The 50 most-cited articles in The Mathematics Enthusiast, 2021

Ethnomodelling can be considered as the association between ethnomathematics and mathematical modelling that enables members of distinct cultural groups to perceive a different reality in relation to the nature of mathematical knowledge. It also provides insights into many diverse forms of mathematics developed locally. Thus, ethnomodelling is defined as the study of mathematical phenomena that adds cultural components to the modelling process. The development of this connection is conducted through three cultural approaches: local, global, and glocal, which are used in the conduction of ethnomodelling investigations that aim to work against colonialism in order to value and respect sociocultural diversity of members of distinct cultural groups. Because ethnomodelling seeks to promote the development of understanding of differences through dialogue; it is important to argue for its inclusion as a translational process for systems taken from the reality of the members of diverse cultures. In this article we argue that ethnomodelling creates a firm foundation that allows for the…

This paper identifies the knowledge that emerges from collaborative situations, in an initial teachers’ education with integration of digital technology for teaching of function, in a blended and collaborative modality course, aiming to emerge Technological, Pedagogical and Content Knowledge (TPACK) in pre-service teachers. The method comprised the design of a teachers’ collaborative training experiment. The experiment had five phases, from a theoretical discussion to plan an Instrumental Orchestration to teach function with digital resources. Collaboratively, students worked in groups. The participants were students from a Mathematics teachers’ education. The experiment took place in the Teaching Methodology of Mathematics discipline at a public university in Brazil. This article focuses on the analysis of students’ knowledge in the planning stage, carried out collaboratively. The data analysis pointed out the three types of Technological, Pedagogical and Content knowledge were identified in different phases of the experiment, sometimes collectively, sometimes individually. In…

Digital Technologies are increasingly present in our activities. Many things we do we are not even able to imagine how they would be done, if we did not have the technological resources at hand. However, perhaps in the opposite direction of this, in the school context, or in teaching and learning, the discussion about the potential and the viability of these resources is still subject of a non-consensual discussion. When this context is Higher Education, specifically in undergraduate courses, the situation is even worse, as stated by research that we bring in this text. In disciplines such as Differential and Integral Calculus, Digital Technologies (DT) can contribute to a treatment in which aspects related to research and visualization are explored. Apps such as GeoGebra Augmented Reality, enhance the exploration of function graphs, for example, and, through movement, allow the analysis of invariants, favoring conceptual understanding. As we saw in the context of an activity[1]proposed for students of a Mathematics Degree course, the app allows for interaction between students and…

Sampling is one of the most fundamental concepts in statistics, as the quality and accuracy of the statistical inferences made, heavily depend on the method used to obtain the sample and the sample’s ability to represent the population of inference. Despite being a simple concept, sampling presents researchers with many challenges. Generally, due to monetary and time constraints, researchers must take a smaller sample size than they would ideally use. Using statistics from these small samples, estimates for population parameters can be made, typically in the form of a confidence interval. However, the validity of these confidence intervals depends on three basic assumptions that are difficult to meet with small sample sizes. This paper traces the development of the sampling method known as bootstrapping that helps small samples to meet these assumptions. The paper touches on previous methods used before the development of bootstrapping and shows how bootstrapping has evolved over the last four decades and become widely used in the field of statistics.

This first of a pair of articles describes a professional development project that prepared four non-Indigenous mathematics teachers (Grades 5-12) to implement Canada’s Truth and Reconciliation Commission’s (TRC, 2016) notion of reconciliation: cross-cultural respect through mutual understanding. The researchers collaboratively mentored the teachers to enhance their mathematics teaching with Indigenous mathematizing3. The teachers’ focus was on developing and revising lesson plans for other teachers to teach. This process explicitly and implicitly revealed precise supports that need to be in place for a teacher to succeed at innovating with this Indigenous culture-based school mathematics (ICBSM). Part I is a template for scaling up the development of much needed Indigenous resources for mathematics teachers. Part II reports on the research results of this year-long research project.

