“A valuable monument of mathematical genius”: The Ladies' Diary (1704–1840)

Our purpose is to view the mathematical contribution of The Ladies' Diary as a whole. We shall range from the state of mathematics in England at the beginning of the 18th century to the transformations of the mathematics that was published in The Diary over 134 years, including the leading role The Ladies' Diary played in the early development of British mathematics periodicals, to finally an account of how progress in mathematics and its journals began to overtake The Diary in Victorian Britain.     

Locally logical mathematics: An emerging teacher honoring both students and mathematics

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.     

The Elizabethan mathematics of everything: John Dee's ‘Mathematicall praeface’ to Euclid's Elements

This article considers John Dee's famous classification and justification of ‘the Sciences, and Artes Mathematicall’ in his Mathematicall praeface to Henry Billingsley's Elements of geometrie of Euclid of Megara (1570), the first English translation of Euclid. It is a revised version of a lecture presented to the British Society for the History of Mathematics Autumn Meeting, October 2010, under the title ‘John Dee and the Elizabethan Mathematics of Everything’.     

Measurement as relational,intensive and analogical: Towards a minor mathematics

Minor mathematics refers to the mathematical practices that are often erased by state-sanctioned curricular images of mathematics. We use the idea of a minor mathematics to explore alternative measurement practices. We argue that minor measurement practices have been buried by a ‘major’ settler mathematics, a process of erasure that distributes ‘sensibility’ and formulates conditions of mathematics dis/ability. We emphasize how measuring involves the making and mixing of analogies, and that this involves attending to intensive relationships rather than extensive properties. Our philosophical and historical approach moves from the archeological origins of human measurement activity, to pivotal developments in modern mathematics, to configurations of curriculum. We argue that the project of proliferating multiple mathematics is required in order to disturb narrow (and perhaps white, western, male) images of mathematics—and to open up opportunities for a more pluralist and inclusive school mathematics.     

Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks

We explore the conjecture that engaging teachers with activities which feature mathematical practices from the past (history-focused tasks) and in today’s mathematics classrooms (mathtasks) can promote teachers’ problematizing of mathematics and its pedagogy. Here, we sample evidence of discursive shifts observed as twelve mathematics teachers engage with a set of problematizing activities (PA) – three rounds of history-focused and mathtask combinations – during a four–month postgraduate course. We trace how the commognitive conflicts orchestrated in the PA triggered changes in the teachers’ narratives about: mathematical objects (such as what a function is); how mathematical objects come to be (such as what led to the emergence of the function object); and, pedagogy (such as what value may lie in listening to students or in trialing innovative assessment practices). Our study explores a hitherto under-researched capacity of the commognitive framework to steer the design, evidence identification and impact evaluation of pedagogical interventions.     

Interaction analysis of mathematical communication in primary teaching: The epistemological perspective

Communication between students and teacher in the mathematics classroom is a form of social interaction which focuses on a specific topic:mathematical knowledge. This knowledge cannot be introduced into classroom interaction “from the outside”, but grows through the communicative process, in the course of interactive exchanges between the participants of discussion. Although mathematical communication must be seen and analysed in the same way as any other form of communication, the particularity of interactive constructions of mathematical knowledge and its specificsocial epistemology within the context of teaching processes has to be taken into consideration. Also, the institutional influences of school institutions and those of teaching (analysed in the frame of general socio-interactive research approaches) must be considered. An epistemology-oriented interaction research approaches the specificity of amathematical classroom and communication culture in its analyses.     

Playing the believing game: Enhancing productive discourse and mathematical understanding

In mathematics classrooms, the practice of doubt pervades. However, Elbow (1986, 2006) contended that teachers must balance their practices of methodological doubt and methodological belief. The study reported here builds upon previous research which revealed the professor played the believing game (Elbow) and students were motivated to do mathematics. We addressed the question: How does a teacher (professor) play the believing game in a mathematics classroom? Videotapes, interviews, and field notes from an entire semester were collected and analyzed qualitatively. Although the professor was not consciously attempting to believe or doubt, we reveal when and under what circumstances they occurred. A temporal continuum of believing and doubting existed for the professor’s practice. Reserved believing and reserved doubting prevailed for the professor when she heard answers or comments she deemed incorrect and rich mathematical conversations transpired as she opened herself up to a deeper understanding of mathematics.     

Logical form,mathematical practice,and Frege's Begriffsschrift

For over a century we have been reading Frege's Begriffsschrift notation as a variant of standard notation. But Frege's notation can also be read differently, in a way enabling us to understand how reasoning in Begriffsschrift is at once continuous with and a significant advance beyond earlier mathematical practices of reasoning within systems of signs. It is this second reading that I outline here, beginning with two preliminary claims. First, I show that one does not reason in specially devised systems of signs of mathematics as one reasons in natural language; the signs are not abbreviations of words. Then I argue that even given a system of signs within which to reason in mathematics, there are two ways one can read expressions involving those signs, either mathematically or mechanically. These two lessons are then applied to a reading of Frege's proof of Theorem 133 in Part III of his 1879 logic, a proof that Frege claims is at once strictly deductive and ampliative, a real extension of our knowledge. In closing, I clarify what this might mean, and how it might be possible.     

Mathematics and Dialectics in the Soviet Union: The Pre-Stalin Period

During the 1920s, Soviet Marxist theorists paid less attention to developments in mathematics in their own country than to various manifestations of “mathematical idealism” in the West. Their criticism concentrated on set-theoretical studies, the theory of probability, mathematical logic, and the foundations of mathematics. Because of their disunity, the Marxist scholars did not present an obstacle to the work of mathematicians, dominated by the much-heralded Moscow school of mathematics, strong in the theory of functions of a real variable and its applications to topology and several other branches of mathematics. The end of the decade was marked by the beginning of Stalinist pressure to establish full ideological control over all branches of mathematics.Copyright 1999 Academic Press.MSC 1991 subject classifications: 01A60; 01A72; 01A74; 01A80.     

