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Science and Mathematics Education

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Articles 151 - 180 of 18673

Full-Text Articles in Education

Using Smartphones To Innovate Laboratories In Introductory Physics Courses, E. Bernardini, M. Carli, M.Y. Elkhashab, Andrea Ferroglia, Miguel C. N. Fiolhais, L. Gabelli, H. Jessen Munch, Darya Krym, P. Mastrolia, Giovanni Ossola, O. Pantano, J. Postiglione, J.S. Poveda Correa, C. Sirignano, F. Soramel Apr 2024

Using Smartphones To Innovate Laboratories In Introductory Physics Courses, E. Bernardini, M. Carli, M.Y. Elkhashab, Andrea Ferroglia, Miguel C. N. Fiolhais, L. Gabelli, H. Jessen Munch, Darya Krym, P. Mastrolia, Giovanni Ossola, O. Pantano, J. Postiglione, J.S. Poveda Correa, C. Sirignano, F. Soramel

Publications and Research

The SmartPhysics project involved two higher education institutions, one in Italy and one in the US, with the aim of exploring the use of smartphones for laboratory experiments in introductory Physics courses. Here we present and discuss two experiments that were developed in the project: the pendulum experiment, consisting in the measurement of the gravitational constant using a proximity stopwatch, and the ‘bouncing ball’ experiment, aimed at measuring the energy lost in inelastic collisions with a hard surface. Both experiments were tested with the students in the academic year 2021-22. Our results and didactical reflections contain suggestions for the use …


An Evaluation Of The Equitable Practices For Stem Education To Prepare All Students For The 21st Century, Elizabeth Simpson Apr 2024

An Evaluation Of The Equitable Practices For Stem Education To Prepare All Students For The 21st Century, Elizabeth Simpson

Dissertations

We are almost a quarter of the way through the 21st Century, yet most education institutions do not provide STEM education for all students. Science, technology, engineering, and math (STEM) have been a special focus of both industry and federal education initiatives in recent years. STEM is a proven way of building student skills needed for the workforce of tomorrow. The purpose of this study is to determine why education decision-makers resist implementing STEM programs equitably for all students. The context of this inquiry is district and school-based leaders throughout the United States, from elementary to high school, who have …


Caminatas Comunitarias Para Enseñar Matemáticas En La Costa Caribe Colombiana: Un Enfoque Rural, Ricela Feliciano-Semidei, Kevin A. Palencia Infante, Jonathan A. Cervantes Barraza Apr 2024

Caminatas Comunitarias Para Enseñar Matemáticas En La Costa Caribe Colombiana: Un Enfoque Rural, Ricela Feliciano-Semidei, Kevin A. Palencia Infante, Jonathan A. Cervantes Barraza

Faculty Books & Book Chapters

Este libro recopila la experiencia vivida por docentes de matemáticas e investigadores que participaron del proyecto Pedagogías basadas en la comunidad rural para la enseñanza de las matemáticas en áreas rurales de la costa caribe colombiana. Los autores contextualizan estas experiencias para que puedan ser replicadas por otros docentes. Para lograrlo, se guía al lector con teorías educativas, políticas educativas, reflexiones y ejemplos alineados al currículo colombiano actual. Los ejemplos sirven de modelo y motivación para crear otras lecciones utilizando estas teorías. Este libro invita a los lectores a transformar las experiencias de los estudiantes de áreas rurales en recursos …


Mutualism As Mutual Trust: An Ethnographic Case Study On An Elementary-School Teacher-Team Participation In A Science Pd Program, Olga Gould-Yakovleva, Xiufeng Liu Apr 2024

Mutualism As Mutual Trust: An Ethnographic Case Study On An Elementary-School Teacher-Team Participation In A Science Pd Program, Olga Gould-Yakovleva, Xiufeng Liu

The Qualitative Report

This grounded in social constructivism yearlong ethnographic case study was conducted at the final stage of a larger, longitudinal, multisite, and multi-year project. The current research focused on a group of urban public elementary schoolteachers who volunteered to participate in the National Science Foundation (NSF)-funded interdisciplinary science and engineering partnership (ISEP) project. These teachers were enrolled in the professional development (PD) college courses in physics and engineering design, summer research in university laboratories, and interdisciplinary science inquiry (ISI) pedagogy sessions. This research sought to understand the factors which contributed to successful teamwork at the elementary school under investigation. Additionally, this …


