Tracing Robert's use of representations to solve counting problems: A 16-year case study

This study describes how Robert used his external representations (formulae, notations, sketches, models, and figures) to solve progressively more challenging counting tasks over a 16-year period. In his explorations of counting tasks, Robert discovered intricate connections between solutions to problems that looked different on the surface. Using video data from Robert's problem solving, analyses of his solutions are presented that shed light on how he built new ideas from existing ideas and how he modified external representations to make new mathematical discoveries and provide justifications for his solutions.     

Accommodation in the formal world of mathematical thinking

ABSTRACT

In this study, we examined a mathematician and one of his students’ teaching journals and thought processes concurrently as the class was moving towards the proof of the Fundamental Theorem of Galois Theory. We employed Tall's framework of three worlds of mathematical thinking as well as Piaget's notion of accommodation to theoretically study the narratives. This paper reveals the pedagogical challenges of proving an elegant theory as the events unfolded. Although the mathematician was conscious of the students’ abilities as he carefully made the path accessible, the disparity between the mind of the mathematician and the student became apparent.     

Characterization of the Relations in Grzegorczyk's Hierarchy Revisited

In his 1953's paper, Grzegorczyk proved that a certain kind of relation classes of Grzegorczyk's hierarchy could be characterized inductively. We give a simpler version of this characterization.     

Beginner’s mind and the middle years mathematics student

ABSTRACT

The Zen concept of beginner's mind describes how one's level of awareness can open one's mind to growth and possibilities, an attitude that would be beneficial for many mathematics students. In this naturalistic case study, two small groups of middle years students engage in the same mathematical task, one group demonstrating the characteristics of beginner's mind and the other not. One group focuses on making sure that all its members can follow the steps of one member's proposed solution. The other group, which displays characteristics of beginner's mind, explores the task thoroughly, and in its openness to ideas it notices mathematically salient details in the task that the first group overlooks completely. This paper suggests that through beginner's mind, students may develop habits of mind that enable them to attend to and think about mathematical possibilities deeply.     

Pure mathematics applied in early twentieth-century America: The case of T.H. Gronwall,consulting mathematician

Thomas Hakon Gronwall (1877–1932) was a Swedish-American mathematician with a broad range of interests in mathematical analysis, physics, and engineering. Though he was primarly known for his results in pure mathematics, his career as a “consulting mathematician” in America from 1912 to his death in 1932 provides a backdrop against which one can discuss contemporary issues involved in the increasing application of mathematics to engineering, industrial, and scientific problems. This paper attempts a summary of his major mathematical contributions to industrial, governmental, and academic institutions while relating his often difficult life during these years.     

Ky Fan (1914�C2010), he spent every waking moment thinking about mathematics

This survey article on Dr. Ky Fan summarizes his versatile achievements and fundamental contributions in the fields of topological groups, nonlinear and convex analysis, operator theory, linear algebra and matrix theory, mathematical programming, and approximation theory, etc., and as well reveals Fan’s exemplary mathematical formation opening up the beauty of pure mathematics, with natural conditions, concise statements and elegant proofs. This article contains a brief biography of Dr. Fan and epitomizes his life. He was not only a great mathematician, but also a very serious teacher known to be extremely strict to his students. He loved his motherland and made generous donations for promoting mathematical development in China. He devoted his life to mathematics, continued his research and published papers till 85 years old.     

Johann Heinrich Lambert,mathematician and scientist, 1728 – 1777

1977 is the two hundredth anniversary of the death of Johann Heinrich Lambert, a little known but nonetheless intriguing figure in 18th century science. In the general histories of science and mathematics Lambert's contributions are often described piecemeal, with each discovery and invention usually divorced both from the method by which he arrived at it and from the totality of his intellectual endeavour. To the student of optics he is remembered for his cosine law in photometry, to the astronomer for his work on comets, to the meteorologist for his design of a gut hygrometer, and to the mathematician for his work on non-Euclidean geometry and his demonstration of the irrationality of π and e. There is no doubt that each of these contributions had a definite importance of its own; but it is not the aim of the present article to enumerate in this way the high points of Lambert's scientific and mathematical work, rather to describe it for once as a unified whole, and to relate it to the contemporary intellectual outlook.     

In celebration of Prof. Dr. Rolf Klötzler 65th birthday

On January 11, 1996, Professor Klötzler celebrated his 65th birthday. This special issue of the journal “Optimization”, consisting of contributions by his friends, students and colleagues, is dedicated to him. Prof. Klötzler can proudly look back on forty successful years as a university teacher and mathematician. After positions at the Academy of Science, the Hochschule für Bauwesen in Leipzig and the Martin-Luther-University in Halle, where he was head of the Mathematics Institute from 1965 to 1971, he returned to the University of Leipzig in 1972 as a full Professor. He was a student there from 1949 to 1953 and had attended lectures by Ernst Hölder and Herbert Beckert. His scientific treatises are extremely extensive and have made great contributions to many branches of calculus of variations, optimal control and mathematical programming. The common thread running through his work is the solution of Hilbert’s problems concerning the calculus of variations and its development,control theory. Coming from the school of calculus of variations founded by Leon Lichtenstein and Otto Hölder in Leipzig, his scientific interest first focussed on questions concerning extension of field theory of calculus of variations and existence theory of global geodesic fields, from which optimality criteria could be derived. In addition, he was able to establish eigen value criteria for weak optimality of extremals of regular variational problems with multiple integrals. Fundamental results from these works appear in the frequently quoted textbook “MehrdimensionaleVariationsrechnung”, published in 1969.     

