Mathematical Writings Research Articles

A Study on the Taxonomy of Evaluation Objectives for Mathematical Writings and Examples

A Study on the Taxonomy of Evaluation Objectives for Mathematical Writings and Examples

An investigation of pedagogical techniques in Descartes'La géométrie

Much research has been conducted about the philosophy and mathematical writings of René Descartes, but that which focuses on pedagogy does so in a holistic manner. The present study uses a systematic approach to identify pedagogical techniques within each sentence of Descartes' La géométrie. Next, the study provides an analysis of La géométrie based on the techniques identified, their frequencies, and patterns of use within the text. The results of this analysis indicate that Descartes placed a high value on the use of demonstration, particularly in conjunction with deductive reasoning and multiple representations; that Descartes believed his method of approaching mathematical problems was superior to other methods; and that Descartes was in fact concerned with whether his readers understood his ideas or not.

STEM Literacies: Integrating Reading, Writing, and Technology in Science and Mathematics

Research into Practice Karen Wood The standards for science, mathematics, technology, and English language arts have strands in common, highlighting the need for educators to emphasize curriculum integration, curricular relevancy, and the digital skills needed to function in the 21st century (International Society for Technology in Education, 2007; National Council of Teachers of English/ International Reading Association, 1996; National Council of Teachers of Mathematics, 2000; National Research Council, 1996). This means educators must view the curriculum holistically; develop thematic curricular lessons and units, strategically making connections across subject areas; and engage in team planning. By doing these things, educators can help achieve a longstanding goal of the middle school movement: that the curriculum taught in our schools be challenging, exploratory, integrative, and relevant (National Middle School Association, 2010, p. 17). To do any less results in what Perkins (1992) called fragile knowledge-an artifact within a fragmented curriculum, characterized by a cursory, surface level of understanding. Fogarty (1991) and Fogarty and Pete (2009) organized models of curriculum integration according to four categories of connections: (1) connections made with other content areas within a single classroom (the fragmented, connected, and nested models), (2) connections made across the disciplines (the sequenced, shared, webbed, threaded, and integrated models), (3) connections made within learners (the immersed model), and (4) connections made across learners (the networked model). In this article, we take you into the classrooms of three teachers who have mastered literacy and technology integration in STEM areas, and we indicate which of Fogarty's models of integration they appear to be implementing in their classrooms. Ryan, a seventh grade science teacher, uses a vocabulary and comprehension strategy that involves graphing terms selected from an online text related to the unit of study. This form of integration represents the sequenced model, in which the teacher topically connects his or her content to another discipline. We describe an example of team planning at work as Janet, an English language arts (ELA) teacher, and her teammate Carlos, a science teacher, work together to help their students build the necessary skills to write a research paper (in both print and digital formats). This represents an example of the shared model, which involves mutual planning as the teachers help students learn how to research and write a paper using multiple sources-a skill needed in all of the subject areas. In the third classroom, Amy uses the Big 6 model to help students engage in research and problem solving in her mathematics class. This example fits most closely with the threaded model, because the school-wide emphasis is on teaching students to use higher-order thinking skills to improve their understanding of all areas of the curriculum. Ryan's seventh grade science class Ryan's seventh grade science students were having difficulty with content area vocabulary, struggling with words in isolation and in the context of their reading. This impacted both the students' word knowledge and their comprehension of the content area text. To increase students' understanding of content area vocabulary and improve their comprehension of science texts, Ryan employed the Ten Important Words Plus strategy during his unit on the human body systems (Yopp & Yopp, 2007). This strategy encourages students to interact with key vocabulary through multiple exposures in a meaningful context. Students determine which words will be chosen for study, giving them voice and a sense of ownership of their own learning, and this may help to motivate and engage students of all ability levels in Ryan's classroom (Dunston & Gambrell, 2009; Wood, Harmon, Kissel, & Hedrick, in press). First, Ryan gave his students access to an online text about the circulatory system (Johnson, n. …

‘Mathematics Made No Contribution to the Public Weal’: Why Jean Fernel (1497-1558) Became a Physician