Identification Of Quality Indicators Of Visual Based Learning Material In Technology Education Programs For Grades 6th - 9th In Greek Schools In Cyclades Aegean Islands, Spyridoula Tsouganatou Apr 2024

Identification Of Quality Indicators Of Visual Based Learning Material In Technology Education Programs For Grades 6th - 9th In Greek Schools In Cyclades Aegean Islands, Spyridoula Tsouganatou

STEMPS Theses & Dissertations

The purpose of this dissertation is to identify quality indicators for visual based learning material for technology education classes for grades 6th to 9th. This is a three round Delphi study aimed in answering the following research question. RQ: What are the quality indicators that contribute to the successful selection of visual-based learning material that contribute effectively to transmitting pedagogical information in a technology education classroom for grades 6th -9th? The quality indicators were determined by the consensus reached by technology education expert panel/educators that have been identified for the contributions and expertise in the field of …


From The Editors, Michele H. Koomen, Thomastine A. Sarchet-Maher, Jessica Williams Mar 2024

From The Editors, Michele H. Koomen, Thomastine A. Sarchet-Maher, Jessica Williams

Journal of Science Education for Students with Disabilities

JSESD remains a venue for the dissemination of research and practice related to the education of students with disabilities in the science classroom and laboratory since 1998. Volumes #1 through 11 were published in a print format. Starting with Volume #12, the journal has been published online and Open Access. Having JSESD in the Open Access format maximizes access for readers and authors and allows the journal to remain economically sustainable. JSESD is proud to now be publishing articles in both PDF and HTML formats (the HTML versions can be accessed through a link from the main articles’ web-page).


An Analysis And Ontology Of Teaching Methods In Cybersecurity Education, Sarah Buckley Mar 2024

An Analysis And Ontology Of Teaching Methods In Cybersecurity Education, Sarah Buckley

LSU Master's Theses

The growing cybersecurity workforce gap underscores the urgent need to address deficiencies in cybersecurity education: the current education system is not producing competent cybersecurity professionals, and current efforts are not informing the non-technical general public of basic cybersecurity practices. We argue that this gap is compounded by a fundamental disconnect between cybersecurity education literature and established education theory. Our research addresses this issue by examining the alignment of cybersecurity education literature concerning educational methods and tools with education literature.

In our research, we endeavor to bridge this gap by critically analyzing the alignment of cybersecurity education literature with education theory. …


An Argument For Expanding Research On Quantitative Pedagogical Content Knowledge, Randall E. Groth Mar 2024

An Argument For Expanding Research On Quantitative Pedagogical Content Knowledge, Randall E. Groth

Numeracy

Pedagogical content knowledge (PCK) traditionally has been used in research concerning prospective and practicing classroom teachers. This essay argues that PCK is also relevant to other professions including those advancing quantitative reasoning (QR). To illustrate, the case of PCK for teaching QR is considered. Those in fields such as public health, journalism, meteorology, and government increasingly find themselves responsible for helping the public understand an ever-growing amount of quantitative information that has a bearing on societal well-being. Several examples illustrate how such professionals’ responsibilities require knowing prevalent QR patterns in society, strategies for fostering sound reasoning, and the general nature …


A Mathematics Textbook Analysis Of Content And Representation, Salam R. Ahmad Mar 2024

A Mathematics Textbook Analysis Of Content And Representation, Salam R. Ahmad

USF Tampa Graduate Theses and Dissertations

In this dissertation study, I recognized the crucial role of mathematics educators’ utilization of instructional materials for effective teaching. I highlighted the significance of content that is taught to students through relevant mathematical tasks and representations. Through a detailed examination, I analyzed the similarities and differences in two editions of a precalculus textbook published by the same publishing company. Following a state-wide mathematics textbook ban in Florida, the precalculus textbook was placed on the Florida Department of Education’s (FLDOE) Instructional Materials Not-Recommended List in 2020. It was later revised and subsequently included to the FLDOE Instructional Materials Adopted List document …


Development Of An Experiential Learning Lab Activity On Skeletal Muscle Physiology In Undergraduate Md Pre-Clerkship Curriculum, Meyer B. Maddox, Angela Martinez, Daniel Salinas, Kelsey Potter-Baker, Amin Shah Mar 2024

