https://www.openbookpublishers.com/product/979 Please note that our site uses cookies to provide basic functionality and feedback. By continuing to use our site you accept this use. More information is provided in our privacy page. [cross] Don't show me this again. Due to the current COVID-19 situation, our customers may experience some delivery delays. We apologise for any inconvenience this may cause. Open Book Publishers [open_ac] [ ] View my basket & checkoutMy Basket (0) Login / Register [facebook_s] [twitter-sm] [Blog-small] Currency [British Pounds ] [panel] Browse by Categories * Anthropology, Archaeology and Religion * Art and Music * Cinema and Photography * Classics * Digital Humanities * Economics, Politics and Sociology * Education * Environmental Studies * Health * History and Biography * History of the Book * History of Science * International Relations * Law * Literature, Language and Culture * Mathematics * Media Studies and Journalism * Open Field Guides * Philosophy * Sciences * Theatre * Women and Gender Studies * Textbooks and Learning Guides * All Books * Forthcoming Titles * SUBMISSIONS More on OBP * About * Our Vision * Information for New Authors * Prizewinning Books * Innovative Publications * Accessibility * What People Say About Us * Media Room * Support us * Sponsor Us * Contact Us * Donate * Vacancies, Training & Volunteering * FAQs Partnerships * Publishing Services and Partnerships * Library Membership Service Information * Ordering * Distribution and Retail * OBP Customise Beyond Publishing * Open Source Software * How We Collect our Readership Statistics Join Our Newsletter Please fill in your email address below to join our newsletter mailing list. [ ] [Join Now!] [twitter] [facebook] Homepage - Mathematics - The Essence of Mathematics Through Elementary Problems The Essence of Mathematics Through Elementary Problems The Essence of Mathematics Through Elementary Problems Alexandre Borovik and Tony Gardiner https://orcid.org/0000-0003-0808-8687 Paperback ISBN: 978-1-78374-699-6 PS18.95 Add to Cart Hardback ISBN: 978-1-78374-700-9 PS32.95 Add to Cart PDF ISBN: 978-1-78374-701-6 PS0.00 Download Click here to read the PDF online for free Click here to read the HTML online for free * Description * Contents * Authors * Copyright * Comments If you believe that offering your pupils an insight into what elementary mathematics, as encountered at school, is really about, then this book should be compulsory reading. - Mathematical Gazette, July 2020. It is clear in the text that the essence of mathematics is in the doing of mathematics rather than in particular subjects or concepts, which is not to say that concepts and ideas are not important. [... This] is certainly not a revolutionary claim [...]. And yet, I was often surprised by how this book made me rethink pretty elementary, even standard problems, and how it often led me to deeper matters in a very natural way. [... S]uch elementary problems have this power if they are set in the right context and, I might add, if they land in the lap of the right person. [...] [D]eeper lessons are [...] embedded in these very elementary problems. [... O]ne person might get more out of the book than another, but anyone reading it reflectively will be bound to get something out of it. [...] Who are the intended readers of this book? [... Anyone] interested in mathematics [... especially] mathematics educators. - Michael N. Fried (2020): From another tribe of mathematics educators, Mathematical Thinking and Learning,available here. This book is a marvel! From the outset [it] sets out its stall ... as a tool for developing ... rich mathematical understanding and experiences for students ... throughout their upper school mathematics. ... The book is a key tool in developing student conceptions of mathematics ... . [I]n addition the problems are ideal for teachers wishing to deepen [their own] understanding through a mixture of the novel and the classic. The problems have already found their way into both my classroom and my departmental meetings, but ... this is not a book it's possible to read without digging into the maths yourself! - Mathematics in School, May 2020 It is increasingly clear that the shapes of reality whether of the natural world, or of the built environment are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself tick , and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside. The Essence of Mathematics consists of a sequence of 270 problems with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline. The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works. The Essence of Mathematics Through Elementary Problems Alexandre Borovik and Tony Gardiner | June 2019 OBP Series in Mathematics, vol. 3 | ISSN: 2397-1126 (Print); 2397-1134 (Online) 389 pp. | 6.14" x 9.21" (234 x 156 mm) ISBN Paperback: 9781783746996 ISBN Hardback: 9781783747009 ISBN Digital (PDF): 9781783747016 DOI: 10.11647/OBP.0168 BIC subject codes: PB (Mathematics) | YQM (Educational: Mathematics and numeracy) | PBK (Calculus and mathematical analysis) | PBM (Geometry) | PBT (Probability and statistics); BISAC subject codes: MAT034000 (MATHEMATICS / Mathematical Analysis), MAT005000 (MATHEMATICS / Calculus), MAT012000, (MATHEMATICS / Geometry / General), MAT029000 (MATHEMATICS / Probability & Statistics / General); OCLC number: 1107447359 Preface About this text 1. Mental Skills 1. Mental arithmetic and algebra 1. Times tables 2. Squares, cubes, and powers of 2 3. Primes 4. Common factors and common multiples 5. The Euclidean algorithm 6. Fractions and ratio 7. Surds 2. Direct and inverse procedures 1. Factorisation 3. Structural arithmetic 4. Pythagoras' Theorem 1. Pythagoras' Theorem, trig for special angles, and CAST 2. Converses and Pythagoras' Theorem 3. Pythagorean triples 4. Sums of two squares 5. Visualisation 6. Trigonometry and radians 1. Sine Rule 2. Radians and spherical triangles 3. Polar form and sin(A+B) 7. Regular polygons and regular polyhedra 1. Regular polygons are cyclic 2. Regular polyhedra 8. Chapter 1: Comments and solutions 2. Arithmetic 1. Place value and decimals: basic structure 2. Order and factors 3. Standard written algorithms 4. Divisibility tests 5. Sequences 1. Triangular numbers 2. Fibonacci numbers 6. Commutative, associative and distributive laws 7. Infnite decimal expansions 8. The binary numeral system 9. The Prime Number Theorem 10. Chapter 2: Comments and solutions 3. Word Problems 1. Twenty problems which embody "3 - 1 = 2" 2. Some classical examples 3. Speed and acceleration 4. Hidden connections 5. Chapter 3: Comments and solutions 4. Algebra 1. Simultaneous linear equations and symmetry 2. Inequalities and modulus 1. Geometrical interpretation of modulus, of inequalities, and of modulus inequalities 2. Inequalities 3. Factors, roots, polynomials and surds 1. Standard factorisations 2. Quadratic equations 4. Complex numbers 5. Cubic equations 6. An extra 7. Chapter 4: Comments and solutions 5. Geometry 1. Comparing geometry and arithmetic 2. Euclidean geometry: a brief summary 3. Areas, lengths and angles 4. Regular and semi-regular tilings in the plan 5. Ruler and compasses constructions for regular polygons 6. Regular and semi-regular polyhedra 7. The Sine Rule and the Cosine Rule 8. Circular arcs and circular sectors 9. Convexity 10. Pythagoras' Theorem in three dimensions 11. Loci and conic sections 12. Cubes in higher dimensions 13. Chapter 5: Comments and solutions 6. Infinity: recursion, induction, infinite descent 1. Proof by mathematical induction 2. 'Mathematical induction' and 'scientific induction' 3. Proof by mathematical induction II 4. Infinite geometric series 5. Some classical inequalities 6. The harmonic series 7. Induction in geometry, combinatorics and number theory 8. Two problems 9. Infinite descent 10. Chapter 6: Comments and solutions Alexandre Borovik is a Professor of Pure Mathematics at the University of Manchester, United Kingdom. His research interests include the philosophy of mathematics, mathematics education, and popularisation of mathematics. Tony Gardiner held the position of Reader in Mathematics and Mathematics Education at the University of Birmingham until December 2011. In 2011, Tony was elected Education Secretary of the London Mathematical Society. (c) 2019 Alexandre Borovik and Tony Gardiner, CC BY 4.0 [CCBYlogo] This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work). Alexandre Borovik and Tony Gardiner, The Essence of Mathematics through Elementary Problems. Cambridge, UK: Open Book Publishers, 2019. https://doi.org/10.11647/OBP.0168 [BiBTeX] [EndNote] [RefMan] Copyright (c) 2007-2021 Open Book Publishers. CC BY 4.0 International License Blog | Information Policy | Web Design