This paper offers a caution that emphasis upon the importance of mathematics in recent historiography is in danger of obscuring the historical fact that, for the most part, mathematics was not seen as important in the pre-modern period. The paper proceeds by following a single case study, and in so doing offers the first account of the mathematical writings of Jean Fernel (1497-1558), better known as a leading medical innovator of the 16th century. After establishing Fernel's early commitment to mathematics, and attempt to forge a career as a cosmographer, it goes on to explain his abandonment of mathematics for a career in medicine. The 'mathematization of the world picture' is usually explained in terms of the perceived usefulness of mathematics, but Fernel's case shows that for many pre-modern thinkers, mathematics was not regarded as a useful pursuit. The paper should serve as a reminder, therefore, that the take-up of mathematics by natural philosophers was by no means inevitable, but had to be carefully managed by early modern mathematical practitioners. The case of Fernel indicates that perhaps he was not the only would-be mathematical practitioner to abandon mathematics in favor of a calling that was more appreciated by contemporaries.

Ludic proof: Greek mathematics and the Alexandrian aesthetic

This book represents a new departure in science studies: an analysis of a scientific style of writing, situating it within the context of the contemporary style of literature. Its philosophical significance is that it provides a novel way of making sense of the notion of a scientific style. For the first time, the Hellenistic mathematical corpus - one of the most substantial extant for the period - is placed centre-stage in the discussion of Hellenistic culture as a whole. Professor Netz argues that Hellenistic mathematical writings adopt a narrative strategy based on surprise, a compositional form based on a mosaic of apparently unrelated elements, and a carnivalesque profusion of detail. He further investigates how such stylistic preferences derive from, and throw light on, the style of Hellenistic poetry. This important book will be welcomed by all scholars of Hellenistic civilization as well as historians of ancient science and Western mathematics.

Changes and Continuities in the Use of Diagrams Tu in Chinese Mathematical Writings (Third Century to Fourteenth Century) [I

This article aims at highlighting a radical change in the materiality of tu between the time when they were first mentioned in Chinese mathematical texts in the third century commentaries on Canons and the thirteenth century, from which there are abundant illustrations in treatises. Moreover, it intends to highlight that the meaning of the word tu 圖, as used in mathematical writings, greatly changed over the same time span. It argues that third century tu 圖 were material objects, cut in paper with squared-grid, and worked out in specific ways. They probably always displayed particular dimensions and only represented objects for plane geometry. Their areas, and not their points, were marked, and they were marked by characters or colors. Areas were cut into pieces and rearranged. Such is the contribution mathematical texts can offer for capturing the nature of tu for these early periods. In contrast to this, thirteenth century tu 圖 to which mathematical texts refer were included in the texts themselves and hence articulated with the discourse on the surface of the page. Moreover, the extension of what could be represented in a tu 圖 increased tremendously. However, as I show in part II of this paper, in the thirteenth century, several traditions must be distinguished, regarding the nature of tu 圖 and the way in which they were integrated into the text. Moreover, part II shows that despite this break in the nature of tu 圖, some thirteenth century mathematicians inherited ways of working with tu 圖 from earlier times. I argue that this occurred within the framework of a specific mathematical domain, that is, a given subtradition. The mathematicians operating within this framework brought into play the same markers (colors, characters) for areas and adapted the operations onto paper. However, what is most interesting is that they made use of these traditional ways of working with figures while bestowing new mathematical meanings upon them. This thus presents an interesting case of continuity and rupture within a given tradition. All these uses of figures, in their variety, are specific to China and differ from the way in which other traditions used figures in mathematics.