Development Of An Experiential Learning Lab Activity On Skeletal Muscle Physiology In Undergraduate Md Pre-Clerkship Curriculum, Meyer B. Maddox, Angela Martinez, Daniel Salinas, Kelsey Potter-Baker, Amin Shah

Research Symposium

Background: Experiential learning is an important part of the medical education curriculum. Due to the clinical relevance of skeletal muscle strength in evaluating patients’ complaints of muscle weakness or imbalance, skeletal muscle physiology is an important concept with extensive potential for experiential learning opportunities. Our goal was to establish an experiential learning skeletal muscle physiology lab activity that would improve undergraduate MD pre-clerkship students’ skill to collect muscle force measurements using a hand-held dynamometer (HHD) and understanding of core physiological concepts.

Methods: As part of the Musculoskeletal and Dermatology (MSKD) Module in the undergraduate MD pre-clerkship curriculum, we developed a …


In-Person, Hybrid, Or Remote: Exploring The Effectiveness Of Teacher Practices On Student Performance During The Covid-19 Pandemic, Ashley M. Segalla Mar 2024

In-Person, Hybrid, Or Remote: Exploring The Effectiveness Of Teacher Practices On Student Performance During The Covid-19 Pandemic, Ashley M. Segalla

LSU Doctoral Dissertations

Schools and universities had to make unexpected changes beginning in Spring 2020 because of the COVID-19 pandemic. In Spring 2021, one university in the southern region of the United States implemented hybrid teaching formats in College Algebra courses, where students attended half of the classes in-person, and the remaining half of classes were attended synchronously and remotely. The purpose of this mixed methods study was to determine if students’ attendance methods (virtual or in-person) impacted their academic performance in the course. Additionally, the study sought to determine how students and the instructor felt student performance was impacted by hybrid attendance. …


“Educational Facials”: A Healing Tool For The Beautiful Struggle, Kari Kokka, Michelle Cody Mar 2024

“Educational Facials”: A Healing Tool For The Beautiful Struggle, Kari Kokka, Michelle Cody

Teaching and Learning Faculty Research

Critical educators of Color often work to support their students to work toward justice. However, because we live and work in a society imbued with white supremacy, cisheteropatriarchy, and additional systems of oppression, students and colleagues may resist efforts toward equity and racial justice, especially in mathematics education with women of Color instructors. In this paper, two mathematics educators, a Japanese American woman and a Black woman, elaborate a theory of educational facials, first coined by the second author in 2015. The theory of educational facials is an analytic tool for healing from and navigating harmful school climates. The authors …


Overcoming The Barriers To Teaching Teamwork To Undergraduates In Stem, Gregory R. Goldsmith, Miranda L. Aiken, Hector M. Camarillo-Abad, Kamal Diki, Daniel L. Gardner, Mario Stipčić, Javier F. Espeleta Mar 2024

Overcoming The Barriers To Teaching Teamwork To Undergraduates In Stem, Gregory R. Goldsmith, Miranda L. Aiken, Hector M. Camarillo-Abad, Kamal Diki, Daniel L. Gardner, Mario Stipčić, Javier F. Espeleta

Biology, Chemistry, and Environmental Sciences Faculty Articles and Research

There is widespread recognition that undergraduate students in the life sciences must learn how to work in teams. However, instructors who wish to incorporate teamwork into their classrooms rarely have formal training in how to teach teamwork. This is further complicated by the application of synonymous and often ambiguous terminology regarding teamwork that is found in literature spread among many different disciplines. There are significant barriers for instructors wishing to identify and implement best practices. We synthesize key concepts in teamwork by considering the knowledge, skills, and attitudes (KSAs) necessary for success, the pedagogies and curricula for teaching those KSAs, …


Variations In Student Approaches To Problem Solving In Undergraduate Biology Education, Jeremy L. Hsu, Rou-Jia Sung, Su L. Swarat, Alexandra J. Gore, Stephanie Kim, Stanley M. Lo Mar 2024

Variations In Student Approaches To Problem Solving In Undergraduate Biology Education, Jeremy L. Hsu, Rou-Jia Sung, Su L. Swarat, Alexandra J. Gore, Stephanie Kim, Stanley M. Lo

