Mathematical Arguments Research Articles

CONTENT ANALYSIS OF STUDENTS' ARGUMENTS BASED ON MATHEMATICAL LITERACY AND CREATION ABILITY

This study aims to show the content of students' argumentation in building their creative reasoning and the relationship between argumentation content and students' mathematical literacy skills. The type of research applied is descriptive qualitative research involving 27 junior high school students in Toli-Toli City, Central Sulawesi. The research instrument used was a like-mathematical literacy test and an assessment rubric. The collected data were analyzed using qualitative content analysis which was used to analyze text data. The results of the research show that 1) The content of students' arguments in building their creative reasoning is supported by using simple statements, complex statements, completed by syntax or no syntax in building mathematical arguments, and 2) the ability of students to show their argumentation in the problem solving is the level of their capacity to interpret and to represent their knowledge and learning experience related with the problem. The breadth and depth of content of students' mathematical literacy give them the flexibility to argue. Students' mathematical literacy ability by simply giving simple statements has a different breadth and depth than those who are able to give complex statements. Likewise with the shrewdness of using syntax in constructing problem-solving plans.

LINGUISTIC ASPECT OF MODALITY IN MODERN MATH DISCOURSE IN ENGLISH

The article reveals linguistic aspect of modality in modern math discourse in English, critically outlines a number of actual problematic issues in the area, such as the distinction between epistemic modality and evidentiality marked by formal logics philosophical grounding. General reference to previous scholarly activity in math modality research proves that it is largely based on propositional aspects of meaning. The math text corpus analysis aims to extract a set of modalities that are indispensable for formulating modal deductive reasoning. However, from a linguistic perspective academic math discourse requires natural language premise selection in the processes of mathematical reasoning and argumentation. It is presumed that two different self-attention cognition layers are focused at the same time on the proper classical symbolic logic and mathematical elements (formal language), while the other attends to natural language. Defining the semantic meanings of math discourse modality markers involves the interpretation phase. Thus, objectivity is generally associated with evidential adverbs which are markers of the evidence verification concerning the speaker’s assessment of the truth value of the proposition. Modal auxiliaries of high, medium and low modality, semimodal verbs and conditionals involve ascribing a justification value in the set of possible logical inference making. The formal logical structure of mathematical reasoning explains the non-intuitive possibility of a deductive proof. It has been grounded that a linguistic category of modality in math discourse indispensably presupposes the universal truth of knowledge, high level of logical formalization in propositional verification status, formulaic nature of the argumentation, i.e. synthesis of hypothetical preconditions, theoretical knowledge and subjectivity of reasoning leading to a new hypothesis verification and visual exemplification of the empirical deductive processes in particular by linguistic means of modality expression.

Understanding and Appreciation of the Concept of Shirk according to Mathematical Theory

Shirk means associating any partners with Allah SWT. Tawhid clearly teaches us that Allah SWT does not share His Rububiah (Lordship) or divine attributes with anything or any partner. A clear example of shirk is idolatry or polytheism, that is worship of deity, god or anything other than Allah. Shirk is a paramount sin in Islam, the one unforgivable sin. However, this term is in the holy verses of the Qur'an that are "difficult" for humans to understand and appreciate. Usually, the interpretation of the word shirk will be given a description of its meaning and consequences to the perpetrator. The common interpretation is that whoever associates anything with Allah, then he has indeed committed a great sin. Muslims are also warned that indeed Allah SWT will not forgive the sin of shirk while Allah SWT will forgive all sins other than shirk, for whom he wills. In effect, all doors of paradise are shut to those who commit shirk. This article is related to the debate on shirk and its significance according to Islamic law and mathematical measurements. The analysis and findings of the study in this article are obtained through the methods of content analysis, document analysis and vectors concept in mathematics. The vectors concept in mathematics has been chosen as a tool to decipher this term to the maximum extent possible. It was chosen because a vector has elastic properties, is easy to set up and is representative of the matter relating to shirk. Therefore, it can be used to explain the definition, interpretation and consequences of shirk. The results of the study show that shirk is not only placed in the category of major sins in Islamic law but also the position of the perpetrators of shirk is considered as insulting Allah SWT and the religion of Islam. This position can not only be assessed from the Islamic law viewpoint but also through mathematical measurements, formulas and arguments. Through mathematical measurements it is impossible for us to assign partners to Allah, except by those who are very stubborn and intend to insult Allah. In this paper, the arguments are given based on mathematical knowledge that indeed human beings should not commit shirk not only because of its consequences, but also because it involves Allah as the Creator of all creatures as well as all other creation.

