Mathematical Arguments Research Articles

Bridging Informal Reasoning and Formal Proving: The Role of Argumentation in Proof-Events

AbstractThis paper explores the relationship between informal reasoning, creativity in mathematics, and problem solving. It underscores the importance of environments that promote interaction, hypothesis generation, examination, refutation, derivation of new solutions, drawing conclusions, and reasoning with others, as key factors in enhancing mathematical creativity. Drawing on argumentation logic, the paper proposes a novel approach to uncover specific characteristics in the development of formalized proving using “proof-events.” Argumentation logic can offer reasoning mechanisms that facilitate these environments. This paper proposes how argumentation can be implemented to discover certain characteristics in the development of formalized proving with “proof-events”. The concept of a proof-event was introduced by Goguen who described mathematical proof as a multi-agent social event involving not only “classical” formal proofs, but also other informal proving actions such as deficient or alleged proofs. Argumentation is an integral component of the discovery process for a mathematical proof since a proof necessitates a dialogue between provers and interpreters to clarify and resolve gaps or assumptions. By formalizing proof-events through argumentation, this paper demonstrates how informal reasoning and conflicts arising during the proving process can be effectively simulated. The paper presents an extended version of the proof-events calculus, rooted in argumentation theories, and highlights the intricate relationships among proof, human reasoning, cognitive processes, creativity, and mathematical arguments.

Low-temperature small-angle electron-electron scattering rate in Fermi metals

Abstract Low-temperature elementary conductivity formula in Fermi metals is reviewed starting from Ohm’s law. This provides the background needed to understand why and how the equations exploited are complicated due to effective mass effect and complex scattering rate even in the presence of small-angle electron-electron scattering at low temperatures. Using the mathematical conditions and physical arguments exploited to derive the Drude conductivity formula, we arrive at our main result—the analytic scattering rate formula at low temperatures that gives rise to the famous T 2 dependence without any ad hoc constants. Our derivation formally proves that the formula, 1 / τ = ( A / ℏ ) ( k B T ) 2 / E F $1/\tau =\left(A/\hslash \right){\left({k}_{\text{B}}T\right)}^{2}/{E}_{\text{F}}$ first guessed by Ashcroft and Mermin to be correct where A = N impurity/4π 2 and N impurity is the number of impurities (or scattering centers) present in a given sample. We also highlight the discovery of a new fundamental physical constant, λ Arulsamy = 3 ℏ 2 ( 4 π ϵ 0 ) 2 / m el e 4 ${\lambda }_{\text{Arulsamy}}=\left[3{\hslash }^{2}{\left(4\pi {{\epsilon}}_{0}\right)}^{2}\right]/\left[{m}_{\text{el}}{e}^{4}\right]$ that associates quantum mechanical energy with that of thermal energy, and is also related to Rydberg constant.

Types of mathematical tasks in lower secondary classrooms in Germany

AbstractTasks play an important role in mathematics education, as they provide opportunities for students to develop their competencies and to cognitively engage with the mathematical content. The potential for cognitive activation as a central feature of a mathematical task has been considered in numerous studies, mostly as a didactical analysis by means of existing classification systems. In this study, which has been carried out in the frame of the TEDS research program, a different approach is taken by which we aim to identify the cognitive demand level of tasks used in ordinary mathematics lessons. Thus, the correlation between general mathematical competencies and the cognitive mathematical activities required to solve the task was analyzed and common properties of groupings of tasks were empirically extracted. In detail, 2490 tasks from mathematics lessons in Germany were analyzed by means of a rational task analysis with regard to their potential for fostering general mathematical competencies, namely modelling, problem solving, reasoning and argumentation, use of representations, use of symbols and operations, and communication. Latent class analysis revealed six classes of mathematical tasks with varying potential for the different competencies. In accordance with previous studies on mathematical tasks in Germany, the biggest class focused solely on the use of symbols and operations, while other classes showed different foci. Post hoc analyses revealed that the classes of tasks differ with regard to the level of cognitive activity they require. The results of the study highlight that the potential for cognitive activation of the tasks used in the classrooms of this more recent study has not improved in the last decades, despite many reform activities in German mathematics education, and that many mathematical tasks used are still more or less calculation oriented.