Biology, Chemistry, and Environmental Sciences Faculty Articles and Research

Existing research has investigated student problem-solving strategies across science, technology, engineering, and mathematics; however, there is limited work in undergraduate biology education on how various aspects that influence learning combine to generate holistic approaches to problem solving. Through the lens of situated cognition, we consider problem solving as a learning phenomenon that involves the interactions between internal cognition of the learner and the external learning environment. Using phenomenography as a methodology, we investigated undergraduate student approaches to problem solving in biology through interviews. We identified five aspects of problem solving (including knowledge, strategy, intention, metacognition, and mindset) that define three …


Level Of Including The Fields Of Science, Technology, Engineering, Art, And Mathematics (Steam) In The Omani Science Curricula Content For Grades 5-8, Ahmed M. Al-Shukaili, Mohamed A. Shahat, Sameh S. Ismail Mar 2024

Level Of Including The Fields Of Science, Technology, Engineering, Art, And Mathematics (Steam) In The Omani Science Curricula Content For Grades 5-8, Ahmed M. Al-Shukaili, Mohamed A. Shahat, Sameh S. Ismail

An-Najah University Journal for Research - B (Humanities)

This study aimed to determine the level of including the fields of Science, Technology, Engineering, Art, and Mathematics (STEAM) in the Omani science curricula content for Grades 5-8. To achieve the objectives of the study, a content analysis card was prepared by using the educational literature of previous studies and research. It consisted of six main areas with (34) indicators. To ensure the content analysis card's validity, it was presented to a group of experts specialized in the field of education. Cohen Kappa equation was used to ensure the instrument's reliability, which was (0.87) across individuals and (0.92) over time. …


Help Or Hindrance: Latinx College Graduates’ From Stem Majors Perceptions Of University Culture Following Vincent Tinto’S Theoretical Framework Of Student Persistence, Carmen Gonzalez, Carmen Lucia Gonzalez Mar 2024

Help Or Hindrance: Latinx College Graduates’ From Stem Majors Perceptions Of University Culture Following Vincent Tinto’S Theoretical Framework Of Student Persistence, Carmen Gonzalez, Carmen Lucia Gonzalez

Dissertations

Purpose: The purpose of this phenomenological study was to explore how Latinx college graduates who have achieved STEM degrees perceive the culture of their university and the STEM program they were enrolled in as supporting or not supporting a Latinx college student’s persistence to graduation.

Methodology: A qualitative phenomenological approach was the methodology selected for this study. This phenomenological research captured and described the lived experiences of study participants through interviews and artifacts.

Findings: The findings of this research illustrate (a) Research participants all perceived that creating relationships with STEM program faculty was important in supporting them toward degree completion, …


Publications And Other Works By R. F. Diffendal, Jr., Robert F. Diffendal Jr. Mar 2024

Publications And Other Works By R. F. Diffendal, Jr., Robert F. Diffendal Jr.

School of Natural Resources: Faculty Publications

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Updated March 2024


Secondary Math Teachers’ Responses To Errors In The Classroom, Richie Vicinanza Mar 2024

Secondary Math Teachers’ Responses To Errors In The Classroom, Richie Vicinanza

Honors College Theses

This study was conducted in an effort to find out how modern-day secondary math teachers respond to errors in their classrooms and the reasonings behind their responses. To fulfill the purpose of this study, various methods of research were used to gather information. Methods included looking into the scholarly sources surrounding the topic as well as a primary-conducted study in the field. Participants of this study were high school math teachers. The participants engaged in two semi-structured interviews (an initial and a debrief) as well as an observation that was conducted during one of their classes. Through the initial interview, …


Acer’S Mathematics Anxiety And Engagement Strategy (Maes): A Framework, Sarah Buckley, Kate Reid, Catherine Pearn, Shani Sniedze Mar 2024

Acer’S Mathematics Anxiety And Engagement Strategy (Maes): A Framework, Sarah Buckley, Kate Reid, Catherine Pearn, Shani Sniedze