In this article, we present a decolonial stance in Mathematics Education, which is not understood as a qualification attributed to particular actions or practices as opposed to others, nor as a tendency that theoretically or methodologically constrains research production, but as a political and epistemic position of permanent transgression and insurgency concerning the patterns of world power established by the myth of Western modernity. From this understanding and towards a political agenda in Mathematics Education, we propose, with no pretensions of totality, a set of situated actions: in Mathematics, in its ontological, epistemological and methodological perspectives, problematizing the naturalization of practices and conceptions on the discipline and its teaching, and setting it in a movement of political-epistemic disobedience; in collective memories linked to Mathematics and Mathematics Education, deconstructing Eurocentric narratives which invibilize bodies, knowledges, and ways of being in the world; in Mathematics teachers’ education processes, incorporating and…

This article illustrates how videos generated through capturing touches on screens (screenrecording) can assist in analyses related to manipulation and reasoning by primary education learners in solving geometry tasks. We are presenting two protocols as ways to elucidate the use of application AZ Screen Recorder in learning about parallel lines intersected by a transversal line. In the first protocol, the interactions and the ways adopted by the subjects when analyzing the relation between corresponding angles are highlighted. In the second protocol, the relevant aspects are related to the particularities of the atmosphere of dynamic geometry used and the strategies adopted by the learners in the study of corresponding angles and collateral internal angles. The screen recording allowed us to (re)view the students’ touches in two simultaneous sources (audio and interactions on screen), using the same application. The analyses point to the fact that screenrecording made it possible to identify interactions on screen, especially through synchronicity, which contributed to track…

Research on textbooks has been carried out in different countries and with different purposes, including a set of studies that seek to know what those books convey in their texts. This type of investigation gives a view of the textbook as something that gives materiality to the proposed curriculum as a way to approach the knowledge taught and/or to identify elements that contribute to understanding reasons for some learning difficulties and persistent errors. Since the early 2000s, I have guided research studies that seeks to understand what kind of mathematics is proposed in textbooks intended for Brazilian basic education and how it is proposed. After a few years of this work, I outlined a model for textbook analysis based on elements of the anthropological theory of the didactic. This model consists of the following, not necessarily linear, steps: constituting the corpus for the analysis, the modellation of mathematical praxeologies present in the Course part of the material, doing the mathematical analysis of the Activities Proposed, the modellation of Didactic Praxeologies, and…

This second part of two related articles reports the answers to the research question: What precise supports must be in place for Grades 5 to 12 teachers to enhance their mathematics classes in a sustainable way with Indigenous mathematizing and Indigenous worldview perspectives? In addition to various logistical supports, two other types of supports were identified: supports for learning and unlearning ways of perceiving the world generally and perceiving Western mathematics specifically. These needed supports came to light when we mentored the teachers. On the one hand, the co-researching teachers learned, or had already learned: (a) the plurality of mathematical systems; (b) the perspective of Western mathematics as a human endeavor along with its values, ideologies, and definitions; (c) the mere inclusion of Indigenous mathematizing in a lesson is not enough; and (d) the goal of two-eyed seeing. On the other hand, the co-researching teachers unlearned, or had already unlearned: (a) pure mathematics’ claim to be value-free, (b) all students have a predilection to excel at…

Mathematical techniques in proof writing can be narrowed down to specific proof styles. Simply put, proofs can be direct or indirect- the latter using the Law of the Excluded Middle from logic as well as the axiom of Choice, to prove existence of mathematical objects. However, the thinking skills involved in writing indirect proofs are prone to errors, especially from novice proof writers such as prospective teachers. Creativity in mathematics entails the use of both direct and indirect approaches to determine the validity of a statement. In this article, I shed some light on this relationship, by reporting on some findings from a study on how students comprehend and validate direct and indirect proofs. Furthermore, I use the constructs of fixation and flexibility from creativity research to examine student approaches to direct and indirect proofs.

The aim of this article is to discuss aspects of statistical literacy from research on statistics education conducted with teachers and developed by the Research Group on Mathematics and Statistics Education (GPEME) at the Universidade Federal de Pernambuco (UFPE). These investigations were influenced by international discussions on a perspective of statistical literacy that focuses not only on knowledge and technical procedures of curriculum content in statistics, but also on developing critical attitudes towards approaching data. The reflections are based on the analysis of research with teachers of different levels and teaching modalities, who engaged in problem-solving activities associated with various topics in statistics. The article problematizes the challenges of addressing statistical literacy in the initial and continuing education of teachers who learn and teach statistics. Discussions are included on research that has methodologically addressed teacher participation in groups that involve cooperative and collaborative processes and that associate teacher education with…

Drawing on the Semiotic-cultural approach and the Anthropological theory of the didactic, this paper discusses how exploration of historically framed conceptualizations of mathematical objects can establish bridges between different mathematical areas such as calculus and Euclidean geometry. A classroom intervention study in two secondary mathematics class-rooms involving dynamic geometry software tools to support the construction of a parabola and its tangent and aiming at the development of representational flexibility between algebraic/ functional and geometrical approaches, illustrates how students may benefit from participation in such explorations.

