Mathematical Discourse Research Articles

고등학생들의 직관적인 무한소 담론과 학교수학 담론의 commognitive 갈등 사례 연구

This study was conducted to identify high school students' thought of infinitesimals and the resulting commognitive conflict with school mathematics discourse. Data collection was conducted in two stages: an open-ended questionnaire survey and follow-up interviews. The open-ended questionnaire was composed of questions to identify students' thoughts on the existence of infinitesimals and their opinions on conflict situations related to infinitesimals. Interviews were conducted with students who showed infinitesimal opinion in answering open-ended questionnaires. As a result of the study, some students revealed the idea that infinitesimals and infinitely large numbers exist, and showed conflicts by presenting narratives that contradicted the narratives of school mathematics. In addition, some students expressed their disagreement with the meta-rules of school mathematics, which adopts an intuitive explanation in the justification method of 0.999...=1 and the expression of the relationship between points, lines, and planes. The appearance of students experiencing conflict and the phenomenon of refuting and questioning the meta-rules of school mathematics could be interpreted as a commognitive conflict between students' discourse and school mathematics discourse. The specific appearance of commogntive conflict that emerged as a result of this study can serve as basic data for the exploration of teaching and learning methods that help students recognize the need for rigorous mathematical definitions and justifications and promote the transition to new discourse.

Effect of Sophorolipid Adsorption on the Coal Microstructure: Experimental and Wettability Mathematical Model Discussion.

Green biosurfactants are emerging as a promising area of research. However, there is a limited focus on the adsorption and wetting characteristics of biosurfactants on coal dust. This study explores the effects of sophorolipid (SL) biosurfactants on the microstructure and wettability of different coalification degree coal. The microstructure parameters of SL adsorbed on coal dust were measured using a surface tensiometer, contact angle analyzer, and particle size analyzer. The results indicate that SL has the lowest critical surface tension, leading to a 9.25° decrease in the contact angle for low-rank bituminous coal (YZ-LRBC). Furthermore, SL significantly altered the particle size distribution of lignite (NM-LC) and YZ-LRBC. The pore size structure of SL-infiltrated coal dust was quantified using a specific surface area analyzer, revealing a decrease in the specific surface area and an increase in the average pore size. The infrared analysis demonstrated that SL permeation significantly increased the percentage of hydrophilic functional groups (hydroxyl structures) while reducing the hydrophobic functional groups (aliphatic hydrocarbon and aromatic structure). Based on the measured microstructure parameters, a regression equation for contact angle was established: [contact angle (°)] = 73.800 - 0.860 × [D10 (nm)] + 4.280 × [specific surface area (m2/g)]. Notably, the characteristic particle size D10 had a significant negative effect on the contact angle, while the specific surface area had a significant positive effect. These findings provide a theoretical foundation for the application of biosurfactants in water injection to reduce dust and improve the wetting efficiency.

Multimodality and the expression of mathematical ideas through videos by Mathematics undergraduates

The research reported in this article sought to understand how Mathematics undergraduates in Distance Education combine semiotic resources when using videos to express mathematical ideas. Adopting the qualitative methodology, the research was developed with students of the Analytical Geometry discipline and the participant observation in forums of the virtual learning environment was used in the monitoring of the video productions by the participants. Functional Systemic — Multimodal Discourse Analysis supported the analysis of the strategies used in the combinations of semiotic resources in the produced videos. In this article, the analysis of one of the videos produced in the research is discussed. As a result, it was found that the multimodal nature of the video encourages the use of contextualization in the construction of digital mathematical discourse. However, the concepts are formalized from a recontextualization referring to the classroom, which shows the influence of the experience in this environment in the construction of the mathematical discourse shared in the online environment in the context of this research.