Changes and Continuities in the Use of Diagrams Tu in Chinese Mathematical Writings (Third Century to Fourteenth Century) [I

This article aims at highlighting a radical change in the materiality of tu 圖 between the time when they were first mentioned in Chinese mathematical texts in the third century commentaries on Canons and the thirteenth century, from which there are abundant illustrations in treatises. Moreover, it intends to highlight that the meaning of the word tu 圖, as used in mathematical writings, greatly changed over the same time span. It argues that third century tu 圖 were material objects, cut in paper with squared-grid, and worked out in specific ways. They probably always displayed particular dimensions and only represented objects for plane geometry. Their areas, and not their points, were marked, and they were marked by characters or colors. Areas were cut into pieces and rearranged. Such is the contribution mathematical texts can offer for capturing the nature of tu for these early periods. In contrast to this, thirteenth century tu 圖 to which mathematical texts refer were included in the texts themselves and hence articulated with the discourse on the surface of the page. Moreover, the extension of what could be represented in a tu 圖 increased tremendously. However, as I show in part II of this paper, in the thirteenth century, several traditions must be distinguished, regarding the nature of tu 圖 and the way in which they were integrated into the text. Moreover, part II shows that despite this break in the nature of tu 圖, some thirteenth century mathematicians inherited ways of working with tu 圖 from earlier times. I argue that this occurred within the framework of a specific mathematical domain, that is, a given subtradition. The mathematicians operating within this framework brought into play the same markers (colors, characters) for areas and adapted the operations onto paper. However, what is most interesting is that they made use of these traditional ways of working with figures while bestowing new mathematical meanings upon them. This thus presents an interesting case of continuity and rupture within a given tradition. All these uses of figures, in their variety, are specific to China and differ from the way in which other traditions used figures in mathematics.

Beyond templates 2: exploring students’ approaches to learning on a mathematical writing course

This is the second of two articles on Developing Students’ Learning through Mathematical Writing (a Maths, Stats & OR Network (MSOR Network) funded mini-project) at Queen Mary, University of London (QMUL). These articles report on the Mathematical Writing (MW) course at QMUL, devised by Professor Franco Vivaldi. The investigation into student learning on the MW course was carried out by staff in the Thinking Writing (TW) initiative at QMUL which works to develop the teaching and assessment of writing (see http://www.thinkingwriting.qmul.ac.uk/).

Imagination and Layered Ontology in Greek Mathematics

This essay is a study in the routine use of imagination in Greek mathematical writings. By "routine" is meant both that this use is indeed very common, and that it is ultimately mundane. No flights of fancy, no poetical-like imaginative licenses are at stake. The issue rather is the systematic use made of the ability to imagine a virtual presence, and to refer to this virtual presence as if it were on equal footing with the real. This process is not merely routine in Greek mathematics: it may well be considered one of its chief characteristics. In this essay, I describe some features of this practice in detail, and then briefly offer some interpretative comments on the history and philosophy of this practice. The emphasis, however, is on the description, and I concentrate mainly on one key element—the use of the Greek verb noein . Thus the structure of the essay is as follows: part 1 is an analysis of noein in Greek mathematical writings; part 2 is a less-detailed description of other practices of Greek mathematics that involve "imagination" or, in general, a "layered" reality; and part 3 is a tentative interpretation.

Clavius, Proclus, and the Limits of Interpretation: Snapshot-Idealization Versus Projectionism

It is hard to overestimate the role played by Christopher Clavius (1538–1612), professor of mathematics at the Collegio Romano from 1565 until his death, in the establishment of mathematics as an incontestable part of the Jesuit curriculum and body of thought. Besides his direct influence on the constitution of the mathematical part(s) of the Ratio studiorum, Clavius’s mathematical writings, ranging from introductory textbooks to commentaries to polemical pamphlets, almost immediately began to serve as standard references, despite their — occasionally — controversial character. Today, Clavius’s contentious stand towards the older school at the Collegio embodied in the person of Benedictus Pereira, who held an extremely negative view on mathematics, is regarded as a decisive factor in the sixteenth-century renaissance of mathematics, both within and outside the Society. Nowadays, most scholars seem to agree that the philosophical framework of Clavius’s institutional struggle can be found in the Prolegomena to his edition of Euclid’s Elements, first published in Rome in 1574. According to some, the general outlines of this preface reflect a philosophy of mathematics that is largely indebted to the one presented by the Neoplatonist Proclus (410/412 – 485) in the double prologue of his commentary on the first book of Euclid’s Elements. Without wanting to obscure the importance of Clavius’s revolutionary project, I will try to sketch a different picture of Clavius’s philosophy of mathematics, both on the ontological and on the epistemological level. In this re-evaluation, the limits of both Proclus’s influence and the interpretation of his commentary on Euclid will be my primary focus. Clavius’s views on the nature of mathematical objects will be shown to be incompatible with the two conventional modes of interpretation found in the ancient commentators, namely the projectionism maintained by Proclus on the one hand, and the constructive abstractionism of writers such as Ammonius and Philoponus on the other. For Proclus, mathematical objects are extended and divisible projections of the unextended and indivisible ideas with which our understanding is equipped. The imagination acts both as the faculty that projects these ideas and as the imaginary screen on which they are projected. The constructive abstractionism read in Ammonius, for example, sees mathematical entities as existing only in the mind of the mathematician who in thought has separated, i.e. abstracted, them from matter. Clavius’s position will lay bare the need to invoke a third category, namely that of a ‘snapshot-idealization’: in mathematics, according to Clavius, sensible objects are indeed considered free from matter, but they are not attributed any properties that they do not perfectly instantiate as sensible objects. As in a photograph, certain

