Sandbox stuff

modular group

Applications to Number Theory

[...include here at least some mention of quadratic forms, the fundamental domain (modular curve) and modular forms...]

Relationship to Lattices

The lattice Δ_τ is the set of points in the complex plane generated by the values 1 and τ (where τ is not real). So

$\Delta _{\tau }=\{m+n\tau :m,n\in Z\}$

Clearly τ and -τ generate the same lattice i.e. Δ_τ=Δ_-τ. In addition, τ and τ' generate the same lattice if they are related by a fractional linear transformation that is a member of the modular group.

Relationship to Quadratic Forms

The set of values taken by the positive definite binary quadratic form am²+bmn+cn² is related to the lattice Δ_τ as follows :-

$\{am^{2}+bmn+cn^{2}:m,n\in Z\}=\{a|z|^{2}:z\in \Delta _{\tau }\}$

where τ is chosen such that Re(τ)=b/2a and |τ|² = c/a.

Congruence Subgroups

[...brief mention and definition of congruence subgroup, this really deserves its own article independent of Γ]

Julia set

Examples

Investigation of the shape and structure of the Julia sets of iterated maps on the complex plane requires advanced mathematics. However, examples of Julia sets arise from simpler maps of the real line to itself.

Example 1

x_{n+1}=x_{n}^{2}

The behavour of x_n depends on the initial value x₀ as follows :

x_{0}<-1\Rightarrow x_{n}\rightarrow -\infty

x_{0}>1\Rightarrow x_{n}\rightarrow \infty

Mathematics education

History

Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, a formal education was only available to male children with a sufficently high status, wealth or caste.

In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.

The first mathematics textbooks to be written in English were published by Robert Recorde, beginning with The Grounde of Artes in 1540.

In the Renaissance the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy.

This trend was somewhat reversed in the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics, established by the University of Cambridge in 1662. However, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661.

In the eighteenth and nineteenth centuries the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age.

By the twentieth century mathematics was part of the core curriculum in all developed countries. However, diverse and changing ideas about the purpose of mathemtical education led to little overall consistency in the content or methods that were adopted.

Objectives

At different times and in different cultures and countries, mathematical education has attampted to achieve a variety of different objectives. These objectives have included :-

The teaching of basic numeracy skills to all pupils
The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry) to most pupils, to equip them to follow a trade or craft
The teaching of abstract mathematical concepts (such as set and function) at an early age
The teaching of selected areas of mathematics (such as Euclidean geometry) as an example of an axiomatic system and a model of deductive reasoning
The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the modern world
The teaching of advanced mathematics to those pupils who wish to follow a career in science

Methods of teaching mathematics have varied in line with changing objectives.

Standards

Levels

Methods

Methods of teaching mathematics include the following:

Rote learning - the teaching of mathematical results, definitions and concepts by repetition and memorisation. Typically used to teach multiplication tables.
Exercises - the teaching of mathematical skills by completing large numbers of exercises of a similar type, such as adding fractions or solving quadratic equations.
New math - a method of teaching mathematics which focusses on abstract concepts rather than practical applications.

The method or methods used in any particular situation are largely determined by the obectives that the relevant educational system is trying to achieve.

Cultural differences

Mathematics teachers

The following people all taught mathematics at some stage in their lives, although they are better known for other things :-

Lewis Carroll, pen name of British author Charles Dodgson, lectured in mathematics at Christ Church, Oxford
John Dalton, British chemist and physicist, taught mathematics in schools and colleges in Manchester, Oxford and York
Tom Lehrer, American songwriter and satirist, taught mathematics at Harvard and MIT
Georg Joachim Rheticus, Austrian cartographer, taught mathematics at the University of Wittenberg
Edmund Rich, Archbishop of Canterbury in the 13th century, lectured on mathematics at the universities of Oxford and Paris
Archie Williams, American athlete and Olympic gold medallist, taught mathematics at high schools in California

External links

History of Mathematical Education

Scraps

Philosophy of mathematics education

The philosophy of mathematics education could be imagined to be a dry and overly academic domain. But there are issues central to it that have sparked great controversy within it in the past decade or two. Within the five clusters of questions identified above some selected controversies are as follows.

1. Philosophy of Mathematics

As one of the oldest sciences, and as the paradigm of certain and cumulative knowledge, mathematics and its philosophy seems an unlikely area for controversy. But currently the so-called ‘Science Wars’ are raging, mostly in USA, but also in other English-speaking countries, about philosophical views of science and mathematics. Although primarily ignited between realist and social constructivist or science studies accounts of the nature of science, the heated debate has also spilled over into the domain of mathematics. Foundationalists and absolutists, on the one hand, want to maintain that mathematics is certain, cumulative and untouched by social interests or developments beyond the normal patterns of historical growth. Fallibilists, humanists, relativists and social constructivists, on the other hand, have been arguing that mathematics is through and through historical and social, and that there are cultural limitations to its claims of certainty, universality and absoluteness. This controversy can become very heated and even emotionally charged, as correspondence in the American Mathematical Monthly and Mathematical Intelligencer illustrates. Barnard and Saunders (1994) illustrate the negative reaction of some British mathematicians to the claims of fallibilism in philosophy of mathematics.

