Mathematical Narrative Research Articles

Mathematical Symbols in Academic Writing: The Case of Incorporating Mathematical Ideals in Academic Writing for Education Researchers

Mathematical symbols, such as those embodying quantum concepts, are indispensable for conveying complex ideas and relationships in academic writing. However, some education researchers and students keep a distance from anything mathematical: algebraic equations, geometrical reasoning, or statistical symbols. How to lower the access threshold for this type of mathematical narrative and reveal the meanings of a range of quantum conceptions to modern educators thus becomes a real problem. Using the pendulum motion equation as a reference point, I argue in this article for the advantages of academic English or French writing genres that fuse a range of mathematical symbols of quantum concepts and conceptual change. Such writings help demonstrate how incorporating the idea of probability (a) refines the debate among conceptual, verbal, and mathematical academic writing; (b) allows new conceptions that draw on the insights from quantum cognition-supported theories; (c) helps explain students’ understanding of mathematical symbols; and (d) offers a new taxonomy for categorizing academic writings.

고등학생들의 직관적인 무한소 담론과 학교수학 담론의 commognitive 갈등 사례 연구

This study was conducted to identify high school students' thought of infinitesimals and the resulting commognitive conflict with school mathematics discourse. Data collection was conducted in two stages: an open-ended questionnaire survey and follow-up interviews. The open-ended questionnaire was composed of questions to identify students' thoughts on the existence of infinitesimals and their opinions on conflict situations related to infinitesimals. Interviews were conducted with students who showed infinitesimal opinion in answering open-ended questionnaires. As a result of the study, some students revealed the idea that infinitesimals and infinitely large numbers exist, and showed conflicts by presenting narratives that contradicted the narratives of school mathematics. In addition, some students expressed their disagreement with the meta-rules of school mathematics, which adopts an intuitive explanation in the justification method of 0.999...=1 and the expression of the relationship between points, lines, and planes. The appearance of students experiencing conflict and the phenomenon of refuting and questioning the meta-rules of school mathematics could be interpreted as a commognitive conflict between students' discourse and school mathematics discourse. The specific appearance of commogntive conflict that emerged as a result of this study can serve as basic data for the exploration of teaching and learning methods that help students recognize the need for rigorous mathematical definitions and justifications and promote the transition to new discourse.

The stories we tell: Why unpacking narratives of mathematics is important for teacher conocimiento

This article articulates how mathematics (e.g., what they are, how they can be used, who they are by and for, who is able to do and understand them) are a social construct, just like racial or gender identities are social constructs. The authors describe how Political Conocimiento in Teaching Mathematics–a relational knowing that involves the entanglement of mathematics, pedagogies, students, and politics–can be used as a lens to reveal narratives about mathematics re(told) through stories (e.g., “Mathematics is culture-free, objective, and universal”). Drawing on their work with teachers, the authors offer an example scenario/activity and a teacher discussion that unpacks where mathematical stories are being told, from a dominant perspective, as well as how focusing on healthier narratives can help teachers work toward liberatory futures. Implications for teaching, teacher education, and future research are described.

Theorizing Mathematical Narrative through Machine Learning

Theorizing Mathematical Narrative through Machine Learning

‘몇 시 몇 분 전’으로 시각 읽기에 얽힌 수학적 활동의 탐색

This article arose from the question ‘how can the relative time reading be regarded as doing mathematics?’ To answer the question, I explored the interplay of time words relative time reading and mediation of tools which reflect two different time representational systems from a sociocultural description on learning that includes explicit treatments of word meaning and use. I conducted the clock reading test with 126 3rd graders and interviews with students who proved their ability for reading time in a relative way. The findings were as follows: (1) Participating students were approaching the relative time reading in a different manner as absolute time reading. The differences in performance upon tools were also identified. (2) Students were endorsing various mathematical narratives to read time in a relative way. (3) Errors in the relative time reading were related to the structural properties of time tools. Draw on the finding, the word 전(jeon) in relative time reading is dynamically interwoven with doing mathematics and interacting with tools reflect representational systems. The fact that the use and meaning of words become doing mathematics indicates that words themselves are dynamically interwoven with activity. The meaning of doing mathematics mediated by tool properties which are possible only in time learning and implications for the chance to increase mathematical learning opportunities in time contents in elementary mathematics were discussed.

Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization.

