Mathematical Work Research Articles

On resolving the tension between visiting mathematical works and questioning the quantum world

For the past 40 years, researchers in didactics of mathematics have developed and utilised didactic engineering as a methodology for design-based research. In this article we discuss its second step, the so-called a priori analysis. In the framework of the Anthropological Theory of the Didactic, engineering is in the form of the implementation of a study and research path (SRP), an inquiry-based course. SRPs may be either finalised, that is aimed at the teaching of a particular piece of knowledge, or rather open, that is solely aimed at answering a given question. This tension between sometimes competing goals especially shows in the a priori analysis of SRPs, a phenomenon we address in this paper. Our study is based on pieces of evidence collected by setting up an SRP two years in a row at the interface between mathematics and quantum mechanics, in third year of bachelor's degree. We take a particular care in explicating the rationale behind the a priori analysis and its connection with the preliminary analysis. We bring out the significance of spelling out one's epistemological stance, as didactic engineer, emphasising the value of the notion of reference epistemological models to do so.

Application of functionally graded graphene nanoplatelets to improve transient vibrations of the car’s roof under mechanical loading: Introducing machine learning algorithm for transient problems

Doubly curved structures under external force due to their applicable structure can be used in the roofs of various structures. Good knowledge about the transient dynamics of this kind of structure under external shock loading is very important in the engineering field. So, in the current work, steady and transient vibrations of composite doubly curved structures under time-dependent external force for the first time are presented. The current structure is reinforced by graphene nanoplatelets with high efficiency to improve the mechanical properties of the system in the thickness direction due to external shock loading in this direction. The peak pulse pressure, the Laplace transform, and some mathematical work to get time-dependent vibration responses of the composite doubly curved structures under external force. With the aid of different pre-processing of machine learning algorithms (Z-Score, Rrobust, Min-Max, and Quantile transformation), the results are predicted. The optimization algorithm, neural network structure, learning rate, mean of square error, and other training data are obtained for the current machine learning algorithm. The outputs of the machine learning method show that instead of simulating the complex structures in various situations with the aid of mathematical simulation, the researchers can use machine learning algorithms to correctly simulate this kind of system in different situations and material properties to obtain optimum conditions for doubly curved structure structures. The findings in the results section suggest that the time-dependent displacement and stress fields of the system are significantly influenced by the geometrical and physical characteristics of the current composite curved structure.

Deinstitutionalized Careers: Intersectionality of Gender Pay Gap among Women Knowledge Workers

The purpose of this study is to explore women knowledge workers’ experiences of the intersectionality of the gender pay gap and to scrutinize the reasons behind gender pay gap. The study problematizes the gender pay gap phenomenon by using intersectionality theory. The study describes how the gender pay gap varies based on ethnical differences besides being women and reveals how this situation deinstitutionalizes women’s careers. The study utilized a descriptive qualitative research design employing thematic analysis conducted in Germany and Türkiye with a purposively selected sample of participants. The study’s empirical material was generated through semi-structured interviews with women knowledge workers who differ in ethnicity and generally work in science, technology, engineering, and mathematics (STEM) fields. The study’s findings indicated that ethnicity should be considered among other intersectional factors to analyze the gender pay gap comprehensively. The reasons behind the gender pay gap are found to be the lack of transparency in the workplace, gender-based occupational segregation, culture, the glass ceiling, the existence of male-dominated informal communication channels, and the fact that women workers are more easily preferred in organizational downsizing initiatives.

“Searching for order at all levels”. Antonio Lima-de-Faria (July 4, 1921 – December 27, 2023)

“Searching for order at all levels”. Antonio Lima-de-Faria (July 4, 1921 – December 27, 2023)

The impact of temperature variation on the algae–zooplankton dynamics with size-selective disturbance

The phenomenon of algal blooms caused by global warming frequently occurs in global water bodies, however, the mechanism by which temperature affects algal growth is still unclear. In this paper, we investigate the dynamics of a deterministic algae–zooplankton model with water temperature, as well as its corresponding stochastic version. Mathematical theoretical work mainly analyzes the stability, persistence and Hopf bifurcation of the deterministic model, and performs a survival analysis on the stochastic model and shows the existence of a unique ergodic stationary distribution, which in turn provides a theoretical basis for numerical simulations. Numerical analysis indicates that the variation of temperature can generate complex impacts on the fundamental mechanisms underlying the growth of algae, whether in deterministic or stochastic environment. One of the most interesting results indicates that the decrease of temperature can destabilize the algae–zooplankton interactions, but increase their permanence; the increase of temperature can stabilize the algae–zooplankton interactions, but reduce their permanence.