A Relativistic Theory of Consciousness

In recent decades, the scientific study of consciousness has significantly increased our understanding of this elusive phenomenon. Yet, despite critical development in our understanding of the functional side of consciousness, we still lack a fundamental theory regarding its phenomenal aspect. There is an “explanatory gap” between our scientific knowledge of functional consciousness and its “subjective,” phenomenal aspects, referred to as the “hard problem” of consciousness. The phenomenal aspect of consciousness is the first-person answer to “what it’s like” question, and it has thus far proved recalcitrant to direct scientific investigation. Naturalistic dualists argue that it is composed of a primitive, private, non-reductive element of reality that is independent from the functional and physical aspects of consciousness. Illusionists, on the other hand, argue that it is merely a cognitive illusion, and that all that exists are ultimately physical, non-phenomenal properties. We contend that both the dualist and illusionist positions are flawed because they tacitly assume consciousness to be an absolute property that doesn’t depend on the observer. We develop a conceptual and a mathematical argument for a relativistic theory of consciousness in which a system either has or doesn’t have phenomenal consciousness with respect to some observer. Phenomenal consciousness is neither private nor delusional, just relativistic. In the frame of reference of the cognitive system, it will be observable (first-person perspective) and in other frame of reference it will not (third-person perspective). These two cognitive frames of reference are both correct, just as in the case of an observer that claims to be at rest while another will claim that the observer has constant velocity. Given that consciousness is a relativistic phenomenon, neither observer position can be privileged, as they both describe the same underlying reality. Based on relativistic phenomena in physics we developed a mathematical formalization for consciousness which bridges the explanatory gap and dissolves the hard problem. Given that the first-person cognitive frame of reference also offers legitimate observations on consciousness, we conclude by arguing that philosophers can usefully contribute to the science of consciousness by collaborating with neuroscientists to explore the neural basis of phenomenal structures.

K‐fold cross‐validation for complex sample surveys

Although K‐fold cross‐validation (CV) is widely used for model evaluation and selection, there has been limited understanding of how to perform CV for non‐iid data, including those from sampling designs with unequal selection probabilities. We introduce CV methodology that is appropriate for design‐based inference from complex survey sampling designs. For such data, we claim that we will tend to make better inferences when we choose the folds and compute the test errors in ways that account for the survey design features such as stratification and clustering. Our mathematical arguments are supported with simulations, and our methods are illustrated on real survey data.

MP10-14 MODELING THE UROLOGY RESIDENCY APPLICATION AND MATCH PROCESS: A MATHEMATICAL ARGUMENT FOR APPLICATION AND INTERVIEW LIMITS