The aim of this study is to analyse the actions of prospective mathematics teachers (PMTs), triggered in preparation for their teaching practicum (TP), to foster their autonomy when seeking their sense of agency in the face of a vulnerability, in the movement to construct their professional identity. To this end, we have carried out a qualitative research about the practicum lesson plans and the simulations of those lessons by PMTs who attended the same class in two disciplines of the Supervised Curricular Teaching Practice, in the last two years of the mathematics teaching degree course of the State University of Londrina (UEL). The identified vulnerability experiences enabled the search for the PMTs’ sense of agency regarding the following actions: defining concepts with which the students had already had contact and problematising them to systematise a mathematical idea; resorting to previous experiences worked with potential basic education students; having support from colleagues for possible actions to be developed in the classroom; problematising the answer to a problem;…

In this theoretical essay we problematize the statement “To teach mathematics one must know a specific mathematics for teaching” of the discourse of specific mathematical knowledge to teach. The study was inspired by Foucaultian concepts and the analysis focused on studies in the area of Mathematical Education that mobilize discourses regarding Mathematical Knowledge for Teaching, Mathematics Teacher’s Specialized Knowledge and Mathematics for Teaching. The analysis showed that the discourses operate under two truth regimes about the mathematical knowledge needed to teach. Analyzing the enunciative intertwining, we observe an approximation with: i) the educational discourse, which deals with the inseparability between theory and practice; and ii) the capitalist economic discourse, which associates of economic development with the mathematical proficiency of a country.

This paper introduces the model of instrumental meta-orchestration (IMO), as an extension of the model of instrumental orchestration. This IMO model is defined as a systematic and intentional monitoring, by a teacher educator, of artefacts and teachers (or pre-service teachers) for facing a Meta-situation, defined as a composition of situations of different natures and difficulty levels. An IMO, in itself, is a composition of instrumental orchestrations (sequenced or interwoven). This paper presents and discusses an instrumental meta-orchestration experienced in a class of undergraduate mathematics teachers. The development of the initial model results in: new concepts, such as ad hoc reaction, didactic meta-configuration, exploitation modes and didactic meta-performance; new features, such as flexibility and interactivity; and new phenomena, such as cascade effects. It also reveals the importance of unexpected events that occur between orchestrations. Moreover, it expanded the forms of destination for evaluating specific online documents, which are named webdocs. Finally, the paper…

In this article, we bring forward a research with ethnomathematical aspects that shows the socio-cultural relations experienced in a quilombola community registered as Santa Luzia do Maruanum, in the rural region of the capital Macapá, in the state of Amapá, Brazil. Such research was based on the activities developed by 12 ceramic craftswomen in the community, members of the Ceramic Craftswomen Association of Maruanum - ALOMA (Associação de Louceiras do Maruanum in portuguese). It was possible, through field research, with a qualitative approach, to understand the social context of ceramic craftswomen, to verify the spoken mathematical knowledge learned out of school in the production of ceramic tableware, in family farming, and consequently in daily habits, such as Marabaixo culture. Through participant observation, it was possible to verify aspects in the relationship between the contents of school mathematics and the activities developed by the ceramic craftswomen, which brings us possibilities of a bridge for teaching mathematics in quilombola schools. The results presented here…

This study investigates 48 preservice middle school teachers’ formal textbook definitions, formulas, and graphs of the directly and inversely proportional relationships. The connections among these three types of representations and how preservice teachers’ representations differ according to the two relationships are also examined. The data of the study included the preservice teachers’ responses to a paper-and-pencil test with two items. An explanatory research design model was followed when developing this study. The study findings indicated the preservice teachers’ over-attention to the simultaneous increases and/or decreases when representing the directly and inversely proportional relationships. Only four preservice teachers stated in their definitions that the ratio was constant in the directly proportional relationship. Whereas none of them stated constant product in their inversely proportional definition. Although the preservice teachers were slightly better at writing the direct proportion formula than the inverse proportion formula, this difference was much greater in…