2. Aims of Mathematics Education

The aims of mathematics education can be a hotly contested area, especially when new curricula are being developed or disseminated through a national or regional education system. In Ernest (1991) I identified the aims of five different groups contesting the nature and aims of the British (for England Wales) National Curriculum in mathematics, during the late 1980s and early 1990s. In summary, these five groups and their aims were as follows:

1. Industrial Trainer aims - 'back-to-basics': numeracy and social training in obedience (authoritarian),

2. Technological Pragmatist aims - useful mathematics to the appropriate level and knowledge and skill certification (industry-centred),

3. Old Humanist aims - transmission of the body of mathematical knowledge (mathematics-centred),

4. Progressive Educator aims - creativity, self- realisation through mathematics (child-centred),

5. Public Educator aims - critical awareness and democratic citizenship via mathematics (social justice centred).

These aims are best understood as part of an overall ideological framework that includes views of knowledge, values, society, human nature as well as education. In Britain, the contestation between these groups was largely behind the scenes, although sometimes it spilled over into the public arena when interest groups sought to gain public support for their positions. My analysis suggested that the first three interest groups formed a powerful and largely victorious alliance in 1980s and 1990s Britain. This forced the aims of group 4 (Progressive Educators) to be compromised and filtered through those of group 2 (Technological Pragmatist) in order to have an impact on the curriculum. The aims of group 5 (Public Educators) were eliminated in this struggle, and had no impact at all. Similar struggles and contestations have been noted in other countries too.

3. Theories of Learning Mathematics

Elements of constructivist theories of learning can be traced back to antiquity, although perhaps the most influential modern source is Jean Piaget. None of this is very controversial. However, in the Summer of 1987 constructivism burst onto the international scene at the exciting and controversial Eleventh International Conference on the Psychology of Mathematics Education in Montreal. A number of distinguished speakers attempted a critique of radical constructivism, most notably the strong version due to Ernst von Glasersfeld (1995). As he had noted “To introduce epistemological considerations into a discussion of education has always been dynamite” (Glasersfeld 1983: 41). Ironically, the attacks on radical constructivism at that conference, which were intended to expose the weaknesses of the position fatally, served instead as a platform from which it was launched to widespread international acceptance and approbation. This is not without continuing strong critiques of constructivism from mathematicians and others (e.g., Barnard and Saunders 1994).

Since then, yet further controversy has erupted between different versions of constructivism, most notably radical constructivism versus social constructivism (Ernest 1994a), as well as powerful critiques of constructivism learning theory both within science and mathematics education, and from without.

4. Mathematics Teaching

The teaching of mathematics is also an area in which there can be heated and controversial clashes of philosophy or ideologies. Among the ‘hot’ areas and issues are the following

1. Mathematical pedagogy - problem solving and investigational approaches to mathematics versus traditional, routine or expository approaches? Such oppositions go back, at least, to the controversies surrounding discovery methods in the 1960s.

2. Technology in mathematics teaching – should electronic calculators be permitted or do they interfere with the learning of number and the rules of computation? Should computers be used as electronic skills tutors or as the basis of open learning? Can computers replace teachers, as Seymour Papert has suggested?

3. Mathematics and symbolisation – should mathematics be taught as a formal symbolic system or should emphasis be put on oral, mental and intuitive mathematics including child methods?

4. Mathematics and culture – should traditional mathematics with its formal tasks and problems be the basis of the curriculum, or should it be presented in realistic, authentic, or ethnomathematical contexts?

Each of these issues and oppositions has been the basis of heated debate and contestation world wide, and rests on philosophical issues and assumptions.

Georgy Fedoseevich Voronoy

Born: 28 April 1868 in Zhuravka, Poltava guberniya, Russia (now Ukraine) Died: 20 Nov 1908 in Warsaw, Poland

Georgy Voronoy studied at the gymnasium in Priluki graduating in 1885. He then entered the University of St Petersburg, joining the Faculty of Physics and Mathematics.

After graduating from St Petersburg in 1889, Voronoy decided to remain there and work for his teaching qualification. He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.

Voronoy lectured at Warsaw University, being appointed professor of pure mathematics there. He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg. In fact both Voronoy's master's thesis and his doctoral thesis were of such high quality that they were awarded the Bunyakovsky prize by the St Petersburg Academy of Sciences.

Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers. He extended work by Zolotarev and his work was the starting point for Vinogradov's investigations. His methods were also used by Hardy and Littlewood.

In 1904 Voronoy attended the Third International Congress of Mathematicians at Heidelberg. There he met Minkowski and they discovered that they were each working on similar topics.

The concept of Voronoi diagrams first appeared in works of Descartes as early as 1644. Descartes used Voronoi-like diagrams to show the disposition of matter in the solar system and its environs.

The first man who studied the Voronoi diagram as a concept was a German mathematician G. L. Dirichlet. He studied the two- and three dimensional case and that is why this concept is also known as Dirichlet tessellation. However it is much better known as a Voronoi diagram because another German mathematician M. G. Voronoi in 1908 studied the concept and defined it for a more general n-dimensional case.

Very soon after it was defined by Voronoi it was developed independently in other areas like meteorology and crystalography. Thiessen developed it in meteorology in 1911 as an aid to computing more accurate estimates of regional rainfall averages. In the field of crystalography German researchers dominated and Niggli in 1927 introduced the term Wirkungsbereich (area of influence) as a reference to a Voronoi diagram.

http://www.comp.lancs.ac.uk/~kristof/research/notes/voronoi/voronoi.pdf