You have accessJournal of UrologyCME1 May 2022MP10-14 MODELING THE UROLOGY RESIDENCY APPLICATION AND MATCH PROCESS: A MATHEMATICAL ARGUMENT FOR APPLICATION AND INTERVIEW LIMITS Sameer Farooq, and Akanksha Mehta Sameer FarooqSameer Farooq More articles by this author , and Akanksha MehtaAkanksha Mehta More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000002532.14AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: The competitive nature of the Urology match has led to an untenable environment for applicants, who feel pressured to apply to >70 programs, and for residency programs, which receive hundreds of applications for a limited number of training positions. We used a mathematical model to investigate the effect of an application and interview cap on the urology match success rate. METHODS: Using 2020-2021 AUA data, we created a mathematical model in MATLAB that simulated 481 applicants applying to a total of 357 training positions distributed across 143 urology residency programs. Applicants were divided into top (1/5), middle (3/5), and bottom (1/5) strata. Programs were divided into three equally sized strata (1/3, 1/3, 1/3) based on Doximity rankings. The goal was to optimize the match rate and fill all available training positions.We made the following assumptions:1) all programs stratify applicants in the same way, and vice versa 2) all applicants/programs within a specific stratum are considered equivalent 3) all applicants apply to the maximum number of programs.In the model, we modified the maximum number of applications and interviews per applicant, created pseudo-random rank lists that preferentially ordered applicants and programs based on top >middle >bottom strata, and ran a simulated match. Each scenario was repeated 1000 times. RESULTS: Application caps and interview caps for applicants were associated with programs’ ability to interview enough candidates to fill all available training spots. However, this effect disappeared at an application cap of 50 and interview cap of 20When looking at match results with an interview cap of 20, the preference distribution of applicants remained stable when changing the application cap, indicating that above a certain threshold, more applications do not provide additional benefit. Conversely, as the caps were further reduced, the preference distribution was negatively affected. CONCLUSIONS: Based on the results of our simulated model, match parameters were optimized at a maximum of 50 applications and 20 interviews per applicant. Such limits would reduce cost and time burden for both programs and applicants and likely result in no deleterious effects, while still reducing average applications by at least 35% from the 2020-2021 cycle. Source of Funding: None © 2022 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 207Issue Supplement 5May 2022Page: e152 Advertisement Copyright & Permissions© 2022 by American Urological Association Education and Research, Inc.MetricsAuthor Information Sameer Farooq More articles by this author Akanksha Mehta More articles by this author Expand All Advertisement PDF DownloadLoading ...

Flood frequency analysis – Part 2: Development of a modified plotting position

The original plotting position concept was suggested more than a century ago. Since then, many alternative plotting position approaches have been developed. Despite a general lack of agreement around which plotting position is theoretically ‘correct’ and the ‘best’ to use, all plotting positions fail to adequately address outliers and data of similar magnitude. Hydrologists generally fail to acknowledge that the plotting position primarily offers an informative display of data, against which distributions can be compared, rather than an absolute measure of probability. This paper does not intend to challenge any of the many lengthy theoretical mathematical arguments, utilised to ‘prove’ why one plotting position is superior to the others. These theoretical arguments may very well be valid for a ‘population’ of flood peaks – the reality, however, is that hydrologists are confronted with the challenge of analysing very limited ‘samples’ of the population. Consequently, the plotting position issue demands a more pragmatic approach, rather than a purely theoretical approach. This paper illustrates various problems with existing plotting position techniques in use and offers an alternative approach and a more sensible plotting position technique, using Z-scores and referred to as the Z-set PP, against which distributions can be checked. The study further illustrates how effectively the Z‑set PP deals with outliers and its robustness with various record lengths. Although derived from a study of flood peak data obtained from South African flow-gauging sites, it is deemed that it will be universally applicable.

Progressions in young learners’ understandings of parity arguments

ABSTRACT Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K–1 students’ thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K–1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students’ practice of argumentation from the start of formal schooling.

Epistemic phase transitions in mathematical proofs

Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows exponentially as the argument expands. When a mathematician encounters a proof, how does she come to believe it? Here we show that, under a cognitively-plausible belief formation mechanism combining deductive and abductive reasoning, belief in mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with five hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.

‘Powerless to separate from the clouds’: Badiou, mathematics and geography

ABSTRACT Alain Badiou’s work provides an important opportunity for critical human geographers to enhance our grasp of a range of abstract mathematical concepts while clarifying that toward which we must remain critical. Yet the geographical encounter with Badiou thus far has been needlessly hampered by, and has itself reinforced, a certain pessimism about the ability of qualitatively-trained geographers to deal in any meaningful way with his mathematical arguments. Challenging this pessimism, the present paper argues that if we trust ourselves a bit more to think through, with, and against mathematical concepts, we can in fact learn a great deal from Badiou. To illustrate this claim, the paper draws upon mathematical dimensions of Badiou’s arguments – as well as some ideas from Gestalt theory – to highlight failures both of his ontological and of his phenomenological projects. In light of the latter failure, however, Badiou’s mathematical concepts suggest the possibility of an analysis of qualitative geographical phenomena that both retains a place for subjectivity and leaves space for the recognition of ‘proto-quantitative relations’. The paper closes by suggesting how Badiou’s abstract mathematical concept of the ‘transcendental’ can help to understand the manipulative production of space in twenty-first century capitalism.

Students' dialogic and justifying moves during dialogic argumentation in mathematics and physics

In this study, we focus on dialogic argumentation among students in a whole class setting in mathematics and physics in lower secondary school. By drawing on previous studies on the structure of argumentation and dialogic interactions, we suggest that transparency of student reasoning and students' engagement with each other's ideas are two key aspects in dialogic argumentation. We examine what levels might exist in these key aspects and how they can exist simultaneously. We collected data by video recording mathematics and physics lessons in lower secondary school, and created a coding scheme for students' dialogic and justifying moves. By using the coding scheme, we recognized different kinds of argumentation with respect to how dialogic student–student interaction is and how transparently student reasoning is expressed. Furthermore, we found different ways how students can provide justifications as a dialogic reaction to others' ideas.

A reaction–convection–diffusion model for PEM fuel cells

In this paper, we present a novel 1D singularly perturbed reaction–convection–diffusion mathematical model, with non-linear coefficients (SP-RCD model), for the physical modeling of a fuel cell. The model is a generalization of the macro-homogeneous model, revisited from the point of view of singularly perturbed differential equations. To solve the system of coupled second-order differential equations, we propose a numerical scheme based on vanishing the artificial diffusion of the finite element method within an iterative fixed-point algorithm. We also propose an adaptive Shishkin mesh, as a function of the derivative of the current density in the subdomain with a fast-growing slope. Results of the proposed SP-RCD model are comparable to those of the macro-homogeneous model. In addition, it describes the oxygen concentration profiles in the thickness of the cathode catalytic layer under different operating currents and represents, with enough precision, the experimental polarization curve reported in the literature. • We introduce a novel RCD mathematical model for the numerical simulation of PEM fuel cells. • We introduce a method for stabilizing the numerical solution, it introduces artificial diffusion (AD), until stabilization, then AD is removed. • We introduce a numerical strategy for adapting a Shishkin mesh, avoiding user-made tuning. • Mathematical arguments about convergence are provided. • We validate the algorithm with reported cases and other model.

Oxidative metabolisms catalyzed Earth\u2019s oxygenation

The burial of organic carbon, which prevents its remineralization via oxygen-consuming processes, is considered one of the causes of Earth’s oxygenation. Yet, higher levels of oxygen are thought to inhibit burial. Here we propose a resolution of this conundrum, wherein Earth’s initial oxygenation is favored by oxidative metabolisms generating partially oxidized organic matter (POOM), increasing burial via interaction with minerals in sediments. First, we introduce the POOM hypothesis via a mathematical argument. Second, we reconstruct the evolutionary history of one key enzyme family, flavin-dependent Baeyer–Villiger monooxygenases, that generates POOM, and show the temporal consistency of its diversification with the Proterozoic and Phanerozoic atmospheric oxygenation. Finally, we propose that the expansion of oxidative metabolisms instigated a positive feedback, which was amplified by the chemical changes to minerals on Earth’s surface. Collectively, these results suggest that Earth’s oxygenation is an autocatalytic transition induced by a combination of biological innovations and geological changes.