Procedural Explanations in Mathematics Writing

This study analyzes the procedural explanations written by remedial college mathematics students. Relevant literatures suggest that six communication activities might be key in effective procedural explanations in mathematics writing: (a) orienting the learner, (b) providing kernels or definitions of concepts and procedures, (c) using exemplars or worked examples, (d) providing descriptions of the process or procedure, (e) solidifying learner understanding, and (f) facilitating linguistic control of mathematical terms. Using this framework, 18 practices or types of difficulties were discovered in students' written explanations. Independent experts consistently evaluated student explanations more highly when the explanations contained arithmetic or algebraic exemplars, described specific actions and their meanings, linked new with prior knowledge, and used descriptive language; experts evaluated student explanations more negatively when students displayed difficulties reasoning with kernels, reasoning with exemplars, or with describing processes.

Beyond templates: exploring mathematical concepts through writing

Beyond templates: exploring mathematical concepts through writing Volume/Issue Vol 8, Issue 3 Date Wednesday, 1 October, 2008 Journal Name MSOR Connections Author(s) Teresa McConlogue Sally Mitchell Will Clavering Abstract This is the first of two articles on the Developing Students’ Learning through Mathematical Writing project at Queen Mary, University of London (QMUL) funded by the Maths, Stats & OR Network (MSOR Network). This project aims to evaluate the Mathematical Writing (MW) course at QMUL and the postgraduate course for teaching assistants which prepares post graduate teaching assistants (PGTAs) for tutoring and marking on the MW course. The evaluation is being carried out by the staff of the Thinking Writing (TW) initiative which works with academics at QMUL to explore ways of helping students learn through writing and become better writers. In this short piece, we give a brief introduction to the issues we are exploring in the evaluation; the later article will provide a fuller account and make suggestions for the development of the mathematical writing approach in future years. DOI doi: 10.11120/msor.2008.08030029 File upload

Understanding Students' Problem-Solving Knowledge through Their Writing

The Curriculum and Evaluation Standards for School Mathematics, published in 1989 by NCTM, placed an emphasis on the importance of communication for learning and doing mathematics. Writing is one way to achieve this valuable communication in the classroom. In 2000, Principles and Standards for School Mathematics emphasized writing as being an important aspect of communication: “Writing in mathematics can also help students consolidate their thinking because it requires them to reflect on their work and clarify their thoughts about the ideas developed in the lesson' (NCTM 2000, p. 61). Several studies have also shown ways that writing can be an effective tool for improving students' learning of mathematics (Bell and Bell 1985; Pugalee 2001; Steele 2005). Pugalee (2001) found in his work with students that writing could potentially help students increase their problemsolving ability in mathematics.

Logic for Mathematical Writing

In the School of Mathematical Sciences at Queen Mary in the University of London we have been running a module that teaches the students to write good mathematical English. The module is for second-year undergraduates and has been running for three years. It is based on logic, but the logic—though mathematically precise—is informal and doesn't use logical symbols. Some theory of definitions is taught in order to give a structure for mathematical descriptions, and some natural deduction rules form a basis for writing mathematical arguments. Alongside this logical material, the students have weekly exercises that involve writing informal explanations of simple mathematical ideas for non-mathematicians.