Analyzing elementary students’ access to cognitive-oriented positions in mathematics

The effectiveness of group problem-solving in mathematics depends on the extent to which meaningful participation is distributed across all group members. One way to explore how participation is distributed within groups is by examining how students are positioned within group interactions. In this study, we explore the social instructional factors that cause elementary students to move in and out of positions that support cognitive engagement during collaborative problem-solving in mathematics. Using a case study analysis of three elementary students working in a group, we found five social instructional factors that caused students to move in and out of cognitive-oriented positions during group work in mathematics: (1) building an ally through common language, (2) physical access to the chalkboard and resources, (3) tone of voice, (4) teacher intervention, and (5) contestation from peers. The findings promote implications for effectively facilitating group work in mathematics.

Intermittency and Lower Dimensional Dissipation in Incompressible Fluids

In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as “intermittency”, and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents ζp=p/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _p={p}/{3}$$\\end{document} might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in every p-th order structure function for p>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>3$$\\end{document}. By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor’s frozen turbulence hypothesis, our strongest upper bound on ζp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _p$$\\end{document} coincides with the β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally, we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on the p-th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.

The Use of Mother Tongue in Communicating Mathematics: Implication for Teaching and Learning of Mathematics

Mathematics as a school subject is designed to solve the everyday problem created by human activities; these problems are domesticated in the different cultures of man all over the world. However, the performance of students keeps dwindling due to the westernize approach of teaching and learning mathematics. In other to solve this precarious situation this paper advocated the use of mother tongue in the teaching of mathematics. Mother tongue approach here refers to first-language education as a medium of instruction. The use of mother tongue in the teaching and learning of mathematics is meant to address the high functional illiteracy of Nigeria where language plays a significant factor. Using mother tongue to teach the basic concepts of numbers, operations and other concepts of mathematics helps build a strong foundation for the understanding and learning of higher mathematics. This approach is effective not only in getting the interest of students in the lesson but as a springboard in teaching new mathematical concepts, principles and in deepening student understanding on how mathematical operations or processes work. The mother tongue concepts of teaching and learning mathematics in this work shall be based on the following sub-topics: Numeration/Counting System, Basic Arithmetic Operations, Rhymes and Geometric Concepts. The paper completely reduced the Eurocentric mathematics beliefs of students and make them discover how best mathematics can be learning from their mother tongue. The paper recommended amongst others that teachers of mathematics should adopt the use of mother tongue in teaching mathematics in order to improve students understanding of the subject.

Strongly didactic contracts and mathematical work

Strongly didactic contracts and mathematical work

«МЕТОДОЛОГИЧЕСКИЙ АНАРХИЗМ» ПОЛА ФЕЙЕРАБЕНДА И ЭПИСТЕМОЛОГИЧЕСКИЙ СТАТУС ДЕЯТЕЛЬНОСТИ УЧЕНОГО В ПОЛИТАРНУЮ ЭПОХУ

The relevance of the article is determined by the importance of comprehension and theoretical explication of the origins and forms of specifically scientific cognition of objective reality, which originated in the civilizations of the ancient and medieval East (ancient Egypt and Babylon, ancient and medieval India and China, the medieval Arab-Muslim East), generally designated in the article as «politary societies». The purpose of the article is to show the possibility and effectiveness of applying the concept of the so-called «methodological anarchism» developed by the American philosopher Paul Feyerabend (1924–1994) to the epistemological activity that took place over the course of about three millennia in societies of the politary type. As the main method of research, the author uses the ascent from the abstract to the concrete: from the abstract generalized description of the methodological principles proclaimed by Feyerabend, the author moves on to the concrete realities of the epistemological discourse of the politary epoch, demonstrating the relevance of these principles in this respect and the fruitfulness of the coverage of politary science from this perspective. Three principles of Feyerabend’s were chosen as the supporting pillars of the study: 1) «Leave science to scientists!»; 2) «Anything goes!»; 3) separation of science from the state. The unity of the logical and the historical acts as an auxiliary method: the concrete-historical features of natural-scientific and mathematical work under the conditions of politarism appear as more or less pronounced individualized specifications of those general, universal criteria of scientific activity which modern philosophy of science tries to logically reconstruct, and which (as Feyerabend never tired of showing) are still very far from perfection. As a result of the study, the author draws conclusions that have a novelty. First of all, the very possibility of successful application of the concepts developed by Feyerabend on the material of the history of Western science, far beyond the socio-cultural sphere with which the American thinker directly dealt is substantiated. Secondly, it has been demonstrated that the reasons that ensured, on the one hand, the success of scientific and cognitive research, and, on the other hand, hindered it and harmed it, were, in fact, of the same type both on the European continent and in the politary East. Thirdly, it is shown that it is the acquisition by the research community of subjectness, the right to independently form research priorities, that is the condition for obtaining significant scientific innovations, and a condition that works in any culture and in any era - from antiquity to the present. The practical significance of the work carried out by the author of the article is seen in the fact that a methodological basis is created for a stricter demarcation between the concepts of «science» and «proto- (pre-) science» and for qualified theoretical and moral assessments of the intellectual and creative work that was done by scientists of the ancient and medieval East.

Giuseppe Peano in Germany, His Connections with German Mathematicians, and the First English Translation of Gottlob Frege’s Unpublished Letter to Adolph Mayer on Giuseppe Peano’s Mathematical Logic

The article aims to answer whether Gottlob Frege’s letter to Adolph Mayer, dated 8 July 1896, could help German mathematicians get acquainted with Giuseppe Peano’s mathematical work, including his mathematical logic. It is the first publication of this letter in English. At the beginning, I present the main characters of this story. Next, I refer to the letters concerning Peano and his mathematical results. Thus, I show the background of Frege’s letter to Mayer. In the last part, I collect information about Peano’s contacts with German mathematicians – where he was quoted and by whom, who was quoted by Peano, and in which period of his life. I conclude that Peano was known in Germany before Frege wrote to Mayer in 1896. However, the letter could have helped publish five of Peano’s articles in Germany, where Peano’s mathematical logic was hardly known then. Undoubtedly, the letter promoted that knowledge.

Dynamical complexity driven by water temperature in a size-dependent phytoplankton–zooplankton model with environmental variability

In this paper, we propose a stochastic phytoplankton–zooplankton (PZ) system modeling the growth dynamic of phytoplankton, where the death rate of zooplankton and the growth rate of phytoplankton are considered to be dependent on water temperature, and random environmental perturbation is characterized by mean-reverting Ornstein–Uhlenbeck process. Mathematical theoretical work mainly studies the stability, persistence, and Hopf bifurcation of deterministic PZ model, as well as analyzes the stochastic extinction, persistence in mean, and ergodic stationary distribution of stochastic PZ model, which provides a theoretical basis for numerical simulations. Numerical simulation work mainly reveals that water temperature plays a crucial role in influencing the dynamic of size-dependent phytoplankton–zooplankton interactions, whether in deterministic or random environments. An important conclusion of this study is that the increase of water temperature is able to induce bubbling effect, however, the small phytoplankton cell size can weaken this bubbling effect, and the large phytoplankton cell size can strengthen this bubbling effect, which supports the experimental conclusion that body size plays a key role in mitigating the negative impact of warming on predator–prey interactions. Further, it is found that the plankton with large body size has the advantages of stable and sustainable survival in the cold environment, while the plankton with small body size has advantages of stable and sustainable survival in the warm environment. Therefore, it can be inferred that plankton will evolve into large body size in the cold environment, but they will evolve into small body size in the warm environment.

Marrying the radical, the conventional, and the mystical: Mathematics, gender and religion in the lives of William Kingdon and Lucy Lane Clifford

The avowed atheist, evolutionary naturalist and mathematician William Kingdon Clifford is often remembered for his essay, “The Ethics of Belief,” in which he opposed organized religion in any form. As a mathematician, Clifford was an early advocate of non-Euclidean geometry in England. Combining William Rowan Hamilton’s work on quaternions with Hermann Grassmann’s theory of linear extension, he invented an original system of geometric algebra. Breaking with conservative traditionalism in his philosophical and mathematical work, Clifford’s marriage to the children’s writer, novelist, and dramatist Lucy Lane was a relatively conventional, if brief, Victorian marriage. After his untimely death from consumption in 1879, Lucy outlived her husband by fifty years. Raising their two daughters and supporting herself after his passing, Lucy refashioned Clifford’s posthumous reputation to temper his philosophical radicalism. Her collaboration with Clifford’s publisher and editor reveal Lucy’s concern that Clifford not be remembered as someone ruled by passion in his mathematical work. Her efforts to expunge writings suggestive of William’s weakness, excitability, or inconstancy from the public record demonstrates her desire to craft an image of her husband in alignment with gendered expectations of masculinity. This paper argues that Lucy fashioning of William’s memory conformed, rather than departed from, normative parameters of gender as defined by Victorian society.

Bringing the history of mathematics home: Entangled practices of domesticity, gender, and mathematical work

Although much scholarship on nineteenth and twentieth century mathematics has focused on processes of professionalization, historical mathematicians themselves rarely experienced their lives as neatly divisible into the professional and the private. Taking marriage as a focal point, this introduction brings the fruitful historiography of gender, collaborative couples, and domesticity in science into a broader conversation with the history of mathematics. By historicizing marriage and its relationship to mathematical careers, we lay the groundwork for the special issue which uncovers the myriad ways in which spousal collaboration and support have been central to mathematical work.

The plight of Modified Work and Study Program (MWSP) students in learning mathematics

The Philippine educational system established alternative delivery modes of education, such as the Modified Work and Study Program, to eradicate student dropout incidence. This phenomenological study aims to probe the challenges and explore the coping mechanism of MWSP students in studying mathematics. The study was conducted at a public high school offering the program. Six students were chosen to participate in the study. Focus group discussion was utilized for the data gathering of the study. Results showed that mediocrity of resources and conspicuous learning environment were the leading challenges that MWSP students encounter. Moreover, the coping mechanisms include good classroom management, utilization of concrete examples, time scheduling, steward support, motivational encouragement, and recreational and body relaxation. It is recommended that MWSP mathematics teachers integrate innovative strategies, provide more concrete contextualized examples, intensify the development of learning kits, and reinforce teaching and learning through supplementary materials. Keywords: Mathematics Learning; Modified Work and Study Program; Phenomenology

Uncovering the reciprocal relationship between domain-specific and domain-general skills: Combined numerical and working memory training improves children’s mathematical knowledge

The acquisition of domain-specific number knowledge and domain-general cognitive processes, including working memory, have shown to contribute to math learning and achievement. Correlational work has emphasized a reciprocal relationship between these processes; however, little is known about their relative and causal contributions to math learning. Here, we test the individual and additive benefits of playing tablet-based games targeting domain-specific and domain-general skills to improve mathematical knowledge and working memory in a diverse population of kindergarten children (N = 235, 50 % female, 45 % Hispanic/Latino, 35 % African American/Black, 21 % White, 6 % biracial/mixed race, 51 % annual household income less than $30,000). Our results provide causal evidence for the reciprocal relations between numerical knowledge and working memory. Specifically, we found that playing a working memory game led to improvements in mathematical knowledge as compared to an active control and after taking into account baseline performance. Importantly, we also found that children who played both a working memory game and a numerical knowledge game showed the most pronounced and sustained benefits in mathematical knowledge, with benefits remaining present several weeks after intervention completion. Our findings illustrate the importance of targeting both domain-general and domain-specific skills in order to maximize the benefits of interventions to improve mathematical knowledge. Yet, given the inconsistent results across various studies, we argue that it is critical to further investigate underlying mechanisms and training features, as even subtle variations seem to dramatically affect training outcomes.

Comparing perceived and observed instructional practices and their predictive power for student mathematics achievement: An analysis of Shanghai data from OECD global teaching inSights

This study examined the alignment and predictive power of instructional practices as reported by teachers, students, and external raters by using the Shanghai data that included 85 teachers and 2,613 students who participated in the Global Teaching Insights study. Results from exploratory and confirmatory factor analysis along with ordinary least square regression indicate that the same four conceptual components including classroom discourse (e.g., allowing students to explain their ideas and engage in peer discussions), meaning-making (e.g., explaining why a mathematical procedure works), cognitive activation (e.g., encouraging students’ critical thinking in solving complex tasks), and clarity instruction (e.g., teachers’ giving clear explanation of subject matter) were identified in the instructional practices reported by teachers and their students. The cognitive activation factor in the data reported by teachers emerged as the most significant predictor of students’ post-test scores, whereas the classroom discourse factor in the data reported by students accounted for the largest portion of variance in students’ post-test scores. Furthermore, our analysis revealed that the alignment between ratings reported by students and external raters was the highest, and student ratings of their mathematics teachers’ instructional practices demonstrated the highest predictive power for students’ post-test scores. Results of this study provide important empirical evidence for the merit of cognitive activation and classroom discourse in mathematics teaching and inspire researchers, practitioners, and policy-makers to pay careful attention to student-reported instructional practices that can serve as a better source of data in measuring mathematics teaching quality.

Mathematics self-concept moderates the relation between cognitive functions and mathematical skills in primary school children

In this paper, we investigated the contribution of both cognitive and affective factors to mathematical skills. In particular, we looked at the protective role of self-concept for mathematical learning and performance. In a field study, we tested the relation of math self-concept and short-term visuo-spatial working memory to the mathematical abilities of second- grade primary school children in Italy (N = 105). Measures included the “Test for the evaluation of calculating and problem-solving abilities” (AC-MT 6–11), the backward Corsi blocks test (Battery for Visuo-Spatial Memory), and the mathematics self-concept sub-scale of the Self-Description Questionnaire-I (SDQ-I, Italian version). As expected, correlation and moderated regression analyses showed that mathematics self-concept and working memory both positively predict mathematical operations and numeracy, but not accuracy. Simple slope analysis confirmed our moderation hypothesis, with working memory predicting mathematical abilities at low levels of math self-concept, but not at medium and high levels. The theoretical and practical implications are discussed.

Conceptualizing Flexibility in Programming-Based Mathematical Problem-Solving

Computational thinking (CT) has received much attention in mathematics education in recent years, and researchers have begun to experiment with the integration of CT into mathematics education to promote students’ CT and mathematical thinking (MT) development. However, there is a lack of empirical evidence and new theoretical perspectives on the mechanisms of interaction between CT and MT. To address this research gap, this study analyses the participants’ thinking processes in solving programming-based mathematical problems from a flexibility perspective, focusing on the interplay between computational and mathematical thinking, that is, how CT and MT work together to influence and determine the problem-solver’s choice of solution strategy. Using data collected from a large design-based study, we summarise two types of flexibility and six subtypes of flexibility demonstrated by participants in the programming-based mathematical problem-solving process using thematic analysis. These different types of flexibility provide researchers and mathematics educators with new theoretical perspectives to examine the interplay of CT and MT. Findings will also contribute toward student learning characteristics in programming-based mathematical problem-solving to sketch the big picture of how CT and MT emerge in complementary or mismatching ways.

‘Izz al-Dīn al-Zanjānī and His Scientific Contribution to Mathematics and Astronomy

In this research paper, light is shed on unknown life of a forgotten mathematician. He achieved a fame through his writings and literary works that highlighted his scientific personality. He lived in a tense and dark period of Islamic history in Islamic East. This mathematician is ʿIzz al-Dīn al-Zanjānī, who lived in the first half of seventh century AH/ thirteenth century AD during Mongol invasion of Islamic East. We also try to enumerate his various works in mathematics and astronomy with their locations in different libraries of the world, and refer to books of his authorship that were attributed to others.