Characterizing mathematical discourse according to teacher and student interactions: The core of mathematical discourse

Characterizing mathematical discourse according to teacher and student interactions: The core of mathematical discourse

Professional learning tasks and mathematical knowledge involving the algebraic structure of Groups: an experience in the degree in Mathematics teaching

Considering the teacher education of prospective teachers, the objective of this article is to understand how professional learning tasks are carried out in classes of a course of Algebra in the teacher education. Using Design-Based Research cycles, documents and video recordings were collected throughout the course of the classes. Data were analyzed in order to interpret how future teachers mobilize School-Related Content Knowledge and how teacher educators generate professional learning opportunities for them. The results indicate the possibility of approaching academic mathematics in specific-content disciplines of the teacher education, focusing on teacher training and the changes that future teachers’ knowledge may undergo during training. It is concluded that the use of resources that explore connections between academic and school mathematics, as was the case with formative tasks, seems to be a way to promote mathematical and didactic discussions in the teaching degree.

Developing mathematical whole-class discussions: An exploratory study of teachers’ learning paths

In this study, we aimed to map out teachers' learning in a developmental teacher-researcher collaboration that focused on developing lessons with whole-class discussions in design cycles of preparation, enactment, and reflection. Analysis of data produced during these cycles resulted in detailed learning paths to account for how two teachers' teaching developed toward exemplary lessons in which students presented multiple solution methods and also engaged with each other's thinking. Our findings suggest teachers' learning evolves across multiple settings – including individual lesson preparation and reflection, is personal, and involves three interrelated subpractices: teacher moves, teacher noticing, and teacher mathematical thinking.

Exploring early mathematics through picturebooks: A case study in the context of Head Start

The purpose of the study was to better understand the phenomenon of exploring early mathematics through book reading. The study centers on Head Start and lays on Bronfenbrenner’s bioecological framework). Two sub-questions guided the qualitative single case study of six Head Start adult participants (teachers, parents, administrators): (1) What are Head Start participants’ experiences in exploring early mathematics through picturebooks? and (2) What do Head Start participants say about exploring early mathematics through picturebooks? Findings showed that participants expressed interest toward exploring early mathematics through picturebooks. Picturebooks were commonly used in the classroom but also accessible for families. Participants provided evidence of mathematics practices and discussions around mathematics in the school and in the home. The participants’ sayings and experiences in exploring early mathematics through picturebooks aligned with child development and contexts of learning, two cornerstones of Developmentally Appropriate Practice (DAP), but potential obstacles emerged. Implications for researchers and practitioners are discussed.

Internalism and the Determinacy of Mathematics

Abstract A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse.

Structuralism and informal provability

Mathematical structuralism can be understood as a theory of mathematical ontology, of the objects that mathematics is about. It can also be understood as a theory of the semantics for mathematical discourse, of how and to what mathematical terms refer. In this paper we propose an epistemological interpretation of mathematical structuralism. According to this interpretation, the main epistemological claim is that mathematical knowledge is purely structural in character; mathematical statements contain purely structural information. To make this more precise, we invoke a notion that is central to mathematical epistemology, the notion of (informal) proof. Appealing to the notion of proof, an epistemological version of the structuralist thesis can be formulated as: Every mathematical statement that is provable expresses purely structural information. We introduce a bi-modal framework that formalizes the notions of structural information and informal provability in order to draw connections between them and confirm that the epistemological structuralist thesis holds.

Correlation between parameters in the microstructural vector theory and Hill's plastic potential

The dissipation inequality provides a useful method for specifying constitutive equations of plastic deformation. There are two main ways to use the dissipation inequality to determine the plastic deformation path: by utilizing a plastic potential with the normality rule, or by not utilizing the plastic potential to determine the plastic flow according to the elastic distortion of the material coordinate axis (microstructural vectors). While both methods satisfy the inequality condition and predict experimental results well, they have been developed separately without discussion on relationship. This study presents a physics-based mathematical discussion on the relative dependence between the two methods of determining plastic flow and their point of connection, using Hill's quadratic plastic potential. For this, the conversion of the distortional length and angle changes in microstructural vectors to deviatoric strain tensor under small elastic deformation conditions is mathematically demonstrated. Using these equations, the rate of inelastic deformation derived from microstructural vectors exhibited a similar form to that derived from Hill's quadratic potential. Moreover, the model parameters of the microstructural vectors can be calibrated using the Hill's function. The plastic flow derived from the microstructural vectors and Hill's function exhibited a remarkably similar trend under the assumption of small elastic deformation, although not identical. The newly defined relationships were implemented numerically and validated through simulations. The parameters of the microstructural vector theory derived from Hill's plastic potential can be utilized in materials science and engineering.