Early mathematical education in narrative contexts and the local environment

The article describes the implementation of the project that took the form of a teaching experiment. The “Myśleć jak architect” [en. “Think like an Architect”] project was aimed at developing children's spatial, mathematical and creative skills, integrating the social environment: academic, educational, family and cultural, and promoting science. Its essence was to design and conduct a mathematical and creative didactic cycle in the university space and in the local cultural space. The essence of educational meetings were: play, tasks, games and mathematical problems undertaken and solved by children in active and creative action, also using innovative teaching materials. The workshop didactic program was designed as a cycle of 12 hours of didactic classes, in which there was a deliberate use of rhymes, poems, fairy tales and mathematical narratives. The project lasted from November 2018 to May 2019. The beneficiaries of the project were the oldest pre-schoolers and younger students from grades I-III, in 9 groups, each consisting of 12 people. The study confirmed the effectiveness and attractiveness of integrating literature with mathematics and learning mathematics in various areas. The study has shown that the strategies for teacher management of construction play using narration are an important and effective ways of supporting the development of a child's mathematical thinking.

The difficulties of the elementary school students in solving the mathematical narrative test items

Mathematical learning is based on the context of everyday life and is reflected in the evaluation questions given to students. One type of question that is proper for the teacher to use is a word problem. Word problems are more difficult than other models. This type does not present problems with mathematical models directly but with narrative stories that students should understand first. This research was phenomenological research that aimed to find out what difficulties were experienced by students in solving mathematical problems presented narratively. The participants of this study were 121 fifth grade students from five elementary schools in the Special Region of Yogyakarta, Indonesia. Student errors were analyzed using the Newman Analysis Method. The Newman Analysis Method grouped students' mistakes into five categories: reading errors, comprehension errors, transformation errors, process skills, and encoding errors. The results of this study indicate that students have comprehension errors were 44.04%, transformation errors were 40.22%, process skills errors were 57.44%, and encoding errors were 57, 44%. Most errors are in problem number three with the type of reading errors. One of the reasons is the ability of students who are still lacking to understand can cause problems at a later stage.

Graphical Forms: The Adaptation of Sherin’s Symbolic Forms for the Analysis of Graphical Reasoning Across Disciplines

In this paper, we introduce and discuss a construct called graphical forms, an extension of Sherin’s symbolic forms. In its original conceptualization, symbolic forms characterize the ideas students associate with patterns in a mathematical expression. To expand symbolic forms beyond only characterizing mathematical equations, we use the general term registration to describe structural features attended to by individuals (parts of an equation or regions in a graph). When mathematical ideas are assigned to registrations in a graph, we characterize this as reasoning using graphical forms. As an analytic framework, graphical forms provide the language to discuss intuitive mathematical ideas associated with features in a graph, but we are also interested in engagement in modeling. Our approach to investigating graphical reasoning involves conceptualizing modeling as discussing mathematical narratives. This affords the language to describe reasoning about the process (or “story”) that could give rise to a graph; in practice, this occurs when mathematical reasoning (i.e. reasoning using graphical forms) is integrated with context-specific ideas. In this work we describe graphical forms as an extension of symbolic forms and emphasize its utility for analyzing graphical reasoning. In order to illustrate how the framework could be applied, we provide examples of interpretations of graphs across disciplines, using graphs selected from introductory biology, calculus, chemistry, and physics textbooks.

The realization tree assessment tool – examining explorative participation in mathematics lessons

In this paper we offer a new tool for assessing explorative participation in mathematics lessons, namely participation where students focus on constructing narratives about mathematical objects based on their own authority and on previously established mathematical narratives. This tool, named the Realization Tree Assessment tool (RTA), aims to record opportunities for objectification and the level of students' authority in whole classroom discussions. Inspired by the commognitive theory of mathematical objectification, the RTA visualizes the realizations of mathematical objects that arise during classroom discussions and the extent to which students authored the different realizations. We exemplify the tool on 14 lessons, all of which were based on an identical task – the Hexagons Task. The exemplification shows how the tool captures narratives about realizations and connections between these realizations offered by students and the teacher. We discuss the affordances of the tool for analysing and evaluating instruction that aims towards explorative participation.

Covariational reasoning and mathematical narratives: investigating students’ understanding of graphs in chemical kinetics

Graphical representations are an important tool used to model abstract processes in fields such as chemistry. Successful interpretation of a graph involves a combination of mathematical expertise and discipline-specific content to reason about the relationship between the variables and to describe the phenomena represented. In this work, we studied students’ graphical reasoning as they responded to a chemical kinetics prompt. Qualitative data was collected and analyzed for a sample of 70 students through the use of an assessment involving short-answer test items administered in a first-year, non-majors chemistry course at a Swedish university. The student responses were translated from Swedish to English and subsequently coded to analyze the chemical and mathematical ideas students attributed to the graph. Mathematical reasoning and ideas related to covariation were analyzed using graphical forms and the shape thinking perspective of graphical reasoning. Student responses were further analyzed by focusing on the extent to which they integrated chemistry and mathematics. This was accomplished by conceptualizing modeling as discussing mathematical narratives, characterizing how students described the “story” communicated by the graph. Analysis provided insight into students’ understanding of mathematical models of chemical processes.