Random Quantum Circuits Anticoncentrate in Log Depth

We consider quantum circuits consisting of randomly chosen two-local gates and study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anti-concentrated, roughly meaning that the probability mass is not too concentrated on a small number of measurement outcomes. Understanding the conditions for anti-concentration is important for determining which quantum circuits are difficult to simulate classically, as anti-concentration has been in some cases an ingredient of mathematical arguments that simulation is hard and in other cases a necessary condition for easy simulation. Our definition of anti-concentration is that the expected collision probability, that is, the probability that two independently drawn outcomes will agree, is only a constant factor larger than if the distribution were uniform. We show that when the 2-local gates are each drawn from the Haar measure (or any two-design), at least $\Omega(n \log(n))$ gates (and thus $\Omega(\log(n))$ circuit depth) are needed for this condition to be met on an $n$ qudit circuit. In both the case where the gates are nearest-neighbor on a 1D ring and the case where gates are long-range, we show $O(n \log(n))$ gates are also sufficient, and we precisely compute the optimal constant prefactor for the $n \log(n)$. The technique we employ relies upon a mapping from the expected collision probability to the partition function of an Ising-like classical statistical mechanical model, which we manage to bound using stochastic and combinatorial techniques.

Adaptable scaffolding of mathematical argumentation skills: The role of self-regulation when scaffolded with CSCL scripts and heuristic worked examples

Collaboration scripts and heuristic worked examples have been described as powerful scaffolds to support skill acquisition in CSCL. While CSCL scripts particularly facilitate argumentative discourse within groups, heuristic worked examples provide heuristics and worked out pathways to solve domain-specific tasks. Yet, both scripts and heuristic worked examples are often designed in a one-size-fits-all fashion. Granting learners the opportunity to adapt these scaffolds to their self-perceived needs might be a way to further enhance their effects. We tested this assumption in two experiments. In experiment 1, we compared the effects of learning with adaptable and non-adaptable CSCL scripts. In experiment 2, we compared the effects of learning with adaptable and non-adaptable heuristic worked examples. University students (N = 167) learned repeatedly in pairs with either adaptable or non-adaptable scaffolding in the context of mathematical conjecture problems. Results show that adaptable CSCL scripts were partly helpful for students with higher levels of self-regulation skills. Non-adaptable maximal scaffolding supported learning of distinctive skill components. Social-discursive components were best facilitated by maximal heuristic worked examples through content knowledge scaffolds. In contrast, CSCL scripts best facilitated domain-specific skill components by scaffolding learners’ engagement in social discourse about domain knowledge. The study provides recommendations for designing adaptable scaffolding by taking into account the relation between the targeted skill component and the activities scaffolded in the learning process. We suggest conducting future studies on adaptable scaffolding with a focus on supporting learning regulation and group awareness to improve learners’ success in CSCL environments.

Energy conserving successive multi-stage method for the linear wave equation

We propose a new high-order multi-stage method to solve the linear wave equation in an unconditionally energy stable manner. This Successive Multi-Stage (SMS) method is extended from the Crank–Nicolson method and unconditional energy conservation is guaranteed. We develop up to the sixth-order SMS method using the order conditions for Runge–Kutta methods and provide mathematical arguments showing that the SMS method is a different branch from well-known high order energy preserving methods for Hamiltonian systems. We present a proof of the unique solvability and numerically demonstrate the accuracy and stability of the proposed methods compared with comparisons.