UV-based advanced oxidation processes in photoreactors with reflective sleeves

UV-based advanced oxidation processes (UV-based AOPs) via photolysis of radical precursor chemicals (RPCs) are among the most effective technologies for eliminating trace organic contaminants in water treatment facilities. However, mathematical discussions on the kinetics and energy consumption for the engineering photoreactor set-up of UV-based AOPs, especially the newly proposed UV-based AOPs (e.g., UV/persulfate, UV/chlorine and UV/chloramine), are scarce. In this study, a photoreactor with reflective sleeves was mathematically characterized and validated using clean and complex water matrices, and approaches for measuring effective light absorbance by RPC (IRPC) and the reflectance of the photoreactor wall (η) were proposed. By applying IRPC and η, and employing geosmin as a model compound, energy optimization of the UV-based AOPs was conducted based on an updated kinetic model. A new equation was derived for UV/H2O2 to obtain the most cost-effective RPC dose regarding EE/O (electric energy required for one order of contaminant removal) (Copt-EE/O). This equation was then successfully extended to other UV-based AOPs (including UV/persulfate, UV/chlorine, and UV/chloramine). The balance between the treatment goals and operational costs was also discussed. Overall, the new concepts and approaches proposed in this study may aid in the energy optimization and reduce RPC consumption of various UV-based AOPs for engineering applications.

Productive Mathematical Whole-Class Discussions: a Mixed-Method Approach Exploring the Potential of Multiple-Choice Tasks Supported by a Classroom Response System

In recent decades, research has stressed the prominence of mathematics classroom discussions in productive instructional practices in mathematics instruction. In this context, problem-solving activities have been a common focus of research. Research shows that teachers need to deal with prerequisites and challenges such as norms, design of tasks, activating students, and leading students’ discussion to achieve a productive whole-class discussion. However, another promising activity for achieving productive discussions involves using a classroom response system and implementing different task types in a multiple-choice format. There is little knowledge about whole-class discussions using this approach. To meet this need, this paper presents results from a mixed-method approach that characterizes whole-class discussions to explore the potential of multiple-choice tasks supported by a classroom response system to achieve productive whole-class discussions. Three types of multiple-choice tasks were implemented in the classrooms of twelve mathematics teachers at secondary schools. The lessons, including 35 whole-class discussions, were video-recorded, transcribed, and analyzed. The results summarize the characteristics of these whole-class discussions, including measures of students’ opportunities to talk and teacher and student actions. These results can help us develop a more profound understanding of whether and how multiple-choice tasks supported by a classroom response system can support teachers in achieving productive whole-class discussions.

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.

Mathematical foundation, discussion and suggestion on penalty parameter setting of penalty-based boundary intersection method for many-objective optimization problems

Mathematical foundation, discussion and suggestion on penalty parameter setting of penalty-based boundary intersection method for many-objective optimization problems

Households and 1-World Economy: A Mathematical Discourse

We examine several options available for households to withstand the challenges posed by long held positions about nation-state macroeconomics and mathematically argue outlets for showcasing the foundational essence of household economy in the coming mainstreaming of 1-World economy without the necessary drawbacks of the failing nation-state macroeconomics which its attendant inadequacies and inefficiencies. We brought out the core importance of the zeroing in on household economics as we evolve further virtual economics as these are the main frames as regards our quest for survival of the human race, and showed that the household must be the core for this new world to work. This paper suggests perhaps another perspective to the other known conventional systems that can be used to assess the impact of internet enabled economic activities on households within a setting without economic borders or barriers. Finally, we relate two the concepts to the world economy.

From Collaborative Construction, Through Whole-Class Presentation, to a Posteriori Reflection: Proof Progression in a Topology Classroom

Coming from a social perspective, we introduce a classroom organizational frame, where students’ proofs progress from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. This design is informed by a view on proofs as successive social processes in the mathematics community. To illustrate opportunities for mathematics learning of proof progressions, we present a commognitive analysis of a single proof from a small course in topology. The analysis illuminates the processes through which students’ proof was restructured, developed previously unarticulated elements, and became more formal and elaborate. Within this progression, the provers developed their mathematical discourses and the course teacher seized valuable teachable moments. The findings are discussed in relation to key themes within the social perspective on proof.