Student learning processes

The Mathematics Anxiety and Engagement Strategy (MAES) was established by the Australian Council for Educational Research (ACER) in 2018 in response to declining mathematics achievement and participation among Australian students. Findings from the 2022 cycle of the Programme for International Student Assessment (PISA) show that 15-year-old students in Australia had significantly lower levels of mathematical literacy than students in 2003 (De Bortoli et al, 2023). The 2022 results also showed that male students outperformed female students and only 30 percent of students from more disadvantaged backgrounds achieved the National Proficient Standard in mathematics compared to 72 percent of students from …


Examination Of The Delphi Method For Establishing A Quantitative Literacy Framework In Undergraduate Mathematics Course, Emily Elrod, Joo Young Park Feb 2024

Examination Of The Delphi Method For Establishing A Quantitative Literacy Framework In Undergraduate Mathematics Course, Emily Elrod, Joo Young Park

Numeracy

The increased relevancy and importance of quantitative literacy (QL) have called for educational reform in undergraduate mathematics course offerings. However, lack of clear guidelines has led to wide variability in the content and outcomes of math courses meant to address QL. Therefore, having an expert consensus regarding content and outcomes for an undergraduate mathematics course focusing on QL would advance the quantitative literacy initiative in higher education. The authors propose the Delphi Method to aid in developing a framework based on expert opinions. The authors explain the Method and suggest the result of this process should yield a research-based framework …


Family-Centered Theory Of Change: A Conceptual Framework For Improving Teaching And Learning In Undergraduate Stem Courses, Juan Salinas, Parwinder Grewal, Jose J. Gutierrez, Nicolas A. Pereyra, Dagoberto Ramirez, Elizabeth Salinas, Griselda Salinas, Virginia Santanas, Can Saygin Feb 2024

Family-Centered Theory Of Change: A Conceptual Framework For Improving Teaching And Learning In Undergraduate Stem Courses, Juan Salinas, Parwinder Grewal, Jose J. Gutierrez, Nicolas A. Pereyra, Dagoberto Ramirez, Elizabeth Salinas, Griselda Salinas, Virginia Santanas, Can Saygin

School of Integrative Biological and Chemical Sciences Faculty Publications and Presentations

Hispanic-Serving Institutions (HSIs) are often characterized as Hispanic enrolling (rather than serving) that practice deficit-based systems that continue to marginalize Latinx and other underrepresented students, especially in STEM fields. Extant research on HSIs stresses the importance of investigating the value of grassroots advocacy groups as external influencers of institutional servingness through deeper engagement with the Latinx community. Using a novel family-centered theory of change that addresses diversity, equity, and inclusion, we integrated intersectionality and servingness into a family-integrated education serving and transforming academic framework. We investigated the potential transformational impact of this framework on students, families, faculty, and administrators at …


Steminism: Analyzing Factors That Improve Retention Of Women In Stem, Kira Carter, Jane Kelley, Jason Vasser-Elong, Rc Patterson Feb 2024

Steminism: Analyzing Factors That Improve Retention Of Women In Stem, Kira Carter, Jane Kelley, Jason Vasser-Elong, Rc Patterson

Dissertations

Our co-authored research ‘Steminism: Analyzing Factors That Improve Retention for Women as STEM Majors’ analyzed factors that contributed to the retention of women in science, technology, engineering, and mathematics (STEM) programs at Missouri University of Science & Technology (Missouri S&T). Women make up half of the US population, and while careers in (STEM) are an integral part of the US economy, women are underrepresented in these career fields. The purpose of our dissertation is to address the underrepresentation of women in STEM majors. Our methodology included homogeneous sampling to collect qualitative data. More specifically, we consulted with academic advisors and …


Using Trees To See A Forest: Leveraging Machine Learning To Classify Student Thinking, Darryl Chamberlain, Patrick Mcguinness, Emily Faulconer, Beverly Wood Feb 2024

Using Trees To See A Forest: Leveraging Machine Learning To Classify Student Thinking, Darryl Chamberlain, Patrick Mcguinness, Emily Faulconer, Beverly Wood

Publications

ChatGPT has caught the world’s attention and has researchers considering the ways artificial intelligence can be integrated into human endeavors (Kasneci et al., 2023). This methodological poster will present a systematic method to apply a particular machine learning classification model, Decision Tree (Song & Lu, 2015), to perform and extend the scale of qualitative analysis in mathematics education research.