Learning to argue while arguing to learn: Students’ emotional experiences during argumentation for graphing real-life functions

Argumentation in school mathematics is an important but demanding practice that supports important learning goals. Much of the research to date has focused on students’ cognition but researchers are paying attention to students’ affective experiences that influence their learning, and particularly in socio-cultural settings. In this qualitative case study, we drew on an emotion coding scheme to investigate students’ emotions during a cyclic sequence of small-group argumentation tasks on graphing and critiquing graphs of real-life situations. Insights into the emotional experiences of argument construction, critique, and reaching agreement were gained by analyzing observations, written reflections, video data, and interviews of six 9th-grade students (aged 14-15 years). A wide range of emotions were observed and expressed throughout the sequence. We found examples of emotions, including tension and frustration, playing a productive role in the students’ mathematics learning because of their experience of argumentation. The emotion of tension experienced by some students when receiving and giving critique of their mathematical ideas seemed to trigger productive attention to misconceptions. The emotion of frustration experienced by some students seemed to trigger productive co-construction of arguments and ‘aha’ moments of mathematical understanding. We also found examples of non-deliberative argumentation (disrespectful interactions, lack of final consensus) influencing emotions in the moment (distress) and in the longer term after the lesson sequence (disappointment). Implications and considerations for future research on argumentation for learning are discussed.

Students' mathematical argumentation ability when proving mathematical statements based on self-efficacy

Argumentation as an aspect of problem-solving has been studied in mathematics education. However, mathematical proof still needs to be resolved further. This study investigates students' mathematical argumentation skills when proving mathematical statements based on their self-efficacy. The research subjects were 43 mathematics education students at a university in Aceh Province who had taken a number theory course. The study used a qualitative approach with a case study design: students’ mathematical proving self-efficacy. Data was obtained using self-efficacy questionnaires and mathematical proof test instruments that experts have validated, while the data triangulation used was an in-depth interview. The results of this study reveal that students' mathematical argumentation skills in proving mathematical statements have differences based on their self-efficacy. The mathematical argumentation ability of students with high self-efficacy involves all aspects of argumentation well so that the construction of the proof is scientifically correct. Meanwhile, the argumentation ability of students with moderate or low self-efficacy still does not involve essential aspects of argumentation. So, the proof results are not scientifically correct because they have not arrived at the proper conclusion.

Analisis Peningkatan Level Berpikir Geometri Mahasiswa Berdasarkan Teori Van Hiele dengan Pendekatan Konstruktivisme

The provision of problem-solving, analytical, systematic, critical, and creative skills is important to improve human resources that can be obtained from learning mathematics. Geometry is a field of mathematics that aims to develop logical thinking skills, develop spatial intuition, and interpret mathematical arguments. Several studies show that students' understanding of geometry material is still at a low level, so it is necessary to develop or improve the geometry learning process. The purpose of this study is to describe the increase in students' geometric thinking levels based on Van Hiele's theory with a constructivism approach. The method used in this research is descriptive qualitative with data analysis techniques using the Miles & Huberman model in the form of data reduction, data presentation, and drawing conclusions. The results showed that before the learning process using a constructivist approach, students' geometric thinking skills were at levels 1 to 3 or the analysis stage to a formal deduction with the average being at the informal deduction and pre-formal deduction stages. Meanwhile, after the learning process using a constructivist approach was carried out, geometric thinking skills increased at levels 2 to 4 or the informal deduction stage to the rigor with an average increase in the formal deduction stage.

Formula omitted]-sets for kernel-based spaces

The concept of H-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and H-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, H-sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of H-sets to Linear Programming is studied. But on the downside, it is explained why H-sets have a very limited range of applicability in the times of large-scale computing.

How Do Students' Method of Solving Proofs Change When Working in Groups?

This study aimed to describe the shift in students’ mathematical arguments from working on a proof individually to working on it in a group. This descriptive-exploratory study used a qualitative approach. Nine undergraduate students from a private university in Jombang, Indonesia, in the 5th semester were the subjects of the study. They were selected because they had already learned the concepts of math logic, argumentation, the theory of numbers, and analysis. Hence, they were ready to construct arguments in dialogue and non-dialogue forms. A task of argumentation and an interview were used to collect data. The study had several stages. First, the students solved the given argumentation task individually. Second, they had to make small groups of 3 members and discuss the same task. The result found that four of them had a complementary shift in their arguments, while the other five students had a reconstructive shift in their views. The complementary change happened because they reconsidered their initial thoughts, complementing their previous thinking structure. On the other hand, the reconstructive shift occurred due to group discussion (i.e., dialogue), which made them reconstruct or even entirely change their previous thoughts. Furthermore, they initially used inductive reasoning and then shifted their reasoning to a deductive one.
 Keywords: Mathematical Argumentation, Proving, Reconstruct, Complementary shift, Reconstructive shift

Exploring mathematical connections in the context of proof and mathematical argumentation: A new proposal of networking of theories

Extended theory of mathematical connections (ETC) and theory of mathematical argumentation (TMA) based on Toulmin’s (1984) model were articulated for the study of mathematical connections activated in the argumentation process. For this purpose, a “networking of theories” was made to obtain the complementarities between both theories. Then, a class episode was selected that dealt with the demonstration of the continuity theorem of functions of real variable “<i>if a function is derivable at a point then it is continuous at that point</i>”, made by an in-service mathematics teacher of differential calculus, who participated in a non-participant observation, where his classes were videotaped. The arguments of this episode were analyzed through with Toulmin’s (1984) model, after with thematic analysis method to identify mathematical connections, and, finally, the connections in the proof and mathematical argumentation were analyzed. The main result of the research reveals that the mathematical connections play a fundamental role in the argumentation process of the episode, given that, connection is important for the establishment and identification the argument and the warrant that supports it. In addition, complementarities were found between both theories, which makes this networking a useful tool for a better analysis of mathematical argumentation processes.

