Mathematical Discourse Research Articles

Two contrasting and complementary views of Göttingen: letters from the first female Norwegian Ph.D. in mathematics, Elizabeth Stephansen, to her compatriot Carl Størmer in the winter semester of 1902/3

Late in 1902 and early in 1903 Elizabeth Stephansen, the first female Norwegian Ph.D. in mathematics (1902), wrote two letters from Göttingen to her colleague and compatriot Carl Størmer. The letters can be partly read as a reaction to some criticism which Størmer had made of the mathematical atmosphere in Göttingen before he left in December 1902 after a stay of only two months. The letters are discussed in this paper and published in the appendix in their entirety both in original Norwegian and in English translation. The article provides an insight into some aspirations and social conditions in Scandinavian, in particular Norwegian, mathematics at the beginning of the twentieth century. It also helps to broaden and deepen the historical discussion of mathematics in Göttingen around the turn of the twentieth century.

The Effect of Training Program Based on Instructional Practices According to the (NCTM) Standards on Improving the Pedagogical Knowledge among Mathematics Teachers in Jordan

The current study aims at revealing the effect of a training program based on teaching practices according to NCTM standards in improving the pedagogical knowledge of mathematics teachers. The study followed the semi- experimental method. To achieve the aims of the research, a training program that is based on the standards of NCTM was designed. The program consists of eight packages which are (Establish mathematics goals to focus learning, implement tasks that promote reasoning and problem solving, Use and connect mathematical representations, facilitate meaningful mathematical discourse, pose purposeful questions, build procedural FLuency from conceptual understanding, support productive struggle in learning mathematics, Elicit and use evidence of student thinking). After confirming the validity of the tool, a sample of (20) male and female teachers from Deir Ala Directorate of Education in the academic year 2020-2021.To reveal the effectiveness of the program, a test in pedagogical knowledge was prepared. The test consists of six respects which are identification of mathematics content, the pedagogical knowledge of the mathematical content and knowing students' thinking. The test validity and reliability were checked. The results of the study showed that there were statistically significant differences between measuring averages in the post application of the test. These results were pro- sample in all dimensions and in the total degree. Depending on these results, the study asserted the necessity for the training department in the Jordanian Ministry of Education to benefit from the suggested and used program in the study to focus more on the teaching practices.

Mathematics discourse in instruction during functions lessons in grade 10 rural classrooms

Despite the global significance of functions in mathematics education, there has been limited research on how South African rural teachers address specific mathematical concepts related to functions. This qualitative multiple case study, conducted within a post-structural paradigm, delved into the discourses of Grade 10 teachers during functions lessons in rural classrooms. The study focused on the teaching of mathematical concepts related to functions and involved five mathematics teachers from rural school sites in Acornhoek, Mpumalanga Province, South Africa. These teachers were selected based on their experience in teaching mathematics. The research employed semi-structured interviews, unstructured classroom observations, and Video-Stimulated Recall Interviews with all five teachers. By narrowing the focus to one teacher's lesson, this paper analyses the presentation of functions concepts using the Mathematical Discourse in Instruction (MDI) framework. The study revealed that clear teaching materials and learner-engaging instructional methods enable meaningful engagement and concept internalisation. Through the visible use of the components of the MDI framework, the teacher successfully engaged learners, clarified misconceptions, and encouraged learners' participation, ultimately enhancing the coherence and understanding of mathematical concepts related to functions in the Grade 10 classroom.

Cultivating Students’ Agency by Infusing Number Talks with Notice and Wonder

ABSTRACT Instructional routines, like Number Talks and Notice and Wonder, engage students in doing and then talking about their mathematical thinking. Number Talk routines are short mathematical discussions where students mentally solve and share the strategies they used to make sense of the problem. In comparison, the Notice and Wonder routine asks students to reflect on a high-interest visual prompt with multiple entry points, share their noticings and wonderings, and then the teacher steers the discussion toward a mathematical question to explore further. But what happens when the Notice and Wonder routine is used in conjunction with a Number Talk? In this descriptive case study, we analyze whether and how this variation of a Number Talk provides a class of third-grade students opportunities to exercise agency. Findings from this study yielded two major themes: 1) agency is built when authority is shared during a mathematical discussion, and 2) Notice and Wonder is an instructional routine that supports sharing authority. We then discuss why this matters and next steps for research and practice.