Building Connections: The Power Of Embedding Literacy And Math Content Into Science And Social Studies Contexts, Rachel Hallett-Njuguna Edd Feb 2024

Building Connections: The Power Of Embedding Literacy And Math Content Into Science And Social Studies Contexts, Rachel Hallett-Njuguna Edd

Constellations: Online STEM Teacher Education Journal

Nationally, the lack of improvement in literacy scores continues to baffle experts. Instructional leaders from math, science, and social studies in one district knew the value of leveraging their subject areas to support literacy achievement in secondary students. Starting with an engaging STEM-related novel, the group of curriculum experts developed meaningful literacy connected tasks for their teacher and teacher leader participants. Working through the activities as their students would, the group found a new appreciation for the importance of leveraging the relevance of science and social studies content and the usefulness of math content when creating literacy lessons. The group’s …


Table Of Contents Feb 2024

Table Of Contents

Early College Folio

(2023) "Table of Contents," Early College Folio: Vol. 3: Iss. 1, Article 1. Available at: https://digitalcommons.bard.edu/earlycollegefolio/vol3/iss1/1


Place-Based Education: A Pedagogy Of Community Feb 2024

Place-Based Education: A Pedagogy Of Community

The International Journal of Ecopsychology (IJE)

Every place, no matter how seemingly mundane or described as ‘exotic,’ has a story—a human story involving culture (history, language, migration, etc.); a natural story involving land and water, plants and animals, and other natural phenomena; and a story that combines the two—humans in the natural space and their impact on that space. The degree to which we know the story of a place informs our connection to that place, be it positive, negative, or neutral. The incorporation of these stories—the cultural, historical, and ecological stories of a place—into school curricula is known as place-based education (PBE) and is the …


“It’S Just Lines”: A Qualitative Analysis Of Emergent Structures And Experiences Within Steam Education Initiatives For Secondary-Level Students, John O'Meara Feb 2024

“It’S Just Lines”: A Qualitative Analysis Of Emergent Structures And Experiences Within Steam Education Initiatives For Secondary-Level Students, John O'Meara

LASER Journal

This qualitative analysis is oriented around the experiences and discourse of high school students participating in LASER (Linking Art and Science through Education and Research), an initiative at Montclair State University in Montclair, NJ that seeks mathematics and physics education reform through an immersive and innovative approach to STEAM (Science, Technology, Engineering, Art, and Mathematics) experiences. This work focuses on the concluding task of a one-day workshop wherein students are led on a campus walk to observe the local campus architecture, and then must recreate one of the observed structures within the Desmos graphing utility. Student approaches and narratives reveal …


Cognitive Load Scale In Learning Formal Definition Of Limit: A Rasch Model Approach, Rina Oktaviyanthi, Ria Noviana Agus, Mark Lester B. Garcia, Kornkanok Lertdechapat Feb 2024

Cognitive Load Scale In Learning Formal Definition Of Limit: A Rasch Model Approach, Rina Oktaviyanthi, Ria Noviana Agus, Mark Lester B. Garcia, Kornkanok Lertdechapat

Mathematics Faculty Publications

Constructing proofs for the limit using the formal definition induces a high cognitive load. Common assessment tools, like cognitive load scales, lack specificity for the concept of limits. This research aims to validate an instrument tailored to assess cognitive load in students focused on the formal definition of limits, addressing the need for diverse strategies in education. The research employs a quantitative survey design with a Rasch model approach, utilizing a data collection instrument in the form of a questionnaire. Subsequently, the data are analyzed by focusing on three aspects: (1) item fit to the Rasch model, (2) unidimensionality, and …


Polygon Quadrature And Dodecagonal Tessellation With Pattern Blocks, Gunhan Caglayan, Ben Kamau Jan 2024

Polygon Quadrature And Dodecagonal Tessellation With Pattern Blocks, Gunhan Caglayan, Ben Kamau

Journal of Humanistic Mathematics

The age-old challenge of polygon quadrature involves converting a polygon into a square of equal area. In this educational resource, we utilize pattern blocks, commonly employed instructional aids in K-12 education across the United States, to visually demonstrate the transformation of different equilateral and regular pattern block polygons into squares. This is achieved through the application of the area conservation principle and geometric congruence/similarity reasoning.


Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett Jan 2024

Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett

Journal of Humanistic Mathematics

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.