Comparing metrics for mixed quantum states: Sjöqvist and Bures

It is known that there are infinitely many distinguishability metrics for mixed quantum states. This freedom, in turn, leads to metric-dependent interpretations of physically meaningful geometric quantities such as complexity and volume of quantum states. In this paper, we first present an explicit and unabridged mathematical discussion on the relation between the Sjoqvist metric and the Bures metric for arbitrary nondegenerate mixed quantum states, using the notion of decompositions of density operators by means of ensembles of pure quantum states. Then, to enhance our comprehension of the difference between these two metrics from a physics standpoint, we compare the formal expressions of these two metrics for arbitrary thermal quantum states specifying quantum systems in equilibrium with a reservoir at non-zero temperature. For illustrative purposes, we show the difference between these two metrics in the case of a simple physical system characterized by a spin-qubit in an arbitrarily oriented uniform and stationary external magnetic field in thermal equilibrium with a finite-temperature bath. Finally, we compare the Bures and Sjoqvist metrics in terms of their monotonicity property.

The History of Fractions and Teaching Insights

The history of mathematics, as the history of mathematical development, is a guide for teaching and has significant implications for contemporary teaching. If teachers can use the resources of the history of mathematical discourse wisely in teaching, they can enable students to experience the wisdom of their predecessors and the charm of mathematical culture in the process of learning new knowledge. As early as around 3000 B.C., human beings began to understand fractions. Since then, the representation of fractions, the application of fractions, and the perspective of understanding fractions have been constantly improved, which have profound implications for the teaching of fractions.

A conceptual synthesis on approximations of practice in mathematics teacher education

ABSTRACT Approximations of practice are opportunities for preservice teachers (PSTs) to engage in components of teaching. However, how the varying levels of complexity or authenticity of teaching create different learning opportunities for PSTs is underexplored. Forty-one studies related to approximations of mathematics teaching were reviewed to explore how approximations are embedded in PST education, how authenticity represents itself in approximations, and how mathematics teacher educators report approximations facilitating PSTs’ learning. The review identified eight types of approximations used, indicating that PSTs developed aspects of specific practices such as professional noticing, conducting a meaningful mathematics discussion, as well as generic lesson planning and implementation. We propose teacher education programs consider a continuum of authenticity to capture the complexity of approximations of practice as well as the recomposition of decomposed practices. We discuss how varying levels of authenticity in approximations of practice creates different opportunities for PSTs.

Investigating the Mathematical Discourses of Secondary School 8th Grade Students: Case of Linear Equations

Cebir öğretiminde denklem sistemlerinin, kavramsal olarak zorluğu ve karmaşık yapısının dikkate alınması gerekmektedir. Diğer taraftan, öğrencileri kuralları ezberlemeye teşvik eden, çoğunlukla uygulamaya dönük, gündelik hayatla ilişkili bulunmayan, yalnızca sembol ve işlemi kapsayan uygulamaları içeren cebir öğretimi, bu yapının açığa çıkmasında engel oluşturmaktadır. Doğrusal denklemler kapsamında öğrencilerin gelişimsel öğrenmelerini ve öğretmenlerin doğrusal denklemler konusunun öğretimini sosyo-kültürel bakış açısından ele alan öğrenme ile öğretimi birlikte inceleyen çalışmaların gerekliliği öngörülmektedir. Bu doğrultuda araştırmada, ortaokul öğrencilerinin doğrusal denklemlerle ilgili söylemlerinin değişimini ve öğretimle olan bağını ortaya çıkarmak hedeflenmiştir. Bu hedef doğrultusunda yedi öğrenciden oluşan katılımcılarla yarı yapılandırılmış klinik görüşme ve sınıf gözlemleri gerçekleştirilmiştir. Bulgular değerlendirildiğinde öğrencilerin doğrusal denklemlerle ilgili matematiksel söylemleri oluştururken güçlük çektikleri belirlenmiştir.