Mathematics

This article presents the findings of a case study which introduced online quizzes as a form of assessment in pure mathematics. Rather than being designed as an assessment of learning, these quizzes were designed to be an assessment for learning; they aimed to academically support students in their transition from A-Level mathematics to university-level pure mathematics by providing an extrinsic motivation to engage them with their learning material early on and to emphasise the small details within proofs, such as defining notation, which are not necessarily emphasised by written homework assignments. The results obtained during the two-year study using online quizzes show e-assessment to be a powerful complementary tool to traditional written homework assignments.

In this note, we show that each positive rational number can be written as φ( m 2 )/φ( n 2 ) , where φ is Euler's totient function and m and n are positive integers.

How were surfaces evaluated before the invention of the sexagesimal place value notation in Mesopotamia? This chapter examines a group of five tablets containing tables for surfaces of squares and rectangles dated to the Early Dynastic period (ca. 2600-2350 BCE) and unearthed in southern Mesopotamia. In order to capture the methods used by ancient scribes to quantify surfaces, special attention is paid to the layout and organization of the tables, as well as to the way in which measurement values are written down. It is argued that these methods vary according to the dimensions of the squares or rectangles concerned: the quantification of small surfaces does not use the same mathematical tools as the quantification of large parcels of land. The chapter shows a reciprocal influence between the metrological systems adopted by the ancient scribes and the methods of calculation of surface they implemented. Some methods may reflect ancient land-surveying practices, and others may testify the emergence of new mathematical concepts applied to all kinds of surfaces, large or small. Ultimately, several different conceptualizations of the notion of surface emerge from the examination of these tables.

In this short paper we show that with a small change of the common ruler and compass construction of the regular pentadecagon, we can produce more regular polygons

We give a transcription of a letter from Eisenstein's parents to Gauss, and an unpublished proof of the quadratic reciprocity law by Eisenstein using the tangent function.

We present a study of the implementation of the Electronic Preparatory Test for beginning undergraduates reading mathematics. The Test comprises two elements: diagnostic and self-learning. The diagnostic element identifies gaps in the background knowledge, whilst the self-learning element guides students through an upcoming material. The Test lends itself to an early identification of weak and strong students coming from a wide range of background, allowing follow-ups to be made on a topic-specific basis. The results from the Tests, collected over three years, correlate positively with end-of-year examination results. We show that such a Preparatory Test can be a better predictor of success in the first-year examination in comparison with university entry qualifications alone.

This is a textbook about elementary number theory, with emphasis on classical topics around the Euklidean Algorithm.

This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important mathematical ideas and theories. The materials can be used by pupils and students for self-study, and by teachers. This is an abridged pre-copyedit version of the published book submitted with the permission of the publisher. Each included individual material is self-contained and ready-for-use. Solutions to problems are not included intentionally. This collection consolidates updates of several arXiv submissions, e.g., arXiv:1305.2598.

The Fields Medal, often referred as the Nobel Prize of mathematics, is awarded to no more than four mathematician under the age of 40, every four years. In recent years, its conferral has come under scrutiny of math historians, for rewarding the existing elite rather than its original goal of elevating mathematicians from under-represented communities. Prior studies of elitism focus on citational practices and sub-fields; the structural forces that prevent equitable access remain unclear. Here we show the flow of elite mathematicians between countries and lingo-ethnic identity, using network analysis and natural language processing on 240,000 mathematicians and their advisor-advisee relationships. We found that the Fields Medal helped integrate Japan after WWII, through analysis of the elite circle formed around Fields Medalists. Arabic, African, and East Asian identities remain under-represented at the elite level. Through analysis of inflow and outflow, we rebuts the myth that minority communities create their own barriers to entry. Our results demonstrate concerted efforts by international academic committees, such as prize-giving, are a powerful force to give equal access. We anticipate our methodology of academic genealogical analysis can serve as a useful diagnostic for equality within academic fields.

The article presents simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertexes of the cones is a hyperbola. The hyperbola has focuses which coincidence with the ellipse vertexes. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the focuses of the ellipse are located in the hyperbola's vertexes. These two relationships create kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is perfect example of mathematical beauty which may be shown by use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.