To Fall in Love with Math, Do This

In the viral New York Times essay, “To Fall in Love with Anyone, Do This,” Mandy Len Catron details an experience she and an acquaintance had as they shared responses to psychologist Arthur Aron’s thirty-six questions intended to make participants fall in love. She notes that, “we all have a narrative of ourselves that we offer up to strangers and acquaintances, but Dr. Aron’s questions make it impossible to rely on that narrative.” In this paper, we claim that we also have narratives of our relationship to mathematics that we offer up to ourselves and others. Following, we offer a mathematical version on Aron’s thirty-six questions, designed to break out of our personal mathematical narratives and foster intimacy with mathematics.

초등학생의 수학 이야기에 나타난 수학적 상상 연구

초등학생의 수학 이야기에 나타난 수학적 상상 연구

The problem of certainty in mathematics

Two questions about certainty in mathematics are asked. First, is mathematical knowledge known with certainty? Second, why is the belief in the certainty of mathematical knowledge so widespread and where does it come from? This question is little addressed in the literature. In explaining the reasons for these beliefs, both cultural-historical and individual psychological factors are identified. The cultural development of mathematics contributes four factors: (1) the invariance and conservation of number and the reliability of calculation; (2) the emergence of numbers as abstract entities with apparently independent existence; (3) the emergence of proof with its goal of convincing readers of certainty of mathematical results; (4) the engulfment of historical contradictions and uncertainties and their incorporation into the mathematical narrative of certainty. Individual learners of mathematics internalize ideas of invariance, reliability and certainty through their classroom experiences and exposure to such cultural factors. Lastly, with regard to the first question, it is concluded that mathematics can be known with a certainty circumscribed by the limits of human knowing.

A thermodynamic theory of ecology: Helmholtz theorem for Lotka–Volterra equation, extended conservation law, and stochastic predator–prey dynamics

We carry out mathematical analyses, à la Helmholtz’s and Boltzmann’s 1884 studies of monocyclic Newtonian dynamics, for the Lotka–Volterra (LV) equation exhibiting predator–prey oscillations. In doing so, a novel ‘thermodynamic theory’ of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g. the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz’s theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness θ , quantities characterizing dynamic ranges of populations A and α , and the ecological force F α . The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ‘equation of ecological states’. Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology .

Building the wall brick by brick: one prospective teacher’s experiences with mathematics anxiety

Mathematics education researchers have investigated mathematics anxiety in prospective elementary teachers. While many of these studies have focused on the bodily sensations and emotions of mathematics anxiety, particularly those felt in assessment situations, opportunities remain to investigate how prospective elementary teachers interpret their experiences with mathematics anxiety and connect them over time to compose personal histories of mathematics anxiety. Currently, over 90 % of elementary teachers in US schools are women, and women have been shown to suffer more from mathematics anxiety than do men. In this article, I analyze how one woman prospective elementary teacher described, explained, and related her experiences of mathematics anxiety across her personal narratives of learning mathematics as a K-12 student and of learning to teach mathematics as a college student in a teacher preparation program. My research demonstrates that experiences of mathematics anxiety may persist beyond assessment situations to influence women prospective elementary teachers’ larger mathematical histories. I also show that women prospective elementary teachers may interpret mathematics anxiety as specific fears (e.g., loss of opportunities for social participation) and may develop particular coping strategies related to those fears. Finally, I point out that while a coping strategy may be used consistently across K-12 mathematics learning and undergraduate teacher preparation, and may even offer a woman prospective elementary teacher some relief from mathematics anxiety, it may also limit her mathematics learning and professional development. To conclude, I present implications of my research for mathematics teacher educators.