Conceptualizing inquiry-based education in mathematics

The terms inquiry-based learning and inquiry-based education have appeared with increasing frequency in educational policy and curriculum documents related to mathematics and science education over the past decade, indicating a major educational trend. We go back to the origin of inquiry as a pedagogical concept in the work of Dewey (e.g. 1916, 1938) to analyse and discuss its migration to science and mathematics education. For conceptualizing inquiry-based mathematics education (IBME) it is important to analyse how this concept resonates with already well-established theoretical frameworks in mathematics education. Six such frameworks are analysed from the perspective of inquiry: the problem-solving tradition, the theory of didactical situations, the realistic mathematics education programme, the mathematical modelling perspective, the anthropological theory of didactics, and the dialogical and critical approach to mathematics education. In an appendix these frameworks are illustrated with paradigmatic examples of teaching activities with inquiry elements. The paper is rounded off with a list of ten concerns for the development and implementation of IBME.     

Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

The Mathematics Attitudes and Perceptions Survey: an instrument to assess expert-like views and dispositions among undergraduate mathematics students

One goal of an undergraduate education in mathematics is to help students develop a productive disposition towards mathematics. A way of conceiving of this is as helping mathematical novices transition to more expert-like perceptions of mathematics. This conceptualization creates a need for a way to characterize students' perceptions of mathematics in authentic educational settings. This article presents a survey, the Mathematics Attitudes and Perceptions Survey (MAPS), designed to address this need. We present the development of the MAPS instrument and its validation on a large (N = 3411) set of student data. Results from various MAPS implementations corroborate results from analogous instruments in other STEM disciplines. We present these results and highlight some in particular: MAPS scores correlate with course grades; students tend to move away from expert-like orientations over a semester or year of taking a mathematics course; and interactive-engagement type lectures have less of a negative impact, but no positive impact, on students' overall orientations than traditional lecturing. We include the MAPS instrument in this article and suggest ways in which it may deepen our understanding of undergraduate mathematics education.     

Tracing the early history of algebra: Testimonies on Diophantus in the Greek-speaking world (4th–7th century CE)

The transmission and reception of the mathesis carried by Diophantus' Arithmetica has not attracted much attention among historians of Greek mathematics, who have devoted their scholarly activity almost exclusively to questions about the proper understanding of the character of the mathematical undertaking of the Alexandrian mathematician. As a result, the common belief is that Diophantus' Arithmetica is presented as an isolated, and thus uncontextualized phenomenon in the history of ancient Greek mathematics. The aim of this paper is to investigate testimonies and other piece of evidence suggesting that Diophantus' heritage was present in intellectual milieus of the Greek-speaking world during the late antique and early medieval times. Special emphasis is given to a number of scholia to the arithmetical epigrams of the Palatine Anthology which witness the persistence of the method of problem solving taught by Diophantus in the late antique world.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

The discursive footprint of learning across mathematical domains: The case of the tangent line

This paper employs the commognitive frame (Sfard, 2008) to investigate how experiences with tangents across mathematical domains leave their marks on students’ subsequent work with tangents. To this aim, I introduce the notion of the discursive footprint of tangents and its characteristics by reviewing how tangents are used across mathematical domains in school textbooks. Manifestations of this footprint were sought in 182 undergraduate mathematics students’ responses to a questionnaire about tangents by labelling their responses and by identifying patterns in the endorsed narratives. Manifestations include the identification of characteristics of sole (and combination of) discourses (geometry, algebra, calculus, mathematical analysis) in student responses. Five themes emerged from the analysis: apparent replication of word use in different narratives; geometry-local hybrid discourse; endorsement of conflicting narratives; enrichment of familiar narratives with new words; and, mathematical analysis as a subsuming discourse. Finally, I discuss the potency of the discursive footprint in research and teaching.     

Historia Mathematica: 25 Years/Context and Content

This year marks the completion of the first quarter-century ofHistoria Mathematica, which initially appeared exactly 25 years ago, in February of 1974, under the editorship of Kenneth O. May. The brief survey presented here is intended to illuminate the context of the journal within the history of periodicals for the history of mathematics (dating back to 1855 when Olry Terquem launched theBulletin de bibliographie, d'histoire et de biographie mathématiquesin Paris). The origins ofHistoria Mathematicaand its connections to the international community of historians of mathematics on the one hand and to mathematicians on the other are also discussed.Copyright 1999 Academic Press.La revue,Historia Mathematica(HM), apparut en février 1974, il y a vingt-cinq ans cette année, sous la direction de Kenneth O. May. L'étude présentée ci-dessous cherche à situerHMdans le contexte du développement historique des revues dévouées à l'histoire des mathématiques, dont la première fut leBulletin de bibliographie, d'histoire et de biographie mathématiquesfondé à Paris en 1855 par Olry Terquem. Les origines d'HMy sont discutées aussi, ainsi que les liens entre la revue et la communauté internationale d'historiens des mathématiques d'une part et la communauté mathématique internationale d'autre part.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Mathematics and Science Teachers' Use of and Confidence in Empirical Reasoning: Implications for STEM Teacher Preparation

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, particularly between the status of inductive and deductive modes of reasoning in mathematics. Through a specific set of mathematical conjectures, researchers explored differences between mathematics (n = 24) and science (n = 23) teachers' reasoning schemes, as well as the confidence they had in their justifications. Results from the study indicate differences between the two groups in terms of their levels of mathematical proof, as well as correlational trends that inform their confidence across these levels. Implications particularly for teacher training and preparation within the context of an integrated STEM education model are discussed.