the function object); and, pedagogy (such as what value may lie in listening to students or in trialing innovative assessment practices). Our study explores a hitherto under-researched capacity of the commognitive framework to steer the design, evidence identification and impact evaluation of pedagogical interventions.