Course-based Undergraduate Research Experiences (CUREs) have been well developed in the hard sciences, but math CUREs are all but absent from the literature. Like biology and chemistry, math programs suffer from a lack of research experiences and many students are not able to participate in programs like REUs (Research Experiences for Undergraduates). CUREs are a great alternative, but the current definition of CURE (see [1]) has potential barriers when applied to mathematics (e.g. time, novelty of project). Our solution to these barriers was to develop a math CURE pathway in which students complete "Math CUREs" in targeted courses. After finishing the pathway (or part of the pathway), students complete a research project in at least one of the following areas: Lie theory, representation theory, or combinatorics. The focus of this paper is the math CURE implemented in a discrete mathematics course for math and computer science majors. We share our experiences with the development and implementation of this CURE over several iterations as well as the impact of the CURE on students…

This article aims to identify and analyze power relations in the negotiation of meanings in a Community of Practice, called the “Observatório da Educação Matemática” da Bahia (OEM-BA), formed by academics from the university and teachers of Basic Education, who came together to produce educational math curriculum materials. The research carried out was of a qualitative nature and the data were produced through observations of the activities carried out by OEM-BA. To support the analysis, we used Etienne Wenger's Social Theory of Learning and Basil Bernstein's Theory of Codes. By analyzing the data, we identify power relations between academic and non-academic discourses, as well as interdisciplinary and interdisciplinary discourses. This happened when members negotiated meanings relative to teaching and learning content operations with whole numbers and quantities directly and inversely proportional. The results of this research gave visibility to variations in the principles and hierarchies underlying the meeting between academics and scholars.

The Articulation Principle explains how mathematical expressions can be given a clear and unambiguous meaning by the way in which they are spoken and heard. Leaving short pauses in speech, sub-expressions can be interpreted as operations to be carried out in time or as mental objects that can be manipulated at a more sophisticated level. This offers a fundamental foundation for the growth of meaningful mathematical thinking at all levels from young children to the wide array of adult mathematics. This paper sets the Articulation Principle in a wider long-term learning framework and provides empirical evidence for its use in meaningful interpretation of mathematical expressions. In this paper, we investigate how two teachers who have learned to operate routinely with expressions react when they are presented with expressions spoken and written in different ways. This reveals how the experience enables them to give meaning to mathematical expressions that previously had only been learned by rote. The Articulation Principle has the advantage that it can be introduced at any level to…

This paper reports on Data Analytics Research (DAR), a course-based undergraduate research experience (CURE) in which undergraduate students conduct data analysis research on open real- world problems for industry, university, and community clients. We describe how DAR, offered by the Mathematical Sciences Department at Rensselaer Polytechnic Institute (RPI), is an essential part of an early low-barrier pipeline into data analytics studies and careers for diverse students. Students first take a foundational course, typically Introduction to Data Mathematics, that teaches linear algebra, data analytics, and R programming simultaneously using a project-based learning (PBL) approach. Then in DAR, students work in teams on open applied data analytics research problems provided by the clients. We describe the DAR organization which is inspired in part by agile software development practices. Students meet for coaching sessions with instructors multiple times a week and present to clients frequently. In a fully remote format during the pandemic, the students continued to be highly…

The Education for All proposal is an international document substantiating some approved national laws aimed at providing high-quality education for all: peasants, quilombolas, indigenous, landless individuals, women, people with disabilities, among others. However, when it comes to inclusion and School Mathematics , regardless of all advances in the Mathematical Education field, this subject remains formally introduced and taught, it still disregards the concept of teaching all, since it is inaccessible to most students. The aim of the current article is to reason on cultural , normality and difference issues, as well as on practices observed in the School Mathematics that contribute to promote exclusion in school environment. It was done by problematizing three contexts: Special, Afro-Brazilian and Indigenous education, and its inherent inclusion advances and challenges.

This article describes a course-based research experience in a cross-listed mathematical problem solving course for middle school pre-service and in-service teachers. The course was conducted entirely online due to statewide and university restrictions during the coronavirus pandemic. This study examines 4 undergraduate and 5 graduate students’ understanding of mathematical modeling. Undergraduate and graduate students were paired in groups to develop open-ended model-eliciting activities for their classmates to solve, and groups conducted educational action research to analyze their classmates’ work on the activities. We present one group’s activities and their results as an example of how action research can be conducted in a virtual teaching environment.

