On the Structure of Bispecial Sturmian Words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of \emph{all} Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167

Episturmian words: a survey

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more reference

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction

A geometrical approach to Palindromic Factors of Standard Billiard Words

Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes

A study of Scottish teachers' beliefs about the interplay of problem solving and problem posing in mathematics education

The Scottish Curriculum for Excellence (CfE) advocates that the learning and teaching of mathematical problem solving is no longer compartmentalised but is an overarching feature designed to improve higher order thinking skills at all levels by focusing on conceptual understanding. Comitantly, a growing body of literature acknowledges the interrelated educational benefits of mathematical problem posing within classrooms. Teachers’ beliefs are considered powerful indicators of professional practice and can articulate the positionality of teachers with regards to curricula reform. Despite their significance, research into the implementation of mathematical problem solving and mathematical problem posing is, as yet, under-researched particularly in Scotland. The main purpose of this study was to investigate Scottish teachers’ beliefs and espoused instructional practices of mathematical problem solving and mathematical problem posing. More prosaically, it explored beliefs regarding the nature of mathematics, the learning of mathematics and the teaching of mathematics. A mixed methods explanatory design consisting of an online questionnaire followed by semi-structured interviews was selected as the instruments to measure and capture espoused beliefs and reported practices. This study involved a representative sample of 478 participants (229 primary and 249 secondary mathematics practitioners respectively) generated from 21 local education authorities in Scotland. A supplementary feature of the online questionnaire, which harvested 87 volunteered comments, augmented the data collection process. Descriptive and inferential statistics were employed to analyse quantitative data with thematic coding used to organise and interrogate qualitative data. Factor analysis identified three distinct belief systems consistent with a dominant learner-centred approach (i.e. social constructivist, problem solving and collaborative orientation), mainly learner-centred approach (i.e. social constructivist, problem solving and static transmission orientation) and dominant teacher-centred approach (i.e. static and mechanistic transmission orientation). In other words, teachers’ deep-rooted beliefs do not align to one particular group of belief systems but are embedded mutually within a cluster. A mixture of positive, negative and inconsistent beliefs is reported. Significant dissonance exists between the sectors. Characteristics impacting on beliefs include grade and highest level of qualification in the field of education. This study suggests that the conceptualisation and operationalisation of mathematical problem solving and problem posing may be circumscribed in practice and that primary teachers hold stronger mathematical beliefs than secondary mathematics teachers. Several reasons help to illuminate these findings including a lack of pedagogical content knowledge, ineffective manifestations of mathematical creativity, low mathematics teaching self-efficacy and an over dominant national assessment culture. Implications and recommendations for policy and ITE are discussed