Collective Argumentation in Integrated Contexts: A Typology of Warrants Contributed in Mathematics and Coding Arguments

It is important to understand how students reason in K-12 integrated STEM settings to better prepare teachers to engage their students in integrated STEM tasks. To understand the reasoning that occurs in these settings, we used the lens of collective argumentation, specifically attending to the types of warrants elementary students and their teachers provided and accepted in integrated STEM contexts and how teachers supported students in providing these warrants. We watched 103 h of classroom instruction from 10 elementary school teachers and analyzed warrants that occurred in arguments in mathematics, coding, and integrated contexts to develop a typology of warrants contributed in mathematics and coding arguments. We found that these students made their warrants explicit the majority of the time, regardless of the teacher’s presence or absence. When teachers were present, they supported argumentation in various ways; however, they offered less support in integrated contexts. Additionally, we found students relied more on visual observations in coding contexts than in mathematics or integrated contexts, where they often provided warrants based on procedures required to accomplish a task. These findings have implications for improving integrated STEM instruction through engaging students in argumentation.

Justified Epistemic Exclusions in Mathematics

Abstract Who gets to contribute to knowledge production of an epistemic community? Scholarship has focussed on unjustified forms of exclusion. Here I study justified forms of exclusion by investigating the phenomenon of so-called ‘cranks’ in mathematics. I argue that workload-management concerns justify the exclusion of these outsiders from mathematical knowledge-making practices. My discussion reveals three insights. There are reasons other than incorrect mathematical argument that justify exclusions from mathematical practices. There are instances in which mathematicians are justified in rejecting even correct mathematical arguments. Finally, the way mathematicians spot mathematical crankery does not support the pejorative connotations of the ‘crank’ terminology.

Modified Stückelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent.

Genuine multifractality in time series is due to temporal correlations.

Based on the mathematical arguments formulated within the multifractal detrended fluctuation analysis (MFDFA) approach it is shown that, in the uncorrelated time series from the Gaussian basin of attraction, the effects resembling multifractality asymptotically disappear for positive moments when the length of time series increases. A hint is given that this applies to the negative moments as well and extends to the Lévy stable regime of fluctuations. The related effects are also illustrated and confirmed by numerical simulations. This documents that the genuine multifractality in time series may only result from the long-range temporal correlations, and the fatter distribution tails of fluctuations may broaden the width of the singularity spectrum only when such correlations are present. The frequently asked question of what makes multifractality in time series-temporal correlations or broad distribution tails-is thus ill posed. In the absence of correlations only the bifractal or monofractal cases are possible. The former corresponds to the Lévy stable regime of fluctuations while the latter to the ones belonging to the Gaussian basin of attraction in the sense of the central limit theorem.

Path dependence in an evolving system: a modeling perspective

AbstractIn recent years, path dependence has gained increasing scientific attention in many disciplines, leading to various new concepts and notations, such as path creation or path plasticity. However, if mathematical arguments are used, they are based on the early works by Brian W. Arthur and Paul A. David, usually referring to the mathematical concept of ergodicity. We extend their mathematical framework and develop a graphical representation of systems that allows for a metaphorical discussion of system behaviors beyond the original cases, especially in evolving systems, and the inclusion of the recently developed concepts within path dependence. Visualizations are used to explain the definition and characteristics of seven types of path dependence: lock-in, path-breaking, path-furrowing, path plasticity, path formation, path creation, and path selection. Although these visualizations are explained verbally and can be understood without a mathematical expertise, a mathematical model is used to generate them. The deduction of the metaphorical concept from a mathematical model guarantees the completeness of the identified processes and the rigor in their categorization as well as the identification of respective characteristics for their distinction. However, the aim of the paper is to provide an illustrative concept that allows researchers to classify and structure the various path-dependent processes they observe in their application. While five of the identified processes are in line with concepts from the literature and are defined accordingly, we also detect a sixth process that is new to the literature so far: path-furrowing. Moreover, slightly deviating from the literature, we define path selection as the possibility to choose a path intentionally, thereby focusing on the mindful choice of options.