This paper presents a study using a discursive perspective to develop a theoretical model of Mathematics for Teaching of the function concept, employing the following sources: a systematic review of the research literature, two series of textbooks and a discussion study with a group of teachers. The model presents a descriptive language with a theoretical structure that relies fundamentally on the realization and recognition rules inspired in Basil Bernstein's theory. Also, the model is based on categories of realizations (landscapes) of the concept of function. The landscapes that make up the model are the tabular, diagram, algebraic, transformation machine, graphic, pattern generalization and formal landscapes. The model provides a discursive transparency for the communication about function, which may inform curriculum development and curriculum material design for students and teachers as well as planning strategies to address this topic in educational contexts.

Our research team surveyed members of the mathematics education scholarly community to gain insight into the community’s perceptions of the two-column proof and its role in high school geometry. We created an email survey, which we sent to email addresses gathered from public lists of conference participants and speakers from recent mathematics education conferences. We asked study participants to describe the value of the two-column proof and discuss whether they would be in favor of eliminating it from the high school geometry curriculum. There was a wide range of diversity in the responses, and we present several themes that we observed. We conclude with a discussion regarding the implications for research.

We quantify flattening the curve under the assumption of a soft quarantine in the spread of a contagious viral disease in a society. In particular, the maximum daily infection rate is expected to drop by twice the percentage drop in the virus reproduction number. The same percentage drop is expected for the maximum daily hospitalization or fatality rate. A formula for the expected maximum daily fatality rate is given.

The present work presents some examples of data originated from a set of investigations developed in Brazil, at the Federal Institute of Education, Science and Technology of the State of Ceará - IFCE, in the context of teacher education, and the consideration of elements of a historical, mathematical, and evolutionary nature. Thus, from a perspective of theoretical complementarity involving the research elements derived from the notion of the didactic engineering for development and some assumptions of the French aspect of the professional didactics, two sets of data are discussed. The first set involved research carried out between 2017 and 2020. The second set considered involved studies still under development in Brazil, considering the period of 2020 - 2023. The historical landscape of investigation rests on considering the recurring number sequences of Fibonacci, Lucas, Pell, Jacobsthal, Coordonier or Padovan, Perrin, Mersenne, Oresme, Naraynna, and Leonardo. Furthermore, in a broad sense, the study aims to provide a learning scenario for the teacher, affected by a perspective…

This article aims to describe discursive regularities of theses and dissertations by researchers from the Group “Research, Education, Language, Cultural Practices” - Phala and their effects for / on Mathematics Education. Due to the nature of the proposed study, three axes of description emerged: (1) starting cultural archive and the literatures-traces triggered with respect to Mathematics; (2) therapeutic analog movements operated in research, and (3) aesthetics and writing style. We identified that the archives, taking deconstructionist therapy as a research lens, expand the debates about the understanding of Mathematics as a fixed, linear, compartmentalized, abstract and hierarchical set of knowledge, shaped to meet cognitive, structuralist and neoliberal school learning projects; as well as they broaden the debates about the ways of practicing research in a hermeneutic-analytical way, addressing problems that involve modern schooling, teacher training, indigenous and non-indigenous education, assessment of learning and historiographic and non-historiographical aspects of…

In the present article, we undertook a systematic study of bibliographic review based on the discussions, in the academic scope, about the contribution of the Anthropological Theory of Didactic – ATD (Chevallard, 1992) linked to Mathematics teaching and learning processes. The purpose was to investigate the current appropriation of ATD in Stricto sensu researches in Brazil. It is worth mentioning that the present study refers to a part of a PhD thesis. Thus, we aim to present an overview of thesis and dissertations answering the following question: "What are the main programs, institutions and researchers in Brazil that have been working on ATD?" In addition, we developed a study on the investigation axes of the researches, classified as: I) Teaching: focusing on the teacher, II) Learning: the object being the student's learning and III) Analysis of documents: books, lesson plans, course projects. For this purpose, we selected as a corpus 69 dissertations and 37 thesis defended in the period from 2005 to 2017.

This paper describes the use of scientific practices in the Passion-Driven Statistics CURE and presents the results of surveys from the implementation of this CURE at three different colleges. Overall, students experienced positive changes in thinking and working like a scientist, personal gains related to research, and gains in research skills, attitudes and behaviors. The Passion-Driven Statistics CURE aims to equip the future STEM workforce with the data analysis skills and reasoning needed across industries.

