This work deals with chess and mathematical thinking. In the last years the interest in chess activity by educational agencies notably increased. Chess is an historical strategy game, played over the world with the same rules. The International Chess Federation (FIDE) has 181 member countries. In this book, chess is seen from different points of view: cognitive, epistemological and historical. Chess and mathematics have several common features, in particular about logic and geometrical concepts. Is chess a useful tool for Education, in particular for Mathematics Education? This book tries to give a response to this question, but, as a consequence of reflections about the nature of the teaching/learning processes and about this experimental work, it could be more correct to reformulate the question in a different way: What conditions, methods and approaches are advisable to make chess a useful practice for Education, in particular for Mathematics Education?
In the book, the authors provided broad information about the mathematical proof, its types, teaching mathematical proof, as well as about the essential role of the mathematical proof in development of the logical thinking. Besides, the proof by the method of analysis and synthesis, proof by contradiction, and proof by analogy had been widely analysed and presented to the reader. The authors came to the conclusion that the mathematical proof is an important mean in development of the logical thinking.
Focusing on the environment of learning of a PD course and on teacher’s classroom work, the aim of this study was to find links between what teachers learned about Mathematical Thinking Activities (Wiskundige Denkactiviteiten – WDA) with their work in classroom. Therefore, the study attempted to identify similarities, changes or empowerment on teacher’s pedagogical content knowledge (knowledge about how to teach a subject) and teacher’s professional competence (person’s ability to transfer knowledge into action) (Maass, K.; Doorman, M.; pg. 890; 2013). In order to carry on the research, the study was developed in Netherlands under the two research questions: • In terms of changes in teachers' practices when working with a WDA activity, how are the empowerments of teaching approaches? Which changes emerge on these approaches? • To what extent does a WDA course contribute to the empowerment of teachers’ competence in working with inquiry learning? On this study, WDA specification is related with an Inquiry – Based way of teaching and learning. Thus, it was expected that teachers could recognize and/or develop WDA tasks with characteristics of Inquiry Based Learning (IBL) Theory.
Drawing on instructive stories, this book reveals the strategic ways of thinking that always give a player - in life as in chess - the edge. It also reveals how and why the game of chess is a fitting and powerful teacher of how to be prepared for, and how to win in, even the most competitive situations.
"Chess and Maths" is an investigation that discovers links between mathematics and a game of chess. Algorithms, Theories and its examples considered in the book prove that when making a move you're actually solving a maths problem that you aren't aware of. Also it was shown how a computer functioned by a machine programming plays chess via mathematic data. Investigations with computer programming led to imagine the way computer sees the chess board and how it thinks via numbers. Part II analyses the most productive and efficient openings currently in a game of chess by usage of maths. Mathematical methods of set theory were used to find a probability of success of the openings, while heuristic evaluation function was used to examine productiveness of the openings.
Analyzing mathematical thinking while solving problems has been an interest for many researchers although it is rather challenging even in paper-and-pencil environments. Researchers, could not find a direct method to monitor and analyze mathematical thinking, have developed indirect methods such as thinking/talking aloud, keeping log files, and eye-tracking to analyze mathematical thinking. The Frame Analysis method was developed by Dr. Karadag to add a new approach to the field. The method is basically based on recording student activities in computer screen by employing a screen capture software and analyzing the captured data frame by frame. It is a qualitative analysis approach to the thick data collected from the field. The work done was completed under the supervision of Professor Douglas McDougall. I deeply acknowledge his contribution as the supervisor. It would not be made possible without his support.
Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. Mathematical Thinking and Writing teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point. Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure. After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas.* Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction* Explains identification of techniques and how they are applied in the specific problem* Illustrates how to read written proofs with many step by step examples* Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter* The Instructors Guide and Solutions Manual points out which exercises simply must be either assigned or at least discussed because they undergird later results
The construct of mathematical identity has recently been widely used in mathematics education with the intention to understand how students relate to and engage (or disengage) with mathematics. Mathematical identity is defined as the students’ knowledge, abilities, skills, beliefs, dispositions, attitudes and emotions, that relates to mathematics and mathematics learning. A key part of this relationship that students have with mathematics is the students’ evolving sense of self to understand how mathematics fits with this self. Research shows that students’ identity has many facets or multiple identities that are formed throughout their life history; engagement with their peers, family, and teachers; as well as engagement with mathematical tasks. Doing mathematics can be viewed as mathematical activity that involves integrating mathematical thinking by using mathematical facts and knowledge, and requires active student learning. Mathematical tasks used in the classroom form the basis for student’s learning and different tasks are used to develop different types of skills and thinking. These tasks often appear in curricular or instructional materials in textbooks.
Creative and critical thinking has always been the driving force of compositional activity. Combined with advanced signal processing, an integrative and creative framework can be flourished, leading to the origination, conception and discovery of musical ideas. In this book, the way mathematical-based approaches, such as non-linear dynamics, stochastic analysis, game theory and neural networks, could be used as organisational basis of the sound characteristics is explored, creating abstractions (both in macro- and micro-structural level) from which potential structural relationships are made. To this end, sophisticated control of effective pre-compositional planning, by employing computer-based programming, is presented, in an attempt to develop models and strategies with a high degree of generalisation, directly transferable into a music-generating outcome. The realisation procedure of these models and strategies into actual musical works unveils hidden procedural relations and behaviours and controls effectively the statistical characteristics of the compositional building units, such as their distribution in time, frequency and space domains.
Thinking is the most precious cognitive ability with which man is elevated among all animal.Thinking has several kinds and each kind has several components.This book has given a clear analysis of the different kinds of thinking and focuses on the Impact of Critical Thinking Skills on achievement in Mathematics at secondary school.Critical Thinking Skills in mathematics is the ability and disposition to incorporate prior knowledge,mathematical reasoning and cognitive strategies to generalise,prove or evaluate unfamiliar mathematical situations in a classroom for reflective manner.Students must be stimulated to think critically on their own to resolve dilemmas,take stands on issues,judge propositions about knowledge or ideas at school level.Successful mathematics teaching and learning process involves practice of critical thinking skills through Mathematics.The mathematics teacher should make sincere and consistent effort in acquiring and developing abilities and skills by learners in the classrooms.
The mental power one gets from learning mathematics is the acquisition of the art of proper thinking and effective reasoning. An epidemic model was formulated include a memory term which gives information on the past history of the disease that helped improved our medical understanding of infectious diseases.
The aim of this research was to investigate the level of conceptual and procedural understanding of linear algebra concepts among first and second year university students. An initial pilot study provided enough evidence, to convince the researcher to design a theoretical framework to pursue an alternative approach for the teaching and learning of a group of linear algebra concepts. Based on the methodology a number of case studies were carried out and the outcome from the tests, interviews and concept maps were analysed. It is important to note that, it is not the intention of this study to generalise these findings for all the mathematics students of this, or any other university. However, the results indicate strong reasons for proposing further investigation in teaching and learning of linear algebra concepts.
Chess Metaphors – Artificial Intelligence and the Human Mind
Machers and Rockers – Chess Records and the Business of Rock and Roll