L'ALGÈBRE ARABE DANS LES TEXTES HÉBRAÏQUES (II). DANS L'ITALIE DES XVe ET XVIe SIÈCLES, SOURCES ARABES ET SOURCES VERNACULAIRES

Until the end of the 14th century, the sources of Hebrew mathematical writings were almost exclusively in Arabic. This was particularly true of texts that contained elements of algebra or algebraic developments. The testimonies we present and analyze here are due to Jewish authors living in Italy, primarily in the 15th century, who made use of the most varied sources, in addition to Arabic: in Castilian, in Italian, and perhaps in Latin. These testimonies constitute both an indication, and a product, of the circulation of Arab algebraic traditions in Renaissance Italy. Simon Moṭoṭ’s book on The Calculation of Algebra stems from the Italian tradition of ‘‘treatises on the abacus’’. Mordekhay Finzi of Mantua is the author of a Hebrew version of the great work on algebra by Abū Kāmil (9th century), as well as of a version, distinct from the preceding, of the Arabic scholar’s introductory exposition. Beginning in 1473, Finzi also translated from Italian to Hebrew the important treatise on algebra by Maestro Dardi of Pisa (1344). We also indicate some 16th century continuations of Hebrew mathematical production, which contain algebraic developments.

Aiming for Understanding: Lessons Learned about Writing in Mathematics: Reflect and Discuss

Finding ways to help students effectively communicate their mathematical thinking is a challenge for teachers across the country. Through a Professional Development Schools (PDS) partnership between the University of Maryland and several elementary schools in a neighboring district, school and university partners worked together to explore, discuss, and reflect on the challenges of helping students write about mathematics. Along with gaining insights into effective instructional strategies, partners from both institutions gained a deeper appreciation for the benefits of school-university dialogue.

Bibliography

Bibliography

Remarks on Mathematical Terminology in Medieval Latin : Greek and Arabic Influences

The development of mathematical terminology in the Latin Middle Ages has not yet been systematically investigated ; there is in print no report on the whole field, only three general articles 1 and occasional specialized information on a special text. Accordingly, my talk is based on my own work on mathematical texts of this time. They show that there were two different traditions. One of these came from Greek mathematical writings which were known in Latin by translations and reworkings made in Roman times and late antiquity they were a formative influence on medieval culture. The second tradition begins with the translations from Arabic into Latin in the 12th century : thereby a large number of Greek writings became known and at the same time Indian and Arabic mathematics, which in part belonged to the same tradition. In the following I should like to describe these traditions and to characterize their principal trends with the help of a few examples.

Research, Reflection, Practice: Writing to Reflect in a Mathematics Methods Course

For more than a decade, several authors have highlighted the benefits to students of writing to learn mathematics. Writing is an important component of communication in the classroom. As Principles and Standards for School Mathematics (NCTM 2000) notes, “Writing in mathematics can also help students consolidate their thinking because it requires them to reflect on their work and clarify their thoughts about the ideas developed in the lesson” (p. 61). Teachers probably will not use this tool, however, unless they have had the experience themselves of writing in relation to mathematics. This article presents a brief review of the benefits of students writing to learn mathematics. In the second part of the article, we invite the reader to consider another possible use of writing: as a tool to help preservice teachers reflect on their own growth as they learn to teach mathematics. We discuss some of the benefits that writing has for prospective teachers and present examples of preservice elementary teachers' writing that were collected in several one-semester undergraduate mathematics methods courses that the first author taught. The second author participated as a student in one of the courses. In a second article to be published in this journal, we will focus on the process of writing and writing for an audience.

Mathematical reasoning: writing and proof

1. Introduction to Writing in Mathematics. 2. Logical Reasoning. 3. Constructing and Writing Proofs in Mathematics. 4. Set Theory. 5. Mathematical Induction. 6. Functions. 7. Equivalence Relations. 8. Topics in Number Theory. 9. Topics in Set Theory. A: Guidelines for Writing Mathematical Proofs. B: Answers and Hints for Selected Exercises. C: List of Symbols.