Knowledge of mathematics teachers in initial training regarding mathematical proofs: Logic-mathematical aspects in the evaluation of arguments

The objective of this study is to characterize the knowledge of mathematics teachers in initial training (MTITs) at the Universidad Nacional (Costa Rica) on the logic-syntactic and mathematical aspects involved in proving, when evaluating mathematical arguments. The research is positioned in the interpretive paradigm and has a qualitative approach. It consists of two empirical phases: in the first, a questionnaire regarding logic-syntactic aspects was applied to 25 subjects, during the months of September and October 2018 and; in the second phase, a second questionnaire covering mathematical aspects was applied to 19 subjects, during the months of May and June 2019. For the analysis of the information, knowledge indicators were proposed. Knowledge indicators are understood as phrases to determine evidence of knowledge in the responses of the subjects. It was appreciated that the vast majority of future mathematics teachers show knowledge to discriminate when a mathematical argument corresponds or not to a proof by virtue of the logic and syntactic aspects, and of mathematical elements associated with propositions with the structure of universal implication. In general, subjects displayed greater evidence of knowledge on the logic-syntactic aspects than on the mathematical aspects. Specifically, they evidenced that consideration of a particular case or the proof of the reciprocal proposition does not prove the result; likewise, subjects evidenced knowledge about the direct and indirect proof of the universal implication. In the case of the mathematical aspects considered as hypotheses, axioms, definitions and theorems, it was appreciated that subjects could have different levels of difficulties to understand a proof.

ANALISIS KEMAMPUAN BERPIKIR GEOMETRI BERDASARKAN TAHAP BERPIKIR VAN HIELE PADA PEMBELAJARAN MATEMATIKA SISWA SMP

Geometry is a mathematical branch that deals with the shape of an object, the spatial relationship between various objects, and its properties. Learning geometry is essential to supporting logical thinking ability, perceptual intuition, and reading and interpreting mathematical arguments. But many middle school students do not reach the stage of deduction and precision. The research was to analyze a high school student's geometry ability based on Van Hiele's stages of thinking. They consist of five levels of visualizing, analysis, sequence, deduction, and precision. It is a type of qualitative study with a descriptive method. The sample consisted of 20 eighth-graders in the located in the Karawang district. Data collection is carried out using a written questionnaire and test. Research shows that stage-0 (visualization) is the student's most achievable point from a questionnaire at 55% and a written test of 65%. The third stage (deduction) and fourth (accuracy) are the most difficult for students to achieve, based on written tests only 9% of students reach the third stage and only 6% of students reach the fourth. Student difficulties in learning this geometry may be due to several factors in their understanding of prerequisite materials, ability to concepts and pale-building qualities is lacking, and limited skill in using geometry ideas to solve mathematical problems.

Fake News Phenomenon: Formation of Beliefs under Pragmatic Optics and Mathematical Education

Background: Few studies in mathematics education have been looking at the new dynamics of communication with focus on the analysis of speeches propagated by false news. This social phenomenon is worrying: news that uses mathematical arguments to guide, shape, and reflect public opinion and popular thinking based on misleading information. Objectives: The aim of this investigation was to understand the role of mathematics education in the process of strengthening democracy. Data collection and analysis: A video published on a YouTube channel, in which mathematical arguments are used to convince public opinion about a certain point of view, was the object of the study whose result is presented in this paper. Design: With theoretical-methodological procedures that are based on the pragmatic theory of fixation of beliefs and on the verification of the mathematical content, the analysis adopted a qualitative approach. Setting and Participants: This is a study about a video considered false by news agencies, aired by a famous Brazilian journalist. Results: The study of fake news and desinformation scenario has allowed us to observe that mathematical speeches linked to the ideology of mathematical certainty are responsible for the establishment of beliefs and the formation of opinions by authority and tenacity methods. Conclusions: We conclude that mathematical models and mathematical discourses used in virtual communication environments can be responsible for camouflaging the human factor in political decisions and obscuring the visibility of ethical and morality variables.

Symmetry In Mathematics and the Mathematics of Symmetry

On the one hand, symmetry is frequently used in mathematical arguments as an effective tool to produce simple and elegant proofs. On the other hand, the mathematical concept of group is an important tool to classify objects with respect to symmetry.