Towards mmWave Altimetry for UAS: Exploring the Potential of 77 GHz Automotive Radars

Precise altitude data are indispensable for flight navigation, particularly during the autonomous landing of unmanned aerial systems (UASs). Conventional light and barometric sensors employed for altitude estimation are limited by poor visibility and temperature conditions, respectively, whilst global positioning system (GPS) receivers provide the altitude from the mean sea level (MSL) marred with a slow update rate. To cater to the landing safety requirements, UASs necessitate precise altitude information above ground level (AGL) impervious to environmental conditions. Radar altimeters, a mainstay in commercial aviation for at least half a century, realize these requirements through minimum operational performance standards (MOPSs). More recently, the proliferation of 5G technology and interference with the universally allocated band for radar altimeters from 4.2 to 4.4 GHz underscores the necessity to explore novel avenues. Notably, there is no dedicated MOPS tailored for radar altimeters of UASs. To gauge the performance of a radar altimeter offering for UASs, existing MOPSs are the de facto choice. Historically, frequency-modulated continuous wave (FMCW) radars have been extensively used in a broad spectrum of ranging applications including radar altimeters. Modern monolithic millimeter wave (mmWave) automotive radars, albeit designed for automotive applications, also employ FMCW for precise ranging with a cost-effective and compact footprint. Given the technology maturation with excellent size, weight, and power (SWaP) metrics, there is a growing trend in industry and academia to explore their efficacy beyond the realm of the automotive industry. To this end, their feasibility for UAS altimetry remains largely untapped. While the literature on theoretical discourse is prevalent, a specific focus on mmWave radar altimetry is lacking. Moreover, clutter estimation with hardware specifications of a pure look-down mmWave radar is unreported. This article argues the applicability of MOPSs for commercial aviation for adaptation to a UAS use case. The theme of the work is a tutorial based on a simplified mathematical and theoretical discussion on the understanding of performance metrics and inherent intricacies. A systems engineering approach for deriving waveform specifications from operational requirements of a UAS is offered. Lastly, proposed future research directions and insights are included.

An energy estimate and a stabilized Lagrange–Galerkin scheme for a multiphase flow model

Multiphase flow models are commonly employed for understanding complex fluid flows, while few mathematical discussions exist. For a general multiphase flow model in Gidaspow (1994), an energy decay property is proved. A stabilized Lagrange–Galerkin scheme for the model and its stability property are presented. Here, a hyperbolic tangent transformation is employed to preserve the boundedness of the volume fraction. A novel artificial term is added to obtain the stability property. Two-dimensional numerical examples exhibit the experimental order of convergence and applicability in modelling sedimentation phenomena.

Transforming sociomathematical norms in a South African grade 11 classroom to enhance learners’ mathematical proficiency

In this paper we use parts of qualitative data from the first author’s doctoral study to explore how transforming existing sociomathematical norms enhance learners’ mathematical proficiency. The study was conducted in a grade 11 mathematics classroom comprising of 23 learners, facilitated by the first author as learners engaged in a mathematical discourse on analytical geometry. Data were gathered through video recording, documents and researcher journal. We adopted Yackel and Cobb’s (1996) interpretive framework and Kilpatrick et al.’s (2001) notion of mathematical proficiency as lenses, which guided the data analysis. We analyzed the data following Polkinghorne’s (1995) narrative analysis method. We found that transforming existing sociomathematical norms to enhance learners’ mathematical proficiency involved a three-stages process: negotiating entry into learners’ existing sociomathematical norms, disrupting learners’ existing sociomathematical norms and constituting ‘new’ sociomathematical norms. As learners developed new taken-as-shared meanings regarding acceptable mathematics explanations, justifications and mathematically different solutions they enhanced their <i>conceptual understanding, procedural fluency, adaptive reasoning</i>,<i> </i>and<i> strategic competence</i>.

Investigating the relationship between math literacy and linguistic synchrony in online mathematical discussions through large‐scale data analytics

AbstractPrevious literature has associated math literacy with linguistic factors such as verbal ability and phonological skills. However, few studies have investigated linguistic synchrony, shown in mathematical discussions. This study modelled math literacy and examined the relationship of math literacy with linguistic synchrony between students and facilitators. We retrieved data from 20,776 online mathematical discussion threads at a secondary school level. First, we assessed students' math literacy based on their discussions and classified them into high‐ and low‐math literacy groups. Then, we conducted Cross‐Recurrence Quantification Analysis (CRQA) to calculate linguistic synchrony within each thread. The result implies that students with high math literacy are more likely to share common words (eg, mathematical terms) with facilitators. At the same time, they would paraphrase the facilitators' words rather than blindly mimic them as the exact sentences or phrases. On the other hand, students with low math literacy tend to use overlapping words with facilitators less frequently and are more likely to repeat the exact same phrases from the facilitators. The findings provide an empirical data analysis and insights into mathematical discussions and linguistic synchrony. In addition, this paper implies the directions to improve online mathematical discussions and foster math literacy. Practitioner notesWhat is already known about this topic Mathematical discussions are known to be an effective way to promote math literacy. Math literacy and linguistic skills have a strong link. Linguistic synchrony is related to better collaboration and common knowledge building. What this paper adds Reveals the relationship between math literacy and linguistic synchrony and deepens the understanding of digital communication in online learning environments. Provides empirical analysis of natural language data in group discussions using CRQA. Conceptualizes linguistic synchrony with three sub‐concepts: linguistic concurrence, predictability, and complexity. Implications for practice and/or policy Educators and practitioners could utilize the automatic formative assessment of math literacy based on the student's language use in mathematical discussions. Educational technology researchers and designers could include CRQA indices and recurrence plots in the dashboard design to provide information to support teachers and learners. Teachers would be able to provide real‐time interventions to promote effective mathematical communication and foster math literacy throughout mathematical discussions.

Intelligent Networking for Energy Harvesting Powered IoT Systems

As the next-generation battery substitute for IoT system, energy harvesting (EH) technology revolutionizes the IoT industry with environmental friendliness, ubiquitous accessibility, and sustainability, which enables various self-sustaining IoT applications. However, due to the weak and intermittent nature of EH power, the performance of EH-powered IoT systems as well as its collaborative routing mechanism can severely deteriorate, rendering unpleasant data package loss during each power failure. Such a phenomenon makes conventional routing policies and energy allocation strategies impractical. Given the complexity of the problem, reinforcement learning (RL) appears to be one of the most promising and applicable methods to address this challenge. Nevertheless, although the energy allocation and routing policy are jointly optimized by the RL method, due to the energy restriction of EH devices, the inappropriate configuration of multi-hop network topology severely degrades the data collection performance. Therefore, this article first conducts a thorough mathematical discussion and develops the topology design and validation algorithm under energy harvesting scenarios. Then, this article develops DeepIoTRouting , a distributed and scalable deep reinforcement learning (DRL)-based approach, to address the routing and energy allocation jointly for the energy harvesting powered distributed IoT system. The experimental results show that with topology optimization, DeepIoTRouting achieves at least 38.71% improvement on the amount of data delivery to sink in a 20-device IoT network, which significantly outperforms state-of-the-art methods.

Exploring and developing a framework for analysing whole-class discussions in mathematics

ABSTRACT Leading whole-class discussions in mathematics instruction in a productive way is identified as a core practice. However, productive discussions are rarely seen in classrooms. Thus, we need more knowledge about how different teacher actions in response to students’ comments can affect the students’ learning. If research is to be able to produce such knowledge, detailed analytical frameworks and concepts are required. Thus, this study sought to propose concepts and refine definitions of existing categories of teacher actions to help develop Drageset’s framework ([2014]. Redirecting, progressing, and focusing actions – a framework for describing how teachers use students’ comments to work with mathematics. Educational Studies in Mathematics, 85(2), 281–304.). This was done by analysing 35 mathematical whole-class discussions, generated from multiple-choice tasks supported by classroom response systems, in secondary schools in Sweden and the UK. The results suggest five new categories of teacher actions and enrichments of two existing categories in Drageset’s framework.

Interactional practices of inviting minoritized students to whole-class mathematics discussions

Interactional practices of inviting minoritized students to whole-class mathematics discussions

Rituals and Explorations in Students’ Mathematical Discourses: The Case of Polynomial Inequalities

The study is a qualitative case study that seeks to determine whether students’ mathematical discourses in solving polynomial inequalities are more ritualistic or explorative. A comprehensive analysis of students’ routines was conducted through the observations of what they said and did (write, draw, and so on) around task situations in a small group. This study’s participants were five 11th-grade students from a public high school. These participants were chosen using the maximum diversity method of sampling. The data for this study were obtained through small-group work. The small-group interactions lasted 80 minutes and were video-recorded with two cameras. The commognitive approach was used to analyze the student routines in this study. The criteria for analyzing routines were the performers’ agentivity /external authority, focus on the goal or the procedure, and flexibility. The findings of this study revealed that the students’ routines were neither purely ritualistic nor sheer explorative. Even those whose routines were ritualistic in all task situations thought about the procedure and asked logical questions about the task. In addition, the findings indicate that teachers can play an important role in encouraging students to engage in more exploratory mathematical discourse. This study contributes to the future research on students’ discourse in the context of inequality.

EKONOMI MATEMATIKA: BAGAIMANA FUNGSI LINEAR MENENTUKAN KESIMBANGAN HARGA PASAR

The primary objective of this scientific inquiry is to thoroughly investigate and expound upon the practical application and implementation of linear function equations within the domain of economic mathematics. The central focus lies in dissecting the intricate modeling of market equilibrium prices, with a specific emphasis on the assessment of consumer and producer surpluses associated with essential commodities, such as rice. The core methodology of this study involves an in-depth literature review, which serves as the backbone for understanding the nuances of market equilibrium price models and the nuanced application of linear equations within economic mathematics. The economic and business disciplines are often considered semi-precise sciences, demanding a nuanced blend of qualitative and quantitative analyses to achieve a comprehensive understanding of their multifaceted dynamics. In this context, quantitative analysis in economics and business invariably necessitates the utilization of mathematics as an indispensable tool. Hence, this study consciously adopts a quantitative research approach, a methodological process rooted in the systematic acquisition of knowledge using numerical data as a tool to dissect and analyze crucial information relevant to the subject matter. Delving into the intricacies of supply and demand laws through the lens of linear functions, the study ultimately concludes that the adoption of mathematical models grounded in linear functions holds the potential to effectively determine the demand and supply functions, especially concerning staple commodities like rice. Moreover, it proves instrumental in identifying and comprehending market equilibrium, furnishing invaluable insights that can significantly contribute to a nuanced understanding and effective management of economic phenomena. In essence, this research aims not only to contribute to the theoretical discourse in economic mathematics but also to provide practical insights that can inform decision-making processes within the realms of economics and business.

학교수학에서 무리수 도입으로 인한 담론 변화 분석

This study examined the method of discovering incommensurable quantities, the origin of irrational numbers, and the changes in Greek mathematics resulting from the discovery of these quantities from Sfard's commognitive perspective. In addition, in light of the history of mathematics, this study looked into changes in the discourse of school mathematics due to the introduction of irrational numbers. As a result of the analysis, the discovery of incommensurable quantities in the history of mathematics was an opportunity to change from empirical mathematics to formal and theoretical mathematics, and was an event that brought about changes in various mathematical concepts, proportion theory and roles of mathematical tools. In other words, the discovery of incommensurable magnitude transformed Greek mathematics into an incommensurable discourse. These changes in the history of mathematics appear similarly in the changes in the discourse of school mathematics due to the introduction of irrational numbers. The introduction of irrational numbers into school mathematics resulted in the application of formal definitions of numbers, changed the concepts of number and area, and led to the emergence of narratives that seemed to contradict previous discourses. It is understood that the introduction of irrational numbers has changed the discourse of school mathematics into a discourse that is incommensurable with the previous one. The analysis of this study could be basic data for the design of teaching and learning methods for students facing the task of transitioning to the incommensurable discourse of irrational number learning.

Impediments to learning problem-solving in Malawian lower secondary mathematics textbooks

Background: In a culture where teachers follow the textbook prescriptively, Malawian students perform low in mathematics, and no students reach the problem-solving levels.Aim: To explore reasons for students’ low performance, this study aims at investigating opportunities to learn problem-solving in Malawian mathematics textbooks.Setting: This study focuses on Malawian mathematics textbooks in the lower secondary grades focusing on the areas of linear equations and simultaneous linear equations. These areas have a particular emphasis on problem-solving.Methods: Four textbooks from two of the most widely used series of mathematics textbooks in Malawian secondary school were analysed. The Mathematics Discourse in Instruction framework was used to analyse examples and tasks in the four textbooks.Results: Analysis indicates that the textbooks provide relatively few opportunities to learn problem-solving, and most of the opportunities are given through word problems. These word problems are typically presented towards the end of the chapters, and students are thus stimulated to apply already learned procedures to solve the problems rather than learn through problem-solving.Conclusion: Limitations in opportunities to learn problem-solving are particularly challenging in a context like Malawi, where teacher–textbook compliance is high, where there is a shortage of qualified mathematics teachers and where few students have access to their own textbook.Contribution: This study provides an overview of impediments to learning problem-solving in Malawian mathematics textbook, and knowledge about such impediments is necessary for change.

Let’s Give Them Something to Talk About

Different types of open tasks can be used as a tool to promote rigorous student mathematical discourse and considerations for facilitation.

Peer-Delivered Modified Schema-Based Instruction in Word Problem-Solving for High-School Students with Intellectual Disability

There is a growing body of evidence on the effectiveness of modified schema-based instruction (MSBI) for teaching word problem-solving alongside other important skills, such as self-determination and mathematical discourse and reasoning. Yet the majority of studies have not been conducted in naturalistic instructional arrangements. We report single-case design quantitative data from a mixed methods study that sought to evaluate the effectiveness and social validity of peer-delivered MSBI with high school students who were engaged in an established peer mentoring program. Results from the single-case multiple probe across participants design demonstrate a functional relation between peer-delivered MSBI and math problem-solving behaviors. All mentees with intellectual disability independently solved multiplicative comparison word problems following intervention and most maintained and generalized to a socially significant degree. Mentors delivered the intervention as designed and supported their mentees in expressing reasoning to justify their solutions. We provide recommendations for research and practice aimed at increasing expectations and opportunities for secondary students with intellectual disability regarding the use of peer-mediated interventions for academic skills.

Pedagogical listening: understanding how teachers listen to student struggle during mathematical sense-making discussions

ABSTRACT Research suggests that students’ verbalisation of how they are struggling to understand something new is important for learning mathematics with conceptual understanding. However, less is known about how teachers listen while students ‘think aloud’ through struggle. In this study, we sought to answer the following research question: What types of listening do teachers enact when students are verbalising struggle during mathematical sense-making discussions? We detail how we created and applied a Framework for Pedagogical Listening, which extends previous theoretical and empirical research on teacher listening to identify and differentiate between five types of teacher listening: empathic, supportive, educative, self-reflective and generative. Our study involved nine teachers and their students in the US and Scotland, contexts which are focused on reform-efforts towards inquiry-oriented mathematics instruction that engages students in sense-making discussions. Our findings suggest that the five types of teacher listening in our framework are present when students verbalise struggle during sense-making discussions, and that our Pedagogical Listening Framework is a useful tool for identifying and documenting the complex ways teachers listen when students verbalise struggle. We present three vignettes of classroom interaction during mathematical discussion that illustrate the five pedagogical listening types.

Facing and challenging language ideologies towards a more inclusive understanding of language in mathematics education research—the case of sign languages

Research on language in mathematics education is largely dominated by a ‘normalcy’ of spoken languages. This modal hegemony does not only affect a whole group of learners in failing to provide access that is epistemologically equitable—those using sign language as their preferred mode for mathematical discourse—it also obscures our view on the roles language can play in mathematical thinking and learning. As a field, we can only win from seeking to understand Deaf learners of mathematics beyond a disability, as learners of mathematics with a specific linguistic background that influences mathematical thinking and learning in peculiar ways. In this contribution, I suggest a shift in mindset towards a more inclusive view on language in mathematics education research and practice. I propose basic principles to inform a perspective for reconsidering the role of language in mathematics thinking and learning, inspired by work of philosopher Francois Jullien. This perspective counters a perspective that merely integrates sign language into existing research and instead searches for dialogue between linguistic modalities in learning mathematics, looking beyond language as spoken or written. This approach will be illustrated by the case of the modal affordance of iconicity foregrounded in signed mathematical discourse, its role in Deaf students’ mathematics thinking and learning, and how this can inform existing research and practice dealing with language in mathematics education.

Detection of Banking Financial Frauds Using Hyper-Parameter Tuning of DL in Cloud Computing Environment

When income, assets, sales, and profits are inflated while expenditures, debts, and losses are artificially lowered, the outcome is a set of fraudulent financial statements (FFS). Manual auditing and inspections are time-consuming, inefficient, and expensive options for spotting these false statements. Auditors will find great assistance from the use of intelligent methods in the analysis of several financial declarations. Now more than ever, victims of financial fraud are at risk since more and more individuals are using the Internet to conduct their financial transactions. And the frauds are getting more complex, evading the protections that banks have put in place. In this paper, we offer a new-fangled method for detecting fraud using NLP models: an ensemble model comprising Feedforward neural networks (FNNs) and Long Short-Term Memories (LSTMs). The Spotted Hyena Optimizer is a unique metaheuristic optimization technique used to choose weights and biases for LSTM (SHO). The proposed method takes inspiration from the law of gravity and is meant to mimic the group dynamics of spotted hyenas. Mathematical models and discussions of the three fundamental phases of SHO — searching for prey, encircling prey, and at-tacking prey — are presented. We build a model of the user’s spending habits and look for suspicious outliers to identify fraud. We do this by using the ensemble mechanism, which helps us predict and make the most of previous trades. Based on our analysis of real-world data, we can confidently say that our model provides superior performance compared to state-of-the-art approaches in a variety of settings, with respect to both precision and.