Mathematics is a dynamic field of knowledge of human creation and invention that is in continuous expansion. However, the mathematics is presented as a ready and finished field of knowledge in school systems, and there is no concern with the development of cognitive processes other than memorization and symbolic manipulation, therefore, there is no concern with stimulating the development of cognitive processes such as creativity, present in the development of mathematics. On the other hand, the concepts of scientifical mathematics as infinite and infinitely small that are related to the understanding of the dynamics present in phenomena, do not have didactic treatment to be presented to students of basic education. In this work, I will propose an approach of mathematics by mean of the fundamental concepts from the first school levels, approaching mathematics as a semiotic activity based on the possibilities of interpretations of Zeno's aporias, as metaphors of the infinite, to develop creativity in the didactics of mathematics. This work will present description of didactic…

This paper describes a research aimed at problematizing cases of indication to the use of Base Ten Blocks (BTB) and Green Chips (GC) for the teaching and learning of the Arabic numeral system and of arithmetic operations. The cases analyzed are present in the five collections of Elementary School mathematics textbooks selected by the National Program of Textbook (PNLD), which were the most purchased by the Ministry of Education (MEC) in 2016, to supply the system of public education in Brazil during the triennium 2016-2018. The identified indications were classified in three types, and two of them were particularly problematized for presenting some type of limitation or confusion. Only one of the five collections studied did not present problems regarding manipulatives.

This article proposes an exercise of problematization and denaturalization of a math classroom that is born from research carried out with a cartographic approach, in which, when occupying the elementary school math classroom, questions are asked: what happens in a math classroom? Which mathematics happens in a classroom? The event is here in affinity with the philosophy of the event of Gilles Deleuze, in which event is that which cuts time off, drags it, breaks it. Events are possible worlds that erupt in the things that happen. To do so, the writing of this paper takes place in relation to an episode in a classroom, evoking events that involve mathematical production, "mistakes" and learning, experiencing engendering of learning in the learning process. This writing involves two problematizing movements: the first one unfolds the problem of connection between learning and teaching in mathematical education; the second one in which, by the force of the event in the math classroom, the connection between learning and teaching as something necessary is under suspicion since it…

In order to analyze advances in the resolution of combinatorial situations, due to the identification, conversion and treatment of semiotic registers, two studies were carried out. In the first study, 5th grade students identified, from problems in natural language, registers in trees of possibilities, lists and numerical expressions. The second study, carried out with 5th, 7th and 9th grade students, was configured as an intervention study in which trees or lists were used as an intermediate representation of the departure register (natural language) to the arrival register (numerical expression). The results of the studies confirmed the hypothesis that the conversion to numerical expression is more complex than the conversion to trees or lists. It was also confirmed that trees are more congruent, than lists, with registers in numerical expression. It is concluded that the use of intermediate representations, such as trees or systematic lists, is a good teaching strategy for advances in the combinatorial reasoning of students in the early and middle years of schooling.

While legal education unquestionably hones students’ critical thinking skills, it also privileges students who are faster readers and have prior background knowledge or larger working memories. According to the prevailing mythology of law school pedagogy, students learn by struggling to find their way out of chaos. Only then is their learning deep enough to permit them to engage in critical thinking and legal reasoning. Learning theory and research suggest this type of “inquiry” learning is not an effective way to introduce novice learners to a subject. Lacking basic substantive and procedural knowledge, students’ struggles are often unproductive and dispiriting. Initial explicit instruction early in a student’s learning more predictably creates stable, accurate knowledge. Because higher-order thinking depends on having some knowledge, ensuring students have a strong foundation of substantive and procedural knowledge increases the likelihood that they will develop critical thinking skills. However, legal education uniformly dismisses anything that looks like “spoon-feeding.” If the…

Stemming from an idea put forward by Loo-Keng Hua in 1962, this article describes an original way to perform arithmetic directly on infinite decimals. This approach leads to a new and elementary construction of the real number system via decimal representation. Based on the least upper bound property, a definition of trigonometric functions is also included, which settles an issue that Godfrey H. Hardy called \a fatal defect" in his Course of Pure Mathematics.