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.     

How mathematical impossibility changed welfare economics: A history of Arrow's impossibility theorem

During the 20th century, impossibility theorems have become an important part of mathematics. Arrow's impossibility theorem (1950) stands out as one of the first impossibility theorems outside of pure mathematics. It states that it is impossible to design a welfare function (or a voting method) that satisfies some rather innocent looking requirements. Arrow's theorem became the starting point of social choice theory that has had a great impact on welfare economics. This paper will analyze the history of Arrow's impossibility theorem in its mathematical and economic contexts. It will be argued that Arrow made a radical change of the mathematical model of welfare economics by connecting it to the theory of voting and that this change was preconditioned by his deep knowledge of the modern axiomatic approach to mathematics and logic.     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

On Farkas' lemma and related propositions in BISH

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.     

Robert Richard Anstice (1813–1853): a Hertfordshire bicentenary

R R Anstice was a little known English clergyman who made significant contributions to Combinatorics, anticipating results in modern design theory. The year 2013 is the 200th anniversary of his birth. Anstice took the first steps in treating Combinatorial Design Theory as part of Algebra, in retrospect anticipating many later developments. In particular he touched on an important structure now known as a ‘Room square’, named after the twentieth century Anglo-Australian mathematician T G Room. Here we review Anstice's short life and give an account of his mathematics.     

John Reynolds of the Mint: A mathematician in the service of king and commonwealth

In his youth, John Reynolds showed a talent for arithmetic and was destined for a career as a mathematician at the Tower Mint in London. He became skilled in the algorithms needed to determine the correct relationship between the weight and purity of coins and their values. This was a matter of national importance, and his work came to the attention of King James I, who reigned from 1603 to 1625, and his chief ministers, including Robert Cecil and Francis Bacon. It seemed that John might attain high office himself, but the murky administration of the early Stuart period cast its shadow over his career. Nevertheless, for the next forty years he continued to play a major part in the nation's affairs. He produced books of tables for the valuation of coins in the commercial world, and for the highly technical work of the assayers. Also, he was actively involved in the production of standard measures and instruments used by the excise officers. His life and works illustrate how mathematical ideas were employed by the English government in the period of the early Stuart kings and the Commonwealth.     

An Axiomatisation of the Conditionals of Post's Many Valued Logics

The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's (m, u)-conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values (cf. [5]). The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic (cf. [2]).     

Mathematical Literacy and Abstraction in the 21st Century

Ed Dubinsky received his Ph.D. in mathematics from the University of Michigan in 1962 and for the next 25 years engaged in research in theoretical mathematics. In the mid‐80′s he became interested in mathematics education and has worked exclusively in the area since then. In his research, he tries to understand how a person's mind might be working when he or she tries to understand mathematical concepts at the postsecondary level. Based on this research, he has conducted large‐scale curriculum development projects in calculus, discrete mathematics, abstract algebra and cooperative learning. He has been editor or co‐editor of UME Trends, Research in Collegiate Mathematics Education, and the Journal of Computers in Mathematics and Science Teaching. He has held faculty positions at 8 universities in 5 countries on 3 continents: Fourah Bey College (Sierra Leone), University of Ghana, Tulane University, McMaster University, Polish Academy of Sciences, Clarkson University, Purdue University, and Georgia State University. Dr. Dubinsky is presently retired and consults with several universities on education matters.     

Teaching Mathematics with a New Curriculum: Changes to Classroom Organization and Interactions

This report describes a high school mathematics teacher's decisions about classroom organization and interactions during his first two years using a new curriculum intended to support teachers' development of student-centered, contributive classroom discourse. In year one, the teacher conducted class and interacted with students primarily in small groups. In year two, he conducted more whole-class instruction. In both years, teacher-student interactions contained univocal and contributive discourse, but in year two the teacher sustained contributive discourse with students for longer periods. The teacher facilitated the most significant changes to classroom discourse in the instructional format with which he had the greatest experience (whole-class instruction). Over the period of this study, two key factors appeared to affect the teacher's decisions about classroom organization and interactions: his perception of students' expectations about mathematics classroom roles and activity, and his own discomfort associated with using a new curriculum. These areas are important candidates for future research about teachers' use of innovative mathematics curricula.     

W.F. Sheppard's correspondence with Karl Pearson and the development of his tables and moment estimates

As a new statistician, W.F. Sheppard wrote a series of letters to the leading statistician of the day, Karl Pearson, for his advice. Written a century ago and spanning three decades, the letters provide a glimpse into the development of two significant contributions to statistics: the normal probability tables, and the corrections of moment estimates. Sheppard's normal probability tables were the first set of modern tables for the standard normal distribution and have been widely used since the twentieth century. We provide an examination of the statistical research carried out by Sheppard and Pearson in the context of their correspondence.     

A Priori knowledge contextualised and Benacerraf’s dilemma

In this article, I discuss Hawthorne’s contextualist solution to Benacerraf’s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne’s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge.