Mathematical Work Research Articles

Peirce Polymath

We present succinctly Peirce’s extremely diversified works in mathematics and logic, trying to underline both their differential specificities and their integral connections. As a result, the global understanding of Peirce as a polymath is also reflected in its local mathematical architecture.

Deciding finiteness of bosonic dynamics with tunable interactions

Abstract In this work we are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie algebras to be finite dimensional. We consider cases where each free Hamiltonian term is itself an element of the generated Lie algebra. In our approach, we develop new tools to systematically divide skew-hermitian bosonic operators into appropriate subspaces, and construct specific sequences of skew-hermitian operators that are used to gauge the dimensionality of the Lie algebras themselves. The significance of our result relies on conditions that constrain only the independently controlled generators in a particular Hamiltonian, thereby providing an effective algorithm for verifying the finiteness of the generated Lie algebra. In addition, our results are tightly connected to mathematical work where the polynomials of creation and annihilation operators are known as the Weyl algebra. Our work paves the way for better understanding factorization of bosonic dynamics relevant to quantum control and quantum technology.

The work of a prospective high school teacher in pre-professional training in highlighting mathematical knowledge

This article aims to identify the knowledge that enables a prospective secondary school teacher to mobilize number sense components during a pre-professional practicum. Using a qualitative case study approach, four classroom episodes focused on fractions, percentages, and algebraic language are analyzed. The study involves one prospective teacher and 23 students from a multi-grade school in central-southern Chile. The analysis integrates teacher’s specialized knowledge and mathematical working spaces to explore how these frameworks inform the teacher’s knowledge. The findings reveal that the teacher employs personal theories, real-life examples, and graphical aids to enhance students’ understanding, though there is a reliance on rules and algorithms. While specialized knowledge aids in teaching, there is less emphasis on fostering students’ conceptual understanding and judgement. The study recommends improving teacher training programs to better equip prospective teachers in grasping number sense components, highlighting the need for more focused training on its development.

Context of Contextualized Teaching Situations in the Initial Training of Mathematics Teachers at the Popular University of Cesar

The purpose of this research is to reflect on the role of the context environment of the contextualized situations of teaching school mathematics, which make significant contributions to the initial training of the mathematics teacher at the Universidad Popular del Cesar-Valledupar-Colombia, where the informants were the twelve students enrolled in the subjects of Elective III, Mathematics Didactics, History and Epistemology of Mathematics and Degree Work II, of the VI, VII, VIII and IX semesters respectively, of the Bachelor's Degree in Mathematics program; assigned to the Faculty of Education, of the Universidad Popular del Cesar. The study was situated in the qualitative paradigm, under the epistemological approach of sociocultural constructivism, with the recapitulations of the foundations and orientations of the phenomenological-transcendental methodology. It is concluded that it is feasible to change the mathematics master class for a space where students can interact in the teaching-learning process, with the necessary knowledge to understand the messages of the context of the habitat of the subject who learns; and for this reason, the relationships of the trio: Teacher-School Mathematical Knowledge-Students and these with the context can always be placed in the context of mathematics education.

Detection and Signal Processing for Near‐Field Nanoscale Fourier Transform Infrared Spectroscopy

AbstractResearchers from a broad spectrum of scientific and engineering disciplines are increasingly using scattering‐type near‐field infrared spectroscopic techniques to characterize materials non‐destructively with nanoscale spatial resolution. However, a sub‐optimal understanding of a technique's implementation can complicate data interpretation and act as a barrier to entering the field. Here the key detection and processing steps involved in producing scattering‐type near‐field nanoscale Fourier transform infrared spectra (nano‐FTIR) are outlined. The self‐contained mathematical and experimental work derives and explains: i) how normalized complex‐valued nano‐FTIR spectra are generated, ii) why the real and imaginary components of spectra qualitatively relate to dispersion and absorption respectively, iii) a new and generally valid equation for spectra which can be used as a springboard for additional modeling of the scattering processes, and iv) an algebraic expression that can be used to extract an approximation to the sample's local extinction coefficient from nano‐FTIR. The algebraic model for weak oscillators is validated with nano‐FTIR and attenuated total reflectance Fourier transform infrared (ATR‐FTIR) spectra on samples of polystyrene and Kapton and further provides a pedagogical pathway to cementing some of the technique's key qualitative attributes.

Metacognitive strategies in mathematical modelling: a case study with engineering students

During the last years, mathematical modelling has been entering the field of engineering education because it is a bridge between mathematics and industry that also promotes the development of skills that require the engineer of the twenty-first century. Nevertheless, the modelling process is involved and requires the deployment of cognitive skills of superior level, particularly metacognitive skills. This contribution presents the results of a multiple qualitative case study on metacognitive strategies used by groups of engineering students associated with each transition of the mathematical modelling process. During the development of the course, observations and retrospective interviews were carried out, which evidenced the use of metacognitive planning strategies from simplification to interpretation; metacognitive monitoring strategies throughout the modelling process, metacognitive regulation strategies from mathematization to mathematical work and metacognitive evaluation strategies during simplification, mathematization and validation. The results expand the classification framework of metacognitive strategies for engineering students.

STEM-It-Yourself: Increasing Participation of Girls and Minorities in STEM

Have you ever wondered why, when browsing the internet looking for scientists, technologists, engineers, or mathematicians, most of the results show men? Or have you ever wondered if there was someone out there who might look like you doing work in science, technology, engineering, and mathematics (STEM)? Our research aimed to better understand why STEM fields are not diverse in gender or race, and to empower the next generation of STEM leaders through a project called STEM-It-Yourself (SIY). Through SIY, we introduced middle school girls to women in STEM who are representative of the girls’ races. By hearing about the STEM journeys of the presenters, who were women, and Black, Indigenous, and people of Color (BIPOC), the girls participating in SIY could cultivate STEM identities, which could motivate them to stay in STEM and help diversify the STEM workforce.

Interpreting the modern history of finance theory from Henri Poincaré’s perspective

Henri Poincaré is well known for his work in mathematics, physics, and the philosophy of science. From 1900 onwards, Poincaré had the opportunity to become a forerunner of the Efficient Market Hypothesis (EMH), a concept that was formally developed 65 years later, but he never seised it. We show that Poincaré’s view on randomness was incompatible with the EMH framework. Although he did not work directly on financial topics, his work has helped to understand the limits of random walks in asset pricing; a way to reconcile the mathematician with the field of finance.

New pharmacodynamic parameters linked with ibrutinib responses in chronic lymphocytic leukemia: Prospective study in real-world patients and mathematical modeling.

One of the first clinical observations of ibrutinib activity in the treatment of chronic lymphocytic leukemia (CLL) is a rapid decline in lymph nodes size. This phenomenon is accompanied by an hyperlymphocytosis, either transient or prolonged, which is associated with distinct clinical responses and thus has an impact on long-term outcomes. Understanding which factors determine distinct disease courses upon ibrutinib treatment remains a scientific challenge. From 2016 to 2021, we conducted a longitudinal and observational study in 2 cohorts of patients with chronic lymphocytic leukemia (CLL) (cohort 1, n = 41; cohort 2, n = 81). These cohorts reflect the well-known clinical features of CLL patients, such as Male/Female sex ratio of 2/1, a median age of 70 years at diagnosis, and include patients in first-line therapy (27%) or relapsed/refractory patients (73%). Blood cell counts were followed for each patient during 2 years of ibrutinib treatment. In addition, immunophenotyping and whole-body magnetic resonance imaging (MRI) were assessed in patients from cohort 1. These data were integrated in a newly built mathematical model, inspired by previous mathematical works on CLL treatment and combining dynamical and statistical models, leading to the identification of biological mechanisms associated with the 2 types of clinical responses. This multidisciplinary approach allowed to identify baseline parameters that dictated lymphocytes kinetics upon ibrutinib treatment. Indeed, ibrutinib-induced lymphocytosis defined 2 CLL patient subgroups, transient hyperlymphocytosis (tHL) or prolonged hyperlymphocytosis (pHL), that can be discriminated, before the treatment, by absolute counts of CD4+ T lymphocytes (p = 0.026) and regulatory CD4 T cells (p = 0.007), programmed cell death protein 1 PD1 (p = 0.022) and CD69 (p = 0.03) expression on B leukemic cells, CD19/CD5high/CXCR4low level (p = 0.04), and lymph node cellularity. We also pinpointed that the group of patients identified by the transient hyperlymphocytosis has lower duration response and a poor clinical outcome. The mathematical approach led to the reproduction of patient-specific dynamics and the estimation of associated patient-specific biological parameters, and highlighted that the differences between the 2 groups were mainly due to the production of leukemic B cells in lymph node compartments, and to a lesser extent to T lymphocytes and leukemic B cell egress into bloodstream. Access to additional data, especially longitudinal MRI data, could strengthen the conclusions regarding leukemic B cell dynamics in lymph nodes and the relevance of 2 distinct groups of patients. Altogether, our multidisciplinary study provides a better understanding of ibrutinib response and highlights new pharmacodynamic parameters before and along ibrutinib treatment. Since our results highlight a reduced duration response and outcome in patients with transient hyperlymphocytosis, our approach provides support for managing ibrutinib therapy after 3 months of treatment. ClinicalTrials.gov NCT02824159.

Contribution of Laxmi Pati Pandey: A Comprehensive Study of Nepali Mathematician and Astrologer

This article presents a comprehensive study of the life and contributions of Pandit. Laxmi Pati Pandey, a distinguished mathematician and astrologer from 18th-century Nepal. The historical context is explored to understand the intertwining of mathematics and astronomy in Nepalese culture, with a focus on the significance of mathematical induction and the evolution of mathematical thought in Shah era of Nepal. It outlines the historical relationship between mathematics and astronomy in Indian culture, emphasizing the close connection between the two disciplines. It also highlights the emergence of mathematical techniques in Siddhiintas, focusing on their application to solve astronomical problems. The study delves into Laxmi Pati Pandey's mathematical works, including his notable translation of book "Bhaswati," and Sanskrit Text "Ratna Deep". The study adopts a qualitative research design, incorporating historical analysis, content analysis, and interviews. Archival research, literature review, and interviews with descendants and experts are conducted to gather information on Laxmi Pati Pandey's background, contributions, and educational practices. Ethical considerations are highlighted to ensure the respectful treatment of information. Thematic analysis is employed to interpret collected data, identifying key themes related to Laxmi Pati Pandey's contributions and educational practices. The analysis involves a comparison of his work with historical sources to identify potential influences and contributions to Nepali mathematics. The findings reveal Laxmi Pati Pandey's significant contributions to mathematics and astronomy, including the translation of Sanskrit slokas into Nepali and the writing of mathematical books. His influence on educational practices and the development of mathematical education in Nepal is evident. The study uncovers his intellectual legacy through his descendants, such as Lila Nath Pandey and Gopal Pandey.

Mathematical Work of a Future Teacher in Teaching the Box Plot Diagram

Objective: The study aims to characterize the teaching proposals of the future Mathematics teacher in the context of their practical training in the final year of their university studies. Theoretical Framework: To achieve this, we consider the theory of Mathematical Workspaces, which allows for the analysis of both the mathematical activity that an individual engages in while solving a mathematical task, and the activity that is promoted during teaching. Method: A qualitative methodology is adopted through the design of an instrumental case study. The case pertains to a future teacher conducting a class on constructing a box plot. This class was observed and transcribed for analysis in light of the proposed mathematical work. Results and Discussion: The mathematical work exhibited by the future teacher includes a strong semiotic component and the use of non-material artifacts for quartile calculations. Students' prior knowledge is utilized in this context, with procedural aspects taking precedence over statistical thinking. Implications of the Research: The study raises concerns about the statistical education of mathematics teachers and its impact on future teaching proposals. Originality/Value: This research contributes to the study of statistics and initial teacher training in their influence on the future practices of Mathematics teachers. It provides a characterization of the mathematical work promoted by a future teacher and offers insights into concerns regarding the development of statistical thinking.

Innovative Mathematical Insights through Artificial Intelligence (AI): Analysing Ramanujan Series and the Relationship between e and π\pi

This paper presents a comprehensive case study on the application of Artificial Intelligence (AI) in mathematics, focusing on the Ramanujan series and the intricate relationship between the mathematical constants e and π. The study explores how AI, particularly machine learning and pattern recognition techniques, can be harnessed to discover new mathematical series and patterns, thereby extending the pioneering work of the legendary mathematician Srinivasa Ramanujan. The paper begins with an overview of the Ramanujan series, illustrating their significance and applications in mathematical computations. It then delves into the specifics of AI methodologies employed to unearth new series for e and π, highlighting the algorithms and models used. Through detailed analysis and experimentation, the study demonstrates how AI can generate new series expansions for e and π, offering enhanced convergence rates and computational efficiencies. Furthermore, the paper examines the relationship between these two constants, providing insights into their interconnected nature through AI-discovered series and patterns. Practical applications of these new series in fields such as numerical methods, cryptography, and theoretical physics are also discussed. By showcasing the successful integration of AI in the realm of mathematical research, this case study underscores the potential of AI to revolutionize traditional mathematical approaches, fostering the discovery of new knowledge and the refinement of existing theories. The findings contribute to a deeper understanding of the interplay between e and π, reinforcing the profound impact of Ramanujan's work in modern mathematics and the transformative power of AI in advancing this legacy.

ANALYSIS OF HIGH SCHOOL STUDENT'S KNOWLEDGE OF RECOGNIZING EUCLIDEAN ELEMENTS

Euclid's Elements is a work written by an ancient Greek mathematician named Euclid in the 3rd century BC. It is one of history's most famous and influential mathematical works, serving as a significant basis for studying geometry and number theory for centuries. This research aims to describe Euclid's Elements and determine the extent of high school students' knowledge about Euclid and his work and their level of understanding in solving mathematical problems. This qualitative research utilizes case study methods and literature reviews. Data were collected through problem-solving exercises and interviews. Thematic data analysis was conducted to elucidate Euclid's Elements and students' level of knowledge. The case study involved presenting mathematics problems and interviewing five high school students, using assessment instruments incorporating problem-solving and open interviews. The findings reveal that many students lack familiarity with the elements of Euclid's Elements but demonstrate proficiency in solving mathematical problems related to it.

Teachers’ mathematical work based on examples presented in the teaching of algebra in secondary education

Both international examination results and scientific evidence related to the teaching and learning of school algebra point to the necessity for analyzing mathematical practice in the classroom. In this context, we have formulated the research objective of characterizing the mathematical work that teachers favor based on the examples that they implement in the teaching of topics within the domain of secondary school algebra. To this end, this study is situated within the interpretive paradigm utilizing an instrumental collective case study approach. The study design entails the selection of two representative cases of mathematics teachers in Chilean schools, with the objective of understanding and describing their mathematical work in the secondary school classroom based on the examples they present. The results show that the first teacher utilizes varied examples that activate distinct forms of mathematical thinking, promoting the activation of both instrumental and discursive genesis. In contrast, the second teacher follows a more traditional approach, with the use of examples to illustrate and motivate, and a direct presentation of knowledge, activating primarily semiotic genesis, in which a transition between the numerical and the algebraic is lacking. Ultimately, the results highlight the importance of deeper consideration of the choice of examples in the teaching of algebra and how these examples can influence students’ learning. Furthermore, additional ideas are put forward for future work in this line of research associated with the use of examples in the classroom.

Dienes block media assisted by mathematical worksheets to improve student learning outcomes: Teacher action research

The development of science has shifted the process of learning mathematics from previously only involving numbers to numbers that have meaning, one of which is in the form of stories. However, many problems still need to be solved with the concept of addition and subtraction in the form of story problems. This research aims to improve student learning outcomes in addition and subtraction material through story problems. This research used the Teacher Action Research (TAR) approach with 14 second-grade students at Ciren Elementary School, Indonesia as participants. This research also used two data collection methods: providing mathematical worksheets and observations carried out by special observers. Data analysis was carried out by looking at the mean student scores, reflecting through FGDs with observers, and filling in observation sheets by the teacher. Furthermore, the results of this research are presented in the form of teacher observation sheet scores and mathematical worksheet work results. The score results on the observation sheet show an increase in scores from learning in cycle 1 to cycle 2. In addition, the average score of students' mathematical worksheets after intervention in the form of dienes block media and mathematical worksheets always increases with pre-cycle details (48), cycle I (80), and cycle II (84). Thus, using dienes block media assisted by mathematical worksheets can have direct implications for improving mathematics learning in addition and subtraction operation material.

Advances and trends in research on mathematical working spaces: A systematic review

This study addresses the research developed around the theory of mathematical working spaces (MWS). To this end, factors such as researchers’ country of origin are considered, as well as methodological characteristics, proposed objectives in each study, specific theoretical factors, and the links between MWS and other theoretical perspectives. This systematic review follows the guidelines stated by reporting items for systematic reviews and meta-analyses for documenting each stage of the review. In total, 102 studies were considered from databases including Dialnet, SciElo, Scopus, and Web of Science. The results help to illustrate research trends in MWS theory, as well as the possibility of fertile ideas for new studies and future lines of research for those interested in this theoretical approach.

School work performance in sub-Saharan Africa: A bivariate multilevel analysis

The objective of this study is to evaluate the internal performance of education systems in sub-Saharan Africa. The choice of the academic performance of student work in reading and mathematics (defined as the ratio between the score obtained in a subject and the investment in learning that subject) as the approach to internal performance constitutes the originality of this research. Based on 2019 data from the Confemen Programme for the Analysis of Educational Systems (PASEC) covering 14 countries in sub-Saharan Africa, the choice of bivariate modelling allows the hierarchical structure of the data to be taken into account and the performance of schoolwork in reading and mathematics to be treated simultaneously. The results show that the influence of learner and school characteristics on performance in reading and mathematics converges. Nevertheless, certain personal attributes of the pupils or certain aspects specific to the school attended generate differentiated impacts depending on the subject considered. Sensitivity and robustness analyses confirm the stability of the results, offering avenues for understanding and improving students' academic performance.

Women in mathematics: Stories of overcoming

This article explores the remarkable contribution of women to the field of mathematics throughout history. Traditionally underestimated and underrepresented in this field, women have played crucial roles in the advancement of mathematical knowledge. This paper examines the lives and accomplishments of some notable women in mathematics, highlighting their achievements and overcoming gender barriers. The role of women in the history of mathematics is often underestimated, but their contribution is vast and significant. This article explores the beauty of this history, highlighting remarkable women who have defied gender norms to leave a lasting mark on the field. From Hippatia, a 4th century mathematician and philosopher, to Ada Lovelace, history's first programmer, women have played crucial roles in the evolution of mathematics. The article also addresses the long period in which women were excluded from academic institutions, even when they made substantial contributions. The narrative highlights the resilience and determination of women who persisted despite the barriers. The modern era has witnessed an increase in the recognition of women's mathematical work, with figures such as Mary Cartwright and Emmy Noether standing out. The article highlights not only individual achievements, but also collective efforts to promote gender equality in mathematics. By exploring this history, the aim is to inspire a renewed appreciation for the role of women in mathematics and to encourage future generations of women to pursue their mathematical interests, thus contributing to a more inclusive and diverse discipline.

Students’ dynamic communication while transforming mathematical representations in a dynamic geometry environment

In mathematics, students’ abilities to make transformations between mathematical representations are fundamental. The recent implementation of digital technologies, such as Dynamic Geometry Environments (DGEs), have changed students’ access to mathematical representations by providing a variety of different representations, available just by pressing a button. Students use of a DGE may change their mathematical communication to become more dynamic and active. However, it is not clear how the use of DGEs and the change in communication style relate to more epistemological aspects of students’ mathematical work. This study, therefore, investigates the interplay between students’ dynamic mathematical communication and their transformations of mathematical representations when using a DGE from a competency perspective. Based on analyses of instances of lower secondary school students’ dynamic mathematical communication, findings indicate that students’ mathematical communication reflect the new ways of engaging with mathematical representations when using a DGE. Hence, the communication becomes dynamic. Theoretically, the complexity of having “continuous” transformations of mathematical representations may reduce the readiness of the mathematical communication competency. The “continuous” transformation may in fact cause students to unintentionally ascribe dynamic properties to mathematical representation.

The teaching of mathematical modelling and the mathematical work: A dynamic that is problematic

The teaching of mathematical modelling and the mathematical work: A dynamic that is problematic