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proven that ⇡ was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space, the impossible becomes possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well. As a bonus, we have added a solution to the impossibility of Doubling the Cube. As a double bonus, we have also Boxed the…

It is often the case that high quality mathematics education professional development involves enhancing both teachers’ content knowledge and their pedagogical skills, specifically for teaching mathematics. However, when teachers are immersed in their own learning of mathematics, they are often unaware of the facilitators’ instructional decisions and moves that influence their own learning outcomes, as well as how these might apply to their future teaching. Thus, while the teachers can appreciate their new understanding of content, they may not have added significantly to their understanding of the instructor’s pedagogical moves that facilitated their growth. As a result, teachers may leave even high quality professional development without assurance that they will be able to adjust their teaching in ways that support their own students’ meaningful learning of mathematics. In this work we describe one way in which professional development can both enhance teachers’ subject matter knowledge and help to transform these new understandings into pedagogical content knowledge; the…

It is necessary to expand the analysis about affective processes in mathematic learning in different contexts. In Brazil this research area is still in the embryonic phase, but it is a space that has strong potential to respond the gaps that still exist in the field, leading to the understanding affective processes in mathematic learning. It is in this way that the present research aims to analyze how the mathematic learning difficulty is produced in the subjectivity of a child from the elementary school in Brazil. To understand how the student's development occurs in the school space, and how the difficulty is build, we took as a theoretical basis the Theory of Subjectivity by Fernando González Rey. The case study was a resource used in the constructive-interpretative process. The research was carried out in a public school in the Distrito Federal, Brazil. The mathematic learn difficulty, in this work, was produced by the research participant in different contexts, including, family, school and friendships. Several experiences and influences of the person generate subjective senses…

In this work, we present an issue we have observed over the last several years in the first year of Spanish science degrees’ (such as mathematics, physics or engineering) calculus courses. It is related to the study of differentiability of piecewise one real variable real functions at a point. We have analyzed how students study the differentiability of piecewise functions at a point, explaining the students’ work and reasoning and commenting about the common misunderstandings we have found. Then, we have researched about how to help the students in their learning process related to this calculus concept, and we have also used several activities or ways to work: student meeting groups to talk about their worries and misunderstandings, mathematical definitions, explanations and theory and also working out specific examples.

In the first months of life, babies develop visual perception. The notions of space evolve in the everyday of experiences, the recognition of the self through your body, and relationships with others. The topological notions developed by babies correspond to closeness, proximity, continuity and separation. As babies grow, their skills are developed both in the projective space and in the geometric space. These even influence the baby's development in an integral way. This article intends to present results of the topological notions of closure, proximity separation and projections in the baby’s space. This qualitative research is developed under a descriptive perspective with interdisciplinary contributions. Data collection was made from cartography, photographic and filmic records of babies in different cities in Brazil and Colombia. The reflections developed point to the development of perception from the offer of multiple experiences since the first months. In addition, it is evident that the understanding of important mathematical concepts happens since the beginning of life,…

Course-based undergraduate research experiences (CUREs) have benefits for course instructors, undergraduate STEM students, and underrepresented students. CUREs consist of five essential components which include the use of science practices, discovery, broadly relevant or important work, collaboration, and iteration (Auchincloss et al., 2014). Over the past decade, a broad range of CUREs have been developed within and across STEM disciplines. However, resources to assist instructors to develop CUREs in mathematics are limited. The purpose of this paper is to provide an overview of the extant literature on course-based undergraduate research experiences (CUREs), especially in mathematics, to provide future directions for reforming mathematics courses to incorporate research experiences through CUREs, and to provide recommendations for developing CUREs in mathematics.

This study aims to analyze the perceptions of undergraduates in Mathematics Teaching on the evaluation practices of their teachers of specific content subjects and the possible implications of this practice in their training as teachers. It is a qualitative research, expositive and explicative, developed in a Mathematics Teaching degree in a Brazilian Federal University. We used random non-probabilistic sample to select the course and the students. To produce the data, we used: documental analysis, class observation, interviews, group talks, and social networks – Facebook and WhatsApp groups, followed by content analysis. The main intervening which emerged from the data were indications of possible dispositions composing the habitus of mathematics teachers. From these dispositions, undergraduates indicated, directly and indirectly, the legitimizing role of the evaluation practices in the subjects of mathematics content as a way of symbolic violence, when mentioning the tension between the bachelor’s and the teaching degrees, and criticizing or defending the expositive classes with…