Thermodynamics of the general diffusion process: Equilibrium supercurrent and nonequilibrium driven circulation with dissipation

Unbalanced probability circulation, which yields cyclic motions in phase space, is the defining characteristics of a stationary diffusion process without detailed balance. In over-damped soft matter systems, such behavior is a hallmark of the presence of a sustained external driving force accompanied with dissipations. In an under-damped and strongly correlated system, however, cyclic motions are often the consequences of a conservative dynamics. In the present paper, we give a novel interpretation of a class of diffusion processes with stationary circulation in terms of a Maxwell-Boltzmann equilibrium in which cyclic motions are on the level set of stationary probability density function thus non-dissipative, e.g., a supercurrent. This implies an orthogonality between stationary circulation $J^{ss}(x)$ and the gradient of stationary probability density $f^{ss}(x)>0$. A sufficient and necessary condition for the orthogonality is a decomposition of the drift $b(x)=j(x)+ D(x)\nabla\varphi(x)$ where $\nabla\cdot j(x)=0$ and $j(x)$ $\cdot\nabla\varphi(x)=0$. Stationary processes with Maxwell-Boltzmann equilibrium has an underlying conservative dynamics $\dot{x}= j(x)\equiv$ $\big(f^{ss}(x)\big)^{-1}J^{ss}(x)$, and a first integral $\varphi(x)\equiv-\ln f^{ss}(x)=$ const, akin to a Hamiltonian system. At all time, an instantaneous free energy balance equation exists for a given diffusion system; and an extended energy conservation law among a family of diffusion processes with different parameter $\alpha$ can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven $\varphi$-ascending steps and spontaneous $\varphi$-descending movements, alternated with iso-$\varphi$ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics.

Latina/o Youth's Perspectives on Race, Language, and Learning Mathematics

In this article, the author employs critical race theory (CRT) and Latino Critical Theory (LatCrit) to examine Latina/o students' narratives of learning mathematics in a multi-lingual, urban high school. Intersectionality as a tenet of LatCrit is introduced as an important way to understand how students talk about the roles of race, language, and other central identities in their mathematics identity development as well as how they believe race may or may not matter in other people's mathematics achievement. The author's analysis illustrates how mathematics identities are co-constructed in relation to racial, linguistic, and gendered narratives of Latina/o youth. In general, the study adds empirical evidence to previous research on the difficulties that high school students encounter when articulating how race matters to their own identities in academic subjects and highlights the nuanced ways Latina/o students make connections between race, mathematical achievement, and schooling experiences in and through narratives of school success and failure.

The Writing’s on the Board

This article reports on an international study of the teaching of undergraduate mathematics in seven countries. Informed by rhetorical genre theory, activity theory, and the notion of Communities of Practice, this study explores a pedagogical genre at play in university mathematics lecture classrooms. The genre is mediational in that it is a tool employed in the activity of teaching. The data consist of audio/video-recorded lectures, observational notes, semistructured interviews, and written artifacts collected from 50 participants who differed in linguistic, cultural, and educational backgrounds; teaching experience; and languages of instruction. The study suggests that chalk talk, namely, writing out a mathematical narrative on the board while talking aloud, is the central pedagogical genre of the undergraduate mathematics lecture classroom. Pervasive pedagogical genres, like chalk talk, which develop within global disciplinary communities of practice, appear to override local differences across contexts of instruction. Better understanding these genres may lead to new insights regarding academic literacies and teaching.

Temporal Logics of Agency

Time is the grand stage where human activities take place (rational or otherwise). And the view of a branching temporal universe, or tree of possible events, with our actual history linearly advancing through it, is a widely shared cultural idea, not confined to Academia (cf. Borges brilliant 1941 essayEl Jardin de senderos que se bifurcan). Even though logic is often considered a study of timeless propositions, temporal languages and logics over tree structures have a long tradition. Philosophers have studied temporal structure and temporal reasoning since the 1950s, from the ‘tense logic’ of Prior (1967) to the ‘STIT’ system of Belnap et al. (2001). Moreover, starting from 1970s, computer scientists have joined in, and developed many further flavours of temporal logic, with major strands such as Pnueli on program correctness (Manna and Pnueli 1991), Emerson and Clarke on process specification and verification (Emerson and Clarke 1980, 1982), Reiter on the situation calculus in AI (Reiter 2001), and Thomas on the automata-theoretic foundations of computing (Thomas 1990). In addition, the pure firstand second-order logic of tree-like structures, starting from Rabin’s classic decidability result (Rabin 1969), provides deeper background (cf. Gradel et al. (2002)). The chapter by Hodkinson and Reynolds on ‘Temporal Logic’ in the Handbook of Modal Logic (Hodkinson and Reynolds 2006) brings together many of these trends in one mathematical narrative. But the field of reasoning in, and about, time can be mapped out in many further ways: Van Benthems chapter ‘Temporal Logic’ in the