Guidelines for the prediction of species interactions through binary classification

Abstract The prediction of species interactions is gaining momentum as a way to circumvent limitations in data volume. Yet, ecological networks are challenging to predict because they are typically small and sparse. Dealing with extreme class imbalance is a challenge for most binary classifiers, and there are currently no guidelines as to how predictive models can be trained for this specific problem. Using simple mathematical arguments and numerical experiments in which a variety of classifiers (for supervised learning) are trained on simulated networks, we develop a series of guidelines related to the choice of measures to use for model selection, and the ways to assemble the training dataset. Neither classifier accuracy nor the area under the receiver operating characteristic curve (ROC‐AUC) are informative measures for the performance of interaction prediction. The area under the precision‐recall curve (PR‐AUC) is a fairer assessment of performance. In some cases, even standard measures can lead to selecting a more biased classifier because the effect of connectance is strong. The amount of correction to apply to the training dataset depends on network connectance, on the measure to be optimized, and only weakly on the classifier. These results reveal that training machines to predict networks is a challenging task, and that in virtually all cases, the composition of the training set needs to be fine‐tuned before performing the actual training. We discuss these consequences in the context of the low volume of data.

PASCAL'S WAGER THEORY AND THE AUTHENTICITY OF IMAM AL-SADIQ (PEACE BE UPON HIM)

We discuss in our research here two theories, which are the theory of Pascal's bet, and the theory of the argument of Imam Al-Sadiq. Both theories aim at a noble cause, which is belief in God - the Almighty - but we see similarities and differences in each of the two theories, and each of them is due to the culture emerging from both sides (Western and Eastern), and (Christian and Islamic), but regardless of these differences, we find that each theory in its content is a curriculum that is keen to call to God according to its own way, using mathematical arguments and logical proofs to prove its claims.

Editorial for the Special Issue “GNSS, Space Weather and TEC Special Features”

For high-quality scientific communication in the field of technical and natural sciences, it is of utmost importance to ensure clarity of the text, logical mathematical argumentation, and the possibility of verifying the obtained theoretical results using appropriate experiments [...]

Subtraction, Decomposition, and Argumentation

We describe how mathematical argumentation supports curiosity and exploration by sharing a first-grade lesson in which students explored decomposition with subtraction. We also reflect on the conditions that supported the inclusion of mathematical argumentation.

Collapse of corroded pipes under external pressure considering corrosion asymmetry, non-uniformity, and initial ovality

This paper presents an analytical investigation of elastic collapse of corroded pipes considering the corrosion asymmetry, non-uniformity and initial ovality and one novelty is that corrosion asymmetry effect is considered by introducing a misalignment parameter accounting for co-existing internal corrosion and external corrosion. Governing differential equations are derived including the effect of corrosion asymmetry, initial ovality and membrane extensibility. By perturbative techniques, the interaction between corrosion asymmetry and initial ovality is extensively discussed by explicitly calculating the critical initial ovality governing the mode shape changes during initial deformation and collapse deformation phase. One novelty is the introduction of mode shape [Formula: see text] and mode shape 8 characterized by the sign of sum of displacements of two points independent of rigid-body motion. The theoretical solution is agreeing well with the FEA results. Furthermore, for general non-uniform corrosion, unconditionally convergent power series solutions are formulated by Lyapunov artificial small parameter method. By using Holder’s inequality, an explicit estimation method of the needed truncation number is presented for good computation accuracy by a minimization process over Holder’s exponent thus enabling early truncation of the series properly in advance. An improved estimation method is proposed by coupling with Rayleigh quotient techniques and the domain interval partition approach is introduced to accelerate convergence by thoroughly discussing the special uniformly corroded case by mathematical arguments. As a novelty, this paper illustrates why this improvement might be not evident when corrosion is rather narrow theoretically. Finally by using Chebyshev polynomials and the mathematical algebraic ring structure (generated by polynomial and trigonometric functions) closed under addition, multiplication and integration operations, some calculations for rings with corrosion profile in a non-polynomial mode are presented. This paper is intending to enhance the understanding of the effect of corrosion defect on collapse of pipes under external pressure in offshore industry.

The shape effect and its consequences for polar surfaces and for heterogeneous catalysis.

In this paper we develop the shape effect, which is relevant for crystalline materials whose size is larger than that of the thermodynamic limit. According to this effect the electronic properties of one surface of a crystal depend upon all of its surfaces, i.e. on the overall shape. At first, qualitative mathematical arguments are presented for the existence of this effect based on the conditions for the stability of polar surfaces. Our treatment explains why such surfaces are observed even though earlier theory indicated that they should not exist. Then, models are developed from which it is found computationally that changing the shape of a polar crystal can substantially alter the magnitude of its surface charges. Apart from surface charges, it follows that the crystal shape will also significantly affect bulk properties, most notably polarization and piezoelectric responses. Additional model calculations show a strong shape effect on the activation energy for heterogeneous catalysis primarily through local surface charges rather than a non-local/long range electrostatic potential.