This paper discusses a part of a doctoral study based on the theoretical pillars of the Anthropological Theory of the Didactic (ATD). Supported by this theory, our research interest followed a path that led us to the teaching of the mathematical object vector and its institutional configuration in a Mathematics Teaching course at Universidade Estadual de Feira de Santana (UEFS), in the state of Bahia, Brazil, centered on two dichotomous aspects. The first one refers to the possibility of application and institutional relevance of this mathematical object. The second aspect is related to the obstacles encountered in the didactical scope, which have impact on the teaching/learning of this mathematical object. In this discussion, the question that guided our investigation emerged: how to harness praxeological recombinations that can promote mediation between personal and institutional relationships in the scope of vector knowledge in the Mathematics Teaching course at UEFS? Based on this context, we aim at analyzing students’ productions in terms of solution paths to a vector task in a…

In this article, I carry out a qualitative metasynthesis of six doctoral theses that were directed by me, identifying the theoretical and methodological frameworks that support these researches, and how the articulations were made between different theories chosen. I also revisit the results achieved by the studies to highlight their relevance to the Mathematics Education area. This study, although not exhaustive, provides a vision of possible dialogs of Mathematics Education (or education) theories related to teaching and learning of mathematics and the Didactics of Mathematics. The diversity of theories and the specificities of each of them confirm the idea that a single theoretical tendency, or a single model, hardly ever explains and makes explicit all the phenomena involved in the teaching and learning processes of mathematical concepts. All the research works analyzed were used to study the epistemological, ecological, and economic dimensions, to identify the different forms of conceptions of a given mathematical object to help them in the didactic analysis of the findings.…

Textbooks have been subject of national and international research and, in the Brazilian scenario, this thematic has been drawing researchers’ attention since the beginning of the 21st century. Focusing on the scope of geometry, the objective of this paper is to the share results of a study about the presence of Geometry in Mathematics Textbooks, based on the map of the research conducted by the Theorem - Reflections on Geometry Mathematical Education research group, which take the mathematics textbook as subject of study. For this, the use of textbooks is contextualized and a literature review is subsequently presented, addressing studies that discuss geometry in general and in its particularities, and, lastly, we present the theoretical foundation of the investigations conducted by the group. The results of these analyses are divided into two fronts: investigations regarding geometric contents and concepts and development of other thematics associated with the presence of geometry in mathematics textbooks. Lastly, the section quilting and projecting revises the group’s research…

What shapes do you get when you spin a cube on one of its vertices? This article explores the algebraic aspects of a spinning cube and establishes equations for the two cones and the hyperboloid using secondary school mathematics. Both the analysis and the verification take advantage of dynamic mathematics learning technologies.

The exact time when the mathematician Omar Khayyam lived is not well-defined, but it is generally agreed upon that he lived from the end of the 11th century to the beginning of the 12th century C.E. in Nishapur, which is in modern-day Iran and Afghanistan (Struik, 1958). Other than mathematics, Omar Khayyam also made considerable contributions to other fields, such as astronomy, philosophy, and poetry (Struik, 1958). He is probably most famous for his poem titled Rubaiyat of Omar Khayyam, which was translated by Edward Fitzgerald (Struik, 1958). Although famous for his poetry, he was professionally inclined to astronomy and mathematics. In mathematics, he is well-known for being the first individual to find positive root solutions to multiple cubic equations, and he is also known for furthering understanding of the parallel axiom (Eves, 1958, p. 285; Struik, 1958). In this report, details of Omar Khayyam’s life will be mentioned, but the focus will be on his contributions to mathematics and his role in the history of mathematics.

When COVID-19 triggered a historic economic crisis, millions of Americans lost income and fell behind on debts, and complaints to the Consumer Financial Protection Bureau (CFPB) hit record highs. This article argues that the agency, created after the 2008 crisis to protect consumers, largely failed to respond. Through early 2021, it filed no public enforcement actions tied to the pandemic, announced it would relax oversight of rules, and brought smaller cases with less consumer harm, redress, and penalties. The author recommends faster, visible enforcement, partnering with states, and clear guidance on unfair practices during the pandemic, signaling consumers remain protected.

This week’s guest is Craig Hurlbert, co-founder and co-CEO of Local Bounti, an AgTech start up headquartered in Hamilton, Montana. The company’s ambitious mission is to scale production and distribution of high quality produce grown indoors, using 90% less water and land than traditional farming. Craig is a seasoned entrepreneur with deep experience in the energy sector and private equity. In our conversation, we delve into the agricultural challenges Local Bounti hopes to address, what being a special purpose acquisition company means, and the changing economies that effect how consumers get food to the table.